1 correspond to steady, shrinking or expanding breathers, respectively.\n\n(ii) A breather satisfying the above properties for all pairs of t 1 and t 2 of real numbers is called a torsion soliton.\n\nIdeas of Perelman [Pe] (see also [Ca] and [Li]) can be combined with Theorem 1.10, Theorem 1.13 and Theorem 1.14 to show the following classification result. Corollary 1.9. Let (M, J, θ) be a closed pseudohermitian 3-manifold. Then (i) there is no closed steady torsion soliton other than the one which admits zero Tanaka-Webster curvature and vanishing pseudohermitian torsion up to a contact transformation.\n\n(ii) there is no closed expanding torsion soliton other than the one which admits negative Tanaka-Webster curvature and vanishing pseudohermitian torsion.\n\n(iii) there is no closed shrinking torsion soliton other than the one which admits positive Tanaka-Webster curvature and vanishing pseudohermitian torsion.\n\n1.2. Some monotonicity results for Perelman-type functionals. The statements in this section are of a more technical nature: we will derive the CR analogue of Perelman's monotonicity formulas for the so-called coupled torsion flows (1.4), (1.6) and (1.7) in dimension 3.\n\nIn Section 6.1 we define the CR analogue of Perelman's F-functional by\nF(J (t) , θ (t) , ϕ (t) ) = M (W + |∇ b ϕ| 2 J,θ )e −ϕ dµ with the constraint M e −ϕ dµ = 1.\nUnder the flow (1.3), this is equivalent to M (ϕ t − 4η (t) )e −ϕ dµ = 0. Therefore, the following coupled torsion flow is natural,\n(1.4) ∂ t J (t) = 2E, ∂ t θ (t) = 2η (t) θ (t) , ∂ t ϕ (t) = 4η (t) , with E 11 = e ϕ (A 11 − iϕ 11 − iϕ 1 ϕ 1 ) and η (t) = e ϕ (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W ). Theorem 1.10. Let (M, J, θ) be a closed 3-dimensional pseudohermitian manifold and J (t) , θ (t) , ϕ (t) be a solution of the coupled torsion flow (1.4) on M × [0, T ). Then d dt F(J (t) , θ (t) , ϕ (t) ) = −2 M (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W ) 2 dµ −2 M |A 11 − iϕ 11 − iϕ 1 ϕ 1 | 2 dµ ≤ 0.\nThe monotonicity formula is strict unless\nA 11 − iϕ 11 − iϕ 1 ϕ 1 = 0 and 2∆ b ϕ − |∇ b ϕ| 2 J,θ + W = 0.\nThat is, up to a contact transformation θ = e −ϕ θ A 11 = 0 and W = 0.\n\nRemark 1.11. Observe that for θ = e −ϕ θ,\n(1.5) A 11 = e ϕ (A 11 − iϕ 11 − iϕ 1 ϕ 1 ), W = e ϕ (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W ).\nThen the coupled torsion flow (1.4) on (M, J (t) , θ (t) ) is equivalent to the following system of coupled torsion flows on (M,\nJ (t) , θ (t) ) F( J (t) , θ (t) , ϕ (t) ) = E( J (t) , θ (t) ) = M W d µ and ∂ t J (t) = 2A J (t) , θ (t) , ∂ t θ (t) = −2 W (t) θ (t) , ∂ t ϕ (t) = 4 W (t) .\nIn Section 6.2 we define two functionals analogous to Perelman's W-functional, namely the W + -functional,\nW + (J (t) , θ (t) , ϕ (t) , τ (t) ) = M [τ (W + |∇ b ϕ| 2 J,θ ) + 1 2 ϕ − 1](4πτ ) −2 e −ϕ dµ and the W − -functional W − (J (t) , θ (t) , ϕ (t) , τ (t) ) = M [τ (W + |∇ b ϕ| 2 J,θ ) − 1 2 ϕ + 1](4πτ ) −2 e −ϕ dµ. Remark 1.12. Note that W + and W − are invariant under the rescaling τ −→ cτ and θ −→ cθ. Furthermore, we have W ± (J, θ, ϕ, τ ) = W ± (Φ * J, Φ * θ, ϕ • Φ, τ ) for a contact diffeomorphism Φ : M → M.\nIn view of Theorem 1.10, we first study the monotonicity property of W + -functional. By the same discussion as before, the constraint M (4πτ ) −2 e −ϕ dµ = 1 is equivalent to another constraint, namely M (ϕ t + 2τ −1 dτ dt − 4η (t) )(4πτ ) −2 e −ϕ dµ = 0 under the flow (1.3). Therefore we consider the following coupled torsion flow:\n(1.6) ∂ t J (t) = 2E, ∂ t θ (t) = 2η (t) θ (t) , ∂ t ϕ (t) = 4(η (t) − τ −1 ), ∂ t τ = 2, with E 11 = (A 11 − iϕ 11 − iϕ 1 ϕ 1 ) and η (t) = (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W ).\nTheorem 1.13. Let (M, J, θ) be a closed 3-dimensional pseudohermitian manifold and J (t) , θ (t) , ϕ (t) and τ (t) be a solution of the coupled torsion flow (1.6). Then\nd dt W + (J (t) , θ (t) , ϕ (t) , τ (t) ) = −2τ M (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W − τ −1 ) 2 (4πτ ) −2 e −ϕ dµ −2τ M |A 11 − iϕ 11 − iϕ 1 ϕ 1 | 2 (4πτ ) −2 e −ϕ dµ ≤ 0.\nThe monotonicity formula is strict unless\nA 11 − iϕ 11 − iϕ 1 ϕ 1 = 0 and 2∆ b ϕ − |∇ b ϕ|\nunder the flow (1.3). Therefore we consider the following coupled torsion flow :\n\n(1.7)\n ∂ t J (t) = 2E, ∂ t θ (t) = 2η (t) θ (t) , ∂ t ϕ (t) = 4(η (t) + τ −1 ), ∂ t τ = −2, with E 11 = (A 11 − iϕ 11 − iϕ 1 ϕ 1 ) and η (t) = (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W )\n. Theorem 1.14. Let (M, J, θ) be a closed 3-dimensional pseudohermitian manifold and J (t) , θ (t) , ϕ (t) and τ (t) be a solution of the coupled torsion flow (1.7). Then\nd dt W − (J (t) , θ (t) , ϕ (t) , τ (t) ) = −2τ M (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W + τ −1 ) 2 (4πτ ) −2 e −ϕ dµ −2τ M |A 11 − iϕ 11 − iϕ 1 ϕ 1 | 2 (4πτ ) −2 e −ϕ dµ ≤ 0.\nThe monotonicity formula is strict unless\nA 11 − iϕ 11 − iϕ 1 ϕ 1 = 0 and 2∆ b ϕ − |∇ b ϕ| 2 J,θ + W + τ −1 = 0.\nThat is, up to a contact transformation θ = e −ϕ θ A 11 = 0 and W + τ −1 e ϕ = 0.\n\nRemark 1.15. Note that for θ (t) = e −ϕ θ (t) , we may reparametrize the time t by the formula\nt = t 0 e −ϕ (s) (x(s)) ds,. We have d t dt = e −ϕ (t) (x(t))\n, so the coupled torsion flows (1.6) and (1.7) on (M, J (t) , θ (t) ) are equivalent to the following coupled torsion flows on (M, J ( t) , θ ( t) ), respectively :\n ∂ t J ( t) = 2A J ( t) , θ ( t) , ∂ t θ ( t) = −2 W θ ( t) , ∂ t ϕ ( t) = 4( W − τ −1 e ϕ ), ∂ t τ = 2, and ∂ t J ( t) = 2A J ( t) , θ ( t) , ∂ t θ ( t) = −2 W θ ( t) , ∂ t ϕ ( t) = 4( W + τ −1 e ϕ ), ∂ t τ = −2.\nRecall that X f is called a contact vector field or an infinitesimal contact diffeomorphism if the Lie derivative L X f θ = ηθ for some function η. Such a contact vector field has the form\nX f = if 1 Z 1 − if 1 Z 1 − f T for some smooth function f on M . Furthermore (1.8) L X f J = 2B J f := (f 11 + iA 11 f )θ 1 ⊗ Z 1 + (f 11 − iA 11 f )θ 1 ⊗ Z 1 so (1.4) is equivalent to (1.9) ∂ t J (t) = 2JB J f = JL X f J, ∂ t θ (t) = 2η (t) θ (t) , ∂ t ϕ (t) = 4η (t) .\nwith f = e ϕ . Similar results hold for (1.6) and (1.7).\n\n\nPreliminaries and definitions\n\nIn this section we introduce some basic notions from pseudohermitian geometry. We learned many of these notions from [L1, L2], and we refer to these papers for proofs and more references.\n\nDefinition 2.1. Let M be a smooth manifold and ξ ⊂ T M a subbundle. A CR structure on ξ consists of an endomorphism J : ξ → ξ with J 2 = − Id such that the following integrability condition holds.\n(1) if X, Y ∈ ξ, then so is [JX, Y ] + [X, JY ]. (2) J([JX, Y ] + [X, JY ]) = [JX, JY ] − [X, Y ].\nThe CR structure J can be extended to ξ ⊗ C, which we can then decompose into the direct sum of eigenspaces of J. The eigenvalues of J are i and −i, and the corresponding eigenspaces will be denoted by T 1,0 and T 0,1 , respectively. The integrability condition can then be reformulated as\nX, Y ∈ T 1,0 implies [X, Y ] ∈ T 1,0 .\nNow consider a closed 2n + 1-manifold M with a cooriented contact structure ξ = ker θ. This means that θ ∧ dθ n = 0. The Reeb vector field of θ is the vector field T uniquely determined by the equations (2.1) θ(T ) = 1, and dθ(T, ·) = 0.\nDefinition 2.2. A pseudohermitian manifold is a triple (M 2n+1 , θ, J) where • θ is a contact form on M .\n• J is a CR structure on ker θ.\n\nDefinition 2.3. The Levi form , is the Hermitian form on T 1,0 defined by\nH(Z, W ) = Z, W = −i dθ, Z ∧ W .\nWe can extend this Hermitian form , to T 0,1 by defining Z, W = Z, W for all Z, W ∈ T 1,0 . Furthermore, the Levi form naturally induces a Hermitian form on the dual bundle of T 1,0 , and hence on all induced tensor bundles.\n\nWe now restrict ourselves to strictly pseudoconvex manifolds, or in other words compatible complex structures J. This means that the Levi form induces a Hermitian metric ·, · J,θ by V, U J,θ = dθ(V, JU ).\n\nThe associated norm is defined as usual: |V | 2 J,θ = V, V J,θ . It follows that H also gives rise to a Hermitian metric for T 1,0 , and hence we obtain Hermitian metrics on all induced tensor bundles. By integrating this Hermitian metric over M with respect to the volume form dµ = θ ∧ dθ n , we get an L 2 -inner product on the space of sections of each tensor bundle.\n\nDefinition 2.4. Let (M, J, ξ = ker θ 0 ) be a CR 3-manifold that is the smooth boundary of a bounded, strictly pseudoconvex domain in a complete Stein manifold V 4 . We shall call such a CR 3-manifold Stein fillable.\n\n2.1. Pseudohermitian connection. Let T, Z α , Zβ be a frame of T M ⊗ C, where {Z α } is any local frame of T 1,0 , and Zβ = Z β ∈ T 0,1 . Then θ, θ α , θβ , the coframe dual to T, Z α , Zβ , satisfies\ndθ = ih αβ θ α ∧ θβ,\nwhere h αβ is a positive definite matrix. By the Gram-Schmidt process we can always choose Z α such that h αβ = δ αβ ; throughout this paper, we shall take such a frame. The pseudohermitian connection or Tanaka-Webster connection of (J, θ) is the connection ∇ on T M ⊗ C (and extended to tensors) given in terms of a local frame {Z α } for T 1,0 by\n∇Z α = ω α β ⊗ Z β , ∇Zᾱ = ωᾱβ ⊗ Zβ, ∇T = 0,\nwhere ω α β is the 1-form uniquely determined by the following equations:\ndθ β = θ α ∧ ω α β + θ ∧ τ β τ α ∧ θ α = 0 ω α β + ωβᾱ = 0. (2.2)\nHere τ α is called the pseudohermitian torsion, which we can also write as\nτ α = A αβ θ β .\nThe components A αβ satisfy A αβ = A βα . We often consider the torsion tensor given by\nA J,θ = A αβ Z α ⊗ θβ + Aᾱ β Zᾱ ⊗ θ β .\nThe following remark gives some geometric meaning to the pseudohermitian torsion.\n\nRemark 2.5. Let X f be the contact vector field for the real-valued function f ∈ C 2 (M ). Then we have [CL1,Lemma 3.4]. In particular, we have X f = T for f = 1, and the above equation reduces to L T J = 2JA J,θ , so we see that the torsion tensor measures to what extend the complex structure J is invariant under the Reeb flow.\nL X f θ = −(T f )θ, and L X f J = 2B J f := 2(fβ α + iA αβ )θ α ⊗ Zβ + 2(f βᾱ − iAᾱ β )θᾱ ⊗ Z β See for instance\nWe now consider the curvature of the Tanaka-Webster connection in terms of the coframe {θ = θ 0 , θ α , θβ}. The second structure equation gives\nΩ β α = Ωβᾱ = dω β α − ω β γ ∧ ω γ α , Ω 0 α = Ω α 0 = Ω 0β = Ωβ 0 = Ω 0 0 = 0.\nIn [We,Formulas 1.33 and 1.35], Webster showed that the curvature Ω β α can be written as\n(2.3) Ω β α = R β α ρσ θ ρ ∧ θσ + W β α ρ θ ρ ∧ θ − W α βρ θρ ∧ θ + iθ β ∧ τ α − iτ β ∧ θ α , where the coefficients satisfy R βᾱρσ = R αβσρ = Rᾱ βσρ = R ρᾱβσ , W βᾱγ = W γᾱβ .\nIn addition, by [L2, (2.4)] the coefficients W α β ρ are determined by the torsion,\nW α β ρ = A αρ, β .\nContraction of (2.3) yields\nΩ α α = dω α α = R ρσ θ ρ ∧ θσ + W α α ρ θ ρ ∧ θ − W α αρ θρ ∧ θ = R ρσ θ ρ ∧ θσ + A αρ α θ ρ ∧ θ − Aᾱρᾱθρ ∧ θ (2.4)\nDefinition 2.6. The pseudohermitian Ricci tensor is the tensor with components R ρσ . Its trace W := R ρ ρ is called the Webster curvature. If the pseudohermitian Ricci tensor is a scalar multiple of the Levi form, then we say that the pseudohermitian structure is pseudo-Einstein.\n\nRemark 2.7. From the definition it is clear that the Webster curvature is the analogue of the scalar curvature in Riemannian geometry, and we also see that the pseudo-Einstein condition mimics the Einstein condition. Unlike the Riemannian case, pseudo-Einstein structures do not necessarily have constant Webster curvature, even in higher dimensions.\n\nWe will denote components of covariant derivatives by indices preceded by a comma. For instance, we write A αβ,γ . Here the indices {0, α,β} indicate derivatives with respect to {T, Z α , Zβ}. For derivatives of a scalar function, we will often omit the comma. For example, ϕ α = Z α ϕ, ϕ αβ = ZβZ α ϕ − ω α γ (Zβ)Z γ ϕ, ϕ 0 = T ϕ for a (smooth) function ϕ. For a real-valued function ϕ, the subgradient ∇ b ϕ is defined as the unique vector field ∇ b ϕ ∈ ξ such that Z, ∇ b ϕ = dϕ(Z) for all vector fields Z tangent to the contact distribution ξ. Locally\n∇ b ϕ = ϕ α Z α + ϕᾱZᾱ. Define the sublaplacian ∆ b by (2.5) ∆ b ϕ = ϕ α α + ϕᾱᾱ.\nFinally, we also need commutation relations, sometimes called Ricci identities. We list a few of them. These cover 3-dimensional CR-manifolds, but we also need higher-dimensional versions which we do not list here. For the idea of the derivation, see [L2,Equation 2.15].\n\n(2.6) C I,01 − C I,10 = C I,1 A 11 − kC I, A 11,1 , C I,01 − C I,10 = C I,1 A 11 + kC I, A 11,1 , C I,11 − C I,11 = iC I,0 + kW C I .\n\nHere C I denotes a coefficient of a tensor with multi-index I consisting of only 1 and 1, and k is the number of 1's minus the number of 1's in I.\n\nNext we mention some curvature identities. The ones that are relevant for us are the contracted CR Bianchi identities from [L2, Lemma 2.2].\nR ρσ,γ − R γσ,ρ = iA αγ, α h ρσ − iA αρ, α h γσ , W ,γ − R γσ,σ = i(n − 1)A αγ, α , R ρσ,0 = A αρ, ασ + Aβσ ,β ρ , W ,0 = A αρ, αρ + Aβσ ,βσ .\n(2.7)\n\n2.2. Volume-normalized contact forms and pseudo-Einstein CR-manifolds. To give our results some geometric meaning, we recall some results concerning volume-normalized contact forms and related notions. First recall the following definitions, taken from [L1].\n\nDefinition 2.8. Let (M, θ, J) be a closed pseudohermitian 3-manifold. Using Chern-Weil theory, we define the real first Chern class c 1 (T 1,0 (M )) as\nc 1 (T 1,0 M ) = i 2π [dω α α ]\nwhere ω α β is the connection form of the Tanaka-Webster connection in some local frame. Let (M 2n+1 , θ, J) be a (strictly pseudoconvex) CR-manifold. The canonical bundle of (M 2n+1 , θ, J) is the complex line bundle\nΛ n+1,0 (M ) = {η ∈ Ω n+1 ⊗ C | i V η = 0 for all V ∈ T 0,1 M }.\nA contact form θ in a closed pseudohermitian 3-manifold M is said to be volume-normalized with respect to a nowhere vanishing section ζ of the canonical bundle Λ n+1,0 (M ) if\nθ ∧ dθ n = i n 2 n!θ ∧ (T ζ) ∧ (T ζ)\nwhere T is the Reeb vector field with respect to θ.\n\nWe say that a CR-manifold (M, J, ξ = ker θ 0 ) has a transverse symmetry if there exists a contact form θ in the same conformal class as θ 0 such that J is invariant under the flow of the Reeb vector field T θ of θ,\nL T θ J = 0.\nEquivalently A J,θ = 0.\n\n2.2.1. Notions specific to dimension 3. For the remainder of this section we specialize to 3dimensional CR-manifolds.\n\nProposition 2.9. (i) ( [Hi]) Let (M, θ, J) be a closed pseudohermitian 3-manifold. A contact form θ is volume-normalized with respect to a non-vanishing closed section ζ in a closed pseudohermitian 3-manifold (M, J, θ) if and only if dω 1 1 = −id (W θ).\n\nThat is,\nW 1 − iA1 1,1 = 0.\n(ii) ( [L1]) If M is a closed CR embedding of a hypersurface in C 2 , then θ = e 2f θ 0 is volumenormalized with respect to a closed section ζ = j * (dz 1 ∧ dz 2 ) for j : (M, θ 0 , J) → C 2 and some smooth function f (x).\n\nRemark 2.10. The condition W 1 − iA1 1,1 = 0 corresponds to the pseudo-Einstein condition for higher dimensional CR-manifolds. Indeed, suppose that n ≥ 2. Then according to [L2,Lemma 4.1], a contact form θ on a pseudohermitian manifold (M 2n+1 , J, ker θ) is pseudo-Einstein if for every admissible coframe, we have\ndω α α = − i n d(W θ).\nBy writing out this equation, we see that the following holds for a higher-dimensional pseudo-Einstein manifold,\n(2.8) W ,α − inA βα, β = 0.\nHowever, in dimension 3 any CR-manifold is pseudo-Einstein, since the metric tensor has only one component. For us, a modification of this condition will serve as an integrability condition for the torsion flow.\n\nDefine the tensor\nR 1 θ 1 := (W 1 − iA1 1,1 )θ 1 .\nUse this to define the CR Q-curvature, see [Hi, CCC],\n(2.9) Q := −R ,1 1 = − 1 2 (∆ b W − 2 Im A1 ,1 1,1 ).\nThe last equality follows from [Hi,Lemma 5.4]. Clearly, vanishing of R 1 guarantees vanishing of the CR Q-curvature. For a converse, we already mentioned Proposition 1.2.\n\n\nShort time Existence\n\nIn this section we give some results concerning the short-time existence of the torsion flow. As mentioned, the motivation for the integrability condition comes from higher-dimensional CRmanifolds, so we give proofs that also hold for these high-dimensional cases. We thank Jih-Hsin Cheng for valuable contributions in early computations in the 3-dimensional case, in particular involving the variation formulas and linearized operator. See also his paper for a related flow, [C].\n\nLet θ (t) be a family of smooth contact forms and J (t) be a family of CR structures on (M, J, θ). We consider the following flow on a closed pseudohermitian (2n + 1)-manifold (M, J, θ) × [0, T ):\n(3.1) ∂ t J (t) = 2E, ∂ t θ (t) = 2η (t) θ (t) .\nHere\nJ = iθ α ⊗ Z α − iθ α ⊗ Z α and E = E α β θ α ⊗ Z β + E α β θ α ⊗ Z β .\nWe start by deriving some evolution equations under the general flow (3.1) before specifying to the torsion flow, for which E = A J (the torsion tensor), and η = −W (the Webster curvature). All computations will be done in a local frame. Fix a unit-length local frame {Z α } and let {θ α } be its dual admissible 1-form. Let Z α(t) , θ α (t) denote a unit-length frame and dual admissible 1-form with respect to (J (t) , θ (t) ). Since θ α (Z β(t) ) is a positive real function, we can write\nZ α = F α β Z β +G α β Z β\nwhere F α β are real and G α β are complex. The fact that Z α(t) is an orthonormal frame means that\nδ αβ = −idθ (t) (Z α(t) ∧ Z β(t) ).\nBy differentiating and substituting the above expression for Z α , we obtain F α β = −ηδ β α . By differentiating J (t) Z α(t) = iZ α(t) we find\n0 = JZ α + J Z α − i Z α = 2E α β Z β − 2iG α β Z β , so (3.2) Z α = −ηZ α − iE α β Z β .\n\nNow differentiate the identities\ndθ (t) = ih αβ θ α (t) ∧ θ β (t) , θ α (t) (Z β(t) ) = δ α β , and θ α (t) (Z β(t) ) = 0, to deduce that (3.3) θ α = 2iη α θ + ηθ α − iE α β θ β . Now we differentiate (2.2) to obtain (3.4) d θ α = θ γ ∧ω γ α + θ γ ∧ ω γ α + A αγ θ∧θ γ + A αγ θ∧θ γ + A αγ θ∧ θ γ .\nSince we will derive an identity involving tensors, we will take an adapted frame satisfying ω γ α = 0 at a point. Plug in (3.3) and consider the θ∧θ γ terms to obtain\n(3.5) A αγ = −2(iη αγ + ηA αγ ) − iE αγ , 0 .\nOn the other hand, contracting (3.4) with Z β and then contracting with h βα , computing modulo θ γ yields\nω α α = i(A α γ E γ α + A α γ E γ α + η α α )θ − [(n + 2)η α + iE γα , γ ]θ α mod θ γ .\nSince ω α α is pure imaginary, we have\n(3.6) ω α α = i(A α γ E γ α + A α γ E γ α + ∆ b η)θ +[(n + 2)η α − iE γα , γ ]θ α − [(n + 2)η α + iE γα , γ ]θ α .\nDifferentiate the structure equation (2.4) with respect to t and consider only the θ ρ ∧ θσ terms. This gives\n(3.7) R ρσ = −(A α γ E γ α + A α γ E γ α + ∆ b η)h ρσ − 2ηR ρσ −[(n + 2)η ρ − iE γρ , γ ], σ −[(n + 2)η σ + iE γσ , γ ], ρ\nAfter contracting with h ρσ we get\n(3.8) W = i(E γα , γα −E γα , γα ) − n(A α γ E γ α + A α γ E γ α ) −[2(n + 1)∆ b η + 2W η] = 2 Re iE γα , γα −nA α γ E γ α − [2(n + 1)∆ b η + 2W η].\nRecall that the transformation law of the connection under a change of pseudohermitian structure was computed in [L1,Sec. 5]. Letθ = e 2f θ be another pseudohermitian structure. Then we can define an admissible coframe byθ α = e f (θ α + 2if α θ). With respect to this coframe, the connection 1-form and the pseudohermitian torsion are given by\nω β α = ω β α + 2(f β θ α − f α θ β ) + δ α β (f γ θ γ − f γ θ γ ) + i(f α β + f β α + 4δ α β f γ f γ )θ,(3.9)\nand (3.10)\nA αβ =e −2f (A αβ + 2if αβ − 4if α f β ),\nrespectively. Thus the Webster curvature transforms as\n(3.11) W = e −2f (W − 2(n + 1)∆ b f − 4n(n + 1)f γ f γ ).\nHere covariant derivatives on the right side are taken with respect to the pseudohermitian structure θ and an admissible coframe θ α . Note also that the dual frame of {θ,θ α ,θ α } is given by\n{ T , Z α , Z α }, where T = e −2f (T + 2if γ Z γ − 2if γ Z γ ), Z α = e −f Z α .\nWe need one more ingredient before we can prove Theorem 1.4. Let f be a smooth function, and define the CR pluriharmonic operator P α f by\nP α f = f γ γ α + inA αγ f γ .\nDefine the CR Paneitz real operator by\nP θ f = (P α f ), α .\nWe have the following lemma.\n\nLemma 3.1. Let (M 2n+1 , J, ξ = kerθ) be a pseudohermitian manifold, and suppose, if n = 1, that Conjecture 1.1 holds for this manifold. Then there is a smooth function f such that\n(3.12) W α − inA αβ , β −2(n + 2)P α f = 0,\nand thus W α α − inA αβ , βα −2(n + 2)P θ f = 0.\n\nProof. By Conjecture 1.1 in case n = 1 and Proposition 1.3 in case n ≥ 2 there is a smooth function f such the rescaled contact form θ = e 2f θ is volume-normalized. By Proposition 2.9 and Remark 2.10, this volume-normalized contact form satisfies\nW α − in A αβ , β = 0.\nThe second contracted Bianchi identity from (2.7) implies then that\n(3.13) 0 = n−1 n ( W α − in A αβ , β ) = ( R αβ − W n h αβ ), β . Also by [L2, page 172], we have (3.14) (R αβ − W n h αβ ) − 2(n + 2)(f αβ − 1 n f γ γ h αβ ) = R αβ − W n h αβ = 0.\nFollowing the same computation as in the proof of Lemma 5.4 in [Hi], we find using (3.9), (3.10) and (3.11), that\nW α = Z α W = e −f Z α e −2f (W − 2(n + 1)∆ b f − 2n(n + 1)|∇ b f | 2 ) = e −3f [W α − 2W f α + 4(n + 1)(∆ b f + n|∇ b f | 2 )f α − 2(n + 1)(f γ γ α + f γ γ α ) − 4n(n + 1)(f γα f γ + f γ f γ α )], i A αβ , γ = i( Z γ A αβ − ω α l ( Z γ ) A βl − ω β l ( Z γ ) A αl ) = ie −f [(Z γ + 2f γ ) A αβ + 2(δ αγ A βl + δ βγ A αl )f l ] = ie −f (Z γ + 2f γ )e −2f (A αβ + 2if αβ − 4if α f β ) +2e −3f [δ βγ (iA αl − 2f αl + 4f α f l ) + δ αγ (iA βl − 2f βl + 4f β f l )]f l = e −3f [iA αβ,γ − 2f αβγ + 4(f αγ f β + f α f βγ )] +2e −3f [δ βγ (iA αl − 2f αl + 4f α f l ) + δ αγ (iA βl − 2f βl + 4f β f l )]f l .\nContract the second equation with respect to the Levi metric h γβ = h γβ . This yields\ni A αβ , β = e −3f [iA αβ , β − 2f αβ β + 4(f α β f β + f α f β β ) +2(n + 1)(iA αβ −2f αβ + 4f α f β )f β ].\nThus we get\nW α − in A αβ , β = e −3f [W α − inA αβ , β − 2(n + 1)(f β β α + f β β α ) + 2nf αβ β − 2W f α − 2n(n + 1)iA αβ f β + 4(n + 1)(f β β + f β β )f α − 4n(n + 1)f β α f β − 4n(f α β f β + f β β f α )].\nWe will apply the following commutation relations, which are similar to those in (2.6)\n−2(n + 1)f β β α + 2nf αβ β = −2f β β α + 2nR αβ f β − 2niA αβ f β , f αβ − f βα = ih αβ f 0 , and 2n(R αβ − W n h αβ )f β = 4n(n + 2)(f α β f β − 1 n f β β f α ).\nThe latter relation follows from (3.14). Together with the above we obtain the following transformation law\nW α − in A αβ , β = e −3f [W α − inA αβ , β − 2(n + 2)(f β β α + niA αβ f β )] = e −3f [W α − inA αβ , β −2(n + 2)P α f ].\nApply (3.13) to complete the proof of (3.12).\n\nThe following lemma is needed to prove Corollary 1.5. We use the observation that CR 3manifolds with transverse symmetry are Stein fillable.\n\nLemma 3.2. Suppose that (M 3 , J 0 , ξ = ker θ 0 ) has a transverse symmetry and θ 0 is volumenormalized with respect to a non-vanishing closed section ζ. Assume furthermore that (J(t), θ(t) ) is a solution of the torsion flow (1.1) on the interval [0, T ]. Then for each t ∈ [0, T ], there exists a volume-normalized contact form θ (t) for (ξ, J(t) ).\n\nProof. Since (M, J (t) , ξ = ker θ (t) ) is Stein fillable, there exists a contact form θ (t) with vanishing CR Q-curvature by (3) \nθ (t) = e 2f θ 0\nwhere θ 0 is volume-normalized. By [Hi,Lemma 5.4,formula 5.7\n] the CR Q-curvature of θ (t) is given byQ = e 4f (Q 0 − 6P θ0 f )\nand, because Q 0 = 0 by (2) of Proposition 1.2 (also a result of Hirachi), we find\nP θ0 f = 0.\nSince M has a transverse symmetry, it follows from [Hi,Proposition 7.3] that f is a CR pluriharmonic function, and P 1 f = 0. This implies θ (t) is volume-normalized.\n\nProof of Theorem 1.4. We follow the arguments of Hamilton, [H1, Section 4], and adapt them to our setup. First rewrite the torsion flow (3.1) as\n(3.15) ∂ ∂t (J ⊕ θ) = 2(A J,θ ⊕ −W θ) = 2Q(J, θ), with Q(J, θ) = A J,θ ⊕ −W θ.\nStep 1: determining the linearization of the torsion flow We use δJ, δθ to denote the variations of J and θ, respectively. Set δJ = 2E and δθ = 2hθ,\n\nwhere E is an endomorphism: ξ → ξ satisfying J • E + E • J = 0 and h is a smooth function. Let δ J and δ θ denote the linearization operators with respect to J and θ. From (3.5) and (3.8), we see that\n(3.16) δ J A αβ = iE αβ , 0 δ θ A αβ = 2(ih αβ − hA αβ ) and (3.17) δ J W = i(E γα , γα −E γα , γα ) − n(A α γ E γ α + A α γ E γ α ) δ θ W = −2(n + 1)∆ b h − 2W h.\nWe first compute the linearization of 2A J,θ . Let O m denote an operator of weight ≤ m (see [CL1,page 234]), and define the total variation as δ = δ J + δ θ . PutẼ := E ⊕ hθ. With (3.16) we compute the variation of the torsion as\nδ(2A J,θ ) = (2δA α β )θ α ⊗ Z β + 2A α β (δθ α ) ⊗ Z β +2A α β θ α ⊗ (δZ β ) + conjugate (3.18) = 2 Re [2iE α β , 0 +4ih α β ]θ α ⊗ Z β + O 1 ( E).\nThe linearization of −2W θ can be computed with (3.17),\nδ(−2W θ) = −2(δW )θ − 2W (δθ) = {2i(E γα , γα −E γα , γα ) + 2n(A α γ E γ α + A α γ E γ α ) (3.19) + 4(n + 1)∆ b h}θ\nWith these linearizations (3.18) and (3.19) we write the highest weight operator L J,θ ( E) of the linearization δ(2Q(J, θ)) as\nL J,θ ( E) = 2 Re [2iE α β , 0 +4ih α β ]θ α ⊗ Z β (3.20) ⊕[2i(E βα , βα −E βα , βα ) + 4(n + 1)∆ b h]θ.\nStep 2: finding integrability conditions Following Hamilton's ideas from ([H1, section 5]), we find an integrability condition of the form H(J, θ)Q(J, θ) = 0 for the torsion flow (3.15). By Lemma 3.1 we find a function f satisfying equation (3.12). We then define O 0 (A J,θ ⊕ −W θ) := 2(n + 2)P α f . We use this function to get an integrability condition. Define the linear operator H(J, θ) by\n(3.21) H(J, θ)( E) := h α + inE αβ , β +O 0 ( E).\nThis operator has degree 1 in E := E ⊕ hθ; the term O 0 has lower order. By (3.12) we see that Q(J, θ) satisfies the integrability condition\nH(J, θ)Q(J, θ) = −W α + inA αβ , β +O 0 (A J,θ ⊕ −W θ) = 0.\nBy taking first variations, we find\nH(J, θ)δQ(J, θ)( E) + (δH(J, θ))Q(J, θ)( E) = 0.\nWe continue to argue as in [H1, section 4]. The operator in E given by δH(J, θ) has degree 1, so its symbol, which we denote by σH, of maximal degree vanishes. The other term, H(J, θ)δQ(J, θ), has maximal degree since H has degree 1 and δQ has positive degree. The symbols of linear differential operators H(J, θ) and Q(J, θ) therefore satisfy (3.22) [σH(J, θ)](·) • σ[δQ(J, θ)](·)( E) = 0, so we conclude that the image of σ[δQ(J, θ)] must lie in the kernel of σH(J, θ). We now want to show that the restriction of the symbol σδ2Q(J, θ)(ξ) to ker σH has only positive eigenvalues when ξ = 0. To see this, we first need additional commutation relations, again similar to those in (2.6), namely\nE αβ , β γ − E αγ , β β = i(n − 1)E αγ,0 + R ασ E σ γ − R α β γσ E σ β , E αγ , β β − E αγ , β β = inE αγ,0 + R γ σ E ασ + R α σ E σγ .\nFrom these relations we derive that on the subspace\n(3.23) ker H(J, θ)( E) = { E = E ⊕ hθ]|h α + inE αβ , β +O 0 ( E) = 0}\nthe following identities hold true,\n(3.24) ∆ b h = in(E αβ , βα −E αβ , βα ) + O 1 ( E), ∆ b E α β = i(2 − n)E α β , 0 + 2 n ih α β + O 1 ( E).\nTherefore it follows from (3.20) that the highest weight operator L J,θ ( E) of the linearization δ(2Q(J, θ)) is\n(3.25) L J,θ ( E) = {4n Re[(L α E α β )θ α ⊗ Z β ] ⊕ 2(2n + 2 + 1 n )(∆ b h)\nθ} on ker H(J, θ)( E). Note here that both the Folland-Stein operator L α , given for n ≥ 2 by\nL α = ∆ b − iαT, α = − (n − 1) 2 n = −(n − 2 + 1 n ),\nand the sublaplacian ∆ b are subelliptic, see [FS]. Therefore it follows from (3.22) and (3.25) that all eigenvalues of the eigenspaces of σ[δQ(J, θ)] in ker σH(J, θ) are strictly positive.\n\nAlternatively, note that the proof of Lemma 3.1 can also be used to show that there are no degeneracies for the torsion flow (3.1) other than those implied by the second contracted Bianchi identity from (2.7).\n\nStep 3: solving the linearized equations and handling the non-linear problem\n\nTo complete the argument, we follow [H1, Section 5 and Section 6]. The linearized equations can be solved using the regularization trick from [H1, Section 6]. Theorem 5.1 of [H1] can then be used to obtain short-time existence of the torsion flow (3.1). The main ingredient is the Nash-Moser inverse function theorem ( [H3]).\n\n\nContact 3-manifolds with a global frame and pseudohermitian structures\n\nWe now specialize to 3-dimensional pseudohermitian manifolds. Let (M, ξ = ker θ) be a cooriented contact 3-manifold. Denote the Reeb field by T . Furthermore, in this section and in the next, Section 5, we shall assume that ξ admits a global symplectic trivialization, i.e. there are vector fields U, V such that ξ = Span(U, V ) and dθ(U, V ) = 1. Proof. The contact structure ξ admits the structure of a symplectic vector bundle (ξ, dθ). By choosing a compatible complex structure J, we obtain a complex line bundle (ξ, J). It is wellknown that smooth complex line bundles are trivial if and only if their first Chern class vanishes, see [W,Chapter III,Section 4].\n\nThe last step needs the first Chern class with integer coefficients. Chern-Weil theory will not suffice in general. Henceforth, we shall assume that the globally defined vector fields U, V form a symplectic basis of (ξ, dθ). Consider the coframe θ, α, β dual to T, U, V . Then dθ(U, V ) = 1, so (4.1) dθ = α∧β.\n\nLemma 4.2. Let J be a compatible complex structure for the symplectic vector bundle (ξ = ker θ, dθ). Then there are smooth functions a : M → R and c : M → R >0 such that, with respect to the frame U, V , the complex structure J is represented by the matrix\nJ = a − 1+a 2 c c −a .\nProof. With respect to the global frame U, V , the endomorphism J is represented by a 2 × 2matrix. Writing out the condition J 2 = −id shows that the matrix representation for J has the above form. The compatibility condition means that dθ(·, J·) is a metric, so it is represented by a positive definite matrix. Writing out this matrix shows that c is a positive function.\n\nThe following is motivated by our goal to convert the torsion flow (a PDE for tensors) into a PDE for functions. Choose real-valued functions a, b and c where b and c are positive. We attach super-and subscripts to indicate the dependence on these functions. In order to keep track of deformations of the contact form, we express all data in the given frame T, U, V . Define This can be checked by plugging in the vector field into the defining equations (2.1). The second assertion is obtained by writing out the terms.\nθ b = b 2 θ, α b = bα − 2V (b)θ, β b = bβ + 2U (b)θ, T b = 1 b 2 T + 2V (b) b 3 U − 2U (b) b 3 V, U b = 1 b U, V b = 1 b V.\nRemark 4.4. If b is a constant function, then the deformation from (4.2) corresponds to a ξhomothetic deformation as defined in [Bl,Section 10.4]. We take b 2 in θ b to have fewer expressions with square roots. Define a complex structure by\nJ abc (U b ) = aU b + cV b , J abc (V b ) = − 1 + a 2 c U b − aV b\nBy Lemma 4.2, this is the most general choice.\n\nRemark 4.5. In higher dimensions, strictly pseudoconvex CR-manifolds require an integrability condition, see Definition 2.1, which is trivially satisfied in dimension 3.\n\nWe now compute the Tanaka-Webster connection as in Section 2.1. We use the coframe θ, θ 1 , θ1, where\nθ 1 abc = 2c(a 2 + 1) −i 2(a − i) α b + i 2c β b , θ1 abc = 2c(a 2 + 1) i 2(a + i) α b − i 2c β b . (4.3)\nThis satisfies our normalization condition\ndθ b = α b ∧β b = iθ 1 b ∧θ1 b . The corresponding eigenvectors of J abc are Z abc 1 = 1 2c(a 2 + 1) (a 2 + 1)U b + c(a − i)V b , Z abc 1 = 1 2c(a 2 + 1) a 2 + 1)U b + c(a + i)V b .\n(4.4) 4.1. Converting the torsion flow into a system of PDE's for the functions a, b, c. To write down the equations of the torsion flow, we need the work out the torsion tensor. We have\nZ abc 1 ⊗θ 1 abc = −i(a + i) 2 U b ⊗α b + i(a 2 + 1) 2c U b ⊗β b + −i(a + i)c 2(a − i) V b ⊗α b + i(a + i) 2 V b ⊗β b , so we find A abc J,θ = A abc 11 Z abc 1 ⊗ θ 1 abc + A abc 11 Z abc 1 ⊗ θ1 abc = +2 Re(A abc 11 Z abc 1 ⊗ θ 1 abc ) = + Re(A abc 11 ) + a Im(A abc 11 ) U b ⊗ α b − Im(A abc 11 )(a 2 + 1) c U b ⊗ β b − Re(A abc 11 ) −2ac a 2 + 1 + Im(A abc 11 ) (1 − a 2 )c a 2 + 1 V b ⊗ α b − Re(A abc 11 ) + a Im(A abc 11 ) V b ⊗ β b .\nHence the first equation of the torsion flow (1.1),J abc = 2A abc J,θ is equivalent to the systeṁ a = 2 Re(A abc 11 ) + a Im(A abc 11 )\nċ = −2 Re(A abc 11 ) −2ac a 2 + 1 + Im(A abc 11 ) (1 − a 2 )c a 2 + 1 d dt − (1 + a 2 ) c = −2 Im(A abc 11 ) a 2 + 1 c d dt (−a) = −2 Re(A abc 11 ) + a Im(A abc 11 ) (4.5) Indeed, modulo θ we have U b ⊗ α b ≡ U ⊗ α U b ⊗ β b ≡ U ⊗ β V b ⊗ α b ≡ V ⊗ α V b ⊗ β b ≡ V ⊗ β,\nso we obtain the above system by looking at the coefficients of U ⊗ α, U ⊗ β, V ⊗ α, and V ⊗ β. This works since these tensors are time-independent.\n\nLemma 4.6 (A smaller system for the J-part of the torsion flow). The system given by (4.5) is equivalent to system given bẏ a = 2 Re(A abc 11 ) + a Im(A abc 11 ) ċ = −2 Re(A abc 11 ) −2ac a 2 + 1 + Im(A abc 11 )\n\n(1 − a 2 )c a 2 + 1 . (4.6)\n\nProof. The first equation of (4.5) implies the fourth. We now verify that the first and second equation of (4.5) imply the third.\nd dt − 1 + a 2 c = − 2aȧ c + 1 + a 2 c 2ċ = − 4a c Re(A abc 11 ) − 4a 2 c Im(A abc 11 ) + 1 + a 2 c 2 Re(A abc 11 ) 4ac a 2 + 1 − 1 + a 2 c 2 Im(A abc 11 ) 2c(1 − a 2 ) a 2 + 1 = −2 Im(A abc 11 ) 2a 2 c + 1 − a 2 c = −2 Im(A abc 11 ) a 2 + 1 c .\nOn the other hand, the second equation of the torsion flow (1.1) reduces to\nd dt (θ b ) = d dt (b 2 )θ = −2W abc b 2 θ,\nso we can reduce the torsion flow to a system of PDE's for the functions a, b, c, giving us the following proposition. (1 − a 2 )c a 2 + 1 d dt b 2 = −2W abc b 2 .\n\n(4.7)\n\nRemark 4.8. Spatial derivatives of a, b, c come in via the definition of torsion and Webster curvature.\n\n\n3-manifolds with constant structure constants and the Tanaka connection\n\nIn this section we consider manifolds M 3 that admit global 1-forms ω 1 , ω 2 , ω 3 such that (1) ω 1 , ω 2 , ω 3 form a basis of T * M .\n\n(2) The structure coefficients are constant, i.e.\ndω i = j<k c i jk ω j ∧ω k with c i jk constant. (3) There is a constant form θ of the form θ = i c i ω i ,\nwhere c i are constant. We shall call such a contact manifold a homogeneous contact manifold. This terminology is not standard, but it serves a useful purpose in this note. Let us point out that a related, but not equivalent notion, also referred to as homogeneous contact, was used by Perrone, [Pr].\n\nRemark 5.1. The structure coefficients are the structure constants of some 3-dimensional Liealgebra. Indeed, the dual frame {X 1 , X 2 , X 3 } satisfies\n[X j , X k ] = − i c i jk X i ,\nthe Lie bracket on vector fields satisfies the Jacobi identity. We shall call this the Lie algebra of a homogeneous contact manifold.\n\nFrom Lemma 4.1 we get immediately.\n\nLemma 5.2. Homogeneous contact manifolds have trivial Chern class.\n\nBefore we define a CR structure on such manifolds, we use the following lemma to provide a better coframe. In many cases, this lemma can be improved upon, but this version is sufficiently convenient.\n\nLemma 5.3. Let (M, {ω i } i , θ) be a homogeneous contact manifold. Then there is a basis {ω i } i such that\n\n•ω 1 is contact, and c 1 23 = 1. • The structure coefficients dω i = j,k c i jkω j ∧ω k satisfy c 1 12 = c 1 13 = c 2 12 = c 3 13 = 0. Proof. Choose a compatible complex structure J for (ξ, dθ). Consider the operator h := 1 2 L T J. From [Bl,Lemma 6.2] we see that h is self-adjoint with respect to the metric dθ(·, J·), and we also get the identity\n0 = 1 2 L T J 2 = Jh + hJ.\nSince h is self-adjoint, we can find a basis of eigenvectors X, Y of h for ξ. If the eigenvalue of X is λ, then JX is an eigenvector with eigenvalue −λ,\nhJX = −JhX = −λJX,\nso we can assume that Y = JX. We consider the Levi-Civita connection ∇ for g = θ ⊗θ +dθ(·, J·). Take the basis e 1 = T, e 2 = X, e 3 = JX, where T is the Reeb field of θ. Then [e 1 , e 2 ] = [T, X] = ∇ T X − ∇ X T = +JX + λJX − µJX.\n\nfor some µ ∈ R. In the last step we have used the identity (see [Bl,Lemma 6.2] )\n∇ X T = −JX − JhX.\nThe same steps work for [e 1 , e 3 ], so we conclude that there are constants C 1 , C 2 such that [e 1 , e 2 ] = C 1 e 3 [e 1 , e 3 ] = C 2 e 2 .\n\nConsider the dual basis {ω i } i . Then ω 1 is a contact form, and since T is the Reeb field, we have 0 = i e1 dω 1 = c 1 12 ω 2 + c 1 13 ω 3 . Hence c 1 12 = c 1 13 = 0, and from Lie bracket computations we see that c 2 12 = c 3 13 = 0. By rescaling, we see that the claim holds.\n\nWe assume now that ω 1 , ω 2 , ω 3 is a basis that is provided by this lemma. Take the basis X 1 , X 2 , X 3 that is dual to ω 1 , ω 2 , ω 3 , i.e. ω i (X j ) = δ i j . We have i X1 ω 1 = 1 and i X1 dω 1 = k c 1 1k ω k = 0, so X 1 is the Reeb vector field for ω 1 . Note that ξ = ker ω 1 = Span(X 2 , X 3 ). Also, if c 1 23 = 1, then dω 1 = ω 2 ∧ω 3 , so we have the right normalization convention for the setup of the Tanaka connection described in Formula (4.1) with θ = ω 1 , α = ω 2 and β = ω 3 .\n\nChoose constants a ∈ R and b, c > 0, and define a complex structure on ξ (or CR-structure on M ) following Lemma 4.2 by\n(5.1) J abc = aX 2 ⊗ ω 2 + cX 3 ⊗ ω 2 − 1 + a 2 c X 2 ⊗ ω 3 − aX 3 ⊗ ω 3 .\nWe call such an endomorphism a homogeneous complex structure, and we refer to a homogeneous contact manifold together with the above complex structure as a homogeneous CRmanifold or a homogeneous CR structure. By direct computation, we obtain the following result for the Tanaka-Webster connection, its torsion and the Webster curvature.\n\nProposition 5.4. Let (M, {ω i } i , θ) be a homogeneous contact manifold with basis provided by Lemma 5.3. Fix a ∈ R and b, c > 0 and define J abc as in formula (5.1). Then the connection form for the Tanaka connection of the pseudohermitian manifold (M, θ b = b 2 θ, J abc ) is given by\nω 1 1 = i b 2 − a 2 + 1 2c c 3 12 + c 2 c 2 13 θ b + i b c · c 2 23 − a · c 3 23 α b + i b a 2 + 1 c c 3 23 − a · c 2 23 β b\nIts torsion is given by\nA 11 abc = 1 b 2 i a 2 + 1 2c c 3 12 − i c(a + i) 2(a − i) c 2 13 ,\nand its Webster curvature by\nW abc = 1 b 2 a 2 + 1 2c c 3 12 − c 2 c 2 13 − c · c 2 23 2 + 2a · c 2 23 c 3 23 − a 2 + 1 c c 3 23 2 .\nProof. See the computations in the appendix. Alternatively, these computations are essentially also contained in [Pr]. Note that Perrone uses a J-basis, that is e 1 = T, e 2 , e 3 = Je 2 .\n\nWe can now reduce the torsion flow for homogeneous CR-manifolds to an ODE by plugging in the results of Proposition 5.4 into Proposition 4.7. The general system is fairly complicated, so we will work out some interesting case in Section 5.2.\n\nWe shall also consider the normalized torsion flow which, in general, is given by the system\n(5.2) ∂ t J (t) = 2A J (t) ,θ (t) , ∂ t θ (t) = −2(W −W )θ (t) ,\nwhereW = M W θ∧dθ/ M θ∧dθ. Since the Webster curvature is constant in space for a homogeneous CR-manifold, then second equation of the normalized is flow is trivial. Inserting the result of Proposition 5.4 into the explicit system provided by Proposition 4.7 gives the following.\n\nProposition 5.5 (Normalized torsion flow for homogeneous CR-manifolds). Let (M, {ω i } i , θ) be a homogeneous contact manifold with θ = ω 1 . Set b = 1, and let a t , c t be real valued functions Then for a complex structure J atbct as defined in (5.1), the normalized torsion flow satisfies the ODEȧ\nt = c 2 13 a t c t − c 3 12 a 2 t + 1 c t a ṫ c t = c 2 13 c 2 t + c 3 12 (1 − a 2 t ).\n(5.3)\n\n\nHomogeneous CR-manifolds and vanishing torsion.\n\nObserve that homogeneous CRmanifolds need not be compact. In Section 5.2 we give some compact examples, but we start out by characterizing homogeneous CR-manifolds with vanishing torsion directly in terms of the structure coefficients and coefficients for the CR-structure a, b, c.\n\nProposition 5.6. Let (M, {ω i } i , θ) be a homogeneous contact manifold with θ = ω 1 and basis provided by Lemma 5.3. For a ∈ R, b = 1 and c > 0, a complex structure J abc as defined in (5.1) has vanishing torsion precisely when one of the following conditions is satisfied.\n\n1 if a = 0, c 2 12 = 0, and c = − 2 if a = 0, c 2 13 = c 2 12 = c 3 12 = 0. Proposition 5.7 (Convergence of the torsion flow). Let (M, {ω i } i , θ) be a homogeneous contact manifold with θ = ω 1 and a basis provided by Lemma 5.3. Take a ∈ R, b = 1 and c > 0, and let J abc be a complex structure as defined in (5.1).\n\nA If c 3 12 > 0 and c 2 13 < 0, then the normalized torsion flow converges to the unique torsion free complex structure of Proposition 5.6 ) in the upper half-plane with coordinates (a, c). The phase diagram for the repelling (R) case is given in Figure 1 The phase diagram for the attracting (A) case is similar, but the arrows are reversed. It is clear that the attracting case ), we make two observations.\n\n• any initial condition (a 0 , c 0 ) that starts in the set given by |c 2 13 |c 2 0 + |c 3 12 |a 2 0 − |c 3 12 | > 0 blows up in finite time. Indeed, the c-coordinate is strictly increasing in that case, so there is t 0 such that c t0 > 1 or the solution blows up before t 0 . If c t0 > 1 then the solution blows up in finite by the following argument. We havė c = |c 3 12 |c 2 + |c 2 13 |(a 2 − 1) ≥ |c 3 12 |c 2 − |c 2 13 |, which must blow up in finite time. with c 0 > 0 exits the half-disk {|c 2 13 |c 2 0 + |c 3 12 |a 2 0 − |c 3 12 | ≤ 0} in some finite time t 1 . Then either c t1 = 0, meaning that a blows up in finite time, or we can reduce to the first case.\n\nAn alternative proof can be given by starting with a J-basis. Then a 0 = 0 and c 0 = 1. Such a basis is not preserved under the torsion flow, but the condition a t = 0 holds. The resulting ODE is simpler to analyze. This method is applied in Section 5.2.\n\nRemark 5.8. By Proposition 5.4 there is a relation between the Webster curvature and the sign of c 3 12 . However, it is not true that repelling is equivalent to negative curvature.\n\n5.1.1. Homogeneous CR structures on unimodular Lie groups. We recall the definitions involved.\n\nDefinition 5.9. A Lie group G is called unimodular if the left-invariant Haar measure is also right invariant. A Lie algebra (g, [·, ·]) is called unimodular if T r ad X = 0 for all X ∈ g.\n\nNote that the Lie algebra of a unimodular Lie group is unimodular. The unimodularity of a Lie group can be used to simplify the structure coefficients of the Lie algebra.\n\nLemma 5.10. Let G be a unimodular, 3-dimensional Lie group admitting a homogeneous contact structure ξ = ker θ. Then there is a basis {e i } i of the Lie algebra g such that • e 1 is the Reeb vector field, and e 2 , e 3 lie in the contact structure ξ.\n\n• There are λ, µ ∈ R such that [e 1 , e 2 ] = λe 3 , [e 1 , e 3 ] = µe 2 . Furthermore, [e 2 , e 3 ] = e 1 .\n\nProof. First apply Lemma 5.3. Note that unimodularity implies that j c j ij = 0 for all i. We conclude that c 1 i1 + c 2 i2 + c 3 i3 = 0. By putting i = 2, 3 we deduce that c 2 32 = c 3 23 = 0, so [e 2 , e 3 ] = −c 1 23 e 1 = −e 1 .\n\nHere is a table with all possible unimodular Lie groups admitting a homogeneous contact structure,\nc 2 31 = −c 2 13 c 3 12 Geometry ∃J with A J,θ = 0 + + SU (2) = S 3 yes - - SL(2, R) yes - + SL(2, R) no 0 + E(2) no 0 - E(1, 1) no 0 0\nHeisenberg yes Remark 5.11. We point out that the topology or geometry of the underlying CR-manifold does not uniquely determine the underlying contact structure. In particular, for some compact quotients of SL(2, R) the above contact structures are not isomorphic. Explicit examples are given in Sections 5.2.1 and 5.2.2.\n\nTheorem 5.12 (Convergence to torsion free CR structure). Let (M, {ω i } i , θ = ω 1 ) be a homogeneous contact manifold whose Lie algebra is isomorphic to su(2). Then there is a unique homogeneous complex structure J a∞,b=1,c∞ that is torsion free. Moreover, for any choice of homogeneous complex structure J a,b=1,c , the normalized torsion flow converges to this unique CRstructure (ker θ, J a∞,b=1,c∞ ).\n\nIn particular, for any choice of homogeneous complex structure on SU (2), the normalized torsion flow converges to the standard CR-structure. \nf : C 2 −→ R z −→ 1 2 z 2 .\nLet S 3 := f −1 ( 1 2 ), and put ω 1 = −df • i, ω 2 = −df • j, ω 3 = −df • k, where i, j, k are the standard quaternions. With θ = ω 1 , this gives S 3 the structure of a homogeneous contact manifold. Its structure constants are c 1 23 = 1, c 2 13 = −1 and c 3 12 = 1. Recall that the standard CR structure on S 3 is then given by (θ, J = i). Put θ 1 := 1 √ 2 ω 2 + iω 3 . Then {θ, θ 1 , θ1} is an admissible frame. Following [CCY] we define the CR structure via the deformed coframe\nθ 1 t = 1 √ 1 − t 2 θ 1 − tθ1 .\nWriting this out gives\nθ 1 t = 1 2 √ 1 − t √ 1 + t α + i √ 1 + t √ 1 − t β .\nComparing this with Equation (4.3) shows that the examples of Rossi are homogeneous CR structures with\nJ abc = J a=0,b=1/ √ 2,c= 1−t 1+t = 0 − 1+t 1−t 1−t 1+t 0 .\nFor t > 0, these CR-manifolds are not embeddable.\n\nOn the other hand, Theorem 5.12 applies to Rossi's examples, so we have.\n\nCorollary 5.14. Under the normalized torsion flow, Rossi's examples flow to the standard CR structure on S 3 , which is embeddable.\n\n\n5.2.\n\nExamples of compact homogeneous CR-manifolds: different CR structures on ST * Σ. We describe the torsion flow on several geometries, namely SU (2), E(2), SL(2, R) and Heisenberg geometry.\n\nAs an explicit, compact model covering the first three cases we consider a compact orientable surface Riemann surface (Σ, g). According to a standard theorem in Riemannian geometry, the unit cotangent bundle ST * Σ admits a canonical coframe ω 1 , ω 2 , ω 3 (see for instance [BCS], Chapter 4.4 for the more general Finsler case with a different ordering of the coframe) satisfying\ndω 1 = −ω 2 ∧ω 3 dω 2 = −ω 3 ∧ω 1 dω 3 = −Kω 1 ∧ω 2 , (5.4)\nwhere K is the Gauss curvature of (Σ, g). Assume that g is a metric of constant Gauss curvature. Then these manifolds provide models of homogeneous contact manifolds. 5.2.1. Homogeneous contact structure associated with the canonical contact structure \"pdq\". We consider the standard contact structure (\"pdq\") on the unit cotangent bundle of (Σ, g). With respect to the canonical coframe (5.4), the defining form for this contact structure is ω 1 .\n\nConsider time-dependent functions a t , b t , c t that are constant in space, and define the coframe\nθ bt = b t 2 ω 1 , α bt = b t ω 3 , β bt = b t ω 2 .\nWith this ordering, we obtain the structure coefficients c 1 23 = 1, c 2 13 = −K, and c 3 12 = 1 (take b 0 = 1), and all other coefficients vanish. With the standard choice of complex structure J abc , we obtain a pseudohermitian manifold (ST * Σ, θ bt , J atbtct ). We compute the torsion and Webster curvature with the formulas from Proposition 5.4:\nA 11 atbtct = i b 2 t a 2 t + 1 2c t + c t 2 a t + i a t − i K = − a t c t a 2 t + 1 K + i a 2 t + 1 2c t c 3 12 + c t K 2 a 2 t − 1 a 2 t + 1 , W atbtct = 1 b 2 t a 2 t + 1 2c t + c t 2 K .\n(5.5)\n\nWe specialize to the case that a = 0 and substitute B(t) = b(t) 2 . By Proposition 4.7 the (unnormalized) torsion flow reduces to ODĖ c = − 2ac (a 2 + 1)\n2 K b 2 + 1 b 2 a 2 + 1 c + c a 2 − 1 a 2 + 1 K (1 − a 2 )c a 2 + 1 = 1 − c 2 K Ḃ B = −cK − 1 c c(0) = c 0 B(0) = (b 0 ) 2 .\nIf Kc 2 0 = 1, then the solution to this system is given by\nc(t) = c 0 e (1−Kc 2 0 )t b 2 0 c 0 B(t) = Kc 2 0 e (1−Kc 2 0 )t b 2 0 c 0 2 − 1 (Kc 2 0 − 1)e (1−Kc 2 0 )t b 2 0 c 0 (b 0 ) 2 .\nIf Kc 2 0 = 1, which can only happen if K > 0, then\nc(t) = c 0 B(t) = b 2 0 − t c 0 c 2 0 · K + 1 .\nWe draw some conclusions:\n\n• For K ≤ 0, the solution exists for all time. For K = 0 (torus case), one has the curious property that the Webster curvature is constant. The torsion is also constant in that case, when measured in our coframe θ b , α b , β b . For all K ≤ 0, the torsion flow skews the complex structure more and more. The limit lim t→∞ c (t) = ∞, so in the limit, the complex structure blows up. • For K > 0, the solution blows up in finite time because of shrinking: b(t) = B(t) reaches 0 in finite time. The special case Kc 2 0 = 1 corresponds to vanishing torsion.\n\nRemark 5.15. We point out that, with its canonical contact structure \"pdq\", only the unit cotangent bundle of S 2 admits a complex structure for which the torsion vanishes. Indeed, all other unit cotangent bundles of surfaces with constant Gauss curvature are not K-contact, which is a necessary requirement by the appendix of Weinstein in [CH].\n\nIn this specialized case a = 0, the volume-normalized flow is particularly simple. We havė\nc (t) = 1 − Kc 2 (t) .\nWe see the following\n\n(1) if K > 0, then there exists a torsion free complex structure, namely for c ∞ = 1/ √ K. We see that the torsion flow exists for all time, and that it converges to this torsion free complex structure.\n\n(2) if K = 0 (the torus case), then c increases linearly. The flow exists for all time, but the complex structure does not converge. (3) if K < 0, then c blows up in finite time. Geometrically, we see by (5.5) that torsion grows in norm, and the Webster curvature becomes more and more negative. Accordingly, the complex structure blows up.\n\nNote that ST * S 2 ∼ = SO(3), so alternatively we can apply Proposition 5.7 to the case K > 0.\n\n\n5.2.2.\n\nPrequantization structures on ST * Σ. We consider again the canonical coframe on the unit cotangent bundle with structure coefficients as in (5.4) for a surface with constant Gauss curvature. If Σ is not a torus, then we define the following coframe\nθ b = b 2 ω 2 , α b = −bKω 1 , β b = bω 3 .\nThe resulting contact manifold is known as a prequantization bundle, a circle bundle over a symplectic manifold (here Σ) whose fibers are periodic Reeb orbits. The corresponding structure coefficients are now c 1 23 = 1, c 2 13 = −K, c 3 12 = 1 K , and all other coefficients vanish. By defining J abc as before, we obtain a pseudohermitian manifold (ST * Σ, θ b , J abc ). Its torsion and Webster curvature are given by\nA 11 abc = i b 2 a 2 + 1 2c 1 K + c(a + i) 2(a − i) K , W abc = 1 b 2 a 2 + 1 2c 1 K + c 2 K .\nRemark 5.16. If Σ = S 2 , then the resulting contact structure is not contactomorphic to the \"pdq\"-structure from the previous section. Also, the contact structure is now K-contact, and we can choose a complex structure with vanishing torsion. Indeed, choose a = 0, and c = 1 |K| , and the torsion tensor will vanish.\n\nAs in the previous section we specialize to the case that a = 0 and substitute B(t) = b(t) 2 . By If we start the flow at a 0 = 0 and c 0 = 1 |K| , then we have vanishing torsion, and the torsion flow just contracts or expands depending on the sign of the Webster curvature. We have\nB(t) = B 0 − 2|K| K t.\nThe normalized torsion for these homogeneous contact manifolds are covered by Proposition 5.7. 5.2.3. Heisenberg geometry. As an explicit, compact example with Heisenberg geometry, consider the 2-torus with standard symplectic form (T 2 , Ω = dφ 1 ∧ dφ 2 ). There is a principal circle bundle p : P → T 2 whose connection form θ satisfies dθ = p * Ω. We see that (P, θ, α = dφ 1 , β = dφ 2 ) is a homogeneous contact manifold of Heisenberg type. Indeed, all structure coefficients except c 1 23 vanish. Hence any homogeneous CR-structure has vanishing torsion and Webster curvature. It follows that the torsion flow is constant, so this is an explicit example of a torsion soliton, namely a steady breather, see Corollary 1.9, case (i).\n\n\nEntropy functionals\n\nThe following section discusses entropy functionals on a closed 3-dimensional pseudohermitian manifold (M, J, θ).\n\n6.1. The Entropy F-Functional. Let (M, J, θ) be a closed pseudohermitian 3-manifold. In this section, we study the monotonicity property of the F-functional F(J (t) , θ (t) , ϕ (t) ) = M (W + |∇ b ϕ| 2 J,θ )e −ϕ dµ with the constraint M e −ϕ dµ = 1 under the coupled torsion flow (1.4).\n\nProof. We compute ∂ ∂t ∇ b φ with Equation (3.2) and use the result to obtain (6.1) ∂ ∂t |∇ b ϕ| 2 J,θ = 4 Re (iE 11 ϕ 1 ϕ 1 ) + 2 ∇ b ϕ, ∇ b ϕ t J,θ − 2η (t) |∇ b ϕ| 2 J,θ .\n\nBy (1.4) we find (6.2) ∂ ∂t dµ = 4η (t) dµ.\n\nUse these formulas together with Equation (3.8) to compute the variation of the F-functional,\n(6.3) − 1 2 d dt F(J (t) , θ (t) , ϕ (t) ) = − M η (t) [W + |∇ b ϕ| 2 J,θ ]e −ϕ dµ + 2 M (∆ b η (t) )e −ϕ dµ + 1 2 M (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W )ϕ t e −ϕ dµ −2 M Re (iE 11 ϕ 1 ϕ 1 ) e −ϕ dµ − M Re iE 11,11 − A 11 E 11 e −ϕ dµ = M ( 1 2 ϕ t − η (t) )(2∆ b ϕ − |∇ b ϕ| 2 J,θ + W )e −ϕ dµ + M Re[(A 11 − iϕ 11 − iϕ 1 ϕ 1 ) E 11 ]e −ϕ dµ = M η (t) (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W )e −ϕ dµ + M Re[(A 11 − iϕ 11 − iϕ 1 ϕ 1 ) E 11 ]e −ϕ dµ.\nWe first set E 11 = e ϕ (A 11 − iϕ 11 − iϕ 1 ϕ 1 ) and η (t) = e ϕ (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W ), then − 1 2 d dt F(J (t) , θ (t) , ϕ (t) ) = M (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W ) 2 dµ + M |A 11 − iϕ 11 − iϕ 1 ϕ 1 | 2 dµ ≥ 0.\n\nThe monotonicity formula is strict unless A 11 − iϕ 11 − iϕ 1 ϕ 1 = 0 and 2∆ b ϕ − |∇ b ϕ| 2 J,θ + W = 0.\n\nMoreover, up to a contact transformation θ = e −ϕ θ A 11 = 0, W = 0.\n\nThis completes the proof of Theorem 1.10.\n\n6.2. The Entropy W ± -Functionals. We study the monotonicity property of the W + -functional Proof. Following the same computations as in the proof of Theorem \n1 b X 2 , V b = 1 b X 3 .\nWe first work out some general formulas. Given the second equation of (2.2), we can assume without loss of generality that (7.1) ω 1 1 = ic θ θ + ic Z θ 1 + ic Z θ 1 , with c θ a real function. Now we insert our frame to get explicit equations dθ 1 (Z 1 , Z1) = ω 1 1 (Z1) = ic Z dθ 1 (T, Z 1 ) = −ic θ dθ 1 (T, Z1) = A 11 .\n\n\n(7.2)\n\nThe Webster curvature is determined by the second structure equation, dω 1 1 = W θ 1 ∧θ1 + 2i Im(A1 1,1 θ 1 ∧θ). Wedging this equation with θ gives θ∧dω 1 1 = W θ∧θ 1 ∧θ1, which can be rewritten by using (7.1) to write out dω 1 1 . We find θ∧dω 1 1 = −c θ − 2|c Z | 2 − 2 Im(Z 1 (c Z ) ) θ∧θ 1 ∧θ1, and conclude (7.3) W = −c θ − 2|c Z | 2 − 2 Im(Z 1 (c Z ) ).\n\nWe now determine these coefficients. For c abc θ , we use the second equation from (7.2) to get\n−ic abc θ b = − i(a + i) 2 dα b (T b , U b ) + a 2 + 1 2c idβ b (T b , U b ) − ci 2 dα b (T b , V b ) + i 2 (a − i)dβ b (T b , V b )\n= − i(a + i) 2b 2 c 2 12 + i a 2 + 1 2cb 2 c 3 12 − ci 2b 2 c 2 13 + i 2b 2 (a − i)c 3 13 .\n\nUse Lemma 5.3 to see that c 2 12 = c 3 13 = 0. For the c Z -component, we use the first equation from (7.2) we compute ic Z = dθ 1 (Z 1 , Z1) = 1 2c(a 2 + 1) − i 2(a − i) dα b (U b , V b ) (a 2 + 1)c(a + i) − c(a − i)(a 2 + 1)\n+ i 2c dβ b (U b , V b ) (a 2 + 1)c(a + i) − c(a − i)(a 2 + 1) = 1\n2c(a 2 + 1) c(a + i)c 2 23 − (a 2 + 1)c 3 23 .\n\nBy the above formulas, these coefficients determine the connection form ω 1 1 and the curvature W . For the torsion we use formulas (4.4), (4.3) and (7.2), and find\nA 11 = dθ 1 b (T b , Z b,1 ) = i 2(a − i) dα b (T b (a 2 + 1)U b + c(a + i)V b ) + i 2c dβ b (T b , (a 2 + 1)U b + c(a + i)V b )\n= i 2(a − i)b 2 (a 2 + 1)c 2 12 + c(a + i)c 2 13 + i 2cb 2 (a 2 + 1)c 3 12 + c(a + i)c 3 13 .\n\nCombining this with c 2 12 = c 3 13 = 0 gives the desired expression for the torsion.\n\nLemma 4 . 1 .\n41Let (M, ξ = ker θ) be a contact 3-manifold. Then there is a global trivialization U, V of its contact structure if and only if c 1 (ξ) = 0.\n\n. 3 .\n3The vector field T b is the Reeb vector field for θ b . Furthermore, we have dθ b = α b ∧β b .\n\nProposition 4. 7 .\n7Let (M 3 , θ, J) be a CR-manifold with c 1 (ξ, J) = 0. Then there exists a basis of T * M , and functions a, b, c such that• the complex structure J can be written as J abc .• the torsion flow (1.1) is equivalent to the systeṁ a = 2 Re(A abc 11 ) + a Im(\n\n.\n0, then the normalized torsion flow blows up in finite time and the complex structure does not converge unless a 0 = 0 and c In the latter case, the torsion vanishes and the torsion flow is constant.Proof. In both cases, the ODE describing the normalized torsion flow has a unique\n\nFigure 1 .\n1Phase Indeed, the function |a| is decreasing, and the only limit can be a ∞ = 0.From the second equation we then deduce that the function c converges to c ∞ = − that solutions in the repelling case blow up unless (a 0 , c 0 ) =\n\nExample 5. 13 (\n13Rossi's examples). We recall Rossi's examples of non-embeddable CR-manifolds. Define the strictly plurisubharmonic function\n\nW\n+ (J (t) , θ (t) , ϕ (t) , τ (t) ) = M [τ (W + |∇ b ϕ| 2 J,θ ) + 1 2 ϕ − 1](4πτ ) −2 e −ϕ dµwith the constraint M (4πτ ) −2 e −ϕ dµ = 1 under the coupled torsion flow (1.6).\n\n\nof Proposition 1.2. Write\n\n\n1.10, we can derive that d dt M [τ (W + |∇ b ϕ| Proof. Following the same computations as in the proof of Theorem 1.13, we can derive that d dt M [τ (W + |∇ b ϕ|\nDepartment of Mathematics and Taida Institute for Mathematical Sciences, National Taiwan Uni-\nAcknowledgements We thank Jih-Hsin Cheng for valuable discussions, and we are in particular grateful to his contributions on the entropy formulas and short time existence, where he suggested the correct formulation of entropy; he also helped us out with the linearized operator in the current proof of short time existence.The research of SCC and CTW is supported in part by NSC of Taiwan. 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Perrone, Homogeneous contact Riemannian three-manifolds, Illinois J. Math. 42 (1998), no. 2, 243256.\n\nN Tanaka, A Differential Geometric Study on Strongly Pseudoconvex Manifolds. TokyoKinokuniya Co. LtdN. Tanaka, A Differential Geometric Study on Strongly Pseudoconvex Manifolds, 1975, Kinokuniya Co. Ltd., Tokyo\n\nPseudohermitian structures on a real hypersurface. S M Webster, J. Diff. Geom. 13S. M. Webster, Pseudohermitian structures on a real hypersurface, J. Diff. Geom. 13 (1978), 25-41.\n\nDifferential analysis on complex manifolds. Prentice-Hall Series in Modern Analysis. R O WellsJr, Prentice-Hall, Inc252Englewood Cliffs, N.J.R. O. Wells, Jr. , Differential analysis on complex manifolds. Prentice-Hall Series in Modern Analysis. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. x+252 pp.\n"},"corpus_id":{"kind":"number","value":119309615,"string":"119,309,615"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1305.5391\",\n \"mag\": 2129172854,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1305.5391v2.pdf\"\n ],\n \"pdf_sha\": \"e486d0cfa016026532801bab53c7e036ffb97df2\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"THE TORSION FLOW ON A CLOSED PSEUDOHERMITIAN 3-MANIFOLD"},"authors":{"kind":"list like","value":["Shu-Cheng Chang","Otto Van Koert","Chin-Tung Wu"],"string":"[\n \"Shu-Cheng Chang\",\n \"Otto Van Koert\",\n \"Chin-Tung Wu\"\n]","isConversational":false},"abstract":{"kind":"string","value":"In this paper we define the torsion flow, a CR analogue of the Ricci flow. We show that the torsion flow on a closed, strictly pseudoconvex pseudohermitian manifold has short time existence for some favorable choice of initial conditions. For homogeneous CR manifolds we give explicit solutions to the torsion flow illustrating various kinds of behavior. We also derive monotonicity formulas for CR entropy functionals. As an application, we classify torsion breathers.2010 Mathematics Subject Classification. 53C21, 32G07.1.1. Motivation for the torsion flow and statement of results. For the basic definitions and notions involved, we refer the reader to Section 2. Consider a closed 2n + 1-manifold M , with a smooth family of pseudohermitian structures (J (t) , θ (t) ) for which J (t) is compatible with dθ (t) : this means thatforms a hermitian metric on the complex vector bundle (ξ (t) = ker θ (t) , J (t) ). Furthermore, H (t) induces a metric on all tensor fields. We shall use these metrics and the induced norms without explicitly referring to H (t) . Throughout the paper, we only consider a fixed contact structure, i.e. ξ (t) = ker θ (t) is independent of t. Henceforth, we just write ξ.Take a local orthonormal frame {T, Z α , Zβ}, where T is the Reeb field, {Z α } is a basis of (ξ ⊗ C) 1,0 , and {Zβ} is a basis of (ξ ⊗ C) 0,1 . Then we write J = iθ α ⊗ Z α − iθ α ⊗ Z α . Define E = E αβ θ α ⊗ Zβ + Eᾱ β θᾱ ⊗ Z β , and consider the general flow on (M, J, θ) × [0, T ) given byThe CR Einstein-Hilbert functional is defined byHere dµ = θ ∧ dθ n is the volume form and W denotes Tanaka-Webster curvature. From computations later on, namely (3.8) and (6.2), it follows thatHere A J,θ := Aβ α Zβ ⊗ θ α + A βᾱ Z β ⊗ θᾱ denotes the torsion tensor. It is therefore natural to consider the torsion flow on M × [0, T ) as defined in (1.1). Unfortunately, we do not know whether a short-time solution to the torsion flow (1.1) exists in general. We use the following conjecture to guarantee a suitable initial condition for the flow.Conjecture 1.1. Let (M, J, θ) be a closed, strictly pseudoconvex CR 3-manifold whose first Chern class vanishes, c 1 (T 1,0 M ) = 0. Then there exists a volume-normalized contact form with respect to a nowhere vanishing closed section ζ of the canonical bundle Λ 2,0 M with the same conformal class as θ.This conjecture is known to hold in the several cases, and in the following proposition we have collected some of them.Proposition 1.2. The following criteria for the existence of volume-normalized contact forms hold true. 2 J,θ + W − τ −1 = 0.That is, up to a contact transformation θ = e −ϕ θ A 11 = 0 and W − τ −1 e ϕ = 0.Next we study the monotonicity property of W − -functional W − (J (t) , θ (t) , ϕ (t) , τ (t) ) = M [τ (W + |∇ b ϕ| 2 J,θ ) − 1 2 ϕ + 1](4πτ ) −2 e −ϕ dµ. By the same discussion as before, the constraint M (4πτ ) −2 e −ϕ dµ = 1 is equivalent to"},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"The Ricci flow, introduced by Hamilton, is a geometric flow for metrics on 3-manifolds, and has played a decisive role in the proof of the Poincaré conjecture and Thurston's geometrization conjecture for 3-manifolds. It is natural to then investigate a corresponding problem for contact 3manifolds. One of way of doing this is to find a CR analogue of the Ricci flow on a pseudohermitian 3-manifold (see Section 2 for definitions and basic notions in pseudohermitian geometry).\nRecall that a strictly pseudoconvex CR structure on a 3-manifold M is given by a cooriented plane field ker θ, where θ is a contact form, together with a compatible complex structure J. This gives rise to a natural metric g = θ ⊗ θ + dθ(·, J·) for M . Given this data, there is a natural connection, the so-called Tanaka-Webster connection or pseudohermitian connection. We denote the torsion of this connection by A J,θ , and the Webster curvature, a kind of scalar curvature, by W . The torsion flow is then the following PDE,\nIt seems to us that the torsion flow (1.1) is the right CR analogue of the Ricci flow.\nThe torsion flow greatly simplifies if the torsion vanishes. This only happens in very special setups. Indeed, CR 3-manifolds with vanishing torsion are K-contact, meaning that the Reeb vector field is a Killing vector field for the metric g. In general, one can still hope that the torsion flow improves properties of the contact manifold underlying the CR-manifold. Like the Ricci flow, the torsion flow is a weakly parabolic PDE, provided some suitable initial conditions hold, and is hard to study directly.\nThe mostly used tools in the study of Hamilton's Ricci flow [H1] consist of maximum principles. Exceptions are formed by Hamilton's entropy formula which holds for closed surfaces with positive Gaussian curvature [H2], and also by Perelman's entropy formulas [Pe]. These formulas can be thought of as monotonicity formulas for integrals of local geometric quantities.\nIn this paper, we try to do the same for the torsion flow by setting up some monotonicity formulas for Perelman-type functionals. We also give a short time existence result which works in any dimension, but unfortunately it requires additional assumptions in dimension 3.\nWe conclude this introduction with a brief plan of the paper.\n• In Section 1.1 and Section 1.2 we motivate the definition of the torsion flow and give more precise statements of our results. • In Section 2 we survey basic notions in CR geometry. • We prove a short time existence result for the torsion flow in Section 3.\n• In Sections 4 and 5 we describe CR manifolds with a global coframe and we also define homogeneous CR manifolds. On the latter class the torsion flow reduces to an ODE if we start with some appropriate initial conditions. These computations illustrate the behavior of the torsion flow in special cases, and in these cases the torsion flow behaves as can be expected from a Ricci-like flow. • Finally, in Section 6 we discuss analogues of Perelman's entropy formulas.\n(1) ( [FH]) Let (S 3 , J, θ) be a closed, strictly pseudoconvex CR 3-sphere. Thenθ = e 2f θ is a contact form with vanishing CR Q-curvature if and only ifθ is a volume-normalized contact form with respect to a nowhere vanishing closed section ζ of the bundle Λ 2,0 (S 3 ).\n(2) ( [Hi]) Suppose that (M, J, ξ = ker θ 0 ) has a transverse symmetry and θ 0 is a volumenormalized contact form. Then a contact form θ for ξ has vanishing CR Q-curvature if and only if θ is a volume-normalized contact form. In particular, this holds if (M, J, θ 0 ) is a closed hypersurface in C 2 with transverse symmetry.\n(3) ( [CC]) Let (M, J, ξ = ker θ 0 ) be a CR 3-manifold that is Stein fillable. Then there is contact form θ for ξ with vanishing CR Q-curvature. (4) For a homogeneous CR 3-manifold (M, J, θ) as defined in Section 5 the homogeneous contact form θ is volume-normalized.\nThe notions in this proposition are defined and discussed in Section 2.9. Further evidence for this conjecture is the following. Proposition 1.3. ( [CCT]) Let (M, J, θ) be a closed, strictly pseudoconvex CR-manifold of dimension 2n+1 satisfying c 1 (T 1,0 M ) = 0 and n ≥ 2. Then there is a smooth real-valued function u such that e 2u θ is a contact form of pseudo-Einstein type. In other words, e 2u θ is a volume-normalized contact form with respect to a nowhere vanishing closed section ζ of the bundle Λ n+1,0 (M ).\nOther related statements are discussed in Section 2.2. Note that the contact structure is fixed under the torsion flow. Since the Chern class of a contact structure is independent of the choice of compatible complex structure, we conclude that the condition c 1 (T 1,0 M ) = 0 is preserved under the torsion flow (1.1). Furthermore, Proposition 1.3 tells us that the higher-dimensional analogue of Conjecture 1.1 holds for a strictly pseudoconvex CR-manifold of dimension 2n + 1 with c 1 (T 1,0 M ) = 0 and n ≥ 2. This leads us to a suitable initial condition that guarantees the following short-time existence result for the torsion flow.\nTheorem 1.4. Let (M, J 0 , θ 0 ) be a closed pseudohermitian 2n + 1 -manifold with vanishing first Chern class, c 1 (T 1,0 M ) = 0. If n = 1, assume that Conjecture 1.1 holds. Then there exists δ > 0 and a unique smooth solution (J(t), θ(t) ) to the torsion flow (1.1) on the interval [0, δ) such that (J(0), θ(0) ) = (J 0 , θ 0 ). From Lemma 3.1 and part (ii) of Proposition 2.9 we deduce the following corollary, Corollary 1.5. Assume that (M, J 0 , θ 0 ) is a closed hypersurface in C 2 (so c 1 (T 1,0 M ) = 0) and has a transverse symmetry. Then there exists δ > 0 and a unique smooth solution (J(t), θ(t) ) to the torsion flow (1.1) on the interval [0, δ) such that (J(0), θ(0) ) = (J 0 , θ 0 ). Also, part (4) of Proposition 1.2 implies the corollary.\nCorollary 1.6. Let (M,Ĵ,θ) be a closed homogeneous pseudohermitian 3-manifold, so c 1 (T 1,0 M ) = 0. Then there exists δ > 0 and a unique smooth solution (J(t), θ(t) ) to the torsion flow (1.1) on the interval [0, δ) such that (J(0), θ(0) ) = (J 0 , θ 0 ).\nFor the homogeneous CR manifolds we define in Section 5, we also show the following convergence result.\nTheorem 1.7 (Convergence to torsion free CR structure). Let (M, {ω i } i , θ = ω 1 ) be a homogeneous contact manifold whose Lie algebra is isomorphic to su(2). Then there is a unique homogeneous complex structure J a∞,b=1,c∞ that is torsion free. Moreover, for any choice of homogeneous complex structure J a,b=1,c , the normalized torsion flow converges to this unique CR-structure (ker θ, J a∞,b=1,c∞ ).\nIn particular, for any choice of homogeneous complex structure on SU (2), the normalized torsion flow converges to the standard CR-structure.\nIn the next section we shall discuss somewhat technical results concerning monotonicity properties of Perelman type functionals. As an application of these monotonicity results from Section 1.2, we classify the torsion breathers and solitons. The classification of torsion solitons is a necessary step in understanding the singularity formation in the torsion flow. Indeed, one expects the torsion soliton solutions to model finite time singularities of the torsion flow. In view of the flow (1.9) and original definition in [Pe], it is natural to define the soliton solutions for torsion flow (1.1) as follows.\nDefinition 1.8. (i) A family J (t) of CR structures on (M, θ, J) evolving by the torsion flow (1.1) is called a breather if for some t 1 < t 2 and δ > 0, the CR structures J (t1) and J (t2) differ only by a contact diffeomorphism Φ with θ (t2) = λΦ * θ (t1) ; the cases λ = 1, λ < 1, λ > 1 correspond to steady, shrinking or expanding breathers, respectively.\n(ii) A breather satisfying the above properties for all pairs of t 1 and t 2 of real numbers is called a torsion soliton.\nIdeas of Perelman [Pe] (see also [Ca] and [Li]) can be combined with Theorem 1.10, Theorem 1.13 and Theorem 1.14 to show the following classification result. Corollary 1.9. Let (M, J, θ) be a closed pseudohermitian 3-manifold. Then (i) there is no closed steady torsion soliton other than the one which admits zero Tanaka-Webster curvature and vanishing pseudohermitian torsion up to a contact transformation.\n(ii) there is no closed expanding torsion soliton other than the one which admits negative Tanaka-Webster curvature and vanishing pseudohermitian torsion.\n(iii) there is no closed shrinking torsion soliton other than the one which admits positive Tanaka-Webster curvature and vanishing pseudohermitian torsion.\n1.2. Some monotonicity results for Perelman-type functionals. The statements in this section are of a more technical nature: we will derive the CR analogue of Perelman's monotonicity formulas for the so-called coupled torsion flows (1.4), (1.6) and (1.7) in dimension 3.\nIn Section 6.1 we define the CR analogue of Perelman's F-functional by\nUnder the flow (1.3), this is equivalent to M (ϕ t − 4η (t) )e −ϕ dµ = 0. Therefore, the following coupled torsion flow is natural,\nThe monotonicity formula is strict unless\nThat is, up to a contact transformation θ = e −ϕ θ A 11 = 0 and W = 0.\nRemark 1.11. Observe that for θ = e −ϕ θ,\nThen the coupled torsion flow (1.4) on (M, J (t) , θ (t) ) is equivalent to the following system of coupled torsion flows on (M,\nIn Section 6.2 we define two functionals analogous to Perelman's W-functional, namely the W + -functional,\nIn view of Theorem 1.10, we first study the monotonicity property of W + -functional. By the same discussion as before, the constraint M (4πτ ) −2 e −ϕ dµ = 1 is equivalent to another constraint, namely M (ϕ t + 2τ −1 dτ dt − 4η (t) )(4πτ ) −2 e −ϕ dµ = 0 under the flow (1.3). Therefore we consider the following coupled torsion flow:\nTheorem 1.13. Let (M, J, θ) be a closed 3-dimensional pseudohermitian manifold and J (t) , θ (t) , ϕ (t) and τ (t) be a solution of the coupled torsion flow (1.6). Then\nThe monotonicity formula is strict unless\nunder the flow (1.3). Therefore we consider the following coupled torsion flow :\n(1.7)\n. Theorem 1.14. Let (M, J, θ) be a closed 3-dimensional pseudohermitian manifold and J (t) , θ (t) , ϕ (t) and τ (t) be a solution of the coupled torsion flow (1.7). Then\nThe monotonicity formula is strict unless\nThat is, up to a contact transformation θ = e −ϕ θ A 11 = 0 and W + τ −1 e ϕ = 0.\nRemark 1.15. Note that for θ (t) = e −ϕ θ (t) , we may reparametrize the time t by the formula\n, so the coupled torsion flows (1.6) and (1.7) on (M, J (t) , θ (t) ) are equivalent to the following coupled torsion flows on (M, J ( t) , θ ( t) ), respectively :\nRecall that X f is called a contact vector field or an infinitesimal contact diffeomorphism if the Lie derivative L X f θ = ηθ for some function η. Such a contact vector field has the form\nwith f = e ϕ . Similar results hold for (1.6) and (1.7)."},{"title":"Preliminaries and definitions","content":"In this section we introduce some basic notions from pseudohermitian geometry. We learned many of these notions from [L1, L2], and we refer to these papers for proofs and more references.\nDefinition 2.1. Let M be a smooth manifold and ξ ⊂ T M a subbundle. A CR structure on ξ consists of an endomorphism J : ξ → ξ with J 2 = − Id such that the following integrability condition holds.\nThe CR structure J can be extended to ξ ⊗ C, which we can then decompose into the direct sum of eigenspaces of J. The eigenvalues of J are i and −i, and the corresponding eigenspaces will be denoted by T 1,0 and T 0,1 , respectively. The integrability condition can then be reformulated as\nNow consider a closed 2n + 1-manifold M with a cooriented contact structure ξ = ker θ. This means that θ ∧ dθ n = 0. The Reeb vector field of θ is the vector field T uniquely determined by the equations (2.1) θ(T ) = 1, and dθ(T, ·) = 0.\n• J is a CR structure on ker θ.\nDefinition 2.3. The Levi form , is the Hermitian form on T 1,0 defined by\nWe can extend this Hermitian form , to T 0,1 by defining Z, W = Z, W for all Z, W ∈ T 1,0 . Furthermore, the Levi form naturally induces a Hermitian form on the dual bundle of T 1,0 , and hence on all induced tensor bundles.\nWe now restrict ourselves to strictly pseudoconvex manifolds, or in other words compatible complex structures J. This means that the Levi form induces a Hermitian metric ·, · J,θ by V, U J,θ = dθ(V, JU ).\nThe associated norm is defined as usual: |V | 2 J,θ = V, V J,θ . It follows that H also gives rise to a Hermitian metric for T 1,0 , and hence we obtain Hermitian metrics on all induced tensor bundles. By integrating this Hermitian metric over M with respect to the volume form dµ = θ ∧ dθ n , we get an L 2 -inner product on the space of sections of each tensor bundle.\nDefinition 2.4. Let (M, J, ξ = ker θ 0 ) be a CR 3-manifold that is the smooth boundary of a bounded, strictly pseudoconvex domain in a complete Stein manifold V 4 . We shall call such a CR 3-manifold Stein fillable.\n2.1. Pseudohermitian connection. Let T, Z α , Zβ be a frame of T M ⊗ C, where {Z α } is any local frame of T 1,0 , and Zβ = Z β ∈ T 0,1 . Then θ, θ α , θβ , the coframe dual to T, Z α , Zβ , satisfies\nwhere h αβ is a positive definite matrix. By the Gram-Schmidt process we can always choose Z α such that h αβ = δ αβ ; throughout this paper, we shall take such a frame. The pseudohermitian connection or Tanaka-Webster connection of (J, θ) is the connection ∇ on T M ⊗ C (and extended to tensors) given in terms of a local frame {Z α } for T 1,0 by\nwhere ω α β is the 1-form uniquely determined by the following equations:\nHere τ α is called the pseudohermitian torsion, which we can also write as\nThe components A αβ satisfy A αβ = A βα . We often consider the torsion tensor given by\nThe following remark gives some geometric meaning to the pseudohermitian torsion.\nRemark 2.5. Let X f be the contact vector field for the real-valued function f ∈ C 2 (M ). Then we have [CL1,Lemma 3.4]. In particular, we have X f = T for f = 1, and the above equation reduces to L T J = 2JA J,θ , so we see that the torsion tensor measures to what extend the complex structure J is invariant under the Reeb flow.\nWe now consider the curvature of the Tanaka-Webster connection in terms of the coframe {θ = θ 0 , θ α , θβ}. The second structure equation gives\nIn [We,Formulas 1.33 and 1.35], Webster showed that the curvature Ω β α can be written as\nIn addition, by [L2, (2.4)] the coefficients W α β ρ are determined by the torsion,\nContraction of (2.3) yields\nDefinition 2.6. The pseudohermitian Ricci tensor is the tensor with components R ρσ . Its trace W := R ρ ρ is called the Webster curvature. If the pseudohermitian Ricci tensor is a scalar multiple of the Levi form, then we say that the pseudohermitian structure is pseudo-Einstein.\nRemark 2.7. From the definition it is clear that the Webster curvature is the analogue of the scalar curvature in Riemannian geometry, and we also see that the pseudo-Einstein condition mimics the Einstein condition. Unlike the Riemannian case, pseudo-Einstein structures do not necessarily have constant Webster curvature, even in higher dimensions.\nWe will denote components of covariant derivatives by indices preceded by a comma. For instance, we write A αβ,γ . Here the indices {0, α,β} indicate derivatives with respect to {T, Z α , Zβ}. For derivatives of a scalar function, we will often omit the comma. For example, ϕ α = Z α ϕ, ϕ αβ = ZβZ α ϕ − ω α γ (Zβ)Z γ ϕ, ϕ 0 = T ϕ for a (smooth) function ϕ. For a real-valued function ϕ, the subgradient ∇ b ϕ is defined as the unique vector field ∇ b ϕ ∈ ξ such that Z, ∇ b ϕ = dϕ(Z) for all vector fields Z tangent to the contact distribution ξ. Locally\nFinally, we also need commutation relations, sometimes called Ricci identities. We list a few of them. These cover 3-dimensional CR-manifolds, but we also need higher-dimensional versions which we do not list here. For the idea of the derivation, see [L2,Equation 2.15].\n(2.6) C I,01 − C I,10 = C I,1 A 11 − kC I, A 11,1 , C I,01 − C I,10 = C I,1 A 11 + kC I, A 11,1 , C I,11 − C I,11 = iC I,0 + kW C I .\nHere C I denotes a coefficient of a tensor with multi-index I consisting of only 1 and 1, and k is the number of 1's minus the number of 1's in I.\nNext we mention some curvature identities. The ones that are relevant for us are the contracted CR Bianchi identities from [L2, Lemma 2.2].\n(2.7)\n2.2. Volume-normalized contact forms and pseudo-Einstein CR-manifolds. To give our results some geometric meaning, we recall some results concerning volume-normalized contact forms and related notions. First recall the following definitions, taken from [L1].\nDefinition 2.8. Let (M, θ, J) be a closed pseudohermitian 3-manifold. Using Chern-Weil theory, we define the real first Chern class c 1 (T 1,0 (M )) as\nwhere ω α β is the connection form of the Tanaka-Webster connection in some local frame. Let (M 2n+1 , θ, J) be a (strictly pseudoconvex) CR-manifold. The canonical bundle of (M 2n+1 , θ, J) is the complex line bundle\nA contact form θ in a closed pseudohermitian 3-manifold M is said to be volume-normalized with respect to a nowhere vanishing section ζ of the canonical bundle Λ n+1,0 (M ) if\nwhere T is the Reeb vector field with respect to θ.\nWe say that a CR-manifold (M, J, ξ = ker θ 0 ) has a transverse symmetry if there exists a contact form θ in the same conformal class as θ 0 such that J is invariant under the flow of the Reeb vector field T θ of θ,\nEquivalently A J,θ = 0.\n2.2.1. Notions specific to dimension 3. For the remainder of this section we specialize to 3dimensional CR-manifolds.\nProposition 2.9. (i) ( [Hi]) Let (M, θ, J) be a closed pseudohermitian 3-manifold. A contact form θ is volume-normalized with respect to a non-vanishing closed section ζ in a closed pseudohermitian 3-manifold (M, J, θ) if and only if dω 1 1 = −id (W θ).\nThat is,\n(ii) ( [L1]) If M is a closed CR embedding of a hypersurface in C 2 , then θ = e 2f θ 0 is volumenormalized with respect to a closed section ζ = j * (dz 1 ∧ dz 2 ) for j : (M, θ 0 , J) → C 2 and some smooth function f (x).\nRemark 2.10. The condition W 1 − iA1 1,1 = 0 corresponds to the pseudo-Einstein condition for higher dimensional CR-manifolds. Indeed, suppose that n ≥ 2. Then according to [L2,Lemma 4.1], a contact form θ on a pseudohermitian manifold (M 2n+1 , J, ker θ) is pseudo-Einstein if for every admissible coframe, we have\nBy writing out this equation, we see that the following holds for a higher-dimensional pseudo-Einstein manifold,\nHowever, in dimension 3 any CR-manifold is pseudo-Einstein, since the metric tensor has only one component. For us, a modification of this condition will serve as an integrability condition for the torsion flow.\nDefine the tensor\nUse this to define the CR Q-curvature, see [Hi, CCC],\nThe last equality follows from [Hi,Lemma 5.4]. Clearly, vanishing of R 1 guarantees vanishing of the CR Q-curvature. For a converse, we already mentioned Proposition 1.2."},{"title":"Short time Existence","content":"In this section we give some results concerning the short-time existence of the torsion flow. As mentioned, the motivation for the integrability condition comes from higher-dimensional CRmanifolds, so we give proofs that also hold for these high-dimensional cases. We thank Jih-Hsin Cheng for valuable contributions in early computations in the 3-dimensional case, in particular involving the variation formulas and linearized operator. See also his paper for a related flow, [C].\nLet θ (t) be a family of smooth contact forms and J (t) be a family of CR structures on (M, J, θ). We consider the following flow on a closed pseudohermitian (2n + 1)-manifold (M, J, θ) × [0, T ):\nHere\nWe start by deriving some evolution equations under the general flow (3.1) before specifying to the torsion flow, for which E = A J (the torsion tensor), and η = −W (the Webster curvature). All computations will be done in a local frame. Fix a unit-length local frame {Z α } and let {θ α } be its dual admissible 1-form. Let Z α(t) , θ α (t) denote a unit-length frame and dual admissible 1-form with respect to (J (t) , θ (t) ). Since θ α (Z β(t) ) is a positive real function, we can write\nwhere F α β are real and G α β are complex. The fact that Z α(t) is an orthonormal frame means that\nBy differentiating and substituting the above expression for Z α , we obtain F α β = −ηδ β α . By differentiating J (t) Z α(t) = iZ α(t) we find"},{"title":"Now differentiate the identities","content":"Since we will derive an identity involving tensors, we will take an adapted frame satisfying ω γ α = 0 at a point. Plug in (3.3) and consider the θ∧θ γ terms to obtain\nOn the other hand, contracting (3.4) with Z β and then contracting with h βα , computing modulo θ γ yields\nSince ω α α is pure imaginary, we have\nDifferentiate the structure equation (2.4) with respect to t and consider only the θ ρ ∧ θσ terms. This gives\nAfter contracting with h ρσ we get\nRecall that the transformation law of the connection under a change of pseudohermitian structure was computed in [L1,Sec. 5]. Letθ = e 2f θ be another pseudohermitian structure. Then we can define an admissible coframe byθ α = e f (θ α + 2if α θ). With respect to this coframe, the connection 1-form and the pseudohermitian torsion are given by\nand (3.10)\nrespectively. Thus the Webster curvature transforms as\nHere covariant derivatives on the right side are taken with respect to the pseudohermitian structure θ and an admissible coframe θ α . Note also that the dual frame of {θ,θ α ,θ α } is given by\nWe need one more ingredient before we can prove Theorem 1.4. Let f be a smooth function, and define the CR pluriharmonic operator P α f by\nDefine the CR Paneitz real operator by\nWe have the following lemma.\nLemma 3.1. Let (M 2n+1 , J, ξ = kerθ) be a pseudohermitian manifold, and suppose, if n = 1, that Conjecture 1.1 holds for this manifold. Then there is a smooth function f such that\nand thus W α α − inA αβ , βα −2(n + 2)P θ f = 0.\nProof. By Conjecture 1.1 in case n = 1 and Proposition 1.3 in case n ≥ 2 there is a smooth function f such the rescaled contact form θ = e 2f θ is volume-normalized. By Proposition 2.9 and Remark 2.10, this volume-normalized contact form satisfies\nThe second contracted Bianchi identity from (2.7) implies then that\nFollowing the same computation as in the proof of Lemma 5.4 in [Hi], we find using (3.9), (3.10) and (3.11), that\nContract the second equation with respect to the Levi metric h γβ = h γβ . This yields\nThus we get\nWe will apply the following commutation relations, which are similar to those in (2.6)\nThe latter relation follows from (3.14). Together with the above we obtain the following transformation law\nApply (3.13) to complete the proof of (3.12).\nThe following lemma is needed to prove Corollary 1.5. We use the observation that CR 3manifolds with transverse symmetry are Stein fillable.\nLemma 3.2. Suppose that (M 3 , J 0 , ξ = ker θ 0 ) has a transverse symmetry and θ 0 is volumenormalized with respect to a non-vanishing closed section ζ. Assume furthermore that (J(t), θ(t) ) is a solution of the torsion flow (1.1) on the interval [0, T ]. Then for each t ∈ [0, T ], there exists a volume-normalized contact form θ (t) for (ξ, J(t) ).\nProof. Since (M, J (t) , ξ = ker θ (t) ) is Stein fillable, there exists a contact form θ (t) with vanishing CR Q-curvature by (3)\nwhere θ 0 is volume-normalized. By [Hi,Lemma 5.4,formula 5.7\nand, because Q 0 = 0 by (2) of Proposition 1.2 (also a result of Hirachi), we find\nSince M has a transverse symmetry, it follows from [Hi,Proposition 7.3] that f is a CR pluriharmonic function, and P 1 f = 0. This implies θ (t) is volume-normalized.\nProof of Theorem 1.4. We follow the arguments of Hamilton, [H1, Section 4], and adapt them to our setup. First rewrite the torsion flow (3.1) as\nStep 1: determining the linearization of the torsion flow We use δJ, δθ to denote the variations of J and θ, respectively. Set δJ = 2E and δθ = 2hθ,\nwhere E is an endomorphism: ξ → ξ satisfying J • E + E • J = 0 and h is a smooth function. Let δ J and δ θ denote the linearization operators with respect to J and θ. From (3.5) and (3.8), we see that\nWe first compute the linearization of 2A J,θ . Let O m denote an operator of weight ≤ m (see [CL1,page 234]), and define the total variation as δ = δ J + δ θ . PutẼ := E ⊕ hθ. With (3.16) we compute the variation of the torsion as\nThe linearization of −2W θ can be computed with (3.17),\nWith these linearizations (3.18) and (3.19) we write the highest weight operator L J,θ ( E) of the linearization δ(2Q(J, θ)) as\nStep 2: finding integrability conditions Following Hamilton's ideas from ([H1, section 5]), we find an integrability condition of the form H(J, θ)Q(J, θ) = 0 for the torsion flow (3.15). By Lemma 3.1 we find a function f satisfying equation (3.12). We then define O 0 (A J,θ ⊕ −W θ) := 2(n + 2)P α f . We use this function to get an integrability condition. Define the linear operator H(J, θ) by\nThis operator has degree 1 in E := E ⊕ hθ; the term O 0 has lower order. By (3.12) we see that Q(J, θ) satisfies the integrability condition\nBy taking first variations, we find\nWe continue to argue as in [H1, section 4]. The operator in E given by δH(J, θ) has degree 1, so its symbol, which we denote by σH, of maximal degree vanishes. The other term, H(J, θ)δQ(J, θ), has maximal degree since H has degree 1 and δQ has positive degree. The symbols of linear differential operators H(J, θ) and Q(J, θ) therefore satisfy (3.22) [σH(J, θ)](·) • σ[δQ(J, θ)](·)( E) = 0, so we conclude that the image of σ[δQ(J, θ)] must lie in the kernel of σH(J, θ). We now want to show that the restriction of the symbol σδ2Q(J, θ)(ξ) to ker σH has only positive eigenvalues when ξ = 0. To see this, we first need additional commutation relations, again similar to those in (2.6), namely\nFrom these relations we derive that on the subspace\nthe following identities hold true,\nTherefore it follows from (3.20) that the highest weight operator L J,θ ( E) of the linearization δ(2Q(J, θ)) is\nθ} on ker H(J, θ)( E). Note here that both the Folland-Stein operator L α , given for n ≥ 2 by\nand the sublaplacian ∆ b are subelliptic, see [FS]. Therefore it follows from (3.22) and (3.25) that all eigenvalues of the eigenspaces of σ[δQ(J, θ)] in ker σH(J, θ) are strictly positive.\nAlternatively, note that the proof of Lemma 3.1 can also be used to show that there are no degeneracies for the torsion flow (3.1) other than those implied by the second contracted Bianchi identity from (2.7).\nStep 3: solving the linearized equations and handling the non-linear problem\nTo complete the argument, we follow [H1, Section 5 and Section 6]. The linearized equations can be solved using the regularization trick from [H1, Section 6]. Theorem 5.1 of [H1] can then be used to obtain short-time existence of the torsion flow (3.1). The main ingredient is the Nash-Moser inverse function theorem ( [H3])."},{"title":"Contact 3-manifolds with a global frame and pseudohermitian structures","content":"We now specialize to 3-dimensional pseudohermitian manifolds. Let (M, ξ = ker θ) be a cooriented contact 3-manifold. Denote the Reeb field by T . Furthermore, in this section and in the next, Section 5, we shall assume that ξ admits a global symplectic trivialization, i.e. there are vector fields U, V such that ξ = Span(U, V ) and dθ(U, V ) = 1. Proof. The contact structure ξ admits the structure of a symplectic vector bundle (ξ, dθ). By choosing a compatible complex structure J, we obtain a complex line bundle (ξ, J). It is wellknown that smooth complex line bundles are trivial if and only if their first Chern class vanishes, see [W,Chapter III,Section 4].\nThe last step needs the first Chern class with integer coefficients. Chern-Weil theory will not suffice in general. Henceforth, we shall assume that the globally defined vector fields U, V form a symplectic basis of (ξ, dθ). Consider the coframe θ, α, β dual to T, U, V . Then dθ(U, V ) = 1, so (4.1) dθ = α∧β.\nLemma 4.2. Let J be a compatible complex structure for the symplectic vector bundle (ξ = ker θ, dθ). Then there are smooth functions a : M → R and c : M → R >0 such that, with respect to the frame U, V , the complex structure J is represented by the matrix\nProof. With respect to the global frame U, V , the endomorphism J is represented by a 2 × 2matrix. Writing out the condition J 2 = −id shows that the matrix representation for J has the above form. The compatibility condition means that dθ(·, J·) is a metric, so it is represented by a positive definite matrix. Writing out this matrix shows that c is a positive function.\nThe following is motivated by our goal to convert the torsion flow (a PDE for tensors) into a PDE for functions. Choose real-valued functions a, b and c where b and c are positive. We attach super-and subscripts to indicate the dependence on these functions. In order to keep track of deformations of the contact form, we express all data in the given frame T, U, V . Define This can be checked by plugging in the vector field into the defining equations (2.1). The second assertion is obtained by writing out the terms.\nRemark 4.4. If b is a constant function, then the deformation from (4.2) corresponds to a ξhomothetic deformation as defined in [Bl,Section 10.4]. We take b 2 in θ b to have fewer expressions with square roots. Define a complex structure by\nBy Lemma 4.2, this is the most general choice.\nRemark 4.5. In higher dimensions, strictly pseudoconvex CR-manifolds require an integrability condition, see Definition 2.1, which is trivially satisfied in dimension 3.\nWe now compute the Tanaka-Webster connection as in Section 2.1. We use the coframe θ, θ 1 , θ1, where\nThis satisfies our normalization condition\n(4.4) 4.1. Converting the torsion flow into a system of PDE's for the functions a, b, c. To write down the equations of the torsion flow, we need the work out the torsion tensor. We have\nHence the first equation of the torsion flow (1.1),J abc = 2A abc J,θ is equivalent to the systeṁ a = 2 Re(A abc 11 ) + a Im(A abc 11 )\nso we obtain the above system by looking at the coefficients of U ⊗ α, U ⊗ β, V ⊗ α, and V ⊗ β. This works since these tensors are time-independent.\nLemma 4.6 (A smaller system for the J-part of the torsion flow). The system given by (4.5) is equivalent to system given bẏ a = 2 Re(A abc 11 ) + a Im(A abc 11 ) ċ = −2 Re(A abc 11 ) −2ac a 2 + 1 + Im(A abc 11 )\n(1 − a 2 )c a 2 + 1 . (4.6)\nProof. The first equation of (4.5) implies the fourth. We now verify that the first and second equation of (4.5) imply the third.\nOn the other hand, the second equation of the torsion flow (1.1) reduces to\nso we can reduce the torsion flow to a system of PDE's for the functions a, b, c, giving us the following proposition. (1 − a 2 )c a 2 + 1 d dt b 2 = −2W abc b 2 .\n(4.7)\nRemark 4.8. Spatial derivatives of a, b, c come in via the definition of torsion and Webster curvature."},{"title":"3-manifolds with constant structure constants and the Tanaka connection","content":"In this section we consider manifolds M 3 that admit global 1-forms ω 1 , ω 2 , ω 3 such that (1) ω 1 , ω 2 , ω 3 form a basis of T * M .\n(2) The structure coefficients are constant, i.e.\nwhere c i are constant. We shall call such a contact manifold a homogeneous contact manifold. This terminology is not standard, but it serves a useful purpose in this note. Let us point out that a related, but not equivalent notion, also referred to as homogeneous contact, was used by Perrone, [Pr].\nRemark 5.1. The structure coefficients are the structure constants of some 3-dimensional Liealgebra. Indeed, the dual frame {X 1 , X 2 , X 3 } satisfies\nthe Lie bracket on vector fields satisfies the Jacobi identity. We shall call this the Lie algebra of a homogeneous contact manifold.\nFrom Lemma 4.1 we get immediately.\nLemma 5.2. Homogeneous contact manifolds have trivial Chern class.\nBefore we define a CR structure on such manifolds, we use the following lemma to provide a better coframe. In many cases, this lemma can be improved upon, but this version is sufficiently convenient.\nLemma 5.3. Let (M, {ω i } i , θ) be a homogeneous contact manifold. Then there is a basis {ω i } i such that\n•ω 1 is contact, and c 1 23 = 1. • The structure coefficients dω i = j,k c i jkω j ∧ω k satisfy c 1 12 = c 1 13 = c 2 12 = c 3 13 = 0. Proof. Choose a compatible complex structure J for (ξ, dθ). Consider the operator h := 1 2 L T J. From [Bl,Lemma 6.2] we see that h is self-adjoint with respect to the metric dθ(·, J·), and we also get the identity\nSince h is self-adjoint, we can find a basis of eigenvectors X, Y of h for ξ. If the eigenvalue of X is λ, then JX is an eigenvector with eigenvalue −λ,\nso we can assume that Y = JX. We consider the Levi-Civita connection ∇ for g = θ ⊗θ +dθ(·, J·). Take the basis e 1 = T, e 2 = X, e 3 = JX, where T is the Reeb field of θ. Then [e 1 , e 2 ] = [T, X] = ∇ T X − ∇ X T = +JX + λJX − µJX.\nfor some µ ∈ R. In the last step we have used the identity (see [Bl,Lemma 6.2] )\nThe same steps work for [e 1 , e 3 ], so we conclude that there are constants C 1 , C 2 such that [e 1 , e 2 ] = C 1 e 3 [e 1 , e 3 ] = C 2 e 2 .\nConsider the dual basis {ω i } i . Then ω 1 is a contact form, and since T is the Reeb field, we have 0 = i e1 dω 1 = c 1 12 ω 2 + c 1 13 ω 3 . Hence c 1 12 = c 1 13 = 0, and from Lie bracket computations we see that c 2 12 = c 3 13 = 0. By rescaling, we see that the claim holds.\nWe assume now that ω 1 , ω 2 , ω 3 is a basis that is provided by this lemma. Take the basis X 1 , X 2 , X 3 that is dual to ω 1 , ω 2 , ω 3 , i.e. ω i (X j ) = δ i j . We have i X1 ω 1 = 1 and i X1 dω 1 = k c 1 1k ω k = 0, so X 1 is the Reeb vector field for ω 1 . Note that ξ = ker ω 1 = Span(X 2 , X 3 ). Also, if c 1 23 = 1, then dω 1 = ω 2 ∧ω 3 , so we have the right normalization convention for the setup of the Tanaka connection described in Formula (4.1) with θ = ω 1 , α = ω 2 and β = ω 3 .\nChoose constants a ∈ R and b, c > 0, and define a complex structure on ξ (or CR-structure on M ) following Lemma 4.2 by\nWe call such an endomorphism a homogeneous complex structure, and we refer to a homogeneous contact manifold together with the above complex structure as a homogeneous CRmanifold or a homogeneous CR structure. By direct computation, we obtain the following result for the Tanaka-Webster connection, its torsion and the Webster curvature.\nProposition 5.4. Let (M, {ω i } i , θ) be a homogeneous contact manifold with basis provided by Lemma 5.3. Fix a ∈ R and b, c > 0 and define J abc as in formula (5.1). Then the connection form for the Tanaka connection of the pseudohermitian manifold (M, θ b = b 2 θ, J abc ) is given by\nIts torsion is given by\nand its Webster curvature by\nProof. See the computations in the appendix. Alternatively, these computations are essentially also contained in [Pr]. Note that Perrone uses a J-basis, that is e 1 = T, e 2 , e 3 = Je 2 .\nWe can now reduce the torsion flow for homogeneous CR-manifolds to an ODE by plugging in the results of Proposition 5.4 into Proposition 4.7. The general system is fairly complicated, so we will work out some interesting case in Section 5.2.\nWe shall also consider the normalized torsion flow which, in general, is given by the system\nwhereW = M W θ∧dθ/ M θ∧dθ. Since the Webster curvature is constant in space for a homogeneous CR-manifold, then second equation of the normalized is flow is trivial. Inserting the result of Proposition 5.4 into the explicit system provided by Proposition 4.7 gives the following.\nProposition 5.5 (Normalized torsion flow for homogeneous CR-manifolds). Let (M, {ω i } i , θ) be a homogeneous contact manifold with θ = ω 1 . Set b = 1, and let a t , c t be real valued functions Then for a complex structure J atbct as defined in (5.1), the normalized torsion flow satisfies the ODEȧ\n(5.3)"},{"title":"Homogeneous CR-manifolds and vanishing torsion.","content":"Observe that homogeneous CRmanifolds need not be compact. In Section 5.2 we give some compact examples, but we start out by characterizing homogeneous CR-manifolds with vanishing torsion directly in terms of the structure coefficients and coefficients for the CR-structure a, b, c.\nProposition 5.6. Let (M, {ω i } i , θ) be a homogeneous contact manifold with θ = ω 1 and basis provided by Lemma 5.3. For a ∈ R, b = 1 and c > 0, a complex structure J abc as defined in (5.1) has vanishing torsion precisely when one of the following conditions is satisfied.\n1 if a = 0, c 2 12 = 0, and c = − 2 if a = 0, c 2 13 = c 2 12 = c 3 12 = 0. Proposition 5.7 (Convergence of the torsion flow). Let (M, {ω i } i , θ) be a homogeneous contact manifold with θ = ω 1 and a basis provided by Lemma 5.3. Take a ∈ R, b = 1 and c > 0, and let J abc be a complex structure as defined in (5.1).\nA If c 3 12 > 0 and c 2 13 < 0, then the normalized torsion flow converges to the unique torsion free complex structure of Proposition 5.6 ) in the upper half-plane with coordinates (a, c). The phase diagram for the repelling (R) case is given in Figure 1 The phase diagram for the attracting (A) case is similar, but the arrows are reversed. It is clear that the attracting case ), we make two observations.\n• any initial condition (a 0 , c 0 ) that starts in the set given by |c 2 13 |c 2 0 + |c 3 12 |a 2 0 − |c 3 12 | > 0 blows up in finite time. Indeed, the c-coordinate is strictly increasing in that case, so there is t 0 such that c t0 > 1 or the solution blows up before t 0 . If c t0 > 1 then the solution blows up in finite by the following argument. We havė c = |c 3 12 |c 2 + |c 2 13 |(a 2 − 1) ≥ |c 3 12 |c 2 − |c 2 13 |, which must blow up in finite time. with c 0 > 0 exits the half-disk {|c 2 13 |c 2 0 + |c 3 12 |a 2 0 − |c 3 12 | ≤ 0} in some finite time t 1 . Then either c t1 = 0, meaning that a blows up in finite time, or we can reduce to the first case.\nAn alternative proof can be given by starting with a J-basis. Then a 0 = 0 and c 0 = 1. Such a basis is not preserved under the torsion flow, but the condition a t = 0 holds. The resulting ODE is simpler to analyze. This method is applied in Section 5.2.\nRemark 5.8. By Proposition 5.4 there is a relation between the Webster curvature and the sign of c 3 12 . However, it is not true that repelling is equivalent to negative curvature.\n5.1.1. Homogeneous CR structures on unimodular Lie groups. We recall the definitions involved.\nDefinition 5.9. A Lie group G is called unimodular if the left-invariant Haar measure is also right invariant. A Lie algebra (g, [·, ·]) is called unimodular if T r ad X = 0 for all X ∈ g.\nNote that the Lie algebra of a unimodular Lie group is unimodular. The unimodularity of a Lie group can be used to simplify the structure coefficients of the Lie algebra.\nLemma 5.10. Let G be a unimodular, 3-dimensional Lie group admitting a homogeneous contact structure ξ = ker θ. Then there is a basis {e i } i of the Lie algebra g such that • e 1 is the Reeb vector field, and e 2 , e 3 lie in the contact structure ξ.\n• There are λ, µ ∈ R such that [e 1 , e 2 ] = λe 3 , [e 1 , e 3 ] = µe 2 . Furthermore, [e 2 , e 3 ] = e 1 .\nProof. First apply Lemma 5.3. Note that unimodularity implies that j c j ij = 0 for all i. We conclude that c 1 i1 + c 2 i2 + c 3 i3 = 0. By putting i = 2, 3 we deduce that c 2 32 = c 3 23 = 0, so [e 2 , e 3 ] = −c 1 23 e 1 = −e 1 .\nHere is a table with all possible unimodular Lie groups admitting a homogeneous contact structure,\nHeisenberg yes Remark 5.11. We point out that the topology or geometry of the underlying CR-manifold does not uniquely determine the underlying contact structure. In particular, for some compact quotients of SL(2, R) the above contact structures are not isomorphic. Explicit examples are given in Sections 5.2.1 and 5.2.2.\nTheorem 5.12 (Convergence to torsion free CR structure). Let (M, {ω i } i , θ = ω 1 ) be a homogeneous contact manifold whose Lie algebra is isomorphic to su(2). Then there is a unique homogeneous complex structure J a∞,b=1,c∞ that is torsion free. Moreover, for any choice of homogeneous complex structure J a,b=1,c , the normalized torsion flow converges to this unique CRstructure (ker θ, J a∞,b=1,c∞ ).\nIn particular, for any choice of homogeneous complex structure on SU (2), the normalized torsion flow converges to the standard CR-structure.\nLet S 3 := f −1 ( 1 2 ), and put ω 1 = −df • i, ω 2 = −df • j, ω 3 = −df • k, where i, j, k are the standard quaternions. With θ = ω 1 , this gives S 3 the structure of a homogeneous contact manifold. Its structure constants are c 1 23 = 1, c 2 13 = −1 and c 3 12 = 1. Recall that the standard CR structure on S 3 is then given by (θ, J = i). Put θ 1 := 1 √ 2 ω 2 + iω 3 . Then {θ, θ 1 , θ1} is an admissible frame. Following [CCY] we define the CR structure via the deformed coframe\nWriting this out gives\nComparing this with Equation (4.3) shows that the examples of Rossi are homogeneous CR structures with\nFor t > 0, these CR-manifolds are not embeddable.\nOn the other hand, Theorem 5.12 applies to Rossi's examples, so we have.\nCorollary 5.14. Under the normalized torsion flow, Rossi's examples flow to the standard CR structure on S 3 , which is embeddable."},{"title":"5.2.","content":"Examples of compact homogeneous CR-manifolds: different CR structures on ST * Σ. We describe the torsion flow on several geometries, namely SU (2), E(2), SL(2, R) and Heisenberg geometry.\nAs an explicit, compact model covering the first three cases we consider a compact orientable surface Riemann surface (Σ, g). According to a standard theorem in Riemannian geometry, the unit cotangent bundle ST * Σ admits a canonical coframe ω 1 , ω 2 , ω 3 (see for instance [BCS], Chapter 4.4 for the more general Finsler case with a different ordering of the coframe) satisfying\nwhere K is the Gauss curvature of (Σ, g). Assume that g is a metric of constant Gauss curvature. Then these manifolds provide models of homogeneous contact manifolds. 5.2.1. Homogeneous contact structure associated with the canonical contact structure \"pdq\". We consider the standard contact structure (\"pdq\") on the unit cotangent bundle of (Σ, g). With respect to the canonical coframe (5.4), the defining form for this contact structure is ω 1 .\nConsider time-dependent functions a t , b t , c t that are constant in space, and define the coframe\nWith this ordering, we obtain the structure coefficients c 1 23 = 1, c 2 13 = −K, and c 3 12 = 1 (take b 0 = 1), and all other coefficients vanish. With the standard choice of complex structure J abc , we obtain a pseudohermitian manifold (ST * Σ, θ bt , J atbtct ). We compute the torsion and Webster curvature with the formulas from Proposition 5.4:\n(5.5)\nWe specialize to the case that a = 0 and substitute B(t) = b(t) 2 . By Proposition 4.7 the (unnormalized) torsion flow reduces to ODĖ c = − 2ac (a 2 + 1)\nIf Kc 2 0 = 1, then the solution to this system is given by\nIf Kc 2 0 = 1, which can only happen if K > 0, then\nWe draw some conclusions:\n• For K ≤ 0, the solution exists for all time. For K = 0 (torus case), one has the curious property that the Webster curvature is constant. The torsion is also constant in that case, when measured in our coframe θ b , α b , β b . For all K ≤ 0, the torsion flow skews the complex structure more and more. The limit lim t→∞ c (t) = ∞, so in the limit, the complex structure blows up. • For K > 0, the solution blows up in finite time because of shrinking: b(t) = B(t) reaches 0 in finite time. The special case Kc 2 0 = 1 corresponds to vanishing torsion.\nRemark 5.15. We point out that, with its canonical contact structure \"pdq\", only the unit cotangent bundle of S 2 admits a complex structure for which the torsion vanishes. Indeed, all other unit cotangent bundles of surfaces with constant Gauss curvature are not K-contact, which is a necessary requirement by the appendix of Weinstein in [CH].\nIn this specialized case a = 0, the volume-normalized flow is particularly simple. We havė\nWe see the following\n(1) if K > 0, then there exists a torsion free complex structure, namely for c ∞ = 1/ √ K. We see that the torsion flow exists for all time, and that it converges to this torsion free complex structure.\n(2) if K = 0 (the torus case), then c increases linearly. The flow exists for all time, but the complex structure does not converge. (3) if K < 0, then c blows up in finite time. Geometrically, we see by (5.5) that torsion grows in norm, and the Webster curvature becomes more and more negative. Accordingly, the complex structure blows up.\nNote that ST * S 2 ∼ = SO(3), so alternatively we can apply Proposition 5.7 to the case K > 0."},{"title":"5.2.2.","content":"Prequantization structures on ST * Σ. We consider again the canonical coframe on the unit cotangent bundle with structure coefficients as in (5.4) for a surface with constant Gauss curvature. If Σ is not a torus, then we define the following coframe\nThe resulting contact manifold is known as a prequantization bundle, a circle bundle over a symplectic manifold (here Σ) whose fibers are periodic Reeb orbits. The corresponding structure coefficients are now c 1 23 = 1, c 2 13 = −K, c 3 12 = 1 K , and all other coefficients vanish. By defining J abc as before, we obtain a pseudohermitian manifold (ST * Σ, θ b , J abc ). Its torsion and Webster curvature are given by\nRemark 5.16. If Σ = S 2 , then the resulting contact structure is not contactomorphic to the \"pdq\"-structure from the previous section. Also, the contact structure is now K-contact, and we can choose a complex structure with vanishing torsion. Indeed, choose a = 0, and c = 1 |K| , and the torsion tensor will vanish.\nAs in the previous section we specialize to the case that a = 0 and substitute B(t) = b(t) 2 . By If we start the flow at a 0 = 0 and c 0 = 1 |K| , then we have vanishing torsion, and the torsion flow just contracts or expands depending on the sign of the Webster curvature. We have\nThe normalized torsion for these homogeneous contact manifolds are covered by Proposition 5.7. 5.2.3. Heisenberg geometry. As an explicit, compact example with Heisenberg geometry, consider the 2-torus with standard symplectic form (T 2 , Ω = dφ 1 ∧ dφ 2 ). There is a principal circle bundle p : P → T 2 whose connection form θ satisfies dθ = p * Ω. We see that (P, θ, α = dφ 1 , β = dφ 2 ) is a homogeneous contact manifold of Heisenberg type. Indeed, all structure coefficients except c 1 23 vanish. Hence any homogeneous CR-structure has vanishing torsion and Webster curvature. It follows that the torsion flow is constant, so this is an explicit example of a torsion soliton, namely a steady breather, see Corollary 1.9, case (i)."},{"title":"Entropy functionals","content":"The following section discusses entropy functionals on a closed 3-dimensional pseudohermitian manifold (M, J, θ).\n6.1. The Entropy F-Functional. Let (M, J, θ) be a closed pseudohermitian 3-manifold. In this section, we study the monotonicity property of the F-functional F(J (t) , θ (t) , ϕ (t) ) = M (W + |∇ b ϕ| 2 J,θ )e −ϕ dµ with the constraint M e −ϕ dµ = 1 under the coupled torsion flow (1.4).\nProof. We compute ∂ ∂t ∇ b φ with Equation (3.2) and use the result to obtain (6.1) ∂ ∂t |∇ b ϕ| 2 J,θ = 4 Re (iE 11 ϕ 1 ϕ 1 ) + 2 ∇ b ϕ, ∇ b ϕ t J,θ − 2η (t) |∇ b ϕ| 2 J,θ .\nBy (1.4) we find (6.2) ∂ ∂t dµ = 4η (t) dµ.\nUse these formulas together with Equation (3.8) to compute the variation of the F-functional,\nWe first set E 11 = e ϕ (A 11 − iϕ 11 − iϕ 1 ϕ 1 ) and η (t) = e ϕ (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W ), then − 1 2 d dt F(J (t) , θ (t) , ϕ (t) ) = M (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W ) 2 dµ + M |A 11 − iϕ 11 − iϕ 1 ϕ 1 | 2 dµ ≥ 0.\nThe monotonicity formula is strict unless A 11 − iϕ 11 − iϕ 1 ϕ 1 = 0 and 2∆ b ϕ − |∇ b ϕ| 2 J,θ + W = 0.\nMoreover, up to a contact transformation θ = e −ϕ θ A 11 = 0, W = 0.\nThis completes the proof of Theorem 1.10.\n6.2. The Entropy W ± -Functionals. We study the monotonicity property of the W + -functional Proof. Following the same computations as in the proof of Theorem\nWe first work out some general formulas. Given the second equation of (2.2), we can assume without loss of generality that (7.1) ω 1 1 = ic θ θ + ic Z θ 1 + ic Z θ 1 , with c θ a real function. Now we insert our frame to get explicit equations dθ 1 (Z 1 , Z1) = ω 1 1 (Z1) = ic Z dθ 1 (T, Z 1 ) = −ic θ dθ 1 (T, Z1) = A 11 ."},{"title":"(7.2)","content":"The Webster curvature is determined by the second structure equation, dω 1 1 = W θ 1 ∧θ1 + 2i Im(A1 1,1 θ 1 ∧θ). Wedging this equation with θ gives θ∧dω 1 1 = W θ∧θ 1 ∧θ1, which can be rewritten by using (7.1) to write out dω 1 1 . We find θ∧dω 1 1 = −c θ − 2|c Z | 2 − 2 Im(Z 1 (c Z ) ) θ∧θ 1 ∧θ1, and conclude (7.3) W = −c θ − 2|c Z | 2 − 2 Im(Z 1 (c Z ) ).\nWe now determine these coefficients. For c abc θ , we use the second equation from (7.2) to get\n= − i(a + i) 2b 2 c 2 12 + i a 2 + 1 2cb 2 c 3 12 − ci 2b 2 c 2 13 + i 2b 2 (a − i)c 3 13 .\nUse Lemma 5.3 to see that c 2 12 = c 3 13 = 0. For the c Z -component, we use the first equation from (7.2) we compute ic Z = dθ 1 (Z 1 , Z1) = 1 2c(a 2 + 1) − i 2(a − i) dα b (U b , V b ) (a 2 + 1)c(a + i) − c(a − i)(a 2 + 1)\n2c(a 2 + 1) c(a + i)c 2 23 − (a 2 + 1)c 3 23 .\nBy the above formulas, these coefficients determine the connection form ω 1 1 and the curvature W . For the torsion we use formulas (4.4), (4.3) and (7.2), and find\n= i 2(a − i)b 2 (a 2 + 1)c 2 12 + c(a + i)c 2 13 + i 2cb 2 (a 2 + 1)c 3 12 + c(a + i)c 3 13 .\nCombining this with c 2 12 = c 3 13 = 0 gives the desired expression for the torsion."},{"title":"Lemma 4 . 1 .","content":""},{"title":". 3 .","content":""},{"title":"Proposition 4. 7 .","content":""},{"title":".","content":""},{"title":"Figure 1 .","content":""},{"title":"Example 5. 13 (","content":""},{"title":"W","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"The Ricci flow, introduced by Hamilton, is a geometric flow for metrics on 3-manifolds, and has played a decisive role in the proof of the Poincaré conjecture and Thurston's geometrization conjecture for 3-manifolds. It is natural to then investigate a corresponding problem for contact 3manifolds. One of way of doing this is to find a CR analogue of the Ricci flow on a pseudohermitian 3-manifold (see Section 2 for definitions and basic notions in pseudohermitian geometry).\\nRecall that a strictly pseudoconvex CR structure on a 3-manifold M is given by a cooriented plane field ker θ, where θ is a contact form, together with a compatible complex structure J. This gives rise to a natural metric g = θ ⊗ θ + dθ(·, J·) for M . Given this data, there is a natural connection, the so-called Tanaka-Webster connection or pseudohermitian connection. We denote the torsion of this connection by A J,θ , and the Webster curvature, a kind of scalar curvature, by W . The torsion flow is then the following PDE,\\nIt seems to us that the torsion flow (1.1) is the right CR analogue of the Ricci flow.\\nThe torsion flow greatly simplifies if the torsion vanishes. This only happens in very special setups. Indeed, CR 3-manifolds with vanishing torsion are K-contact, meaning that the Reeb vector field is a Killing vector field for the metric g. In general, one can still hope that the torsion flow improves properties of the contact manifold underlying the CR-manifold. Like the Ricci flow, the torsion flow is a weakly parabolic PDE, provided some suitable initial conditions hold, and is hard to study directly.\\nThe mostly used tools in the study of Hamilton's Ricci flow [H1] consist of maximum principles. Exceptions are formed by Hamilton's entropy formula which holds for closed surfaces with positive Gaussian curvature [H2], and also by Perelman's entropy formulas [Pe]. These formulas can be thought of as monotonicity formulas for integrals of local geometric quantities.\\nIn this paper, we try to do the same for the torsion flow by setting up some monotonicity formulas for Perelman-type functionals. We also give a short time existence result which works in any dimension, but unfortunately it requires additional assumptions in dimension 3.\\nWe conclude this introduction with a brief plan of the paper.\\n• In Section 1.1 and Section 1.2 we motivate the definition of the torsion flow and give more precise statements of our results. • In Section 2 we survey basic notions in CR geometry. • We prove a short time existence result for the torsion flow in Section 3.\\n• In Sections 4 and 5 we describe CR manifolds with a global coframe and we also define homogeneous CR manifolds. On the latter class the torsion flow reduces to an ODE if we start with some appropriate initial conditions. These computations illustrate the behavior of the torsion flow in special cases, and in these cases the torsion flow behaves as can be expected from a Ricci-like flow. • Finally, in Section 6 we discuss analogues of Perelman's entropy formulas.\\n(1) ( [FH]) Let (S 3 , J, θ) be a closed, strictly pseudoconvex CR 3-sphere. Thenθ = e 2f θ is a contact form with vanishing CR Q-curvature if and only ifθ is a volume-normalized contact form with respect to a nowhere vanishing closed section ζ of the bundle Λ 2,0 (S 3 ).\\n(2) ( [Hi]) Suppose that (M, J, ξ = ker θ 0 ) has a transverse symmetry and θ 0 is a volumenormalized contact form. Then a contact form θ for ξ has vanishing CR Q-curvature if and only if θ is a volume-normalized contact form. In particular, this holds if (M, J, θ 0 ) is a closed hypersurface in C 2 with transverse symmetry.\\n(3) ( [CC]) Let (M, J, ξ = ker θ 0 ) be a CR 3-manifold that is Stein fillable. Then there is contact form θ for ξ with vanishing CR Q-curvature. (4) For a homogeneous CR 3-manifold (M, J, θ) as defined in Section 5 the homogeneous contact form θ is volume-normalized.\\nThe notions in this proposition are defined and discussed in Section 2.9. Further evidence for this conjecture is the following. Proposition 1.3. ( [CCT]) Let (M, J, θ) be a closed, strictly pseudoconvex CR-manifold of dimension 2n+1 satisfying c 1 (T 1,0 M ) = 0 and n ≥ 2. Then there is a smooth real-valued function u such that e 2u θ is a contact form of pseudo-Einstein type. In other words, e 2u θ is a volume-normalized contact form with respect to a nowhere vanishing closed section ζ of the bundle Λ n+1,0 (M ).\\nOther related statements are discussed in Section 2.2. Note that the contact structure is fixed under the torsion flow. Since the Chern class of a contact structure is independent of the choice of compatible complex structure, we conclude that the condition c 1 (T 1,0 M ) = 0 is preserved under the torsion flow (1.1). Furthermore, Proposition 1.3 tells us that the higher-dimensional analogue of Conjecture 1.1 holds for a strictly pseudoconvex CR-manifold of dimension 2n + 1 with c 1 (T 1,0 M ) = 0 and n ≥ 2. This leads us to a suitable initial condition that guarantees the following short-time existence result for the torsion flow.\\nTheorem 1.4. Let (M, J 0 , θ 0 ) be a closed pseudohermitian 2n + 1 -manifold with vanishing first Chern class, c 1 (T 1,0 M ) = 0. If n = 1, assume that Conjecture 1.1 holds. Then there exists δ > 0 and a unique smooth solution (J(t), θ(t) ) to the torsion flow (1.1) on the interval [0, δ) such that (J(0), θ(0) ) = (J 0 , θ 0 ). From Lemma 3.1 and part (ii) of Proposition 2.9 we deduce the following corollary, Corollary 1.5. Assume that (M, J 0 , θ 0 ) is a closed hypersurface in C 2 (so c 1 (T 1,0 M ) = 0) and has a transverse symmetry. Then there exists δ > 0 and a unique smooth solution (J(t), θ(t) ) to the torsion flow (1.1) on the interval [0, δ) such that (J(0), θ(0) ) = (J 0 , θ 0 ). Also, part (4) of Proposition 1.2 implies the corollary.\\nCorollary 1.6. Let (M,Ĵ,θ) be a closed homogeneous pseudohermitian 3-manifold, so c 1 (T 1,0 M ) = 0. Then there exists δ > 0 and a unique smooth solution (J(t), θ(t) ) to the torsion flow (1.1) on the interval [0, δ) such that (J(0), θ(0) ) = (J 0 , θ 0 ).\\nFor the homogeneous CR manifolds we define in Section 5, we also show the following convergence result.\\nTheorem 1.7 (Convergence to torsion free CR structure). Let (M, {ω i } i , θ = ω 1 ) be a homogeneous contact manifold whose Lie algebra is isomorphic to su(2). Then there is a unique homogeneous complex structure J a∞,b=1,c∞ that is torsion free. Moreover, for any choice of homogeneous complex structure J a,b=1,c , the normalized torsion flow converges to this unique CR-structure (ker θ, J a∞,b=1,c∞ ).\\nIn particular, for any choice of homogeneous complex structure on SU (2), the normalized torsion flow converges to the standard CR-structure.\\nIn the next section we shall discuss somewhat technical results concerning monotonicity properties of Perelman type functionals. As an application of these monotonicity results from Section 1.2, we classify the torsion breathers and solitons. The classification of torsion solitons is a necessary step in understanding the singularity formation in the torsion flow. Indeed, one expects the torsion soliton solutions to model finite time singularities of the torsion flow. In view of the flow (1.9) and original definition in [Pe], it is natural to define the soliton solutions for torsion flow (1.1) as follows.\\nDefinition 1.8. (i) A family J (t) of CR structures on (M, θ, J) evolving by the torsion flow (1.1) is called a breather if for some t 1 < t 2 and δ > 0, the CR structures J (t1) and J (t2) differ only by a contact diffeomorphism Φ with θ (t2) = λΦ * θ (t1) ; the cases λ = 1, λ < 1, λ > 1 correspond to steady, shrinking or expanding breathers, respectively.\\n(ii) A breather satisfying the above properties for all pairs of t 1 and t 2 of real numbers is called a torsion soliton.\\nIdeas of Perelman [Pe] (see also [Ca] and [Li]) can be combined with Theorem 1.10, Theorem 1.13 and Theorem 1.14 to show the following classification result. Corollary 1.9. Let (M, J, θ) be a closed pseudohermitian 3-manifold. Then (i) there is no closed steady torsion soliton other than the one which admits zero Tanaka-Webster curvature and vanishing pseudohermitian torsion up to a contact transformation.\\n(ii) there is no closed expanding torsion soliton other than the one which admits negative Tanaka-Webster curvature and vanishing pseudohermitian torsion.\\n(iii) there is no closed shrinking torsion soliton other than the one which admits positive Tanaka-Webster curvature and vanishing pseudohermitian torsion.\\n1.2. Some monotonicity results for Perelman-type functionals. The statements in this section are of a more technical nature: we will derive the CR analogue of Perelman's monotonicity formulas for the so-called coupled torsion flows (1.4), (1.6) and (1.7) in dimension 3.\\nIn Section 6.1 we define the CR analogue of Perelman's F-functional by\\nUnder the flow (1.3), this is equivalent to M (ϕ t − 4η (t) )e −ϕ dµ = 0. Therefore, the following coupled torsion flow is natural,\\nThe monotonicity formula is strict unless\\nThat is, up to a contact transformation θ = e −ϕ θ A 11 = 0 and W = 0.\\nRemark 1.11. Observe that for θ = e −ϕ θ,\\nThen the coupled torsion flow (1.4) on (M, J (t) , θ (t) ) is equivalent to the following system of coupled torsion flows on (M,\\nIn Section 6.2 we define two functionals analogous to Perelman's W-functional, namely the W + -functional,\\nIn view of Theorem 1.10, we first study the monotonicity property of W + -functional. By the same discussion as before, the constraint M (4πτ ) −2 e −ϕ dµ = 1 is equivalent to another constraint, namely M (ϕ t + 2τ −1 dτ dt − 4η (t) )(4πτ ) −2 e −ϕ dµ = 0 under the flow (1.3). Therefore we consider the following coupled torsion flow:\\nTheorem 1.13. Let (M, J, θ) be a closed 3-dimensional pseudohermitian manifold and J (t) , θ (t) , ϕ (t) and τ (t) be a solution of the coupled torsion flow (1.6). Then\\nThe monotonicity formula is strict unless\\nunder the flow (1.3). Therefore we consider the following coupled torsion flow :\\n(1.7)\\n. Theorem 1.14. Let (M, J, θ) be a closed 3-dimensional pseudohermitian manifold and J (t) , θ (t) , ϕ (t) and τ (t) be a solution of the coupled torsion flow (1.7). Then\\nThe monotonicity formula is strict unless\\nThat is, up to a contact transformation θ = e −ϕ θ A 11 = 0 and W + τ −1 e ϕ = 0.\\nRemark 1.15. Note that for θ (t) = e −ϕ θ (t) , we may reparametrize the time t by the formula\\n, so the coupled torsion flows (1.6) and (1.7) on (M, J (t) , θ (t) ) are equivalent to the following coupled torsion flows on (M, J ( t) , θ ( t) ), respectively :\\nRecall that X f is called a contact vector field or an infinitesimal contact diffeomorphism if the Lie derivative L X f θ = ηθ for some function η. Such a contact vector field has the form\\nwith f = e ϕ . Similar results hold for (1.6) and (1.7).\"\n },\n {\n \"title\": \"Preliminaries and definitions\",\n \"content\": \"In this section we introduce some basic notions from pseudohermitian geometry. We learned many of these notions from [L1, L2], and we refer to these papers for proofs and more references.\\nDefinition 2.1. Let M be a smooth manifold and ξ ⊂ T M a subbundle. A CR structure on ξ consists of an endomorphism J : ξ → ξ with J 2 = − Id such that the following integrability condition holds.\\nThe CR structure J can be extended to ξ ⊗ C, which we can then decompose into the direct sum of eigenspaces of J. The eigenvalues of J are i and −i, and the corresponding eigenspaces will be denoted by T 1,0 and T 0,1 , respectively. The integrability condition can then be reformulated as\\nNow consider a closed 2n + 1-manifold M with a cooriented contact structure ξ = ker θ. This means that θ ∧ dθ n = 0. The Reeb vector field of θ is the vector field T uniquely determined by the equations (2.1) θ(T ) = 1, and dθ(T, ·) = 0.\\n• J is a CR structure on ker θ.\\nDefinition 2.3. The Levi form , is the Hermitian form on T 1,0 defined by\\nWe can extend this Hermitian form , to T 0,1 by defining Z, W = Z, W for all Z, W ∈ T 1,0 . Furthermore, the Levi form naturally induces a Hermitian form on the dual bundle of T 1,0 , and hence on all induced tensor bundles.\\nWe now restrict ourselves to strictly pseudoconvex manifolds, or in other words compatible complex structures J. This means that the Levi form induces a Hermitian metric ·, · J,θ by V, U J,θ = dθ(V, JU ).\\nThe associated norm is defined as usual: |V | 2 J,θ = V, V J,θ . It follows that H also gives rise to a Hermitian metric for T 1,0 , and hence we obtain Hermitian metrics on all induced tensor bundles. By integrating this Hermitian metric over M with respect to the volume form dµ = θ ∧ dθ n , we get an L 2 -inner product on the space of sections of each tensor bundle.\\nDefinition 2.4. Let (M, J, ξ = ker θ 0 ) be a CR 3-manifold that is the smooth boundary of a bounded, strictly pseudoconvex domain in a complete Stein manifold V 4 . We shall call such a CR 3-manifold Stein fillable.\\n2.1. Pseudohermitian connection. Let T, Z α , Zβ be a frame of T M ⊗ C, where {Z α } is any local frame of T 1,0 , and Zβ = Z β ∈ T 0,1 . Then θ, θ α , θβ , the coframe dual to T, Z α , Zβ , satisfies\\nwhere h αβ is a positive definite matrix. By the Gram-Schmidt process we can always choose Z α such that h αβ = δ αβ ; throughout this paper, we shall take such a frame. The pseudohermitian connection or Tanaka-Webster connection of (J, θ) is the connection ∇ on T M ⊗ C (and extended to tensors) given in terms of a local frame {Z α } for T 1,0 by\\nwhere ω α β is the 1-form uniquely determined by the following equations:\\nHere τ α is called the pseudohermitian torsion, which we can also write as\\nThe components A αβ satisfy A αβ = A βα . We often consider the torsion tensor given by\\nThe following remark gives some geometric meaning to the pseudohermitian torsion.\\nRemark 2.5. Let X f be the contact vector field for the real-valued function f ∈ C 2 (M ). Then we have [CL1,Lemma 3.4]. In particular, we have X f = T for f = 1, and the above equation reduces to L T J = 2JA J,θ , so we see that the torsion tensor measures to what extend the complex structure J is invariant under the Reeb flow.\\nWe now consider the curvature of the Tanaka-Webster connection in terms of the coframe {θ = θ 0 , θ α , θβ}. The second structure equation gives\\nIn [We,Formulas 1.33 and 1.35], Webster showed that the curvature Ω β α can be written as\\nIn addition, by [L2, (2.4)] the coefficients W α β ρ are determined by the torsion,\\nContraction of (2.3) yields\\nDefinition 2.6. The pseudohermitian Ricci tensor is the tensor with components R ρσ . Its trace W := R ρ ρ is called the Webster curvature. If the pseudohermitian Ricci tensor is a scalar multiple of the Levi form, then we say that the pseudohermitian structure is pseudo-Einstein.\\nRemark 2.7. From the definition it is clear that the Webster curvature is the analogue of the scalar curvature in Riemannian geometry, and we also see that the pseudo-Einstein condition mimics the Einstein condition. Unlike the Riemannian case, pseudo-Einstein structures do not necessarily have constant Webster curvature, even in higher dimensions.\\nWe will denote components of covariant derivatives by indices preceded by a comma. For instance, we write A αβ,γ . Here the indices {0, α,β} indicate derivatives with respect to {T, Z α , Zβ}. For derivatives of a scalar function, we will often omit the comma. For example, ϕ α = Z α ϕ, ϕ αβ = ZβZ α ϕ − ω α γ (Zβ)Z γ ϕ, ϕ 0 = T ϕ for a (smooth) function ϕ. For a real-valued function ϕ, the subgradient ∇ b ϕ is defined as the unique vector field ∇ b ϕ ∈ ξ such that Z, ∇ b ϕ = dϕ(Z) for all vector fields Z tangent to the contact distribution ξ. Locally\\nFinally, we also need commutation relations, sometimes called Ricci identities. We list a few of them. These cover 3-dimensional CR-manifolds, but we also need higher-dimensional versions which we do not list here. For the idea of the derivation, see [L2,Equation 2.15].\\n(2.6) C I,01 − C I,10 = C I,1 A 11 − kC I, A 11,1 , C I,01 − C I,10 = C I,1 A 11 + kC I, A 11,1 , C I,11 − C I,11 = iC I,0 + kW C I .\\nHere C I denotes a coefficient of a tensor with multi-index I consisting of only 1 and 1, and k is the number of 1's minus the number of 1's in I.\\nNext we mention some curvature identities. The ones that are relevant for us are the contracted CR Bianchi identities from [L2, Lemma 2.2].\\n(2.7)\\n2.2. Volume-normalized contact forms and pseudo-Einstein CR-manifolds. To give our results some geometric meaning, we recall some results concerning volume-normalized contact forms and related notions. First recall the following definitions, taken from [L1].\\nDefinition 2.8. Let (M, θ, J) be a closed pseudohermitian 3-manifold. Using Chern-Weil theory, we define the real first Chern class c 1 (T 1,0 (M )) as\\nwhere ω α β is the connection form of the Tanaka-Webster connection in some local frame. Let (M 2n+1 , θ, J) be a (strictly pseudoconvex) CR-manifold. The canonical bundle of (M 2n+1 , θ, J) is the complex line bundle\\nA contact form θ in a closed pseudohermitian 3-manifold M is said to be volume-normalized with respect to a nowhere vanishing section ζ of the canonical bundle Λ n+1,0 (M ) if\\nwhere T is the Reeb vector field with respect to θ.\\nWe say that a CR-manifold (M, J, ξ = ker θ 0 ) has a transverse symmetry if there exists a contact form θ in the same conformal class as θ 0 such that J is invariant under the flow of the Reeb vector field T θ of θ,\\nEquivalently A J,θ = 0.\\n2.2.1. Notions specific to dimension 3. For the remainder of this section we specialize to 3dimensional CR-manifolds.\\nProposition 2.9. (i) ( [Hi]) Let (M, θ, J) be a closed pseudohermitian 3-manifold. A contact form θ is volume-normalized with respect to a non-vanishing closed section ζ in a closed pseudohermitian 3-manifold (M, J, θ) if and only if dω 1 1 = −id (W θ).\\nThat is,\\n(ii) ( [L1]) If M is a closed CR embedding of a hypersurface in C 2 , then θ = e 2f θ 0 is volumenormalized with respect to a closed section ζ = j * (dz 1 ∧ dz 2 ) for j : (M, θ 0 , J) → C 2 and some smooth function f (x).\\nRemark 2.10. The condition W 1 − iA1 1,1 = 0 corresponds to the pseudo-Einstein condition for higher dimensional CR-manifolds. Indeed, suppose that n ≥ 2. Then according to [L2,Lemma 4.1], a contact form θ on a pseudohermitian manifold (M 2n+1 , J, ker θ) is pseudo-Einstein if for every admissible coframe, we have\\nBy writing out this equation, we see that the following holds for a higher-dimensional pseudo-Einstein manifold,\\nHowever, in dimension 3 any CR-manifold is pseudo-Einstein, since the metric tensor has only one component. For us, a modification of this condition will serve as an integrability condition for the torsion flow.\\nDefine the tensor\\nUse this to define the CR Q-curvature, see [Hi, CCC],\\nThe last equality follows from [Hi,Lemma 5.4]. Clearly, vanishing of R 1 guarantees vanishing of the CR Q-curvature. For a converse, we already mentioned Proposition 1.2.\"\n },\n {\n \"title\": \"Short time Existence\",\n \"content\": \"In this section we give some results concerning the short-time existence of the torsion flow. As mentioned, the motivation for the integrability condition comes from higher-dimensional CRmanifolds, so we give proofs that also hold for these high-dimensional cases. We thank Jih-Hsin Cheng for valuable contributions in early computations in the 3-dimensional case, in particular involving the variation formulas and linearized operator. See also his paper for a related flow, [C].\\nLet θ (t) be a family of smooth contact forms and J (t) be a family of CR structures on (M, J, θ). We consider the following flow on a closed pseudohermitian (2n + 1)-manifold (M, J, θ) × [0, T ):\\nHere\\nWe start by deriving some evolution equations under the general flow (3.1) before specifying to the torsion flow, for which E = A J (the torsion tensor), and η = −W (the Webster curvature). All computations will be done in a local frame. Fix a unit-length local frame {Z α } and let {θ α } be its dual admissible 1-form. Let Z α(t) , θ α (t) denote a unit-length frame and dual admissible 1-form with respect to (J (t) , θ (t) ). Since θ α (Z β(t) ) is a positive real function, we can write\\nwhere F α β are real and G α β are complex. The fact that Z α(t) is an orthonormal frame means that\\nBy differentiating and substituting the above expression for Z α , we obtain F α β = −ηδ β α . By differentiating J (t) Z α(t) = iZ α(t) we find\"\n },\n {\n \"title\": \"Now differentiate the identities\",\n \"content\": \"Since we will derive an identity involving tensors, we will take an adapted frame satisfying ω γ α = 0 at a point. Plug in (3.3) and consider the θ∧θ γ terms to obtain\\nOn the other hand, contracting (3.4) with Z β and then contracting with h βα , computing modulo θ γ yields\\nSince ω α α is pure imaginary, we have\\nDifferentiate the structure equation (2.4) with respect to t and consider only the θ ρ ∧ θσ terms. This gives\\nAfter contracting with h ρσ we get\\nRecall that the transformation law of the connection under a change of pseudohermitian structure was computed in [L1,Sec. 5]. Letθ = e 2f θ be another pseudohermitian structure. Then we can define an admissible coframe byθ α = e f (θ α + 2if α θ). With respect to this coframe, the connection 1-form and the pseudohermitian torsion are given by\\nand (3.10)\\nrespectively. Thus the Webster curvature transforms as\\nHere covariant derivatives on the right side are taken with respect to the pseudohermitian structure θ and an admissible coframe θ α . Note also that the dual frame of {θ,θ α ,θ α } is given by\\nWe need one more ingredient before we can prove Theorem 1.4. Let f be a smooth function, and define the CR pluriharmonic operator P α f by\\nDefine the CR Paneitz real operator by\\nWe have the following lemma.\\nLemma 3.1. Let (M 2n+1 , J, ξ = kerθ) be a pseudohermitian manifold, and suppose, if n = 1, that Conjecture 1.1 holds for this manifold. Then there is a smooth function f such that\\nand thus W α α − inA αβ , βα −2(n + 2)P θ f = 0.\\nProof. By Conjecture 1.1 in case n = 1 and Proposition 1.3 in case n ≥ 2 there is a smooth function f such the rescaled contact form θ = e 2f θ is volume-normalized. By Proposition 2.9 and Remark 2.10, this volume-normalized contact form satisfies\\nThe second contracted Bianchi identity from (2.7) implies then that\\nFollowing the same computation as in the proof of Lemma 5.4 in [Hi], we find using (3.9), (3.10) and (3.11), that\\nContract the second equation with respect to the Levi metric h γβ = h γβ . This yields\\nThus we get\\nWe will apply the following commutation relations, which are similar to those in (2.6)\\nThe latter relation follows from (3.14). Together with the above we obtain the following transformation law\\nApply (3.13) to complete the proof of (3.12).\\nThe following lemma is needed to prove Corollary 1.5. We use the observation that CR 3manifolds with transverse symmetry are Stein fillable.\\nLemma 3.2. Suppose that (M 3 , J 0 , ξ = ker θ 0 ) has a transverse symmetry and θ 0 is volumenormalized with respect to a non-vanishing closed section ζ. Assume furthermore that (J(t), θ(t) ) is a solution of the torsion flow (1.1) on the interval [0, T ]. Then for each t ∈ [0, T ], there exists a volume-normalized contact form θ (t) for (ξ, J(t) ).\\nProof. Since (M, J (t) , ξ = ker θ (t) ) is Stein fillable, there exists a contact form θ (t) with vanishing CR Q-curvature by (3)\\nwhere θ 0 is volume-normalized. By [Hi,Lemma 5.4,formula 5.7\\nand, because Q 0 = 0 by (2) of Proposition 1.2 (also a result of Hirachi), we find\\nSince M has a transverse symmetry, it follows from [Hi,Proposition 7.3] that f is a CR pluriharmonic function, and P 1 f = 0. This implies θ (t) is volume-normalized.\\nProof of Theorem 1.4. We follow the arguments of Hamilton, [H1, Section 4], and adapt them to our setup. First rewrite the torsion flow (3.1) as\\nStep 1: determining the linearization of the torsion flow We use δJ, δθ to denote the variations of J and θ, respectively. Set δJ = 2E and δθ = 2hθ,\\nwhere E is an endomorphism: ξ → ξ satisfying J • E + E • J = 0 and h is a smooth function. Let δ J and δ θ denote the linearization operators with respect to J and θ. From (3.5) and (3.8), we see that\\nWe first compute the linearization of 2A J,θ . Let O m denote an operator of weight ≤ m (see [CL1,page 234]), and define the total variation as δ = δ J + δ θ . PutẼ := E ⊕ hθ. With (3.16) we compute the variation of the torsion as\\nThe linearization of −2W θ can be computed with (3.17),\\nWith these linearizations (3.18) and (3.19) we write the highest weight operator L J,θ ( E) of the linearization δ(2Q(J, θ)) as\\nStep 2: finding integrability conditions Following Hamilton's ideas from ([H1, section 5]), we find an integrability condition of the form H(J, θ)Q(J, θ) = 0 for the torsion flow (3.15). By Lemma 3.1 we find a function f satisfying equation (3.12). We then define O 0 (A J,θ ⊕ −W θ) := 2(n + 2)P α f . We use this function to get an integrability condition. Define the linear operator H(J, θ) by\\nThis operator has degree 1 in E := E ⊕ hθ; the term O 0 has lower order. By (3.12) we see that Q(J, θ) satisfies the integrability condition\\nBy taking first variations, we find\\nWe continue to argue as in [H1, section 4]. The operator in E given by δH(J, θ) has degree 1, so its symbol, which we denote by σH, of maximal degree vanishes. The other term, H(J, θ)δQ(J, θ), has maximal degree since H has degree 1 and δQ has positive degree. The symbols of linear differential operators H(J, θ) and Q(J, θ) therefore satisfy (3.22) [σH(J, θ)](·) • σ[δQ(J, θ)](·)( E) = 0, so we conclude that the image of σ[δQ(J, θ)] must lie in the kernel of σH(J, θ). We now want to show that the restriction of the symbol σδ2Q(J, θ)(ξ) to ker σH has only positive eigenvalues when ξ = 0. To see this, we first need additional commutation relations, again similar to those in (2.6), namely\\nFrom these relations we derive that on the subspace\\nthe following identities hold true,\\nTherefore it follows from (3.20) that the highest weight operator L J,θ ( E) of the linearization δ(2Q(J, θ)) is\\nθ} on ker H(J, θ)( E). Note here that both the Folland-Stein operator L α , given for n ≥ 2 by\\nand the sublaplacian ∆ b are subelliptic, see [FS]. Therefore it follows from (3.22) and (3.25) that all eigenvalues of the eigenspaces of σ[δQ(J, θ)] in ker σH(J, θ) are strictly positive.\\nAlternatively, note that the proof of Lemma 3.1 can also be used to show that there are no degeneracies for the torsion flow (3.1) other than those implied by the second contracted Bianchi identity from (2.7).\\nStep 3: solving the linearized equations and handling the non-linear problem\\nTo complete the argument, we follow [H1, Section 5 and Section 6]. The linearized equations can be solved using the regularization trick from [H1, Section 6]. Theorem 5.1 of [H1] can then be used to obtain short-time existence of the torsion flow (3.1). The main ingredient is the Nash-Moser inverse function theorem ( [H3]).\"\n },\n {\n \"title\": \"Contact 3-manifolds with a global frame and pseudohermitian structures\",\n \"content\": \"We now specialize to 3-dimensional pseudohermitian manifolds. Let (M, ξ = ker θ) be a cooriented contact 3-manifold. Denote the Reeb field by T . Furthermore, in this section and in the next, Section 5, we shall assume that ξ admits a global symplectic trivialization, i.e. there are vector fields U, V such that ξ = Span(U, V ) and dθ(U, V ) = 1. Proof. The contact structure ξ admits the structure of a symplectic vector bundle (ξ, dθ). By choosing a compatible complex structure J, we obtain a complex line bundle (ξ, J). It is wellknown that smooth complex line bundles are trivial if and only if their first Chern class vanishes, see [W,Chapter III,Section 4].\\nThe last step needs the first Chern class with integer coefficients. Chern-Weil theory will not suffice in general. Henceforth, we shall assume that the globally defined vector fields U, V form a symplectic basis of (ξ, dθ). Consider the coframe θ, α, β dual to T, U, V . Then dθ(U, V ) = 1, so (4.1) dθ = α∧β.\\nLemma 4.2. Let J be a compatible complex structure for the symplectic vector bundle (ξ = ker θ, dθ). Then there are smooth functions a : M → R and c : M → R >0 such that, with respect to the frame U, V , the complex structure J is represented by the matrix\\nProof. With respect to the global frame U, V , the endomorphism J is represented by a 2 × 2matrix. Writing out the condition J 2 = −id shows that the matrix representation for J has the above form. The compatibility condition means that dθ(·, J·) is a metric, so it is represented by a positive definite matrix. Writing out this matrix shows that c is a positive function.\\nThe following is motivated by our goal to convert the torsion flow (a PDE for tensors) into a PDE for functions. Choose real-valued functions a, b and c where b and c are positive. We attach super-and subscripts to indicate the dependence on these functions. In order to keep track of deformations of the contact form, we express all data in the given frame T, U, V . Define This can be checked by plugging in the vector field into the defining equations (2.1). The second assertion is obtained by writing out the terms.\\nRemark 4.4. If b is a constant function, then the deformation from (4.2) corresponds to a ξhomothetic deformation as defined in [Bl,Section 10.4]. We take b 2 in θ b to have fewer expressions with square roots. Define a complex structure by\\nBy Lemma 4.2, this is the most general choice.\\nRemark 4.5. In higher dimensions, strictly pseudoconvex CR-manifolds require an integrability condition, see Definition 2.1, which is trivially satisfied in dimension 3.\\nWe now compute the Tanaka-Webster connection as in Section 2.1. We use the coframe θ, θ 1 , θ1, where\\nThis satisfies our normalization condition\\n(4.4) 4.1. Converting the torsion flow into a system of PDE's for the functions a, b, c. To write down the equations of the torsion flow, we need the work out the torsion tensor. We have\\nHence the first equation of the torsion flow (1.1),J abc = 2A abc J,θ is equivalent to the systeṁ a = 2 Re(A abc 11 ) + a Im(A abc 11 )\\nso we obtain the above system by looking at the coefficients of U ⊗ α, U ⊗ β, V ⊗ α, and V ⊗ β. This works since these tensors are time-independent.\\nLemma 4.6 (A smaller system for the J-part of the torsion flow). The system given by (4.5) is equivalent to system given bẏ a = 2 Re(A abc 11 ) + a Im(A abc 11 ) ċ = −2 Re(A abc 11 ) −2ac a 2 + 1 + Im(A abc 11 )\\n(1 − a 2 )c a 2 + 1 . (4.6)\\nProof. The first equation of (4.5) implies the fourth. We now verify that the first and second equation of (4.5) imply the third.\\nOn the other hand, the second equation of the torsion flow (1.1) reduces to\\nso we can reduce the torsion flow to a system of PDE's for the functions a, b, c, giving us the following proposition. (1 − a 2 )c a 2 + 1 d dt b 2 = −2W abc b 2 .\\n(4.7)\\nRemark 4.8. Spatial derivatives of a, b, c come in via the definition of torsion and Webster curvature.\"\n },\n {\n \"title\": \"3-manifolds with constant structure constants and the Tanaka connection\",\n \"content\": \"In this section we consider manifolds M 3 that admit global 1-forms ω 1 , ω 2 , ω 3 such that (1) ω 1 , ω 2 , ω 3 form a basis of T * M .\\n(2) The structure coefficients are constant, i.e.\\nwhere c i are constant. We shall call such a contact manifold a homogeneous contact manifold. This terminology is not standard, but it serves a useful purpose in this note. Let us point out that a related, but not equivalent notion, also referred to as homogeneous contact, was used by Perrone, [Pr].\\nRemark 5.1. The structure coefficients are the structure constants of some 3-dimensional Liealgebra. Indeed, the dual frame {X 1 , X 2 , X 3 } satisfies\\nthe Lie bracket on vector fields satisfies the Jacobi identity. We shall call this the Lie algebra of a homogeneous contact manifold.\\nFrom Lemma 4.1 we get immediately.\\nLemma 5.2. Homogeneous contact manifolds have trivial Chern class.\\nBefore we define a CR structure on such manifolds, we use the following lemma to provide a better coframe. In many cases, this lemma can be improved upon, but this version is sufficiently convenient.\\nLemma 5.3. Let (M, {ω i } i , θ) be a homogeneous contact manifold. Then there is a basis {ω i } i such that\\n•ω 1 is contact, and c 1 23 = 1. • The structure coefficients dω i = j,k c i jkω j ∧ω k satisfy c 1 12 = c 1 13 = c 2 12 = c 3 13 = 0. Proof. Choose a compatible complex structure J for (ξ, dθ). Consider the operator h := 1 2 L T J. From [Bl,Lemma 6.2] we see that h is self-adjoint with respect to the metric dθ(·, J·), and we also get the identity\\nSince h is self-adjoint, we can find a basis of eigenvectors X, Y of h for ξ. If the eigenvalue of X is λ, then JX is an eigenvector with eigenvalue −λ,\\nso we can assume that Y = JX. We consider the Levi-Civita connection ∇ for g = θ ⊗θ +dθ(·, J·). Take the basis e 1 = T, e 2 = X, e 3 = JX, where T is the Reeb field of θ. Then [e 1 , e 2 ] = [T, X] = ∇ T X − ∇ X T = +JX + λJX − µJX.\\nfor some µ ∈ R. In the last step we have used the identity (see [Bl,Lemma 6.2] )\\nThe same steps work for [e 1 , e 3 ], so we conclude that there are constants C 1 , C 2 such that [e 1 , e 2 ] = C 1 e 3 [e 1 , e 3 ] = C 2 e 2 .\\nConsider the dual basis {ω i } i . Then ω 1 is a contact form, and since T is the Reeb field, we have 0 = i e1 dω 1 = c 1 12 ω 2 + c 1 13 ω 3 . Hence c 1 12 = c 1 13 = 0, and from Lie bracket computations we see that c 2 12 = c 3 13 = 0. By rescaling, we see that the claim holds.\\nWe assume now that ω 1 , ω 2 , ω 3 is a basis that is provided by this lemma. Take the basis X 1 , X 2 , X 3 that is dual to ω 1 , ω 2 , ω 3 , i.e. ω i (X j ) = δ i j . We have i X1 ω 1 = 1 and i X1 dω 1 = k c 1 1k ω k = 0, so X 1 is the Reeb vector field for ω 1 . Note that ξ = ker ω 1 = Span(X 2 , X 3 ). Also, if c 1 23 = 1, then dω 1 = ω 2 ∧ω 3 , so we have the right normalization convention for the setup of the Tanaka connection described in Formula (4.1) with θ = ω 1 , α = ω 2 and β = ω 3 .\\nChoose constants a ∈ R and b, c > 0, and define a complex structure on ξ (or CR-structure on M ) following Lemma 4.2 by\\nWe call such an endomorphism a homogeneous complex structure, and we refer to a homogeneous contact manifold together with the above complex structure as a homogeneous CRmanifold or a homogeneous CR structure. By direct computation, we obtain the following result for the Tanaka-Webster connection, its torsion and the Webster curvature.\\nProposition 5.4. Let (M, {ω i } i , θ) be a homogeneous contact manifold with basis provided by Lemma 5.3. Fix a ∈ R and b, c > 0 and define J abc as in formula (5.1). Then the connection form for the Tanaka connection of the pseudohermitian manifold (M, θ b = b 2 θ, J abc ) is given by\\nIts torsion is given by\\nand its Webster curvature by\\nProof. See the computations in the appendix. Alternatively, these computations are essentially also contained in [Pr]. Note that Perrone uses a J-basis, that is e 1 = T, e 2 , e 3 = Je 2 .\\nWe can now reduce the torsion flow for homogeneous CR-manifolds to an ODE by plugging in the results of Proposition 5.4 into Proposition 4.7. The general system is fairly complicated, so we will work out some interesting case in Section 5.2.\\nWe shall also consider the normalized torsion flow which, in general, is given by the system\\nwhereW = M W θ∧dθ/ M θ∧dθ. Since the Webster curvature is constant in space for a homogeneous CR-manifold, then second equation of the normalized is flow is trivial. Inserting the result of Proposition 5.4 into the explicit system provided by Proposition 4.7 gives the following.\\nProposition 5.5 (Normalized torsion flow for homogeneous CR-manifolds). Let (M, {ω i } i , θ) be a homogeneous contact manifold with θ = ω 1 . Set b = 1, and let a t , c t be real valued functions Then for a complex structure J atbct as defined in (5.1), the normalized torsion flow satisfies the ODEȧ\\n(5.3)\"\n },\n {\n \"title\": \"Homogeneous CR-manifolds and vanishing torsion.\",\n \"content\": \"Observe that homogeneous CRmanifolds need not be compact. In Section 5.2 we give some compact examples, but we start out by characterizing homogeneous CR-manifolds with vanishing torsion directly in terms of the structure coefficients and coefficients for the CR-structure a, b, c.\\nProposition 5.6. Let (M, {ω i } i , θ) be a homogeneous contact manifold with θ = ω 1 and basis provided by Lemma 5.3. For a ∈ R, b = 1 and c > 0, a complex structure J abc as defined in (5.1) has vanishing torsion precisely when one of the following conditions is satisfied.\\n1 if a = 0, c 2 12 = 0, and c = − 2 if a = 0, c 2 13 = c 2 12 = c 3 12 = 0. Proposition 5.7 (Convergence of the torsion flow). Let (M, {ω i } i , θ) be a homogeneous contact manifold with θ = ω 1 and a basis provided by Lemma 5.3. Take a ∈ R, b = 1 and c > 0, and let J abc be a complex structure as defined in (5.1).\\nA If c 3 12 > 0 and c 2 13 < 0, then the normalized torsion flow converges to the unique torsion free complex structure of Proposition 5.6 ) in the upper half-plane with coordinates (a, c). The phase diagram for the repelling (R) case is given in Figure 1 The phase diagram for the attracting (A) case is similar, but the arrows are reversed. It is clear that the attracting case ), we make two observations.\\n• any initial condition (a 0 , c 0 ) that starts in the set given by |c 2 13 |c 2 0 + |c 3 12 |a 2 0 − |c 3 12 | > 0 blows up in finite time. Indeed, the c-coordinate is strictly increasing in that case, so there is t 0 such that c t0 > 1 or the solution blows up before t 0 . If c t0 > 1 then the solution blows up in finite by the following argument. We havė c = |c 3 12 |c 2 + |c 2 13 |(a 2 − 1) ≥ |c 3 12 |c 2 − |c 2 13 |, which must blow up in finite time. with c 0 > 0 exits the half-disk {|c 2 13 |c 2 0 + |c 3 12 |a 2 0 − |c 3 12 | ≤ 0} in some finite time t 1 . Then either c t1 = 0, meaning that a blows up in finite time, or we can reduce to the first case.\\nAn alternative proof can be given by starting with a J-basis. Then a 0 = 0 and c 0 = 1. Such a basis is not preserved under the torsion flow, but the condition a t = 0 holds. The resulting ODE is simpler to analyze. This method is applied in Section 5.2.\\nRemark 5.8. By Proposition 5.4 there is a relation between the Webster curvature and the sign of c 3 12 . However, it is not true that repelling is equivalent to negative curvature.\\n5.1.1. Homogeneous CR structures on unimodular Lie groups. We recall the definitions involved.\\nDefinition 5.9. A Lie group G is called unimodular if the left-invariant Haar measure is also right invariant. A Lie algebra (g, [·, ·]) is called unimodular if T r ad X = 0 for all X ∈ g.\\nNote that the Lie algebra of a unimodular Lie group is unimodular. The unimodularity of a Lie group can be used to simplify the structure coefficients of the Lie algebra.\\nLemma 5.10. Let G be a unimodular, 3-dimensional Lie group admitting a homogeneous contact structure ξ = ker θ. Then there is a basis {e i } i of the Lie algebra g such that • e 1 is the Reeb vector field, and e 2 , e 3 lie in the contact structure ξ.\\n• There are λ, µ ∈ R such that [e 1 , e 2 ] = λe 3 , [e 1 , e 3 ] = µe 2 . Furthermore, [e 2 , e 3 ] = e 1 .\\nProof. First apply Lemma 5.3. Note that unimodularity implies that j c j ij = 0 for all i. We conclude that c 1 i1 + c 2 i2 + c 3 i3 = 0. By putting i = 2, 3 we deduce that c 2 32 = c 3 23 = 0, so [e 2 , e 3 ] = −c 1 23 e 1 = −e 1 .\\nHere is a table with all possible unimodular Lie groups admitting a homogeneous contact structure,\\nHeisenberg yes Remark 5.11. We point out that the topology or geometry of the underlying CR-manifold does not uniquely determine the underlying contact structure. In particular, for some compact quotients of SL(2, R) the above contact structures are not isomorphic. Explicit examples are given in Sections 5.2.1 and 5.2.2.\\nTheorem 5.12 (Convergence to torsion free CR structure). Let (M, {ω i } i , θ = ω 1 ) be a homogeneous contact manifold whose Lie algebra is isomorphic to su(2). Then there is a unique homogeneous complex structure J a∞,b=1,c∞ that is torsion free. Moreover, for any choice of homogeneous complex structure J a,b=1,c , the normalized torsion flow converges to this unique CRstructure (ker θ, J a∞,b=1,c∞ ).\\nIn particular, for any choice of homogeneous complex structure on SU (2), the normalized torsion flow converges to the standard CR-structure.\\nLet S 3 := f −1 ( 1 2 ), and put ω 1 = −df • i, ω 2 = −df • j, ω 3 = −df • k, where i, j, k are the standard quaternions. With θ = ω 1 , this gives S 3 the structure of a homogeneous contact manifold. Its structure constants are c 1 23 = 1, c 2 13 = −1 and c 3 12 = 1. Recall that the standard CR structure on S 3 is then given by (θ, J = i). Put θ 1 := 1 √ 2 ω 2 + iω 3 . Then {θ, θ 1 , θ1} is an admissible frame. Following [CCY] we define the CR structure via the deformed coframe\\nWriting this out gives\\nComparing this with Equation (4.3) shows that the examples of Rossi are homogeneous CR structures with\\nFor t > 0, these CR-manifolds are not embeddable.\\nOn the other hand, Theorem 5.12 applies to Rossi's examples, so we have.\\nCorollary 5.14. Under the normalized torsion flow, Rossi's examples flow to the standard CR structure on S 3 , which is embeddable.\"\n },\n {\n \"title\": \"5.2.\",\n \"content\": \"Examples of compact homogeneous CR-manifolds: different CR structures on ST * Σ. We describe the torsion flow on several geometries, namely SU (2), E(2), SL(2, R) and Heisenberg geometry.\\nAs an explicit, compact model covering the first three cases we consider a compact orientable surface Riemann surface (Σ, g). According to a standard theorem in Riemannian geometry, the unit cotangent bundle ST * Σ admits a canonical coframe ω 1 , ω 2 , ω 3 (see for instance [BCS], Chapter 4.4 for the more general Finsler case with a different ordering of the coframe) satisfying\\nwhere K is the Gauss curvature of (Σ, g). Assume that g is a metric of constant Gauss curvature. Then these manifolds provide models of homogeneous contact manifolds. 5.2.1. Homogeneous contact structure associated with the canonical contact structure \\\"pdq\\\". We consider the standard contact structure (\\\"pdq\\\") on the unit cotangent bundle of (Σ, g). With respect to the canonical coframe (5.4), the defining form for this contact structure is ω 1 .\\nConsider time-dependent functions a t , b t , c t that are constant in space, and define the coframe\\nWith this ordering, we obtain the structure coefficients c 1 23 = 1, c 2 13 = −K, and c 3 12 = 1 (take b 0 = 1), and all other coefficients vanish. With the standard choice of complex structure J abc , we obtain a pseudohermitian manifold (ST * Σ, θ bt , J atbtct ). We compute the torsion and Webster curvature with the formulas from Proposition 5.4:\\n(5.5)\\nWe specialize to the case that a = 0 and substitute B(t) = b(t) 2 . By Proposition 4.7 the (unnormalized) torsion flow reduces to ODĖ c = − 2ac (a 2 + 1)\\nIf Kc 2 0 = 1, then the solution to this system is given by\\nIf Kc 2 0 = 1, which can only happen if K > 0, then\\nWe draw some conclusions:\\n• For K ≤ 0, the solution exists for all time. For K = 0 (torus case), one has the curious property that the Webster curvature is constant. The torsion is also constant in that case, when measured in our coframe θ b , α b , β b . For all K ≤ 0, the torsion flow skews the complex structure more and more. The limit lim t→∞ c (t) = ∞, so in the limit, the complex structure blows up. • For K > 0, the solution blows up in finite time because of shrinking: b(t) = B(t) reaches 0 in finite time. The special case Kc 2 0 = 1 corresponds to vanishing torsion.\\nRemark 5.15. We point out that, with its canonical contact structure \\\"pdq\\\", only the unit cotangent bundle of S 2 admits a complex structure for which the torsion vanishes. Indeed, all other unit cotangent bundles of surfaces with constant Gauss curvature are not K-contact, which is a necessary requirement by the appendix of Weinstein in [CH].\\nIn this specialized case a = 0, the volume-normalized flow is particularly simple. We havė\\nWe see the following\\n(1) if K > 0, then there exists a torsion free complex structure, namely for c ∞ = 1/ √ K. We see that the torsion flow exists for all time, and that it converges to this torsion free complex structure.\\n(2) if K = 0 (the torus case), then c increases linearly. The flow exists for all time, but the complex structure does not converge. (3) if K < 0, then c blows up in finite time. Geometrically, we see by (5.5) that torsion grows in norm, and the Webster curvature becomes more and more negative. Accordingly, the complex structure blows up.\\nNote that ST * S 2 ∼ = SO(3), so alternatively we can apply Proposition 5.7 to the case K > 0.\"\n },\n {\n \"title\": \"5.2.2.\",\n \"content\": \"Prequantization structures on ST * Σ. We consider again the canonical coframe on the unit cotangent bundle with structure coefficients as in (5.4) for a surface with constant Gauss curvature. If Σ is not a torus, then we define the following coframe\\nThe resulting contact manifold is known as a prequantization bundle, a circle bundle over a symplectic manifold (here Σ) whose fibers are periodic Reeb orbits. The corresponding structure coefficients are now c 1 23 = 1, c 2 13 = −K, c 3 12 = 1 K , and all other coefficients vanish. By defining J abc as before, we obtain a pseudohermitian manifold (ST * Σ, θ b , J abc ). Its torsion and Webster curvature are given by\\nRemark 5.16. If Σ = S 2 , then the resulting contact structure is not contactomorphic to the \\\"pdq\\\"-structure from the previous section. Also, the contact structure is now K-contact, and we can choose a complex structure with vanishing torsion. Indeed, choose a = 0, and c = 1 |K| , and the torsion tensor will vanish.\\nAs in the previous section we specialize to the case that a = 0 and substitute B(t) = b(t) 2 . By If we start the flow at a 0 = 0 and c 0 = 1 |K| , then we have vanishing torsion, and the torsion flow just contracts or expands depending on the sign of the Webster curvature. We have\\nThe normalized torsion for these homogeneous contact manifolds are covered by Proposition 5.7. 5.2.3. Heisenberg geometry. As an explicit, compact example with Heisenberg geometry, consider the 2-torus with standard symplectic form (T 2 , Ω = dφ 1 ∧ dφ 2 ). There is a principal circle bundle p : P → T 2 whose connection form θ satisfies dθ = p * Ω. We see that (P, θ, α = dφ 1 , β = dφ 2 ) is a homogeneous contact manifold of Heisenberg type. Indeed, all structure coefficients except c 1 23 vanish. Hence any homogeneous CR-structure has vanishing torsion and Webster curvature. It follows that the torsion flow is constant, so this is an explicit example of a torsion soliton, namely a steady breather, see Corollary 1.9, case (i).\"\n },\n {\n \"title\": \"Entropy functionals\",\n \"content\": \"The following section discusses entropy functionals on a closed 3-dimensional pseudohermitian manifold (M, J, θ).\\n6.1. The Entropy F-Functional. Let (M, J, θ) be a closed pseudohermitian 3-manifold. In this section, we study the monotonicity property of the F-functional F(J (t) , θ (t) , ϕ (t) ) = M (W + |∇ b ϕ| 2 J,θ )e −ϕ dµ with the constraint M e −ϕ dµ = 1 under the coupled torsion flow (1.4).\\nProof. We compute ∂ ∂t ∇ b φ with Equation (3.2) and use the result to obtain (6.1) ∂ ∂t |∇ b ϕ| 2 J,θ = 4 Re (iE 11 ϕ 1 ϕ 1 ) + 2 ∇ b ϕ, ∇ b ϕ t J,θ − 2η (t) |∇ b ϕ| 2 J,θ .\\nBy (1.4) we find (6.2) ∂ ∂t dµ = 4η (t) dµ.\\nUse these formulas together with Equation (3.8) to compute the variation of the F-functional,\\nWe first set E 11 = e ϕ (A 11 − iϕ 11 − iϕ 1 ϕ 1 ) and η (t) = e ϕ (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W ), then − 1 2 d dt F(J (t) , θ (t) , ϕ (t) ) = M (2∆ b ϕ − |∇ b ϕ| 2 J,θ + W ) 2 dµ + M |A 11 − iϕ 11 − iϕ 1 ϕ 1 | 2 dµ ≥ 0.\\nThe monotonicity formula is strict unless A 11 − iϕ 11 − iϕ 1 ϕ 1 = 0 and 2∆ b ϕ − |∇ b ϕ| 2 J,θ + W = 0.\\nMoreover, up to a contact transformation θ = e −ϕ θ A 11 = 0, W = 0.\\nThis completes the proof of Theorem 1.10.\\n6.2. The Entropy W ± -Functionals. We study the monotonicity property of the W + -functional Proof. Following the same computations as in the proof of Theorem\\nWe first work out some general formulas. Given the second equation of (2.2), we can assume without loss of generality that (7.1) ω 1 1 = ic θ θ + ic Z θ 1 + ic Z θ 1 , with c θ a real function. Now we insert our frame to get explicit equations dθ 1 (Z 1 , Z1) = ω 1 1 (Z1) = ic Z dθ 1 (T, Z 1 ) = −ic θ dθ 1 (T, Z1) = A 11 .\"\n },\n {\n \"title\": \"(7.2)\",\n \"content\": \"The Webster curvature is determined by the second structure equation, dω 1 1 = W θ 1 ∧θ1 + 2i Im(A1 1,1 θ 1 ∧θ). Wedging this equation with θ gives θ∧dω 1 1 = W θ∧θ 1 ∧θ1, which can be rewritten by using (7.1) to write out dω 1 1 . We find θ∧dω 1 1 = −c θ − 2|c Z | 2 − 2 Im(Z 1 (c Z ) ) θ∧θ 1 ∧θ1, and conclude (7.3) W = −c θ − 2|c Z | 2 − 2 Im(Z 1 (c Z ) ).\\nWe now determine these coefficients. For c abc θ , we use the second equation from (7.2) to get\\n= − i(a + i) 2b 2 c 2 12 + i a 2 + 1 2cb 2 c 3 12 − ci 2b 2 c 2 13 + i 2b 2 (a − i)c 3 13 .\\nUse Lemma 5.3 to see that c 2 12 = c 3 13 = 0. For the c Z -component, we use the first equation from (7.2) we compute ic Z = dθ 1 (Z 1 , Z1) = 1 2c(a 2 + 1) − i 2(a − i) dα b (U b , V b ) (a 2 + 1)c(a + i) − c(a − i)(a 2 + 1)\\n2c(a 2 + 1) c(a + i)c 2 23 − (a 2 + 1)c 3 23 .\\nBy the above formulas, these coefficients determine the connection form ω 1 1 and the curvature W . For the torsion we use formulas (4.4), (4.3) and (7.2), and find\\n= i 2(a − i)b 2 (a 2 + 1)c 2 12 + c(a + i)c 2 13 + i 2cb 2 (a 2 + 1)c 3 12 + c(a + i)c 3 13 .\\nCombining this with c 2 12 = c 3 13 = 0 gives the desired expression for the torsion.\"\n },\n {\n \"title\": \"Lemma 4 . 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \". 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 4. 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \".\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Example 5. 13 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"W\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Let (M, ξ = ker θ) be a contact 3-manifold. Then there is a global trivialization U, V of its contact structure if and only if c 1 (ξ) = 0.","content":"Lemma 4 . 1 .\n41Let (M, ξ = ker θ) be a contact 3-manifold. Then there is a global trivialization U, V of its contact structure if and only if c 1 (ξ) = 0."},{"id":"","caption":"The vector field T b is the Reeb vector field for θ b . Furthermore, we have dθ b = α b ∧β b .","content":". 3 .\n3The vector field T b is the Reeb vector field for θ b . Furthermore, we have dθ b = α b ∧β b ."},{"id":"","caption":"Let (M 3 , θ, J) be a CR-manifold with c 1 (ξ, J) = 0. Then there exists a basis of T * M , and functions a, b, c such that• the complex structure J can be written as J abc .• the torsion flow (1.1) is equivalent to the systeṁ a = 2 Re(A abc 11 ) + a Im(","content":"Proposition 4. 7 .\n7Let (M 3 , θ, J) be a CR-manifold with c 1 (ξ, J) = 0. Then there exists a basis of T * M , and functions a, b, c such that• the complex structure J can be written as J abc .• the torsion flow (1.1) is equivalent to the systeṁ a = 2 Re(A abc 11 ) + a Im("},{"id":"","caption":"0, then the normalized torsion flow blows up in finite time and the complex structure does not converge unless a 0 = 0 and c In the latter case, the torsion vanishes and the torsion flow is constant.Proof. In both cases, the ODE describing the normalized torsion flow has a unique","content":".\n0, then the normalized torsion flow blows up in finite time and the complex structure does not converge unless a 0 = 0 and c In the latter case, the torsion vanishes and the torsion flow is constant.Proof. In both cases, the ODE describing the normalized torsion flow has a unique"},{"id":"","caption":"Phase Indeed, the function |a| is decreasing, and the only limit can be a ∞ = 0.From the second equation we then deduce that the function c converges to c ∞ = − that solutions in the repelling case blow up unless (a 0 , c 0 ) =","content":"Figure 1 .\n1Phase Indeed, the function |a| is decreasing, and the only limit can be a ∞ = 0.From the second equation we then deduce that the function c converges to c ∞ = − that solutions in the repelling case blow up unless (a 0 , c 0 ) ="},{"id":"","caption":"Rossi's examples). We recall Rossi's examples of non-embeddable CR-manifolds. Define the strictly plurisubharmonic function","content":"Example 5. 13 (\n13Rossi's examples). We recall Rossi's examples of non-embeddable CR-manifolds. Define the strictly plurisubharmonic function"},{"id":"","caption":"+ (J (t) , θ (t) , ϕ (t) , τ (t) ) = M [τ (W + |∇ b ϕ| 2 J,θ ) + 1 2 ϕ − 1](4πτ ) −2 e −ϕ dµwith the constraint M (4πτ ) −2 e −ϕ dµ = 1 under the coupled torsion flow (1.6).","content":"W\n+ (J (t) , θ (t) , ϕ (t) , τ (t) ) = M [τ (W + |∇ b ϕ| 2 J,θ ) + 1 2 ϕ − 1](4πτ ) −2 e −ϕ dµwith the constraint M (4πτ ) −2 e −ϕ dµ = 1 under the coupled torsion flow (1.6)."},{"id":"","caption":"of Proposition 1.2. Write","content":"of Proposition 1.2. Write"},{"id":"","caption":"1.10, we can derive that d dt M [τ (W + |∇ b ϕ| Proof. Following the same computations as in the proof of Theorem 1.13, we can derive that d dt M [τ (W + |∇ b ϕ|","content":"1.10, we can derive that d dt M [τ (W + |∇ b ϕ| Proof. Following the same computations as in the proof of Theorem 1.13, we can derive that d dt M [τ (W + |∇ b ϕ|"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Let (M, ξ = ker θ) be a contact 3-manifold. Then there is a global trivialization U, V of its contact structure if and only if c 1 (ξ) = 0.\",\n \"content\": \"Lemma 4 . 1 .\\n41Let (M, ξ = ker θ) be a contact 3-manifold. Then there is a global trivialization U, V of its contact structure if and only if c 1 (ξ) = 0.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The vector field T b is the Reeb vector field for θ b . Furthermore, we have dθ b = α b ∧β b .\",\n \"content\": \". 3 .\\n3The vector field T b is the Reeb vector field for θ b . Furthermore, we have dθ b = α b ∧β b .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let (M 3 , θ, J) be a CR-manifold with c 1 (ξ, J) = 0. Then there exists a basis of T * M , and functions a, b, c such that• the complex structure J can be written as J abc .• the torsion flow (1.1) is equivalent to the systeṁ a = 2 Re(A abc 11 ) + a Im(\",\n \"content\": \"Proposition 4. 7 .\\n7Let (M 3 , θ, J) be a CR-manifold with c 1 (ξ, J) = 0. Then there exists a basis of T * M , and functions a, b, c such that• the complex structure J can be written as J abc .• the torsion flow (1.1) is equivalent to the systeṁ a = 2 Re(A abc 11 ) + a Im(\"\n },\n {\n \"id\": \"\",\n \"caption\": \"0, then the normalized torsion flow blows up in finite time and the complex structure does not converge unless a 0 = 0 and c In the latter case, the torsion vanishes and the torsion flow is constant.Proof. In both cases, the ODE describing the normalized torsion flow has a unique\",\n \"content\": \".\\n0, then the normalized torsion flow blows up in finite time and the complex structure does not converge unless a 0 = 0 and c In the latter case, the torsion vanishes and the torsion flow is constant.Proof. In both cases, the ODE describing the normalized torsion flow has a unique\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Phase Indeed, the function |a| is decreasing, and the only limit can be a ∞ = 0.From the second equation we then deduce that the function c converges to c ∞ = − that solutions in the repelling case blow up unless (a 0 , c 0 ) =\",\n \"content\": \"Figure 1 .\\n1Phase Indeed, the function |a| is decreasing, and the only limit can be a ∞ = 0.From the second equation we then deduce that the function c converges to c ∞ = − that solutions in the repelling case blow up unless (a 0 , c 0 ) =\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Rossi's examples). We recall Rossi's examples of non-embeddable CR-manifolds. Define the strictly plurisubharmonic function\",\n \"content\": \"Example 5. 13 (\\n13Rossi's examples). We recall Rossi's examples of non-embeddable CR-manifolds. Define the strictly plurisubharmonic function\"\n },\n {\n \"id\": \"\",\n \"caption\": \"+ (J (t) , θ (t) , ϕ (t) , τ (t) ) = M [τ (W + |∇ b ϕ| 2 J,θ ) + 1 2 ϕ − 1](4πτ ) −2 e −ϕ dµwith the constraint M (4πτ ) −2 e −ϕ dµ = 1 under the coupled torsion flow (1.6).\",\n \"content\": \"W\\n+ (J (t) , θ (t) , ϕ (t) , τ (t) ) = M [τ (W + |∇ b ϕ| 2 J,θ ) + 1 2 ϕ − 1](4πτ ) −2 e −ϕ dµwith the constraint M (4πτ ) −2 e −ϕ dµ = 1 under the coupled torsion flow (1.6).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"of Proposition 1.2. Write\",\n \"content\": \"of Proposition 1.2. Write\"\n },\n {\n \"id\": \"\",\n \"caption\": \"1.10, we can derive that d dt M [τ (W + |∇ b ϕ| Proof. Following the same computations as in the proof of Theorem 1.13, we can derive that d dt M [τ (W + |∇ b ϕ|\",\n \"content\": \"1.10, we can derive that d dt M [τ (W + |∇ b ϕ| Proof. Following the same computations as in the proof of Theorem 1.13, we can derive that d dt M [τ (W + |∇ b ϕ|\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"D Blair, Riemannian geometry of contact and symplectic manifolds. Boston, MABirkhäuser Boston, Inc203D. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203. Birkhäuser Boston, Inc., Boston, MA, 2002. xii+260 pp. ISBN: 0-8176-4261-7.","doi":""},{"text":"An introduction to Riemann-Finsler geometry. D Bao, S.-S Chern, Z Shen, Graduate Texts in Mathematics. New YorkSpringer-Verlag431D. Bao, S.-S. Chern, Z. Shen, An introduction to Riemann-Finsler geometry., Graduate Texts in Mathe- matics, 200. Springer-Verlag, New York, 2000. xx+431 pp. ISBN: 0-387-98948-X","doi":""},{"text":"First eigenvalues of geometric operators under the Ricci flow. X.-D Cao, Math. Ann. 337X.-D. Cao, First eigenvalues of geometric operators under the Ricci flow, Math. Ann., 337 (2007), 435-441.","doi":""},{"text":"Pseudo-Einstein and Q-flat Metrics with Egenvalue Estimates on CR-hypersurfaces. J Cao, S.-C Chang, Indiana Univ. Math. Journal. 566J. Cao and S.-C. Chang, Pseudo-Einstein and Q-flat Metrics with Egenvalue Estimates on CR-hypersurfaces, Indiana Univ. Math. Journal, 56, No.6, (2007), 2839-2857.","doi":""},{"text":"S.-C Chang, J.-H Cheng, H.-L Chiu, The Fourth-order Q-curvature flow on a CR 3-manifold. 56S.-C. Chang, J.-H. Cheng and H.-L. Chiu, The Fourth-order Q-curvature flow on a CR 3-manifold, Indiana Univ. Math. J., Vol. 56, No. 4 (2007), 1793-1826.","doi":""},{"text":"Calabi-Yau Theorem and Hodge-Laplacian Heat Equation in a Closed Strictly Pseudoconvex CR Manifold preprint. D.-C Chang, S.-C Chang, J.-Z Tie, D.-C. Chang, S.-C. Chang and J.-Z. Tie, Calabi-Yau Theorem and Hodge-Laplacian Heat Equation in a Closed Strictly Pseudoconvex CR Manifold preprint.","doi":""},{"text":"The Li-Yau-Hamilton inequality for Yamabe flow on a closed CR 3-manifold. S.-C Chang, H.-L Chiu, C.-T Wu, Transactions of AMS. 362S.-C. Chang, H.-L. Chiu and C.-T. Wu, The Li-Yau-Hamilton inequality for Yamabe flow on a closed CR 3-manifold, Transactions of AMS, Vol 362 (2010), 1681-1698.","doi":""},{"text":"Subgradient Estimate and Liouville-type Theorems for the CR Heat Equation on Heisenberg groups H n. S.-C Chang, J.-Z Tie, C.-T Wu, to appear in Asian J. of MathS.-C. Chang, J.-Z. Tie and C.-T. Wu, Subgradient Estimate and Liouville-type Theorems for the CR Heat Equation on Heisenberg groups H n , to appear in Asian J. of Math.","doi":""},{"text":"Evolution of Yamabe constant under Ricci flow. S.-C Chang, P Lu, AGAG. 31S.-C. Chang and P. Lu, Evolution of Yamabe constant under Ricci flow, AGAG 31 (2007), 147-153.","doi":""},{"text":"S Chanillo, H.-L Chiu, P Yang, arXiv:1007.5020v3Embeddability for Three-Dimensional Cauchy-Riemann Manifolds and CR Yamabe Invariants. S. Chanillo, H.-L. Chiu and P. Yang,Embeddability for Three-Dimensional Cauchy-Riemann Manifolds and CR Yamabe Invariants, arXiv:1007.5020v3","doi":"arXiv:1007.5020v3"},{"text":"J.-H Cheng, arXiv:1305.4451Flows and a tangency condition for embeddable CR structures in dimension 3. J.-H. Cheng, Flows and a tangency condition for embeddable CR structures in dimension 3, arXiv:1305.4451","doi":"arXiv:1305.4451"},{"text":"The Burns-Epstein invariant and deformation of the CR structures. J.-H Cheng, J M Lee, Duke Math. J. 60J.-H. Cheng and J. M. Lee, The Burns-Epstein invariant and deformation of the CR structures, Duke Math. J. 60 (1990), 221-254.","doi":""},{"text":"A Local Slice Theorem for 3-Dimensional CR Structures. J.-H Cheng, J M Lee, American Journal of Mathematics. 1175J.-H. Cheng and J. M. Lee, A Local Slice Theorem for 3-Dimensional CR Structures, American Journal of Mathematics, Vol. 117, No. 5 (1995), 1249-1298.","doi":""},{"text":"On Riemannian metrics adapted to three-dimensional contact manifolds. S.-S Chern, R S Hamilton, Lecture Notes in Math. 279SpringerWith an appendix by Alan WeinsteinS.-S. Chern, R. S. Hamilton, On Riemannian metrics adapted to three-dimensional contact manifolds. With an appendix by Alan Weinstein, Lecture Notes in Math., 1111, Workshop Bonn 1984 (Bonn, 1984), 279, Springer, Berlin, 1985.","doi":""},{"text":"B Chow, S.-C Chu, D Glickenstein, C Guenther, J Isenberg, T Ivey, D Knopf, P Lu, F Luo, L Ni, The Ricci Flow : Techniques and Applications, Part I : Geometric Aspects. AMS135B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Ricci Flow : Techniques and Applications, Part I : Geometric Aspects, Mathematical Surveys and Monographs Vol. 135 (2007), AMS.","doi":""},{"text":"Ambient Metric Construction of Q-Curvature in Conformal and CR Geometries. C Fefferman, K Hirachi, Math. Res. Lett. 105C. Fefferman and K. Hirachi, Ambient Metric Construction of Q-Curvature in Conformal and CR Geome- tries, Math. Res. Lett., 10, No. 5-6 (2003), 819-831.","doi":""},{"text":"Estimates for the ∂ b complex and analysis on the Heisenberg group. G B Folland, E M Stein, Comm. Pure Appl. Math. 27G. B. Folland and and E. M. Stein, Estimates for the ∂ b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522.","doi":""},{"text":"K Hirachi, Scalar pseudohermitian invariants and the Szegö kernel on three-dimensional CR manifolds, Complex Geometry. Dekker143K. Hirachi, Scalar pseudohermitian invariants and the Szegö kernel on three-dimensional CR manifolds, Complex Geometry, Lect. Notes in Pure and Appl. Math. 143, 67-76, Dekker, 1993.","doi":""},{"text":"Three-Manifolds with Positive Ricci Curvature. R S Hamilton, J. Diff. Geom. 17R.S. Hamilton, Three-Manifolds with Positive Ricci Curvature, J. Diff. Geom. 17 (1982), 255-306.","doi":""},{"text":"The Ricci flow on surfaces. R S Hamilton, Contemp. Math. 71Amer. Math. SocR.S. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71 (1988), 237-262, Amer. Math. Soc., Provi- dence, RI.","doi":""},{"text":"The inverse function theorem of Nash and Moser, Bull. (new series). R S Hamilton, Amer. Math. Soc. 7R.S. Hamilton, The inverse function theorem of Nash and Moser, Bull. (new series) Amer. Math. Soc. 7 (1982), 65-222.","doi":""},{"text":"The Fefferman metric and pseudohermitian invariants. J M Lee, Trans. AMS. 296J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. AMS 296 (1986), 411-429","doi":""},{"text":"Pseudo-Einstein metric on CR manifolds. J M Lee, Amer. J. Math. 110J. M. Lee, Pseudo-Einstein metric on CR manifolds, Amer. J. Math. 110 (1988), 157-178.","doi":""},{"text":"Eigenvalues and energy functionals with monotonicity formula under Ricci flow. J.-F Li, Math. Ann. 338J.-F. Li, Eigenvalues and energy functionals with monotonicity formula under Ricci flow, Math. Ann., 338 (2007), 927-946.","doi":""},{"text":"The Entropy Formula for the Ricci Flow and its Geometric Applications. G Perelman, ArXiv. Math. DG/0211159G. Perelman, The Entropy Formula for the Ricci Flow and its Geometric Applications. ArXiv. Math. DG/0211159.","doi":"ArXiv. Math. DG/0211159"},{"text":"Homogeneous contact Riemannian three-manifolds. D Perrone, Illinois J. Math. 422243256D. Perrone, Homogeneous contact Riemannian three-manifolds, Illinois J. Math. 42 (1998), no. 2, 243256.","doi":""},{"text":"N Tanaka, A Differential Geometric Study on Strongly Pseudoconvex Manifolds. TokyoKinokuniya Co. LtdN. Tanaka, A Differential Geometric Study on Strongly Pseudoconvex Manifolds, 1975, Kinokuniya Co. Ltd., Tokyo","doi":""},{"text":"Pseudohermitian structures on a real hypersurface. S M Webster, J. Diff. Geom. 13S. M. Webster, Pseudohermitian structures on a real hypersurface, J. Diff. Geom. 13 (1978), 25-41.","doi":""},{"text":"Differential analysis on complex manifolds. Prentice-Hall Series in Modern Analysis. R O WellsJr, Prentice-Hall, Inc252Englewood Cliffs, N.J.R. O. Wells, Jr. , Differential analysis on complex manifolds. Prentice-Hall Series in Modern Analysis. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. x+252 pp.","doi":""}],"string":"[\n {\n \"text\": \"D Blair, Riemannian geometry of contact and symplectic manifolds. Boston, MABirkhäuser Boston, Inc203D. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203. Birkhäuser Boston, Inc., Boston, MA, 2002. xii+260 pp. ISBN: 0-8176-4261-7.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An introduction to Riemann-Finsler geometry. D Bao, S.-S Chern, Z Shen, Graduate Texts in Mathematics. New YorkSpringer-Verlag431D. Bao, S.-S. Chern, Z. Shen, An introduction to Riemann-Finsler geometry., Graduate Texts in Mathe- matics, 200. Springer-Verlag, New York, 2000. xx+431 pp. ISBN: 0-387-98948-X\",\n \"doi\": \"\"\n },\n {\n \"text\": \"First eigenvalues of geometric operators under the Ricci flow. X.-D Cao, Math. Ann. 337X.-D. Cao, First eigenvalues of geometric operators under the Ricci flow, Math. Ann., 337 (2007), 435-441.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pseudo-Einstein and Q-flat Metrics with Egenvalue Estimates on CR-hypersurfaces. J Cao, S.-C Chang, Indiana Univ. Math. Journal. 566J. Cao and S.-C. Chang, Pseudo-Einstein and Q-flat Metrics with Egenvalue Estimates on CR-hypersurfaces, Indiana Univ. Math. Journal, 56, No.6, (2007), 2839-2857.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S.-C Chang, J.-H Cheng, H.-L Chiu, The Fourth-order Q-curvature flow on a CR 3-manifold. 56S.-C. Chang, J.-H. Cheng and H.-L. Chiu, The Fourth-order Q-curvature flow on a CR 3-manifold, Indiana Univ. Math. J., Vol. 56, No. 4 (2007), 1793-1826.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Calabi-Yau Theorem and Hodge-Laplacian Heat Equation in a Closed Strictly Pseudoconvex CR Manifold preprint. D.-C Chang, S.-C Chang, J.-Z Tie, D.-C. Chang, S.-C. Chang and J.-Z. Tie, Calabi-Yau Theorem and Hodge-Laplacian Heat Equation in a Closed Strictly Pseudoconvex CR Manifold preprint.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Li-Yau-Hamilton inequality for Yamabe flow on a closed CR 3-manifold. S.-C Chang, H.-L Chiu, C.-T Wu, Transactions of AMS. 362S.-C. Chang, H.-L. Chiu and C.-T. Wu, The Li-Yau-Hamilton inequality for Yamabe flow on a closed CR 3-manifold, Transactions of AMS, Vol 362 (2010), 1681-1698.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Subgradient Estimate and Liouville-type Theorems for the CR Heat Equation on Heisenberg groups H n. S.-C Chang, J.-Z Tie, C.-T Wu, to appear in Asian J. of MathS.-C. Chang, J.-Z. Tie and C.-T. Wu, Subgradient Estimate and Liouville-type Theorems for the CR Heat Equation on Heisenberg groups H n , to appear in Asian J. of Math.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Evolution of Yamabe constant under Ricci flow. S.-C Chang, P Lu, AGAG. 31S.-C. Chang and P. Lu, Evolution of Yamabe constant under Ricci flow, AGAG 31 (2007), 147-153.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S Chanillo, H.-L Chiu, P Yang, arXiv:1007.5020v3Embeddability for Three-Dimensional Cauchy-Riemann Manifolds and CR Yamabe Invariants. S. Chanillo, H.-L. Chiu and P. Yang,Embeddability for Three-Dimensional Cauchy-Riemann Manifolds and CR Yamabe Invariants, arXiv:1007.5020v3\",\n \"doi\": \"arXiv:1007.5020v3\"\n },\n {\n \"text\": \"J.-H Cheng, arXiv:1305.4451Flows and a tangency condition for embeddable CR structures in dimension 3. J.-H. Cheng, Flows and a tangency condition for embeddable CR structures in dimension 3, arXiv:1305.4451\",\n \"doi\": \"arXiv:1305.4451\"\n },\n {\n \"text\": \"The Burns-Epstein invariant and deformation of the CR structures. J.-H Cheng, J M Lee, Duke Math. J. 60J.-H. Cheng and J. M. Lee, The Burns-Epstein invariant and deformation of the CR structures, Duke Math. J. 60 (1990), 221-254.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Local Slice Theorem for 3-Dimensional CR Structures. J.-H Cheng, J M Lee, American Journal of Mathematics. 1175J.-H. Cheng and J. M. Lee, A Local Slice Theorem for 3-Dimensional CR Structures, American Journal of Mathematics, Vol. 117, No. 5 (1995), 1249-1298.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On Riemannian metrics adapted to three-dimensional contact manifolds. S.-S Chern, R S Hamilton, Lecture Notes in Math. 279SpringerWith an appendix by Alan WeinsteinS.-S. Chern, R. S. Hamilton, On Riemannian metrics adapted to three-dimensional contact manifolds. With an appendix by Alan Weinstein, Lecture Notes in Math., 1111, Workshop Bonn 1984 (Bonn, 1984), 279, Springer, Berlin, 1985.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"B Chow, S.-C Chu, D Glickenstein, C Guenther, J Isenberg, T Ivey, D Knopf, P Lu, F Luo, L Ni, The Ricci Flow : Techniques and Applications, Part I : Geometric Aspects. AMS135B. Chow, S.-C. Chu, D. Glickenstein, C. Guenther, J. Isenberg, T. Ivey, D. Knopf, P. Lu, F. Luo, and L. Ni, The Ricci Flow : Techniques and Applications, Part I : Geometric Aspects, Mathematical Surveys and Monographs Vol. 135 (2007), AMS.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ambient Metric Construction of Q-Curvature in Conformal and CR Geometries. C Fefferman, K Hirachi, Math. Res. Lett. 105C. Fefferman and K. Hirachi, Ambient Metric Construction of Q-Curvature in Conformal and CR Geome- tries, Math. Res. Lett., 10, No. 5-6 (2003), 819-831.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Estimates for the ∂ b complex and analysis on the Heisenberg group. G B Folland, E M Stein, Comm. Pure Appl. Math. 27G. B. Folland and and E. M. Stein, Estimates for the ∂ b complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429-522.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"K Hirachi, Scalar pseudohermitian invariants and the Szegö kernel on three-dimensional CR manifolds, Complex Geometry. Dekker143K. Hirachi, Scalar pseudohermitian invariants and the Szegö kernel on three-dimensional CR manifolds, Complex Geometry, Lect. Notes in Pure and Appl. Math. 143, 67-76, Dekker, 1993.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Three-Manifolds with Positive Ricci Curvature. R S Hamilton, J. Diff. Geom. 17R.S. Hamilton, Three-Manifolds with Positive Ricci Curvature, J. Diff. Geom. 17 (1982), 255-306.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Ricci flow on surfaces. R S Hamilton, Contemp. Math. 71Amer. Math. SocR.S. Hamilton, The Ricci flow on surfaces, Contemp. Math. 71 (1988), 237-262, Amer. Math. Soc., Provi- dence, RI.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The inverse function theorem of Nash and Moser, Bull. (new series). R S Hamilton, Amer. Math. Soc. 7R.S. Hamilton, The inverse function theorem of Nash and Moser, Bull. (new series) Amer. Math. Soc. 7 (1982), 65-222.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Fefferman metric and pseudohermitian invariants. J M Lee, Trans. AMS. 296J. M. Lee, The Fefferman metric and pseudohermitian invariants, Trans. AMS 296 (1986), 411-429\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pseudo-Einstein metric on CR manifolds. J M Lee, Amer. J. Math. 110J. M. Lee, Pseudo-Einstein metric on CR manifolds, Amer. J. Math. 110 (1988), 157-178.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Eigenvalues and energy functionals with monotonicity formula under Ricci flow. J.-F Li, Math. Ann. 338J.-F. Li, Eigenvalues and energy functionals with monotonicity formula under Ricci flow, Math. Ann., 338 (2007), 927-946.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Entropy Formula for the Ricci Flow and its Geometric Applications. G Perelman, ArXiv. Math. DG/0211159G. Perelman, The Entropy Formula for the Ricci Flow and its Geometric Applications. ArXiv. Math. DG/0211159.\",\n \"doi\": \"ArXiv. Math. DG/0211159\"\n },\n {\n \"text\": \"Homogeneous contact Riemannian three-manifolds. D Perrone, Illinois J. Math. 422243256D. Perrone, Homogeneous contact Riemannian three-manifolds, Illinois J. Math. 42 (1998), no. 2, 243256.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N Tanaka, A Differential Geometric Study on Strongly Pseudoconvex Manifolds. TokyoKinokuniya Co. LtdN. Tanaka, A Differential Geometric Study on Strongly Pseudoconvex Manifolds, 1975, Kinokuniya Co. Ltd., Tokyo\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Pseudohermitian structures on a real hypersurface. S M Webster, J. Diff. Geom. 13S. M. Webster, Pseudohermitian structures on a real hypersurface, J. Diff. Geom. 13 (1978), 25-41.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Differential analysis on complex manifolds. Prentice-Hall Series in Modern Analysis. R O WellsJr, Prentice-Hall, Inc252Englewood Cliffs, N.J.R. O. Wells, Jr. , Differential analysis on complex manifolds. Prentice-Hall Series in Modern Analysis. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. x+252 pp.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":572,"cells":{"text":{"kind":"string","value":"\nImaging, counting, and positioning single interstitial atoms in solids\n\n\nJizhe Cui \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina\n\nHaozhi Sha \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina\n\nLiangze Mao \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina\n\nKang Sun \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina\n\nWenfeng Yang \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina\n\nRong Yu \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina\n\nImaging, counting, and positioning single interstitial atoms in solids\n4957B1D621899D984E5B7EA631A52632\nInterstitial atoms are ubiquitous in solids and they are widely incorporated into materials to tune their lattice structure, electronic transportation, and mechanical properties.Because the distribution of interstitial atoms in matrix materials is usually disordered and most of them are light atoms with weak scattering ability, it remains a challenge to directly image single interstitial atoms and measure their geometrical positions.In this work, direct imaging and measuring of single interstitial atoms have been realized with adaptive-propagator ptychography.The measurement of their three-dimensional coordinates enables quantitative analysis of the pair distribution function of the interstitial atoms and reveals the anisotropic occupation of oxygen in the interstitial sites in titanium.The current work paves the way for the determination of interstitial atoms in materials, and for the correlation between the atomicscale behavior of interstitial atoms and the physical properties of materials.\n\nIntroduction\n\nInterstitial atoms are ubiquitous in materials and play important role in tuning the mechanical and physical properties of semiconductors, superconductors, and alloys [1][2][3][4] .Recently, incorporation of interstitial atoms in alloys has been increasingly employed to improve mechanical properties of alloys [5][6][7][8][9][10][11] .As a paradigm of materials science, understanding the physical properties often requires studying the atomic structure of the materials 12,13 .Therefore, it is imperative to obtain atomic-scale information of interstitial atoms for understanding and controlling the behavior of interstitial atoms in materials.Previous studies on interstitial atoms mostly rely on statistical methods, such as the positron annihilation spectroscopy 14 , X-ray diffuse scattering 15 , internal friction 16 and electron diffraction 17 .While these methods are useful, they often sacrifice spatial resolution and provide only average results.\n\nDirect imaging of interstitial atoms in solids has been challenging due to two primary reasons.\n\nFirst, interstitial atoms are usually light atoms, like carbon, nitrogen, and oxygen; they give very weak electron scattering compared with the matrix atoms.Second, the surfaces of solids are often structurally modified, damaged, or contaminated, forming a layer with different atomic structure than the bulk inside.The electron scattering signals from the surface layers result in strong noise that prevent the detection of interstitial atoms inside the bulk.As a result, direct imaging of interstitial atoms in solids requires very high signal-to-noise ratio in images and the elimination of the surface contributions to the bulk signals.\n\nRecently, oxygen atoms in a TiZrNb alloy of the body-center-cubic structure was shown to occupy the tetrahedral interstitial sites 18 .Since the concentrations (12 at.%) of oxygen in the alloy is high, the interstitial atoms are crowded together with an average spacing of 6 Å.These densely distributed interstitial atoms can provide stronger signals for detection, but at the same time make the detection only provide their average information and not about the behavior of individual interstitial atoms.\n\nInterstitial atoms usually have low concentrations in solids; even a few parts per million of interstitial doping is enough to significantly improve properties 1,19 .It is therefore important to be able to detect single interstitial atoms, even though they can only give weak signals for detection.\n\nHere we show that imaging, counting, and positioning of single interstitial atoms in lightly alloyed solids is possible using adaptive propagator ptychography (APP) 20 .Slightly oxidized pure titanium is used as a model system.We revealed that interstitial O atoms display an anisotropic occupation in the interstitial sites in the matrix, rather than a random arrangement, even when the atomic concentration is below 1%.\n\n\nResults and discussions\n\nIn this study, we investigated the behavior of interstitial oxygen in α-Ti.Using the sample preparation methods described in the Methods section, we heat-treated Pure-phase α-Ti in an oxygen atmosphere.The diffusion of oxygen into the bulk results in a gradient of oxygen in the matrix with interstitial atoms of continuously varying concentration, providing a convenient platform for the current study, as illustrated in Fig. 1a.The direction of the arrow indicates the direction of reducing oxygen content.The labeled regions 1-4 correspond to the areas where ptychographic reconstruction and analysis were performed.\n\nElectron ptychography is a computational imaging method that combines scanning transmission electron microscopy and coherent diffraction imaging (CDI) 21,22 .It is characterized by high dose efficiency, high spatial resolution, high phase accuracy, and depth resolution [23][24][25] .By using multislice electron ptychography with an adaptive propagator 20 , deep-sub-angstrom resolution for defective materials 24 , sub-angstrom resolution for beam-sensitive materials 26 , and lattice-resolved antiferromagnetic imaging 25 have been realized.The multislice electron ptychography with adaptive propagator enables direct imaging of interstitial atoms, thanks to several crucial features.First, it eliminates aberrations during the imaging process, thus improving image quality over aberrationcorrected electron microscopy.Second, it is capable of obtaining high accuracy in the retrieved phase of the electron wave function, making it possible to detect weak signals from interstitial atoms.Third, it has good depth-resolution in the electron beam direction and helps to remove signal interference from surface damage or contamination layers, which is particularly crucial for the observation of discrete and weak interstitial signals.Taking an interstitial oxygen atom in a 20-nm-thick sample as an example.In the projected phase of the 20 nm sample, the interstitial O atom accounts for a proportion of 0.4%, whereas in the 1 nm thick slice, it accounts for a proportion of 8%, much higher than 0.4%.Therefore, the employment of 'multislice' significantly enhances the detectability of single interstitial atoms.\n\nFig. 1 presents a schematic to the experiment analysis, including the sample, the APP method, and the reconstructed phase images.Using Area 4 as an illustrative example, we utilize the STEM mode to perform a point-by-point scan using a convergent electron beam.A direct electron detector (EMPAD) 27 captures the diffraction pattern at each scanning point, as shown in Fig. 1b.The datasets collected serve as input for the multislice APP method, enabling iterative reconstruction and generating a sequence of object slices, as depicted in Fig. 1c.After that, we can effectively eliminate damages and oxidations present on the upper and lower surfaces of the sample (shown in Fig. 1e and 1f).Consequently, we are able to analyze intermediate slices that describe the intrinsic structure inside the bulk.[11-20] zone axis of α-Ti.The orange atoms represent interstitial oxygen atoms occupying octahedral interstices.The stacked atomic layers are denoted as A and B. (e, f, and g) Three typical slices of reconstructed phase images of region 4, where surface damages could be found in the top and bottom layers.(h) The corresponding HAADF image.The phase image (g) clearly exhibits interstitial oxygen atoms located at the octahedral interstices, highlighted with red circles.In (h), the HAADF image only displays the matrix Ti atoms, without observable interstitial atoms.(i) The diffractogram corresponding to (g), showing an information limit of 0.32 Å. (j) The diffractogram of the HAADF image, showing an information limit of 0.75 Å.\n\n\nGradient distribution of oxygen diffusion in the matrix\n\nAPP reconstruction was performed for a series of regions depicted in Fig. 1a, with varying oxygen contents.Combining the EDS results (shown in Extended Data Fig. 1) and the HAADF images (displayed in Extended Data Fig. 2), it is evident that region 1 corresponds to the oxide layer and regions 2-4 represent matrix regions that maintain the α-Ti structure.The EDS signals of Extended Data Fig. 1 indicate lower oxygen content in these regions, which decreases gradually with depth.It seems that even deep inside the bulk, the concentration of O remains higher than 10%, which comes from the surface oxidation of the thin TEM sample.It illustrates again the necessity of utilizing multislice ptychography for the study of intrinsic behavior of interstitial atoms inside the bulk.Extended Data Fig. 2 shows the average phase of regions 2-4.There are numerous interstitial atoms in region 2 at the octahedral sites indicated in Fig. 1d, suggesting high oxygen content at this depth.In region 3, the content of interstitial atoms is noticeably lower than that in region 2. In region 4, the contrast of interstitial atoms becomes weak and difficult to detect.For convenience, we designate region 2 as the High-O area, region 3 as the Mid-O area, and region 4 as the Low-O area.\n\n\nDirect imaging of single interstitial oxygen\n\nWhile the damaged surface layers (Fig. 1e and 1f) are eliminated, weak image spots at octahedral sites can be readily observed in the intermediate slices, as highlighted by red circless in Fig. 1g.As a comparison, only Ti atoms are observable in the HAADF image (Fig. 1h) of the same regions.\n\nThrough quantitative analysis, we will show that the weak spots in the phase images are oxygen interstitial atoms.Fig. 1i and 1j show the diffractograms of an APP phase image and a HAADF image, respectively.The information limit of the APP phase is much higher than that of HAADF, supporting direct observation of single interstitial atoms using the APP method.\n\n\nPositioning interstitial atoms\n\nAs shown above, interstitial atoms appear spindles in depth profiles of the phase images (Fig. 2b and 2e) due to the limited resolution along the electron beam direction (about 2-5 nm).In Fig. 2b, the true position of the interstitial atom is indicated, and the center of the spindle matches well the true position.Therefore, the position of the interstitial atom in the electron beam direction can be determined by measuring the center of the spindle.\n\nIn the [11-20] direction of α-Ti, the atomic spacing is 2.95 Å, which is far smaller than the depth resolution.Consequently, the elongation of atoms due to the limited depth resolution hinders the determination of three-dimensional (3D) configuration of matrix Ti atoms.Fortunately, the determination of 3D configuration of interstitial atoms is possible at the limited depth resolution.This is because the content of interstitial atoms is typically below 1%.Roughly estimating, for randomly distributed interstitial atoms, the spacing between two interstitial atoms along the same atomic column is about 10 nm, much greater than the depth resolution (2.9 nm in this study).The position of an interstitial atom in the depth direction corresponds to the center of the spindle, as indicated by the dashed line in Fig. 2e (at 11.6 nm).\n\nWe first performed simulation test.Fifty oxygen atoms are randomly located at octahedral sites in a supercell of 6 nm × 6 nm × 20 nm (with 20 nm along the [11-20] direction), corresponding to a content of 0.18 at.%.A 4D dataset was generated through multislice simulations and the phase images were reconstructed using the APP method.Extended Data Fig. 3 错误!未找到引用源。 displays the 3D distribution of the interstitial oxygen determined according to above procedure, together with the true positions in the model.All 50 interstitial oxygen atoms are successfully identified.\n\n\n3D distribution of interstitial oxygen\n\nIn the low-O region (region 4) in Fig. 1a, 144 interstitial atoms are identified, corresponding to a content of 0.3 at.%.The interstitial atoms are displayed in Fig. 3a, with their projections shown in Fig. 3b-d.The distribution of interstitial oxygen in the Low-O region seem random and is relatively uniform in the image plane.The uneven distribution in the depth direction may be related to the formation of the surface oxide layers.The pair distribution function (PDF) g(r) (where r represents the distance between interstitials) of the measured oxygen atoms was calculated and compared with that for hypothetical randomlydistributed interstitial atoms, as shown in Fig. 3e.The atomic distances of the PDF peaks and the corresponding atomic environments are depicted in Fig. 3f.\n\nWe note that the experimental PDF curve and that for the hypothetical randomly-distributed interstitial atoms do not coincide, indicating that the distribution of interstitial O in α-Ti is not completely disordered.The most obvious difference is the absence of the R1 peak in the experimental PDF.The R1 peak corresponds to the atomic configuration with neighboring oxygen atoms in the c direction occupying the same in-plane position (Fig. 3f-R1).The absence of the R1 peak indicates that the occupancy of neighboring octahedral sites in the c direction would lead to high energy and is not allowed.The presence of the R2 peak in the experimental results implies that the neighboring octahedral sites in the basal plane are allowed to be occupied simultaneously.Finally, the absence of the R1 peak and the presence of the R2 and R4 peaks indicate a medium-range ordering (MRO) of interstitial O in the α-Ti.In other words, in a local region, interstitial O atoms tends to occupy interstitial sites within same basal plane, forming a planar configuration and leaving the interstitial sites in neighboring planes vacant.This phenomenon will be further demonstrated later in local strain analysis.\n\n\nInterstitial-induced anisotropic lattice expansion\n\nIn the Mid-O region (Region 3 in Fig. 1a), 1000 interstitial atoms are identified, corresponding to a content of 1.1 at.%.As described above, the interstitial O atoms are not completely disordered even in the Low-O region; they avoid occupying neighboring sites in the c direction.As the oxygen content increases, the interstitial atoms form higher degree of ordering.Fig. 4a shows two phase images of the Mid-O region at the depth of 7 nm and 17 nm.Alternating basal planes are labelled by blue and yellow rhombuses.As indicated by the bule and yellow frames, the interstitial atoms form planar configuration in the basal planes, which are occupied alternatively.\n\nThe size of the interstitial platelets is nanometer scale.It is interesting that the occupation would change with depth.At the depth of 7 nm, interstitial atoms occupy the basal plane labelled by the blue rhombus.At the depth of 17 nm, they occupy the basal plane labelled by the yellow rhombus.c, d, g, and h), indicating a bimodal distribution of distances between adjacent basal planes but a unimodal distribution of lattice length within the basal plane at both 7 nm and 17 nm depths.The introduction of interstitial atoms primarily induces deformation perpendicular to the basal plane.(j) The PDF curves of interstitial oxygen in the Mid-O region, with the atomic distribution in three dimensions used for calculating the PDF presented in Extended Data Fig. 4.\n\nThe formation of interstitial platelets is associated with anisotropic lattice expansion, which is about 2% in the c direction (Fig. 4c and 5g), but almost zero inside the basal planes (Fig. 4d and 5h).Fig. 4i shows the histograms of the interplanar spacings as indicated by the insets.\n\nThe PDF curve of the interstitial oxygen atoms in the Mid-O region is shown in Fig. 4j.A minor peak is observed at the R1 position within this region.This appearance of the R1 peak indicates that a small number of O atoms are forced into unfavorable positions due to the increased O content.Fig. 4j reveals a noticeable rightward shift of the R1 and R3 peaks, while the R2 peak remains unaffected, which is consistent with the anisotropic lattice expansion.\n\n\nConclusions\n\nIn summary, we have realized direct imaging, counting, and positioning of single interstitial O atoms in titanium through adaptive propagation ptychography.Based on the spindle feature of the interstitial atoms in the depth direction, the coordinate of the interstitial atoms can be measured.\n\nQuantitative analysis indicates that the interstitial O atoms in α-Ti form local ordering, which depends on the oxygen content.For 0.3% content of oxygen, the interstitial atoms avoid occupying neighboring sites in the c direction.For 1.1% content of oxygen, they form nanoscale platelets with anisotropic lattice expansion.Imaging, counting, and positioning of single interstitial atoms pave the way for a deeper understanding of the behavior of interstitial atoms and fine tuning of the physical, mechanical, and chemical properties of materials.\n\n\nMaterials and methods\n\n\nMaterials and Sample preparation\n\nPure-phase α-Ti was placed in a tube furnace for heating.Throughout the process, oxygen was introduced at a pressure of 0.2 atmospheres, and the sample was maintained at a temperature of 500 C for one hour to facilitate the diffusion of interstitial oxygen.A thin oxide layer, approximately 50 nm thick, formed on the surface of the Ti metal.Oxygen diffusion into the bulk produces a gradient of oxygen concentration within the matrix.\n\n\nExperimental data collection and reconstruction\n\nThe experimental 4D datasets were collected using a probe-aberration-corrected FEI Titan Cubed Themis G2 electron microscope, equipped with a pixelated detector known as EMPAD.The accelerated voltage applied was 300 kV, and the convergence semi-angle was set to 25 mrad.Each diffraction pattern in the dataset had dimensions of 128×128, with a pixel size of 0.342 1/Å.The probe was under-focused by approximately 20 nm, with a scan step size of 0.468 Å.The dataset was collected using a beam current of roughly 20 pA and a dwell time of 0.5 ms.The diffraction patterns were padded to a size of 200×200 during the reconstruction, corresponding to a pixel size of 0.146 Å in real space.\n\n\nSimulation of ptychography dataset\n\nWe used a custom-built multislice simulation program to generate the 4D dataset for ptychographic reconstruction.The program's reliability has been confirmed in our previous studies.\n\nFig. 1 .\n1\nFig. 1.The schematic of the procedure of investigating single interstitial atoms.(a)\n\n\nFig. 2\n2\nFig.2compares the experimental and simulated phase images of single oxygen interstitial atoms in α-Ti.For the simulation dataset, the thickness of Ti is 20 nm, with an O atom occupying an octahedral interstitial site.After multislice electron diffraction simulation and APP reconstruction, the phase images are shown in Fig.2a-c.As shown in Fig.2a, the signal of the interstitial atom was only visible at the depth of 7 nm, but not at 2 nm and 12 nm.Due to the limited depth resolution in the electron beam direction, the interstitial atom is elongated into a spindle in the depth profile of the reconstructed phase images, as shown in Fig.2b.Fig.2cshow the lateral profile cross the center of the spindle in the depth profile.The interstitial oxygen atom is marked.It is worth noting that spots in individual phase images do not necessarily correspond to interstitial atoms; they are possibly noise spots.Only spindles in depth profiles correspond to interstitial atoms.\n\n\nFig. 2 .\n2\nFig. 2. Direct imaging of a single interstitial oxygen.(a, b, and c) A single interstitial oxygen in APP phase images reconstructed using the simulated dataset.(a) the slices at different depth, the single interstitial oxygen can only be seen at the depth of 7 nm.(b) The depth profile along the dashed line in (a), showing the spindle shape of the interstitial atom.The true position of the interstitial oxygen atom is at the center of the spindle, as indicated by the orange spot.(c) The lateral profile across the center of the spindle in (b), with the peak of the interstitial oxygen indicated.(d, e, and f) The experimental results of an interstitial oxygen atom.(d) phase images at various depths.An interstitial O atom is at 11.6 nm.(e) Depth profile along the dashed line in (d).(f) The lateral profile across the center of the spindle in (e), with the interstitial O indicated.\n\n\nFig. 3 .\n3\nFig. 3. 3D distribution of interstitial oxygen in the Low-O region.(a) The 3D distribution of interstitial oxygen atoms obtained in this region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.(e) The pair distribution function g(r) of the interstitial atoms in (a).The x-axis represents the radial distance r between atoms, and the y-axis represents the corresponding proportion of atom pairs at that distance.The labels R1-R6 indicate the distances between interstices, as shown in (f): R1 = 2.34 Å, R2 = 2.95 Å, R3 = 3.77 Å, R4 = 4.68 Å, R5 = 5.10 Å. R6 represents a collection of distances of 5.54 Å and 5.62 Å.In (e), the black dotted line represents the PDF curve of completely disordered octahedral interstices, while the red dashed line represents the experimental result of interstitial oxygen.\n\n\nFig. 4 .\n4\nFig. 4. The medium-range ordering of interstitial oxygen in Mid-O region.(a) Phase image at a depth of 7 nm, with alternating basal planes labeled by blue and yellow rhombuses.Interstitial atoms are located in the basal planes marked by the blue rhombuses, and typical interstitial sites are highlighted with red circles.(b, c, and d) Represent changes in lattice area, variations in the distance between adjacent basal planes, and changes in lattice length within the basal plane, respectively, illustrating anisotropic lattice expansion induced by the interstitials.(e, f, g, and h) Show the phase image and lattice expansion at a depth of 17 nm, where interstitial oxygen atoms occupy the basal planes labeled by the yellow rhombuses.(i) Histograms of the lengths shown in(c, d, g, and h), indicating a bimodal distribution of distances between adjacent basal planes but a unimodal distribution of lattice length within the basal plane at both 7 nm and 17 nm depths.The introduction of interstitial atoms primarily induces deformation perpendicular to the basal plane.(j) The PDF curves of interstitial oxygen in the Mid-O region, with the atomic distribution in three dimensions used for calculating the PDF presented in Extended Data Fig.4.\n\n\n\n\nWe ensured that the simulation parameters-voltage, convergence angle, and pixel size of the diffraction patterns-matched those used in the actual experiments.The simulated model had a total thickness of 20 nm, closely resembling the experimental sample.To mimic experimental conditions, Poisson noise was added to the diffraction patterns, corresponding to a beam current of 20 pA and a dwell time of 0.5 ms.Extended Data FiguresExtended Data Fig. 1.EDS result of FIB sample.The EDS results from the sample's oxide layer to the interior of the bulk sample are shown.(a) the EDS mapping distribution results, revealing the clear interface between the oxide layer and the α-Ti matrix.(b) the line scan EDS results along the orange line marked in (a).In the region of the oxide layer, the atomic ratio of Ti: O is close to 1: 1, and as we go deeper into the bulk sample, the oxygen content stabilizes at around 18%.This contribution of oxygen percentage is primarily due to the surface oxidation of the transmission electron microscopy (TEM) sample cut by FIB.In the experiment, it is necessary to remove the oxide layer on the upper and lower surfaces of the FIB sample to obtain the real information of the sample.Extended Data Fig. 2. APP results of different regions revealing the different O content.Extended Data Fig. 3. Simulation test of positioning single interstitial atoms.(a) 3D distribution of interstitial atoms determined in the APP phase images (color spots) and the true positions of the atoms in the model (black circles).The color of the spots represents the difference between the corresponding circle and spot, which is the positional error in the depth direction.(b) Projection of the atoms along the electron beam direction.(c) Projection of the atoms along the x direction.Extended Data Fig. 4. Three-dimensional distribution of interstitial oxygen in the Mid-O region.(a) The three-dimensional distribution of interstitial O atoms in Mid-O region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nAcknowledgments:In this work we used the resources of the Physical Sciences Center and Center of High-Performance Computing, Tsinghua University.Funding:This work was supported by the National Natural Science Foundation of China (52388201 and 51525102).Competing interests:The authors declare no competing financial interests.\nFundamentals of semiconductors. P Yu, 2005Springer\n\nSuperconductivity: fundamentals and applications. W Buckel, R Kleiner, 2008John Wiley & Sons\n\nInterstitial alloys. H J Goldschmid, 2013Springer\n\nMassive interstitial solid solution alloys achieve near-theoretical strength. 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M W Tate, Microscopy and Microanalysis. 222016\n"},"corpus_id":{"kind":"number","value":270067865,"string":"270,067,865"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2405.18164\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2405.18164v1.pdf\"\n ],\n \"pdf_sha\": \"54dfe98b9413890583e39b9594351d5ff61e468d\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Imaging, counting, and positioning single interstitial atoms in solids"},"authors":{"kind":"list like","value":["Jizhe Cui \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina","Haozhi Sha \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina","Liangze Mao \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina","Kang Sun \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina","Wenfeng Yang \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina","Rong Yu \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina"],"string":"[\n \"Jizhe Cui \\nSchool of Materials Science and Engineering\\nTsinghua University\\n100084BeijingChina\\n\\nKey Laboratory of Advanced Materials of Ministry of Education\\nTsinghua University\\n100084BeijingChina\\n\\nState Key Laboratory of New Ceramics and Fine Processing\\nTsinghua University\\n100084BeijingChina\",\n \"Haozhi Sha \\nSchool of Materials Science and Engineering\\nTsinghua University\\n100084BeijingChina\\n\\nKey Laboratory of Advanced Materials of Ministry of Education\\nTsinghua University\\n100084BeijingChina\\n\\nState Key Laboratory of New Ceramics and Fine Processing\\nTsinghua University\\n100084BeijingChina\",\n \"Liangze Mao \\nSchool of Materials Science and Engineering\\nTsinghua University\\n100084BeijingChina\\n\\nKey Laboratory of Advanced Materials of Ministry of Education\\nTsinghua University\\n100084BeijingChina\\n\\nState Key Laboratory of New Ceramics and Fine Processing\\nTsinghua University\\n100084BeijingChina\",\n \"Kang Sun \\nSchool of Materials Science and Engineering\\nTsinghua University\\n100084BeijingChina\\n\\nKey Laboratory of Advanced Materials of Ministry of Education\\nTsinghua University\\n100084BeijingChina\\n\\nState Key Laboratory of New Ceramics and Fine Processing\\nTsinghua University\\n100084BeijingChina\",\n \"Wenfeng Yang \\nSchool of Materials Science and Engineering\\nTsinghua University\\n100084BeijingChina\\n\\nKey Laboratory of Advanced Materials of Ministry of Education\\nTsinghua University\\n100084BeijingChina\\n\\nState Key Laboratory of New Ceramics and Fine Processing\\nTsinghua University\\n100084BeijingChina\",\n \"Rong Yu \\nSchool of Materials Science and Engineering\\nTsinghua University\\n100084BeijingChina\\n\\nKey Laboratory of Advanced Materials of Ministry of Education\\nTsinghua University\\n100084BeijingChina\\n\\nState Key Laboratory of New Ceramics and Fine Processing\\nTsinghua University\\n100084BeijingChina\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Interstitial atoms are ubiquitous in solids and they are widely incorporated into materials to tune their lattice structure, electronic transportation, and mechanical properties.Because the distribution of interstitial atoms in matrix materials is usually disordered and most of them are light atoms with weak scattering ability, it remains a challenge to directly image single interstitial atoms and measure their geometrical positions.In this work, direct imaging and measuring of single interstitial atoms have been realized with adaptive-propagator ptychography.The measurement of their three-dimensional coordinates enables quantitative analysis of the pair distribution function of the interstitial atoms and reveals the anisotropic occupation of oxygen in the interstitial sites in titanium.The current work paves the way for the determination of interstitial atoms in materials, and for the correlation between the atomicscale behavior of interstitial atoms and the physical properties of materials."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Interstitial atoms are ubiquitous in materials and play important role in tuning the mechanical and physical properties of semiconductors, superconductors, and alloys [1][2][3][4] .Recently, incorporation of interstitial atoms in alloys has been increasingly employed to improve mechanical properties of alloys [5][6][7][8][9][10][11] .As a paradigm of materials science, understanding the physical properties often requires studying the atomic structure of the materials 12,13 .Therefore, it is imperative to obtain atomic-scale information of interstitial atoms for understanding and controlling the behavior of interstitial atoms in materials.Previous studies on interstitial atoms mostly rely on statistical methods, such as the positron annihilation spectroscopy 14 , X-ray diffuse scattering 15 , internal friction 16 and electron diffraction 17 .While these methods are useful, they often sacrifice spatial resolution and provide only average results.\nDirect imaging of interstitial atoms in solids has been challenging due to two primary reasons.\nFirst, interstitial atoms are usually light atoms, like carbon, nitrogen, and oxygen; they give very weak electron scattering compared with the matrix atoms.Second, the surfaces of solids are often structurally modified, damaged, or contaminated, forming a layer with different atomic structure than the bulk inside.The electron scattering signals from the surface layers result in strong noise that prevent the detection of interstitial atoms inside the bulk.As a result, direct imaging of interstitial atoms in solids requires very high signal-to-noise ratio in images and the elimination of the surface contributions to the bulk signals.\nRecently, oxygen atoms in a TiZrNb alloy of the body-center-cubic structure was shown to occupy the tetrahedral interstitial sites 18 .Since the concentrations (12 at.%) of oxygen in the alloy is high, the interstitial atoms are crowded together with an average spacing of 6 Å.These densely distributed interstitial atoms can provide stronger signals for detection, but at the same time make the detection only provide their average information and not about the behavior of individual interstitial atoms.\nInterstitial atoms usually have low concentrations in solids; even a few parts per million of interstitial doping is enough to significantly improve properties 1,19 .It is therefore important to be able to detect single interstitial atoms, even though they can only give weak signals for detection.\nHere we show that imaging, counting, and positioning of single interstitial atoms in lightly alloyed solids is possible using adaptive propagator ptychography (APP) 20 .Slightly oxidized pure titanium is used as a model system.We revealed that interstitial O atoms display an anisotropic occupation in the interstitial sites in the matrix, rather than a random arrangement, even when the atomic concentration is below 1%."},{"title":"Results and discussions","content":"In this study, we investigated the behavior of interstitial oxygen in α-Ti.Using the sample preparation methods described in the Methods section, we heat-treated Pure-phase α-Ti in an oxygen atmosphere.The diffusion of oxygen into the bulk results in a gradient of oxygen in the matrix with interstitial atoms of continuously varying concentration, providing a convenient platform for the current study, as illustrated in Fig. 1a.The direction of the arrow indicates the direction of reducing oxygen content.The labeled regions 1-4 correspond to the areas where ptychographic reconstruction and analysis were performed.\nElectron ptychography is a computational imaging method that combines scanning transmission electron microscopy and coherent diffraction imaging (CDI) 21,22 .It is characterized by high dose efficiency, high spatial resolution, high phase accuracy, and depth resolution [23][24][25] .By using multislice electron ptychography with an adaptive propagator 20 , deep-sub-angstrom resolution for defective materials 24 , sub-angstrom resolution for beam-sensitive materials 26 , and lattice-resolved antiferromagnetic imaging 25 have been realized.The multislice electron ptychography with adaptive propagator enables direct imaging of interstitial atoms, thanks to several crucial features.First, it eliminates aberrations during the imaging process, thus improving image quality over aberrationcorrected electron microscopy.Second, it is capable of obtaining high accuracy in the retrieved phase of the electron wave function, making it possible to detect weak signals from interstitial atoms.Third, it has good depth-resolution in the electron beam direction and helps to remove signal interference from surface damage or contamination layers, which is particularly crucial for the observation of discrete and weak interstitial signals.Taking an interstitial oxygen atom in a 20-nm-thick sample as an example.In the projected phase of the 20 nm sample, the interstitial O atom accounts for a proportion of 0.4%, whereas in the 1 nm thick slice, it accounts for a proportion of 8%, much higher than 0.4%.Therefore, the employment of 'multislice' significantly enhances the detectability of single interstitial atoms.\nFig. 1 presents a schematic to the experiment analysis, including the sample, the APP method, and the reconstructed phase images.Using Area 4 as an illustrative example, we utilize the STEM mode to perform a point-by-point scan using a convergent electron beam.A direct electron detector (EMPAD) 27 captures the diffraction pattern at each scanning point, as shown in Fig. 1b.The datasets collected serve as input for the multislice APP method, enabling iterative reconstruction and generating a sequence of object slices, as depicted in Fig. 1c.After that, we can effectively eliminate damages and oxidations present on the upper and lower surfaces of the sample (shown in Fig. 1e and 1f).Consequently, we are able to analyze intermediate slices that describe the intrinsic structure inside the bulk.[11-20] zone axis of α-Ti.The orange atoms represent interstitial oxygen atoms occupying octahedral interstices.The stacked atomic layers are denoted as A and B. (e, f, and g) Three typical slices of reconstructed phase images of region 4, where surface damages could be found in the top and bottom layers.(h) The corresponding HAADF image.The phase image (g) clearly exhibits interstitial oxygen atoms located at the octahedral interstices, highlighted with red circles.In (h), the HAADF image only displays the matrix Ti atoms, without observable interstitial atoms.(i) The diffractogram corresponding to (g), showing an information limit of 0.32 Å. (j) The diffractogram of the HAADF image, showing an information limit of 0.75 Å."},{"title":"Gradient distribution of oxygen diffusion in the matrix","content":"APP reconstruction was performed for a series of regions depicted in Fig. 1a, with varying oxygen contents.Combining the EDS results (shown in Extended Data Fig. 1) and the HAADF images (displayed in Extended Data Fig. 2), it is evident that region 1 corresponds to the oxide layer and regions 2-4 represent matrix regions that maintain the α-Ti structure.The EDS signals of Extended Data Fig. 1 indicate lower oxygen content in these regions, which decreases gradually with depth.It seems that even deep inside the bulk, the concentration of O remains higher than 10%, which comes from the surface oxidation of the thin TEM sample.It illustrates again the necessity of utilizing multislice ptychography for the study of intrinsic behavior of interstitial atoms inside the bulk.Extended Data Fig. 2 shows the average phase of regions 2-4.There are numerous interstitial atoms in region 2 at the octahedral sites indicated in Fig. 1d, suggesting high oxygen content at this depth.In region 3, the content of interstitial atoms is noticeably lower than that in region 2. In region 4, the contrast of interstitial atoms becomes weak and difficult to detect.For convenience, we designate region 2 as the High-O area, region 3 as the Mid-O area, and region 4 as the Low-O area."},{"title":"Direct imaging of single interstitial oxygen","content":"While the damaged surface layers (Fig. 1e and 1f) are eliminated, weak image spots at octahedral sites can be readily observed in the intermediate slices, as highlighted by red circless in Fig. 1g.As a comparison, only Ti atoms are observable in the HAADF image (Fig. 1h) of the same regions.\nThrough quantitative analysis, we will show that the weak spots in the phase images are oxygen interstitial atoms.Fig. 1i and 1j show the diffractograms of an APP phase image and a HAADF image, respectively.The information limit of the APP phase is much higher than that of HAADF, supporting direct observation of single interstitial atoms using the APP method."},{"title":"Positioning interstitial atoms","content":"As shown above, interstitial atoms appear spindles in depth profiles of the phase images (Fig. 2b and 2e) due to the limited resolution along the electron beam direction (about 2-5 nm).In Fig. 2b, the true position of the interstitial atom is indicated, and the center of the spindle matches well the true position.Therefore, the position of the interstitial atom in the electron beam direction can be determined by measuring the center of the spindle.\nIn the [11-20] direction of α-Ti, the atomic spacing is 2.95 Å, which is far smaller than the depth resolution.Consequently, the elongation of atoms due to the limited depth resolution hinders the determination of three-dimensional (3D) configuration of matrix Ti atoms.Fortunately, the determination of 3D configuration of interstitial atoms is possible at the limited depth resolution.This is because the content of interstitial atoms is typically below 1%.Roughly estimating, for randomly distributed interstitial atoms, the spacing between two interstitial atoms along the same atomic column is about 10 nm, much greater than the depth resolution (2.9 nm in this study).The position of an interstitial atom in the depth direction corresponds to the center of the spindle, as indicated by the dashed line in Fig. 2e (at 11.6 nm).\nWe first performed simulation test.Fifty oxygen atoms are randomly located at octahedral sites in a supercell of 6 nm × 6 nm × 20 nm (with 20 nm along the [11-20] direction), corresponding to a content of 0.18 at.%.A 4D dataset was generated through multislice simulations and the phase images were reconstructed using the APP method.Extended Data Fig. 3 错误!未找到引用源。 displays the 3D distribution of the interstitial oxygen determined according to above procedure, together with the true positions in the model.All 50 interstitial oxygen atoms are successfully identified."},{"title":"3D distribution of interstitial oxygen","content":"In the low-O region (region 4) in Fig. 1a, 144 interstitial atoms are identified, corresponding to a content of 0.3 at.%.The interstitial atoms are displayed in Fig. 3a, with their projections shown in Fig. 3b-d.The distribution of interstitial oxygen in the Low-O region seem random and is relatively uniform in the image plane.The uneven distribution in the depth direction may be related to the formation of the surface oxide layers.The pair distribution function (PDF) g(r) (where r represents the distance between interstitials) of the measured oxygen atoms was calculated and compared with that for hypothetical randomlydistributed interstitial atoms, as shown in Fig. 3e.The atomic distances of the PDF peaks and the corresponding atomic environments are depicted in Fig. 3f.\nWe note that the experimental PDF curve and that for the hypothetical randomly-distributed interstitial atoms do not coincide, indicating that the distribution of interstitial O in α-Ti is not completely disordered.The most obvious difference is the absence of the R1 peak in the experimental PDF.The R1 peak corresponds to the atomic configuration with neighboring oxygen atoms in the c direction occupying the same in-plane position (Fig. 3f-R1).The absence of the R1 peak indicates that the occupancy of neighboring octahedral sites in the c direction would lead to high energy and is not allowed.The presence of the R2 peak in the experimental results implies that the neighboring octahedral sites in the basal plane are allowed to be occupied simultaneously.Finally, the absence of the R1 peak and the presence of the R2 and R4 peaks indicate a medium-range ordering (MRO) of interstitial O in the α-Ti.In other words, in a local region, interstitial O atoms tends to occupy interstitial sites within same basal plane, forming a planar configuration and leaving the interstitial sites in neighboring planes vacant.This phenomenon will be further demonstrated later in local strain analysis."},{"title":"Interstitial-induced anisotropic lattice expansion","content":"In the Mid-O region (Region 3 in Fig. 1a), 1000 interstitial atoms are identified, corresponding to a content of 1.1 at.%.As described above, the interstitial O atoms are not completely disordered even in the Low-O region; they avoid occupying neighboring sites in the c direction.As the oxygen content increases, the interstitial atoms form higher degree of ordering.Fig. 4a shows two phase images of the Mid-O region at the depth of 7 nm and 17 nm.Alternating basal planes are labelled by blue and yellow rhombuses.As indicated by the bule and yellow frames, the interstitial atoms form planar configuration in the basal planes, which are occupied alternatively.\nThe size of the interstitial platelets is nanometer scale.It is interesting that the occupation would change with depth.At the depth of 7 nm, interstitial atoms occupy the basal plane labelled by the blue rhombus.At the depth of 17 nm, they occupy the basal plane labelled by the yellow rhombus.c, d, g, and h), indicating a bimodal distribution of distances between adjacent basal planes but a unimodal distribution of lattice length within the basal plane at both 7 nm and 17 nm depths.The introduction of interstitial atoms primarily induces deformation perpendicular to the basal plane.(j) The PDF curves of interstitial oxygen in the Mid-O region, with the atomic distribution in three dimensions used for calculating the PDF presented in Extended Data Fig. 4.\nThe formation of interstitial platelets is associated with anisotropic lattice expansion, which is about 2% in the c direction (Fig. 4c and 5g), but almost zero inside the basal planes (Fig. 4d and 5h).Fig. 4i shows the histograms of the interplanar spacings as indicated by the insets.\nThe PDF curve of the interstitial oxygen atoms in the Mid-O region is shown in Fig. 4j.A minor peak is observed at the R1 position within this region.This appearance of the R1 peak indicates that a small number of O atoms are forced into unfavorable positions due to the increased O content.Fig. 4j reveals a noticeable rightward shift of the R1 and R3 peaks, while the R2 peak remains unaffected, which is consistent with the anisotropic lattice expansion."},{"title":"Conclusions","content":"In summary, we have realized direct imaging, counting, and positioning of single interstitial O atoms in titanium through adaptive propagation ptychography.Based on the spindle feature of the interstitial atoms in the depth direction, the coordinate of the interstitial atoms can be measured.\nQuantitative analysis indicates that the interstitial O atoms in α-Ti form local ordering, which depends on the oxygen content.For 0.3% content of oxygen, the interstitial atoms avoid occupying neighboring sites in the c direction.For 1.1% content of oxygen, they form nanoscale platelets with anisotropic lattice expansion.Imaging, counting, and positioning of single interstitial atoms pave the way for a deeper understanding of the behavior of interstitial atoms and fine tuning of the physical, mechanical, and chemical properties of materials."},{"title":"Materials and methods","content":""},{"title":"Materials and Sample preparation","content":"Pure-phase α-Ti was placed in a tube furnace for heating.Throughout the process, oxygen was introduced at a pressure of 0.2 atmospheres, and the sample was maintained at a temperature of 500 C for one hour to facilitate the diffusion of interstitial oxygen.A thin oxide layer, approximately 50 nm thick, formed on the surface of the Ti metal.Oxygen diffusion into the bulk produces a gradient of oxygen concentration within the matrix."},{"title":"Experimental data collection and reconstruction","content":"The experimental 4D datasets were collected using a probe-aberration-corrected FEI Titan Cubed Themis G2 electron microscope, equipped with a pixelated detector known as EMPAD.The accelerated voltage applied was 300 kV, and the convergence semi-angle was set to 25 mrad.Each diffraction pattern in the dataset had dimensions of 128×128, with a pixel size of 0.342 1/Å.The probe was under-focused by approximately 20 nm, with a scan step size of 0.468 Å.The dataset was collected using a beam current of roughly 20 pA and a dwell time of 0.5 ms.The diffraction patterns were padded to a size of 200×200 during the reconstruction, corresponding to a pixel size of 0.146 Å in real space."},{"title":"Simulation of ptychography dataset","content":"We used a custom-built multislice simulation program to generate the 4D dataset for ptychographic reconstruction.The program's reliability has been confirmed in our previous studies."},{"title":"Fig. 1 .","content":"Fig. 1.The schematic of the procedure of investigating single interstitial atoms.(a)"},{"title":"Fig. 2","content":"Fig.2compares the experimental and simulated phase images of single oxygen interstitial atoms in α-Ti.For the simulation dataset, the thickness of Ti is 20 nm, with an O atom occupying an octahedral interstitial site.After multislice electron diffraction simulation and APP reconstruction, the phase images are shown in Fig.2a-c.As shown in Fig.2a, the signal of the interstitial atom was only visible at the depth of 7 nm, but not at 2 nm and 12 nm.Due to the limited depth resolution in the electron beam direction, the interstitial atom is elongated into a spindle in the depth profile of the reconstructed phase images, as shown in Fig.2b.Fig.2cshow the lateral profile cross the center of the spindle in the depth profile.The interstitial oxygen atom is marked.It is worth noting that spots in individual phase images do not necessarily correspond to interstitial atoms; they are possibly noise spots.Only spindles in depth profiles correspond to interstitial atoms."},{"title":"Fig. 2 .","content":"Fig. 2. Direct imaging of a single interstitial oxygen.(a, b, and c) A single interstitial oxygen in APP phase images reconstructed using the simulated dataset.(a) the slices at different depth, the single interstitial oxygen can only be seen at the depth of 7 nm.(b) The depth profile along the dashed line in (a), showing the spindle shape of the interstitial atom.The true position of the interstitial oxygen atom is at the center of the spindle, as indicated by the orange spot.(c) The lateral profile across the center of the spindle in (b), with the peak of the interstitial oxygen indicated.(d, e, and f) The experimental results of an interstitial oxygen atom.(d) phase images at various depths.An interstitial O atom is at 11.6 nm.(e) Depth profile along the dashed line in (d).(f) The lateral profile across the center of the spindle in (e), with the interstitial O indicated."},{"title":"Fig. 3 .","content":"Fig. 3. 3D distribution of interstitial oxygen in the Low-O region.(a) The 3D distribution of interstitial oxygen atoms obtained in this region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.(e) The pair distribution function g(r) of the interstitial atoms in (a).The x-axis represents the radial distance r between atoms, and the y-axis represents the corresponding proportion of atom pairs at that distance.The labels R1-R6 indicate the distances between interstices, as shown in (f): R1 = 2.34 Å, R2 = 2.95 Å, R3 = 3.77 Å, R4 = 4.68 Å, R5 = 5.10 Å. R6 represents a collection of distances of 5.54 Å and 5.62 Å.In (e), the black dotted line represents the PDF curve of completely disordered octahedral interstices, while the red dashed line represents the experimental result of interstitial oxygen."},{"title":"Fig. 4 .","content":"Fig. 4. The medium-range ordering of interstitial oxygen in Mid-O region.(a) Phase image at a depth of 7 nm, with alternating basal planes labeled by blue and yellow rhombuses.Interstitial atoms are located in the basal planes marked by the blue rhombuses, and typical interstitial sites are highlighted with red circles.(b, c, and d) Represent changes in lattice area, variations in the distance between adjacent basal planes, and changes in lattice length within the basal plane, respectively, illustrating anisotropic lattice expansion induced by the interstitials.(e, f, g, and h) Show the phase image and lattice expansion at a depth of 17 nm, where interstitial oxygen atoms occupy the basal planes labeled by the yellow rhombuses.(i) Histograms of the lengths shown in(c, d, g, and h), indicating a bimodal distribution of distances between adjacent basal planes but a unimodal distribution of lattice length within the basal plane at both 7 nm and 17 nm depths.The introduction of interstitial atoms primarily induces deformation perpendicular to the basal plane.(j) The PDF curves of interstitial oxygen in the Mid-O region, with the atomic distribution in three dimensions used for calculating the PDF presented in Extended Data Fig.4.\nWe ensured that the simulation parameters-voltage, convergence angle, and pixel size of the diffraction patterns-matched those used in the actual experiments.The simulated model had a total thickness of 20 nm, closely resembling the experimental sample.To mimic experimental conditions, Poisson noise was added to the diffraction patterns, corresponding to a beam current of 20 pA and a dwell time of 0.5 ms.Extended Data FiguresExtended Data Fig. 1.EDS result of FIB sample.The EDS results from the sample's oxide layer to the interior of the bulk sample are shown.(a) the EDS mapping distribution results, revealing the clear interface between the oxide layer and the α-Ti matrix.(b) the line scan EDS results along the orange line marked in (a).In the region of the oxide layer, the atomic ratio of Ti: O is close to 1: 1, and as we go deeper into the bulk sample, the oxygen content stabilizes at around 18%.This contribution of oxygen percentage is primarily due to the surface oxidation of the transmission electron microscopy (TEM) sample cut by FIB.In the experiment, it is necessary to remove the oxide layer on the upper and lower surfaces of the FIB sample to obtain the real information of the sample.Extended Data Fig. 2. APP results of different regions revealing the different O content.Extended Data Fig. 3. Simulation test of positioning single interstitial atoms.(a) 3D distribution of interstitial atoms determined in the APP phase images (color spots) and the true positions of the atoms in the model (black circles).The color of the spots represents the difference between the corresponding circle and spot, which is the positional error in the depth direction.(b) Projection of the atoms along the electron beam direction.(c) Projection of the atoms along the x direction.Extended Data Fig. 4. Three-dimensional distribution of interstitial oxygen in the Mid-O region.(a) The three-dimensional distribution of interstitial O atoms in Mid-O region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Interstitial atoms are ubiquitous in materials and play important role in tuning the mechanical and physical properties of semiconductors, superconductors, and alloys [1][2][3][4] .Recently, incorporation of interstitial atoms in alloys has been increasingly employed to improve mechanical properties of alloys [5][6][7][8][9][10][11] .As a paradigm of materials science, understanding the physical properties often requires studying the atomic structure of the materials 12,13 .Therefore, it is imperative to obtain atomic-scale information of interstitial atoms for understanding and controlling the behavior of interstitial atoms in materials.Previous studies on interstitial atoms mostly rely on statistical methods, such as the positron annihilation spectroscopy 14 , X-ray diffuse scattering 15 , internal friction 16 and electron diffraction 17 .While these methods are useful, they often sacrifice spatial resolution and provide only average results.\\nDirect imaging of interstitial atoms in solids has been challenging due to two primary reasons.\\nFirst, interstitial atoms are usually light atoms, like carbon, nitrogen, and oxygen; they give very weak electron scattering compared with the matrix atoms.Second, the surfaces of solids are often structurally modified, damaged, or contaminated, forming a layer with different atomic structure than the bulk inside.The electron scattering signals from the surface layers result in strong noise that prevent the detection of interstitial atoms inside the bulk.As a result, direct imaging of interstitial atoms in solids requires very high signal-to-noise ratio in images and the elimination of the surface contributions to the bulk signals.\\nRecently, oxygen atoms in a TiZrNb alloy of the body-center-cubic structure was shown to occupy the tetrahedral interstitial sites 18 .Since the concentrations (12 at.%) of oxygen in the alloy is high, the interstitial atoms are crowded together with an average spacing of 6 Å.These densely distributed interstitial atoms can provide stronger signals for detection, but at the same time make the detection only provide their average information and not about the behavior of individual interstitial atoms.\\nInterstitial atoms usually have low concentrations in solids; even a few parts per million of interstitial doping is enough to significantly improve properties 1,19 .It is therefore important to be able to detect single interstitial atoms, even though they can only give weak signals for detection.\\nHere we show that imaging, counting, and positioning of single interstitial atoms in lightly alloyed solids is possible using adaptive propagator ptychography (APP) 20 .Slightly oxidized pure titanium is used as a model system.We revealed that interstitial O atoms display an anisotropic occupation in the interstitial sites in the matrix, rather than a random arrangement, even when the atomic concentration is below 1%.\"\n },\n {\n \"title\": \"Results and discussions\",\n \"content\": \"In this study, we investigated the behavior of interstitial oxygen in α-Ti.Using the sample preparation methods described in the Methods section, we heat-treated Pure-phase α-Ti in an oxygen atmosphere.The diffusion of oxygen into the bulk results in a gradient of oxygen in the matrix with interstitial atoms of continuously varying concentration, providing a convenient platform for the current study, as illustrated in Fig. 1a.The direction of the arrow indicates the direction of reducing oxygen content.The labeled regions 1-4 correspond to the areas where ptychographic reconstruction and analysis were performed.\\nElectron ptychography is a computational imaging method that combines scanning transmission electron microscopy and coherent diffraction imaging (CDI) 21,22 .It is characterized by high dose efficiency, high spatial resolution, high phase accuracy, and depth resolution [23][24][25] .By using multislice electron ptychography with an adaptive propagator 20 , deep-sub-angstrom resolution for defective materials 24 , sub-angstrom resolution for beam-sensitive materials 26 , and lattice-resolved antiferromagnetic imaging 25 have been realized.The multislice electron ptychography with adaptive propagator enables direct imaging of interstitial atoms, thanks to several crucial features.First, it eliminates aberrations during the imaging process, thus improving image quality over aberrationcorrected electron microscopy.Second, it is capable of obtaining high accuracy in the retrieved phase of the electron wave function, making it possible to detect weak signals from interstitial atoms.Third, it has good depth-resolution in the electron beam direction and helps to remove signal interference from surface damage or contamination layers, which is particularly crucial for the observation of discrete and weak interstitial signals.Taking an interstitial oxygen atom in a 20-nm-thick sample as an example.In the projected phase of the 20 nm sample, the interstitial O atom accounts for a proportion of 0.4%, whereas in the 1 nm thick slice, it accounts for a proportion of 8%, much higher than 0.4%.Therefore, the employment of 'multislice' significantly enhances the detectability of single interstitial atoms.\\nFig. 1 presents a schematic to the experiment analysis, including the sample, the APP method, and the reconstructed phase images.Using Area 4 as an illustrative example, we utilize the STEM mode to perform a point-by-point scan using a convergent electron beam.A direct electron detector (EMPAD) 27 captures the diffraction pattern at each scanning point, as shown in Fig. 1b.The datasets collected serve as input for the multislice APP method, enabling iterative reconstruction and generating a sequence of object slices, as depicted in Fig. 1c.After that, we can effectively eliminate damages and oxidations present on the upper and lower surfaces of the sample (shown in Fig. 1e and 1f).Consequently, we are able to analyze intermediate slices that describe the intrinsic structure inside the bulk.[11-20] zone axis of α-Ti.The orange atoms represent interstitial oxygen atoms occupying octahedral interstices.The stacked atomic layers are denoted as A and B. (e, f, and g) Three typical slices of reconstructed phase images of region 4, where surface damages could be found in the top and bottom layers.(h) The corresponding HAADF image.The phase image (g) clearly exhibits interstitial oxygen atoms located at the octahedral interstices, highlighted with red circles.In (h), the HAADF image only displays the matrix Ti atoms, without observable interstitial atoms.(i) The diffractogram corresponding to (g), showing an information limit of 0.32 Å. (j) The diffractogram of the HAADF image, showing an information limit of 0.75 Å.\"\n },\n {\n \"title\": \"Gradient distribution of oxygen diffusion in the matrix\",\n \"content\": \"APP reconstruction was performed for a series of regions depicted in Fig. 1a, with varying oxygen contents.Combining the EDS results (shown in Extended Data Fig. 1) and the HAADF images (displayed in Extended Data Fig. 2), it is evident that region 1 corresponds to the oxide layer and regions 2-4 represent matrix regions that maintain the α-Ti structure.The EDS signals of Extended Data Fig. 1 indicate lower oxygen content in these regions, which decreases gradually with depth.It seems that even deep inside the bulk, the concentration of O remains higher than 10%, which comes from the surface oxidation of the thin TEM sample.It illustrates again the necessity of utilizing multislice ptychography for the study of intrinsic behavior of interstitial atoms inside the bulk.Extended Data Fig. 2 shows the average phase of regions 2-4.There are numerous interstitial atoms in region 2 at the octahedral sites indicated in Fig. 1d, suggesting high oxygen content at this depth.In region 3, the content of interstitial atoms is noticeably lower than that in region 2. In region 4, the contrast of interstitial atoms becomes weak and difficult to detect.For convenience, we designate region 2 as the High-O area, region 3 as the Mid-O area, and region 4 as the Low-O area.\"\n },\n {\n \"title\": \"Direct imaging of single interstitial oxygen\",\n \"content\": \"While the damaged surface layers (Fig. 1e and 1f) are eliminated, weak image spots at octahedral sites can be readily observed in the intermediate slices, as highlighted by red circless in Fig. 1g.As a comparison, only Ti atoms are observable in the HAADF image (Fig. 1h) of the same regions.\\nThrough quantitative analysis, we will show that the weak spots in the phase images are oxygen interstitial atoms.Fig. 1i and 1j show the diffractograms of an APP phase image and a HAADF image, respectively.The information limit of the APP phase is much higher than that of HAADF, supporting direct observation of single interstitial atoms using the APP method.\"\n },\n {\n \"title\": \"Positioning interstitial atoms\",\n \"content\": \"As shown above, interstitial atoms appear spindles in depth profiles of the phase images (Fig. 2b and 2e) due to the limited resolution along the electron beam direction (about 2-5 nm).In Fig. 2b, the true position of the interstitial atom is indicated, and the center of the spindle matches well the true position.Therefore, the position of the interstitial atom in the electron beam direction can be determined by measuring the center of the spindle.\\nIn the [11-20] direction of α-Ti, the atomic spacing is 2.95 Å, which is far smaller than the depth resolution.Consequently, the elongation of atoms due to the limited depth resolution hinders the determination of three-dimensional (3D) configuration of matrix Ti atoms.Fortunately, the determination of 3D configuration of interstitial atoms is possible at the limited depth resolution.This is because the content of interstitial atoms is typically below 1%.Roughly estimating, for randomly distributed interstitial atoms, the spacing between two interstitial atoms along the same atomic column is about 10 nm, much greater than the depth resolution (2.9 nm in this study).The position of an interstitial atom in the depth direction corresponds to the center of the spindle, as indicated by the dashed line in Fig. 2e (at 11.6 nm).\\nWe first performed simulation test.Fifty oxygen atoms are randomly located at octahedral sites in a supercell of 6 nm × 6 nm × 20 nm (with 20 nm along the [11-20] direction), corresponding to a content of 0.18 at.%.A 4D dataset was generated through multislice simulations and the phase images were reconstructed using the APP method.Extended Data Fig. 3 错误!未找到引用源。 displays the 3D distribution of the interstitial oxygen determined according to above procedure, together with the true positions in the model.All 50 interstitial oxygen atoms are successfully identified.\"\n },\n {\n \"title\": \"3D distribution of interstitial oxygen\",\n \"content\": \"In the low-O region (region 4) in Fig. 1a, 144 interstitial atoms are identified, corresponding to a content of 0.3 at.%.The interstitial atoms are displayed in Fig. 3a, with their projections shown in Fig. 3b-d.The distribution of interstitial oxygen in the Low-O region seem random and is relatively uniform in the image plane.The uneven distribution in the depth direction may be related to the formation of the surface oxide layers.The pair distribution function (PDF) g(r) (where r represents the distance between interstitials) of the measured oxygen atoms was calculated and compared with that for hypothetical randomlydistributed interstitial atoms, as shown in Fig. 3e.The atomic distances of the PDF peaks and the corresponding atomic environments are depicted in Fig. 3f.\\nWe note that the experimental PDF curve and that for the hypothetical randomly-distributed interstitial atoms do not coincide, indicating that the distribution of interstitial O in α-Ti is not completely disordered.The most obvious difference is the absence of the R1 peak in the experimental PDF.The R1 peak corresponds to the atomic configuration with neighboring oxygen atoms in the c direction occupying the same in-plane position (Fig. 3f-R1).The absence of the R1 peak indicates that the occupancy of neighboring octahedral sites in the c direction would lead to high energy and is not allowed.The presence of the R2 peak in the experimental results implies that the neighboring octahedral sites in the basal plane are allowed to be occupied simultaneously.Finally, the absence of the R1 peak and the presence of the R2 and R4 peaks indicate a medium-range ordering (MRO) of interstitial O in the α-Ti.In other words, in a local region, interstitial O atoms tends to occupy interstitial sites within same basal plane, forming a planar configuration and leaving the interstitial sites in neighboring planes vacant.This phenomenon will be further demonstrated later in local strain analysis.\"\n },\n {\n \"title\": \"Interstitial-induced anisotropic lattice expansion\",\n \"content\": \"In the Mid-O region (Region 3 in Fig. 1a), 1000 interstitial atoms are identified, corresponding to a content of 1.1 at.%.As described above, the interstitial O atoms are not completely disordered even in the Low-O region; they avoid occupying neighboring sites in the c direction.As the oxygen content increases, the interstitial atoms form higher degree of ordering.Fig. 4a shows two phase images of the Mid-O region at the depth of 7 nm and 17 nm.Alternating basal planes are labelled by blue and yellow rhombuses.As indicated by the bule and yellow frames, the interstitial atoms form planar configuration in the basal planes, which are occupied alternatively.\\nThe size of the interstitial platelets is nanometer scale.It is interesting that the occupation would change with depth.At the depth of 7 nm, interstitial atoms occupy the basal plane labelled by the blue rhombus.At the depth of 17 nm, they occupy the basal plane labelled by the yellow rhombus.c, d, g, and h), indicating a bimodal distribution of distances between adjacent basal planes but a unimodal distribution of lattice length within the basal plane at both 7 nm and 17 nm depths.The introduction of interstitial atoms primarily induces deformation perpendicular to the basal plane.(j) The PDF curves of interstitial oxygen in the Mid-O region, with the atomic distribution in three dimensions used for calculating the PDF presented in Extended Data Fig. 4.\\nThe formation of interstitial platelets is associated with anisotropic lattice expansion, which is about 2% in the c direction (Fig. 4c and 5g), but almost zero inside the basal planes (Fig. 4d and 5h).Fig. 4i shows the histograms of the interplanar spacings as indicated by the insets.\\nThe PDF curve of the interstitial oxygen atoms in the Mid-O region is shown in Fig. 4j.A minor peak is observed at the R1 position within this region.This appearance of the R1 peak indicates that a small number of O atoms are forced into unfavorable positions due to the increased O content.Fig. 4j reveals a noticeable rightward shift of the R1 and R3 peaks, while the R2 peak remains unaffected, which is consistent with the anisotropic lattice expansion.\"\n },\n {\n \"title\": \"Conclusions\",\n \"content\": \"In summary, we have realized direct imaging, counting, and positioning of single interstitial O atoms in titanium through adaptive propagation ptychography.Based on the spindle feature of the interstitial atoms in the depth direction, the coordinate of the interstitial atoms can be measured.\\nQuantitative analysis indicates that the interstitial O atoms in α-Ti form local ordering, which depends on the oxygen content.For 0.3% content of oxygen, the interstitial atoms avoid occupying neighboring sites in the c direction.For 1.1% content of oxygen, they form nanoscale platelets with anisotropic lattice expansion.Imaging, counting, and positioning of single interstitial atoms pave the way for a deeper understanding of the behavior of interstitial atoms and fine tuning of the physical, mechanical, and chemical properties of materials.\"\n },\n {\n \"title\": \"Materials and methods\",\n \"content\": \"\"\n },\n {\n \"title\": \"Materials and Sample preparation\",\n \"content\": \"Pure-phase α-Ti was placed in a tube furnace for heating.Throughout the process, oxygen was introduced at a pressure of 0.2 atmospheres, and the sample was maintained at a temperature of 500 C for one hour to facilitate the diffusion of interstitial oxygen.A thin oxide layer, approximately 50 nm thick, formed on the surface of the Ti metal.Oxygen diffusion into the bulk produces a gradient of oxygen concentration within the matrix.\"\n },\n {\n \"title\": \"Experimental data collection and reconstruction\",\n \"content\": \"The experimental 4D datasets were collected using a probe-aberration-corrected FEI Titan Cubed Themis G2 electron microscope, equipped with a pixelated detector known as EMPAD.The accelerated voltage applied was 300 kV, and the convergence semi-angle was set to 25 mrad.Each diffraction pattern in the dataset had dimensions of 128×128, with a pixel size of 0.342 1/Å.The probe was under-focused by approximately 20 nm, with a scan step size of 0.468 Å.The dataset was collected using a beam current of roughly 20 pA and a dwell time of 0.5 ms.The diffraction patterns were padded to a size of 200×200 during the reconstruction, corresponding to a pixel size of 0.146 Å in real space.\"\n },\n {\n \"title\": \"Simulation of ptychography dataset\",\n \"content\": \"We used a custom-built multislice simulation program to generate the 4D dataset for ptychographic reconstruction.The program's reliability has been confirmed in our previous studies.\"\n },\n {\n \"title\": \"Fig. 1 .\",\n \"content\": \"Fig. 1.The schematic of the procedure of investigating single interstitial atoms.(a)\"\n },\n {\n \"title\": \"Fig. 2\",\n \"content\": \"Fig.2compares the experimental and simulated phase images of single oxygen interstitial atoms in α-Ti.For the simulation dataset, the thickness of Ti is 20 nm, with an O atom occupying an octahedral interstitial site.After multislice electron diffraction simulation and APP reconstruction, the phase images are shown in Fig.2a-c.As shown in Fig.2a, the signal of the interstitial atom was only visible at the depth of 7 nm, but not at 2 nm and 12 nm.Due to the limited depth resolution in the electron beam direction, the interstitial atom is elongated into a spindle in the depth profile of the reconstructed phase images, as shown in Fig.2b.Fig.2cshow the lateral profile cross the center of the spindle in the depth profile.The interstitial oxygen atom is marked.It is worth noting that spots in individual phase images do not necessarily correspond to interstitial atoms; they are possibly noise spots.Only spindles in depth profiles correspond to interstitial atoms.\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"Fig. 2. Direct imaging of a single interstitial oxygen.(a, b, and c) A single interstitial oxygen in APP phase images reconstructed using the simulated dataset.(a) the slices at different depth, the single interstitial oxygen can only be seen at the depth of 7 nm.(b) The depth profile along the dashed line in (a), showing the spindle shape of the interstitial atom.The true position of the interstitial oxygen atom is at the center of the spindle, as indicated by the orange spot.(c) The lateral profile across the center of the spindle in (b), with the peak of the interstitial oxygen indicated.(d, e, and f) The experimental results of an interstitial oxygen atom.(d) phase images at various depths.An interstitial O atom is at 11.6 nm.(e) Depth profile along the dashed line in (d).(f) The lateral profile across the center of the spindle in (e), with the interstitial O indicated.\"\n },\n {\n \"title\": \"Fig. 3 .\",\n \"content\": \"Fig. 3. 3D distribution of interstitial oxygen in the Low-O region.(a) The 3D distribution of interstitial oxygen atoms obtained in this region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.(e) The pair distribution function g(r) of the interstitial atoms in (a).The x-axis represents the radial distance r between atoms, and the y-axis represents the corresponding proportion of atom pairs at that distance.The labels R1-R6 indicate the distances between interstices, as shown in (f): R1 = 2.34 Å, R2 = 2.95 Å, R3 = 3.77 Å, R4 = 4.68 Å, R5 = 5.10 Å. R6 represents a collection of distances of 5.54 Å and 5.62 Å.In (e), the black dotted line represents the PDF curve of completely disordered octahedral interstices, while the red dashed line represents the experimental result of interstitial oxygen.\"\n },\n {\n \"title\": \"Fig. 4 .\",\n \"content\": \"Fig. 4. The medium-range ordering of interstitial oxygen in Mid-O region.(a) Phase image at a depth of 7 nm, with alternating basal planes labeled by blue and yellow rhombuses.Interstitial atoms are located in the basal planes marked by the blue rhombuses, and typical interstitial sites are highlighted with red circles.(b, c, and d) Represent changes in lattice area, variations in the distance between adjacent basal planes, and changes in lattice length within the basal plane, respectively, illustrating anisotropic lattice expansion induced by the interstitials.(e, f, g, and h) Show the phase image and lattice expansion at a depth of 17 nm, where interstitial oxygen atoms occupy the basal planes labeled by the yellow rhombuses.(i) Histograms of the lengths shown in(c, d, g, and h), indicating a bimodal distribution of distances between adjacent basal planes but a unimodal distribution of lattice length within the basal plane at both 7 nm and 17 nm depths.The introduction of interstitial atoms primarily induces deformation perpendicular to the basal plane.(j) The PDF curves of interstitial oxygen in the Mid-O region, with the atomic distribution in three dimensions used for calculating the PDF presented in Extended Data Fig.4.\\nWe ensured that the simulation parameters-voltage, convergence angle, and pixel size of the diffraction patterns-matched those used in the actual experiments.The simulated model had a total thickness of 20 nm, closely resembling the experimental sample.To mimic experimental conditions, Poisson noise was added to the diffraction patterns, corresponding to a beam current of 20 pA and a dwell time of 0.5 ms.Extended Data FiguresExtended Data Fig. 1.EDS result of FIB sample.The EDS results from the sample's oxide layer to the interior of the bulk sample are shown.(a) the EDS mapping distribution results, revealing the clear interface between the oxide layer and the α-Ti matrix.(b) the line scan EDS results along the orange line marked in (a).In the region of the oxide layer, the atomic ratio of Ti: O is close to 1: 1, and as we go deeper into the bulk sample, the oxygen content stabilizes at around 18%.This contribution of oxygen percentage is primarily due to the surface oxidation of the transmission electron microscopy (TEM) sample cut by FIB.In the experiment, it is necessary to remove the oxide layer on the upper and lower surfaces of the FIB sample to obtain the real information of the sample.Extended Data Fig. 2. APP results of different regions revealing the different O content.Extended Data Fig. 3. Simulation test of positioning single interstitial atoms.(a) 3D distribution of interstitial atoms determined in the APP phase images (color spots) and the true positions of the atoms in the model (black circles).The color of the spots represents the difference between the corresponding circle and spot, which is the positional error in the depth direction.(b) Projection of the atoms along the electron beam direction.(c) Projection of the atoms along the x direction.Extended Data Fig. 4. Three-dimensional distribution of interstitial oxygen in the Mid-O region.(a) The three-dimensional distribution of interstitial O atoms in Mid-O region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Fig. 1.The schematic of the procedure of investigating single interstitial atoms.(a)","content":"Fig. 1 .\n1\nFig. 1.The schematic of the procedure of investigating single interstitial atoms.(a)"},{"id":"","caption":"Fig.2compares the experimental and simulated phase images of single oxygen interstitial atoms in α-Ti.For the simulation dataset, the thickness of Ti is 20 nm, with an O atom occupying an octahedral interstitial site.After multislice electron diffraction simulation and APP reconstruction, the phase images are shown in Fig.2a-c.As shown in Fig.2a, the signal of the interstitial atom was only visible at the depth of 7 nm, but not at 2 nm and 12 nm.Due to the limited depth resolution in the electron beam direction, the interstitial atom is elongated into a spindle in the depth profile of the reconstructed phase images, as shown in Fig.2b.Fig.2cshow the lateral profile cross the center of the spindle in the depth profile.The interstitial oxygen atom is marked.It is worth noting that spots in individual phase images do not necessarily correspond to interstitial atoms; they are possibly noise spots.Only spindles in depth profiles correspond to interstitial atoms.","content":"Fig. 2\n2\nFig.2compares the experimental and simulated phase images of single oxygen interstitial atoms in α-Ti.For the simulation dataset, the thickness of Ti is 20 nm, with an O atom occupying an octahedral interstitial site.After multislice electron diffraction simulation and APP reconstruction, the phase images are shown in Fig.2a-c.As shown in Fig.2a, the signal of the interstitial atom was only visible at the depth of 7 nm, but not at 2 nm and 12 nm.Due to the limited depth resolution in the electron beam direction, the interstitial atom is elongated into a spindle in the depth profile of the reconstructed phase images, as shown in Fig.2b.Fig.2cshow the lateral profile cross the center of the spindle in the depth profile.The interstitial oxygen atom is marked.It is worth noting that spots in individual phase images do not necessarily correspond to interstitial atoms; they are possibly noise spots.Only spindles in depth profiles correspond to interstitial atoms."},{"id":"","caption":"Fig. 2. Direct imaging of a single interstitial oxygen.(a, b, and c) A single interstitial oxygen in APP phase images reconstructed using the simulated dataset.(a) the slices at different depth, the single interstitial oxygen can only be seen at the depth of 7 nm.(b) The depth profile along the dashed line in (a), showing the spindle shape of the interstitial atom.The true position of the interstitial oxygen atom is at the center of the spindle, as indicated by the orange spot.(c) The lateral profile across the center of the spindle in (b), with the peak of the interstitial oxygen indicated.(d, e, and f) The experimental results of an interstitial oxygen atom.(d) phase images at various depths.An interstitial O atom is at 11.6 nm.(e) Depth profile along the dashed line in (d).(f) The lateral profile across the center of the spindle in (e), with the interstitial O indicated.","content":"Fig. 2 .\n2\nFig. 2. Direct imaging of a single interstitial oxygen.(a, b, and c) A single interstitial oxygen in APP phase images reconstructed using the simulated dataset.(a) the slices at different depth, the single interstitial oxygen can only be seen at the depth of 7 nm.(b) The depth profile along the dashed line in (a), showing the spindle shape of the interstitial atom.The true position of the interstitial oxygen atom is at the center of the spindle, as indicated by the orange spot.(c) The lateral profile across the center of the spindle in (b), with the peak of the interstitial oxygen indicated.(d, e, and f) The experimental results of an interstitial oxygen atom.(d) phase images at various depths.An interstitial O atom is at 11.6 nm.(e) Depth profile along the dashed line in (d).(f) The lateral profile across the center of the spindle in (e), with the interstitial O indicated."},{"id":"","caption":"Fig. 3. 3D distribution of interstitial oxygen in the Low-O region.(a) The 3D distribution of interstitial oxygen atoms obtained in this region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.(e) The pair distribution function g(r) of the interstitial atoms in (a).The x-axis represents the radial distance r between atoms, and the y-axis represents the corresponding proportion of atom pairs at that distance.The labels R1-R6 indicate the distances between interstices, as shown in (f): R1 = 2.34 Å, R2 = 2.95 Å, R3 = 3.77 Å, R4 = 4.68 Å, R5 = 5.10 Å. R6 represents a collection of distances of 5.54 Å and 5.62 Å.In (e), the black dotted line represents the PDF curve of completely disordered octahedral interstices, while the red dashed line represents the experimental result of interstitial oxygen.","content":"Fig. 3 .\n3\nFig. 3. 3D distribution of interstitial oxygen in the Low-O region.(a) The 3D distribution of interstitial oxygen atoms obtained in this region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.(e) The pair distribution function g(r) of the interstitial atoms in (a).The x-axis represents the radial distance r between atoms, and the y-axis represents the corresponding proportion of atom pairs at that distance.The labels R1-R6 indicate the distances between interstices, as shown in (f): R1 = 2.34 Å, R2 = 2.95 Å, R3 = 3.77 Å, R4 = 4.68 Å, R5 = 5.10 Å. R6 represents a collection of distances of 5.54 Å and 5.62 Å.In (e), the black dotted line represents the PDF curve of completely disordered octahedral interstices, while the red dashed line represents the experimental result of interstitial oxygen."},{"id":"","caption":"Fig. 4. The medium-range ordering of interstitial oxygen in Mid-O region.(a) Phase image at a depth of 7 nm, with alternating basal planes labeled by blue and yellow rhombuses.Interstitial atoms are located in the basal planes marked by the blue rhombuses, and typical interstitial sites are highlighted with red circles.(b, c, and d) Represent changes in lattice area, variations in the distance between adjacent basal planes, and changes in lattice length within the basal plane, respectively, illustrating anisotropic lattice expansion induced by the interstitials.(e, f, g, and h) Show the phase image and lattice expansion at a depth of 17 nm, where interstitial oxygen atoms occupy the basal planes labeled by the yellow rhombuses.(i) Histograms of the lengths shown in(c, d, g, and h), indicating a bimodal distribution of distances between adjacent basal planes but a unimodal distribution of lattice length within the basal plane at both 7 nm and 17 nm depths.The introduction of interstitial atoms primarily induces deformation perpendicular to the basal plane.(j) The PDF curves of interstitial oxygen in the Mid-O region, with the atomic distribution in three dimensions used for calculating the PDF presented in Extended Data Fig.4.","content":"Fig. 4 .\n4\nFig. 4. The medium-range ordering of interstitial oxygen in Mid-O region.(a) Phase image at a depth of 7 nm, with alternating basal planes labeled by blue and yellow rhombuses.Interstitial atoms are located in the basal planes marked by the blue rhombuses, and typical interstitial sites are highlighted with red circles.(b, c, and d) Represent changes in lattice area, variations in the distance between adjacent basal planes, and changes in lattice length within the basal plane, respectively, illustrating anisotropic lattice expansion induced by the interstitials.(e, f, g, and h) Show the phase image and lattice expansion at a depth of 17 nm, where interstitial oxygen atoms occupy the basal planes labeled by the yellow rhombuses.(i) Histograms of the lengths shown in(c, d, g, and h), indicating a bimodal distribution of distances between adjacent basal planes but a unimodal distribution of lattice length within the basal plane at both 7 nm and 17 nm depths.The introduction of interstitial atoms primarily induces deformation perpendicular to the basal plane.(j) The PDF curves of interstitial oxygen in the Mid-O region, with the atomic distribution in three dimensions used for calculating the PDF presented in Extended Data Fig.4."},{"id":"","caption":"We ensured that the simulation parameters-voltage, convergence angle, and pixel size of the diffraction patterns-matched those used in the actual experiments.The simulated model had a total thickness of 20 nm, closely resembling the experimental sample.To mimic experimental conditions, Poisson noise was added to the diffraction patterns, corresponding to a beam current of 20 pA and a dwell time of 0.5 ms.Extended Data FiguresExtended Data Fig. 1.EDS result of FIB sample.The EDS results from the sample's oxide layer to the interior of the bulk sample are shown.(a) the EDS mapping distribution results, revealing the clear interface between the oxide layer and the α-Ti matrix.(b) the line scan EDS results along the orange line marked in (a).In the region of the oxide layer, the atomic ratio of Ti: O is close to 1: 1, and as we go deeper into the bulk sample, the oxygen content stabilizes at around 18%.This contribution of oxygen percentage is primarily due to the surface oxidation of the transmission electron microscopy (TEM) sample cut by FIB.In the experiment, it is necessary to remove the oxide layer on the upper and lower surfaces of the FIB sample to obtain the real information of the sample.Extended Data Fig. 2. APP results of different regions revealing the different O content.Extended Data Fig. 3. Simulation test of positioning single interstitial atoms.(a) 3D distribution of interstitial atoms determined in the APP phase images (color spots) and the true positions of the atoms in the model (black circles).The color of the spots represents the difference between the corresponding circle and spot, which is the positional error in the depth direction.(b) Projection of the atoms along the electron beam direction.(c) Projection of the atoms along the x direction.Extended Data Fig. 4. Three-dimensional distribution of interstitial oxygen in the Mid-O region.(a) The three-dimensional distribution of interstitial O atoms in Mid-O region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.","content":"We ensured that the simulation parameters-voltage, convergence angle, and pixel size of the diffraction patterns-matched those used in the actual experiments.The simulated model had a total thickness of 20 nm, closely resembling the experimental sample.To mimic experimental conditions, Poisson noise was added to the diffraction patterns, corresponding to a beam current of 20 pA and a dwell time of 0.5 ms.Extended Data FiguresExtended Data Fig. 1.EDS result of FIB sample.The EDS results from the sample's oxide layer to the interior of the bulk sample are shown.(a) the EDS mapping distribution results, revealing the clear interface between the oxide layer and the α-Ti matrix.(b) the line scan EDS results along the orange line marked in (a).In the region of the oxide layer, the atomic ratio of Ti: O is close to 1: 1, and as we go deeper into the bulk sample, the oxygen content stabilizes at around 18%.This contribution of oxygen percentage is primarily due to the surface oxidation of the transmission electron microscopy (TEM) sample cut by FIB.In the experiment, it is necessary to remove the oxide layer on the upper and lower surfaces of the FIB sample to obtain the real information of the sample.Extended Data Fig. 2. APP results of different regions revealing the different O content.Extended Data Fig. 3. Simulation test of positioning single interstitial atoms.(a) 3D distribution of interstitial atoms determined in the APP phase images (color spots) and the true positions of the atoms in the model (black circles).The color of the spots represents the difference between the corresponding circle and spot, which is the positional error in the depth direction.(b) Projection of the atoms along the electron beam direction.(c) Projection of the atoms along the x direction.Extended Data Fig. 4. Three-dimensional distribution of interstitial oxygen in the Mid-O region.(a) The three-dimensional distribution of interstitial O atoms in Mid-O region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes."},{"id":"","caption":"","content":""},{"id":"","caption":"","content":""},{"id":"","caption":"","content":""}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Fig. 1.The schematic of the procedure of investigating single interstitial atoms.(a)\",\n \"content\": \"Fig. 1 .\\n1\\nFig. 1.The schematic of the procedure of investigating single interstitial atoms.(a)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig.2compares the experimental and simulated phase images of single oxygen interstitial atoms in α-Ti.For the simulation dataset, the thickness of Ti is 20 nm, with an O atom occupying an octahedral interstitial site.After multislice electron diffraction simulation and APP reconstruction, the phase images are shown in Fig.2a-c.As shown in Fig.2a, the signal of the interstitial atom was only visible at the depth of 7 nm, but not at 2 nm and 12 nm.Due to the limited depth resolution in the electron beam direction, the interstitial atom is elongated into a spindle in the depth profile of the reconstructed phase images, as shown in Fig.2b.Fig.2cshow the lateral profile cross the center of the spindle in the depth profile.The interstitial oxygen atom is marked.It is worth noting that spots in individual phase images do not necessarily correspond to interstitial atoms; they are possibly noise spots.Only spindles in depth profiles correspond to interstitial atoms.\",\n \"content\": \"Fig. 2\\n2\\nFig.2compares the experimental and simulated phase images of single oxygen interstitial atoms in α-Ti.For the simulation dataset, the thickness of Ti is 20 nm, with an O atom occupying an octahedral interstitial site.After multislice electron diffraction simulation and APP reconstruction, the phase images are shown in Fig.2a-c.As shown in Fig.2a, the signal of the interstitial atom was only visible at the depth of 7 nm, but not at 2 nm and 12 nm.Due to the limited depth resolution in the electron beam direction, the interstitial atom is elongated into a spindle in the depth profile of the reconstructed phase images, as shown in Fig.2b.Fig.2cshow the lateral profile cross the center of the spindle in the depth profile.The interstitial oxygen atom is marked.It is worth noting that spots in individual phase images do not necessarily correspond to interstitial atoms; they are possibly noise spots.Only spindles in depth profiles correspond to interstitial atoms.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 2. Direct imaging of a single interstitial oxygen.(a, b, and c) A single interstitial oxygen in APP phase images reconstructed using the simulated dataset.(a) the slices at different depth, the single interstitial oxygen can only be seen at the depth of 7 nm.(b) The depth profile along the dashed line in (a), showing the spindle shape of the interstitial atom.The true position of the interstitial oxygen atom is at the center of the spindle, as indicated by the orange spot.(c) The lateral profile across the center of the spindle in (b), with the peak of the interstitial oxygen indicated.(d, e, and f) The experimental results of an interstitial oxygen atom.(d) phase images at various depths.An interstitial O atom is at 11.6 nm.(e) Depth profile along the dashed line in (d).(f) The lateral profile across the center of the spindle in (e), with the interstitial O indicated.\",\n \"content\": \"Fig. 2 .\\n2\\nFig. 2. Direct imaging of a single interstitial oxygen.(a, b, and c) A single interstitial oxygen in APP phase images reconstructed using the simulated dataset.(a) the slices at different depth, the single interstitial oxygen can only be seen at the depth of 7 nm.(b) The depth profile along the dashed line in (a), showing the spindle shape of the interstitial atom.The true position of the interstitial oxygen atom is at the center of the spindle, as indicated by the orange spot.(c) The lateral profile across the center of the spindle in (b), with the peak of the interstitial oxygen indicated.(d, e, and f) The experimental results of an interstitial oxygen atom.(d) phase images at various depths.An interstitial O atom is at 11.6 nm.(e) Depth profile along the dashed line in (d).(f) The lateral profile across the center of the spindle in (e), with the interstitial O indicated.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 3. 3D distribution of interstitial oxygen in the Low-O region.(a) The 3D distribution of interstitial oxygen atoms obtained in this region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.(e) The pair distribution function g(r) of the interstitial atoms in (a).The x-axis represents the radial distance r between atoms, and the y-axis represents the corresponding proportion of atom pairs at that distance.The labels R1-R6 indicate the distances between interstices, as shown in (f): R1 = 2.34 Å, R2 = 2.95 Å, R3 = 3.77 Å, R4 = 4.68 Å, R5 = 5.10 Å. R6 represents a collection of distances of 5.54 Å and 5.62 Å.In (e), the black dotted line represents the PDF curve of completely disordered octahedral interstices, while the red dashed line represents the experimental result of interstitial oxygen.\",\n \"content\": \"Fig. 3 .\\n3\\nFig. 3. 3D distribution of interstitial oxygen in the Low-O region.(a) The 3D distribution of interstitial oxygen atoms obtained in this region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.(e) The pair distribution function g(r) of the interstitial atoms in (a).The x-axis represents the radial distance r between atoms, and the y-axis represents the corresponding proportion of atom pairs at that distance.The labels R1-R6 indicate the distances between interstices, as shown in (f): R1 = 2.34 Å, R2 = 2.95 Å, R3 = 3.77 Å, R4 = 4.68 Å, R5 = 5.10 Å. R6 represents a collection of distances of 5.54 Å and 5.62 Å.In (e), the black dotted line represents the PDF curve of completely disordered octahedral interstices, while the red dashed line represents the experimental result of interstitial oxygen.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 4. The medium-range ordering of interstitial oxygen in Mid-O region.(a) Phase image at a depth of 7 nm, with alternating basal planes labeled by blue and yellow rhombuses.Interstitial atoms are located in the basal planes marked by the blue rhombuses, and typical interstitial sites are highlighted with red circles.(b, c, and d) Represent changes in lattice area, variations in the distance between adjacent basal planes, and changes in lattice length within the basal plane, respectively, illustrating anisotropic lattice expansion induced by the interstitials.(e, f, g, and h) Show the phase image and lattice expansion at a depth of 17 nm, where interstitial oxygen atoms occupy the basal planes labeled by the yellow rhombuses.(i) Histograms of the lengths shown in(c, d, g, and h), indicating a bimodal distribution of distances between adjacent basal planes but a unimodal distribution of lattice length within the basal plane at both 7 nm and 17 nm depths.The introduction of interstitial atoms primarily induces deformation perpendicular to the basal plane.(j) The PDF curves of interstitial oxygen in the Mid-O region, with the atomic distribution in three dimensions used for calculating the PDF presented in Extended Data Fig.4.\",\n \"content\": \"Fig. 4 .\\n4\\nFig. 4. The medium-range ordering of interstitial oxygen in Mid-O region.(a) Phase image at a depth of 7 nm, with alternating basal planes labeled by blue and yellow rhombuses.Interstitial atoms are located in the basal planes marked by the blue rhombuses, and typical interstitial sites are highlighted with red circles.(b, c, and d) Represent changes in lattice area, variations in the distance between adjacent basal planes, and changes in lattice length within the basal plane, respectively, illustrating anisotropic lattice expansion induced by the interstitials.(e, f, g, and h) Show the phase image and lattice expansion at a depth of 17 nm, where interstitial oxygen atoms occupy the basal planes labeled by the yellow rhombuses.(i) Histograms of the lengths shown in(c, d, g, and h), indicating a bimodal distribution of distances between adjacent basal planes but a unimodal distribution of lattice length within the basal plane at both 7 nm and 17 nm depths.The introduction of interstitial atoms primarily induces deformation perpendicular to the basal plane.(j) The PDF curves of interstitial oxygen in the Mid-O region, with the atomic distribution in three dimensions used for calculating the PDF presented in Extended Data Fig.4.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"We ensured that the simulation parameters-voltage, convergence angle, and pixel size of the diffraction patterns-matched those used in the actual experiments.The simulated model had a total thickness of 20 nm, closely resembling the experimental sample.To mimic experimental conditions, Poisson noise was added to the diffraction patterns, corresponding to a beam current of 20 pA and a dwell time of 0.5 ms.Extended Data FiguresExtended Data Fig. 1.EDS result of FIB sample.The EDS results from the sample's oxide layer to the interior of the bulk sample are shown.(a) the EDS mapping distribution results, revealing the clear interface between the oxide layer and the α-Ti matrix.(b) the line scan EDS results along the orange line marked in (a).In the region of the oxide layer, the atomic ratio of Ti: O is close to 1: 1, and as we go deeper into the bulk sample, the oxygen content stabilizes at around 18%.This contribution of oxygen percentage is primarily due to the surface oxidation of the transmission electron microscopy (TEM) sample cut by FIB.In the experiment, it is necessary to remove the oxide layer on the upper and lower surfaces of the FIB sample to obtain the real information of the sample.Extended Data Fig. 2. APP results of different regions revealing the different O content.Extended Data Fig. 3. Simulation test of positioning single interstitial atoms.(a) 3D distribution of interstitial atoms determined in the APP phase images (color spots) and the true positions of the atoms in the model (black circles).The color of the spots represents the difference between the corresponding circle and spot, which is the positional error in the depth direction.(b) Projection of the atoms along the electron beam direction.(c) Projection of the atoms along the x direction.Extended Data Fig. 4. Three-dimensional distribution of interstitial oxygen in the Mid-O region.(a) The three-dimensional distribution of interstitial O atoms in Mid-O region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.\",\n \"content\": \"We ensured that the simulation parameters-voltage, convergence angle, and pixel size of the diffraction patterns-matched those used in the actual experiments.The simulated model had a total thickness of 20 nm, closely resembling the experimental sample.To mimic experimental conditions, Poisson noise was added to the diffraction patterns, corresponding to a beam current of 20 pA and a dwell time of 0.5 ms.Extended Data FiguresExtended Data Fig. 1.EDS result of FIB sample.The EDS results from the sample's oxide layer to the interior of the bulk sample are shown.(a) the EDS mapping distribution results, revealing the clear interface between the oxide layer and the α-Ti matrix.(b) the line scan EDS results along the orange line marked in (a).In the region of the oxide layer, the atomic ratio of Ti: O is close to 1: 1, and as we go deeper into the bulk sample, the oxygen content stabilizes at around 18%.This contribution of oxygen percentage is primarily due to the surface oxidation of the transmission electron microscopy (TEM) sample cut by FIB.In the experiment, it is necessary to remove the oxide layer on the upper and lower surfaces of the FIB sample to obtain the real information of the sample.Extended Data Fig. 2. APP results of different regions revealing the different O content.Extended Data Fig. 3. Simulation test of positioning single interstitial atoms.(a) 3D distribution of interstitial atoms determined in the APP phase images (color spots) and the true positions of the atoms in the model (black circles).The color of the spots represents the difference between the corresponding circle and spot, which is the positional error in the depth direction.(b) Projection of the atoms along the electron beam direction.(c) Projection of the atoms along the x direction.Extended Data Fig. 4. Three-dimensional distribution of interstitial oxygen in the Mid-O region.(a) The three-dimensional distribution of interstitial O atoms in Mid-O region.(b, c, and d) Projections along the three axes in (a).Each projection distribution graph contains a histogram of the number of interstitial atoms projected onto two coordinate axes.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"\",\n \"content\": \"\"\n },\n {\n \"id\": \"\",\n \"caption\": \"\",\n \"content\": \"\"\n },\n {\n \"id\": \"\",\n \"caption\": \"\",\n \"content\": \"\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Fundamentals of semiconductors. P Yu, 2005Springer","doi":""},{"text":"Superconductivity: fundamentals and applications. W Buckel, R Kleiner, 2008John Wiley & Sons","doi":""},{"text":"Interstitial alloys. H J Goldschmid, 2013Springer","doi":""},{"text":"Massive interstitial solid solution alloys achieve near-theoretical strength. C Liu, Nature Communications. 1311022022","doi":""},{"text":"Origin of dramatic oxygen solute strengthening effect in titanium. Q Yu, Science. 3472015","doi":""},{"text":"The effect of interstitial carbon on the mechanical properties and dislocation substructure evolution in Fe40.4Ni11.3Mn34.8Al7.5Cr6 high entropy alloys. Z Wang, Acta Materialia. 1202016","doi":""},{"text":"Enhanced strength and ductility in a high-entropy alloy via ordered oxygen complexes. Z Lei, Nature. 5632018","doi":""},{"text":"Mechanism of collective interstitial ordering in Fe-C alloys. X Zhang, Nature Materials. 192020","doi":""},{"text":"High oxygen-content titanium and titanium alloys made from powder. S D Luo, Journal of Alloys and Compounds. 8361555262020","doi":""},{"text":"Strong and ductile titanium-oxygen-iron alloys by additive manufacturing. T Song, Nature. 6182023","doi":""},{"text":"Oxygen-dislocation interaction-mediated nanotwinned nanomartensites in ultra-strong and ductile titanium alloys. C Zhang, Materials Today. 2024","doi":""},{"text":"There's plenty of room at the bottom. R Feynman, Resonance: Journal of Science Education. 162011","doi":""},{"text":"Properties of stress-induced super tetragonal phase in epitaxial BiFeO3 thin film. G Cao, T Zhang, Y.-H Chu, J Zhu, R Yu, Applied Physics Letters. 1182429032021","doi":""},{"text":"Positron annihilation spectroscopy for the determination of thickness and defect profile in thin semiconductor layers. A Zubiaga, Physical Review B. 752053052007","doi":""},{"text":"Low temperature bainitic ferrite: Evidence of carbon supersaturation and tetragonality. C Garcia-Mateo, Acta Materialia. 912015","doi":""},{"text":"Internal-friction analysis of dislocationinterstitial carbon interactions in press-hardened 22MnB5 steel. W S Choi, J Lee, B C De Cooman, Materials Science and Engineering: A. 6392015","doi":""},{"text":"First experimental evidence of oxygen ordering in dilute titaniumoxygen alloys. R Poulain, Materials Research. 102022","doi":""},{"text":"Direct Observation of Oxygen Atoms Taking Tetrahedral Interstitial Sites in Medium-Entropy Body-Centered-Cubic Solutions. C Liu, Adv Mater. 35e22099412023","doi":""},{"text":"Elements of metallurgy and engineering alloys. F C Campbell, 2008ASM international","doi":""},{"text":"Deep sub-angstrom resolution imaging by electron ptychography with misorientation correction. H Sha, J Cui, R Yu, Sci Adv. 822752022","doi":""},{"text":"Diffraction in inhomogeneous primary wave fields. 1. Principle of phase determination from electron diffraction interference. W Hoppe, Acta Crystallographica Section a-Crystal Physics Diffraction Theoretical and General Crystallography A. 251969","doi":""},{"text":"Beyond the Conventional Information Limit -the Relevant Coherence Function. P D Nellist, J M Rodenburg, Ultramicroscopy. 541994","doi":""},{"text":"Electron ptychography achieves atomic-resolution limits set by lattice vibrations. Z Chen, Science. 3722021","doi":""},{"text":"Sub-nanometer-scale mapping of crystal orientation and depthdependent structure of dislocation cores in SrTiO. H Sha, Nat Commun. 1431622023","doi":""},{"text":"Antiferromagnetic imaging via ptychographic phase retrieval. J Cui, H Sha, W Yang, R Yu, Science Bulletin. 692024","doi":""},{"text":"Ptychographic measurements of varying size and shape along zeolite channels. H Sha, Sci Adv. 911512023","doi":""},{"text":"High Dynamic Range Pixel Array Detector for Scanning Transmission Electron Microscopy. M W Tate, Microscopy and Microanalysis. 222016","doi":""}],"string":"[\n {\n \"text\": \"Fundamentals of semiconductors. P Yu, 2005Springer\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Superconductivity: fundamentals and applications. W Buckel, R Kleiner, 2008John Wiley & Sons\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Interstitial alloys. H J Goldschmid, 2013Springer\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Massive interstitial solid solution alloys achieve near-theoretical strength. C Liu, Nature Communications. 1311022022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Origin of dramatic oxygen solute strengthening effect in titanium. Q Yu, Science. 3472015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The effect of interstitial carbon on the mechanical properties and dislocation substructure evolution in Fe40.4Ni11.3Mn34.8Al7.5Cr6 high entropy alloys. Z Wang, Acta Materialia. 1202016\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Enhanced strength and ductility in a high-entropy alloy via ordered oxygen complexes. Z Lei, Nature. 5632018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mechanism of collective interstitial ordering in Fe-C alloys. X Zhang, Nature Materials. 192020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"High oxygen-content titanium and titanium alloys made from powder. S D Luo, Journal of Alloys and Compounds. 8361555262020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Strong and ductile titanium-oxygen-iron alloys by additive manufacturing. T Song, Nature. 6182023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Oxygen-dislocation interaction-mediated nanotwinned nanomartensites in ultra-strong and ductile titanium alloys. C Zhang, Materials Today. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"There's plenty of room at the bottom. R Feynman, Resonance: Journal of Science Education. 162011\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Properties of stress-induced super tetragonal phase in epitaxial BiFeO3 thin film. G Cao, T Zhang, Y.-H Chu, J Zhu, R Yu, Applied Physics Letters. 1182429032021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Positron annihilation spectroscopy for the determination of thickness and defect profile in thin semiconductor layers. A Zubiaga, Physical Review B. 752053052007\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Low temperature bainitic ferrite: Evidence of carbon supersaturation and tetragonality. C Garcia-Mateo, Acta Materialia. 912015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Internal-friction analysis of dislocationinterstitial carbon interactions in press-hardened 22MnB5 steel. W S Choi, J Lee, B C De Cooman, Materials Science and Engineering: A. 6392015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"First experimental evidence of oxygen ordering in dilute titaniumoxygen alloys. R Poulain, Materials Research. 102022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Direct Observation of Oxygen Atoms Taking Tetrahedral Interstitial Sites in Medium-Entropy Body-Centered-Cubic Solutions. C Liu, Adv Mater. 35e22099412023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Elements of metallurgy and engineering alloys. F C Campbell, 2008ASM international\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Deep sub-angstrom resolution imaging by electron ptychography with misorientation correction. H Sha, J Cui, R Yu, Sci Adv. 822752022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Diffraction in inhomogeneous primary wave fields. 1. Principle of phase determination from electron diffraction interference. W Hoppe, Acta Crystallographica Section a-Crystal Physics Diffraction Theoretical and General Crystallography A. 251969\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Beyond the Conventional Information Limit -the Relevant Coherence Function. P D Nellist, J M Rodenburg, Ultramicroscopy. 541994\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Electron ptychography achieves atomic-resolution limits set by lattice vibrations. Z Chen, Science. 3722021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sub-nanometer-scale mapping of crystal orientation and depthdependent structure of dislocation cores in SrTiO. H Sha, Nat Commun. 1431622023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Antiferromagnetic imaging via ptychographic phase retrieval. J Cui, H Sha, W Yang, R Yu, Science Bulletin. 692024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ptychographic measurements of varying size and shape along zeolite channels. H Sha, Sci Adv. 911512023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"High Dynamic Range Pixel Array Detector for Scanning Transmission Electron Microscopy. M W Tate, Microscopy and Microanalysis. 222016\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":573,"cells":{"text":{"kind":"string","value":"\nExperimental assessment of mixing layer scaling laws in Rayleigh-Taylor instability\n\n\nMarco De Paoli marco.de.paoli@tuwien.ac.at \nInstitute of Fluid Mechanics and Heat Transfer\n1060Wien, ViennaTUAustria\n\nPhysics of Fluids Group\nUniversity of Twente\n7500AEEnschedeThe Netherlands\n\nDiego Perissutti \nPolytechnic Department\nUniversity of Udine\n33100UdineItaly\n\nCristian Marchioli \nPolytechnic Department\nUniversity of Udine\n33100UdineItaly\n\nAlfredo Soldati \nInstitute of Fluid Mechanics and Heat Transfer\n1060Wien, ViennaTUAustria\n\nPolytechnic Department\nUniversity of Udine\n33100UdineItaly\n\nExperimental assessment of mixing layer scaling laws in Rayleigh-Taylor instability\n(Dated: September 20, 2022)* Author to whom correspondence should be addressed:\nWe assess experimentally the scaling laws that characterize the mixing region produced by the Rayleigh-Taylor instability in a confined porous medium. In particular, we wish to assess experimentally the existence of a superlinear scaling for the growth of the mixing region, which was observed in recent two-dimensional simulations. To this purpose, we use a Hele-Shaw cell. The flow configuration consists of a heavy fluid layer overlying a lighter fluid layer, initially separated by a horizontal, flat interface. When small perturbations of concentration and velocity fields occur at the interface, convective mixing is eventually produced: Perturbations grow and evolve into large finger-like convective structures that control the transition from the initial diffusiondominated phase of the flow to the subsequent convection-dominated phase. As the flow evolves, diffusion acts to reduce local concentration gradients across the interface of the fingers. When the gradients become sufficiently small, the system attains a stably-stratified state and diffusion is again the dominant mixing mechanisms. We employ an optical method to obtain high-resolution measurements of the density fields and we perform experiments for values of the Rayleigh-Darcy number (i.e., the ratio between convection and diffusion) sufficiently large to exhibit all the flow phases just described, which we characterize via the mixing length, a measure of the extension of the mixing region. We are able to confirm that the growth of the mixing length during the convection-dominated phase follows the superlinear scaling predicted by previous simulations.\n\nI. Introduction\n\nWhen two miscible fluids with different densities move under the action of gravity the relative acceleration between the fluids generates an instability at the separation interface. This instability, called Rayleigh-Taylor instability [1,2], grows in time and leads to the formation of convective flow structures of heavy fluid moving downward and of lighter fluid penetrating upward. The dynamics of the flow is first controlled by diffusion, which is responsible for the thickening of the interface, initially flat and horizontal. Diffusion favours mixing by increasing the length of the separation interface and gives rise to a buoyancy-driven flow by inducing local gradients of concentration. Afterwards, when the flow evolution is controlled by convection, instabilities grow and merge into large and more stable structures. The flow persists until a stable configuration is achieved in which the density distribution is ultimately uniform over the entire domain. Because of its relevance in applications characterized by buoyancy-driven flows, the Rayleigh-Taylor mixing process has been widely investigated when the instability leads to a turbulent flow dominated by convection [3]. In contrast, the evolution of the process within porous domains is far less explored [see 4-6, and references therein] in spite of its paramount importance in many geophysical and industrial situations, including water contamination [7,8], ice formation [9,10], salinity inversion [11,12], petroleum migration [13] and carbon sequestration [14][15][16][17][18].\n\nIn porous media flows, viscosity dominates over inertia at the pore scale and the momentum transport can be described by the Darcy law [19]. In this limit, a wavy and evolving interface leads to the formation of characteristic vertical, elongated structures that are usually called fingers. The lateral spreading (hence, the width or wavelength) and the vertical growth (hence, the amplitude) of these fingers is controlled by the interplay between diffusion and convection. The resulting time evolution of the fingers can be characterized by the mixing length, which is defined as the tip-to-rear finger length, and provides a macroscopic measure of the vertical extension of the mixing region. The evolution of porous Rayleigh-Taylor systems has been investigated with solutes (or chemicals) having different properties. With respect to the properties of the species involved, possible flow configurations can be grouped in three main categories: presence of one chemical, presence of two chemicals with different diffusion coefficients, and presence two species that can chemically react [20]. With the aid of numerical simulations and experiments, Lemaigre et al. [21] observed that in absence of chemical reactions the mixing region grows symmetrically with respect to the initial position of the interface: The growth occurs at a velocity that depends on the nature of the fluids involved, i.e. it is controlled by the diffusivity of the species and by the initial density difference [22]. The evolution of the system is different in presence of chemical reactions and moderate density contrast. The flow pattern develops in an asymmetric fashion [21], and chemical reactions act to stabilize the flow when the product is lighter than the reactants [23]: A local decrease of density induced by the fact that the product is less dense creates a non-monotonic density profile [24] with a minimum that hinders the penetration of the fingers into the bulk of the unmixed region. In contrast, a destabilizing effect can be obtained when the product provides a sufficiently larger density increase with respect to both reactants [25]. Chemical reactions can also induce secondary instabilities in time [20], for instance, when a sufficient amount of chemical reaction product triggers the fingered sinking of denser in the less dense reactant [26]. In summary, reactions can stabilize or destabilize convection, but in all cases, they increase the mixing rate within the host phase.\n\nIn this work, we will refer to the simplest cases, consisting of one chemical specie, i.e.\n\n\nI INTRODUCTION\n\nabsence of double diffusion and chemical reactions, which produces a symmetric growth of the mixing region. According to theoretical scaling arguments, the mixing length (that quantifies the extension of the mixing region) is expected to evolve linearly in time, i.e. with scaling exponent equal to 1, as a result of the balance between buoyancy-induced convection and dissipation due to diffusion and viscosity. However, recent two-dimensional simulations of Darcy flows in porous media [4,5] have shown that the growth of the mixing length is superlinear, with scaling exponent equal to 1.2.\n\nWhile a simplified phenomenological model to estimate the amount of mixing induced by a Rayleigh-Taylor instability has been proposed recently [4], the reasons for the superlinear mixing length growth, not observed in the three-dimensional case [5], are yet unclear. In an effort to shed light on these reasons, in this work we provide a first experimental assessment of the superlinear growth, based on accurate measurements of the scaling exponent. We use a Hele-Shaw apparatus, which consists of two parallel transparent plates separated by a narrow gap. The density difference between the fluid layers is induced by the presence of a solute. For sufficiently-small fluid velocity within the gap, a quasi-two-dimensional Poiseuille flow establishes between the plates. This flow is known to approximate closely the Darcy-type flow observed in ideal porous media when the velocity at which the solute diffuses in the direction normal to the plates is much higher than the vertical advective velocity [27,28].\n\nSince the pioneering work of Saffman and Taylor [29], the Hele-Shaw apparatus has been widely used to mimic the behaviour of two-dimensional Darcy flows. However, timedependent and highly-resolved measurements are still challenging in the Rayleigh-Taylor configuration. This is mainly due to the experimental limitations associated with the initialization of the system, which requires to start the flow from a flat horizontal interface between the fluids. The experimental measurements presented in this work have been performed taking care to achieve proper initial condition and minimizing the local perturbations of concentration and velocity fields, which eventually lead to mixing. At the same time, measurements have been performed with a high resolution in space and time, to ensure an accurate reconstruction of the concentration field, and in domains that are relatively large compared to previous literature results and extend the measurements range. The fluids and the cell have been designed to allow a direct comparison with previous numerical simulations [4,30]. We analyzed the evolution of the mixing length and the wavenumber power spectra (i.e., the finger size). Concerning the mixing length, we observe the same superlinear behaviour predicted by the simulations [4,5], and we provide an explanation for this finding in terms of finite size effects of the flow domain. With respect to the fingers evolution, we find again that the measured number of fingers is in very good agreement with that predicted numerically [30]. Overall, we are able to assess successfully the scaling relations provided by numerical simulations. We believe that this result can open new perspectives for future modelling and parameterization of convective flow in confined porous media. This paper is organised as follows. In Sec. II, we formulate the problem, describe the experimental setup and recall the assumptions underlying the referenced numerical simulations. The results are presented in Sec. III both in qualitative (flow phenomenology) and quantitative (mixing length and wavenumbers) terms. Finally, conclusions are discussed in Sec. IV.\n\n\nII. Methodology\n\nOur experiments were specifically designed to mimic the mixing process produced by the Rayleigh-Taylor instability in a two-dimensional, saturated and confined porous medium. In addition, they were carried out imposing a linear dependency of density and solute concentration, and ensuring the occurrence of a Darcy-type flow to allow direct comparison with previous numerical simulations [4,30].\n\n\nA. Problem formulation\n\nThe process of convective dissolution is studied via the Rayleigh-Taylor instability, corresponding to two layers of fluid of different density, initially in an unstable configuration and subject to relative acceleration under the action of gravity [3]. The process is simulated in the frame of porous media flows, mimicked with the aid of a Hele-Shaw cell, i.e. two parallel and transparent plates of height H separated by a narrow gap b, as shown in Fig. 1(a). The cell is initially saturated with two miscible fluids having same viscosity (µ), arranged such that the heavy fluid (density ρ M ) lies on top of the lighter one (density ρ 0 ), as sketched in Fig. 1(b). The maximum density difference within the system is ∆ρ = ρ M − ρ 0 and is induced by the presence of a solute, namely potassium permanganate (KMnO 4 ). The amount of solute is quantified by its concentration C, which is maximum at the upper layer (C = C M ) and minimum at the lower layer (C = 0). The evolution of the system is controlled by the contributions of buoyancy, which tends to bring the fluids in a stable configuration and it is associated with the buoyancy velocity U , and diffusion, which acts to reduce local concentration gradients and to increase the mixing in the domain, and it is quantified by the diffusion coefficient D. The relative importance of these two contributions can be estimated by the Rayleigh-Darcy number, which represents the governing dimensionless parameter of the system, and is defined as:\nRa = U H φD = H ,(1)\nwith:\nU = g∆ρk µ , = φD U ,(2)\nthe buoyancy velocity and the length scale over which advection and diffusion balance, respectively [31]. The buoyancy velocity, is defined here as the combination of buoyancy (controlled by the acceleration due to gravity g, density contrast ∆ρ, and medium permeability k) and dissipation (controlled by fluid viscosity, µ). The length scale is particularly important when a comparison among systems having different Rayleigh numbers is required. Permeability k and porosity φ are the characteristic properties of the porous medium mimicked by the Hele-Shaw cell. In our experiments, k = b 2 /12 and φ = 1. In order to prevent potential three-dimensional effects we followed the indications emerged in the theoretical paper by Letelier et al. [27] and later confirmed experimentally by De Paoli et al. [28] and Alipour et al. [32]. We introduce the anisotropy ratio, which is an additional parameter describing the geometry of the cell, defined as:\n= b √ 12H .(3)\nand we can use this parameter in combination with the Rayleigh number to ensure that the flow will be of the Darcy-type if the condition 2 Ra → 0 is met. In our study, ≈ 10 −3 and 4.4 × 10 −3 < 2 Ra < 5.4 × 10 −2 , indicating that we are very close to the theoretical limit corresponding to the Darcy-type regime.\n\nFinally, we are interested in examining a system with boundaries that are impermeable to both fluid and solute. Hence, the following conditions are imposed along the cell walls:\nu · n = 0 , ∂C ∂n = 0 ,(4)\nwith n the unit vector perpendicular to the boundary.\n\n\nB. Experimental setup\n\nWe used a transparent Hele-Shaw cell (Polymethyl Methacrylate -PMMA -thickness 8 mm). The wall-normal depth of the fluid layer is constant for all the experiments performed and corresponds to the value b = 300 µm. An impermeable rubber (Klinger-sil C-4400, thickness 300 µm), is placed between the two acrylic plates, and is used to seal the region containing the fluid but also as a spacer between the cells surfaces. The sealing is laser-cut so that the domain investigated is a square of side H = 87 mm. On each side, the midpoints are connected to pipes that allow the cell filling/emptying process. The transparent sheets and the gasket are held in place by an outer metal frame and a set of 10 bolts, which are tightened with the same torque, to prevent the formation of a non-uniform film thickness. The experimental apparatus is represented in Fig. 1(a). Backlighting is provided to the cell by a tunable LED system (150 lamps covered by a diffusing glass). Finally, the evolution of the flow is recorded by a digital sCMOS camera (FlowSense USB 2M-165, Dantec Dynamics). The imaging system and settings will be described in detail in Sec. II C. Next, we describe the fluids adopted and the processes of concentration reconstruction.\n\nThe initial condition of the experiments is an unstable configuration in which a layer of heavy fluid sits on top of a layer of lighter fluid, these layers being initially unmixed. To realize this challenging configuration, we employ a circuit made of a pump, valves and pipes, represented in Fig. 2. This approach has already been used in previous studies [see [33][34][35]. The entire system is initially filled with degassed water, to minimize the formation of bubbles. Then the cell is filled with the working fluids, which are driven by a peristaltic pump (Watson-Marlow 502 S) with the valves in \"loading mode\". In this configuration, the pump is able to inject degassed water and a homogeneous solution of KMnO 4 (concentration C M ) from two separated containers into the cell. Specifically, the solution and the water are injected from the top and bottom gates of the cell, indicated with (T) and (B) in Fig. 2. In this configuration, the valves connected to the left and right gates of the cell (L and R, in Fig. 2) are both open and allow the mixture of water and solution to leave the cell and be disposed in a residue container. Therefore, a flow from the top/bottom walls towards the left/right wall takes place, as indicated by the red and black arrows in Fig. 2. As a result, a sharp interface forms between the two liquid layers, while the vertical concentration gradient makes the fluid-fluid diffusive interface smoother. The competition between horizontal advection and vertical diffusion determines the interface thickness, which can be controlled by the flow rate (see Appendix C for further details on the interface thickness). A regulation valve is employed to balance the flow rate between the top (T) and bottom (B) gates to achieve a straight interface. When the initial condition is set, the valves are placed in \"experiment mode\", and the cell is by-passed by the inflow. The buoyancy-driven flow takes place and image acquisition starts.\n\nThe two fluids are fully miscible but have different density. Specifically, we used water for the lower (i.e. lighter) fluid layer and an aqueous solution of potassium permanganate (KMnO 4 ) for the upper (i.e. denser) fluid layer. While water density is nearly constant among the experiments (it is only dependent on temperature), the density of the KMnO 4 solution can be varied by changing the solute concentration. We consider the coordinate system sketched in Fig. 1(b), where x and z are the spatial coordinates in horizontal and vertical direction, respectively. In this reference frame, the initial condition can be expressed in terms of the solute concentration as follows: C(x, z ≤ H/2, t = 0) = 0 (lower half of the cell) and [33][34][35]. A pumping system (peristaltic pump, Watson-Marlow 502 S) is used to inject water (density ρ 0 ) and solution (ρ M ) from two separate containers, through the gates B and T, respectively. To set up the initial condition, the valves are placed in \"loading mode\" and the exceeding solution is collected from gates L and R into a residue container. The flow through the gates, which is used to modulate the shape of the interface between the two fluid layers, is controlled by a regulation valve. Finally, when the initial condition is achieved, the system is set in \"experiment mode\" and the cell is by-passed by the fluids, which are channeled to the residue container.\nC(x, z ≤ H/2, t = 0) = C M (upper half of the cell), with C M the initialρ 0 , C = 0 x z T B L R inflow water (ρ 0 ) inflow solution (ρ M ) outflow water + solution (ρ 0 < ⍴ < ρ M ) outflow water + solution (ρ 0 < ⍴ < ρ M )\nconcentration of the KMnO 4 solution (upper fluid layer). We consider that the dynamic viscosity, µ = 9.2 × 10 −4 Pa·s, is constant and independent of the solute concentration [36]. Similarly, we assume that the diffusion coefficient is not sensibly affected by either the solute concentration or the local values of velocity D = 1.65 × 10 −9 m 2 /s. This value has been measured and reported in literature, and it is in excellent agreement with theoretical predictions based on electrical neutrality of a simple salt at infinite dilution [see 36, and references therein]. The working fluids have been chosen because of the linear dependency of the density that characterizes the resulting solution with respect to the solute concentration. This feature is essential to make reliable comparisons between experiments and simulations, which is one of the main objectives of the present work.\n\nThe density of an aqueous KMnO 4 solution can be written as a function of the fluid temperature, ϑ, the water density at temperature ϑ, ρ 0 , and the KMnO 4 concentration, C [37]. Please note that also the water density can be computed from the temperature of the fluid, and therefore the parameters that need to be measured to determine the solution density are ϑ (which we assume to be uniform over the cell and constant during the experiment) and the local value of C (inferred from optical measurements). The density difference between the two fluid layers is ∆ρ = ρ M − ρ 0 , where ρ M = ρ(C = C M ) and C M are determined prior to each experiment. At a given temperature, the density of the mixture obtained from empirical correlations [37] is well described by a function of C and C 3/2 . However, within the range of mass fraction considered here, this correlation is well approximated by a linear function of the solute concentration (see Appendix A) as:\nρ = ρ M 1 + ∆ρ ρ M C M C − C M .(5)\nThis equation complies with the assumption of linear density-concentration dependency that is typically made in simulations, thus ensuring a reliable comparison. Measurements of the concentration field are inferred from the transmitted light intensities, this process being described in detail in the next section.\n\n\nC. Experimental procedure and quantification of uncertainties\n\nThe results discussed in this paper are based on measurements of the time-dependent concentration field of the solute, C(x, z, t), inside the Hele-Shaw cell. To perform these measurements, we follow the method introduced by Slim et al. [36] and later used in a number of studies [28,32,38,39]. Optical measurements of light intensity distribution, I(x, z), are obtained with the aid of a sCMOS camera (FlowSense USB 2M-165, Dantec Dynamics, exposure time used 1 ms, resolution 2Mpx, frame rate 0.35 Hz to 5 Hz, depending on Ra). The measurements require a calibration process, briefly described in Appendix B, in which the value of light intensity is associated with the value of solute concentration. The greyscale images so obtained, an example of which is reported in Fig. 3(a), are preprocessed to reduce the effect of non-uniformities in the distribution of the light intensity over the cell, [40]). Finally, since the relationship between mass fraction and solute concentration is known (see Fig. 9(b) of Appendix B), the reconstruction of the concentration and density fields is performed. An example of this reconstruction is provided in Fig. 3(b).\n\nAn important issue that must be assessed is the uncertainty on the measured value of the Rayleigh number, Ra, defined as in Eq. (1). The uncertainty value, δRa, is obtained from propagation-of-error analysis [41] and reads as:\nδRa Ra = δg g 2 + δ∆ρ ∆ρ 2 + 2 δb b 2 + δH H 2 + δµ µ 2 + δD D 2 1/2 ,(6)\nwhere δ• indicates the uncertainty associated to the quantity • that represents each of the variables appearing in Eq. (1). We assume that the properties µ, D and g are estimated with constant relative uncertainty equal to 1%. The uncertainty on the domain height, δH, is assumed to be 0.5 mm, primarily due to a possible misplacement of the sealing on the cell rather than to the cutting precision. The accuracy on the gap thickness is assumed to be δb = 10µm. The uncertainties just introduced are common to all the experiments. In contrast, the uncertainty of the density difference δ∆ρ is a function of the Rayleigh number, being derived from Eq. (A1) as:\nδ∆ρ = ∂f ∂ϑ δϑ 2 + ∂f ∂C ρδω 2 1 − ω ∂f ∂C 2 ,(7)\nwhere δϑ is given by the instrument (0.5 • C). To use Eq. (7), the uncertainty on the concentration must be known. This is computed from Eq. (A5) as:\nδC = ω 2 (∆ρ) 2 + ρ 2 (δω) 2 .(8)\nThe mass fraction, ω, defined as the ratio between the mass of solute and the mass of solution, is determined measuring first the mass of water (m w ) and then the overall mass of solution (m sol ), i.e. ω = (m sol − m w )/m sol . As a result, the uncertainty δω reads as:\nδω ω = 1 − ω ω δm sol m sol 2 + δm w m w 2 ,(9)\nwith δm w = δm sol = 0.001 g (high-precision scale, Sartorius Acculab Atilon model ATL-423-I, ±0.001 g). The list of the solution properties, flow parameters and corresponding uncertainties is reported in Tab. I.\n\n\nD. Numerical simulations and dimensionless variables\n\nTo assess the mixing layer scaling laws, we wish to make a direct comparison of the present experimental results with direct numerical simulations of convection in two-dimensional porous media [4,30]. We perform this comparison in the range 347 ≤ Ra ≤ 19953, for which a very good overlap of the governing parameters exists between experiments and simulations. To make the paper self-contained, we recall next the governing equations and the numerical details.\n\nAn isotropic and homogeneous vertical porous slab with permeability k and porosity φ is considered. The porous domain is initially saturated with two miscible fluids having same viscosity (µ) but different density, arranged such that the heavy fluid (density ρ M ) tops the\nE# ϑ ω M C M ∆ρ Ra [ • C] [kg/m 3 ] [kg/m 3 ] E1\n24.0 ± 0. lighter one (density ρ 0 ). This unstable configuration replicates exactly the one sketched in Fig. 1(b). As done in the experiments, the density difference is induced by the presence of a solute, with maximum solute concentration C = C M in the upper layer and minimum solute concentration C = 0 in the lower layer. The flow is assumed to be incompressible and controlled by the Darcy's law:\n∂u ∂x + ∂w ∂z = 0 ,(10)µ k u = − ∂p ∂x , µ k w = − ∂p ∂z − ρg ,(11)\nwhere u and w are the horizontal (x) and vertical (z) velocity components, while p and ρ are the local pressure and density, respectively. We consider that the Oberbeck-Boussinesq approximation applies: This is a reasonable assumption for geophysical problems such as geological carbon dioxide sequestration [42]. In addition, we consider that the density of the mixture is a linear function of the solute concentration C [see Eq. (5)], and that the following transport equation for C holds:\nφ ∂C ∂t + u ∂C ∂x + w ∂C ∂z = φD ∂ 2 C ∂x 2 + ∂ 2 C ∂z 2 ,(12)\nwith t the time and D the solute diffusivity, which we assume to be constant.\n\nTo make Eqs. (10)-(12) dimensionless, we use the buoyancy velocity U and the length scale , defined as in Eq. (2). In addition, the following dimensionless variables are introduced [31,43,44]:\np * = p ∆ρ M g , C * = C C M , t * = U φ t .(13)\nThe resulting governing equations in dimensionless form read as:\n∂u * ∂x * + ∂w * ∂z * = 0,(14)u * = − ∂P * ∂x * , w * = − ∂P * ∂z * − C * ,(15)∂C * ∂t * + u * ∂C * ∂x * + w * ∂C * ∂z * = ∂ 2 C * ∂x * 2 + ∂ 2 C * ∂z * 2 ,(16)\nwhere the superscript * indicates dimensionless variables, P * = p * + z * (ρ M /∆ρ − 1) is the reduced pressure and ∆ρ is the density difference between the upper and the lower fluid layers.\n\nThe controlling parameter of the present system is the Rayleigh-Darcy number Ra, defined as in Eq. (1). Unlike the Hele-Shaw experiments, in which the geometry of the cell is characterised by the anisotropy ratio, two-dimensional Darcy simulations do not need additional dimensionless parameters: The Rayleigh-Darcy number describes completely the fluid properties (∆ρ, µ, D), the porous medium properties (k, φ) and the domain size (H).\n\nAlthough not explicitly appearing in the governing Eqs. (14)- (16), Ra enters the picture as the dimensionless height of the domain. Impermeable boundary conditions (i.e. no-flux) are imposed for both the fluid and the solute at the top and bottom horizontal boundaries, whereas periodicity is applied at the side boundaries (along x). In dimensionless form, these boundary conditions read as:\nw * = 0 , ∂C * ∂z * = 0 for z * = 0 and z * = Ra .(17)\nThe set of equations (14)- (16) has been discretized numerically using a Chebyshev-Tau method. We refer to [4,30,43,45] for further details on the flow solver.\n\n\nIII. Results\n\nThe comparison of the experimental results against numerical simulations is focused on the evolution of the mixing length and on the analysis of the wavenumber power spectra. These spectra are obtained by making a Fourier transform of the instantaneous concentration field and provide a robust characterization of the finger size, as this size can be associated to the wavenumber at which the spectra exhibit a peak. We consider a wide range of Rayleigh numbers: 4.39 × 10 3 ≤ Ra ≤ 5.43 × 10 4 . We refer to Tab. I for the experimental mass fraction and density difference values used to obtain these Ra. Above the upper value Ra max = 5.43 × 10 4 , the density difference between the KMnO 4 -rich solution (heavy fluid) and water (light fluid) induces large vertical velocities, and the flow is affected by non-Darcy effects [27,28]. For lower values of ∆ρ, in contrast, the initial condition is hard to achieve and the duration of the experiments increases, leading to higher chances of inducing density perturbations in the cell from the side and horizontal channels (L, R, T, B in Fig.1). In the following, we describe the experimental results in terms of flow phenomenology (Sec. III A), time evolution of the mixing length (Sec. III B) and wavenumbers (Sec. III C), comparing visa-vis experimental observations against numerical simulations [4,30]. The datasets analyzed in the present study have been deposited in [46]. [4]. The dimensionless time instants (t * ) at which the fields refer are indicated on top of each panel. We remark that the experiments are realised in a square geometry, whereas the simulations are performed in wider domains (aspect ratio π/2). Therefore, for better comparison, only a portion of the numerical domain having unitary aspect ratio is reported. The desired initial position of the interface (located at z = H/2, i.e. z * = Ra /2) is also shown as a dashed black line. See also the movie, Movie S1, in the Supplemental Material [47] for a comparison of the time-dependent evolution of the concentration fields in the experiment and in the simulation.\nC ⇤ = C/C M 1 1 /2 0 C ⇤ = C/C M 1 1 /2 0 (a) t ⇤ = 0.08 ⇥ 10 4 (b) t ⇤ = 0.68 ⇥ 10 4 (c) t ⇤ = 1.57 ⇥ 10 4 (d) t ⇤ = 2.47 ⇥ 10 4 (e) t ⇤ = 0.08 ⇥ 10 4 (f) t ⇤ = 0.68 ⇥ 10 4 (g) t ⇤ = 1.57 ⇥ 10 4 (h) t ⇤ = 2.47 ⇥ 10 4 x ⇤ z ⇤ x ⇤ z ⇤ 0 Ra 0 Ra 0 Ra 0 Ra 0 Ra/2\n\nA. Flow phenomenology\n\nTo analyze in detail the flow phenomenology, we report in Fig. 4 the time-dependent dimensionless concentration field (C * ) obtained experimentally (4a-d) and numerically [4] (4e-h). The experiment (E4 in Tab. I, Ra = 19789) is compared against numerical simulations performed at similar Rayleigh number, Ra = 19953: We consider this value because it is large enough to observe all the main phases of the mixing process for a significant amount of time. On top of each panel, the dimensionless time instants (t * ) at which the fields are taken are provided.\n\nThe concentration field is initially in an unstable configuration (Fig. 4a,e) with the heavy fluid characterised by the dimensionless concentration C * = 1 lying on top of the lighter one (C * = 0). The initial position of the interface, indicated here with the dashed line at z * = Ra /2, is initially flat and horizontal. Achieving such a configuration in the experiment is very difficult, and some discrepancy, though not apparent, always exists between the actual fluid-fluid interface and the desired interface. This is visible for instance in Fig. 4(a), where the upper fluid layer seems to be slightly inclined with respect to the horizontal dashed line. In the experiments, the flow is initially triggered by perturbations of the local density and velocity fields. Indeed, to set the initial flow configuration and generate the fluid-fluid interface, some fluid is removed from the side channels and simultaneously injected from the channels at the horizontal boundaries (see Sec. II C for a thorough description of this process). As a result, the fluid will not be initially still and will be partially affected by the flow circulations induced by the procedure adopted to set the initial condition. The situation is completely controllable in the simulations, where the interface starts as perfectly flat and is made unstable by adding a perturbation directly to the concentration field (Fig. 4e). The amplitude of the perturbation considered, however, is crucial in determining the onset of the instability [30], whereas the late stage dynamics are not sensitive to this parameter. In the present study, we considered simulations in which the perturbations applied to the dimensionless concentration field have amplitude 10 −3 .\n\nThe flow phenomenology associated with the mixing process can be summarised as follows. The mechanism of diffusion is initially responsible for the flow evolution (Fig. 4a,e). Afterwards, the interfacial layer thickens and, due to the unstable density profile, finger-like structures form. These fingers are controlled by the interplay of convection, which makes them grow vertically, and diffusion, which reduces the horizontal concentration gradients and promotes mixing [35]. During this intermediate stage (Fig. 4b-c,f-g), the fingers grow symmetrically with respect to the domain centerline, and eventually merge. Indeed, we observe that the number of fingers reduces from t * = 0.68 × 10 4 (Fig. 4b,f) to t * = 1.57 × 10 4 (Fig. 4c,g). We anticipate here that this dynamics is quantitatively confirmed by the wavenumber measurements reported in Sec. III C. At a later stage (Fig. 4d,h), the number of fingers does not increase further. Eventually, the strength of convection diminishes and the flow evolution is controlled again by diffusion. In this phase (not shown here), the system has achieved a stable configuration in which the average concentration in the lower layer is higher than in the upper layer [4]. This regime, usually referred to as shutdown of convection in view of the reduced strength of the driving forces [43,48], starts after the tips of the fingers reach the horizontal walls and the effect of the boundaries propagates back to the domain centerline. The time required to reach the shutdown is large (e.g., of the order of 1 hour for the lower Ra considered here), and hence hard to achieve experimentally because the flow conditions have to be carefully controlled for a considerable amount of time.\n\nIt is apparent that the flow evolution observed in the experiments is qualitatively similar to that of numerical simulations, albeit with some differences. The growth of the fingers, expressed in terms of their tip-to-rear extension, is similar within the same time interval (Fig. 4b,f and Fig. 4c,g). We also refer the reader to the movie, Movie S1, in the Supplemental Material [47] for a comparison of the time-dependent evolution of the concentration fields in experiments and simulations. However, we also observe a different number of fingers in the two cases (see Sec. III C for further discussion).\n\n\nB. Mixing length\n\nThe evolution of a Rayleigh-Taylor system can be conveniently analysed by looking at the instantaneous vertical extension of the fluid-fluid interface [3,49]. This quantity, defined as mixing length h, can be also interpreted as the distance between the upwarding and downwarding tips of the fingers. To compute the mixing length in a quantitative way, one can use either global [50] (i.e. integral) or local [35,51] flow properties. In this work, we characterize the mixing length by means of the local concentration field [4,51], i.e. h is the portion of the domain where the horizontally-averaged concentration field satisfies the condition ε < C(z, t) < 1 − ε, with:\nC(z, t) = 1 H H 0 C(x, z, t) C M dx(18)\nthe concentration field averaged along the horizontal direction x, ε = 3 × 10 −2 a threshold coefficient, and H the domain width (in x direction). It follows from this definition that 0 ≤ h ≤ H or, in dimensionless terms using the length scale defined in Eq. (2), 0 ≤ h * ≤ Ra.\n\nThe time evolution of the dimensionless mixing length for all the experiments we performed is shown in Fig. 5. As previously observed in simulations in confined domains [4,30], the dynamics consists essentially of three phases in sequence: (i) diffusion-dominated, (ii) convection dominated and (iii) wall-induced shutdown. During the initial phase, the thickness of the mixing layer is initially controlled by diffusion. Assuming an initial stepwise concentration profile, the dimensionless concentration field evolves as:\nC * (z * , t * ) = 1 2 1 + erf z * −Ra /2 √ 4t * ,(19)\nand the corresponding mixing length as:\nh * (t * ) = −2 √ 4t * erf −1 (2ε − 1) ≈ √ 4π(1 − 2ε) √ t *(20)\n(we refer to [4] for a detailed derivation). The same evolution of the mixing length in the initial diffusive regime has been observed in previous numerical works, both in porous media convection [4,35] and in pure fluids with different initial conditions [52]. Despite the seemingly universal behaviour exhibited by the mixing length, we observe that the initial diffusive stage is hardly captured by the experimental measurements. Only the measurements performed at lower Ra, namely 4407 and 7023, are in good agreement with the theoretical predictions (blue dashed line in Fig. 5). While the trend nicely follows the predicted behaviour, namely h * ∼ √ t * , the magnitude slightly differs from the solution derived in Eq. (20). This is due to uncertainty of the experimental measurements, which is comparable to the values of h * at low Ra and early times. When the fluid-fluid interface is sufficiently thick (10 3 < t < 10 4 ), it becomes unstable and eventually finger-like structures form. When observed via the mixing length, the transition to this convection-dominated regime is smooth, similarly to what has been observed in the simulations. However, we find that the transition in experiments occurs at different instants depending on the value of Ra, as a result of the uncertainty on the initial perturbation of the concentration field, which cannot be reproduced faithfully in the laboratory. We also observe that, during the transition, the growth of the mixing length modelled as a power law, i.e. h * ∼ (t * ) α , is characterized by an exponent that is higher than that observed asymptotically, in agreement with previous numerical [4,35] and experimental (α = 2) [53] findings.\n\nThe fluid mixing is more efficient when convection takes place. The formation of fingerlike structures marks the beginning of the convection-dominated phase, more persistent in time compared to the initial diffusive stage. In this phase, fingers grow, merge and finally reach the horizontal boundaries of the cell. The measurements reported in this work reveal that the mixing length grows more than linearly in time, in close agreement with previous simulations (namely, h * ∼ (t * ) 1.208 , [4,5]). This scaling is anomalous: It is not expected in two-dimensional porous media convection, where a linear evolution of the mixing length with time is predicted as a result of the balance between buoyancy (i.e., convection) and friction (i.e. dissipation due to viscosity). The occurrence of a linear growth would imply an asymptotic velocity of the finger tips, which determines also the growth rate of the mixing length, equal to the buoyancy velocity U , defined as in Eq. (2). We remind the reader that U can be interpreted as the velocity at which the contribution of buoyancy (g∆ρk) balances the dissipation (µ).\n\nIn three-dimensional porous media, the growth of the mixing length follows the linear prediction obtained from the physical arguments mentioned above [5]. The behaviour is different in two dimensions, where the fingers are slenderer compared to their three-dimensional counterpart. The superlinear evolution of the mixing length we observed, can be ascribed to multiple reasons: (i) the simulated time is not sufficiently large to achieve the asymptotic regime, or (ii) the Ra number of the system is too low, and the effect of diffusion is still dominant. We speculate that in the present case a combination of these two effects controls the evolution of the flow. Initially, the fluid accelerates from zero velocity, and this corresponds to a finite-time effect. At a later stage, when the initial condition has been absorbed by the system and fluid velocity is non-zero, the finite-size (i.e., finite-Ra) effect comes into play, via the domain boundaries. This non-linear behaviour seems indeed to vanish at higher Ra numbers [6], but in absence of boundaries. We remark here that a linear growth of the mixing length has been observed numerically [35], but only in the limit of lower Rayleigh numbers, namely Ra ≈ 4 × 10 3 . For these values of Ra, also previous numerical simulations [4] and present experiments do not exhibit a clear superlinear behaviour. In addition, the experimental data reported by Wooding [53], which refer to higher Rayleigh numbers (approximately 10 3 ≤ Ra ≤ 7 × 10 4 ) are also well fitted by a linear scaling. However, it must be pointed out that the data have been collected at very low acquisition rate, making the amount of available measurements very scarce for a time-dependent flow analysis. We are not able to explain, in this case, why the superlinear scaling was not observed. One possible reason could be attributed to the method adopted to change the Rayleigh number, consisting of varying the orientation of the cell with respect to gravity. Another possibility is represented by the absence of horizontal boundaries. A conclusive answer on the asymptotic scaling of the mixing length could be obtained with the aid of Darcy simulations or experiments at higher Ra, (≈ 10 6 ), which are beyond the present capabilities.\n\nTo conclude the discussion on the mixing length, we now examine the wall-induced convective shutdown phase [4], which sets in when the high (resp. low) concentration fingers reach the lower (resp. upper) horizontal boundary. In this phase, the local concentration gradients decrease, weakening the density differences that drive the flow within the domain. As a result, convection vanishes and the overall mixing rate, identified for instance by means of the mean scalar dissipation rate, decreases dramatically [30]. This effect is crucial for applications of practical interest (e.g., carbon dioxide sequestration in saline aquifers), in which the fluids are confined by impermeable boundaries that play a key role in the solute mixing process. A simple way to quantify the convective shutdown is to measure the time t * s required for the fingers to reach the horizontal boundaries. The values of t * s obtained as a function of the Rayleigh number are shown in Fig. 6. Experimental and numerical [4] results are reported as open (•) and filled symbols ( ), respectively. The best-fitting power law for present results (red dashed line) is 9.64 ×Ra 0.80 , in excellent agreement with numerical results (14.98 ×Ra 0.78 , black dashed line). This is not surprising, given the close correspondence observed for the evolution of the mixing length. In leading order, we can estimate t * s = t * : h * (t * ) = Ra which gives t * s ∼ Ra 1/1.2 ∼ Ra 0.83 , consistent with the Ra 0.80 scaling provided by data fitting.\n\nDespite the excellent agreement of the scaling exponents, we also observe some discrepancies between simulations and experiments. In particular, for a given Ra, the value of t * s is lower in the experiments, leading to a lower value of the multiplicative coefficient obtained from data fitting. We attribute this discrepancy to the perturbation of the interfacial concentration field. In the experiments, the amplitude of this perturbation is hard to control, and it is likely to be larger compared to the numerical simulations considered, leading to an earlier onset of convective instabilities. Another limitation of our experimental setup is represented by the range of Rayleigh numbers that can be investigated. To be able to achieve a Darcy flow and to control effectively the boundary conditions, the values of domain size (H * ) and the density difference (∆ρ) cannot be too large (see also Sec. II B). The values of Ra considered here are sufficiently large to overlap with and go beyond (up to 2.5 times, approximately) the largest Rayleigh numbers simulated numerically. On the other hand, achieving smaller values of Ra is also challenging. Lower values of ∆ρ correspond to lower velocities, and therefore longer experiments. Smaller values of H * , i.e. shorter domains, would instead required more careful setup of the initial flow configuration.\n\nThe findings discussed in this section prove that, in terms of evolution of the mixing length, two-dimensional Darcy simulations and Hele-Shaw experiments produce similar results when the sameRa is considered. However, the ability of the experimental measurements to capture the initial development of the flow is lower due to the setup used, and numerical simulations should be considered as more reliable to investigate this stage of the flow evolution.\n\n\nC. Wavenumbers\n\nThe flow dynamics is complex and characterised by a continuous change of the fluid-fluid interface position. To provide a quantitative estimate of the flow topology evolution, we analyze here the number of fingers and the wavenumbers at the centerline of the domain (z * = H * /2). The process of flow analysis is illustrated in Fig. 7, where an experiment at Ra = 5.43 × 10 4 is considered for reference. First, we build the space time map at the centerline, where the interface is initially located, i.e. we report in Fig. 7(a) the contour of the time-dependent dimensionless concentration field, C * (x * , z * = Ra /2, t * ). The simplest way to define the number of structures in the system is to count the upwarding and downwarding fingers. To this aim, the spacetime map is first binarised [ C(x * , z * = Ra /2, t * ), Fig. 7(b)] with respect to a threshold value corresponding to the mean concentration (C * = 1/2), and then the number of fingers is counted [ Fig. 7(c)]. The process is iterated during the entire flow evolution, and the number of fingers, N (t), is computed as the number of regions with different binarized concentration values [ Fig. 7(d), black line]. Due to noise in the experimental measurements, spurious fluctuations of the concentration field may occur. To avoid counting these as fingers, we discarded contiguous regions of the binarized map that are smaller than 4 pixel (corresponding to a physical length of 350 µm). We observe in Fig. 7(d) that this threshold is extremely sensitive to variations of the concentration field, i.e. it appears to fluctuate without an apparent physical reason. A more robust characterization of the flow topology is provided by the wavenumbers associated with the power spectra of the concentration field. We employ a Fourier decomposition of the concentration profile at the cell centerline to estimate the number of fingers, and compute the power-averaged mean Measurements of the number of fingers over time for Ra = 5.43 × 10 4 (E6 in Tab. I). Space-time map, consisting of the evolution of the concentration field along the centerline, C * (x * , z * = Ra /2, t * ), is shown in (a). To compute the finger number, the space-time map is binarized, C(x * , z * = Ra /2, t * ), choosing C * = 1/2 as threshold (b). The concentration profile (red line) and the discretized profile (black line) measured along the centerline at t * ≈ 1.1 × 10 5 (indicated by the dashed blue line in panels a, b and d) are also reported as a function of the horizontal coordinate x * (c). Finally (panel d), the time-dependent finger number is computed as from the binarized field C (black line), from power-averaged mean wavenumber as defined in Eq. (21) ( N = L k /π, red line) and from a k peak (green line), i.e. the wavenumber maximizing the spectrum amplitude of C. See also the movie, Movie S2, in the Supplemental Material [54] for the time-dependent evolution of the wavenumber and the corresponding concentration field. wavenumber at each time instant, defined as [51]:\nk = k n |C n (t * )| 2 dk n |C n (t * )| 2 dk n ,(21)\nwith k n the wavenumber and |C n (t * )| 2 the squared amplitude of the n-th basic harmonic function obtained from the Fourier decomposition of the fluctuation of C * (x * , z * = 1/2, t * ) about its mean value. We observe that in this case [red line in Fig. 7(d)] the evolution of the number of fingers, N = L k /π, is smoother compared to the simple finger count, N , and also compared to the peak number, defined as N peak = Lk peak /π [orange line in Fig. 7\n(d)],\nwhere k peak is the wavenumber value for which |C n (t * )| is maximum. Therefore, in the following, we will focus the analysis considering the power-averaged mean finger number N only, which appears similar to but less fluctuating of the finger number N . A comparison of the number of fingers measured in the experiments is reported in the inset of Fig. 8(a). In general, all the experiments are characterized by a similar dynamics. The number of fingers, initially low, increases up to a maximum value. Afterwards, the fingers start to merge and increase in size. This phase, which is also visible from the spacetime maps in Fig. 7(a-b), may start earlier at low Ra and later at high Ra, spanning a relatively long interval (10 3 ≤ t * ≤ 10 4 ) in Fig. 8(a). During growth, N appears to decrease in time at the same rate for all Ra considered. This universal behaviour is more apparent when the results are compared in terms of mean wavenumber k (main panel of Fig. 8a). After entering the convection-dominated regime, all curves exhibit the same trend. A first observation is that the merging rate of the fingers is in agreement with the scaling k ∼ (t * ) −0.6 reported in numerical simulations [30]. In contrast, the behaviour is different during the diffusive phase, when the wavenumber is clearly dependent on Ra.\n\nIn Fig. 8(b), we compare present results (solid lines) against previous numerical findings [30] (dashed lines). For ease of reading, only three values of Ra number are considered. Two differences are apparent: (i) unlike experiments, simulations [30] have been performed on domains of constant dimensionless width (namely, 19953), and (ii) to make reliable comparisons among different Rayleigh numbers, the same initial perturbation has been employed, thus making the early stage flow evolution also universal and independent of Ra. This highlights again a possible limitation of the experimental procedure employed, which makes it hard to precisely control the initial flow conditions. As a result, the initial phase is characterised by large discrepancies between numerical and experimental results (Fig. 8b). In the late stage of the flow dynamics, numerical and experimental measurements are in fair agreement. As a side note, we remark that present results, shown in terms of wavenumbers associated with the concentration field, are compare against the simulations of De Paoli et al. [30], where the power spectra where obtained from the scalar dissipation fields. However, even the experimentally-measured merging rate is well described by the same scaling law, t −0.6 . Present results are in fair agreement with previous experimental findings [35,53], in which the wavenumber is observed to evolve according to (t * ) −1/2 . This scaling has been also derived analytically [31,53] assuming that the fingers are slender, so that vertical derivatives (e.g., diffusion of solute) are negligible compared to the horizontal ones. In the frame of this model, the flow is driven by vertical advection, whereas lateral diffusion acts as dissipative mechanism. We refer to Slim [31] for a detailed derivation of this result. The slight differ-ence in the scaling exponent, compared to the analytical prediction, maybe attributed to the simplified assumptions considered in the model derivation. The scaling k ∼ (t * ) −1/2 has been also suggested by Gopalakrishnan et al. [35], who interpreted this evolution as a diffusion-dominated finger growth, by analyzing the relative contribution of convective and diffusive terms within a finger (but away from the finger tips).\n\n\nIV. Conclusions\n\nIn this work, we analyzed with the aid of high-resolution experiments the Rayleigh-Taylor instability in a Hele-Shaw flow. The system consists of two miscible fluid layers, namely water and an aqueous solution of KMnO 4 , having different density but same viscosity, placed in an initial unstable configuration. The density difference is induced by the presence of the solute (KMnO4), which produces also a colour gradient by which we are able to infer the concentration field. The system is fully described by the Rayleigh number, Ra, which measures the relative strength of convective and dissipative contributions. Present experimental results have been performed taking great care to achieve a proper initial condition and minimizing the local perturbations of the concentration and velocity fields, which trigger the growth of interfacial finger-like instabilities. In addition, a high resolution in space and time has been achieved by considering relatively large domains, and an accurate reconstruction of the concentration field is performed. In this respect, present experimental results go beyond previous measurements published in archival literature.\n\nWe compared present findings against previous numerical measurements in two-dimensional and confined media to which the Darcy equations can be successfully applied, as for instance porous media. [4,30]. The dynamics of the flow is first controlled by diffusion, which is responsible for the thickening of the interface, initially flat and horizontal. Afterwards, the flow evolution is controlled by convection. The unstable fluid-fluid density front promotes the formation of high-wavenumber instabilities, which grow and merge into large and more persistent fingers. Finally, due to the combined action of convective mixing in a confined domain and lateral diffusion across the fingers interface, the solute distribution becomes uniform within the porous slab, with a corresponding stable concentration profile. During this stage, referred to as wall-induced convective shutdown, the flow is again controlled by diffusion. The flow evolution is quantified and compared against theoretical and numerical predictions with the aid of two main observables: the mixing length, h * (t), and the power-averaged mean wavenumber, k .\n\nInitially, the evolution of the mixing length is in fair agreement with the theoretical predictions obtained for flows in the absence of convection (h * (t) ∼ (t * ) 1/2 ). In the intermediate regime, in which convection dominates, h * (t) is observed to grow with a superlinear scaling, h * (t) ∼ (t * ) 1.2 , confirming previous numerical findings [4,5]. This scaling is anomalous, as it differs from the theoretical prediction, h * (t) ∼ t * , possibly as a result of a finite size (i.e., finite Ra) effects. To further compare present experimental findings against previous numerical simulations, we compute the time t * s taken for the fingers to reach the horizontal boundaries of the domain. Also in this case, experiments (t * s ∼ Ra 0.80 ) are in excellent agreement with the simulations (t * s ∼ Ra 0.78 ). Finally, we analysed the flow dynamics by looking at the finger distribution at the domain centerline. The evolution of the characteristic finger size has been quantified by means of the power-averaged mean wavenumber, k . During the convection-dominated phase, the centerline finger dynamics exhibits a universal behaviour that is independent of Ra. Also in A ADDITIONAL DETAILS ON FLUID PROPERTIES this case, we found an excellent agreement with scaling laws derived in numerical simulations [30], namely k ∼ (t * ) −0.6 .\n\nThe analysis carried out in this paper shows that Hele-Shaw experiments can faithfully reproduce the dynamics of convective Rayleigh-Taylor instabilities in two-dimensional and confined porous media. However, it also shows that the proposed experimental approach presents some important limitations, which call for future improvements. First off, the desired initial fluid-fluid interface consists of a thin, flat and horizontal contact line, which is experimentally challenging to achieve. In addition, the instability is initially triggered by local perturbations of the concentration and velocity fields, and hence hard to control. Finally, experiments at low Ra are characterised by low values of vertical velocity, which slow down the mixing process and make the experiments to last longer. As a result, the experimental conditions have to be monitored for longer times, increasing the difficulty of the experimental procedure.\n\nFIG. 9. Given the definition (A5) and the empirical correlations (A1) [37], the relative dependence of density of the solution, ρ, mass fraction, ω, and solute concentration, C, is obtained. We report here C(ω) (panel a), ρ(ω) − ρ 0 (panel b) and ρ(C) − ρ 0 (panel c), being ρ 0 the water density. The profiles shown here correspond to ϑ = 25 • C. We observe that the empirical values (symbols) are very well fitted by linear functions (solid lines). In the instance of density and concentration (panel c), the linear function corresponds to Eq. (5).\n\nby a linear function of the solute concentration [Eq. (5)], as can be seen in Fig. 9c).\n\n\nB. Measurements calibration\n\nWhen the two-dimensional distribution of light intensities is determined, each value of light intensity is assigned a value of mass fraction, ω(x, z). This step is achieved with the aid of calibration curves, an example of which is reported in Fig. 10. For a given illumination condition, the cell is filled with solution with a known value of mass fraction, ω, and the average light intensity over the cell, I cal , is recorded. The higher the concentration of KMnO 4 , the lower the transmitted light intensity [Lambert-Beer law, 55]. Experimental measurements (symbols) are well fitted by a combination of exponential functions, ω( I cal ) = a 0 + a 1 exp(b 1 I cal ) + a 2 exp(b 2 I cal ) (solid line). Further details on the process of calibration are available in [28,32,39].\n\n\nC. Additional details on the initial condition\n\nLarge values of solute concentration (C M ) produce strong concentration gradients across the initial fluid-fluid interface, making a sharp separation between the two fluids hard to obtain. Therefore, to achieve a well-defined initial interface up to the Ra values considered here, the flow rate provided to the cell is increased withRa. In Fig. 11, the initial condition is shown in detail for the experiments E1 and E4 (namely at Ra = 4391 and Ra = 19789), and for both the unprocessed light intensity field, I, (Fig. 11a,d) and the inferred dimensionless concentration field, C * (Figs. 11b,e). We observe that the experimental procedure employed is suitable to obtain a macroscopic sharp and straight interface in correspondence of the domain centerline, represented by the dashed red line in Figs. 11(a,b,d,e). Further information on the local interface shape can be obtained from the close up view of the interfacial region, shown in Figs. 11(c,f), where the iso-contours at C * = 0.3, C * = 0.5 and C * = 0.7 (same\n\nFIG. 1 .\n1(a) Experimental setup. The Hele-Shaw cell is shown, with explicit indication of the position of the connection gates (top, bottom, left and right, indicated as T, B, L, and R respectively), sCMOS camera and backlight illumination. Fluids are supplied to the cell through a system (sketched in Fig. 2) consisting of a pump, pipes and valves. The cell is aligned in the vertical direction, namely parallel to the acceleration due to gravity, g. (b) Sketch of the experimental domain with explicit indication of the boundary conditions (no flux of mass or solute through the walls). The reference frame (x, z) as well as the initial position of the interface (red dashed line) are indicated, with the heavy fluid (density ρ M , concentration C = C M ) initially lying on top of the lighter fluid (ρ 0 , C = 0). The background field consists of one of the images collected, and provides a qualitative picture of the flow for the present physical configuration.\n\nFIG. 2 .\n2Circuital scheme of the experimental setup employed to obtain the desired initial conditions\n\nFIG. 3 .\n3Reconstruction of the concentration field. (a) Raw image consists of a light intensity distribution, I. (b) Upon processing to remove light intensity noises, scratches and non-uniformities of backlight illumination (see [40] for further details), the dimensionless concentration field is obtained. To visualise the interface between the two fluids, corresponding to the fingers, three iso-contours of normalised concentration are shown, corresponding to C * = C/C M = 0.3 (blue), 0.5 (red) and 0.7 (green).\n\nFIG. 4 .\n4Snapshots of the dimensionless concentration field obtained experimentally (a-d) and numerically (e-h). The experiment (E4 in Tab. I,Ra = 19789) is compared vis-á-vis against numerical simulations performed at Ra = 19953 by De Paoli et al.\n\nFIG. 5 .\n5Time evolution of the dimensionless mixing length, h * (t * ) (solid lines), for all the values of the global Rayleigh number considered in this study (and explicitly indicated next to each curve). The evolution of the system, initially controlled by diffusion, nicely follows the analytical solution, h * ≈ (1 − 2ε) √ 4πt * [Eq. (20), dashed blue line]. The intermediate phase, dominated by convection, exhibits the anomalous scaling h * ∼ (t * ) 1.2 first observed by De Paoli et al. [4].\n\nFIG. 6 .\n6Time taken by the fingers to reach the horizontal boundaries (t * s ) as a function of Ra. Experimental (present work) and numerical [4] results are reported as open (•) and filled symbols ( ), respectively. The best-fitting power law for present results (red dashed line) is 9.64 ×Ra 0.80 , in excellent agreement with numerical findings (14.98 ×Ra 0.78 , black dashed line). Experimental results correspond to the mean values obtained for several flow realisations at fixed Ra.\n\n\nFIG. 7. Measurements of the number of fingers over time for Ra = 5.43 × 10 4 (E6 in Tab. I). Space-time map, consisting of the evolution of the concentration field along the centerline, C * (x * , z * = Ra /2, t * ), is shown in (a). To compute the finger number, the space-time map is binarized, C(x * , z * = Ra /2, t * ), choosing C * = 1/2 as threshold (b). The concentration profile (red line) and the discretized profile (black line) measured along the centerline at t * ≈ 1.1 × 10 5 (indicated by the dashed blue line in panels a, b and d) are also reported as a function of the horizontal coordinate x * (c). Finally (panel d), the time-dependent finger number is computed as from the binarized field C (black line), from power-averaged mean wavenumber as defined in Eq. (21) ( N = L k /π, red line) and from a k peak (green line), i.e. the wavenumber maximizing the spectrum amplitude of C. See also the movie, Movie S2, in the Supplemental Material [54] for the time-dependent evolution of the wavenumber and the corresponding concentration field.\n\nFIG. 8 .\n8(a) Experimental measurements. The evolution of the power-averaged mean wavenumber, k (defined in Eq. (21), main panel), and the corresponding finger number N (inset). (b) Comparison of experimental (solid lines) and numerical ([30], dashed lines) measurements. For ease of reading, three values of Rayleigh number are used to compare the behaviour of k .\n\n\nTABLEI. List of flow parameters, solution properties and corresponding uncertainties. Different realizations have been performed for all the Rayleigh numbers, Ra. Fluid temperature (ϑ) is reported, as well as solution mass fraction (ω M ), concentration (C M ) and corresponding density difference (∆ρ). The dimensions of the cell are kept constant for all the experiments (H × H, with H = (87 ± 0.5) mm, thickness b = (300 ± 10) µm). Fluid properties are discussed in Sec. II B.5 (1.41 ± 0.006) × 10 −3 \n1.427 ± 0.006 \n1.132 ± 0.002 \n(4.39 ± 0.08) × 10 3 \nE2 \n19.0 ± 0.5 (2.30 ± 0.006) × 10 −3 \n2.316 ± 0.006 \n1.805 ± 0.002 \n(7.00 ± 0.12) × 10 3 \nE3 \n21.0 ± 0.5 (3.94 ± 0.006) × 10 −3 \n3.949 ± 0.007 \n3.099 ± 0.004 \n(1.20 ± 0.02) × 10 4 \nE4 \n23.5 ± 0.5 (6.42 ± 0.006) × 10 −3 \n6.441 ± 0.008 \n5.101 ± 0.008 \n(1.98 ± 0.03) × 10 4 \nE5 \n21.0 ± 0.5 (1.08 ± 0.001) × 10 −2 10.824 ± 0.010 \n8.495 ± 0.013 \n(3.30 ± 0.06) × 10 4 \nE6 \n21.0 ± 0.5 (1.76 ± 0.001) × 10 −2 17.828 ± 0.015 13.996 ± 0.025 (5.43 ± 0.09) × 10 4 \n\n\nAcknowledgmentsThis research was funded in part by the Austrian Science Fund (FWF) [Grant J-4612]. For the purpose of Open Access, the authors have applied a CC BY public copyright license to any Author Accepted Manuscript (AAM) version arising from this submission. D.P. gratefully acknowledges the financial support provided from Project No. 2020-1-IT02-KA103-078111 funded by UE Erasmus. Mr. Werner Jandl and Mr. Franz Neuwirth are gratefully acknowledged for their help with the experimental work. Eliza Coliban and Mobin Alipour are also acknowledged for their contribution during the initial stage of this work. We are grateful to the anonymous referees for the comments provided.A. Additional details on fluid propertiesFollowing the empirical correlations of Novotný and Söhnel[37], the density of an aqueous solution of KMnO 4 can be expressed as a function of the solute concentration (C) and the temperature of the solution (ϑ) as:where the water density ρ 0 (ϑ) and the temperature-dependent coefficients A 1 (ϑ) and A 2 (ϑ), are given by:A 1 (ϑ) = +1.223 × 10 2 − 1.029 × 10 −1 · ϑ + 8.093 × 10 −3 · ϑ 2 , (A3)Provided that the mass fraction of the solution is defined as:it is possible to determine the respective dependency of density, mass fraction and concentration, which we report inFig. 9. The density of the mixture, ρ(C) is well approximated\nUncertainties on the light intensity and mass fraction are also shown (see Sec. II B for further details). Finally, the mass fraction-light intensity space considered in each experiment, labelled as in Tab. I, is highlighted in red. The measurements are performed where the sensitivity of the method employed is higher, i.e. for low values of mass fraction. as in Fig. 3) are shown. We observe that, despite the large concentration contrast between the two values of Ra considered, the interface thickness (quantified by the vertical spacial separation between the iso-contours in Figs. 11c,f) is similar. Experimental measurements (symbols) are well fitted by a combination of exponential functions, ω( I cal ) = a 0 + a 1 exp(b 1 I cal ) + a 2 exp(b 2 I cal ) (solid line). and it is comparable with the spatial resolution of the measurements (1 pixel ≈ 90 µmaverage value of light intensity, I cal , is used to build the curve. Experimental measurements (symbols) are well fitted by a combination of exponential functions, ω( I cal ) = a 0 + a 1 exp(b 1 I cal ) + a 2 exp(b 2 I cal ) (solid line). Uncertainties on the light intensity and mass fraction are also shown (see Sec. II B for further details). Finally, the mass fraction-light intensity space considered in each experiment, labelled as in Tab. I, is highlighted in red. The measurements are performed where the sensitivity of the method employed is higher, i.e. for low values of mass fraction. as in Fig. 3) are shown. We observe that, despite the large concentration contrast between the two values of Ra considered, the interface thickness (quantified by the vertical spacial separation between the iso-contours in Figs. 11c,f) is similar, and it is comparable with the spatial resolution of the measurements (1 pixel ≈ 90 µm).\n\nInvestigation of the character of the equilibrium of an incompressible heavy fluid of variable density. L Rayleigh, Proc. Lond. Math. Soc. 14L. Rayleigh. Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc., 14:170-177, 1883.\n\nThe instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. 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M De Paoli, V Giurgiu, F Zonta, A Soldati, Phys. Rev. Fluids. 410101501M. De Paoli, V. Giurgiu, F. Zonta, and A. Soldati. Universal behavior of scalar dissipation rate in confined porous media. Phys. Rev. Fluids, 4(10):101501, 2019.\n\nSolutal-convection regimes in a two-dimensional porous medium. A C Slim, J. Fluid Mech. 741A. C. Slim. Solutal-convection regimes in a two-dimensional porous medium. J. Fluid Mech., 741:461-491, 2014.\n\nConcentration-based velocity reconstruction in convective Hele-Shaw flows. M Alipour, M De Paoli, A Soldati, Exp. Fluids. 619M. Alipour, M. De Paoli, and A. Soldati. Concentration-based velocity reconstruction in convective Hele-Shaw flows. Exp. Fluids, 61(9):1-16, 2020.\n\nAcceleration of reaction fronts by hydrodynamic instabilities in immiscible systems. Y Shi, K Eckert, Chem. Eng. Sci. 6117Y. Shi and K. Eckert. Acceleration of reaction fronts by hydrodynamic instabilities in immis- cible systems. Chem. Eng. 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Introduction to error analysis, the study of uncertainties in physical measurements. University Science Books, 1997.\n\nHeat and brine transp. porous media: the Oberbeck-Boussinesq approximation revisited. A J Landman, R J Schotting, Transp. Porous Media. 703A. J. Landman and R. J. Schotting. Heat and brine transp. porous media: the Oberbeck- Boussinesq approximation revisited. Transp. Porous Media, 70(3):355-373, 2007.\n\nDissolution in anisotropic porous media: Modelling convection regimes from onset to shutdown. Phys. Fluids (1994-present). M De Paoli, F Zonta, A Soldati, 2926601M. De Paoli, F. Zonta, and A. Soldati. Dissolution in anisotropic porous media: Modelling convection regimes from onset to shutdown. Phys. Fluids (1994-present), 29(2):026601, 2017.\n\nEffect of permeability anisotropy on buoyancydriven flow for CO2 sequestration in saline aquifers. P Cheng, M Bestehorn, A Firoozabadi, Water Resour. 489P. Cheng, M. Bestehorn, and A. Firoozabadi. Effect of permeability anisotropy on buoyancy- driven flow for CO2 sequestration in saline aquifers. Water Resour., 48(9), 2012.\n\nInfluence of anisotropic permeability on convection in porous media: implications for geological CO2 sequestration. M De Paoli, F Zonta, A Soldati, Phys. Fluids. 28556601M. De Paoli, F. Zonta, and A. Soldati. Influence of anisotropic permeability on convection in porous media: implications for geological CO2 sequestration. Phys. Fluids (1994-present), 28 (5):056601, 2016.\n\nExperimental assessment of mixing layer scaling laws in Rayleigh-Taylor instability. M De Paoli, D Perissutti, C Marchioli, A Soldati, 10.6084/m9.figshare.19761766M. De Paoli, D. Perissutti, C. Marchioli, and A. Soldati. Experimental as- sessment of mixing layer scaling laws in Rayleigh-Taylor instability. Figshare. https://doi.org/10.6084/m9.figshare.19761766. Deposited 25 June 2022.\n\nSee Supplemental Material \"Movie S1.mov\" for a comparison of the evolution of the concentration field in present experiments and simulations by. See Supplemental Material \"Movie S1.mov\" for a comparison of the evolution of the concen- tration field in present experiments and simulations by [4], .\n\nConvective shutdown in a porous medium at high Rayleigh number. D R Hewitt, J A Neufeld, J R Lister, J. Fluid Mech. 719D. R. Hewitt, J. A. Neufeld, and J. R. Lister. Convective shutdown in a porous medium at high Rayleigh number. J. Fluid Mech., 719:551-586, 2013.\n\nSelf-similarity and internal structure of turbulence induced by Rayleigh-Taylor instability. S Dalziel, P Linden, D Youngs, J. Fluid Mech. 399S. Dalziel, P. Linden, and D. Youngs. Self-similarity and internal structure of turbulence induced by Rayleigh-Taylor instability. J. Fluid Mech., 399:1-48, 1999.\n\nThe mixing transition in Rayleigh-Taylor instability. A W Cook, W Cabot, P L Miller, J. Fluid Mech. 511A. W. Cook, W. Cabot, and P. L. Miller. The mixing transition in Rayleigh-Taylor instability. J. Fluid Mech., 511:333-362, 2004.\n\nMiscible density fingering of chemical fronts in porous media: nonlinear simulations. Phys. Fluids (1994-present). A De Wit, 16A. De Wit. Miscible density fingering of chemical fronts in porous media: nonlinear simula- tions. Phys. Fluids (1994-present), 16(1):163-175, 2004.\n\nRayleigh-Taylor turbulence with singular nonuniform initial conditions. L Biferale, G Boffetta, A A Mailybaev, A Scagliarini, Phys. Rev. Fluids. 392601L. Biferale, G. Boffetta, A. A. Mailybaev, and A. Scagliarini. Rayleigh-Taylor turbulence with singular nonuniform initial conditions. Phys. Rev. Fluids, 3:092601, Sep 2018.\n\nGrowth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. R A Wooding, J. Fluid Mech. 3903R. A. Wooding. Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. J. Fluid Mech., 39(03):477-495, 1969.\n\nSee Supplemental Material \"Movie S2.mov\" for the time-dependent evolution of the wavenumber and the corresponding concentration field. See Supplemental Material \"Movie S2.mov\" for the time-dependent evolution of the wavenum- ber and the corresponding concentration field, .\n\nJ D IngleJr, S R Crouch, Spectrochemical analysis. Old Tappan, NJ (USJ. D. Ingle Jr and S. R. Crouch. Spectrochemical analysis. Old Tappan, NJ (US);\n"},"corpus_id":{"kind":"number","value":250073208,"string":"250,073,208"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2206.13363\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevfluids.7.093503\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2206.13363v2.pdf\"\n ],\n \"pdf_sha\": \"9cd766205ebe08bb057044bccd231e3f13e90777\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Experimental assessment of mixing layer scaling laws in Rayleigh-Taylor instability"},"authors":{"kind":"list like","value":["Marco De Paoli marco.de.paoli@tuwien.ac.at \nInstitute of Fluid Mechanics and Heat Transfer\n1060Wien, ViennaTUAustria\n\nPhysics of Fluids Group\nUniversity of Twente\n7500AEEnschedeThe Netherlands","Diego Perissutti \nPolytechnic Department\nUniversity of Udine\n33100UdineItaly","Cristian Marchioli \nPolytechnic Department\nUniversity of Udine\n33100UdineItaly","Alfredo Soldati \nInstitute of Fluid Mechanics and Heat Transfer\n1060Wien, ViennaTUAustria\n\nPolytechnic Department\nUniversity of Udine\n33100UdineItaly"],"string":"[\n \"Marco De Paoli marco.de.paoli@tuwien.ac.at \\nInstitute of Fluid Mechanics and Heat Transfer\\n1060Wien, ViennaTUAustria\\n\\nPhysics of Fluids Group\\nUniversity of Twente\\n7500AEEnschedeThe Netherlands\",\n \"Diego Perissutti \\nPolytechnic Department\\nUniversity of Udine\\n33100UdineItaly\",\n \"Cristian Marchioli \\nPolytechnic Department\\nUniversity of Udine\\n33100UdineItaly\",\n \"Alfredo Soldati \\nInstitute of Fluid Mechanics and Heat Transfer\\n1060Wien, ViennaTUAustria\\n\\nPolytechnic Department\\nUniversity of Udine\\n33100UdineItaly\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We assess experimentally the scaling laws that characterize the mixing region produced by the Rayleigh-Taylor instability in a confined porous medium. In particular, we wish to assess experimentally the existence of a superlinear scaling for the growth of the mixing region, which was observed in recent two-dimensional simulations. To this purpose, we use a Hele-Shaw cell. The flow configuration consists of a heavy fluid layer overlying a lighter fluid layer, initially separated by a horizontal, flat interface. When small perturbations of concentration and velocity fields occur at the interface, convective mixing is eventually produced: Perturbations grow and evolve into large finger-like convective structures that control the transition from the initial diffusiondominated phase of the flow to the subsequent convection-dominated phase. As the flow evolves, diffusion acts to reduce local concentration gradients across the interface of the fingers. When the gradients become sufficiently small, the system attains a stably-stratified state and diffusion is again the dominant mixing mechanisms. We employ an optical method to obtain high-resolution measurements of the density fields and we perform experiments for values of the Rayleigh-Darcy number (i.e., the ratio between convection and diffusion) sufficiently large to exhibit all the flow phases just described, which we characterize via the mixing length, a measure of the extension of the mixing region. We are able to confirm that the growth of the mixing length during the convection-dominated phase follows the superlinear scaling predicted by previous simulations."},"sections":{"kind":"list like","value":[{"title":"I. Introduction","content":"When two miscible fluids with different densities move under the action of gravity the relative acceleration between the fluids generates an instability at the separation interface. This instability, called Rayleigh-Taylor instability [1,2], grows in time and leads to the formation of convective flow structures of heavy fluid moving downward and of lighter fluid penetrating upward. The dynamics of the flow is first controlled by diffusion, which is responsible for the thickening of the interface, initially flat and horizontal. Diffusion favours mixing by increasing the length of the separation interface and gives rise to a buoyancy-driven flow by inducing local gradients of concentration. Afterwards, when the flow evolution is controlled by convection, instabilities grow and merge into large and more stable structures. The flow persists until a stable configuration is achieved in which the density distribution is ultimately uniform over the entire domain. Because of its relevance in applications characterized by buoyancy-driven flows, the Rayleigh-Taylor mixing process has been widely investigated when the instability leads to a turbulent flow dominated by convection [3]. In contrast, the evolution of the process within porous domains is far less explored [see 4-6, and references therein] in spite of its paramount importance in many geophysical and industrial situations, including water contamination [7,8], ice formation [9,10], salinity inversion [11,12], petroleum migration [13] and carbon sequestration [14][15][16][17][18].\nIn porous media flows, viscosity dominates over inertia at the pore scale and the momentum transport can be described by the Darcy law [19]. In this limit, a wavy and evolving interface leads to the formation of characteristic vertical, elongated structures that are usually called fingers. The lateral spreading (hence, the width or wavelength) and the vertical growth (hence, the amplitude) of these fingers is controlled by the interplay between diffusion and convection. The resulting time evolution of the fingers can be characterized by the mixing length, which is defined as the tip-to-rear finger length, and provides a macroscopic measure of the vertical extension of the mixing region. The evolution of porous Rayleigh-Taylor systems has been investigated with solutes (or chemicals) having different properties. With respect to the properties of the species involved, possible flow configurations can be grouped in three main categories: presence of one chemical, presence of two chemicals with different diffusion coefficients, and presence two species that can chemically react [20]. With the aid of numerical simulations and experiments, Lemaigre et al. [21] observed that in absence of chemical reactions the mixing region grows symmetrically with respect to the initial position of the interface: The growth occurs at a velocity that depends on the nature of the fluids involved, i.e. it is controlled by the diffusivity of the species and by the initial density difference [22]. The evolution of the system is different in presence of chemical reactions and moderate density contrast. The flow pattern develops in an asymmetric fashion [21], and chemical reactions act to stabilize the flow when the product is lighter than the reactants [23]: A local decrease of density induced by the fact that the product is less dense creates a non-monotonic density profile [24] with a minimum that hinders the penetration of the fingers into the bulk of the unmixed region. In contrast, a destabilizing effect can be obtained when the product provides a sufficiently larger density increase with respect to both reactants [25]. Chemical reactions can also induce secondary instabilities in time [20], for instance, when a sufficient amount of chemical reaction product triggers the fingered sinking of denser in the less dense reactant [26]. In summary, reactions can stabilize or destabilize convection, but in all cases, they increase the mixing rate within the host phase.\nIn this work, we will refer to the simplest cases, consisting of one chemical specie, i.e."},{"title":"I INTRODUCTION","content":"absence of double diffusion and chemical reactions, which produces a symmetric growth of the mixing region. According to theoretical scaling arguments, the mixing length (that quantifies the extension of the mixing region) is expected to evolve linearly in time, i.e. with scaling exponent equal to 1, as a result of the balance between buoyancy-induced convection and dissipation due to diffusion and viscosity. However, recent two-dimensional simulations of Darcy flows in porous media [4,5] have shown that the growth of the mixing length is superlinear, with scaling exponent equal to 1.2.\nWhile a simplified phenomenological model to estimate the amount of mixing induced by a Rayleigh-Taylor instability has been proposed recently [4], the reasons for the superlinear mixing length growth, not observed in the three-dimensional case [5], are yet unclear. In an effort to shed light on these reasons, in this work we provide a first experimental assessment of the superlinear growth, based on accurate measurements of the scaling exponent. We use a Hele-Shaw apparatus, which consists of two parallel transparent plates separated by a narrow gap. The density difference between the fluid layers is induced by the presence of a solute. For sufficiently-small fluid velocity within the gap, a quasi-two-dimensional Poiseuille flow establishes between the plates. This flow is known to approximate closely the Darcy-type flow observed in ideal porous media when the velocity at which the solute diffuses in the direction normal to the plates is much higher than the vertical advective velocity [27,28].\nSince the pioneering work of Saffman and Taylor [29], the Hele-Shaw apparatus has been widely used to mimic the behaviour of two-dimensional Darcy flows. However, timedependent and highly-resolved measurements are still challenging in the Rayleigh-Taylor configuration. This is mainly due to the experimental limitations associated with the initialization of the system, which requires to start the flow from a flat horizontal interface between the fluids. The experimental measurements presented in this work have been performed taking care to achieve proper initial condition and minimizing the local perturbations of concentration and velocity fields, which eventually lead to mixing. At the same time, measurements have been performed with a high resolution in space and time, to ensure an accurate reconstruction of the concentration field, and in domains that are relatively large compared to previous literature results and extend the measurements range. The fluids and the cell have been designed to allow a direct comparison with previous numerical simulations [4,30]. We analyzed the evolution of the mixing length and the wavenumber power spectra (i.e., the finger size). Concerning the mixing length, we observe the same superlinear behaviour predicted by the simulations [4,5], and we provide an explanation for this finding in terms of finite size effects of the flow domain. With respect to the fingers evolution, we find again that the measured number of fingers is in very good agreement with that predicted numerically [30]. Overall, we are able to assess successfully the scaling relations provided by numerical simulations. We believe that this result can open new perspectives for future modelling and parameterization of convective flow in confined porous media. This paper is organised as follows. In Sec. II, we formulate the problem, describe the experimental setup and recall the assumptions underlying the referenced numerical simulations. The results are presented in Sec. III both in qualitative (flow phenomenology) and quantitative (mixing length and wavenumbers) terms. Finally, conclusions are discussed in Sec. IV."},{"title":"II. Methodology","content":"Our experiments were specifically designed to mimic the mixing process produced by the Rayleigh-Taylor instability in a two-dimensional, saturated and confined porous medium. In addition, they were carried out imposing a linear dependency of density and solute concentration, and ensuring the occurrence of a Darcy-type flow to allow direct comparison with previous numerical simulations [4,30]."},{"title":"A. Problem formulation","content":"The process of convective dissolution is studied via the Rayleigh-Taylor instability, corresponding to two layers of fluid of different density, initially in an unstable configuration and subject to relative acceleration under the action of gravity [3]. The process is simulated in the frame of porous media flows, mimicked with the aid of a Hele-Shaw cell, i.e. two parallel and transparent plates of height H separated by a narrow gap b, as shown in Fig. 1(a). The cell is initially saturated with two miscible fluids having same viscosity (µ), arranged such that the heavy fluid (density ρ M ) lies on top of the lighter one (density ρ 0 ), as sketched in Fig. 1(b). The maximum density difference within the system is ∆ρ = ρ M − ρ 0 and is induced by the presence of a solute, namely potassium permanganate (KMnO 4 ). The amount of solute is quantified by its concentration C, which is maximum at the upper layer (C = C M ) and minimum at the lower layer (C = 0). The evolution of the system is controlled by the contributions of buoyancy, which tends to bring the fluids in a stable configuration and it is associated with the buoyancy velocity U , and diffusion, which acts to reduce local concentration gradients and to increase the mixing in the domain, and it is quantified by the diffusion coefficient D. The relative importance of these two contributions can be estimated by the Rayleigh-Darcy number, which represents the governing dimensionless parameter of the system, and is defined as:\nwith:\nthe buoyancy velocity and the length scale over which advection and diffusion balance, respectively [31]. The buoyancy velocity, is defined here as the combination of buoyancy (controlled by the acceleration due to gravity g, density contrast ∆ρ, and medium permeability k) and dissipation (controlled by fluid viscosity, µ). The length scale is particularly important when a comparison among systems having different Rayleigh numbers is required. Permeability k and porosity φ are the characteristic properties of the porous medium mimicked by the Hele-Shaw cell. In our experiments, k = b 2 /12 and φ = 1. In order to prevent potential three-dimensional effects we followed the indications emerged in the theoretical paper by Letelier et al. [27] and later confirmed experimentally by De Paoli et al. [28] and Alipour et al. [32]. We introduce the anisotropy ratio, which is an additional parameter describing the geometry of the cell, defined as:\nand we can use this parameter in combination with the Rayleigh number to ensure that the flow will be of the Darcy-type if the condition 2 Ra → 0 is met. In our study, ≈ 10 −3 and 4.4 × 10 −3 < 2 Ra < 5.4 × 10 −2 , indicating that we are very close to the theoretical limit corresponding to the Darcy-type regime.\nFinally, we are interested in examining a system with boundaries that are impermeable to both fluid and solute. Hence, the following conditions are imposed along the cell walls:\nwith n the unit vector perpendicular to the boundary."},{"title":"B. Experimental setup","content":"We used a transparent Hele-Shaw cell (Polymethyl Methacrylate -PMMA -thickness 8 mm). The wall-normal depth of the fluid layer is constant for all the experiments performed and corresponds to the value b = 300 µm. An impermeable rubber (Klinger-sil C-4400, thickness 300 µm), is placed between the two acrylic plates, and is used to seal the region containing the fluid but also as a spacer between the cells surfaces. The sealing is laser-cut so that the domain investigated is a square of side H = 87 mm. On each side, the midpoints are connected to pipes that allow the cell filling/emptying process. The transparent sheets and the gasket are held in place by an outer metal frame and a set of 10 bolts, which are tightened with the same torque, to prevent the formation of a non-uniform film thickness. The experimental apparatus is represented in Fig. 1(a). Backlighting is provided to the cell by a tunable LED system (150 lamps covered by a diffusing glass). Finally, the evolution of the flow is recorded by a digital sCMOS camera (FlowSense USB 2M-165, Dantec Dynamics). The imaging system and settings will be described in detail in Sec. II C. Next, we describe the fluids adopted and the processes of concentration reconstruction.\nThe initial condition of the experiments is an unstable configuration in which a layer of heavy fluid sits on top of a layer of lighter fluid, these layers being initially unmixed. To realize this challenging configuration, we employ a circuit made of a pump, valves and pipes, represented in Fig. 2. This approach has already been used in previous studies [see [33][34][35]. The entire system is initially filled with degassed water, to minimize the formation of bubbles. Then the cell is filled with the working fluids, which are driven by a peristaltic pump (Watson-Marlow 502 S) with the valves in \"loading mode\". In this configuration, the pump is able to inject degassed water and a homogeneous solution of KMnO 4 (concentration C M ) from two separated containers into the cell. Specifically, the solution and the water are injected from the top and bottom gates of the cell, indicated with (T) and (B) in Fig. 2. In this configuration, the valves connected to the left and right gates of the cell (L and R, in Fig. 2) are both open and allow the mixture of water and solution to leave the cell and be disposed in a residue container. Therefore, a flow from the top/bottom walls towards the left/right wall takes place, as indicated by the red and black arrows in Fig. 2. As a result, a sharp interface forms between the two liquid layers, while the vertical concentration gradient makes the fluid-fluid diffusive interface smoother. The competition between horizontal advection and vertical diffusion determines the interface thickness, which can be controlled by the flow rate (see Appendix C for further details on the interface thickness). A regulation valve is employed to balance the flow rate between the top (T) and bottom (B) gates to achieve a straight interface. When the initial condition is set, the valves are placed in \"experiment mode\", and the cell is by-passed by the inflow. The buoyancy-driven flow takes place and image acquisition starts.\nThe two fluids are fully miscible but have different density. Specifically, we used water for the lower (i.e. lighter) fluid layer and an aqueous solution of potassium permanganate (KMnO 4 ) for the upper (i.e. denser) fluid layer. While water density is nearly constant among the experiments (it is only dependent on temperature), the density of the KMnO 4 solution can be varied by changing the solute concentration. We consider the coordinate system sketched in Fig. 1(b), where x and z are the spatial coordinates in horizontal and vertical direction, respectively. In this reference frame, the initial condition can be expressed in terms of the solute concentration as follows: C(x, z ≤ H/2, t = 0) = 0 (lower half of the cell) and [33][34][35]. A pumping system (peristaltic pump, Watson-Marlow 502 S) is used to inject water (density ρ 0 ) and solution (ρ M ) from two separate containers, through the gates B and T, respectively. To set up the initial condition, the valves are placed in \"loading mode\" and the exceeding solution is collected from gates L and R into a residue container. The flow through the gates, which is used to modulate the shape of the interface between the two fluid layers, is controlled by a regulation valve. Finally, when the initial condition is achieved, the system is set in \"experiment mode\" and the cell is by-passed by the fluids, which are channeled to the residue container.\nconcentration of the KMnO 4 solution (upper fluid layer). We consider that the dynamic viscosity, µ = 9.2 × 10 −4 Pa·s, is constant and independent of the solute concentration [36]. Similarly, we assume that the diffusion coefficient is not sensibly affected by either the solute concentration or the local values of velocity D = 1.65 × 10 −9 m 2 /s. This value has been measured and reported in literature, and it is in excellent agreement with theoretical predictions based on electrical neutrality of a simple salt at infinite dilution [see 36, and references therein]. The working fluids have been chosen because of the linear dependency of the density that characterizes the resulting solution with respect to the solute concentration. This feature is essential to make reliable comparisons between experiments and simulations, which is one of the main objectives of the present work.\nThe density of an aqueous KMnO 4 solution can be written as a function of the fluid temperature, ϑ, the water density at temperature ϑ, ρ 0 , and the KMnO 4 concentration, C [37]. Please note that also the water density can be computed from the temperature of the fluid, and therefore the parameters that need to be measured to determine the solution density are ϑ (which we assume to be uniform over the cell and constant during the experiment) and the local value of C (inferred from optical measurements). The density difference between the two fluid layers is ∆ρ = ρ M − ρ 0 , where ρ M = ρ(C = C M ) and C M are determined prior to each experiment. At a given temperature, the density of the mixture obtained from empirical correlations [37] is well described by a function of C and C 3/2 . However, within the range of mass fraction considered here, this correlation is well approximated by a linear function of the solute concentration (see Appendix A) as:\nThis equation complies with the assumption of linear density-concentration dependency that is typically made in simulations, thus ensuring a reliable comparison. Measurements of the concentration field are inferred from the transmitted light intensities, this process being described in detail in the next section."},{"title":"C. Experimental procedure and quantification of uncertainties","content":"The results discussed in this paper are based on measurements of the time-dependent concentration field of the solute, C(x, z, t), inside the Hele-Shaw cell. To perform these measurements, we follow the method introduced by Slim et al. [36] and later used in a number of studies [28,32,38,39]. Optical measurements of light intensity distribution, I(x, z), are obtained with the aid of a sCMOS camera (FlowSense USB 2M-165, Dantec Dynamics, exposure time used 1 ms, resolution 2Mpx, frame rate 0.35 Hz to 5 Hz, depending on Ra). The measurements require a calibration process, briefly described in Appendix B, in which the value of light intensity is associated with the value of solute concentration. The greyscale images so obtained, an example of which is reported in Fig. 3(a), are preprocessed to reduce the effect of non-uniformities in the distribution of the light intensity over the cell, [40]). Finally, since the relationship between mass fraction and solute concentration is known (see Fig. 9(b) of Appendix B), the reconstruction of the concentration and density fields is performed. An example of this reconstruction is provided in Fig. 3(b).\nAn important issue that must be assessed is the uncertainty on the measured value of the Rayleigh number, Ra, defined as in Eq. (1). The uncertainty value, δRa, is obtained from propagation-of-error analysis [41] and reads as:\nwhere δ• indicates the uncertainty associated to the quantity • that represents each of the variables appearing in Eq. (1). We assume that the properties µ, D and g are estimated with constant relative uncertainty equal to 1%. The uncertainty on the domain height, δH, is assumed to be 0.5 mm, primarily due to a possible misplacement of the sealing on the cell rather than to the cutting precision. The accuracy on the gap thickness is assumed to be δb = 10µm. The uncertainties just introduced are common to all the experiments. In contrast, the uncertainty of the density difference δ∆ρ is a function of the Rayleigh number, being derived from Eq. (A1) as:\nwhere δϑ is given by the instrument (0.5 • C). To use Eq. (7), the uncertainty on the concentration must be known. This is computed from Eq. (A5) as:\nThe mass fraction, ω, defined as the ratio between the mass of solute and the mass of solution, is determined measuring first the mass of water (m w ) and then the overall mass of solution (m sol ), i.e. ω = (m sol − m w )/m sol . As a result, the uncertainty δω reads as:\nwith δm w = δm sol = 0.001 g (high-precision scale, Sartorius Acculab Atilon model ATL-423-I, ±0.001 g). The list of the solution properties, flow parameters and corresponding uncertainties is reported in Tab. I."},{"title":"D. Numerical simulations and dimensionless variables","content":"To assess the mixing layer scaling laws, we wish to make a direct comparison of the present experimental results with direct numerical simulations of convection in two-dimensional porous media [4,30]. We perform this comparison in the range 347 ≤ Ra ≤ 19953, for which a very good overlap of the governing parameters exists between experiments and simulations. To make the paper self-contained, we recall next the governing equations and the numerical details.\nAn isotropic and homogeneous vertical porous slab with permeability k and porosity φ is considered. The porous domain is initially saturated with two miscible fluids having same viscosity (µ) but different density, arranged such that the heavy fluid (density ρ M ) tops the\n24.0 ± 0. lighter one (density ρ 0 ). This unstable configuration replicates exactly the one sketched in Fig. 1(b). As done in the experiments, the density difference is induced by the presence of a solute, with maximum solute concentration C = C M in the upper layer and minimum solute concentration C = 0 in the lower layer. The flow is assumed to be incompressible and controlled by the Darcy's law:\nwhere u and w are the horizontal (x) and vertical (z) velocity components, while p and ρ are the local pressure and density, respectively. We consider that the Oberbeck-Boussinesq approximation applies: This is a reasonable assumption for geophysical problems such as geological carbon dioxide sequestration [42]. In addition, we consider that the density of the mixture is a linear function of the solute concentration C [see Eq. (5)], and that the following transport equation for C holds:\nwith t the time and D the solute diffusivity, which we assume to be constant.\nTo make Eqs. (10)-(12) dimensionless, we use the buoyancy velocity U and the length scale , defined as in Eq. (2). In addition, the following dimensionless variables are introduced [31,43,44]:\nThe resulting governing equations in dimensionless form read as:\nwhere the superscript * indicates dimensionless variables, P * = p * + z * (ρ M /∆ρ − 1) is the reduced pressure and ∆ρ is the density difference between the upper and the lower fluid layers.\nThe controlling parameter of the present system is the Rayleigh-Darcy number Ra, defined as in Eq. (1). Unlike the Hele-Shaw experiments, in which the geometry of the cell is characterised by the anisotropy ratio, two-dimensional Darcy simulations do not need additional dimensionless parameters: The Rayleigh-Darcy number describes completely the fluid properties (∆ρ, µ, D), the porous medium properties (k, φ) and the domain size (H).\nAlthough not explicitly appearing in the governing Eqs. (14)- (16), Ra enters the picture as the dimensionless height of the domain. Impermeable boundary conditions (i.e. no-flux) are imposed for both the fluid and the solute at the top and bottom horizontal boundaries, whereas periodicity is applied at the side boundaries (along x). In dimensionless form, these boundary conditions read as:\nThe set of equations (14)- (16) has been discretized numerically using a Chebyshev-Tau method. We refer to [4,30,43,45] for further details on the flow solver."},{"title":"III. Results","content":"The comparison of the experimental results against numerical simulations is focused on the evolution of the mixing length and on the analysis of the wavenumber power spectra. These spectra are obtained by making a Fourier transform of the instantaneous concentration field and provide a robust characterization of the finger size, as this size can be associated to the wavenumber at which the spectra exhibit a peak. We consider a wide range of Rayleigh numbers: 4.39 × 10 3 ≤ Ra ≤ 5.43 × 10 4 . We refer to Tab. I for the experimental mass fraction and density difference values used to obtain these Ra. Above the upper value Ra max = 5.43 × 10 4 , the density difference between the KMnO 4 -rich solution (heavy fluid) and water (light fluid) induces large vertical velocities, and the flow is affected by non-Darcy effects [27,28]. For lower values of ∆ρ, in contrast, the initial condition is hard to achieve and the duration of the experiments increases, leading to higher chances of inducing density perturbations in the cell from the side and horizontal channels (L, R, T, B in Fig.1). In the following, we describe the experimental results in terms of flow phenomenology (Sec. III A), time evolution of the mixing length (Sec. III B) and wavenumbers (Sec. III C), comparing visa-vis experimental observations against numerical simulations [4,30]. The datasets analyzed in the present study have been deposited in [46]. [4]. The dimensionless time instants (t * ) at which the fields refer are indicated on top of each panel. We remark that the experiments are realised in a square geometry, whereas the simulations are performed in wider domains (aspect ratio π/2). Therefore, for better comparison, only a portion of the numerical domain having unitary aspect ratio is reported. The desired initial position of the interface (located at z = H/2, i.e. z * = Ra /2) is also shown as a dashed black line. See also the movie, Movie S1, in the Supplemental Material [47] for a comparison of the time-dependent evolution of the concentration fields in the experiment and in the simulation."},{"title":"A. Flow phenomenology","content":"To analyze in detail the flow phenomenology, we report in Fig. 4 the time-dependent dimensionless concentration field (C * ) obtained experimentally (4a-d) and numerically [4] (4e-h). The experiment (E4 in Tab. I, Ra = 19789) is compared against numerical simulations performed at similar Rayleigh number, Ra = 19953: We consider this value because it is large enough to observe all the main phases of the mixing process for a significant amount of time. On top of each panel, the dimensionless time instants (t * ) at which the fields are taken are provided.\nThe concentration field is initially in an unstable configuration (Fig. 4a,e) with the heavy fluid characterised by the dimensionless concentration C * = 1 lying on top of the lighter one (C * = 0). The initial position of the interface, indicated here with the dashed line at z * = Ra /2, is initially flat and horizontal. Achieving such a configuration in the experiment is very difficult, and some discrepancy, though not apparent, always exists between the actual fluid-fluid interface and the desired interface. This is visible for instance in Fig. 4(a), where the upper fluid layer seems to be slightly inclined with respect to the horizontal dashed line. In the experiments, the flow is initially triggered by perturbations of the local density and velocity fields. Indeed, to set the initial flow configuration and generate the fluid-fluid interface, some fluid is removed from the side channels and simultaneously injected from the channels at the horizontal boundaries (see Sec. II C for a thorough description of this process). As a result, the fluid will not be initially still and will be partially affected by the flow circulations induced by the procedure adopted to set the initial condition. The situation is completely controllable in the simulations, where the interface starts as perfectly flat and is made unstable by adding a perturbation directly to the concentration field (Fig. 4e). The amplitude of the perturbation considered, however, is crucial in determining the onset of the instability [30], whereas the late stage dynamics are not sensitive to this parameter. In the present study, we considered simulations in which the perturbations applied to the dimensionless concentration field have amplitude 10 −3 .\nThe flow phenomenology associated with the mixing process can be summarised as follows. The mechanism of diffusion is initially responsible for the flow evolution (Fig. 4a,e). Afterwards, the interfacial layer thickens and, due to the unstable density profile, finger-like structures form. These fingers are controlled by the interplay of convection, which makes them grow vertically, and diffusion, which reduces the horizontal concentration gradients and promotes mixing [35]. During this intermediate stage (Fig. 4b-c,f-g), the fingers grow symmetrically with respect to the domain centerline, and eventually merge. Indeed, we observe that the number of fingers reduces from t * = 0.68 × 10 4 (Fig. 4b,f) to t * = 1.57 × 10 4 (Fig. 4c,g). We anticipate here that this dynamics is quantitatively confirmed by the wavenumber measurements reported in Sec. III C. At a later stage (Fig. 4d,h), the number of fingers does not increase further. Eventually, the strength of convection diminishes and the flow evolution is controlled again by diffusion. In this phase (not shown here), the system has achieved a stable configuration in which the average concentration in the lower layer is higher than in the upper layer [4]. This regime, usually referred to as shutdown of convection in view of the reduced strength of the driving forces [43,48], starts after the tips of the fingers reach the horizontal walls and the effect of the boundaries propagates back to the domain centerline. The time required to reach the shutdown is large (e.g., of the order of 1 hour for the lower Ra considered here), and hence hard to achieve experimentally because the flow conditions have to be carefully controlled for a considerable amount of time.\nIt is apparent that the flow evolution observed in the experiments is qualitatively similar to that of numerical simulations, albeit with some differences. The growth of the fingers, expressed in terms of their tip-to-rear extension, is similar within the same time interval (Fig. 4b,f and Fig. 4c,g). We also refer the reader to the movie, Movie S1, in the Supplemental Material [47] for a comparison of the time-dependent evolution of the concentration fields in experiments and simulations. However, we also observe a different number of fingers in the two cases (see Sec. III C for further discussion)."},{"title":"B. Mixing length","content":"The evolution of a Rayleigh-Taylor system can be conveniently analysed by looking at the instantaneous vertical extension of the fluid-fluid interface [3,49]. This quantity, defined as mixing length h, can be also interpreted as the distance between the upwarding and downwarding tips of the fingers. To compute the mixing length in a quantitative way, one can use either global [50] (i.e. integral) or local [35,51] flow properties. In this work, we characterize the mixing length by means of the local concentration field [4,51], i.e. h is the portion of the domain where the horizontally-averaged concentration field satisfies the condition ε < C(z, t) < 1 − ε, with:\nthe concentration field averaged along the horizontal direction x, ε = 3 × 10 −2 a threshold coefficient, and H the domain width (in x direction). It follows from this definition that 0 ≤ h ≤ H or, in dimensionless terms using the length scale defined in Eq. (2), 0 ≤ h * ≤ Ra.\nThe time evolution of the dimensionless mixing length for all the experiments we performed is shown in Fig. 5. As previously observed in simulations in confined domains [4,30], the dynamics consists essentially of three phases in sequence: (i) diffusion-dominated, (ii) convection dominated and (iii) wall-induced shutdown. During the initial phase, the thickness of the mixing layer is initially controlled by diffusion. Assuming an initial stepwise concentration profile, the dimensionless concentration field evolves as:\nand the corresponding mixing length as:\n(we refer to [4] for a detailed derivation). The same evolution of the mixing length in the initial diffusive regime has been observed in previous numerical works, both in porous media convection [4,35] and in pure fluids with different initial conditions [52]. Despite the seemingly universal behaviour exhibited by the mixing length, we observe that the initial diffusive stage is hardly captured by the experimental measurements. Only the measurements performed at lower Ra, namely 4407 and 7023, are in good agreement with the theoretical predictions (blue dashed line in Fig. 5). While the trend nicely follows the predicted behaviour, namely h * ∼ √ t * , the magnitude slightly differs from the solution derived in Eq. (20). This is due to uncertainty of the experimental measurements, which is comparable to the values of h * at low Ra and early times. When the fluid-fluid interface is sufficiently thick (10 3 < t < 10 4 ), it becomes unstable and eventually finger-like structures form. When observed via the mixing length, the transition to this convection-dominated regime is smooth, similarly to what has been observed in the simulations. However, we find that the transition in experiments occurs at different instants depending on the value of Ra, as a result of the uncertainty on the initial perturbation of the concentration field, which cannot be reproduced faithfully in the laboratory. We also observe that, during the transition, the growth of the mixing length modelled as a power law, i.e. h * ∼ (t * ) α , is characterized by an exponent that is higher than that observed asymptotically, in agreement with previous numerical [4,35] and experimental (α = 2) [53] findings.\nThe fluid mixing is more efficient when convection takes place. The formation of fingerlike structures marks the beginning of the convection-dominated phase, more persistent in time compared to the initial diffusive stage. In this phase, fingers grow, merge and finally reach the horizontal boundaries of the cell. The measurements reported in this work reveal that the mixing length grows more than linearly in time, in close agreement with previous simulations (namely, h * ∼ (t * ) 1.208 , [4,5]). This scaling is anomalous: It is not expected in two-dimensional porous media convection, where a linear evolution of the mixing length with time is predicted as a result of the balance between buoyancy (i.e., convection) and friction (i.e. dissipation due to viscosity). The occurrence of a linear growth would imply an asymptotic velocity of the finger tips, which determines also the growth rate of the mixing length, equal to the buoyancy velocity U , defined as in Eq. (2). We remind the reader that U can be interpreted as the velocity at which the contribution of buoyancy (g∆ρk) balances the dissipation (µ).\nIn three-dimensional porous media, the growth of the mixing length follows the linear prediction obtained from the physical arguments mentioned above [5]. The behaviour is different in two dimensions, where the fingers are slenderer compared to their three-dimensional counterpart. The superlinear evolution of the mixing length we observed, can be ascribed to multiple reasons: (i) the simulated time is not sufficiently large to achieve the asymptotic regime, or (ii) the Ra number of the system is too low, and the effect of diffusion is still dominant. We speculate that in the present case a combination of these two effects controls the evolution of the flow. Initially, the fluid accelerates from zero velocity, and this corresponds to a finite-time effect. At a later stage, when the initial condition has been absorbed by the system and fluid velocity is non-zero, the finite-size (i.e., finite-Ra) effect comes into play, via the domain boundaries. This non-linear behaviour seems indeed to vanish at higher Ra numbers [6], but in absence of boundaries. We remark here that a linear growth of the mixing length has been observed numerically [35], but only in the limit of lower Rayleigh numbers, namely Ra ≈ 4 × 10 3 . For these values of Ra, also previous numerical simulations [4] and present experiments do not exhibit a clear superlinear behaviour. In addition, the experimental data reported by Wooding [53], which refer to higher Rayleigh numbers (approximately 10 3 ≤ Ra ≤ 7 × 10 4 ) are also well fitted by a linear scaling. However, it must be pointed out that the data have been collected at very low acquisition rate, making the amount of available measurements very scarce for a time-dependent flow analysis. We are not able to explain, in this case, why the superlinear scaling was not observed. One possible reason could be attributed to the method adopted to change the Rayleigh number, consisting of varying the orientation of the cell with respect to gravity. Another possibility is represented by the absence of horizontal boundaries. A conclusive answer on the asymptotic scaling of the mixing length could be obtained with the aid of Darcy simulations or experiments at higher Ra, (≈ 10 6 ), which are beyond the present capabilities.\nTo conclude the discussion on the mixing length, we now examine the wall-induced convective shutdown phase [4], which sets in when the high (resp. low) concentration fingers reach the lower (resp. upper) horizontal boundary. In this phase, the local concentration gradients decrease, weakening the density differences that drive the flow within the domain. As a result, convection vanishes and the overall mixing rate, identified for instance by means of the mean scalar dissipation rate, decreases dramatically [30]. This effect is crucial for applications of practical interest (e.g., carbon dioxide sequestration in saline aquifers), in which the fluids are confined by impermeable boundaries that play a key role in the solute mixing process. A simple way to quantify the convective shutdown is to measure the time t * s required for the fingers to reach the horizontal boundaries. The values of t * s obtained as a function of the Rayleigh number are shown in Fig. 6. Experimental and numerical [4] results are reported as open (•) and filled symbols ( ), respectively. The best-fitting power law for present results (red dashed line) is 9.64 ×Ra 0.80 , in excellent agreement with numerical results (14.98 ×Ra 0.78 , black dashed line). This is not surprising, given the close correspondence observed for the evolution of the mixing length. In leading order, we can estimate t * s = t * : h * (t * ) = Ra which gives t * s ∼ Ra 1/1.2 ∼ Ra 0.83 , consistent with the Ra 0.80 scaling provided by data fitting.\nDespite the excellent agreement of the scaling exponents, we also observe some discrepancies between simulations and experiments. In particular, for a given Ra, the value of t * s is lower in the experiments, leading to a lower value of the multiplicative coefficient obtained from data fitting. We attribute this discrepancy to the perturbation of the interfacial concentration field. In the experiments, the amplitude of this perturbation is hard to control, and it is likely to be larger compared to the numerical simulations considered, leading to an earlier onset of convective instabilities. Another limitation of our experimental setup is represented by the range of Rayleigh numbers that can be investigated. To be able to achieve a Darcy flow and to control effectively the boundary conditions, the values of domain size (H * ) and the density difference (∆ρ) cannot be too large (see also Sec. II B). The values of Ra considered here are sufficiently large to overlap with and go beyond (up to 2.5 times, approximately) the largest Rayleigh numbers simulated numerically. On the other hand, achieving smaller values of Ra is also challenging. Lower values of ∆ρ correspond to lower velocities, and therefore longer experiments. Smaller values of H * , i.e. shorter domains, would instead required more careful setup of the initial flow configuration.\nThe findings discussed in this section prove that, in terms of evolution of the mixing length, two-dimensional Darcy simulations and Hele-Shaw experiments produce similar results when the sameRa is considered. However, the ability of the experimental measurements to capture the initial development of the flow is lower due to the setup used, and numerical simulations should be considered as more reliable to investigate this stage of the flow evolution."},{"title":"C. Wavenumbers","content":"The flow dynamics is complex and characterised by a continuous change of the fluid-fluid interface position. To provide a quantitative estimate of the flow topology evolution, we analyze here the number of fingers and the wavenumbers at the centerline of the domain (z * = H * /2). The process of flow analysis is illustrated in Fig. 7, where an experiment at Ra = 5.43 × 10 4 is considered for reference. First, we build the space time map at the centerline, where the interface is initially located, i.e. we report in Fig. 7(a) the contour of the time-dependent dimensionless concentration field, C * (x * , z * = Ra /2, t * ). The simplest way to define the number of structures in the system is to count the upwarding and downwarding fingers. To this aim, the spacetime map is first binarised [ C(x * , z * = Ra /2, t * ), Fig. 7(b)] with respect to a threshold value corresponding to the mean concentration (C * = 1/2), and then the number of fingers is counted [ Fig. 7(c)]. The process is iterated during the entire flow evolution, and the number of fingers, N (t), is computed as the number of regions with different binarized concentration values [ Fig. 7(d), black line]. Due to noise in the experimental measurements, spurious fluctuations of the concentration field may occur. To avoid counting these as fingers, we discarded contiguous regions of the binarized map that are smaller than 4 pixel (corresponding to a physical length of 350 µm). We observe in Fig. 7(d) that this threshold is extremely sensitive to variations of the concentration field, i.e. it appears to fluctuate without an apparent physical reason. A more robust characterization of the flow topology is provided by the wavenumbers associated with the power spectra of the concentration field. We employ a Fourier decomposition of the concentration profile at the cell centerline to estimate the number of fingers, and compute the power-averaged mean Measurements of the number of fingers over time for Ra = 5.43 × 10 4 (E6 in Tab. I). Space-time map, consisting of the evolution of the concentration field along the centerline, C * (x * , z * = Ra /2, t * ), is shown in (a). To compute the finger number, the space-time map is binarized, C(x * , z * = Ra /2, t * ), choosing C * = 1/2 as threshold (b). The concentration profile (red line) and the discretized profile (black line) measured along the centerline at t * ≈ 1.1 × 10 5 (indicated by the dashed blue line in panels a, b and d) are also reported as a function of the horizontal coordinate x * (c). Finally (panel d), the time-dependent finger number is computed as from the binarized field C (black line), from power-averaged mean wavenumber as defined in Eq. (21) ( N = L k /π, red line) and from a k peak (green line), i.e. the wavenumber maximizing the spectrum amplitude of C. See also the movie, Movie S2, in the Supplemental Material [54] for the time-dependent evolution of the wavenumber and the corresponding concentration field. wavenumber at each time instant, defined as [51]:\nwith k n the wavenumber and |C n (t * )| 2 the squared amplitude of the n-th basic harmonic function obtained from the Fourier decomposition of the fluctuation of C * (x * , z * = 1/2, t * ) about its mean value. We observe that in this case [red line in Fig. 7(d)] the evolution of the number of fingers, N = L k /π, is smoother compared to the simple finger count, N , and also compared to the peak number, defined as N peak = Lk peak /π [orange line in Fig. 7\nwhere k peak is the wavenumber value for which |C n (t * )| is maximum. Therefore, in the following, we will focus the analysis considering the power-averaged mean finger number N only, which appears similar to but less fluctuating of the finger number N . A comparison of the number of fingers measured in the experiments is reported in the inset of Fig. 8(a). In general, all the experiments are characterized by a similar dynamics. The number of fingers, initially low, increases up to a maximum value. Afterwards, the fingers start to merge and increase in size. This phase, which is also visible from the spacetime maps in Fig. 7(a-b), may start earlier at low Ra and later at high Ra, spanning a relatively long interval (10 3 ≤ t * ≤ 10 4 ) in Fig. 8(a). During growth, N appears to decrease in time at the same rate for all Ra considered. This universal behaviour is more apparent when the results are compared in terms of mean wavenumber k (main panel of Fig. 8a). After entering the convection-dominated regime, all curves exhibit the same trend. A first observation is that the merging rate of the fingers is in agreement with the scaling k ∼ (t * ) −0.6 reported in numerical simulations [30]. In contrast, the behaviour is different during the diffusive phase, when the wavenumber is clearly dependent on Ra.\nIn Fig. 8(b), we compare present results (solid lines) against previous numerical findings [30] (dashed lines). For ease of reading, only three values of Ra number are considered. Two differences are apparent: (i) unlike experiments, simulations [30] have been performed on domains of constant dimensionless width (namely, 19953), and (ii) to make reliable comparisons among different Rayleigh numbers, the same initial perturbation has been employed, thus making the early stage flow evolution also universal and independent of Ra. This highlights again a possible limitation of the experimental procedure employed, which makes it hard to precisely control the initial flow conditions. As a result, the initial phase is characterised by large discrepancies between numerical and experimental results (Fig. 8b). In the late stage of the flow dynamics, numerical and experimental measurements are in fair agreement. As a side note, we remark that present results, shown in terms of wavenumbers associated with the concentration field, are compare against the simulations of De Paoli et al. [30], where the power spectra where obtained from the scalar dissipation fields. However, even the experimentally-measured merging rate is well described by the same scaling law, t −0.6 . Present results are in fair agreement with previous experimental findings [35,53], in which the wavenumber is observed to evolve according to (t * ) −1/2 . This scaling has been also derived analytically [31,53] assuming that the fingers are slender, so that vertical derivatives (e.g., diffusion of solute) are negligible compared to the horizontal ones. In the frame of this model, the flow is driven by vertical advection, whereas lateral diffusion acts as dissipative mechanism. We refer to Slim [31] for a detailed derivation of this result. The slight differ-ence in the scaling exponent, compared to the analytical prediction, maybe attributed to the simplified assumptions considered in the model derivation. The scaling k ∼ (t * ) −1/2 has been also suggested by Gopalakrishnan et al. [35], who interpreted this evolution as a diffusion-dominated finger growth, by analyzing the relative contribution of convective and diffusive terms within a finger (but away from the finger tips)."},{"title":"IV. Conclusions","content":"In this work, we analyzed with the aid of high-resolution experiments the Rayleigh-Taylor instability in a Hele-Shaw flow. The system consists of two miscible fluid layers, namely water and an aqueous solution of KMnO 4 , having different density but same viscosity, placed in an initial unstable configuration. The density difference is induced by the presence of the solute (KMnO4), which produces also a colour gradient by which we are able to infer the concentration field. The system is fully described by the Rayleigh number, Ra, which measures the relative strength of convective and dissipative contributions. Present experimental results have been performed taking great care to achieve a proper initial condition and minimizing the local perturbations of the concentration and velocity fields, which trigger the growth of interfacial finger-like instabilities. In addition, a high resolution in space and time has been achieved by considering relatively large domains, and an accurate reconstruction of the concentration field is performed. In this respect, present experimental results go beyond previous measurements published in archival literature.\nWe compared present findings against previous numerical measurements in two-dimensional and confined media to which the Darcy equations can be successfully applied, as for instance porous media. [4,30]. The dynamics of the flow is first controlled by diffusion, which is responsible for the thickening of the interface, initially flat and horizontal. Afterwards, the flow evolution is controlled by convection. The unstable fluid-fluid density front promotes the formation of high-wavenumber instabilities, which grow and merge into large and more persistent fingers. Finally, due to the combined action of convective mixing in a confined domain and lateral diffusion across the fingers interface, the solute distribution becomes uniform within the porous slab, with a corresponding stable concentration profile. During this stage, referred to as wall-induced convective shutdown, the flow is again controlled by diffusion. The flow evolution is quantified and compared against theoretical and numerical predictions with the aid of two main observables: the mixing length, h * (t), and the power-averaged mean wavenumber, k .\nInitially, the evolution of the mixing length is in fair agreement with the theoretical predictions obtained for flows in the absence of convection (h * (t) ∼ (t * ) 1/2 ). In the intermediate regime, in which convection dominates, h * (t) is observed to grow with a superlinear scaling, h * (t) ∼ (t * ) 1.2 , confirming previous numerical findings [4,5]. This scaling is anomalous, as it differs from the theoretical prediction, h * (t) ∼ t * , possibly as a result of a finite size (i.e., finite Ra) effects. To further compare present experimental findings against previous numerical simulations, we compute the time t * s taken for the fingers to reach the horizontal boundaries of the domain. Also in this case, experiments (t * s ∼ Ra 0.80 ) are in excellent agreement with the simulations (t * s ∼ Ra 0.78 ). Finally, we analysed the flow dynamics by looking at the finger distribution at the domain centerline. The evolution of the characteristic finger size has been quantified by means of the power-averaged mean wavenumber, k . During the convection-dominated phase, the centerline finger dynamics exhibits a universal behaviour that is independent of Ra. Also in A ADDITIONAL DETAILS ON FLUID PROPERTIES this case, we found an excellent agreement with scaling laws derived in numerical simulations [30], namely k ∼ (t * ) −0.6 .\nThe analysis carried out in this paper shows that Hele-Shaw experiments can faithfully reproduce the dynamics of convective Rayleigh-Taylor instabilities in two-dimensional and confined porous media. However, it also shows that the proposed experimental approach presents some important limitations, which call for future improvements. First off, the desired initial fluid-fluid interface consists of a thin, flat and horizontal contact line, which is experimentally challenging to achieve. In addition, the instability is initially triggered by local perturbations of the concentration and velocity fields, and hence hard to control. Finally, experiments at low Ra are characterised by low values of vertical velocity, which slow down the mixing process and make the experiments to last longer. As a result, the experimental conditions have to be monitored for longer times, increasing the difficulty of the experimental procedure.\nFIG. 9. Given the definition (A5) and the empirical correlations (A1) [37], the relative dependence of density of the solution, ρ, mass fraction, ω, and solute concentration, C, is obtained. We report here C(ω) (panel a), ρ(ω) − ρ 0 (panel b) and ρ(C) − ρ 0 (panel c), being ρ 0 the water density. The profiles shown here correspond to ϑ = 25 • C. We observe that the empirical values (symbols) are very well fitted by linear functions (solid lines). In the instance of density and concentration (panel c), the linear function corresponds to Eq. (5).\nby a linear function of the solute concentration [Eq. (5)], as can be seen in Fig. 9c)."},{"title":"B. Measurements calibration","content":"When the two-dimensional distribution of light intensities is determined, each value of light intensity is assigned a value of mass fraction, ω(x, z). This step is achieved with the aid of calibration curves, an example of which is reported in Fig. 10. For a given illumination condition, the cell is filled with solution with a known value of mass fraction, ω, and the average light intensity over the cell, I cal , is recorded. The higher the concentration of KMnO 4 , the lower the transmitted light intensity [Lambert-Beer law, 55]. Experimental measurements (symbols) are well fitted by a combination of exponential functions, ω( I cal ) = a 0 + a 1 exp(b 1 I cal ) + a 2 exp(b 2 I cal ) (solid line). Further details on the process of calibration are available in [28,32,39]."},{"title":"C. Additional details on the initial condition","content":"Large values of solute concentration (C M ) produce strong concentration gradients across the initial fluid-fluid interface, making a sharp separation between the two fluids hard to obtain. Therefore, to achieve a well-defined initial interface up to the Ra values considered here, the flow rate provided to the cell is increased withRa. In Fig. 11, the initial condition is shown in detail for the experiments E1 and E4 (namely at Ra = 4391 and Ra = 19789), and for both the unprocessed light intensity field, I, (Fig. 11a,d) and the inferred dimensionless concentration field, C * (Figs. 11b,e). We observe that the experimental procedure employed is suitable to obtain a macroscopic sharp and straight interface in correspondence of the domain centerline, represented by the dashed red line in Figs. 11(a,b,d,e). Further information on the local interface shape can be obtained from the close up view of the interfacial region, shown in Figs. 11(c,f), where the iso-contours at C * = 0.3, C * = 0.5 and C * = 0.7 (same"},{"title":"FIG. 1 .","content":""},{"title":"FIG. 2 .","content":""},{"title":"FIG. 3 .","content":""},{"title":"FIG. 4 .","content":""},{"title":"FIG. 5 .","content":""},{"title":"FIG. 6 .","content":""},{"title":"FIG. 8 .","content":""}],"string":"[\n {\n \"title\": \"I. Introduction\",\n \"content\": \"When two miscible fluids with different densities move under the action of gravity the relative acceleration between the fluids generates an instability at the separation interface. This instability, called Rayleigh-Taylor instability [1,2], grows in time and leads to the formation of convective flow structures of heavy fluid moving downward and of lighter fluid penetrating upward. The dynamics of the flow is first controlled by diffusion, which is responsible for the thickening of the interface, initially flat and horizontal. Diffusion favours mixing by increasing the length of the separation interface and gives rise to a buoyancy-driven flow by inducing local gradients of concentration. Afterwards, when the flow evolution is controlled by convection, instabilities grow and merge into large and more stable structures. The flow persists until a stable configuration is achieved in which the density distribution is ultimately uniform over the entire domain. Because of its relevance in applications characterized by buoyancy-driven flows, the Rayleigh-Taylor mixing process has been widely investigated when the instability leads to a turbulent flow dominated by convection [3]. In contrast, the evolution of the process within porous domains is far less explored [see 4-6, and references therein] in spite of its paramount importance in many geophysical and industrial situations, including water contamination [7,8], ice formation [9,10], salinity inversion [11,12], petroleum migration [13] and carbon sequestration [14][15][16][17][18].\\nIn porous media flows, viscosity dominates over inertia at the pore scale and the momentum transport can be described by the Darcy law [19]. In this limit, a wavy and evolving interface leads to the formation of characteristic vertical, elongated structures that are usually called fingers. The lateral spreading (hence, the width or wavelength) and the vertical growth (hence, the amplitude) of these fingers is controlled by the interplay between diffusion and convection. The resulting time evolution of the fingers can be characterized by the mixing length, which is defined as the tip-to-rear finger length, and provides a macroscopic measure of the vertical extension of the mixing region. The evolution of porous Rayleigh-Taylor systems has been investigated with solutes (or chemicals) having different properties. With respect to the properties of the species involved, possible flow configurations can be grouped in three main categories: presence of one chemical, presence of two chemicals with different diffusion coefficients, and presence two species that can chemically react [20]. With the aid of numerical simulations and experiments, Lemaigre et al. [21] observed that in absence of chemical reactions the mixing region grows symmetrically with respect to the initial position of the interface: The growth occurs at a velocity that depends on the nature of the fluids involved, i.e. it is controlled by the diffusivity of the species and by the initial density difference [22]. The evolution of the system is different in presence of chemical reactions and moderate density contrast. The flow pattern develops in an asymmetric fashion [21], and chemical reactions act to stabilize the flow when the product is lighter than the reactants [23]: A local decrease of density induced by the fact that the product is less dense creates a non-monotonic density profile [24] with a minimum that hinders the penetration of the fingers into the bulk of the unmixed region. In contrast, a destabilizing effect can be obtained when the product provides a sufficiently larger density increase with respect to both reactants [25]. Chemical reactions can also induce secondary instabilities in time [20], for instance, when a sufficient amount of chemical reaction product triggers the fingered sinking of denser in the less dense reactant [26]. In summary, reactions can stabilize or destabilize convection, but in all cases, they increase the mixing rate within the host phase.\\nIn this work, we will refer to the simplest cases, consisting of one chemical specie, i.e.\"\n },\n {\n \"title\": \"I INTRODUCTION\",\n \"content\": \"absence of double diffusion and chemical reactions, which produces a symmetric growth of the mixing region. According to theoretical scaling arguments, the mixing length (that quantifies the extension of the mixing region) is expected to evolve linearly in time, i.e. with scaling exponent equal to 1, as a result of the balance between buoyancy-induced convection and dissipation due to diffusion and viscosity. However, recent two-dimensional simulations of Darcy flows in porous media [4,5] have shown that the growth of the mixing length is superlinear, with scaling exponent equal to 1.2.\\nWhile a simplified phenomenological model to estimate the amount of mixing induced by a Rayleigh-Taylor instability has been proposed recently [4], the reasons for the superlinear mixing length growth, not observed in the three-dimensional case [5], are yet unclear. In an effort to shed light on these reasons, in this work we provide a first experimental assessment of the superlinear growth, based on accurate measurements of the scaling exponent. We use a Hele-Shaw apparatus, which consists of two parallel transparent plates separated by a narrow gap. The density difference between the fluid layers is induced by the presence of a solute. For sufficiently-small fluid velocity within the gap, a quasi-two-dimensional Poiseuille flow establishes between the plates. This flow is known to approximate closely the Darcy-type flow observed in ideal porous media when the velocity at which the solute diffuses in the direction normal to the plates is much higher than the vertical advective velocity [27,28].\\nSince the pioneering work of Saffman and Taylor [29], the Hele-Shaw apparatus has been widely used to mimic the behaviour of two-dimensional Darcy flows. However, timedependent and highly-resolved measurements are still challenging in the Rayleigh-Taylor configuration. This is mainly due to the experimental limitations associated with the initialization of the system, which requires to start the flow from a flat horizontal interface between the fluids. The experimental measurements presented in this work have been performed taking care to achieve proper initial condition and minimizing the local perturbations of concentration and velocity fields, which eventually lead to mixing. At the same time, measurements have been performed with a high resolution in space and time, to ensure an accurate reconstruction of the concentration field, and in domains that are relatively large compared to previous literature results and extend the measurements range. The fluids and the cell have been designed to allow a direct comparison with previous numerical simulations [4,30]. We analyzed the evolution of the mixing length and the wavenumber power spectra (i.e., the finger size). Concerning the mixing length, we observe the same superlinear behaviour predicted by the simulations [4,5], and we provide an explanation for this finding in terms of finite size effects of the flow domain. With respect to the fingers evolution, we find again that the measured number of fingers is in very good agreement with that predicted numerically [30]. Overall, we are able to assess successfully the scaling relations provided by numerical simulations. We believe that this result can open new perspectives for future modelling and parameterization of convective flow in confined porous media. This paper is organised as follows. In Sec. II, we formulate the problem, describe the experimental setup and recall the assumptions underlying the referenced numerical simulations. The results are presented in Sec. III both in qualitative (flow phenomenology) and quantitative (mixing length and wavenumbers) terms. Finally, conclusions are discussed in Sec. IV.\"\n },\n {\n \"title\": \"II. Methodology\",\n \"content\": \"Our experiments were specifically designed to mimic the mixing process produced by the Rayleigh-Taylor instability in a two-dimensional, saturated and confined porous medium. In addition, they were carried out imposing a linear dependency of density and solute concentration, and ensuring the occurrence of a Darcy-type flow to allow direct comparison with previous numerical simulations [4,30].\"\n },\n {\n \"title\": \"A. Problem formulation\",\n \"content\": \"The process of convective dissolution is studied via the Rayleigh-Taylor instability, corresponding to two layers of fluid of different density, initially in an unstable configuration and subject to relative acceleration under the action of gravity [3]. The process is simulated in the frame of porous media flows, mimicked with the aid of a Hele-Shaw cell, i.e. two parallel and transparent plates of height H separated by a narrow gap b, as shown in Fig. 1(a). The cell is initially saturated with two miscible fluids having same viscosity (µ), arranged such that the heavy fluid (density ρ M ) lies on top of the lighter one (density ρ 0 ), as sketched in Fig. 1(b). The maximum density difference within the system is ∆ρ = ρ M − ρ 0 and is induced by the presence of a solute, namely potassium permanganate (KMnO 4 ). The amount of solute is quantified by its concentration C, which is maximum at the upper layer (C = C M ) and minimum at the lower layer (C = 0). The evolution of the system is controlled by the contributions of buoyancy, which tends to bring the fluids in a stable configuration and it is associated with the buoyancy velocity U , and diffusion, which acts to reduce local concentration gradients and to increase the mixing in the domain, and it is quantified by the diffusion coefficient D. The relative importance of these two contributions can be estimated by the Rayleigh-Darcy number, which represents the governing dimensionless parameter of the system, and is defined as:\\nwith:\\nthe buoyancy velocity and the length scale over which advection and diffusion balance, respectively [31]. The buoyancy velocity, is defined here as the combination of buoyancy (controlled by the acceleration due to gravity g, density contrast ∆ρ, and medium permeability k) and dissipation (controlled by fluid viscosity, µ). The length scale is particularly important when a comparison among systems having different Rayleigh numbers is required. Permeability k and porosity φ are the characteristic properties of the porous medium mimicked by the Hele-Shaw cell. In our experiments, k = b 2 /12 and φ = 1. In order to prevent potential three-dimensional effects we followed the indications emerged in the theoretical paper by Letelier et al. [27] and later confirmed experimentally by De Paoli et al. [28] and Alipour et al. [32]. We introduce the anisotropy ratio, which is an additional parameter describing the geometry of the cell, defined as:\\nand we can use this parameter in combination with the Rayleigh number to ensure that the flow will be of the Darcy-type if the condition 2 Ra → 0 is met. In our study, ≈ 10 −3 and 4.4 × 10 −3 < 2 Ra < 5.4 × 10 −2 , indicating that we are very close to the theoretical limit corresponding to the Darcy-type regime.\\nFinally, we are interested in examining a system with boundaries that are impermeable to both fluid and solute. Hence, the following conditions are imposed along the cell walls:\\nwith n the unit vector perpendicular to the boundary.\"\n },\n {\n \"title\": \"B. Experimental setup\",\n \"content\": \"We used a transparent Hele-Shaw cell (Polymethyl Methacrylate -PMMA -thickness 8 mm). The wall-normal depth of the fluid layer is constant for all the experiments performed and corresponds to the value b = 300 µm. An impermeable rubber (Klinger-sil C-4400, thickness 300 µm), is placed between the two acrylic plates, and is used to seal the region containing the fluid but also as a spacer between the cells surfaces. The sealing is laser-cut so that the domain investigated is a square of side H = 87 mm. On each side, the midpoints are connected to pipes that allow the cell filling/emptying process. The transparent sheets and the gasket are held in place by an outer metal frame and a set of 10 bolts, which are tightened with the same torque, to prevent the formation of a non-uniform film thickness. The experimental apparatus is represented in Fig. 1(a). Backlighting is provided to the cell by a tunable LED system (150 lamps covered by a diffusing glass). Finally, the evolution of the flow is recorded by a digital sCMOS camera (FlowSense USB 2M-165, Dantec Dynamics). The imaging system and settings will be described in detail in Sec. II C. Next, we describe the fluids adopted and the processes of concentration reconstruction.\\nThe initial condition of the experiments is an unstable configuration in which a layer of heavy fluid sits on top of a layer of lighter fluid, these layers being initially unmixed. To realize this challenging configuration, we employ a circuit made of a pump, valves and pipes, represented in Fig. 2. This approach has already been used in previous studies [see [33][34][35]. The entire system is initially filled with degassed water, to minimize the formation of bubbles. Then the cell is filled with the working fluids, which are driven by a peristaltic pump (Watson-Marlow 502 S) with the valves in \\\"loading mode\\\". In this configuration, the pump is able to inject degassed water and a homogeneous solution of KMnO 4 (concentration C M ) from two separated containers into the cell. Specifically, the solution and the water are injected from the top and bottom gates of the cell, indicated with (T) and (B) in Fig. 2. In this configuration, the valves connected to the left and right gates of the cell (L and R, in Fig. 2) are both open and allow the mixture of water and solution to leave the cell and be disposed in a residue container. Therefore, a flow from the top/bottom walls towards the left/right wall takes place, as indicated by the red and black arrows in Fig. 2. As a result, a sharp interface forms between the two liquid layers, while the vertical concentration gradient makes the fluid-fluid diffusive interface smoother. The competition between horizontal advection and vertical diffusion determines the interface thickness, which can be controlled by the flow rate (see Appendix C for further details on the interface thickness). A regulation valve is employed to balance the flow rate between the top (T) and bottom (B) gates to achieve a straight interface. When the initial condition is set, the valves are placed in \\\"experiment mode\\\", and the cell is by-passed by the inflow. The buoyancy-driven flow takes place and image acquisition starts.\\nThe two fluids are fully miscible but have different density. Specifically, we used water for the lower (i.e. lighter) fluid layer and an aqueous solution of potassium permanganate (KMnO 4 ) for the upper (i.e. denser) fluid layer. While water density is nearly constant among the experiments (it is only dependent on temperature), the density of the KMnO 4 solution can be varied by changing the solute concentration. We consider the coordinate system sketched in Fig. 1(b), where x and z are the spatial coordinates in horizontal and vertical direction, respectively. In this reference frame, the initial condition can be expressed in terms of the solute concentration as follows: C(x, z ≤ H/2, t = 0) = 0 (lower half of the cell) and [33][34][35]. A pumping system (peristaltic pump, Watson-Marlow 502 S) is used to inject water (density ρ 0 ) and solution (ρ M ) from two separate containers, through the gates B and T, respectively. To set up the initial condition, the valves are placed in \\\"loading mode\\\" and the exceeding solution is collected from gates L and R into a residue container. The flow through the gates, which is used to modulate the shape of the interface between the two fluid layers, is controlled by a regulation valve. Finally, when the initial condition is achieved, the system is set in \\\"experiment mode\\\" and the cell is by-passed by the fluids, which are channeled to the residue container.\\nconcentration of the KMnO 4 solution (upper fluid layer). We consider that the dynamic viscosity, µ = 9.2 × 10 −4 Pa·s, is constant and independent of the solute concentration [36]. Similarly, we assume that the diffusion coefficient is not sensibly affected by either the solute concentration or the local values of velocity D = 1.65 × 10 −9 m 2 /s. This value has been measured and reported in literature, and it is in excellent agreement with theoretical predictions based on electrical neutrality of a simple salt at infinite dilution [see 36, and references therein]. The working fluids have been chosen because of the linear dependency of the density that characterizes the resulting solution with respect to the solute concentration. This feature is essential to make reliable comparisons between experiments and simulations, which is one of the main objectives of the present work.\\nThe density of an aqueous KMnO 4 solution can be written as a function of the fluid temperature, ϑ, the water density at temperature ϑ, ρ 0 , and the KMnO 4 concentration, C [37]. Please note that also the water density can be computed from the temperature of the fluid, and therefore the parameters that need to be measured to determine the solution density are ϑ (which we assume to be uniform over the cell and constant during the experiment) and the local value of C (inferred from optical measurements). The density difference between the two fluid layers is ∆ρ = ρ M − ρ 0 , where ρ M = ρ(C = C M ) and C M are determined prior to each experiment. At a given temperature, the density of the mixture obtained from empirical correlations [37] is well described by a function of C and C 3/2 . However, within the range of mass fraction considered here, this correlation is well approximated by a linear function of the solute concentration (see Appendix A) as:\\nThis equation complies with the assumption of linear density-concentration dependency that is typically made in simulations, thus ensuring a reliable comparison. Measurements of the concentration field are inferred from the transmitted light intensities, this process being described in detail in the next section.\"\n },\n {\n \"title\": \"C. Experimental procedure and quantification of uncertainties\",\n \"content\": \"The results discussed in this paper are based on measurements of the time-dependent concentration field of the solute, C(x, z, t), inside the Hele-Shaw cell. To perform these measurements, we follow the method introduced by Slim et al. [36] and later used in a number of studies [28,32,38,39]. Optical measurements of light intensity distribution, I(x, z), are obtained with the aid of a sCMOS camera (FlowSense USB 2M-165, Dantec Dynamics, exposure time used 1 ms, resolution 2Mpx, frame rate 0.35 Hz to 5 Hz, depending on Ra). The measurements require a calibration process, briefly described in Appendix B, in which the value of light intensity is associated with the value of solute concentration. The greyscale images so obtained, an example of which is reported in Fig. 3(a), are preprocessed to reduce the effect of non-uniformities in the distribution of the light intensity over the cell, [40]). Finally, since the relationship between mass fraction and solute concentration is known (see Fig. 9(b) of Appendix B), the reconstruction of the concentration and density fields is performed. An example of this reconstruction is provided in Fig. 3(b).\\nAn important issue that must be assessed is the uncertainty on the measured value of the Rayleigh number, Ra, defined as in Eq. (1). The uncertainty value, δRa, is obtained from propagation-of-error analysis [41] and reads as:\\nwhere δ• indicates the uncertainty associated to the quantity • that represents each of the variables appearing in Eq. (1). We assume that the properties µ, D and g are estimated with constant relative uncertainty equal to 1%. The uncertainty on the domain height, δH, is assumed to be 0.5 mm, primarily due to a possible misplacement of the sealing on the cell rather than to the cutting precision. The accuracy on the gap thickness is assumed to be δb = 10µm. The uncertainties just introduced are common to all the experiments. In contrast, the uncertainty of the density difference δ∆ρ is a function of the Rayleigh number, being derived from Eq. (A1) as:\\nwhere δϑ is given by the instrument (0.5 • C). To use Eq. (7), the uncertainty on the concentration must be known. This is computed from Eq. (A5) as:\\nThe mass fraction, ω, defined as the ratio between the mass of solute and the mass of solution, is determined measuring first the mass of water (m w ) and then the overall mass of solution (m sol ), i.e. ω = (m sol − m w )/m sol . As a result, the uncertainty δω reads as:\\nwith δm w = δm sol = 0.001 g (high-precision scale, Sartorius Acculab Atilon model ATL-423-I, ±0.001 g). The list of the solution properties, flow parameters and corresponding uncertainties is reported in Tab. I.\"\n },\n {\n \"title\": \"D. Numerical simulations and dimensionless variables\",\n \"content\": \"To assess the mixing layer scaling laws, we wish to make a direct comparison of the present experimental results with direct numerical simulations of convection in two-dimensional porous media [4,30]. We perform this comparison in the range 347 ≤ Ra ≤ 19953, for which a very good overlap of the governing parameters exists between experiments and simulations. To make the paper self-contained, we recall next the governing equations and the numerical details.\\nAn isotropic and homogeneous vertical porous slab with permeability k and porosity φ is considered. The porous domain is initially saturated with two miscible fluids having same viscosity (µ) but different density, arranged such that the heavy fluid (density ρ M ) tops the\\n24.0 ± 0. lighter one (density ρ 0 ). This unstable configuration replicates exactly the one sketched in Fig. 1(b). As done in the experiments, the density difference is induced by the presence of a solute, with maximum solute concentration C = C M in the upper layer and minimum solute concentration C = 0 in the lower layer. The flow is assumed to be incompressible and controlled by the Darcy's law:\\nwhere u and w are the horizontal (x) and vertical (z) velocity components, while p and ρ are the local pressure and density, respectively. We consider that the Oberbeck-Boussinesq approximation applies: This is a reasonable assumption for geophysical problems such as geological carbon dioxide sequestration [42]. In addition, we consider that the density of the mixture is a linear function of the solute concentration C [see Eq. (5)], and that the following transport equation for C holds:\\nwith t the time and D the solute diffusivity, which we assume to be constant.\\nTo make Eqs. (10)-(12) dimensionless, we use the buoyancy velocity U and the length scale , defined as in Eq. (2). In addition, the following dimensionless variables are introduced [31,43,44]:\\nThe resulting governing equations in dimensionless form read as:\\nwhere the superscript * indicates dimensionless variables, P * = p * + z * (ρ M /∆ρ − 1) is the reduced pressure and ∆ρ is the density difference between the upper and the lower fluid layers.\\nThe controlling parameter of the present system is the Rayleigh-Darcy number Ra, defined as in Eq. (1). Unlike the Hele-Shaw experiments, in which the geometry of the cell is characterised by the anisotropy ratio, two-dimensional Darcy simulations do not need additional dimensionless parameters: The Rayleigh-Darcy number describes completely the fluid properties (∆ρ, µ, D), the porous medium properties (k, φ) and the domain size (H).\\nAlthough not explicitly appearing in the governing Eqs. (14)- (16), Ra enters the picture as the dimensionless height of the domain. Impermeable boundary conditions (i.e. no-flux) are imposed for both the fluid and the solute at the top and bottom horizontal boundaries, whereas periodicity is applied at the side boundaries (along x). In dimensionless form, these boundary conditions read as:\\nThe set of equations (14)- (16) has been discretized numerically using a Chebyshev-Tau method. We refer to [4,30,43,45] for further details on the flow solver.\"\n },\n {\n \"title\": \"III. Results\",\n \"content\": \"The comparison of the experimental results against numerical simulations is focused on the evolution of the mixing length and on the analysis of the wavenumber power spectra. These spectra are obtained by making a Fourier transform of the instantaneous concentration field and provide a robust characterization of the finger size, as this size can be associated to the wavenumber at which the spectra exhibit a peak. We consider a wide range of Rayleigh numbers: 4.39 × 10 3 ≤ Ra ≤ 5.43 × 10 4 . We refer to Tab. I for the experimental mass fraction and density difference values used to obtain these Ra. Above the upper value Ra max = 5.43 × 10 4 , the density difference between the KMnO 4 -rich solution (heavy fluid) and water (light fluid) induces large vertical velocities, and the flow is affected by non-Darcy effects [27,28]. For lower values of ∆ρ, in contrast, the initial condition is hard to achieve and the duration of the experiments increases, leading to higher chances of inducing density perturbations in the cell from the side and horizontal channels (L, R, T, B in Fig.1). In the following, we describe the experimental results in terms of flow phenomenology (Sec. III A), time evolution of the mixing length (Sec. III B) and wavenumbers (Sec. III C), comparing visa-vis experimental observations against numerical simulations [4,30]. The datasets analyzed in the present study have been deposited in [46]. [4]. The dimensionless time instants (t * ) at which the fields refer are indicated on top of each panel. We remark that the experiments are realised in a square geometry, whereas the simulations are performed in wider domains (aspect ratio π/2). Therefore, for better comparison, only a portion of the numerical domain having unitary aspect ratio is reported. The desired initial position of the interface (located at z = H/2, i.e. z * = Ra /2) is also shown as a dashed black line. See also the movie, Movie S1, in the Supplemental Material [47] for a comparison of the time-dependent evolution of the concentration fields in the experiment and in the simulation.\"\n },\n {\n \"title\": \"A. Flow phenomenology\",\n \"content\": \"To analyze in detail the flow phenomenology, we report in Fig. 4 the time-dependent dimensionless concentration field (C * ) obtained experimentally (4a-d) and numerically [4] (4e-h). The experiment (E4 in Tab. I, Ra = 19789) is compared against numerical simulations performed at similar Rayleigh number, Ra = 19953: We consider this value because it is large enough to observe all the main phases of the mixing process for a significant amount of time. On top of each panel, the dimensionless time instants (t * ) at which the fields are taken are provided.\\nThe concentration field is initially in an unstable configuration (Fig. 4a,e) with the heavy fluid characterised by the dimensionless concentration C * = 1 lying on top of the lighter one (C * = 0). The initial position of the interface, indicated here with the dashed line at z * = Ra /2, is initially flat and horizontal. Achieving such a configuration in the experiment is very difficult, and some discrepancy, though not apparent, always exists between the actual fluid-fluid interface and the desired interface. This is visible for instance in Fig. 4(a), where the upper fluid layer seems to be slightly inclined with respect to the horizontal dashed line. In the experiments, the flow is initially triggered by perturbations of the local density and velocity fields. Indeed, to set the initial flow configuration and generate the fluid-fluid interface, some fluid is removed from the side channels and simultaneously injected from the channels at the horizontal boundaries (see Sec. II C for a thorough description of this process). As a result, the fluid will not be initially still and will be partially affected by the flow circulations induced by the procedure adopted to set the initial condition. The situation is completely controllable in the simulations, where the interface starts as perfectly flat and is made unstable by adding a perturbation directly to the concentration field (Fig. 4e). The amplitude of the perturbation considered, however, is crucial in determining the onset of the instability [30], whereas the late stage dynamics are not sensitive to this parameter. In the present study, we considered simulations in which the perturbations applied to the dimensionless concentration field have amplitude 10 −3 .\\nThe flow phenomenology associated with the mixing process can be summarised as follows. The mechanism of diffusion is initially responsible for the flow evolution (Fig. 4a,e). Afterwards, the interfacial layer thickens and, due to the unstable density profile, finger-like structures form. These fingers are controlled by the interplay of convection, which makes them grow vertically, and diffusion, which reduces the horizontal concentration gradients and promotes mixing [35]. During this intermediate stage (Fig. 4b-c,f-g), the fingers grow symmetrically with respect to the domain centerline, and eventually merge. Indeed, we observe that the number of fingers reduces from t * = 0.68 × 10 4 (Fig. 4b,f) to t * = 1.57 × 10 4 (Fig. 4c,g). We anticipate here that this dynamics is quantitatively confirmed by the wavenumber measurements reported in Sec. III C. At a later stage (Fig. 4d,h), the number of fingers does not increase further. Eventually, the strength of convection diminishes and the flow evolution is controlled again by diffusion. In this phase (not shown here), the system has achieved a stable configuration in which the average concentration in the lower layer is higher than in the upper layer [4]. This regime, usually referred to as shutdown of convection in view of the reduced strength of the driving forces [43,48], starts after the tips of the fingers reach the horizontal walls and the effect of the boundaries propagates back to the domain centerline. The time required to reach the shutdown is large (e.g., of the order of 1 hour for the lower Ra considered here), and hence hard to achieve experimentally because the flow conditions have to be carefully controlled for a considerable amount of time.\\nIt is apparent that the flow evolution observed in the experiments is qualitatively similar to that of numerical simulations, albeit with some differences. The growth of the fingers, expressed in terms of their tip-to-rear extension, is similar within the same time interval (Fig. 4b,f and Fig. 4c,g). We also refer the reader to the movie, Movie S1, in the Supplemental Material [47] for a comparison of the time-dependent evolution of the concentration fields in experiments and simulations. However, we also observe a different number of fingers in the two cases (see Sec. III C for further discussion).\"\n },\n {\n \"title\": \"B. Mixing length\",\n \"content\": \"The evolution of a Rayleigh-Taylor system can be conveniently analysed by looking at the instantaneous vertical extension of the fluid-fluid interface [3,49]. This quantity, defined as mixing length h, can be also interpreted as the distance between the upwarding and downwarding tips of the fingers. To compute the mixing length in a quantitative way, one can use either global [50] (i.e. integral) or local [35,51] flow properties. In this work, we characterize the mixing length by means of the local concentration field [4,51], i.e. h is the portion of the domain where the horizontally-averaged concentration field satisfies the condition ε < C(z, t) < 1 − ε, with:\\nthe concentration field averaged along the horizontal direction x, ε = 3 × 10 −2 a threshold coefficient, and H the domain width (in x direction). It follows from this definition that 0 ≤ h ≤ H or, in dimensionless terms using the length scale defined in Eq. (2), 0 ≤ h * ≤ Ra.\\nThe time evolution of the dimensionless mixing length for all the experiments we performed is shown in Fig. 5. As previously observed in simulations in confined domains [4,30], the dynamics consists essentially of three phases in sequence: (i) diffusion-dominated, (ii) convection dominated and (iii) wall-induced shutdown. During the initial phase, the thickness of the mixing layer is initially controlled by diffusion. Assuming an initial stepwise concentration profile, the dimensionless concentration field evolves as:\\nand the corresponding mixing length as:\\n(we refer to [4] for a detailed derivation). The same evolution of the mixing length in the initial diffusive regime has been observed in previous numerical works, both in porous media convection [4,35] and in pure fluids with different initial conditions [52]. Despite the seemingly universal behaviour exhibited by the mixing length, we observe that the initial diffusive stage is hardly captured by the experimental measurements. Only the measurements performed at lower Ra, namely 4407 and 7023, are in good agreement with the theoretical predictions (blue dashed line in Fig. 5). While the trend nicely follows the predicted behaviour, namely h * ∼ √ t * , the magnitude slightly differs from the solution derived in Eq. (20). This is due to uncertainty of the experimental measurements, which is comparable to the values of h * at low Ra and early times. When the fluid-fluid interface is sufficiently thick (10 3 < t < 10 4 ), it becomes unstable and eventually finger-like structures form. When observed via the mixing length, the transition to this convection-dominated regime is smooth, similarly to what has been observed in the simulations. However, we find that the transition in experiments occurs at different instants depending on the value of Ra, as a result of the uncertainty on the initial perturbation of the concentration field, which cannot be reproduced faithfully in the laboratory. We also observe that, during the transition, the growth of the mixing length modelled as a power law, i.e. h * ∼ (t * ) α , is characterized by an exponent that is higher than that observed asymptotically, in agreement with previous numerical [4,35] and experimental (α = 2) [53] findings.\\nThe fluid mixing is more efficient when convection takes place. The formation of fingerlike structures marks the beginning of the convection-dominated phase, more persistent in time compared to the initial diffusive stage. In this phase, fingers grow, merge and finally reach the horizontal boundaries of the cell. The measurements reported in this work reveal that the mixing length grows more than linearly in time, in close agreement with previous simulations (namely, h * ∼ (t * ) 1.208 , [4,5]). This scaling is anomalous: It is not expected in two-dimensional porous media convection, where a linear evolution of the mixing length with time is predicted as a result of the balance between buoyancy (i.e., convection) and friction (i.e. dissipation due to viscosity). The occurrence of a linear growth would imply an asymptotic velocity of the finger tips, which determines also the growth rate of the mixing length, equal to the buoyancy velocity U , defined as in Eq. (2). We remind the reader that U can be interpreted as the velocity at which the contribution of buoyancy (g∆ρk) balances the dissipation (µ).\\nIn three-dimensional porous media, the growth of the mixing length follows the linear prediction obtained from the physical arguments mentioned above [5]. The behaviour is different in two dimensions, where the fingers are slenderer compared to their three-dimensional counterpart. The superlinear evolution of the mixing length we observed, can be ascribed to multiple reasons: (i) the simulated time is not sufficiently large to achieve the asymptotic regime, or (ii) the Ra number of the system is too low, and the effect of diffusion is still dominant. We speculate that in the present case a combination of these two effects controls the evolution of the flow. Initially, the fluid accelerates from zero velocity, and this corresponds to a finite-time effect. At a later stage, when the initial condition has been absorbed by the system and fluid velocity is non-zero, the finite-size (i.e., finite-Ra) effect comes into play, via the domain boundaries. This non-linear behaviour seems indeed to vanish at higher Ra numbers [6], but in absence of boundaries. We remark here that a linear growth of the mixing length has been observed numerically [35], but only in the limit of lower Rayleigh numbers, namely Ra ≈ 4 × 10 3 . For these values of Ra, also previous numerical simulations [4] and present experiments do not exhibit a clear superlinear behaviour. In addition, the experimental data reported by Wooding [53], which refer to higher Rayleigh numbers (approximately 10 3 ≤ Ra ≤ 7 × 10 4 ) are also well fitted by a linear scaling. However, it must be pointed out that the data have been collected at very low acquisition rate, making the amount of available measurements very scarce for a time-dependent flow analysis. We are not able to explain, in this case, why the superlinear scaling was not observed. One possible reason could be attributed to the method adopted to change the Rayleigh number, consisting of varying the orientation of the cell with respect to gravity. Another possibility is represented by the absence of horizontal boundaries. A conclusive answer on the asymptotic scaling of the mixing length could be obtained with the aid of Darcy simulations or experiments at higher Ra, (≈ 10 6 ), which are beyond the present capabilities.\\nTo conclude the discussion on the mixing length, we now examine the wall-induced convective shutdown phase [4], which sets in when the high (resp. low) concentration fingers reach the lower (resp. upper) horizontal boundary. In this phase, the local concentration gradients decrease, weakening the density differences that drive the flow within the domain. As a result, convection vanishes and the overall mixing rate, identified for instance by means of the mean scalar dissipation rate, decreases dramatically [30]. This effect is crucial for applications of practical interest (e.g., carbon dioxide sequestration in saline aquifers), in which the fluids are confined by impermeable boundaries that play a key role in the solute mixing process. A simple way to quantify the convective shutdown is to measure the time t * s required for the fingers to reach the horizontal boundaries. The values of t * s obtained as a function of the Rayleigh number are shown in Fig. 6. Experimental and numerical [4] results are reported as open (•) and filled symbols ( ), respectively. The best-fitting power law for present results (red dashed line) is 9.64 ×Ra 0.80 , in excellent agreement with numerical results (14.98 ×Ra 0.78 , black dashed line). This is not surprising, given the close correspondence observed for the evolution of the mixing length. In leading order, we can estimate t * s = t * : h * (t * ) = Ra which gives t * s ∼ Ra 1/1.2 ∼ Ra 0.83 , consistent with the Ra 0.80 scaling provided by data fitting.\\nDespite the excellent agreement of the scaling exponents, we also observe some discrepancies between simulations and experiments. In particular, for a given Ra, the value of t * s is lower in the experiments, leading to a lower value of the multiplicative coefficient obtained from data fitting. We attribute this discrepancy to the perturbation of the interfacial concentration field. In the experiments, the amplitude of this perturbation is hard to control, and it is likely to be larger compared to the numerical simulations considered, leading to an earlier onset of convective instabilities. Another limitation of our experimental setup is represented by the range of Rayleigh numbers that can be investigated. To be able to achieve a Darcy flow and to control effectively the boundary conditions, the values of domain size (H * ) and the density difference (∆ρ) cannot be too large (see also Sec. II B). The values of Ra considered here are sufficiently large to overlap with and go beyond (up to 2.5 times, approximately) the largest Rayleigh numbers simulated numerically. On the other hand, achieving smaller values of Ra is also challenging. Lower values of ∆ρ correspond to lower velocities, and therefore longer experiments. Smaller values of H * , i.e. shorter domains, would instead required more careful setup of the initial flow configuration.\\nThe findings discussed in this section prove that, in terms of evolution of the mixing length, two-dimensional Darcy simulations and Hele-Shaw experiments produce similar results when the sameRa is considered. However, the ability of the experimental measurements to capture the initial development of the flow is lower due to the setup used, and numerical simulations should be considered as more reliable to investigate this stage of the flow evolution.\"\n },\n {\n \"title\": \"C. Wavenumbers\",\n \"content\": \"The flow dynamics is complex and characterised by a continuous change of the fluid-fluid interface position. To provide a quantitative estimate of the flow topology evolution, we analyze here the number of fingers and the wavenumbers at the centerline of the domain (z * = H * /2). The process of flow analysis is illustrated in Fig. 7, where an experiment at Ra = 5.43 × 10 4 is considered for reference. First, we build the space time map at the centerline, where the interface is initially located, i.e. we report in Fig. 7(a) the contour of the time-dependent dimensionless concentration field, C * (x * , z * = Ra /2, t * ). The simplest way to define the number of structures in the system is to count the upwarding and downwarding fingers. To this aim, the spacetime map is first binarised [ C(x * , z * = Ra /2, t * ), Fig. 7(b)] with respect to a threshold value corresponding to the mean concentration (C * = 1/2), and then the number of fingers is counted [ Fig. 7(c)]. The process is iterated during the entire flow evolution, and the number of fingers, N (t), is computed as the number of regions with different binarized concentration values [ Fig. 7(d), black line]. Due to noise in the experimental measurements, spurious fluctuations of the concentration field may occur. To avoid counting these as fingers, we discarded contiguous regions of the binarized map that are smaller than 4 pixel (corresponding to a physical length of 350 µm). We observe in Fig. 7(d) that this threshold is extremely sensitive to variations of the concentration field, i.e. it appears to fluctuate without an apparent physical reason. A more robust characterization of the flow topology is provided by the wavenumbers associated with the power spectra of the concentration field. We employ a Fourier decomposition of the concentration profile at the cell centerline to estimate the number of fingers, and compute the power-averaged mean Measurements of the number of fingers over time for Ra = 5.43 × 10 4 (E6 in Tab. I). Space-time map, consisting of the evolution of the concentration field along the centerline, C * (x * , z * = Ra /2, t * ), is shown in (a). To compute the finger number, the space-time map is binarized, C(x * , z * = Ra /2, t * ), choosing C * = 1/2 as threshold (b). The concentration profile (red line) and the discretized profile (black line) measured along the centerline at t * ≈ 1.1 × 10 5 (indicated by the dashed blue line in panels a, b and d) are also reported as a function of the horizontal coordinate x * (c). Finally (panel d), the time-dependent finger number is computed as from the binarized field C (black line), from power-averaged mean wavenumber as defined in Eq. (21) ( N = L k /π, red line) and from a k peak (green line), i.e. the wavenumber maximizing the spectrum amplitude of C. See also the movie, Movie S2, in the Supplemental Material [54] for the time-dependent evolution of the wavenumber and the corresponding concentration field. wavenumber at each time instant, defined as [51]:\\nwith k n the wavenumber and |C n (t * )| 2 the squared amplitude of the n-th basic harmonic function obtained from the Fourier decomposition of the fluctuation of C * (x * , z * = 1/2, t * ) about its mean value. We observe that in this case [red line in Fig. 7(d)] the evolution of the number of fingers, N = L k /π, is smoother compared to the simple finger count, N , and also compared to the peak number, defined as N peak = Lk peak /π [orange line in Fig. 7\\nwhere k peak is the wavenumber value for which |C n (t * )| is maximum. Therefore, in the following, we will focus the analysis considering the power-averaged mean finger number N only, which appears similar to but less fluctuating of the finger number N . A comparison of the number of fingers measured in the experiments is reported in the inset of Fig. 8(a). In general, all the experiments are characterized by a similar dynamics. The number of fingers, initially low, increases up to a maximum value. Afterwards, the fingers start to merge and increase in size. This phase, which is also visible from the spacetime maps in Fig. 7(a-b), may start earlier at low Ra and later at high Ra, spanning a relatively long interval (10 3 ≤ t * ≤ 10 4 ) in Fig. 8(a). During growth, N appears to decrease in time at the same rate for all Ra considered. This universal behaviour is more apparent when the results are compared in terms of mean wavenumber k (main panel of Fig. 8a). After entering the convection-dominated regime, all curves exhibit the same trend. A first observation is that the merging rate of the fingers is in agreement with the scaling k ∼ (t * ) −0.6 reported in numerical simulations [30]. In contrast, the behaviour is different during the diffusive phase, when the wavenumber is clearly dependent on Ra.\\nIn Fig. 8(b), we compare present results (solid lines) against previous numerical findings [30] (dashed lines). For ease of reading, only three values of Ra number are considered. Two differences are apparent: (i) unlike experiments, simulations [30] have been performed on domains of constant dimensionless width (namely, 19953), and (ii) to make reliable comparisons among different Rayleigh numbers, the same initial perturbation has been employed, thus making the early stage flow evolution also universal and independent of Ra. This highlights again a possible limitation of the experimental procedure employed, which makes it hard to precisely control the initial flow conditions. As a result, the initial phase is characterised by large discrepancies between numerical and experimental results (Fig. 8b). In the late stage of the flow dynamics, numerical and experimental measurements are in fair agreement. As a side note, we remark that present results, shown in terms of wavenumbers associated with the concentration field, are compare against the simulations of De Paoli et al. [30], where the power spectra where obtained from the scalar dissipation fields. However, even the experimentally-measured merging rate is well described by the same scaling law, t −0.6 . Present results are in fair agreement with previous experimental findings [35,53], in which the wavenumber is observed to evolve according to (t * ) −1/2 . This scaling has been also derived analytically [31,53] assuming that the fingers are slender, so that vertical derivatives (e.g., diffusion of solute) are negligible compared to the horizontal ones. In the frame of this model, the flow is driven by vertical advection, whereas lateral diffusion acts as dissipative mechanism. We refer to Slim [31] for a detailed derivation of this result. The slight differ-ence in the scaling exponent, compared to the analytical prediction, maybe attributed to the simplified assumptions considered in the model derivation. The scaling k ∼ (t * ) −1/2 has been also suggested by Gopalakrishnan et al. [35], who interpreted this evolution as a diffusion-dominated finger growth, by analyzing the relative contribution of convective and diffusive terms within a finger (but away from the finger tips).\"\n },\n {\n \"title\": \"IV. Conclusions\",\n \"content\": \"In this work, we analyzed with the aid of high-resolution experiments the Rayleigh-Taylor instability in a Hele-Shaw flow. The system consists of two miscible fluid layers, namely water and an aqueous solution of KMnO 4 , having different density but same viscosity, placed in an initial unstable configuration. The density difference is induced by the presence of the solute (KMnO4), which produces also a colour gradient by which we are able to infer the concentration field. The system is fully described by the Rayleigh number, Ra, which measures the relative strength of convective and dissipative contributions. Present experimental results have been performed taking great care to achieve a proper initial condition and minimizing the local perturbations of the concentration and velocity fields, which trigger the growth of interfacial finger-like instabilities. In addition, a high resolution in space and time has been achieved by considering relatively large domains, and an accurate reconstruction of the concentration field is performed. In this respect, present experimental results go beyond previous measurements published in archival literature.\\nWe compared present findings against previous numerical measurements in two-dimensional and confined media to which the Darcy equations can be successfully applied, as for instance porous media. [4,30]. The dynamics of the flow is first controlled by diffusion, which is responsible for the thickening of the interface, initially flat and horizontal. Afterwards, the flow evolution is controlled by convection. The unstable fluid-fluid density front promotes the formation of high-wavenumber instabilities, which grow and merge into large and more persistent fingers. Finally, due to the combined action of convective mixing in a confined domain and lateral diffusion across the fingers interface, the solute distribution becomes uniform within the porous slab, with a corresponding stable concentration profile. During this stage, referred to as wall-induced convective shutdown, the flow is again controlled by diffusion. The flow evolution is quantified and compared against theoretical and numerical predictions with the aid of two main observables: the mixing length, h * (t), and the power-averaged mean wavenumber, k .\\nInitially, the evolution of the mixing length is in fair agreement with the theoretical predictions obtained for flows in the absence of convection (h * (t) ∼ (t * ) 1/2 ). In the intermediate regime, in which convection dominates, h * (t) is observed to grow with a superlinear scaling, h * (t) ∼ (t * ) 1.2 , confirming previous numerical findings [4,5]. This scaling is anomalous, as it differs from the theoretical prediction, h * (t) ∼ t * , possibly as a result of a finite size (i.e., finite Ra) effects. To further compare present experimental findings against previous numerical simulations, we compute the time t * s taken for the fingers to reach the horizontal boundaries of the domain. Also in this case, experiments (t * s ∼ Ra 0.80 ) are in excellent agreement with the simulations (t * s ∼ Ra 0.78 ). Finally, we analysed the flow dynamics by looking at the finger distribution at the domain centerline. The evolution of the characteristic finger size has been quantified by means of the power-averaged mean wavenumber, k . During the convection-dominated phase, the centerline finger dynamics exhibits a universal behaviour that is independent of Ra. Also in A ADDITIONAL DETAILS ON FLUID PROPERTIES this case, we found an excellent agreement with scaling laws derived in numerical simulations [30], namely k ∼ (t * ) −0.6 .\\nThe analysis carried out in this paper shows that Hele-Shaw experiments can faithfully reproduce the dynamics of convective Rayleigh-Taylor instabilities in two-dimensional and confined porous media. However, it also shows that the proposed experimental approach presents some important limitations, which call for future improvements. First off, the desired initial fluid-fluid interface consists of a thin, flat and horizontal contact line, which is experimentally challenging to achieve. In addition, the instability is initially triggered by local perturbations of the concentration and velocity fields, and hence hard to control. Finally, experiments at low Ra are characterised by low values of vertical velocity, which slow down the mixing process and make the experiments to last longer. As a result, the experimental conditions have to be monitored for longer times, increasing the difficulty of the experimental procedure.\\nFIG. 9. Given the definition (A5) and the empirical correlations (A1) [37], the relative dependence of density of the solution, ρ, mass fraction, ω, and solute concentration, C, is obtained. We report here C(ω) (panel a), ρ(ω) − ρ 0 (panel b) and ρ(C) − ρ 0 (panel c), being ρ 0 the water density. The profiles shown here correspond to ϑ = 25 • C. We observe that the empirical values (symbols) are very well fitted by linear functions (solid lines). In the instance of density and concentration (panel c), the linear function corresponds to Eq. (5).\\nby a linear function of the solute concentration [Eq. (5)], as can be seen in Fig. 9c).\"\n },\n {\n \"title\": \"B. Measurements calibration\",\n \"content\": \"When the two-dimensional distribution of light intensities is determined, each value of light intensity is assigned a value of mass fraction, ω(x, z). This step is achieved with the aid of calibration curves, an example of which is reported in Fig. 10. For a given illumination condition, the cell is filled with solution with a known value of mass fraction, ω, and the average light intensity over the cell, I cal , is recorded. The higher the concentration of KMnO 4 , the lower the transmitted light intensity [Lambert-Beer law, 55]. Experimental measurements (symbols) are well fitted by a combination of exponential functions, ω( I cal ) = a 0 + a 1 exp(b 1 I cal ) + a 2 exp(b 2 I cal ) (solid line). Further details on the process of calibration are available in [28,32,39].\"\n },\n {\n \"title\": \"C. Additional details on the initial condition\",\n \"content\": \"Large values of solute concentration (C M ) produce strong concentration gradients across the initial fluid-fluid interface, making a sharp separation between the two fluids hard to obtain. Therefore, to achieve a well-defined initial interface up to the Ra values considered here, the flow rate provided to the cell is increased withRa. In Fig. 11, the initial condition is shown in detail for the experiments E1 and E4 (namely at Ra = 4391 and Ra = 19789), and for both the unprocessed light intensity field, I, (Fig. 11a,d) and the inferred dimensionless concentration field, C * (Figs. 11b,e). We observe that the experimental procedure employed is suitable to obtain a macroscopic sharp and straight interface in correspondence of the domain centerline, represented by the dashed red line in Figs. 11(a,b,d,e). Further information on the local interface shape can be obtained from the close up view of the interfacial region, shown in Figs. 11(c,f), where the iso-contours at C * = 0.3, C * = 0.5 and C * = 0.7 (same\"\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 {\n \"title\": \"FIG. 8 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"(a) Experimental setup. The Hele-Shaw cell is shown, with explicit indication of the position of the connection gates (top, bottom, left and right, indicated as T, B, L, and R respectively), sCMOS camera and backlight illumination. Fluids are supplied to the cell through a system (sketched in Fig. 2) consisting of a pump, pipes and valves. The cell is aligned in the vertical direction, namely parallel to the acceleration due to gravity, g. (b) Sketch of the experimental domain with explicit indication of the boundary conditions (no flux of mass or solute through the walls). The reference frame (x, z) as well as the initial position of the interface (red dashed line) are indicated, with the heavy fluid (density ρ M , concentration C = C M ) initially lying on top of the lighter fluid (ρ 0 , C = 0). The background field consists of one of the images collected, and provides a qualitative picture of the flow for the present physical configuration.","content":"FIG. 1 .\n1(a) Experimental setup. The Hele-Shaw cell is shown, with explicit indication of the position of the connection gates (top, bottom, left and right, indicated as T, B, L, and R respectively), sCMOS camera and backlight illumination. Fluids are supplied to the cell through a system (sketched in Fig. 2) consisting of a pump, pipes and valves. The cell is aligned in the vertical direction, namely parallel to the acceleration due to gravity, g. (b) Sketch of the experimental domain with explicit indication of the boundary conditions (no flux of mass or solute through the walls). The reference frame (x, z) as well as the initial position of the interface (red dashed line) are indicated, with the heavy fluid (density ρ M , concentration C = C M ) initially lying on top of the lighter fluid (ρ 0 , C = 0). The background field consists of one of the images collected, and provides a qualitative picture of the flow for the present physical configuration."},{"id":"","caption":"Circuital scheme of the experimental setup employed to obtain the desired initial conditions","content":"FIG. 2 .\n2Circuital scheme of the experimental setup employed to obtain the desired initial conditions"},{"id":"","caption":"Reconstruction of the concentration field. (a) Raw image consists of a light intensity distribution, I. (b) Upon processing to remove light intensity noises, scratches and non-uniformities of backlight illumination (see [40] for further details), the dimensionless concentration field is obtained. To visualise the interface between the two fluids, corresponding to the fingers, three iso-contours of normalised concentration are shown, corresponding to C * = C/C M = 0.3 (blue), 0.5 (red) and 0.7 (green).","content":"FIG. 3 .\n3Reconstruction of the concentration field. (a) Raw image consists of a light intensity distribution, I. (b) Upon processing to remove light intensity noises, scratches and non-uniformities of backlight illumination (see [40] for further details), the dimensionless concentration field is obtained. To visualise the interface between the two fluids, corresponding to the fingers, three iso-contours of normalised concentration are shown, corresponding to C * = C/C M = 0.3 (blue), 0.5 (red) and 0.7 (green)."},{"id":"","caption":"Snapshots of the dimensionless concentration field obtained experimentally (a-d) and numerically (e-h). The experiment (E4 in Tab. I,Ra = 19789) is compared vis-á-vis against numerical simulations performed at Ra = 19953 by De Paoli et al.","content":"FIG. 4 .\n4Snapshots of the dimensionless concentration field obtained experimentally (a-d) and numerically (e-h). The experiment (E4 in Tab. I,Ra = 19789) is compared vis-á-vis against numerical simulations performed at Ra = 19953 by De Paoli et al."},{"id":"","caption":"Time evolution of the dimensionless mixing length, h * (t * ) (solid lines), for all the values of the global Rayleigh number considered in this study (and explicitly indicated next to each curve). The evolution of the system, initially controlled by diffusion, nicely follows the analytical solution, h * ≈ (1 − 2ε) √ 4πt * [Eq. (20), dashed blue line]. The intermediate phase, dominated by convection, exhibits the anomalous scaling h * ∼ (t * ) 1.2 first observed by De Paoli et al. [4].","content":"FIG. 5 .\n5Time evolution of the dimensionless mixing length, h * (t * ) (solid lines), for all the values of the global Rayleigh number considered in this study (and explicitly indicated next to each curve). The evolution of the system, initially controlled by diffusion, nicely follows the analytical solution, h * ≈ (1 − 2ε) √ 4πt * [Eq. (20), dashed blue line]. The intermediate phase, dominated by convection, exhibits the anomalous scaling h * ∼ (t * ) 1.2 first observed by De Paoli et al. [4]."},{"id":"","caption":"Time taken by the fingers to reach the horizontal boundaries (t * s ) as a function of Ra. Experimental (present work) and numerical [4] results are reported as open (•) and filled symbols ( ), respectively. The best-fitting power law for present results (red dashed line) is 9.64 ×Ra 0.80 , in excellent agreement with numerical findings (14.98 ×Ra 0.78 , black dashed line). Experimental results correspond to the mean values obtained for several flow realisations at fixed Ra.","content":"FIG. 6 .\n6Time taken by the fingers to reach the horizontal boundaries (t * s ) as a function of Ra. Experimental (present work) and numerical [4] results are reported as open (•) and filled symbols ( ), respectively. The best-fitting power law for present results (red dashed line) is 9.64 ×Ra 0.80 , in excellent agreement with numerical findings (14.98 ×Ra 0.78 , black dashed line). Experimental results correspond to the mean values obtained for several flow realisations at fixed Ra."},{"id":"","caption":"FIG. 7. Measurements of the number of fingers over time for Ra = 5.43 × 10 4 (E6 in Tab. I). Space-time map, consisting of the evolution of the concentration field along the centerline, C * (x * , z * = Ra /2, t * ), is shown in (a). To compute the finger number, the space-time map is binarized, C(x * , z * = Ra /2, t * ), choosing C * = 1/2 as threshold (b). The concentration profile (red line) and the discretized profile (black line) measured along the centerline at t * ≈ 1.1 × 10 5 (indicated by the dashed blue line in panels a, b and d) are also reported as a function of the horizontal coordinate x * (c). Finally (panel d), the time-dependent finger number is computed as from the binarized field C (black line), from power-averaged mean wavenumber as defined in Eq. (21) ( N = L k /π, red line) and from a k peak (green line), i.e. the wavenumber maximizing the spectrum amplitude of C. See also the movie, Movie S2, in the Supplemental Material [54] for the time-dependent evolution of the wavenumber and the corresponding concentration field.","content":"FIG. 7. Measurements of the number of fingers over time for Ra = 5.43 × 10 4 (E6 in Tab. I). Space-time map, consisting of the evolution of the concentration field along the centerline, C * (x * , z * = Ra /2, t * ), is shown in (a). To compute the finger number, the space-time map is binarized, C(x * , z * = Ra /2, t * ), choosing C * = 1/2 as threshold (b). The concentration profile (red line) and the discretized profile (black line) measured along the centerline at t * ≈ 1.1 × 10 5 (indicated by the dashed blue line in panels a, b and d) are also reported as a function of the horizontal coordinate x * (c). Finally (panel d), the time-dependent finger number is computed as from the binarized field C (black line), from power-averaged mean wavenumber as defined in Eq. (21) ( N = L k /π, red line) and from a k peak (green line), i.e. the wavenumber maximizing the spectrum amplitude of C. See also the movie, Movie S2, in the Supplemental Material [54] for the time-dependent evolution of the wavenumber and the corresponding concentration field."},{"id":"","caption":"(a) Experimental measurements. The evolution of the power-averaged mean wavenumber, k (defined in Eq. (21), main panel), and the corresponding finger number N (inset). (b) Comparison of experimental (solid lines) and numerical ([30], dashed lines) measurements. For ease of reading, three values of Rayleigh number are used to compare the behaviour of k .","content":"FIG. 8 .\n8(a) Experimental measurements. The evolution of the power-averaged mean wavenumber, k (defined in Eq. (21), main panel), and the corresponding finger number N (inset). (b) Comparison of experimental (solid lines) and numerical ([30], dashed lines) measurements. For ease of reading, three values of Rayleigh number are used to compare the behaviour of k ."},{"id":"","caption":"TABLEI. List of flow parameters, solution properties and corresponding uncertainties. Different realizations have been performed for all the Rayleigh numbers, Ra. Fluid temperature (ϑ) is reported, as well as solution mass fraction (ω M ), concentration (C M ) and corresponding density difference (∆ρ). The dimensions of the cell are kept constant for all the experiments (H × H, with H = (87 ± 0.5) mm, thickness b = (300 ± 10) µm). Fluid properties are discussed in Sec. II B.","content":"TABLEI. List of flow parameters, solution properties and corresponding uncertainties. Different realizations have been performed for all the Rayleigh numbers, Ra. Fluid temperature (ϑ) is reported, as well as solution mass fraction (ω M ), concentration (C M ) and corresponding density difference (∆ρ). The dimensions of the cell are kept constant for all the experiments (H × H, with H = (87 ± 0.5) mm, thickness b = (300 ± 10) µm). Fluid properties are discussed in Sec. II B.5 (1.41 ± 0.006) × 10 −3 \n1.427 ± 0.006 \n1.132 ± 0.002 \n(4.39 ± 0.08) × 10 3 \nE2 \n19.0 ± 0.5 (2.30 ± 0.006) × 10 −3 \n2.316 ± 0.006 \n1.805 ± 0.002 \n(7.00 ± 0.12) × 10 3 \nE3 \n21.0 ± 0.5 (3.94 ± 0.006) × 10 −3 \n3.949 ± 0.007 \n3.099 ± 0.004 \n(1.20 ± 0.02) × 10 4 \nE4 \n23.5 ± 0.5 (6.42 ± 0.006) × 10 −3 \n6.441 ± 0.008 \n5.101 ± 0.008 \n(1.98 ± 0.03) × 10 4 \nE5 \n21.0 ± 0.5 (1.08 ± 0.001) × 10 −2 10.824 ± 0.010 \n8.495 ± 0.013 \n(3.30 ± 0.06) × 10 4 \nE6 \n21.0 ± 0.5 (1.76 ± 0.001) × 10 −2 17.828 ± 0.015 13.996 ± 0.025 (5.43 ± 0.09) × 10 4"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"(a) Experimental setup. The Hele-Shaw cell is shown, with explicit indication of the position of the connection gates (top, bottom, left and right, indicated as T, B, L, and R respectively), sCMOS camera and backlight illumination. Fluids are supplied to the cell through a system (sketched in Fig. 2) consisting of a pump, pipes and valves. The cell is aligned in the vertical direction, namely parallel to the acceleration due to gravity, g. (b) Sketch of the experimental domain with explicit indication of the boundary conditions (no flux of mass or solute through the walls). The reference frame (x, z) as well as the initial position of the interface (red dashed line) are indicated, with the heavy fluid (density ρ M , concentration C = C M ) initially lying on top of the lighter fluid (ρ 0 , C = 0). The background field consists of one of the images collected, and provides a qualitative picture of the flow for the present physical configuration.\",\n \"content\": \"FIG. 1 .\\n1(a) Experimental setup. The Hele-Shaw cell is shown, with explicit indication of the position of the connection gates (top, bottom, left and right, indicated as T, B, L, and R respectively), sCMOS camera and backlight illumination. Fluids are supplied to the cell through a system (sketched in Fig. 2) consisting of a pump, pipes and valves. The cell is aligned in the vertical direction, namely parallel to the acceleration due to gravity, g. (b) Sketch of the experimental domain with explicit indication of the boundary conditions (no flux of mass or solute through the walls). The reference frame (x, z) as well as the initial position of the interface (red dashed line) are indicated, with the heavy fluid (density ρ M , concentration C = C M ) initially lying on top of the lighter fluid (ρ 0 , C = 0). The background field consists of one of the images collected, and provides a qualitative picture of the flow for the present physical configuration.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Circuital scheme of the experimental setup employed to obtain the desired initial conditions\",\n \"content\": \"FIG. 2 .\\n2Circuital scheme of the experimental setup employed to obtain the desired initial conditions\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Reconstruction of the concentration field. (a) Raw image consists of a light intensity distribution, I. (b) Upon processing to remove light intensity noises, scratches and non-uniformities of backlight illumination (see [40] for further details), the dimensionless concentration field is obtained. To visualise the interface between the two fluids, corresponding to the fingers, three iso-contours of normalised concentration are shown, corresponding to C * = C/C M = 0.3 (blue), 0.5 (red) and 0.7 (green).\",\n \"content\": \"FIG. 3 .\\n3Reconstruction of the concentration field. (a) Raw image consists of a light intensity distribution, I. (b) Upon processing to remove light intensity noises, scratches and non-uniformities of backlight illumination (see [40] for further details), the dimensionless concentration field is obtained. To visualise the interface between the two fluids, corresponding to the fingers, three iso-contours of normalised concentration are shown, corresponding to C * = C/C M = 0.3 (blue), 0.5 (red) and 0.7 (green).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Snapshots of the dimensionless concentration field obtained experimentally (a-d) and numerically (e-h). The experiment (E4 in Tab. I,Ra = 19789) is compared vis-á-vis against numerical simulations performed at Ra = 19953 by De Paoli et al.\",\n \"content\": \"FIG. 4 .\\n4Snapshots of the dimensionless concentration field obtained experimentally (a-d) and numerically (e-h). The experiment (E4 in Tab. I,Ra = 19789) is compared vis-á-vis against numerical simulations performed at Ra = 19953 by De Paoli et al.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Time evolution of the dimensionless mixing length, h * (t * ) (solid lines), for all the values of the global Rayleigh number considered in this study (and explicitly indicated next to each curve). The evolution of the system, initially controlled by diffusion, nicely follows the analytical solution, h * ≈ (1 − 2ε) √ 4πt * [Eq. (20), dashed blue line]. The intermediate phase, dominated by convection, exhibits the anomalous scaling h * ∼ (t * ) 1.2 first observed by De Paoli et al. [4].\",\n \"content\": \"FIG. 5 .\\n5Time evolution of the dimensionless mixing length, h * (t * ) (solid lines), for all the values of the global Rayleigh number considered in this study (and explicitly indicated next to each curve). The evolution of the system, initially controlled by diffusion, nicely follows the analytical solution, h * ≈ (1 − 2ε) √ 4πt * [Eq. (20), dashed blue line]. The intermediate phase, dominated by convection, exhibits the anomalous scaling h * ∼ (t * ) 1.2 first observed by De Paoli et al. [4].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Time taken by the fingers to reach the horizontal boundaries (t * s ) as a function of Ra. Experimental (present work) and numerical [4] results are reported as open (•) and filled symbols ( ), respectively. The best-fitting power law for present results (red dashed line) is 9.64 ×Ra 0.80 , in excellent agreement with numerical findings (14.98 ×Ra 0.78 , black dashed line). Experimental results correspond to the mean values obtained for several flow realisations at fixed Ra.\",\n \"content\": \"FIG. 6 .\\n6Time taken by the fingers to reach the horizontal boundaries (t * s ) as a function of Ra. Experimental (present work) and numerical [4] results are reported as open (•) and filled symbols ( ), respectively. The best-fitting power law for present results (red dashed line) is 9.64 ×Ra 0.80 , in excellent agreement with numerical findings (14.98 ×Ra 0.78 , black dashed line). Experimental results correspond to the mean values obtained for several flow realisations at fixed Ra.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 7. Measurements of the number of fingers over time for Ra = 5.43 × 10 4 (E6 in Tab. I). Space-time map, consisting of the evolution of the concentration field along the centerline, C * (x * , z * = Ra /2, t * ), is shown in (a). To compute the finger number, the space-time map is binarized, C(x * , z * = Ra /2, t * ), choosing C * = 1/2 as threshold (b). The concentration profile (red line) and the discretized profile (black line) measured along the centerline at t * ≈ 1.1 × 10 5 (indicated by the dashed blue line in panels a, b and d) are also reported as a function of the horizontal coordinate x * (c). Finally (panel d), the time-dependent finger number is computed as from the binarized field C (black line), from power-averaged mean wavenumber as defined in Eq. (21) ( N = L k /π, red line) and from a k peak (green line), i.e. the wavenumber maximizing the spectrum amplitude of C. See also the movie, Movie S2, in the Supplemental Material [54] for the time-dependent evolution of the wavenumber and the corresponding concentration field.\",\n \"content\": \"FIG. 7. Measurements of the number of fingers over time for Ra = 5.43 × 10 4 (E6 in Tab. I). Space-time map, consisting of the evolution of the concentration field along the centerline, C * (x * , z * = Ra /2, t * ), is shown in (a). To compute the finger number, the space-time map is binarized, C(x * , z * = Ra /2, t * ), choosing C * = 1/2 as threshold (b). The concentration profile (red line) and the discretized profile (black line) measured along the centerline at t * ≈ 1.1 × 10 5 (indicated by the dashed blue line in panels a, b and d) are also reported as a function of the horizontal coordinate x * (c). Finally (panel d), the time-dependent finger number is computed as from the binarized field C (black line), from power-averaged mean wavenumber as defined in Eq. (21) ( N = L k /π, red line) and from a k peak (green line), i.e. the wavenumber maximizing the spectrum amplitude of C. See also the movie, Movie S2, in the Supplemental Material [54] for the time-dependent evolution of the wavenumber and the corresponding concentration field.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) Experimental measurements. The evolution of the power-averaged mean wavenumber, k (defined in Eq. (21), main panel), and the corresponding finger number N (inset). (b) Comparison of experimental (solid lines) and numerical ([30], dashed lines) measurements. For ease of reading, three values of Rayleigh number are used to compare the behaviour of k .\",\n \"content\": \"FIG. 8 .\\n8(a) Experimental measurements. The evolution of the power-averaged mean wavenumber, k (defined in Eq. (21), main panel), and the corresponding finger number N (inset). (b) Comparison of experimental (solid lines) and numerical ([30], dashed lines) measurements. For ease of reading, three values of Rayleigh number are used to compare the behaviour of k .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"TABLEI. List of flow parameters, solution properties and corresponding uncertainties. Different realizations have been performed for all the Rayleigh numbers, Ra. Fluid temperature (ϑ) is reported, as well as solution mass fraction (ω M ), concentration (C M ) and corresponding density difference (∆ρ). The dimensions of the cell are kept constant for all the experiments (H × H, with H = (87 ± 0.5) mm, thickness b = (300 ± 10) µm). Fluid properties are discussed in Sec. II B.\",\n \"content\": \"TABLEI. List of flow parameters, solution properties and corresponding uncertainties. Different realizations have been performed for all the Rayleigh numbers, Ra. Fluid temperature (ϑ) is reported, as well as solution mass fraction (ω M ), concentration (C M ) and corresponding density difference (∆ρ). The dimensions of the cell are kept constant for all the experiments (H × H, with H = (87 ± 0.5) mm, thickness b = (300 ± 10) µm). Fluid properties are discussed in Sec. II B.5 (1.41 ± 0.006) × 10 −3 \\n1.427 ± 0.006 \\n1.132 ± 0.002 \\n(4.39 ± 0.08) × 10 3 \\nE2 \\n19.0 ± 0.5 (2.30 ± 0.006) × 10 −3 \\n2.316 ± 0.006 \\n1.805 ± 0.002 \\n(7.00 ± 0.12) × 10 3 \\nE3 \\n21.0 ± 0.5 (3.94 ± 0.006) × 10 −3 \\n3.949 ± 0.007 \\n3.099 ± 0.004 \\n(1.20 ± 0.02) × 10 4 \\nE4 \\n23.5 ± 0.5 (6.42 ± 0.006) × 10 −3 \\n6.441 ± 0.008 \\n5.101 ± 0.008 \\n(1.98 ± 0.03) × 10 4 \\nE5 \\n21.0 ± 0.5 (1.08 ± 0.001) × 10 −2 10.824 ± 0.010 \\n8.495 ± 0.013 \\n(3.30 ± 0.06) × 10 4 \\nE6 \\n21.0 ± 0.5 (1.76 ± 0.001) × 10 −2 17.828 ± 0.015 13.996 ± 0.025 (5.43 ± 0.09) × 10 4\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Uncertainties on the light intensity and mass fraction are also shown (see Sec. II B for further details). Finally, the mass fraction-light intensity space considered in each experiment, labelled as in Tab. I, is highlighted in red. The measurements are performed where the sensitivity of the method employed is higher, i.e. for low values of mass fraction. as in Fig. 3) are shown. We observe that, despite the large concentration contrast between the two values of Ra considered, the interface thickness (quantified by the vertical spacial separation between the iso-contours in Figs. 11c,f) is similar. Experimental measurements (symbols) are well fitted by a combination of exponential functions, ω( I cal ) = a 0 + a 1 exp(b 1 I cal ) + a 2 exp(b 2 I cal ) (solid line). and it is comparable with the spatial resolution of the measurements (1 pixel ≈ 90 µmaverage value of light intensity, I cal , is used to build the curve. Experimental measurements (symbols) are well fitted by a combination of exponential functions, ω( I cal ) = a 0 + a 1 exp(b 1 I cal ) + a 2 exp(b 2 I cal ) (solid line). Uncertainties on the light intensity and mass fraction are also shown (see Sec. II B for further details). Finally, the mass fraction-light intensity space considered in each experiment, labelled as in Tab. I, is highlighted in red. The measurements are performed where the sensitivity of the method employed is higher, i.e. for low values of mass fraction. as in Fig. 3) are shown. We observe that, despite the large concentration contrast between the two values of Ra considered, the interface thickness (quantified by the vertical spacial separation between the iso-contours in Figs. 11c,f) is similar, and it is comparable with the spatial resolution of the measurements (1 pixel ≈ 90 µm).","doi":""},{"text":"Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. L Rayleigh, Proc. Lond. Math. Soc. 14L. Rayleigh. Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. 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In panels (c, f), three iso-contours of dimensionless concentration (C * = 0.3 -blue, C * = 0.5 -red, and C * = 0.7 -green) are used to identify the interface shape in a close up view of the interfacial region. We observe that at both macroscopic (panels a, b, d, e) and interfacial (panels c, f) levels, the fluid-fluid separation region is well defined within a narrow areaFIG. 11. Initial conditions for two different experiments, namely E1 and E4, characterized by Ra = 4391 (panels a, b and c) and Ra = 19789 (panels d, e and f), respectively. The domain centerline (red dashed line) is indicated on the unprocessed light intensity fields I (panels a, d) and on the normalized inferred concentration fields C * (panels b,e). In panels (c, f), three iso-contours of dimensionless concentration (C * = 0.3 -blue, C * = 0.5 -red, and C * = 0.7 -green) are used to identify the interface shape in a close up view of the interfacial region. 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Water Resour., 48(9), 2012.","doi":""},{"text":"Influence of anisotropic permeability on convection in porous media: implications for geological CO2 sequestration. M De Paoli, F Zonta, A Soldati, Phys. Fluids. 28556601M. De Paoli, F. Zonta, and A. Soldati. Influence of anisotropic permeability on convection in porous media: implications for geological CO2 sequestration. Phys. Fluids (1994-present), 28 (5):056601, 2016.","doi":""},{"text":"Experimental assessment of mixing layer scaling laws in Rayleigh-Taylor instability. M De Paoli, D Perissutti, C Marchioli, A Soldati, 10.6084/m9.figshare.19761766M. De Paoli, D. Perissutti, C. Marchioli, and A. Soldati. Experimental as- sessment of mixing layer scaling laws in Rayleigh-Taylor instability. Figshare. https://doi.org/10.6084/m9.figshare.19761766. Deposited 25 June 2022.","doi":"10.6084/m9.figshare.19761766"},{"text":"See Supplemental Material \"Movie S1.mov\" for a comparison of the evolution of the concentration field in present experiments and simulations by. See Supplemental Material \"Movie S1.mov\" for a comparison of the evolution of the concen- tration field in present experiments and simulations by [4], .","doi":""},{"text":"Convective shutdown in a porous medium at high Rayleigh number. D R Hewitt, J A Neufeld, J R Lister, J. Fluid Mech. 719D. R. Hewitt, J. A. Neufeld, and J. R. Lister. Convective shutdown in a porous medium at high Rayleigh number. J. Fluid Mech., 719:551-586, 2013.","doi":""},{"text":"Self-similarity and internal structure of turbulence induced by Rayleigh-Taylor instability. S Dalziel, P Linden, D Youngs, J. Fluid Mech. 399S. Dalziel, P. Linden, and D. Youngs. Self-similarity and internal structure of turbulence induced by Rayleigh-Taylor instability. J. Fluid Mech., 399:1-48, 1999.","doi":""},{"text":"The mixing transition in Rayleigh-Taylor instability. A W Cook, W Cabot, P L Miller, J. Fluid Mech. 511A. W. Cook, W. Cabot, and P. L. Miller. The mixing transition in Rayleigh-Taylor instability. J. Fluid Mech., 511:333-362, 2004.","doi":""},{"text":"Miscible density fingering of chemical fronts in porous media: nonlinear simulations. Phys. Fluids (1994-present). A De Wit, 16A. De Wit. Miscible density fingering of chemical fronts in porous media: nonlinear simula- tions. Phys. Fluids (1994-present), 16(1):163-175, 2004.","doi":""},{"text":"Rayleigh-Taylor turbulence with singular nonuniform initial conditions. L Biferale, G Boffetta, A A Mailybaev, A Scagliarini, Phys. Rev. Fluids. 392601L. Biferale, G. Boffetta, A. A. Mailybaev, and A. Scagliarini. Rayleigh-Taylor turbulence with singular nonuniform initial conditions. Phys. Rev. Fluids, 3:092601, Sep 2018.","doi":""},{"text":"Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. R A Wooding, J. Fluid Mech. 3903R. A. Wooding. Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. J. Fluid Mech., 39(03):477-495, 1969.","doi":""},{"text":"See Supplemental Material \"Movie S2.mov\" for the time-dependent evolution of the wavenumber and the corresponding concentration field. See Supplemental Material \"Movie S2.mov\" for the time-dependent evolution of the wavenum- ber and the corresponding concentration field, .","doi":""},{"text":"J D IngleJr, S R Crouch, Spectrochemical analysis. Old Tappan, NJ (USJ. D. Ingle Jr and S. R. Crouch. Spectrochemical analysis. Old Tappan, NJ (US);","doi":""}],"string":"[\n {\n \"text\": \"Uncertainties on the light intensity and mass fraction are also shown (see Sec. II B for further details). Finally, the mass fraction-light intensity space considered in each experiment, labelled as in Tab. I, is highlighted in red. The measurements are performed where the sensitivity of the method employed is higher, i.e. for low values of mass fraction. as in Fig. 3) are shown. We observe that, despite the large concentration contrast between the two values of Ra considered, the interface thickness (quantified by the vertical spacial separation between the iso-contours in Figs. 11c,f) is similar. Experimental measurements (symbols) are well fitted by a combination of exponential functions, ω( I cal ) = a 0 + a 1 exp(b 1 I cal ) + a 2 exp(b 2 I cal ) (solid line). and it is comparable with the spatial resolution of the measurements (1 pixel ≈ 90 µmaverage value of light intensity, I cal , is used to build the curve. Experimental measurements (symbols) are well fitted by a combination of exponential functions, ω( I cal ) = a 0 + a 1 exp(b 1 I cal ) + a 2 exp(b 2 I cal ) (solid line). Uncertainties on the light intensity and mass fraction are also shown (see Sec. II B for further details). Finally, the mass fraction-light intensity space considered in each experiment, labelled as in Tab. I, is highlighted in red. The measurements are performed where the sensitivity of the method employed is higher, i.e. for low values of mass fraction. as in Fig. 3) are shown. We observe that, despite the large concentration contrast between the two values of Ra considered, the interface thickness (quantified by the vertical spacial separation between the iso-contours in Figs. 11c,f) is similar, and it is comparable with the spatial resolution of the measurements (1 pixel ≈ 90 µm).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. L Rayleigh, Proc. Lond. Math. Soc. 14L. Rayleigh. Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc., 14:170-177, 1883.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. G I Taylor, Proc. R. Soc. Lond. A. R. Soc. Lond. A201G. I. Taylor. The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A, 201(1065):192-196, 1950.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Incompressible Rayleigh-Taylor turbulence. G Boffetta, A Mazzino, Ann. Rev. Fluid Mech. 491G. Boffetta and A. Mazzino. Incompressible Rayleigh-Taylor turbulence. Ann. Rev. Fluid Mech., 49(1), 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rayleigh-Taylor convective dissolution in confined porous media. M De Paoli, F Zonta, A Soldati, Phys. Rev. Fluids. 423502M. De Paoli, F. Zonta, and A. Soldati. Rayleigh-Taylor convective dissolution in confined porous media. Phys. Rev. Fluids, 4:023502, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Scaling of Rayleigh-Taylor mixing in porous media. G Boffetta, M Borgnino, S Musacchio, Phys. Rev. Fluids. 5662501G. Boffetta, M. Borgnino, and S. Musacchio. Scaling of Rayleigh-Taylor mixing in porous media. Phys. Rev. Fluids, 5(6):62501, 2020.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dimensional effects in Rayleigh-Taylor mixing. G Boffetta, S Musacchio, Philos. Trans. Royal Soc. 38020210084G. Boffetta and S. Musacchio. Dimensional effects in Rayleigh-Taylor mixing. Philos. Trans. Royal Soc., 380(2219):20210084, 2022.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sewage plume in a sand and gravel aquifer. D R Leblanc, US Geological Survey. 2218D. R. LeBlanc. Sewage plume in a sand and gravel aquifer, Cape Cod, Massachusetts, volume 2218. US Geological Survey, 1984.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Ra = 19789 (panels d, e and f), respectively. The domain centerline (red dashed line) is indicated on the unprocessed light intensity fields I (panels a, d) and on the normalized inferred concentration fields C * (panels b,e). FIG. 11. Initial conditions for two different experiments, namely E1 and E4, characterized by Ra = 4391. In panels (c, f), three iso-contours of dimensionless concentration (C * = 0.3 -blue, C * = 0.5 -red, and C * = 0.7 -green) are used to identify the interface shape in a close up view of the interfacial region. We observe that at both macroscopic (panels a, b, d, e) and interfacial (panels c, f) levels, the fluid-fluid separation region is well defined within a narrow areaFIG. 11. Initial conditions for two different experiments, namely E1 and E4, characterized by Ra = 4391 (panels a, b and c) and Ra = 19789 (panels d, e and f), respectively. The domain centerline (red dashed line) is indicated on the unprocessed light intensity fields I (panels a, d) and on the normalized inferred concentration fields C * (panels b,e). In panels (c, f), three iso-contours of dimensionless concentration (C * = 0.3 -blue, C * = 0.5 -red, and C * = 0.7 -green) are used to identify the interface shape in a close up view of the interfacial region. We observe that at both macroscopic (panels a, b, d, e) and interfacial (panels c, f) levels, the fluid-fluid separation region is well defined within a narrow area.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transport of solutes in soils and aquifers. W Van Der Molen, H C Van Ommen, J. Hydrol. 1001W. Van Der Molen and H. C. Van Ommen. Transport of solutes in soils and aquifers. J. Hydrol., 100(1):433 -451, 1988.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sea ice is a mushy layer. D L Feltham, N Untersteiner, J S Wettlaufer, M G Worster, Geophys. Res. Lett. 3314D. L. Feltham, N. Untersteiner, J. S. 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PAMM, 19(1):e201900236, 2019.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Introduction to error analysis, the study of uncertainties in physical measurements. J Taylor, University Science Books. J. Taylor. Introduction to error analysis, the study of uncertainties in physical measurements. University Science Books, 1997.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Heat and brine transp. porous media: the Oberbeck-Boussinesq approximation revisited. A J Landman, R J Schotting, Transp. Porous Media. 703A. J. Landman and R. J. Schotting. Heat and brine transp. porous media: the Oberbeck- Boussinesq approximation revisited. Transp. Porous Media, 70(3):355-373, 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Dissolution in anisotropic porous media: Modelling convection regimes from onset to shutdown. Phys. Fluids (1994-present). M De Paoli, F Zonta, A Soldati, 2926601M. De Paoli, F. Zonta, and A. Soldati. Dissolution in anisotropic porous media: Modelling convection regimes from onset to shutdown. Phys. Fluids (1994-present), 29(2):026601, 2017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Effect of permeability anisotropy on buoyancydriven flow for CO2 sequestration in saline aquifers. P Cheng, M Bestehorn, A Firoozabadi, Water Resour. 489P. Cheng, M. Bestehorn, and A. Firoozabadi. Effect of permeability anisotropy on buoyancy- driven flow for CO2 sequestration in saline aquifers. Water Resour., 48(9), 2012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Influence of anisotropic permeability on convection in porous media: implications for geological CO2 sequestration. M De Paoli, F Zonta, A Soldati, Phys. Fluids. 28556601M. De Paoli, F. Zonta, and A. Soldati. Influence of anisotropic permeability on convection in porous media: implications for geological CO2 sequestration. Phys. Fluids (1994-present), 28 (5):056601, 2016.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Experimental assessment of mixing layer scaling laws in Rayleigh-Taylor instability. M De Paoli, D Perissutti, C Marchioli, A Soldati, 10.6084/m9.figshare.19761766M. De Paoli, D. Perissutti, C. Marchioli, and A. Soldati. Experimental as- sessment of mixing layer scaling laws in Rayleigh-Taylor instability. Figshare. https://doi.org/10.6084/m9.figshare.19761766. Deposited 25 June 2022.\",\n \"doi\": \"10.6084/m9.figshare.19761766\"\n },\n {\n \"text\": \"See Supplemental Material \\\"Movie S1.mov\\\" for a comparison of the evolution of the concentration field in present experiments and simulations by. See Supplemental Material \\\"Movie S1.mov\\\" for a comparison of the evolution of the concen- tration field in present experiments and simulations by [4], .\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convective shutdown in a porous medium at high Rayleigh number. D R Hewitt, J A Neufeld, J R Lister, J. Fluid Mech. 719D. R. Hewitt, J. A. Neufeld, and J. R. Lister. Convective shutdown in a porous medium at high Rayleigh number. J. Fluid Mech., 719:551-586, 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Self-similarity and internal structure of turbulence induced by Rayleigh-Taylor instability. S Dalziel, P Linden, D Youngs, J. Fluid Mech. 399S. Dalziel, P. Linden, and D. Youngs. Self-similarity and internal structure of turbulence induced by Rayleigh-Taylor instability. J. Fluid Mech., 399:1-48, 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The mixing transition in Rayleigh-Taylor instability. A W Cook, W Cabot, P L Miller, J. Fluid Mech. 511A. W. Cook, W. Cabot, and P. L. Miller. The mixing transition in Rayleigh-Taylor instability. J. Fluid Mech., 511:333-362, 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Miscible density fingering of chemical fronts in porous media: nonlinear simulations. Phys. Fluids (1994-present). A De Wit, 16A. De Wit. Miscible density fingering of chemical fronts in porous media: nonlinear simula- tions. Phys. Fluids (1994-present), 16(1):163-175, 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Rayleigh-Taylor turbulence with singular nonuniform initial conditions. L Biferale, G Boffetta, A A Mailybaev, A Scagliarini, Phys. Rev. Fluids. 392601L. Biferale, G. Boffetta, A. A. Mailybaev, and A. Scagliarini. Rayleigh-Taylor turbulence with singular nonuniform initial conditions. Phys. Rev. Fluids, 3:092601, Sep 2018.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. R A Wooding, J. Fluid Mech. 3903R. A. Wooding. Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. J. Fluid Mech., 39(03):477-495, 1969.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"See Supplemental Material \\\"Movie S2.mov\\\" for the time-dependent evolution of the wavenumber and the corresponding concentration field. See Supplemental Material \\\"Movie S2.mov\\\" for the time-dependent evolution of the wavenum- ber and the corresponding concentration field, .\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J D IngleJr, S R Crouch, Spectrochemical analysis. Old Tappan, NJ (USJ. D. Ingle Jr and S. R. Crouch. Spectrochemical analysis. Old Tappan, NJ (US);\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":574,"cells":{"text":{"kind":"string","value":"\nORGANISATION EUROP ÉENNE POUR LA RECHERCHE NUCL ÉAIRE (CERN) Laboratoire Européen pour la Physique des Particules\n18 April 2002\n\nThe Aleph Collaboration \nA Heister \nS Schael \nR Barate \nR Brunelière \nI De Bonis \nD Decamp \nC Goy \nS Jezequel \nJ.-P Lees \nF Martin \nE Merle \nM.-N Minard \nB Pietrzyk \nB Trocmé \nG Boix \nNow at McKinsey and Compagny\nAvenue Louis Casal 181203GenevaSwitzerland\n\nS Bravo \nM P Casado \nM Chmeissani \nJ M Crespo \nE Fernandez \nM Fernandez-Bosman \nLl Garrido \nE Graugés \nJ Lopez \nM Martinez \nG Merino \nR Miquel \nNow at LBNL\n94720BerkeleyCAU.S.A\n\nLl M Mir \nNow at LBNL\n94720BerkeleyCAU.S.A\n\nA Pacheco \nD Paneque \nH Ruiz \nA Colaleo \nD Creanza \nN De Filippis \nM De Palma \nG Iaselli \nG Maggi \nM Maggi \nS Nuzzo \nA Ranieri \nG Raso \nAlso at Dipartimento di Fisica e Tecnologie Relative\nUniversità di Palermo\nPalermoItaly\n\nF Ruggieri \nG Selvaggi \nL Silvestris \nP Tempesta \nA Tricomi \nAlso at Dipartimento di Fisica di Catania and INFN Sezione di Catania\n95129CataniaItaly\n\nG Zito \nX Huang \nJ Lin \nQ Ouyang \nT Wang \nY Xie \nR Xu \nS Xue \nJ Zhang \nL Zhang \nW Zhao \nD Abbaneo \nP Azzurri \nT Barklow \nNow at SLAC\n94309StanfordCAU.S.A\n\nO Buchmüller \nNow at SLAC\n94309StanfordCAU.S.A\n\nM Cattaneo \nF Cerutti \nB Clerbaux \nNow at Institut Inter-universitaire des hautes Energies (IIHE)\nUniversité Libre de Bruxelles\n230, 1050BruxellesCPBelgique\n\nH Drevermann \nR W Forty \nM Frank \nF Gianotti \nT C Greening \nNow at Honeywell\nPhoenixAZU.S.A\n\nJ B Hansen \nJ Harvey \nD E Hutchcroft \nP Janot \nB Jost \nM Kado \nNow at Fermilab\nPO Box 500352, 60510BataviaMS, ILUSA\n\nP Mato \nA Moutoussi \nF Ranjard \nL Rolandi \nD Schlatter \nG Sguazzoni \nW Tejessy \nF Teubert \nA Valassi \nI Videau \nJ J Ward \nF Badaud \nS Dessagne \nA Falvard \nNow at Groupe d' Astroparticules de Montpellier\nUniversité de Montpellier II\n34095MontpellierFrance\n\nD Fayolle \nP Gay \nJ Jousset \nB Michel \nS Monteil \nD Pallin \nJ M Pascolo \nP Perret \nJ D Hansen \nJ R Hansen \nP H Hansen \nB S Nilsson \nA Kyriakis \nC Markou \nE Simopoulou \nA Vayaki \nK Zachariadou \nA Blondel \nNow at Departement de Physique Corpusculaire\nUniversité de Genève\n1211 Genève 4Switzerland\n\nJ.-C Brient \nF Machefert \nA Rougé \nM Swynghedauw \nR Tanaka \nH Videau \nV Ciulli \nE Focardi \nG Parrini \nA Antonelli \nM Antonelli \nG Bencivenni \nF Bossi \nG Capon \nV Chiarella \nP Laurelli \nG Mannocchi \nAlso Istituto di Cosmo-Geofisica del C.N.R\nTorinoItaly\n\nG P Murtas \nL Passalacqua \nJ Kennedy \nJ G Lynch \nP Negus \nV O'shea \nA S Thompson \nS Wasserbaech \nR Cavanaugh \nDepartment of Physics\nNow at University of Florida\n32611-8440GainesvilleFloridaUSA\n\nS Dhamotharan \nNow at BNP Paribas\n60325Frankfurt am MainzGermany\n\nC Geweniger \nP Hanke \nV Hepp \nE E Kluge \nG Leibenguth \nA Putzer \nH Stenzel \nK Tittel \nM Wunsch \nNow at SAP AG\n69185WalldorfGermany\n\nR Beuselinck \nW Cameron \nG Davies \nP J Dornan \nM Girone \nAlso at CERN\n1211 Geneva 23Switzerland\n\nR D Hill \nN Marinelli \nJ Nowell \nS A Rutherford \nJ K Sedgbeer \nJ C Thompson \nR White \nV M Ghete \nP Girtler \nE Kneringer \nD Kuhn \nG Rudolph \nE Bouhova-Thacker \nC K Bowdery \nD P Clarke \nG Ellis \nA J Finch \nF Foster \nG Hughes \nR W L Jones \nM R Pearson \nN A Robertson \nM Smizanska \nO Van Der Aa \nC Delaere \nV Lemaitre \nU Blumenschein \nF Hölldorfer \nK Jakobs \nF Kayser \nK Kleinknecht \nA.-S Müller \nG Quast \nNow at Institut für Experimentelle Kernphysik\nUniversität Karlsruhe\n76128KarlsruheGermany\n\nB Renk \nH.-G Sander \nS Schmeling \nH Wachsmuth \nC Zeitnitz \nT Ziegler \nA Bonissent \nP Coyle \nC Curtil \nA Ealet \nD Fouchez \nP Payre \nA Tilquin \nF Ragusa \nA David \nH Dietl \nG Ganis \nDipartimento di Fisica\nINFN Sezione di Roma II\nUniversità di Roma Tor Vergata\n00133RomaItaly\n\nK Hüttmann \nG Lütjens \nW Männer \nH.-G Moser \nR Settles \nG Wolf \nG Bagliesi \nT Boccali \nL Foà \nA Giammanco \nA Giassi \nF Ligabue \nA Messineo \nF Palla \nG Sanguinetti \nA Sciabà \nR Tenchini \nAlso at CERN\n1211 Geneva 23Switzerland\n\nA Venturi \nAlso at CERN\n1211 Geneva 23Switzerland\n\nP G Verdini \nO Awunor \nG A Blair \nG Cowan \nA Garcia-Bellido \nM G Green \nL T Jones \nT Medcalf \nA Misiejuk \nJ A Strong \nP Teixeira-Dias \nR W Clifft \nT R Edgecock \nP R Norton \nI R Tomalin \nB Bloch-Devaux \nD Boumediene \nP Colas \nB Fabbro \nE Lançon \nM.-C Lemaire \nE Locci \nP Perez \nJ Rander \nB Tuchming \nB Vallage \nN Konstantinidis \nA M Litke \nG Taylor \nC N Booth \nS Cartwright \nF Combley \nP N Hodgson \nM Lehto \nL F Thompson \nK Affholderbach \nA Böhrer \nS Brandt \nC Grupen \nJ Hess \nA Ngac \nG Prange \nU Sieler \nC Borean \nG Giannini \nH He \nJ Putz \nJ Rothberg \nS R Armstrong \nK Berkelman \nK Cranmer \nD P S Ferguson \nY Gao \nAlso at Department of Physics\nTsinghua University\nBeijingThe People's Republic of China\n\nS González \nO J Hayes \nH Hu \nS Jin \nJ Kile \nP A Mcnamara III\nJ Nielsen \nY B Pan \nJ H Von Wimmersperg-Toeller \nW Wiedenmann \nJ Wu \nSau Lan Wu \nX Wu \nG Zobernig \nG Dissertori \n\nPhysikalisches Institut das RWTH-Aachen\nD-52056AachenGermany\n\n\nLaboratoire de Physique des Particules (LAPP)\nIN 2 P 3 -CNRSF-74019Annecy-le-Vieux CedexFrance\n\n\nInstitut de F ísica d'Altes Energies\nUniversitat Autònoma de Barcelona\nE-08193Bellaterra (Barcelona)Spain\n\n\nDipartimento di Fisica\nINFN Sezione di Bari\nI-70126BariItaly\n\n\nInstitute of High Energy Physics\nAcademia Sinica\nBeijingThe People's Republic of China\n\n\nEuropean Laboratory for Particle Physics (CERN)\nCH-1211Geneva 23Switzerland\n\n\nLaboratoire de Physique Corpusculaire\nUniversité Blaise Pascal\nIN 2 P 3 -CNRSF-63177Clermont-Ferrand, AubièreFrance\n\n\nNiels Bohr Institute\n2100CopenhagenDKDenmark\n\n\nNuclear Research Center Demokritos (NRCD)\nGR-15310AttikiGreece\n\n\nLaboratoire de Physique Nucléaire et des Hautes Energies\nEcole Polytechnique\nIN 2 P 3 -CNRSF-91128Palaiseau CedexFrance\n\n\nDipartimento di Fisica\nUniversità di Firenze\nINFN Sezione di Firenze\nI-50125FirenzeItaly\n\n\nLaboratori Nazionali dell'INFN (LNF-INFN)\nI-00044FrascatiItaly\n\n\nDepartment of Physics and Astronomy\nUniversity of Glasgow\nG12 8QQGlasgowUnited Kingdom\n\n\nDepartment of Physics\nHaverford College\n19041-1392HaverfordPAU.S.A\n\n\nKirchhoff-Institut für Physik\nUniversität Heidelberg\nD-69120HeidelbergGermany\n\n\nDepartment of Physics\nImperial College\nSW7 2BZLondonUnited Kingdom\n\n\nInstitut für Experimentalphysik\nUniversität Innsbruck\nA-6020InnsbruckAustria\n\n\nDepartment of Physics\nUniversity of Lancaster\nLA1 4YBLancasterUnited Kingdom\n\n\nInstitut de Physique Nucléaire\nDépartement de Physique\nUniversité Catholique de Louvain\n1348Louvain-la-NeuveBelgium\n\n\nInstitut für Physik\nUniversität Mainz\nD-55099MainzGermany\n\n\nCentre de Physique des Particules de Marseille\nUniv Méditerranée\nIN 2 P 3 -CNRSF-13288MarseilleFrance\n\n\nDipartimento di Fisica\nUniversità di Milano\n\n\n\nINFN Sezione di Milano\nI-20133MilanoItaly\n\n\nMax-Planck-Institut für Physik\nWerner-Heisenberg-Institut\nD-80805MünchenGermany\n\n\nLaboratoire de l'Accélérateur Linéaire\nUniversité de Paris-Sud\nIN 2 P 3 -CNRSF-91898Orsay CedexFrance\n\n\nDipartimento di Fisica dell'Università\nINFN Sezione di Pisa\n\n\n\nScuola Normale Superiore\nI-56010PisaItaly\n\n\nDepartment of Physics\nRoyal Holloway & Bedford New College\nUniversity of London\nTW20 OEXEghamSurreyUnited Kingdom\n\n\nParticle Physics Dept\nOX11 OQXRutherford Appleton Laboratory, Chilton, DidcotOxonUnited Kingdom\n\n\nDAPNIA/Service de Physique des Particules\nCEA\nCE-Saclay\nF-91191Gif-sur-Yvette CedexFrance\n\n\nInstitute for Particle Physics\nUniversity of California at Santa Cruz\n95064Santa CruzCAUSA\n\n\nDepartment of Physics\nUniversity of Sheffield\nS3 7RHSheffieldUnited Kingdom\n\n\nFachbereich Physik\nUniversität Siegen\nD-57068SiegenGermany\n\n\nDipartimento di Fisica\nUniversità di Trieste\n\n\n\nINFN Sezione di Trieste\nI-34127TriesteItaly\n\n\nExperimental Elementary Particle Physics\nUniversity of Washington\n98195SeattleWAU.S.A\n\n\nDepartment of Physics\nUniversity of Wisconsin\n53706MadisonWIUSA\n\n\nInstitute for Particle Physics\nETH Hönggerberg\n8093ZürichSwitzerland\n\n\nSupported by CICYT\nSpain\n\n\nUniversitat de Barcelona\n08208BarcelonaSpain\n\n\nSupported by Bundesministerium für Bildung und Forschung\nGermany\n\n\nSupported by the Direction des Sciences de la Matière\nC.E.A\n\n\nDepartment of Energy\nNow at Skyguide\nSwissair Navigation Services\nFG03-92ER40689. 23GenevaSwitzerland\n\n\nNow at Centre de Physique des Particules de Marseille\nUniv Méditerranée\nF-13288MarseilleFrance\n\n\nAlso at Groupe d' Astroparticules de Montpellier\nUniversité de Montpellier II\n34095MontpellierFrance\n\nORGANISATION EUROP ÉENNE POUR LA RECHERCHE NUCL ÉAIRE (CERN) Laboratoire Européen pour la Physique des Particules\n18 April 20028FA22DC0D11A7053E2C1C4543D039988arXiv:hep-ex/0204036v1Submitted to Physics Letters B)\nSearches for scalar top, scalar bottom and mass-degenerate scalar quarks are performed in the data collected by the ALEPH detector at LEP, at centre-of-mass energies up to 209 GeV, corresponding to an integrated luminosity of 675 pb −1 .No evidence for the production of such particles is found in the decay channels t → c/u χ, t → bℓν, b → bχ, q → qχ or in the stop four-body decay channel t → bχf f′ studied for the first time at LEP.The results of these searches yield improved mass lower limits.In particular, an absolute lower limit of 63 GeV/c 2 is obtained for the stop mass, at 95% confidence level, irrespective of the stop lifetime and decay branching ratios.\n\nIntroduction\n\nThe results of searches for scalar quarks with the data collected in the year 2000 by the ALEPH detector at LEP are presented in this letter.The energies and integrated luminosities of the analysed data samples are given in Table 1.Previous results obtained with lower energy data have been reported by ALEPH in Refs.[1][2][3][4][5] and by the other LEP collaborations in Refs.[6][7][8].The theoretical framework for these studies is the supersymmetric extension of the Standard Model [9], with R-parity conservation.The lightest supersymmetric particle (LSP) is assumed to be the lightest neutralino χ or the sneutrino ν.Such an LSP is stable and weakly interacting.Each chirality state of the Standard Model fermions has a scalar supersymmetric partner.The scalar quarks (squarks) qL and qR are the supersymmetric partners of the left-handed and right-handed quarks, respectively.The mass eigenstates are orthogonal combinations of the weak interaction eigenstates qL and qR .The mixing angle θ q is defined in such a way that q = qL cos θ q + qR sin θ q is the lighter squark.The off-diagonal terms of the mass matrix, responsible for mixing, read, with standard notation: m q (A q − µκ), with κ = tan β for downtype and κ = 1/ tan β for up-type quarks.Since the size of this mixing term is proportional to the mass of the Standard Model partner, it could well be that the lightest supersymmetric charged particle is the lighter scalar top (stop, t) or, in particular for large tan β values, the lighter scalar bottom (sbottom, b).Squarks could be produced at LEP in pairs, e + e − → qq , via s-channel exchange of a virtual photon or Z.The production cross section [10] depends on θ q when mixing is relevant, i.e., for stops and sbottoms.\n\nThe searches for stops described here assume that all supersymmetric particles except the lightest neutralino χ and possibly the sneutrino ν are heavier than the stop.Under these assumptions, the allowed decay channels are t → c/u χ, t → bχf f′ and t → bℓν [10,11].The corresponding diagrams are shown in Fig. 1.The decay t → cχ (Fig. 1a) proceeds only via loops and has a very small width, of the order of 0.01-1 eV [10], depending on the mass difference ∆M between the stop and the neutralino, and on the masses and field content of the particles involved in the loops.For low enough ∆M values (∆M < ∼ 6 GeV/c 2 ), the stop lifetime becomes sizeable, and must be taken into account in the searches for stop production.If ∆M is so small that the t → cχ channel is kinematically closed, the dominant decay mode becomes t → uχ, and the stop can then be considered as stable for practical purposes.For stop masses accessible at LEP, i.e., < ∼ 100 GeV/c 2 , the decay mode t → bχf f′ is mediated by virtual chargino and W (Fig. 1b) or sfermion (Fig. 1c) exchange.It is therefore of the same order in perturbation theory as the loop-induced t → cχ decay, and can be substantially enhanced if charginos have masses not much larger than their present experimental bounds, and could even dominate for light sfermions [11].The four-body decay channel yields topologies with b-jets, additional jets and/or leptons, and with missing mass and missing energy.A new multi-jet analysis, hereafter called MJ, has been designed to cope with these final states.\n\nThe t → bℓν channel proceeds via virtual chargino exchange (Fig. 1d) and has a width of the order of 0.1-10 keV [10].This decay channel dominates when it is kinematically allowed, i.e., if the lightest ν is lighter than the stop.If the lightest neutralino is the LSP, the sneutrino decays invisibly into χν without any change in the experimental topology.\n\nUnder the assumption that the b is lighter than all supersymmetric particles except the χ, the b will decay as b → bχ (Fig. 1e).Compared to the t, the b decay width is larger, of the order of 10-100 MeV.\n\nThe supersymmetric partners of the light quarks are generally expected to be quite heavy.If they are light enough to be within the reach of LEP, their dominant decay mode is expected to be q → qχ.\n\nThe final state topologies addressed by the searches presented in this letter are summarised in Table 2, together with the related signal processes and with the references where analysis details can be found.This letter is organised as follows.In Section 2, the ALEPH detector and the simulated samples used for the analyses are described.Section 3 is dedicated to the selection algorithms with emphasis on the new search for four-body stop decays.In Section 4 the results of the searches are given, along with their interpretation in the theoretical framework.The conclusions of the letter are given in Section 5.\n\n\nALEPH detector and event simulation\n\nA thorough description of the ALEPH detector and of its performance, as well as of the standard reconstruction and analysis algorithms, can be found in Refs.[12,13].Only a brief summary is given here.\n\nThe trajectories of charged particles are measured by a silicon vertex detector (VDET), a cylindrical multi-wire drift chamber (ITC) and a large time projection chamber (TPC).These detectors are immersed in an axial magnetic field of 1.5 T provided by a superconducting solenoidal coil.The VDET consists of two cylindrical layers of silicon microstrip detectors; it performs precise measurements of the impact parameter in space, yielding powerful short-lived particle tags, as described in Ref. [14].\n\nThe electromagnetic calorimeter (ECAL), placed between the TPC and the coil, is a highlysegmented sandwich of lead planes and proportional wire chambers.It consists of a barrel and two endcaps.The hadron calorimeter (HCAL) consists of the iron return yoke of the magnet instrumented with streamer tubes.It is surrounded by two double layers of streamer tubes, the muon chambers.The luminosity monitors (LCAL and SiCAL) extend the calorimeter hermeticity down to 34 mrad from the beam axis.\n\nThe energy flow algorithm described in Ref. [13] combines the measurements of the tracking detectors and of the calorimeters into \"objects\" classified as charged particles, photons, and neutral hadrons.The energy resolution achieved with this algorithm is (0.6 √ E + 0.6) GeV (E in GeV).Electrons are identified by comparing the energy deposit in ECAL to the momentum measured in the tracking system, by using the shower profile in the electromagnetic calorimeter, and by the measurement of the specific ionization energy loss in the TPC.The identification of muons makes use of the hit pattern in HCAL and of the muon chambers.\n\nSignal event samples were simulated with the generator described in Ref. [1] for t → cχ, b → bχ, q → qχ and t → bℓν.A modified version of this generator was designed to simulate the channel t → bχf f′ , where the final state is modelled according to phase space and including parton shower development.The generation of t → c/u χ with lifetime follows the procedure described in Ref. [4].\n\nTo simulate the relevant Standard Model background processes, several Monte Carlo generators were used: BHWIDE [15] for Bhabha scattering, KORALZ [16] for µ + µ − and τ + τ − production, PHOT02 [17] for γγ interactions, KORALW [18] for WW production, and PYTHIA [19] for the other processes (e + e − → qq(γ), Weν, Zee, ZZ, Zν ν).The sizes of the simulated samples typically correspond to ten times the integrated luminosity of the data.\n\nAll background and signal samples were processed through the full detector simulation.\n\n\nEvent selections\n\nSeveral selection algorithms have been developed to search for the topologies given in Table 2.All these channels are characterised by missing energy.The event properties depend significantly on ∆M, the mass difference between the decaying squark and the χ (or the ν in the case of t → bℓν).When ∆M is large, there is a substantial amount of visible energy, and the signal events tend to look like WW, Weν, ZZ, and qq(γ) events.When ∆M is small, the visible energy is small, and the signal events are therefore similar to γγ events.In order to cope with the different signal topologies and background situations, each analysis employs selections dependent on the ∆M range.The stop lifetime may become sizeable at small ∆M, in which case the signal final state topology depends strongly on the t decay length λ t; three different selections are used, each designed to cope with a specific λ t range [4].\n\nThe optimisation of the selection criteria as well as the best combination of selections as a function of ∆M and λ t were obtained according to the N 95 prescription [20], i.e., by minimisation of the 95% C.L. cross section upper limit expected in the absence of a signal.The selections are mostly independent of the centre-of-mass energy except for an appropriate rescaling of the cuts with √ s when relevant.The selections applied to the year 2000 data follow closely those described in Refs.[1][2][3][4][5] except for the new analysis developed to address the t → bχf f′ decay, hereafter described in some detail.\n\n\nSearch for t → bχf f ′\n\nThe MJ analysis consists of a small, a large and a very large ∆M selection.These selections are designed to address simultaneously all bχqq ′ and bχℓν final states, independently of the decay branching ratios.The selections use several anti-γγ criteria, reported in Table 3.The cuts are derived from the AJ selection, described in Ref. [1] as well as the variables used.Only the relevant differences are discussed in the following.\n\nIn the t → bχf f′ channel, the b quark in the final state produces a visible mass higher than in the t → cχ channel.Therefore, for the small ∆M selection, the cut on the number of charged particle tracks N ch is reinforced by requiring N ch > 10, and both the visible mass, M vis , and the visible mass computed excluding the leading lepton, M ex ℓ 1 vis , are required to be greater than 10 GeV/c 2 .These tighter cuts allow others to be loosened: the transverse momentum p t and that calculated excluding the neutral hadrons, p ex NH t , must be greater than 0.005 √ s and 0.01 √ s, respectively.The remaining background is reduced in the small ∆M selection by requiring the thrust to be smaller than 0.875, and by the cut E vis < 0.26 √ s.\n\nFor the large ∆M selections, the multi-jet signature is addressed by requiring y 45 , as calculated with the DURHAM algorithm [21], to be greater than 0.001.The level of the WW, ZZ and Weν background is reduced by taking advantage of the b-quark content in the Table 3: Criteria used in the MJ selections to address the backgrounds from (A) γγ → qq, (B) γγ → qq with spurious calorimetric objects and (C) γγ → τ + τ − .The † indicates that the cut is applied when the azimuthal angle of the missing momentum φ p miss is within 15 • of the vertical plane.\n\n\nSmall ∆M\n\nLarge and Very large ∆M\nA N ch > 10 M vis > 10 GeV/c 2 E 12 • = 0 < 0.05 √ s E 30 • < 0.25E vis < 0.3E vis Φ acop (acoplanarity) < 172.5 • < 174 • Φ acop T (transverse acop.) < 175 • < 175 • p t / √ s > 0.005 (> 0.01) † > 0.05 (> 0.075) † p t /E vis > 1.305 − 0.00725Φ acop > 0.2 M miss < 25.0E vis θ point > 15 • > 5 • if θ scat < 15 • | cos θ p miss | < 0.8 < 0.95 | cos θ thrust | < 0.75 M ex ℓ 1 vis > 10 GeV/c 2 B p ex NH t > 0.01 √ s < 0.03 √ s if E NH vis > 0.45E vis p ch t > 0.005 √ s E NH 1 < 0.3 √ s E NH vis < 0.3E vis E(φ p miss ± 15 • ) < 0.075 √ s C N ch jet i , i = 1, 2 > 4 m jet i , i = 1, 2 > 4 GeV/c 2\nt → bχf f′ final state.The value of − log 10 P uds is required to be greater than 0.5, where P uds is the b-tag event probability introduced in Ref. [14].This background is further suppressed by a missing mass cut, the location of which is a function of the ∆M of the signal considered.For example, for ∆M = 20, 30 and 40 GeV/c 2 the optimal cuts are M miss / √ s > 0.75, 0.70 and 0.65, respectively.\n\nThe region where the very large ∆M selection applies is characterised by a higher visible mass.The sliding cut on the missing mass is looser than that in the large ∆M selection.For ∆M = 20, 40 and 60 GeV/c 2 the optimal cuts are M miss / √ s > 0.58, 0.34 and 0.10.Other cuts are then necessary to reduce the background mainly due to WW events.Similarly to the large ∆M case, − log 10 P uds and y 45 are required to be greater than 0.5 and 0.003, respectively.The mean momentum of all reconstructed charged particle tracks must be less than 0.007 √ s.To reduce background from semileptonic W decays, the fraction of visible energy due to charged objects excluding the leading lepton is required to be greater than 0.5, and the leading lepton, if present, must not be isolated, i.e., the additional energy deposited in a 30 • cone around its direction must be at least 50% of its energy.At this level, the remaining background consists of WW events with energy lost in the beam pipe, responsible for the missing mass.These events are rejected by requiring |p z | < 0.1 √ s and the energy E 12 • deposited at polar angle smaller than 12 • to be less than 0.015 √ s.\n\nThe efficiencies of the three selections were parametrised as a function of ∆M for each stop pair final state that may result from the decay channels considered ( t → cχ, t → bχℓν, t → bχqq ′ ).This allows the signal efficiency to be parametrised as a function of the branching ratios.The efficiencies were checked to be practically independent of the lepton flavour (e, µ, τ ) in the t → bχℓν decay.The small, large and very large ∆M selections are combined using the N 95 procedure as a function of ∆M and of the branching ratios.The background to the small ∆M selection is dominated by γγ → qq events and has a total expectation of 5.0 fb, while the backgrounds of the large and very large ∆M selections, dominated by WW and other four-fermion processes amount to 3.5 and 4.4 fb, respectively.\n\n\nSystematic uncertainties\n\nThe efficiencies of the MJ analysis may be affected by uncertainties regarding the assumptions on the stop hadron physics and by uncertainties related to the detector response.The results of the systematic studies are summarised in Table 4 for the three selections.\n\nThe systematic effects from the assumptions on the stop hadron physics were assessed by varying the parameters of the model implemented in the generator as in Ref. [1].The uncertainties from the stop hadron mass were evaluated by varying the effective spectator mass M eff , set to 0.5 GeV/c 2 in the analysis, in the range between 0.3 and 1.0 GeV/c 2 .The efficiencies of the large and very large ∆M selections are almost insensitive to this change.The 9% effect found for the small ∆M selection reflects the variation in the invariant mass available for the hadronic system.\n\nThe systematic error due to the uncertainty on the stop fragmentation was evaluated by varying ǫ t by an order of magnitude, where ǫ t is the parameter of the Peterson fragmentation function [22].The effect on the efficiency is very small (∼ 2%).The amount of initial state radiation in stop pair production depends on the value of the stop coupling to the Z boson, which is controlled by the stop mixing angle.A variation of θ t from 56 • to 0 • , i.e., from minimal to maximal coupling, was applied.The effect was found to be small in all selections, at the level of 1 to 3%.\n\nDetector effects have been studied for the variables used in the selections.The distributions of all relevant variables show good agreement with the simulation.In particular, the b-tagging performance was checked on hadronic events collected at the Z resonance.The systematic errors associated to detector effects and to the reconstruction procedure were found to be negligible.\n\nBeam-related background, not included in the event simulation, may affect the E 12 • variable.Its effect on the selection efficiency was determined from data collected at random beam crossings.The net effect is a relative decrease of the signal efficiency by about 5%.The uncertainty on this correction is negligible.\n\nFinally, an additional uncertainty of 3% due to the limited Monte Carlo statistics was added.The total systematic uncertainty is at the level of 10% for the small ∆M selection.It is dominated by the limited knowledge of the stop hadron physics, and results from rather extreme changes in the model parameters.The systematic uncertainties for the large and very large ∆M selections are at the level of 4-5%.\n\nThe systematic uncertainties in the selections other than for the t → bχf f′ channel are essentially identical to those reported in Refs.[4,5].\n\n\nResults and interpretation\n\nThe numbers of candidate events selected and background events expected are reported in Table 5 for all the data samples used to derive the results below.An overall agreement is observed.In particular, a total of six candidate events is selected by the new MJ analysis, with 8.5 events expected from background processes; two events are found by each of the selections, in agreement with predictions of 3.3, 2.3 and 2.9 background events at small, large and very large ∆M, respectively.In the framework of the supersymmetric extension of the Standard Model [9], the outcome of these searches can be translated into constraints in the space of the relevant parameters.In this process the systematic uncertainties on the selection efficiencies were included according to the method described in Ref. [23], and no background subtraction was applied.The constraints discussed below, derived from the results given in Table 5, are at 95% confidence level.\n\nThe regions excluded in the plane (M t, M χ ) under the hypothesis of a dominant t → c/u χ decay are shown in Fig. 2a for two values of the t mixing angle θ t, 0 • and 56 • , corresponding to maximal and vanishing tt Z coupling, respectively.For 8 GeV/c 2 < ∆M < M W + M b , and using also CDF results [24], the lower limit on M t is 92 GeV/c 2 , independent of θ t.\n\nThe very small ∆M corridor is partially covered by the \"long-lived hadrons\" analysis as indicated by the plain dark region in Fig. 2a.The stop mass lower limit provided by the \"longlived hadrons\" analysis is shown in Fig. 2b as a function of log(cτ t/cm) for various ∆M values.The smallest ∆M value considered is 1.6 GeV/c 2 , corresponding to the \"effective\" kinematic limit for the decay t → cχ [4].Below that ∆M value, the stop decay mode is t → uχ, and the limit is 95 GeV/c 2 , given by the large lifetime selection.The absolute mass lower limit obtained is 63 GeV/c 2 .It is reached for ∆M = 1.6 GeV/c 2 and for a cτ t value of ∼ 1 cm.In that configuration of parameters, the \"AJ small ∆M\" and the \"long-lived hadrons intermediate lifetime\" selections are combined.\n\nIn the MSSM [9], more restrictive constraints on the stop mass can be obtained since ∆M and the stop lifetime are related.The mass lower limit obtained by scanning over the relevant model parameters as in Ref. [4] is shown in Fig. 2c as a function of tan β.For any tan β, the stop mass limit is 65 GeV/c 2 , reached for tan β ∼ 2.7.\n\nUnder the hypothesis that the decay t → bχf f′ is dominant, the regions excluded in the plane (M t, M χ ) are shown in Fig. 3a, for relative proportions of the possible f f′ final states as in W * decays.In Fig. 3b the leptonic modes bχℓν (with equal branching ratios for ℓ = e, µ and τ ) are assumed to be dominant.The excluded regions are given for θ t = 0 • and θ t = 56 • .For ∆M > 8 GeV/c 2 , the θ t-independent lower limits on M t are 78 GeV/c 2 and 80 GeV/c 2 , for the two cases of W * and leptonic final state dominance, respectively.\n\nThe combination of the AJ and MJ analyses allows constraints to be set under the more general hypothesis that both the t → cχ and t → bχf f′ decay channels contribute to stop decays.The excluded regions in the plane (M t, M χ ) are shown in Fig. 4a for θ t = 0 • and θ t = 56 • .This result was obtained by arbitrarily varying the t → cχ branching ratio and the leptonic fraction in the t → bχf f′ decay, and by using the N 95 prescription to determine the appropriate combination of selections.The stop mass limit is shown in Fig. 4b as a function of the branching ratio BR ( t → cχ) for several fixed ∆M values and for θ t = 56 • .The smallest ∆M value considered is 5 GeV/c 2 , corresponding to the threshold for the production of a b quark in the final state.The lowest limit obtained is 63 GeV/c 2 ; it is reached for ∆M = 5 GeV/c 2 , BR ( t → cχ) = 0.22, and BR ( t → bχℓν) = 0.55.\n\nUnder the assumption that the t → bℓν decay mode is dominant, with equal branching ratios for ℓ = e, µ and τ , the excluded region in the plane (M t, M ν ) is shown in Fig. 5a.If ∆M > 8 GeV/c 2 , and using the LEP1 limit on the sneutrino mass and D0 results [25], the lower limit on M t is 97 GeV/c 2 , independent of θ t.The lower limit is 82 GeV/c 2 if the t → bτ ντ decay mode is dominant and ∆M > 8 GeV/c 2 , independent of θ t.\n\nThe excluded region in the plane (M b, M χ ) is shown in Fig. 5b under the assumption of a dominant b → bχ decay.Taking also the CDF exclusion [24] into account, a lower limit of 89 GeV/c 2 is set on M b, for any b mixing angle and ∆M > 8 GeV/c 2 .The region excluded for θ b = 0 • , for which the bb Z coupling is maximal, is also shown.\n\nAs discussed in detail in Ref. [2], the results of the search for acoplanar jets, with or without b tagging, can also be translated into constraints on the mass of degenerate squarks.In order to compare these results with those obtained at the Tevatron [24,25], limits have been evaluated within the MSSM [9] under the following assumptions: a degenerate mass M q for all left-handed and right-handed ũ, d, c, s, b squarks; lowest order GUT relation between the soft supersymmetry breaking gaugino mass terms, allowing the gluino and neutralino masses to be related; tan β = 4 and µ = −400 GeV.The results in the plane (M g, M q) are shown in Fig. 6.Improved constraints are obtained in the region of small q to χ mass differences.\n\n\nConclusions\n\nSearches for signals of pair-produced scalar partners of quarks have been performed in the data sample of 207 pb −1 collected in the year 2000 with the ALEPH detector at LEP, at centreof-mass energies ranging from 204 to 209 GeV.The final state topologies studied arise from the decays t → c/u χ, t → bχf f′ , t → bℓν, b → bχ, and q → qχ.The four-body stop decay channel was analysed for the first time at LEP, and the corresponding selections were extended to the 675 pb −1 of data collected by ALEPH at centre-of-mass energies of 183 GeV and above.All numbers of candidate events observed are consistent with the backgrounds expected from Standard Model processes.The results of these searches, combined with earlier ones obtained with data collected from 1997 to 1999, have been translated into improved mass lower limits, of which relevant examples are given in Table 6.In particular, a 95% C.L. lower limit of 63 GeV/c 2 has been set on the stop mass, irrespective of its lifetime and decay branching ratios.\n\n\nALEPH\n\nq → q χ Figure 6: Excluded regions at 95% C.L. from the search for generic q pairs, assuming five massdegenerate q flavours.The results are shown in the gluino-squark mass plane for tan β = 4 and µ = −400 GeV, together with results from experiments at pp colliders.\n\n1\n1\n\n\n\n1 :\n1\nSquark decay diagrams considered in this letter: (a) t → c/u χ; (b) t → bχf f′ via W exchange and (c) t → bχf f′ via sfermion exchange; (d) t → bℓν; (e) b → bχ.\n\n\nFigure 2 :\n2\nFigure2: (a) Excluded regions at 95% C.L. in the M χ vs M t plane from t → c/uχ searches; the excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.The dark region in the small ∆M corridor is excluded by the \"long-lived hadrons\" analysis.The CDF experiment result is also indicated.(b) Stop mass lower limit at 95% C.L. from the \"long-lived hadrons\" as a function of log(cτ t/cm) for several ∆M values, without any assumption on the relation between ∆M and the stop lifetime τ t.(c) Stop mass lower limit at 95% C.L. from the \"long-lived hadrons\" analysis as a function of tan β, independent of the other MSSM parameters.\n\n\n→ b χ l ν Figure 3 :Figure 4 :Figure 5 :\n345\nFigure 4: (a) Branching ratio independent excluded regions at 95% C.L. in the M χ vs M t plane, from t → bχf f′ and t → cχ searches.The excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.(b) Limit on the stop mass at 95% C.L. as a function of BR ( t → cχ) for various ∆M values.The limits are given for θ t = 56 • .\n\n\nTable 1 :\n1\nIntegrated luminosities, centre-of-mass energy ranges and mean centre-of-mass energy values for the data collected by the ALEPH detector in the year 2000.\nLuminosity [ pb −1 ] Energy range [GeV]√ s [GeV]9.4 122.6 75.3207 − 209 206 − 207 204 − 206208.0 206.6 205.2\n\nTable 2 :\n2\nTopologies studied in the different scenarios.\nProduction Decay modeTopology/AnalysisReferencestt tt tt tt bb qqt → cχ (∆M > ∼ 6 GeV/c 2 ) t → c/u χ (∆M < ∼ 6 GeV/c 2 ) Long-lived hadrons [4] Acoplanar jets (AJ) [1, 2, 3, 5] Multi-jets (MJ) This letter t → bχf f′ AJ plus leptons [1, 2, 3, 5] t → bℓν AJ plus b tagging [1, 2, 3, 5] b → bχ q → qχ AJ [2, 3, 5]\n\nTable 4 :\n4\nSummary of the relative systematic uncertainties on the efficiencies of the MJ analysis.\nSystematic uncertainties (%)MJ selectionsSmall ∆M Large ∆M Very large ∆MM eff (0.3-1.0 GeV)923ǫ t (10 −5 -10 −4 )222θ t (0 • -56 • )311Detector and reconstruction212Monte Carlo statistics333TOTAL1045\n\nTable 5 :\n5\nNumbers of candidate events observed (N obs ) and expected from background (N exp ) for the different selections.Also given are the sizes ( Ldt) and the average centre-of-mass energies √ s of the samples analysed.exp N obs N exp N obs N exp N obs N exp\nYear1997199819992000SampleLdt [ pb −1 ] √ s [ GeV]57.0 182.7173.6 188.6236.9 197.6207.3 206.2Analysis N obs N AJ Selection small ∆M 1 1.535.522.422.1large ∆M43.554.087.3118.6Long-livedsmall λ t.AJ, small ∆Mhadronsintermediate λ t.--00.300.500.4large λ t.--10.400.600.6MJsmall ∆M00.310.711.201.1large ∆M10.200.600.810.7very large ∆M00.200.811.010.9AJsmall ∆M10.801.932.602.4plus leptonslarge ∆M00.120.421.431.6AJsmall ∆M01.133.312.222.0plus b tagginglarge ∆M10.600.910.701.2\n\nTable 6 :\n6\nLower limits on stop and sbottom masses in some relevant cases.All limits are valid for any value of the mixing angle.\n95% C.L.∆MDominantMass LimitrangedecaySquark [ GeV/c 2 ] [ GeV/c 2 ] channel(s) Commentst92 78 97 63> 8 > 8 > 8 anyt → cχ t → bχW * t → bℓν anyCDF result [24] used LEP 1, D0 result [25] used any branching ratios, any lifetimeb89> 8b → bχCDF result [24] used\nAcknowledgementsWe wish to congratulate our colleagues from the accelerator divisions for the successful operation of LEP at high energies.We would also like to express our gratitude to the engineers and support people at our home institutes without whom this work would not have been possible.Those of us from non-member states wish to thank CERN for its hospitality and support.8Supported by the National Science Foundation of China.9Supported by the Danish Natural Science Research Council.10Supported by the UK Particle Physics and Astronomy Research Council.11Supported by the US Department of Energy, grant DE-FG0295-ER40896.13Supported by the Commission of the European Communities, contract ERBFMBICT982874.14Supported by the Leverhulme Trust.\nSearches for scalar top and scalar bottom quarks at LEP2. 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Lett. 2001\n"},"corpus_id":{"kind":"number","value":262630558,"string":"262,630,558"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"hep-ex/0204036\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/hep-ex/0204036v1.pdf\"\n ],\n \"pdf_sha\": \"a2da7c63135809292e1d910ae8bf04e170a5f98a\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"ORGANISATION EUROP ÉENNE POUR LA RECHERCHE NUCL ÉAIRE (CERN) Laboratoire Européen pour la Physique des Particules"},"authors":{"kind":"list like","value":["The Aleph Collaboration","A Heister","S Schael","R Barate","R Brunelière","I De Bonis","D Decamp","C Goy","S Jezequel","J.-P Lees","F Martin","E Merle","M.-N Minard","B Pietrzyk","B Trocmé","G Boix \nNow at McKinsey and Compagny\nAvenue Louis Casal 181203GenevaSwitzerland","S Bravo","M P Casado","M Chmeissani","J M Crespo","E Fernandez","M Fernandez-Bosman","Ll Garrido","E Graugés","J Lopez","M Martinez","G Merino","R Miquel \nNow at LBNL\n94720BerkeleyCAU.S.A","Ll M Mir \nNow at LBNL\n94720BerkeleyCAU.S.A","A Pacheco","D Paneque","H Ruiz","A Colaleo","D Creanza","N De Filippis","M De Palma","G Iaselli","G Maggi","M Maggi","S Nuzzo","A Ranieri","G Raso \nAlso at Dipartimento di Fisica e Tecnologie Relative\nUniversità di Palermo\nPalermoItaly","F Ruggieri","G Selvaggi","L Silvestris","P Tempesta","A Tricomi \nAlso at Dipartimento di Fisica di Catania and INFN Sezione di Catania\n95129CataniaItaly","G Zito","X Huang","J Lin","Q Ouyang","T Wang","Y Xie","R Xu","S Xue","J Zhang","L Zhang","W Zhao","D Abbaneo","P Azzurri","T Barklow \nNow at SLAC\n94309StanfordCAU.S.A","O Buchmüller \nNow at SLAC\n94309StanfordCAU.S.A","M Cattaneo","F Cerutti","B Clerbaux \nNow at Institut Inter-universitaire des hautes Energies (IIHE)\nUniversité Libre de Bruxelles\n230, 1050BruxellesCPBelgique","H Drevermann","R W Forty","M Frank","F Gianotti","T C Greening \nNow at Honeywell\nPhoenixAZU.S.A","J B Hansen","J Harvey","D E Hutchcroft","P Janot","B Jost","M Kado \nNow at Fermilab\nPO Box 500352, 60510BataviaMS, ILUSA","P Mato","A Moutoussi","F Ranjard","L Rolandi","D Schlatter","G Sguazzoni","W Tejessy","F Teubert","A Valassi","I Videau","J J Ward","F Badaud","S Dessagne","A Falvard \nNow at Groupe d' Astroparticules de Montpellier\nUniversité de Montpellier II\n34095MontpellierFrance","D Fayolle","P Gay","J Jousset","B Michel","S Monteil","D Pallin","J M Pascolo","P Perret","J D Hansen","J R Hansen","P H Hansen","B S Nilsson","A Kyriakis","C Markou","E Simopoulou","A Vayaki","K Zachariadou","A Blondel \nNow at Departement de Physique Corpusculaire\nUniversité de Genève\n1211 Genève 4Switzerland","J.-C Brient","F Machefert","A Rougé","M Swynghedauw","R Tanaka","H Videau","V Ciulli","E Focardi","G Parrini","A Antonelli","M Antonelli","G Bencivenni","F Bossi","G Capon","V Chiarella","P Laurelli","G Mannocchi \nAlso Istituto di Cosmo-Geofisica del C.N.R\nTorinoItaly","G P Murtas","L Passalacqua","J Kennedy","J G Lynch","P Negus","V O'shea","A S Thompson","S Wasserbaech","R Cavanaugh \nDepartment of Physics\nNow at University of Florida\n32611-8440GainesvilleFloridaUSA","S Dhamotharan \nNow at BNP Paribas\n60325Frankfurt am MainzGermany","C Geweniger","P Hanke","V Hepp","E E Kluge","G Leibenguth","A Putzer","H Stenzel","K Tittel","M Wunsch \nNow at SAP AG\n69185WalldorfGermany","R Beuselinck","W Cameron","G Davies","P J Dornan","M Girone \nAlso at CERN\n1211 Geneva 23Switzerland","R D Hill","N Marinelli","J Nowell","S A Rutherford","J K Sedgbeer","J C Thompson","R White","V M Ghete","P Girtler","E Kneringer","D Kuhn","G Rudolph","E Bouhova-Thacker","C K Bowdery","D P Clarke","G Ellis","A J Finch","F Foster","G Hughes","R W L Jones","M R Pearson","N A Robertson","M Smizanska","O Van Der Aa","C Delaere","V Lemaitre","U Blumenschein","F Hölldorfer","K Jakobs","F Kayser","K Kleinknecht","A.-S Müller","G Quast \nNow at Institut für Experimentelle Kernphysik\nUniversität Karlsruhe\n76128KarlsruheGermany","B Renk","H.-G Sander","S Schmeling","H Wachsmuth","C Zeitnitz","T Ziegler","A Bonissent","P Coyle","C Curtil","A Ealet","D Fouchez","P Payre","A Tilquin","F Ragusa","A David","H Dietl","G Ganis \nDipartimento di Fisica\nINFN Sezione di Roma II\nUniversità di Roma Tor Vergata\n00133RomaItaly","K Hüttmann","G Lütjens","W Männer","H.-G Moser","R Settles","G Wolf","G Bagliesi","T Boccali","L Foà","A Giammanco","A Giassi","F Ligabue","A Messineo","F Palla","G Sanguinetti","A Sciabà","R Tenchini \nAlso at CERN\n1211 Geneva 23Switzerland","A Venturi \nAlso at CERN\n1211 Geneva 23Switzerland","P G Verdini","O Awunor","G A Blair","G Cowan","A Garcia-Bellido","M G Green","L T Jones","T Medcalf","A Misiejuk","J A Strong","P Teixeira-Dias","R W Clifft","T R Edgecock","P R Norton","I R Tomalin","B Bloch-Devaux","D Boumediene","P Colas","B Fabbro","E Lançon","M.-C Lemaire","E Locci","P Perez","J Rander","B Tuchming","B Vallage","N Konstantinidis","A M Litke","G Taylor","C N Booth","S Cartwright","F Combley","P N Hodgson","M Lehto","L F Thompson","K Affholderbach","A Böhrer","S Brandt","C Grupen","J Hess","A Ngac","G Prange","U Sieler","C Borean","G Giannini","H He","J Putz","J Rothberg","S R Armstrong","K Berkelman","K Cranmer","D P S Ferguson","Y Gao \nAlso at Department of Physics\nTsinghua University\nBeijingThe People's Republic of China","S González","O J Hayes","H Hu","S Jin","J Kile","P A Mcnamara III","J Nielsen","Y B Pan","J H Von Wimmersperg-Toeller","W Wiedenmann","J Wu","Sau Lan Wu","X Wu","G Zobernig","G Dissertori","Physikalisches Institut das RWTH-Aachen\nD-52056AachenGermany","Laboratoire de Physique des Particules (LAPP)\nIN 2 P 3 -CNRSF-74019Annecy-le-Vieux CedexFrance","Institut de F ísica d'Altes Energies\nUniversitat Autònoma de Barcelona\nE-08193Bellaterra (Barcelona)Spain","Dipartimento di Fisica\nINFN Sezione di Bari\nI-70126BariItaly","Institute of High Energy Physics\nAcademia Sinica\nBeijingThe People's Republic of China","European Laboratory for Particle Physics (CERN)\nCH-1211Geneva 23Switzerland","Laboratoire de Physique Corpusculaire\nUniversité Blaise Pascal\nIN 2 P 3 -CNRSF-63177Clermont-Ferrand, AubièreFrance","Niels Bohr Institute\n2100CopenhagenDKDenmark","Nuclear Research Center Demokritos (NRCD)\nGR-15310AttikiGreece","Laboratoire de Physique Nucléaire et des Hautes Energies\nEcole Polytechnique\nIN 2 P 3 -CNRSF-91128Palaiseau CedexFrance","Dipartimento di Fisica\nUniversità di Firenze\nINFN Sezione di Firenze\nI-50125FirenzeItaly","Laboratori Nazionali dell'INFN (LNF-INFN)\nI-00044FrascatiItaly","Department of Physics and Astronomy\nUniversity of Glasgow\nG12 8QQGlasgowUnited Kingdom","Department of Physics\nHaverford College\n19041-1392HaverfordPAU.S.A","Kirchhoff-Institut für Physik\nUniversität Heidelberg\nD-69120HeidelbergGermany","Department of Physics\nImperial College\nSW7 2BZLondonUnited Kingdom","Institut für Experimentalphysik\nUniversität Innsbruck\nA-6020InnsbruckAustria","Department of Physics\nUniversity of Lancaster\nLA1 4YBLancasterUnited Kingdom","Institut de Physique Nucléaire\nDépartement de Physique\nUniversité Catholique de Louvain\n1348Louvain-la-NeuveBelgium","Institut für Physik\nUniversität Mainz\nD-55099MainzGermany","Centre de Physique des Particules de Marseille\nUniv Méditerranée\nIN 2 P 3 -CNRSF-13288MarseilleFrance","Dipartimento di Fisica\nUniversità di Milano","INFN Sezione di Milano\nI-20133MilanoItaly","Max-Planck-Institut für Physik\nWerner-Heisenberg-Institut\nD-80805MünchenGermany","Laboratoire de l'Accélérateur Linéaire\nUniversité de Paris-Sud\nIN 2 P 3 -CNRSF-91898Orsay CedexFrance","Dipartimento di Fisica dell'Università\nINFN Sezione di Pisa","Scuola Normale Superiore\nI-56010PisaItaly","Department of Physics\nRoyal Holloway & Bedford New College\nUniversity of London\nTW20 OEXEghamSurreyUnited Kingdom","Particle Physics Dept\nOX11 OQXRutherford Appleton Laboratory, Chilton, DidcotOxonUnited Kingdom","DAPNIA/Service de Physique des Particules\nCEA\nCE-Saclay\nF-91191Gif-sur-Yvette CedexFrance","Institute for Particle Physics\nUniversity of California at Santa Cruz\n95064Santa CruzCAUSA","Department of Physics\nUniversity of Sheffield\nS3 7RHSheffieldUnited Kingdom","Fachbereich Physik\nUniversität Siegen\nD-57068SiegenGermany","Dipartimento di Fisica\nUniversità di Trieste","INFN Sezione di Trieste\nI-34127TriesteItaly","Experimental Elementary Particle Physics\nUniversity of Washington\n98195SeattleWAU.S.A","Department of Physics\nUniversity of Wisconsin\n53706MadisonWIUSA","Institute for Particle Physics\nETH Hönggerberg\n8093ZürichSwitzerland","Supported by CICYT\nSpain","Universitat de Barcelona\n08208BarcelonaSpain","Supported by Bundesministerium für Bildung und Forschung\nGermany","Supported by the Direction des Sciences de la Matière\nC.E.A","Department of Energy\nNow at Skyguide\nSwissair Navigation Services\nFG03-92ER40689. 23GenevaSwitzerland","Now at Centre de Physique des Particules de Marseille\nUniv Méditerranée\nF-13288MarseilleFrance","Also at Groupe d' Astroparticules de Montpellier\nUniversité de Montpellier II\n34095MontpellierFrance"],"string":"[\n \"The Aleph Collaboration\",\n \"A Heister\",\n \"S Schael\",\n \"R Barate\",\n \"R Brunelière\",\n \"I De Bonis\",\n \"D Decamp\",\n \"C Goy\",\n \"S Jezequel\",\n \"J.-P Lees\",\n \"F Martin\",\n \"E Merle\",\n \"M.-N Minard\",\n \"B Pietrzyk\",\n \"B Trocmé\",\n \"G Boix \\nNow at McKinsey and Compagny\\nAvenue Louis Casal 181203GenevaSwitzerland\",\n \"S Bravo\",\n \"M P Casado\",\n \"M Chmeissani\",\n \"J M Crespo\",\n \"E Fernandez\",\n \"M Fernandez-Bosman\",\n \"Ll Garrido\",\n \"E Graugés\",\n \"J Lopez\",\n \"M Martinez\",\n \"G Merino\",\n \"R Miquel \\nNow at LBNL\\n94720BerkeleyCAU.S.A\",\n \"Ll M Mir \\nNow at LBNL\\n94720BerkeleyCAU.S.A\",\n \"A Pacheco\",\n \"D Paneque\",\n \"H Ruiz\",\n \"A Colaleo\",\n \"D Creanza\",\n \"N De Filippis\",\n \"M De Palma\",\n \"G Iaselli\",\n \"G Maggi\",\n \"M Maggi\",\n \"S Nuzzo\",\n \"A Ranieri\",\n \"G Raso \\nAlso at Dipartimento di Fisica e Tecnologie Relative\\nUniversità di Palermo\\nPalermoItaly\",\n \"F Ruggieri\",\n \"G Selvaggi\",\n \"L Silvestris\",\n \"P Tempesta\",\n \"A Tricomi \\nAlso at Dipartimento di Fisica di Catania and INFN Sezione di Catania\\n95129CataniaItaly\",\n \"G Zito\",\n \"X Huang\",\n \"J Lin\",\n \"Q Ouyang\",\n \"T Wang\",\n \"Y Xie\",\n \"R Xu\",\n \"S Xue\",\n \"J Zhang\",\n \"L Zhang\",\n \"W Zhao\",\n \"D Abbaneo\",\n \"P Azzurri\",\n \"T Barklow \\nNow at SLAC\\n94309StanfordCAU.S.A\",\n \"O Buchmüller \\nNow at SLAC\\n94309StanfordCAU.S.A\",\n \"M Cattaneo\",\n \"F Cerutti\",\n \"B Clerbaux \\nNow at Institut Inter-universitaire des hautes Energies (IIHE)\\nUniversité Libre de Bruxelles\\n230, 1050BruxellesCPBelgique\",\n \"H Drevermann\",\n \"R W Forty\",\n \"M Frank\",\n \"F Gianotti\",\n \"T C Greening \\nNow at Honeywell\\nPhoenixAZU.S.A\",\n \"J B Hansen\",\n \"J Harvey\",\n \"D E Hutchcroft\",\n \"P Janot\",\n \"B Jost\",\n \"M Kado \\nNow at Fermilab\\nPO Box 500352, 60510BataviaMS, ILUSA\",\n \"P Mato\",\n \"A Moutoussi\",\n \"F Ranjard\",\n \"L Rolandi\",\n \"D Schlatter\",\n \"G Sguazzoni\",\n \"W Tejessy\",\n \"F Teubert\",\n \"A Valassi\",\n \"I Videau\",\n \"J J Ward\",\n \"F Badaud\",\n \"S Dessagne\",\n \"A Falvard \\nNow at Groupe d' Astroparticules de Montpellier\\nUniversité de Montpellier II\\n34095MontpellierFrance\",\n \"D Fayolle\",\n \"P Gay\",\n \"J Jousset\",\n \"B Michel\",\n \"S Monteil\",\n \"D Pallin\",\n \"J M Pascolo\",\n \"P Perret\",\n \"J D Hansen\",\n \"J R Hansen\",\n \"P H Hansen\",\n \"B S Nilsson\",\n \"A Kyriakis\",\n \"C Markou\",\n \"E Simopoulou\",\n \"A Vayaki\",\n \"K Zachariadou\",\n \"A Blondel \\nNow at Departement de Physique Corpusculaire\\nUniversité de Genève\\n1211 Genève 4Switzerland\",\n \"J.-C Brient\",\n \"F Machefert\",\n \"A Rougé\",\n \"M Swynghedauw\",\n \"R Tanaka\",\n \"H Videau\",\n \"V Ciulli\",\n \"E Focardi\",\n \"G Parrini\",\n \"A Antonelli\",\n \"M Antonelli\",\n \"G Bencivenni\",\n \"F Bossi\",\n \"G Capon\",\n \"V Chiarella\",\n \"P Laurelli\",\n \"G Mannocchi \\nAlso Istituto di Cosmo-Geofisica del C.N.R\\nTorinoItaly\",\n \"G P Murtas\",\n \"L Passalacqua\",\n \"J Kennedy\",\n \"J G Lynch\",\n \"P Negus\",\n \"V O'shea\",\n \"A S Thompson\",\n \"S Wasserbaech\",\n \"R Cavanaugh \\nDepartment of Physics\\nNow at University of Florida\\n32611-8440GainesvilleFloridaUSA\",\n \"S Dhamotharan \\nNow at BNP Paribas\\n60325Frankfurt am MainzGermany\",\n \"C Geweniger\",\n \"P Hanke\",\n \"V Hepp\",\n \"E E Kluge\",\n \"G Leibenguth\",\n \"A Putzer\",\n \"H Stenzel\",\n \"K Tittel\",\n \"M Wunsch \\nNow at SAP AG\\n69185WalldorfGermany\",\n \"R Beuselinck\",\n \"W Cameron\",\n \"G Davies\",\n \"P J Dornan\",\n \"M Girone \\nAlso at CERN\\n1211 Geneva 23Switzerland\",\n \"R D Hill\",\n \"N Marinelli\",\n \"J Nowell\",\n \"S A Rutherford\",\n \"J K Sedgbeer\",\n \"J C Thompson\",\n \"R White\",\n \"V M Ghete\",\n \"P Girtler\",\n \"E Kneringer\",\n \"D Kuhn\",\n \"G Rudolph\",\n \"E Bouhova-Thacker\",\n \"C K Bowdery\",\n \"D P Clarke\",\n \"G Ellis\",\n \"A J Finch\",\n \"F Foster\",\n \"G Hughes\",\n \"R W L Jones\",\n \"M R Pearson\",\n \"N A Robertson\",\n \"M Smizanska\",\n \"O Van Der Aa\",\n \"C Delaere\",\n \"V Lemaitre\",\n \"U Blumenschein\",\n \"F Hölldorfer\",\n \"K Jakobs\",\n \"F Kayser\",\n \"K Kleinknecht\",\n \"A.-S Müller\",\n \"G Quast \\nNow at Institut für Experimentelle Kernphysik\\nUniversität Karlsruhe\\n76128KarlsruheGermany\",\n \"B Renk\",\n \"H.-G Sander\",\n \"S Schmeling\",\n \"H Wachsmuth\",\n \"C Zeitnitz\",\n \"T Ziegler\",\n \"A Bonissent\",\n \"P Coyle\",\n \"C Curtil\",\n \"A Ealet\",\n \"D Fouchez\",\n \"P Payre\",\n \"A Tilquin\",\n \"F Ragusa\",\n \"A David\",\n \"H Dietl\",\n \"G Ganis \\nDipartimento di Fisica\\nINFN Sezione di Roma II\\nUniversità di Roma Tor Vergata\\n00133RomaItaly\",\n \"K Hüttmann\",\n \"G Lütjens\",\n \"W Männer\",\n \"H.-G Moser\",\n \"R Settles\",\n \"G Wolf\",\n \"G Bagliesi\",\n \"T Boccali\",\n \"L Foà\",\n \"A Giammanco\",\n \"A Giassi\",\n \"F Ligabue\",\n \"A Messineo\",\n \"F Palla\",\n \"G Sanguinetti\",\n \"A Sciabà\",\n \"R Tenchini \\nAlso at CERN\\n1211 Geneva 23Switzerland\",\n \"A Venturi \\nAlso at CERN\\n1211 Geneva 23Switzerland\",\n \"P G Verdini\",\n \"O Awunor\",\n \"G A Blair\",\n \"G Cowan\",\n \"A Garcia-Bellido\",\n \"M G Green\",\n \"L T Jones\",\n \"T Medcalf\",\n \"A Misiejuk\",\n \"J A Strong\",\n \"P Teixeira-Dias\",\n \"R W Clifft\",\n \"T R Edgecock\",\n \"P R Norton\",\n \"I R Tomalin\",\n \"B Bloch-Devaux\",\n \"D Boumediene\",\n \"P Colas\",\n \"B Fabbro\",\n \"E Lançon\",\n \"M.-C Lemaire\",\n \"E Locci\",\n \"P Perez\",\n \"J Rander\",\n \"B Tuchming\",\n \"B Vallage\",\n \"N Konstantinidis\",\n \"A M Litke\",\n \"G Taylor\",\n \"C N Booth\",\n \"S Cartwright\",\n \"F Combley\",\n \"P N Hodgson\",\n \"M Lehto\",\n \"L F Thompson\",\n \"K Affholderbach\",\n \"A Böhrer\",\n \"S Brandt\",\n \"C Grupen\",\n \"J Hess\",\n \"A Ngac\",\n \"G Prange\",\n \"U Sieler\",\n \"C Borean\",\n \"G Giannini\",\n \"H He\",\n \"J Putz\",\n \"J Rothberg\",\n \"S R Armstrong\",\n \"K Berkelman\",\n \"K Cranmer\",\n \"D P S Ferguson\",\n \"Y Gao \\nAlso at Department of Physics\\nTsinghua University\\nBeijingThe People's Republic of China\",\n \"S González\",\n \"O J Hayes\",\n \"H Hu\",\n \"S Jin\",\n \"J Kile\",\n \"P A Mcnamara III\",\n \"J Nielsen\",\n \"Y B Pan\",\n \"J H Von Wimmersperg-Toeller\",\n \"W Wiedenmann\",\n \"J Wu\",\n \"Sau Lan Wu\",\n \"X Wu\",\n \"G Zobernig\",\n \"G Dissertori\",\n \"Physikalisches Institut das RWTH-Aachen\\nD-52056AachenGermany\",\n \"Laboratoire de Physique des Particules (LAPP)\\nIN 2 P 3 -CNRSF-74019Annecy-le-Vieux CedexFrance\",\n \"Institut de F ísica d'Altes Energies\\nUniversitat Autònoma de Barcelona\\nE-08193Bellaterra (Barcelona)Spain\",\n \"Dipartimento di Fisica\\nINFN Sezione di Bari\\nI-70126BariItaly\",\n \"Institute of High Energy Physics\\nAcademia Sinica\\nBeijingThe People's Republic of China\",\n \"European Laboratory for Particle Physics (CERN)\\nCH-1211Geneva 23Switzerland\",\n \"Laboratoire de Physique Corpusculaire\\nUniversité Blaise Pascal\\nIN 2 P 3 -CNRSF-63177Clermont-Ferrand, AubièreFrance\",\n \"Niels Bohr Institute\\n2100CopenhagenDKDenmark\",\n \"Nuclear Research Center Demokritos (NRCD)\\nGR-15310AttikiGreece\",\n \"Laboratoire de Physique Nucléaire et des Hautes Energies\\nEcole Polytechnique\\nIN 2 P 3 -CNRSF-91128Palaiseau CedexFrance\",\n \"Dipartimento di Fisica\\nUniversità di Firenze\\nINFN Sezione di Firenze\\nI-50125FirenzeItaly\",\n \"Laboratori Nazionali dell'INFN (LNF-INFN)\\nI-00044FrascatiItaly\",\n \"Department of Physics and Astronomy\\nUniversity of Glasgow\\nG12 8QQGlasgowUnited Kingdom\",\n \"Department of Physics\\nHaverford College\\n19041-1392HaverfordPAU.S.A\",\n \"Kirchhoff-Institut für Physik\\nUniversität Heidelberg\\nD-69120HeidelbergGermany\",\n \"Department of Physics\\nImperial College\\nSW7 2BZLondonUnited Kingdom\",\n \"Institut für Experimentalphysik\\nUniversität Innsbruck\\nA-6020InnsbruckAustria\",\n \"Department of Physics\\nUniversity of Lancaster\\nLA1 4YBLancasterUnited Kingdom\",\n \"Institut de Physique Nucléaire\\nDépartement de Physique\\nUniversité Catholique de Louvain\\n1348Louvain-la-NeuveBelgium\",\n \"Institut für Physik\\nUniversität Mainz\\nD-55099MainzGermany\",\n \"Centre de Physique des Particules de Marseille\\nUniv Méditerranée\\nIN 2 P 3 -CNRSF-13288MarseilleFrance\",\n \"Dipartimento di Fisica\\nUniversità di Milano\",\n \"INFN Sezione di Milano\\nI-20133MilanoItaly\",\n \"Max-Planck-Institut für Physik\\nWerner-Heisenberg-Institut\\nD-80805MünchenGermany\",\n \"Laboratoire de l'Accélérateur Linéaire\\nUniversité de Paris-Sud\\nIN 2 P 3 -CNRSF-91898Orsay CedexFrance\",\n \"Dipartimento di Fisica dell'Università\\nINFN Sezione di Pisa\",\n \"Scuola Normale Superiore\\nI-56010PisaItaly\",\n \"Department of Physics\\nRoyal Holloway & Bedford New College\\nUniversity of London\\nTW20 OEXEghamSurreyUnited Kingdom\",\n \"Particle Physics Dept\\nOX11 OQXRutherford Appleton Laboratory, Chilton, DidcotOxonUnited Kingdom\",\n \"DAPNIA/Service de Physique des Particules\\nCEA\\nCE-Saclay\\nF-91191Gif-sur-Yvette CedexFrance\",\n \"Institute for Particle Physics\\nUniversity of California at Santa Cruz\\n95064Santa CruzCAUSA\",\n \"Department of Physics\\nUniversity of Sheffield\\nS3 7RHSheffieldUnited Kingdom\",\n \"Fachbereich Physik\\nUniversität Siegen\\nD-57068SiegenGermany\",\n \"Dipartimento di Fisica\\nUniversità di Trieste\",\n \"INFN Sezione di Trieste\\nI-34127TriesteItaly\",\n \"Experimental Elementary Particle Physics\\nUniversity of Washington\\n98195SeattleWAU.S.A\",\n \"Department of Physics\\nUniversity of Wisconsin\\n53706MadisonWIUSA\",\n \"Institute for Particle Physics\\nETH Hönggerberg\\n8093ZürichSwitzerland\",\n \"Supported by CICYT\\nSpain\",\n \"Universitat de Barcelona\\n08208BarcelonaSpain\",\n \"Supported by Bundesministerium für Bildung und Forschung\\nGermany\",\n \"Supported by the Direction des Sciences de la Matière\\nC.E.A\",\n \"Department of Energy\\nNow at Skyguide\\nSwissair Navigation Services\\nFG03-92ER40689. 23GenevaSwitzerland\",\n \"Now at Centre de Physique des Particules de Marseille\\nUniv Méditerranée\\nF-13288MarseilleFrance\",\n \"Also at Groupe d' Astroparticules de Montpellier\\nUniversité de Montpellier II\\n34095MontpellierFrance\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Searches for scalar top, scalar bottom and mass-degenerate scalar quarks are performed in the data collected by the ALEPH detector at LEP, at centre-of-mass energies up to 209 GeV, corresponding to an integrated luminosity of 675 pb −1 .No evidence for the production of such particles is found in the decay channels t → c/u χ, t → bℓν, b → bχ, q → qχ or in the stop four-body decay channel t → bχf f′ studied for the first time at LEP.The results of these searches yield improved mass lower limits.In particular, an absolute lower limit of 63 GeV/c 2 is obtained for the stop mass, at 95% confidence level, irrespective of the stop lifetime and decay branching ratios."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"The results of searches for scalar quarks with the data collected in the year 2000 by the ALEPH detector at LEP are presented in this letter.The energies and integrated luminosities of the analysed data samples are given in Table 1.Previous results obtained with lower energy data have been reported by ALEPH in Refs.[1][2][3][4][5] and by the other LEP collaborations in Refs.[6][7][8].The theoretical framework for these studies is the supersymmetric extension of the Standard Model [9], with R-parity conservation.The lightest supersymmetric particle (LSP) is assumed to be the lightest neutralino χ or the sneutrino ν.Such an LSP is stable and weakly interacting.Each chirality state of the Standard Model fermions has a scalar supersymmetric partner.The scalar quarks (squarks) qL and qR are the supersymmetric partners of the left-handed and right-handed quarks, respectively.The mass eigenstates are orthogonal combinations of the weak interaction eigenstates qL and qR .The mixing angle θ q is defined in such a way that q = qL cos θ q + qR sin θ q is the lighter squark.The off-diagonal terms of the mass matrix, responsible for mixing, read, with standard notation: m q (A q − µκ), with κ = tan β for downtype and κ = 1/ tan β for up-type quarks.Since the size of this mixing term is proportional to the mass of the Standard Model partner, it could well be that the lightest supersymmetric charged particle is the lighter scalar top (stop, t) or, in particular for large tan β values, the lighter scalar bottom (sbottom, b).Squarks could be produced at LEP in pairs, e + e − → qq , via s-channel exchange of a virtual photon or Z.The production cross section [10] depends on θ q when mixing is relevant, i.e., for stops and sbottoms.\nThe searches for stops described here assume that all supersymmetric particles except the lightest neutralino χ and possibly the sneutrino ν are heavier than the stop.Under these assumptions, the allowed decay channels are t → c/u χ, t → bχf f′ and t → bℓν [10,11].The corresponding diagrams are shown in Fig. 1.The decay t → cχ (Fig. 1a) proceeds only via loops and has a very small width, of the order of 0.01-1 eV [10], depending on the mass difference ∆M between the stop and the neutralino, and on the masses and field content of the particles involved in the loops.For low enough ∆M values (∆M < ∼ 6 GeV/c 2 ), the stop lifetime becomes sizeable, and must be taken into account in the searches for stop production.If ∆M is so small that the t → cχ channel is kinematically closed, the dominant decay mode becomes t → uχ, and the stop can then be considered as stable for practical purposes.For stop masses accessible at LEP, i.e., < ∼ 100 GeV/c 2 , the decay mode t → bχf f′ is mediated by virtual chargino and W (Fig. 1b) or sfermion (Fig. 1c) exchange.It is therefore of the same order in perturbation theory as the loop-induced t → cχ decay, and can be substantially enhanced if charginos have masses not much larger than their present experimental bounds, and could even dominate for light sfermions [11].The four-body decay channel yields topologies with b-jets, additional jets and/or leptons, and with missing mass and missing energy.A new multi-jet analysis, hereafter called MJ, has been designed to cope with these final states.\nThe t → bℓν channel proceeds via virtual chargino exchange (Fig. 1d) and has a width of the order of 0.1-10 keV [10].This decay channel dominates when it is kinematically allowed, i.e., if the lightest ν is lighter than the stop.If the lightest neutralino is the LSP, the sneutrino decays invisibly into χν without any change in the experimental topology.\nUnder the assumption that the b is lighter than all supersymmetric particles except the χ, the b will decay as b → bχ (Fig. 1e).Compared to the t, the b decay width is larger, of the order of 10-100 MeV.\nThe supersymmetric partners of the light quarks are generally expected to be quite heavy.If they are light enough to be within the reach of LEP, their dominant decay mode is expected to be q → qχ.\nThe final state topologies addressed by the searches presented in this letter are summarised in Table 2, together with the related signal processes and with the references where analysis details can be found.This letter is organised as follows.In Section 2, the ALEPH detector and the simulated samples used for the analyses are described.Section 3 is dedicated to the selection algorithms with emphasis on the new search for four-body stop decays.In Section 4 the results of the searches are given, along with their interpretation in the theoretical framework.The conclusions of the letter are given in Section 5."},{"title":"ALEPH detector and event simulation","content":"A thorough description of the ALEPH detector and of its performance, as well as of the standard reconstruction and analysis algorithms, can be found in Refs.[12,13].Only a brief summary is given here.\nThe trajectories of charged particles are measured by a silicon vertex detector (VDET), a cylindrical multi-wire drift chamber (ITC) and a large time projection chamber (TPC).These detectors are immersed in an axial magnetic field of 1.5 T provided by a superconducting solenoidal coil.The VDET consists of two cylindrical layers of silicon microstrip detectors; it performs precise measurements of the impact parameter in space, yielding powerful short-lived particle tags, as described in Ref. [14].\nThe electromagnetic calorimeter (ECAL), placed between the TPC and the coil, is a highlysegmented sandwich of lead planes and proportional wire chambers.It consists of a barrel and two endcaps.The hadron calorimeter (HCAL) consists of the iron return yoke of the magnet instrumented with streamer tubes.It is surrounded by two double layers of streamer tubes, the muon chambers.The luminosity monitors (LCAL and SiCAL) extend the calorimeter hermeticity down to 34 mrad from the beam axis.\nThe energy flow algorithm described in Ref. [13] combines the measurements of the tracking detectors and of the calorimeters into \"objects\" classified as charged particles, photons, and neutral hadrons.The energy resolution achieved with this algorithm is (0.6 √ E + 0.6) GeV (E in GeV).Electrons are identified by comparing the energy deposit in ECAL to the momentum measured in the tracking system, by using the shower profile in the electromagnetic calorimeter, and by the measurement of the specific ionization energy loss in the TPC.The identification of muons makes use of the hit pattern in HCAL and of the muon chambers.\nSignal event samples were simulated with the generator described in Ref. [1] for t → cχ, b → bχ, q → qχ and t → bℓν.A modified version of this generator was designed to simulate the channel t → bχf f′ , where the final state is modelled according to phase space and including parton shower development.The generation of t → c/u χ with lifetime follows the procedure described in Ref. [4].\nTo simulate the relevant Standard Model background processes, several Monte Carlo generators were used: BHWIDE [15] for Bhabha scattering, KORALZ [16] for µ + µ − and τ + τ − production, PHOT02 [17] for γγ interactions, KORALW [18] for WW production, and PYTHIA [19] for the other processes (e + e − → qq(γ), Weν, Zee, ZZ, Zν ν).The sizes of the simulated samples typically correspond to ten times the integrated luminosity of the data.\nAll background and signal samples were processed through the full detector simulation."},{"title":"Event selections","content":"Several selection algorithms have been developed to search for the topologies given in Table 2.All these channels are characterised by missing energy.The event properties depend significantly on ∆M, the mass difference between the decaying squark and the χ (or the ν in the case of t → bℓν).When ∆M is large, there is a substantial amount of visible energy, and the signal events tend to look like WW, Weν, ZZ, and qq(γ) events.When ∆M is small, the visible energy is small, and the signal events are therefore similar to γγ events.In order to cope with the different signal topologies and background situations, each analysis employs selections dependent on the ∆M range.The stop lifetime may become sizeable at small ∆M, in which case the signal final state topology depends strongly on the t decay length λ t; three different selections are used, each designed to cope with a specific λ t range [4].\nThe optimisation of the selection criteria as well as the best combination of selections as a function of ∆M and λ t were obtained according to the N 95 prescription [20], i.e., by minimisation of the 95% C.L. cross section upper limit expected in the absence of a signal.The selections are mostly independent of the centre-of-mass energy except for an appropriate rescaling of the cuts with √ s when relevant.The selections applied to the year 2000 data follow closely those described in Refs.[1][2][3][4][5] except for the new analysis developed to address the t → bχf f′ decay, hereafter described in some detail."},{"title":"Search for t → bχf f ′","content":"The MJ analysis consists of a small, a large and a very large ∆M selection.These selections are designed to address simultaneously all bχqq ′ and bχℓν final states, independently of the decay branching ratios.The selections use several anti-γγ criteria, reported in Table 3.The cuts are derived from the AJ selection, described in Ref. [1] as well as the variables used.Only the relevant differences are discussed in the following.\nIn the t → bχf f′ channel, the b quark in the final state produces a visible mass higher than in the t → cχ channel.Therefore, for the small ∆M selection, the cut on the number of charged particle tracks N ch is reinforced by requiring N ch > 10, and both the visible mass, M vis , and the visible mass computed excluding the leading lepton, M ex ℓ 1 vis , are required to be greater than 10 GeV/c 2 .These tighter cuts allow others to be loosened: the transverse momentum p t and that calculated excluding the neutral hadrons, p ex NH t , must be greater than 0.005 √ s and 0.01 √ s, respectively.The remaining background is reduced in the small ∆M selection by requiring the thrust to be smaller than 0.875, and by the cut E vis < 0.26 √ s.\nFor the large ∆M selections, the multi-jet signature is addressed by requiring y 45 , as calculated with the DURHAM algorithm [21], to be greater than 0.001.The level of the WW, ZZ and Weν background is reduced by taking advantage of the b-quark content in the Table 3: Criteria used in the MJ selections to address the backgrounds from (A) γγ → qq, (B) γγ → qq with spurious calorimetric objects and (C) γγ → τ + τ − .The † indicates that the cut is applied when the azimuthal angle of the missing momentum φ p miss is within 15 • of the vertical plane."},{"title":"Small ∆M","content":"Large and Very large ∆M\nt → bχf f′ final state.The value of − log 10 P uds is required to be greater than 0.5, where P uds is the b-tag event probability introduced in Ref. [14].This background is further suppressed by a missing mass cut, the location of which is a function of the ∆M of the signal considered.For example, for ∆M = 20, 30 and 40 GeV/c 2 the optimal cuts are M miss / √ s > 0.75, 0.70 and 0.65, respectively.\nThe region where the very large ∆M selection applies is characterised by a higher visible mass.The sliding cut on the missing mass is looser than that in the large ∆M selection.For ∆M = 20, 40 and 60 GeV/c 2 the optimal cuts are M miss / √ s > 0.58, 0.34 and 0.10.Other cuts are then necessary to reduce the background mainly due to WW events.Similarly to the large ∆M case, − log 10 P uds and y 45 are required to be greater than 0.5 and 0.003, respectively.The mean momentum of all reconstructed charged particle tracks must be less than 0.007 √ s.To reduce background from semileptonic W decays, the fraction of visible energy due to charged objects excluding the leading lepton is required to be greater than 0.5, and the leading lepton, if present, must not be isolated, i.e., the additional energy deposited in a 30 • cone around its direction must be at least 50% of its energy.At this level, the remaining background consists of WW events with energy lost in the beam pipe, responsible for the missing mass.These events are rejected by requiring |p z | < 0.1 √ s and the energy E 12 • deposited at polar angle smaller than 12 • to be less than 0.015 √ s.\nThe efficiencies of the three selections were parametrised as a function of ∆M for each stop pair final state that may result from the decay channels considered ( t → cχ, t → bχℓν, t → bχqq ′ ).This allows the signal efficiency to be parametrised as a function of the branching ratios.The efficiencies were checked to be practically independent of the lepton flavour (e, µ, τ ) in the t → bχℓν decay.The small, large and very large ∆M selections are combined using the N 95 procedure as a function of ∆M and of the branching ratios.The background to the small ∆M selection is dominated by γγ → qq events and has a total expectation of 5.0 fb, while the backgrounds of the large and very large ∆M selections, dominated by WW and other four-fermion processes amount to 3.5 and 4.4 fb, respectively."},{"title":"Systematic uncertainties","content":"The efficiencies of the MJ analysis may be affected by uncertainties regarding the assumptions on the stop hadron physics and by uncertainties related to the detector response.The results of the systematic studies are summarised in Table 4 for the three selections.\nThe systematic effects from the assumptions on the stop hadron physics were assessed by varying the parameters of the model implemented in the generator as in Ref. [1].The uncertainties from the stop hadron mass were evaluated by varying the effective spectator mass M eff , set to 0.5 GeV/c 2 in the analysis, in the range between 0.3 and 1.0 GeV/c 2 .The efficiencies of the large and very large ∆M selections are almost insensitive to this change.The 9% effect found for the small ∆M selection reflects the variation in the invariant mass available for the hadronic system.\nThe systematic error due to the uncertainty on the stop fragmentation was evaluated by varying ǫ t by an order of magnitude, where ǫ t is the parameter of the Peterson fragmentation function [22].The effect on the efficiency is very small (∼ 2%).The amount of initial state radiation in stop pair production depends on the value of the stop coupling to the Z boson, which is controlled by the stop mixing angle.A variation of θ t from 56 • to 0 • , i.e., from minimal to maximal coupling, was applied.The effect was found to be small in all selections, at the level of 1 to 3%.\nDetector effects have been studied for the variables used in the selections.The distributions of all relevant variables show good agreement with the simulation.In particular, the b-tagging performance was checked on hadronic events collected at the Z resonance.The systematic errors associated to detector effects and to the reconstruction procedure were found to be negligible.\nBeam-related background, not included in the event simulation, may affect the E 12 • variable.Its effect on the selection efficiency was determined from data collected at random beam crossings.The net effect is a relative decrease of the signal efficiency by about 5%.The uncertainty on this correction is negligible.\nFinally, an additional uncertainty of 3% due to the limited Monte Carlo statistics was added.The total systematic uncertainty is at the level of 10% for the small ∆M selection.It is dominated by the limited knowledge of the stop hadron physics, and results from rather extreme changes in the model parameters.The systematic uncertainties for the large and very large ∆M selections are at the level of 4-5%.\nThe systematic uncertainties in the selections other than for the t → bχf f′ channel are essentially identical to those reported in Refs.[4,5]."},{"title":"Results and interpretation","content":"The numbers of candidate events selected and background events expected are reported in Table 5 for all the data samples used to derive the results below.An overall agreement is observed.In particular, a total of six candidate events is selected by the new MJ analysis, with 8.5 events expected from background processes; two events are found by each of the selections, in agreement with predictions of 3.3, 2.3 and 2.9 background events at small, large and very large ∆M, respectively.In the framework of the supersymmetric extension of the Standard Model [9], the outcome of these searches can be translated into constraints in the space of the relevant parameters.In this process the systematic uncertainties on the selection efficiencies were included according to the method described in Ref. [23], and no background subtraction was applied.The constraints discussed below, derived from the results given in Table 5, are at 95% confidence level.\nThe regions excluded in the plane (M t, M χ ) under the hypothesis of a dominant t → c/u χ decay are shown in Fig. 2a for two values of the t mixing angle θ t, 0 • and 56 • , corresponding to maximal and vanishing tt Z coupling, respectively.For 8 GeV/c 2 < ∆M < M W + M b , and using also CDF results [24], the lower limit on M t is 92 GeV/c 2 , independent of θ t.\nThe very small ∆M corridor is partially covered by the \"long-lived hadrons\" analysis as indicated by the plain dark region in Fig. 2a.The stop mass lower limit provided by the \"longlived hadrons\" analysis is shown in Fig. 2b as a function of log(cτ t/cm) for various ∆M values.The smallest ∆M value considered is 1.6 GeV/c 2 , corresponding to the \"effective\" kinematic limit for the decay t → cχ [4].Below that ∆M value, the stop decay mode is t → uχ, and the limit is 95 GeV/c 2 , given by the large lifetime selection.The absolute mass lower limit obtained is 63 GeV/c 2 .It is reached for ∆M = 1.6 GeV/c 2 and for a cτ t value of ∼ 1 cm.In that configuration of parameters, the \"AJ small ∆M\" and the \"long-lived hadrons intermediate lifetime\" selections are combined.\nIn the MSSM [9], more restrictive constraints on the stop mass can be obtained since ∆M and the stop lifetime are related.The mass lower limit obtained by scanning over the relevant model parameters as in Ref. [4] is shown in Fig. 2c as a function of tan β.For any tan β, the stop mass limit is 65 GeV/c 2 , reached for tan β ∼ 2.7.\nUnder the hypothesis that the decay t → bχf f′ is dominant, the regions excluded in the plane (M t, M χ ) are shown in Fig. 3a, for relative proportions of the possible f f′ final states as in W * decays.In Fig. 3b the leptonic modes bχℓν (with equal branching ratios for ℓ = e, µ and τ ) are assumed to be dominant.The excluded regions are given for θ t = 0 • and θ t = 56 • .For ∆M > 8 GeV/c 2 , the θ t-independent lower limits on M t are 78 GeV/c 2 and 80 GeV/c 2 , for the two cases of W * and leptonic final state dominance, respectively.\nThe combination of the AJ and MJ analyses allows constraints to be set under the more general hypothesis that both the t → cχ and t → bχf f′ decay channels contribute to stop decays.The excluded regions in the plane (M t, M χ ) are shown in Fig. 4a for θ t = 0 • and θ t = 56 • .This result was obtained by arbitrarily varying the t → cχ branching ratio and the leptonic fraction in the t → bχf f′ decay, and by using the N 95 prescription to determine the appropriate combination of selections.The stop mass limit is shown in Fig. 4b as a function of the branching ratio BR ( t → cχ) for several fixed ∆M values and for θ t = 56 • .The smallest ∆M value considered is 5 GeV/c 2 , corresponding to the threshold for the production of a b quark in the final state.The lowest limit obtained is 63 GeV/c 2 ; it is reached for ∆M = 5 GeV/c 2 , BR ( t → cχ) = 0.22, and BR ( t → bχℓν) = 0.55.\nUnder the assumption that the t → bℓν decay mode is dominant, with equal branching ratios for ℓ = e, µ and τ , the excluded region in the plane (M t, M ν ) is shown in Fig. 5a.If ∆M > 8 GeV/c 2 , and using the LEP1 limit on the sneutrino mass and D0 results [25], the lower limit on M t is 97 GeV/c 2 , independent of θ t.The lower limit is 82 GeV/c 2 if the t → bτ ντ decay mode is dominant and ∆M > 8 GeV/c 2 , independent of θ t.\nThe excluded region in the plane (M b, M χ ) is shown in Fig. 5b under the assumption of a dominant b → bχ decay.Taking also the CDF exclusion [24] into account, a lower limit of 89 GeV/c 2 is set on M b, for any b mixing angle and ∆M > 8 GeV/c 2 .The region excluded for θ b = 0 • , for which the bb Z coupling is maximal, is also shown.\nAs discussed in detail in Ref. [2], the results of the search for acoplanar jets, with or without b tagging, can also be translated into constraints on the mass of degenerate squarks.In order to compare these results with those obtained at the Tevatron [24,25], limits have been evaluated within the MSSM [9] under the following assumptions: a degenerate mass M q for all left-handed and right-handed ũ, d, c, s, b squarks; lowest order GUT relation between the soft supersymmetry breaking gaugino mass terms, allowing the gluino and neutralino masses to be related; tan β = 4 and µ = −400 GeV.The results in the plane (M g, M q) are shown in Fig. 6.Improved constraints are obtained in the region of small q to χ mass differences."},{"title":"Conclusions","content":"Searches for signals of pair-produced scalar partners of quarks have been performed in the data sample of 207 pb −1 collected in the year 2000 with the ALEPH detector at LEP, at centreof-mass energies ranging from 204 to 209 GeV.The final state topologies studied arise from the decays t → c/u χ, t → bχf f′ , t → bℓν, b → bχ, and q → qχ.The four-body stop decay channel was analysed for the first time at LEP, and the corresponding selections were extended to the 675 pb −1 of data collected by ALEPH at centre-of-mass energies of 183 GeV and above.All numbers of candidate events observed are consistent with the backgrounds expected from Standard Model processes.The results of these searches, combined with earlier ones obtained with data collected from 1997 to 1999, have been translated into improved mass lower limits, of which relevant examples are given in Table 6.In particular, a 95% C.L. lower limit of 63 GeV/c 2 has been set on the stop mass, irrespective of its lifetime and decay branching ratios."},{"title":"ALEPH","content":"q → q χ Figure 6: Excluded regions at 95% C.L. from the search for generic q pairs, assuming five massdegenerate q flavours.The results are shown in the gluino-squark mass plane for tan β = 4 and µ = −400 GeV, together with results from experiments at pp colliders."},{"title":"1","content":""},{"title":"1 :","content":"Squark decay diagrams considered in this letter: (a) t → c/u χ; (b) t → bχf f′ via W exchange and (c) t → bχf f′ via sfermion exchange; (d) t → bℓν; (e) b → bχ."},{"title":"Figure 2 :","content":"Figure2: (a) Excluded regions at 95% C.L. in the M χ vs M t plane from t → c/uχ searches; the excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.The dark region in the small ∆M corridor is excluded by the \"long-lived hadrons\" analysis.The CDF experiment result is also indicated.(b) Stop mass lower limit at 95% C.L. from the \"long-lived hadrons\" as a function of log(cτ t/cm) for several ∆M values, without any assumption on the relation between ∆M and the stop lifetime τ t.(c) Stop mass lower limit at 95% C.L. from the \"long-lived hadrons\" analysis as a function of tan β, independent of the other MSSM parameters."},{"title":"→ b χ l ν Figure 3 :Figure 4 :Figure 5 :","content":"Figure 4: (a) Branching ratio independent excluded regions at 95% C.L. in the M χ vs M t plane, from t → bχf f′ and t → cχ searches.The excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.(b) Limit on the stop mass at 95% C.L. as a function of BR ( t → cχ) for various ∆M values.The limits are given for θ t = 56 • ."},{"title":"Table 1 :","content":"Integrated luminosities, centre-of-mass energy ranges and mean centre-of-mass energy values for the data collected by the ALEPH detector in the year 2000."},{"title":"Table 2 :","content":"Topologies studied in the different scenarios."},{"title":"Table 4 :","content":"Summary of the relative systematic uncertainties on the efficiencies of the MJ analysis."},{"title":"Table 5 :","content":"Numbers of candidate events observed (N obs ) and expected from background (N exp ) for the different selections.Also given are the sizes ( Ldt) and the average centre-of-mass energies √ s of the samples analysed.exp N obs N exp N obs N exp N obs N exp"},{"title":"Table 6 :","content":"Lower limits on stop and sbottom masses in some relevant cases.All limits are valid for any value of the mixing angle."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"The results of searches for scalar quarks with the data collected in the year 2000 by the ALEPH detector at LEP are presented in this letter.The energies and integrated luminosities of the analysed data samples are given in Table 1.Previous results obtained with lower energy data have been reported by ALEPH in Refs.[1][2][3][4][5] and by the other LEP collaborations in Refs.[6][7][8].The theoretical framework for these studies is the supersymmetric extension of the Standard Model [9], with R-parity conservation.The lightest supersymmetric particle (LSP) is assumed to be the lightest neutralino χ or the sneutrino ν.Such an LSP is stable and weakly interacting.Each chirality state of the Standard Model fermions has a scalar supersymmetric partner.The scalar quarks (squarks) qL and qR are the supersymmetric partners of the left-handed and right-handed quarks, respectively.The mass eigenstates are orthogonal combinations of the weak interaction eigenstates qL and qR .The mixing angle θ q is defined in such a way that q = qL cos θ q + qR sin θ q is the lighter squark.The off-diagonal terms of the mass matrix, responsible for mixing, read, with standard notation: m q (A q − µκ), with κ = tan β for downtype and κ = 1/ tan β for up-type quarks.Since the size of this mixing term is proportional to the mass of the Standard Model partner, it could well be that the lightest supersymmetric charged particle is the lighter scalar top (stop, t) or, in particular for large tan β values, the lighter scalar bottom (sbottom, b).Squarks could be produced at LEP in pairs, e + e − → qq , via s-channel exchange of a virtual photon or Z.The production cross section [10] depends on θ q when mixing is relevant, i.e., for stops and sbottoms.\\nThe searches for stops described here assume that all supersymmetric particles except the lightest neutralino χ and possibly the sneutrino ν are heavier than the stop.Under these assumptions, the allowed decay channels are t → c/u χ, t → bχf f′ and t → bℓν [10,11].The corresponding diagrams are shown in Fig. 1.The decay t → cχ (Fig. 1a) proceeds only via loops and has a very small width, of the order of 0.01-1 eV [10], depending on the mass difference ∆M between the stop and the neutralino, and on the masses and field content of the particles involved in the loops.For low enough ∆M values (∆M < ∼ 6 GeV/c 2 ), the stop lifetime becomes sizeable, and must be taken into account in the searches for stop production.If ∆M is so small that the t → cχ channel is kinematically closed, the dominant decay mode becomes t → uχ, and the stop can then be considered as stable for practical purposes.For stop masses accessible at LEP, i.e., < ∼ 100 GeV/c 2 , the decay mode t → bχf f′ is mediated by virtual chargino and W (Fig. 1b) or sfermion (Fig. 1c) exchange.It is therefore of the same order in perturbation theory as the loop-induced t → cχ decay, and can be substantially enhanced if charginos have masses not much larger than their present experimental bounds, and could even dominate for light sfermions [11].The four-body decay channel yields topologies with b-jets, additional jets and/or leptons, and with missing mass and missing energy.A new multi-jet analysis, hereafter called MJ, has been designed to cope with these final states.\\nThe t → bℓν channel proceeds via virtual chargino exchange (Fig. 1d) and has a width of the order of 0.1-10 keV [10].This decay channel dominates when it is kinematically allowed, i.e., if the lightest ν is lighter than the stop.If the lightest neutralino is the LSP, the sneutrino decays invisibly into χν without any change in the experimental topology.\\nUnder the assumption that the b is lighter than all supersymmetric particles except the χ, the b will decay as b → bχ (Fig. 1e).Compared to the t, the b decay width is larger, of the order of 10-100 MeV.\\nThe supersymmetric partners of the light quarks are generally expected to be quite heavy.If they are light enough to be within the reach of LEP, their dominant decay mode is expected to be q → qχ.\\nThe final state topologies addressed by the searches presented in this letter are summarised in Table 2, together with the related signal processes and with the references where analysis details can be found.This letter is organised as follows.In Section 2, the ALEPH detector and the simulated samples used for the analyses are described.Section 3 is dedicated to the selection algorithms with emphasis on the new search for four-body stop decays.In Section 4 the results of the searches are given, along with their interpretation in the theoretical framework.The conclusions of the letter are given in Section 5.\"\n },\n {\n \"title\": \"ALEPH detector and event simulation\",\n \"content\": \"A thorough description of the ALEPH detector and of its performance, as well as of the standard reconstruction and analysis algorithms, can be found in Refs.[12,13].Only a brief summary is given here.\\nThe trajectories of charged particles are measured by a silicon vertex detector (VDET), a cylindrical multi-wire drift chamber (ITC) and a large time projection chamber (TPC).These detectors are immersed in an axial magnetic field of 1.5 T provided by a superconducting solenoidal coil.The VDET consists of two cylindrical layers of silicon microstrip detectors; it performs precise measurements of the impact parameter in space, yielding powerful short-lived particle tags, as described in Ref. [14].\\nThe electromagnetic calorimeter (ECAL), placed between the TPC and the coil, is a highlysegmented sandwich of lead planes and proportional wire chambers.It consists of a barrel and two endcaps.The hadron calorimeter (HCAL) consists of the iron return yoke of the magnet instrumented with streamer tubes.It is surrounded by two double layers of streamer tubes, the muon chambers.The luminosity monitors (LCAL and SiCAL) extend the calorimeter hermeticity down to 34 mrad from the beam axis.\\nThe energy flow algorithm described in Ref. [13] combines the measurements of the tracking detectors and of the calorimeters into \\\"objects\\\" classified as charged particles, photons, and neutral hadrons.The energy resolution achieved with this algorithm is (0.6 √ E + 0.6) GeV (E in GeV).Electrons are identified by comparing the energy deposit in ECAL to the momentum measured in the tracking system, by using the shower profile in the electromagnetic calorimeter, and by the measurement of the specific ionization energy loss in the TPC.The identification of muons makes use of the hit pattern in HCAL and of the muon chambers.\\nSignal event samples were simulated with the generator described in Ref. [1] for t → cχ, b → bχ, q → qχ and t → bℓν.A modified version of this generator was designed to simulate the channel t → bχf f′ , where the final state is modelled according to phase space and including parton shower development.The generation of t → c/u χ with lifetime follows the procedure described in Ref. [4].\\nTo simulate the relevant Standard Model background processes, several Monte Carlo generators were used: BHWIDE [15] for Bhabha scattering, KORALZ [16] for µ + µ − and τ + τ − production, PHOT02 [17] for γγ interactions, KORALW [18] for WW production, and PYTHIA [19] for the other processes (e + e − → qq(γ), Weν, Zee, ZZ, Zν ν).The sizes of the simulated samples typically correspond to ten times the integrated luminosity of the data.\\nAll background and signal samples were processed through the full detector simulation.\"\n },\n {\n \"title\": \"Event selections\",\n \"content\": \"Several selection algorithms have been developed to search for the topologies given in Table 2.All these channels are characterised by missing energy.The event properties depend significantly on ∆M, the mass difference between the decaying squark and the χ (or the ν in the case of t → bℓν).When ∆M is large, there is a substantial amount of visible energy, and the signal events tend to look like WW, Weν, ZZ, and qq(γ) events.When ∆M is small, the visible energy is small, and the signal events are therefore similar to γγ events.In order to cope with the different signal topologies and background situations, each analysis employs selections dependent on the ∆M range.The stop lifetime may become sizeable at small ∆M, in which case the signal final state topology depends strongly on the t decay length λ t; three different selections are used, each designed to cope with a specific λ t range [4].\\nThe optimisation of the selection criteria as well as the best combination of selections as a function of ∆M and λ t were obtained according to the N 95 prescription [20], i.e., by minimisation of the 95% C.L. cross section upper limit expected in the absence of a signal.The selections are mostly independent of the centre-of-mass energy except for an appropriate rescaling of the cuts with √ s when relevant.The selections applied to the year 2000 data follow closely those described in Refs.[1][2][3][4][5] except for the new analysis developed to address the t → bχf f′ decay, hereafter described in some detail.\"\n },\n {\n \"title\": \"Search for t → bχf f ′\",\n \"content\": \"The MJ analysis consists of a small, a large and a very large ∆M selection.These selections are designed to address simultaneously all bχqq ′ and bχℓν final states, independently of the decay branching ratios.The selections use several anti-γγ criteria, reported in Table 3.The cuts are derived from the AJ selection, described in Ref. [1] as well as the variables used.Only the relevant differences are discussed in the following.\\nIn the t → bχf f′ channel, the b quark in the final state produces a visible mass higher than in the t → cχ channel.Therefore, for the small ∆M selection, the cut on the number of charged particle tracks N ch is reinforced by requiring N ch > 10, and both the visible mass, M vis , and the visible mass computed excluding the leading lepton, M ex ℓ 1 vis , are required to be greater than 10 GeV/c 2 .These tighter cuts allow others to be loosened: the transverse momentum p t and that calculated excluding the neutral hadrons, p ex NH t , must be greater than 0.005 √ s and 0.01 √ s, respectively.The remaining background is reduced in the small ∆M selection by requiring the thrust to be smaller than 0.875, and by the cut E vis < 0.26 √ s.\\nFor the large ∆M selections, the multi-jet signature is addressed by requiring y 45 , as calculated with the DURHAM algorithm [21], to be greater than 0.001.The level of the WW, ZZ and Weν background is reduced by taking advantage of the b-quark content in the Table 3: Criteria used in the MJ selections to address the backgrounds from (A) γγ → qq, (B) γγ → qq with spurious calorimetric objects and (C) γγ → τ + τ − .The † indicates that the cut is applied when the azimuthal angle of the missing momentum φ p miss is within 15 • of the vertical plane.\"\n },\n {\n \"title\": \"Small ∆M\",\n \"content\": \"Large and Very large ∆M\\nt → bχf f′ final state.The value of − log 10 P uds is required to be greater than 0.5, where P uds is the b-tag event probability introduced in Ref. [14].This background is further suppressed by a missing mass cut, the location of which is a function of the ∆M of the signal considered.For example, for ∆M = 20, 30 and 40 GeV/c 2 the optimal cuts are M miss / √ s > 0.75, 0.70 and 0.65, respectively.\\nThe region where the very large ∆M selection applies is characterised by a higher visible mass.The sliding cut on the missing mass is looser than that in the large ∆M selection.For ∆M = 20, 40 and 60 GeV/c 2 the optimal cuts are M miss / √ s > 0.58, 0.34 and 0.10.Other cuts are then necessary to reduce the background mainly due to WW events.Similarly to the large ∆M case, − log 10 P uds and y 45 are required to be greater than 0.5 and 0.003, respectively.The mean momentum of all reconstructed charged particle tracks must be less than 0.007 √ s.To reduce background from semileptonic W decays, the fraction of visible energy due to charged objects excluding the leading lepton is required to be greater than 0.5, and the leading lepton, if present, must not be isolated, i.e., the additional energy deposited in a 30 • cone around its direction must be at least 50% of its energy.At this level, the remaining background consists of WW events with energy lost in the beam pipe, responsible for the missing mass.These events are rejected by requiring |p z | < 0.1 √ s and the energy E 12 • deposited at polar angle smaller than 12 • to be less than 0.015 √ s.\\nThe efficiencies of the three selections were parametrised as a function of ∆M for each stop pair final state that may result from the decay channels considered ( t → cχ, t → bχℓν, t → bχqq ′ ).This allows the signal efficiency to be parametrised as a function of the branching ratios.The efficiencies were checked to be practically independent of the lepton flavour (e, µ, τ ) in the t → bχℓν decay.The small, large and very large ∆M selections are combined using the N 95 procedure as a function of ∆M and of the branching ratios.The background to the small ∆M selection is dominated by γγ → qq events and has a total expectation of 5.0 fb, while the backgrounds of the large and very large ∆M selections, dominated by WW and other four-fermion processes amount to 3.5 and 4.4 fb, respectively.\"\n },\n {\n \"title\": \"Systematic uncertainties\",\n \"content\": \"The efficiencies of the MJ analysis may be affected by uncertainties regarding the assumptions on the stop hadron physics and by uncertainties related to the detector response.The results of the systematic studies are summarised in Table 4 for the three selections.\\nThe systematic effects from the assumptions on the stop hadron physics were assessed by varying the parameters of the model implemented in the generator as in Ref. [1].The uncertainties from the stop hadron mass were evaluated by varying the effective spectator mass M eff , set to 0.5 GeV/c 2 in the analysis, in the range between 0.3 and 1.0 GeV/c 2 .The efficiencies of the large and very large ∆M selections are almost insensitive to this change.The 9% effect found for the small ∆M selection reflects the variation in the invariant mass available for the hadronic system.\\nThe systematic error due to the uncertainty on the stop fragmentation was evaluated by varying ǫ t by an order of magnitude, where ǫ t is the parameter of the Peterson fragmentation function [22].The effect on the efficiency is very small (∼ 2%).The amount of initial state radiation in stop pair production depends on the value of the stop coupling to the Z boson, which is controlled by the stop mixing angle.A variation of θ t from 56 • to 0 • , i.e., from minimal to maximal coupling, was applied.The effect was found to be small in all selections, at the level of 1 to 3%.\\nDetector effects have been studied for the variables used in the selections.The distributions of all relevant variables show good agreement with the simulation.In particular, the b-tagging performance was checked on hadronic events collected at the Z resonance.The systematic errors associated to detector effects and to the reconstruction procedure were found to be negligible.\\nBeam-related background, not included in the event simulation, may affect the E 12 • variable.Its effect on the selection efficiency was determined from data collected at random beam crossings.The net effect is a relative decrease of the signal efficiency by about 5%.The uncertainty on this correction is negligible.\\nFinally, an additional uncertainty of 3% due to the limited Monte Carlo statistics was added.The total systematic uncertainty is at the level of 10% for the small ∆M selection.It is dominated by the limited knowledge of the stop hadron physics, and results from rather extreme changes in the model parameters.The systematic uncertainties for the large and very large ∆M selections are at the level of 4-5%.\\nThe systematic uncertainties in the selections other than for the t → bχf f′ channel are essentially identical to those reported in Refs.[4,5].\"\n },\n {\n \"title\": \"Results and interpretation\",\n \"content\": \"The numbers of candidate events selected and background events expected are reported in Table 5 for all the data samples used to derive the results below.An overall agreement is observed.In particular, a total of six candidate events is selected by the new MJ analysis, with 8.5 events expected from background processes; two events are found by each of the selections, in agreement with predictions of 3.3, 2.3 and 2.9 background events at small, large and very large ∆M, respectively.In the framework of the supersymmetric extension of the Standard Model [9], the outcome of these searches can be translated into constraints in the space of the relevant parameters.In this process the systematic uncertainties on the selection efficiencies were included according to the method described in Ref. [23], and no background subtraction was applied.The constraints discussed below, derived from the results given in Table 5, are at 95% confidence level.\\nThe regions excluded in the plane (M t, M χ ) under the hypothesis of a dominant t → c/u χ decay are shown in Fig. 2a for two values of the t mixing angle θ t, 0 • and 56 • , corresponding to maximal and vanishing tt Z coupling, respectively.For 8 GeV/c 2 < ∆M < M W + M b , and using also CDF results [24], the lower limit on M t is 92 GeV/c 2 , independent of θ t.\\nThe very small ∆M corridor is partially covered by the \\\"long-lived hadrons\\\" analysis as indicated by the plain dark region in Fig. 2a.The stop mass lower limit provided by the \\\"longlived hadrons\\\" analysis is shown in Fig. 2b as a function of log(cτ t/cm) for various ∆M values.The smallest ∆M value considered is 1.6 GeV/c 2 , corresponding to the \\\"effective\\\" kinematic limit for the decay t → cχ [4].Below that ∆M value, the stop decay mode is t → uχ, and the limit is 95 GeV/c 2 , given by the large lifetime selection.The absolute mass lower limit obtained is 63 GeV/c 2 .It is reached for ∆M = 1.6 GeV/c 2 and for a cτ t value of ∼ 1 cm.In that configuration of parameters, the \\\"AJ small ∆M\\\" and the \\\"long-lived hadrons intermediate lifetime\\\" selections are combined.\\nIn the MSSM [9], more restrictive constraints on the stop mass can be obtained since ∆M and the stop lifetime are related.The mass lower limit obtained by scanning over the relevant model parameters as in Ref. [4] is shown in Fig. 2c as a function of tan β.For any tan β, the stop mass limit is 65 GeV/c 2 , reached for tan β ∼ 2.7.\\nUnder the hypothesis that the decay t → bχf f′ is dominant, the regions excluded in the plane (M t, M χ ) are shown in Fig. 3a, for relative proportions of the possible f f′ final states as in W * decays.In Fig. 3b the leptonic modes bχℓν (with equal branching ratios for ℓ = e, µ and τ ) are assumed to be dominant.The excluded regions are given for θ t = 0 • and θ t = 56 • .For ∆M > 8 GeV/c 2 , the θ t-independent lower limits on M t are 78 GeV/c 2 and 80 GeV/c 2 , for the two cases of W * and leptonic final state dominance, respectively.\\nThe combination of the AJ and MJ analyses allows constraints to be set under the more general hypothesis that both the t → cχ and t → bχf f′ decay channels contribute to stop decays.The excluded regions in the plane (M t, M χ ) are shown in Fig. 4a for θ t = 0 • and θ t = 56 • .This result was obtained by arbitrarily varying the t → cχ branching ratio and the leptonic fraction in the t → bχf f′ decay, and by using the N 95 prescription to determine the appropriate combination of selections.The stop mass limit is shown in Fig. 4b as a function of the branching ratio BR ( t → cχ) for several fixed ∆M values and for θ t = 56 • .The smallest ∆M value considered is 5 GeV/c 2 , corresponding to the threshold for the production of a b quark in the final state.The lowest limit obtained is 63 GeV/c 2 ; it is reached for ∆M = 5 GeV/c 2 , BR ( t → cχ) = 0.22, and BR ( t → bχℓν) = 0.55.\\nUnder the assumption that the t → bℓν decay mode is dominant, with equal branching ratios for ℓ = e, µ and τ , the excluded region in the plane (M t, M ν ) is shown in Fig. 5a.If ∆M > 8 GeV/c 2 , and using the LEP1 limit on the sneutrino mass and D0 results [25], the lower limit on M t is 97 GeV/c 2 , independent of θ t.The lower limit is 82 GeV/c 2 if the t → bτ ντ decay mode is dominant and ∆M > 8 GeV/c 2 , independent of θ t.\\nThe excluded region in the plane (M b, M χ ) is shown in Fig. 5b under the assumption of a dominant b → bχ decay.Taking also the CDF exclusion [24] into account, a lower limit of 89 GeV/c 2 is set on M b, for any b mixing angle and ∆M > 8 GeV/c 2 .The region excluded for θ b = 0 • , for which the bb Z coupling is maximal, is also shown.\\nAs discussed in detail in Ref. [2], the results of the search for acoplanar jets, with or without b tagging, can also be translated into constraints on the mass of degenerate squarks.In order to compare these results with those obtained at the Tevatron [24,25], limits have been evaluated within the MSSM [9] under the following assumptions: a degenerate mass M q for all left-handed and right-handed ũ, d, c, s, b squarks; lowest order GUT relation between the soft supersymmetry breaking gaugino mass terms, allowing the gluino and neutralino masses to be related; tan β = 4 and µ = −400 GeV.The results in the plane (M g, M q) are shown in Fig. 6.Improved constraints are obtained in the region of small q to χ mass differences.\"\n },\n {\n \"title\": \"Conclusions\",\n \"content\": \"Searches for signals of pair-produced scalar partners of quarks have been performed in the data sample of 207 pb −1 collected in the year 2000 with the ALEPH detector at LEP, at centreof-mass energies ranging from 204 to 209 GeV.The final state topologies studied arise from the decays t → c/u χ, t → bχf f′ , t → bℓν, b → bχ, and q → qχ.The four-body stop decay channel was analysed for the first time at LEP, and the corresponding selections were extended to the 675 pb −1 of data collected by ALEPH at centre-of-mass energies of 183 GeV and above.All numbers of candidate events observed are consistent with the backgrounds expected from Standard Model processes.The results of these searches, combined with earlier ones obtained with data collected from 1997 to 1999, have been translated into improved mass lower limits, of which relevant examples are given in Table 6.In particular, a 95% C.L. lower limit of 63 GeV/c 2 has been set on the stop mass, irrespective of its lifetime and decay branching ratios.\"\n },\n {\n \"title\": \"ALEPH\",\n \"content\": \"q → q χ Figure 6: Excluded regions at 95% C.L. from the search for generic q pairs, assuming five massdegenerate q flavours.The results are shown in the gluino-squark mass plane for tan β = 4 and µ = −400 GeV, together with results from experiments at pp colliders.\"\n },\n {\n \"title\": \"1\",\n \"content\": \"\"\n },\n {\n \"title\": \"1 :\",\n \"content\": \"Squark decay diagrams considered in this letter: (a) t → c/u χ; (b) t → bχf f′ via W exchange and (c) t → bχf f′ via sfermion exchange; (d) t → bℓν; (e) b → bχ.\"\n },\n {\n \"title\": \"Figure 2 :\",\n \"content\": \"Figure2: (a) Excluded regions at 95% C.L. in the M χ vs M t plane from t → c/uχ searches; the excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.The dark region in the small ∆M corridor is excluded by the \\\"long-lived hadrons\\\" analysis.The CDF experiment result is also indicated.(b) Stop mass lower limit at 95% C.L. from the \\\"long-lived hadrons\\\" as a function of log(cτ t/cm) for several ∆M values, without any assumption on the relation between ∆M and the stop lifetime τ t.(c) Stop mass lower limit at 95% C.L. from the \\\"long-lived hadrons\\\" analysis as a function of tan β, independent of the other MSSM parameters.\"\n },\n {\n \"title\": \"→ b χ l ν Figure 3 :Figure 4 :Figure 5 :\",\n \"content\": \"Figure 4: (a) Branching ratio independent excluded regions at 95% C.L. in the M χ vs M t plane, from t → bχf f′ and t → cχ searches.The excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.(b) Limit on the stop mass at 95% C.L. as a function of BR ( t → cχ) for various ∆M values.The limits are given for θ t = 56 • .\"\n },\n {\n \"title\": \"Table 1 :\",\n \"content\": \"Integrated luminosities, centre-of-mass energy ranges and mean centre-of-mass energy values for the data collected by the ALEPH detector in the year 2000.\"\n },\n {\n \"title\": \"Table 2 :\",\n \"content\": \"Topologies studied in the different scenarios.\"\n },\n {\n \"title\": \"Table 4 :\",\n \"content\": \"Summary of the relative systematic uncertainties on the efficiencies of the MJ analysis.\"\n },\n {\n \"title\": \"Table 5 :\",\n \"content\": \"Numbers of candidate events observed (N obs ) and expected from background (N exp ) for the different selections.Also given are the sizes ( Ldt) and the average centre-of-mass energies √ s of the samples analysed.exp N obs N exp N obs N exp N obs N exp\"\n },\n {\n \"title\": \"Table 6 :\",\n \"content\": \"Lower limits on stop and sbottom masses in some relevant cases.All limits are valid for any value of the mixing angle.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"","content":"1\n1"},{"id":"","caption":"Squark decay diagrams considered in this letter: (a) t → c/u χ; (b) t → bχf f′ via W exchange and (c) t → bχf f′ via sfermion exchange; (d) t → bℓν; (e) b → bχ.","content":"1 :\n1\nSquark decay diagrams considered in this letter: (a) t → c/u χ; (b) t → bχf f′ via W exchange and (c) t → bχf f′ via sfermion exchange; (d) t → bℓν; (e) b → bχ."},{"id":"","caption":"Figure2: (a) Excluded regions at 95% C.L. in the M χ vs M t plane from t → c/uχ searches; the excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.The dark region in the small ∆M corridor is excluded by the \"long-lived hadrons\" analysis.The CDF experiment result is also indicated.(b) Stop mass lower limit at 95% C.L. from the \"long-lived hadrons\" as a function of log(cτ t/cm) for several ∆M values, without any assumption on the relation between ∆M and the stop lifetime τ t.(c) Stop mass lower limit at 95% C.L. from the \"long-lived hadrons\" analysis as a function of tan β, independent of the other MSSM parameters.","content":"Figure 2 :\n2\nFigure2: (a) Excluded regions at 95% C.L. in the M χ vs M t plane from t → c/uχ searches; the excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.The dark region in the small ∆M corridor is excluded by the \"long-lived hadrons\" analysis.The CDF experiment result is also indicated.(b) Stop mass lower limit at 95% C.L. from the \"long-lived hadrons\" as a function of log(cτ t/cm) for several ∆M values, without any assumption on the relation between ∆M and the stop lifetime τ t.(c) Stop mass lower limit at 95% C.L. from the \"long-lived hadrons\" analysis as a function of tan β, independent of the other MSSM parameters."},{"id":"","caption":"Figure 4: (a) Branching ratio independent excluded regions at 95% C.L. in the M χ vs M t plane, from t → bχf f′ and t → cχ searches.The excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.(b) Limit on the stop mass at 95% C.L. as a function of BR ( t → cχ) for various ∆M values.The limits are given for θ t = 56 • .","content":"→ b χ l ν Figure 3 :Figure 4 :Figure 5 :\n345\nFigure 4: (a) Branching ratio independent excluded regions at 95% C.L. in the M χ vs M t plane, from t → bχf f′ and t → cχ searches.The excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.(b) Limit on the stop mass at 95% C.L. as a function of BR ( t → cχ) for various ∆M values.The limits are given for θ t = 56 • ."},{"id":"","caption":"Integrated luminosities, centre-of-mass energy ranges and mean centre-of-mass energy values for the data collected by the ALEPH detector in the year 2000.","content":"Table 1 :\n1\nIntegrated luminosities, centre-of-mass energy ranges and mean centre-of-mass energy values for the data collected by the ALEPH detector in the year 2000.\nLuminosity [ pb −1 ] Energy range [GeV]√ s [GeV]9.4 122.6 75.3207 − 209 206 − 207 204 − 206208.0 206.6 205.2"},{"id":"","caption":"Topologies studied in the different scenarios.","content":"Table 2 :\n2\nTopologies studied in the different scenarios.\nProduction Decay modeTopology/AnalysisReferencestt tt tt tt bb qqt → cχ (∆M > ∼ 6 GeV/c 2 ) t → c/u χ (∆M < ∼ 6 GeV/c 2 ) Long-lived hadrons [4] Acoplanar jets (AJ) [1, 2, 3, 5] Multi-jets (MJ) This letter t → bχf f′ AJ plus leptons [1, 2, 3, 5] t → bℓν AJ plus b tagging [1, 2, 3, 5] b → bχ q → qχ AJ [2, 3, 5]"},{"id":"","caption":"Summary of the relative systematic uncertainties on the efficiencies of the MJ analysis.","content":"Table 4 :\n4\nSummary of the relative systematic uncertainties on the efficiencies of the MJ analysis.\nSystematic uncertainties (%)MJ selectionsSmall ∆M Large ∆M Very large ∆MM eff (0.3-1.0 GeV)923ǫ t (10 −5 -10 −4 )222θ t (0 • -56 • )311Detector and reconstruction212Monte Carlo statistics333TOTAL1045"},{"id":"","caption":"Numbers of candidate events observed (N obs ) and expected from background (N exp ) for the different selections.Also given are the sizes ( Ldt) and the average centre-of-mass energies √ s of the samples analysed.exp N obs N exp N obs N exp N obs N exp","content":"Table 5 :\n5\nNumbers of candidate events observed (N obs ) and expected from background (N exp ) for the different selections.Also given are the sizes ( Ldt) and the average centre-of-mass energies √ s of the samples analysed.exp N obs N exp N obs N exp N obs N exp\nYear1997199819992000SampleLdt [ pb −1 ] √ s [ GeV]57.0 182.7173.6 188.6236.9 197.6207.3 206.2Analysis N obs N AJ Selection small ∆M 1 1.535.522.422.1large ∆M43.554.087.3118.6Long-livedsmall λ t.AJ, small ∆Mhadronsintermediate λ t.--00.300.500.4large λ t.--10.400.600.6MJsmall ∆M00.310.711.201.1large ∆M10.200.600.810.7very large ∆M00.200.811.010.9AJsmall ∆M10.801.932.602.4plus leptonslarge ∆M00.120.421.431.6AJsmall ∆M01.133.312.222.0plus b tagginglarge ∆M10.600.910.701.2"},{"id":"","caption":"Lower limits on stop and sbottom masses in some relevant cases.All limits are valid for any value of the mixing angle.","content":"Table 6 :\n6\nLower limits on stop and sbottom masses in some relevant cases.All limits are valid for any value of the mixing angle.\n95% C.L.∆MDominantMass LimitrangedecaySquark [ GeV/c 2 ] [ GeV/c 2 ] channel(s) Commentst92 78 97 63> 8 > 8 > 8 anyt → cχ t → bχW * t → bℓν anyCDF result [24] used LEP 1, D0 result [25] used any branching ratios, any lifetimeb89> 8b → bχCDF result [24] used"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"\",\n \"content\": \"1\\n1\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Squark decay diagrams considered in this letter: (a) t → c/u χ; (b) t → bχf f′ via W exchange and (c) t → bχf f′ via sfermion exchange; (d) t → bℓν; (e) b → bχ.\",\n \"content\": \"1 :\\n1\\nSquark decay diagrams considered in this letter: (a) t → c/u χ; (b) t → bχf f′ via W exchange and (c) t → bχf f′ via sfermion exchange; (d) t → bℓν; (e) b → bχ.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure2: (a) Excluded regions at 95% C.L. in the M χ vs M t plane from t → c/uχ searches; the excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.The dark region in the small ∆M corridor is excluded by the \\\"long-lived hadrons\\\" analysis.The CDF experiment result is also indicated.(b) Stop mass lower limit at 95% C.L. from the \\\"long-lived hadrons\\\" as a function of log(cτ t/cm) for several ∆M values, without any assumption on the relation between ∆M and the stop lifetime τ t.(c) Stop mass lower limit at 95% C.L. from the \\\"long-lived hadrons\\\" analysis as a function of tan β, independent of the other MSSM parameters.\",\n \"content\": \"Figure 2 :\\n2\\nFigure2: (a) Excluded regions at 95% C.L. in the M χ vs M t plane from t → c/uχ searches; the excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.The dark region in the small ∆M corridor is excluded by the \\\"long-lived hadrons\\\" analysis.The CDF experiment result is also indicated.(b) Stop mass lower limit at 95% C.L. from the \\\"long-lived hadrons\\\" as a function of log(cτ t/cm) for several ∆M values, without any assumption on the relation between ∆M and the stop lifetime τ t.(c) Stop mass lower limit at 95% C.L. from the \\\"long-lived hadrons\\\" analysis as a function of tan β, independent of the other MSSM parameters.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 4: (a) Branching ratio independent excluded regions at 95% C.L. in the M χ vs M t plane, from t → bχf f′ and t → cχ searches.The excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.(b) Limit on the stop mass at 95% C.L. as a function of BR ( t → cχ) for various ∆M values.The limits are given for θ t = 56 • .\",\n \"content\": \"→ b χ l ν Figure 3 :Figure 4 :Figure 5 :\\n345\\nFigure 4: (a) Branching ratio independent excluded regions at 95% C.L. in the M χ vs M t plane, from t → bχf f′ and t → cχ searches.The excluded regions are given for θ t = 0 • , corresponding to maximum tt Z coupling, and for θ t = 56 • , corresponding to vanishing tt Z coupling.(b) Limit on the stop mass at 95% C.L. as a function of BR ( t → cχ) for various ∆M values.The limits are given for θ t = 56 • .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Integrated luminosities, centre-of-mass energy ranges and mean centre-of-mass energy values for the data collected by the ALEPH detector in the year 2000.\",\n \"content\": \"Table 1 :\\n1\\nIntegrated luminosities, centre-of-mass energy ranges and mean centre-of-mass energy values for the data collected by the ALEPH detector in the year 2000.\\nLuminosity [ pb −1 ] Energy range [GeV]√ s [GeV]9.4 122.6 75.3207 − 209 206 − 207 204 − 206208.0 206.6 205.2\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Topologies studied in the different scenarios.\",\n \"content\": \"Table 2 :\\n2\\nTopologies studied in the different scenarios.\\nProduction Decay modeTopology/AnalysisReferencestt tt tt tt bb qqt → cχ (∆M > ∼ 6 GeV/c 2 ) t → c/u χ (∆M < ∼ 6 GeV/c 2 ) Long-lived hadrons [4] Acoplanar jets (AJ) [1, 2, 3, 5] Multi-jets (MJ) This letter t → bχf f′ AJ plus leptons [1, 2, 3, 5] t → bℓν AJ plus b tagging [1, 2, 3, 5] b → bχ q → qχ AJ [2, 3, 5]\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Summary of the relative systematic uncertainties on the efficiencies of the MJ analysis.\",\n \"content\": \"Table 4 :\\n4\\nSummary of the relative systematic uncertainties on the efficiencies of the MJ analysis.\\nSystematic uncertainties (%)MJ selectionsSmall ∆M Large ∆M Very large ∆MM eff (0.3-1.0 GeV)923ǫ t (10 −5 -10 −4 )222θ t (0 • -56 • )311Detector and reconstruction212Monte Carlo statistics333TOTAL1045\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Numbers of candidate events observed (N obs ) and expected from background (N exp ) for the different selections.Also given are the sizes ( Ldt) and the average centre-of-mass energies √ s of the samples analysed.exp N obs N exp N obs N exp N obs N exp\",\n \"content\": \"Table 5 :\\n5\\nNumbers of candidate events observed (N obs ) and expected from background (N exp ) for the different selections.Also given are the sizes ( Ldt) and the average centre-of-mass energies √ s of the samples analysed.exp N obs N exp N obs N exp N obs N exp\\nYear1997199819992000SampleLdt [ pb −1 ] √ s [ GeV]57.0 182.7173.6 188.6236.9 197.6207.3 206.2Analysis N obs N AJ Selection small ∆M 1 1.535.522.422.1large ∆M43.554.087.3118.6Long-livedsmall λ t.AJ, small ∆Mhadronsintermediate λ t.--00.300.500.4large λ t.--10.400.600.6MJsmall ∆M00.310.711.201.1large ∆M10.200.600.810.7very large ∆M00.200.811.010.9AJsmall ∆M10.801.932.602.4plus leptonslarge ∆M00.120.421.431.6AJsmall ∆M01.133.312.222.0plus b tagginglarge ∆M10.600.910.701.2\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Lower limits on stop and sbottom masses in some relevant cases.All limits are valid for any value of the mixing angle.\",\n \"content\": \"Table 6 :\\n6\\nLower limits on stop and sbottom masses in some relevant cases.All limits are valid for any value of the mixing angle.\\n95% C.L.∆MDominantMass LimitrangedecaySquark [ GeV/c 2 ] [ GeV/c 2 ] channel(s) Commentst92 78 97 63> 8 > 8 > 8 anyt → cχ t → bχW * t → bℓν anyCDF result [24] used LEP 1, D0 result [25] used any branching ratios, any lifetimeb89> 8b → bχCDF result [24] used\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Searches for scalar top and scalar bottom quarks at LEP2. Phys. Lett. B. 4134311997","doi":""},{"text":"Scalar quark searches in e + e − collisions at √ s = 181 − 184 GeV. Phys. Lett. B. 4341891998","doi":""},{"text":"Searches for slepton and squarks in e + e − collisions at 189 GeV. Phys. Lett. B. 4693031999","doi":""},{"text":"Search for a scalar top almost degenerate with the lightest neutralino in e + e − Collisions at √ s up to 202 GeV. Phys. Lett. B. 4882342000","doi":""},{"text":"Search for Supersymmetric Particles in e + e − Collisions at √ s up to 202 GeV and Mass Limit for the Lightest Neutralino. Phys. Lett. B. 499672001","doi":""},{"text":"Limits on the Masses of Supersymmetric Particles at √ s =.","doi":""},{"text":". Gev, Phys. Lett. B. 489382000","doi":""},{"text":"Searches for Scalar Quarks in e + e − Interactions at √ s = 189 GeV. Phys. Lett. B. 4713081999","doi":""},{"text":"Search for Scalar Top and Scalar Bottom Quarks at √ s = 189 GeV at LEP. Phys. Lett. B. 456951999","doi":""},{"text":"Supersymmetry, Supergravity and Particle Physics. H P Nilles, Phys. Rep. 11011984","doi":""},{"text":"The Search For Supersymmetry: Probing Physics Beyond The Standard Model. H E Haber, G L Kane, Phys. Rep. 117751985","doi":""},{"text":"Looking Beyond The Standard Model: The Supersymmetric Option. R Barbieri, Riv. Nuovo Cim. 11411988","doi":""},{"text":"Light scalar top at e + e − colliders. K Hikasa, M Kobayashi, Phys. Rev. D. 367241987","doi":""},{"text":"Scalar top production in e + e − annihilation. M Drees, K Hikasa, Phys. Lett. B. 2521271990","doi":""},{"text":"Decays of the lightest top squark. C Boehm, A Djouadi, Y Mambrini, Phys. Rev. D. 61950062000","doi":""},{"text":"ALEPH: A Detector for Electron-Positron Annihilations at LEP. Nucl. Instrum. and Methods A. 2941211990","doi":""},{"text":"Performance of the ALEPH detector at LEP. Nucl. Instrum. and Methods A. 3604811995","doi":""},{"text":"A Precise Measurement of Γ(Z → b b)/Γ(Z → hadrons). Phys. Lett. B. 3135351993","doi":""},{"text":"BHWIDE 1.00: O(α) YFS exponentiated Monte Carlo for Bhabha scattering at wide angles for LEP1/SLC and LEP2. S Jadach, W Placzek, B F Ward, Phys. Lett. B. 3902981997","doi":""},{"text":"The Monte Carlo program KORALZ, version 4.0, for the lepton or quark pair production at LEP/SLC energies. S Jadach, B F Ward, Z Was, Comput. Phys. Commun. 795031994","doi":""},{"text":"Two Gamma Physics Versus One Gamma Physics And Whatever Lies In Between. J A M Vermaseren, Proceedings of the IVth international Workshop on Gamma Gamma Interactions. G Cochard, P Kessler, the IVth international Workshop on Gamma Gamma InteractionsSpringer Verlag1980","doi":""},{"text":"An experimental study of γγ → hadrons at LEP. Phys. Lett. B. 3135091993","doi":""},{"text":"Monte Carlo program KORALW 1.42 for all four-fermion final states in e + e − collisions. S Jadach, W Placzek, M Skrzypek, B F Ward, Z Was, Comput. Phys. Commun. 1192721999","doi":""},{"text":"High-energy-physics event generation with PYTHIA 5. T Sjöstrand, JETSET 7.4Comput. Phys. Commun. 82741994","doi":""},{"text":"Search for the Standard Model Higgs Boson. Phys. Lett. B. 3132991993","doi":""},{"text":"Complementary analyses and acceptance optimization in new particle searches. J F Grivaz, F Le Diberder, LAL 92-371992","doi":""},{"text":"Hard QCD Working Group -theory summary. W J Stirling, J. Phys. G. 1715671991","doi":""},{"text":"Scaling violations in inclusive e + e − annihilation spectra. C Peterson, D Schlatter, I Schmitt, P M Zerwas, Phys. Rev. D. 271051983","doi":""},{"text":"Incorporating systematic uncertainties into an upper limit. R D Cousins, V L Highland, Nucl. Instrum. and Methods A. 3203311992","doi":""},{"text":"Search for scalar top and scalar bottom quarks in pp collisions at √ s = 1.8 TeV. Phys. Rev. Lett. 8457042000","doi":""},{"text":"Search for Scalar Top Quark Production in pp Collisions at √ s =.","doi":""},{"text":". Tev, Phys. Rev. Lett. 8452732000","doi":""},{"text":"Search for Gluinos and Scalar Quarks in pp collisions at √ s = 1.8 TeV using the Missing Energy plus Multijets Signature. Phys. Rev. Lett. 88418012002","doi":""},{"text":"Search for Squarks and Gluinos in pp Collisions at √ s = 1.8 TeV. Phys. Rev. Lett. 756181995","doi":""},{"text":"Search for Squarks and Gluinos in Events Containing Jets and Large Imbalance in Transverse Energy. Phys. Rev. Lett. 8349371999","doi":""},{"text":"Search for Bottom Squarks in pp Collisions at √ s = 1.8 TeV. Phys.","doi":""},{"text":"A search for the scalar top quark in pp Collisions at √ s = 1.8 TeV\", submitted to. hep-ex/0108018Phys. Rev. Lett. 2001","doi":""}],"string":"[\n {\n \"text\": \"Searches for scalar top and scalar bottom quarks at LEP2. Phys. Lett. B. 4134311997\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Scalar quark searches in e + e − collisions at √ s = 181 − 184 GeV. Phys. Lett. B. 4341891998\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Searches for slepton and squarks in e + e − collisions at 189 GeV. Phys. Lett. B. 4693031999\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for a scalar top almost degenerate with the lightest neutralino in e + e − Collisions at √ s up to 202 GeV. Phys. Lett. B. 4882342000\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for Supersymmetric Particles in e + e − Collisions at √ s up to 202 GeV and Mass Limit for the Lightest Neutralino. Phys. Lett. B. 499672001\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Limits on the Masses of Supersymmetric Particles at √ s =.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Gev, Phys. Lett. B. 489382000\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Searches for Scalar Quarks in e + e − Interactions at √ s = 189 GeV. Phys. Lett. B. 4713081999\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for Scalar Top and Scalar Bottom Quarks at √ s = 189 GeV at LEP. Phys. Lett. B. 456951999\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Supersymmetry, Supergravity and Particle Physics. H P Nilles, Phys. Rep. 11011984\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Search For Supersymmetry: Probing Physics Beyond The Standard Model. H E Haber, G L Kane, Phys. Rep. 117751985\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Looking Beyond The Standard Model: The Supersymmetric Option. R Barbieri, Riv. Nuovo Cim. 11411988\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Light scalar top at e + e − colliders. K Hikasa, M Kobayashi, Phys. Rev. D. 367241987\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Scalar top production in e + e − annihilation. M Drees, K Hikasa, Phys. Lett. B. 2521271990\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Decays of the lightest top squark. C Boehm, A Djouadi, Y Mambrini, Phys. Rev. D. 61950062000\",\n \"doi\": \"\"\n },\n {\n \"text\": \"ALEPH: A Detector for Electron-Positron Annihilations at LEP. Nucl. Instrum. and Methods A. 2941211990\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Performance of the ALEPH detector at LEP. Nucl. Instrum. and Methods A. 3604811995\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Precise Measurement of Γ(Z → b b)/Γ(Z → hadrons). Phys. Lett. B. 3135351993\",\n \"doi\": \"\"\n },\n {\n \"text\": \"BHWIDE 1.00: O(α) YFS exponentiated Monte Carlo for Bhabha scattering at wide angles for LEP1/SLC and LEP2. S Jadach, W Placzek, B F Ward, Phys. Lett. B. 3902981997\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Monte Carlo program KORALZ, version 4.0, for the lepton or quark pair production at LEP/SLC energies. S Jadach, B F Ward, Z Was, Comput. Phys. Commun. 795031994\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Two Gamma Physics Versus One Gamma Physics And Whatever Lies In Between. J A M Vermaseren, Proceedings of the IVth international Workshop on Gamma Gamma Interactions. G Cochard, P Kessler, the IVth international Workshop on Gamma Gamma InteractionsSpringer Verlag1980\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An experimental study of γγ → hadrons at LEP. Phys. Lett. B. 3135091993\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Monte Carlo program KORALW 1.42 for all four-fermion final states in e + e − collisions. S Jadach, W Placzek, M Skrzypek, B F Ward, Z Was, Comput. Phys. Commun. 1192721999\",\n \"doi\": \"\"\n },\n {\n \"text\": \"High-energy-physics event generation with PYTHIA 5. T Sjöstrand, JETSET 7.4Comput. Phys. Commun. 82741994\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for the Standard Model Higgs Boson. Phys. Lett. B. 3132991993\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Complementary analyses and acceptance optimization in new particle searches. J F Grivaz, F Le Diberder, LAL 92-371992\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Hard QCD Working Group -theory summary. W J Stirling, J. Phys. G. 1715671991\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Scaling violations in inclusive e + e − annihilation spectra. C Peterson, D Schlatter, I Schmitt, P M Zerwas, Phys. Rev. D. 271051983\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Incorporating systematic uncertainties into an upper limit. R D Cousins, V L Highland, Nucl. Instrum. and Methods A. 3203311992\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for scalar top and scalar bottom quarks in pp collisions at √ s = 1.8 TeV. Phys. Rev. Lett. 8457042000\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for Scalar Top Quark Production in pp Collisions at √ s =.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Tev, Phys. Rev. Lett. 8452732000\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for Gluinos and Scalar Quarks in pp collisions at √ s = 1.8 TeV using the Missing Energy plus Multijets Signature. Phys. Rev. Lett. 88418012002\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for Squarks and Gluinos in pp Collisions at √ s = 1.8 TeV. Phys. Rev. Lett. 756181995\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for Squarks and Gluinos in Events Containing Jets and Large Imbalance in Transverse Energy. Phys. Rev. Lett. 8349371999\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Search for Bottom Squarks in pp Collisions at √ s = 1.8 TeV. Phys.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A search for the scalar top quark in pp Collisions at √ s = 1.8 TeV\\\", submitted to. hep-ex/0108018Phys. Rev. Lett. 2001\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":575,"cells":{"text":{"kind":"string","value":"\nCooper-Frye sampling with short-range repulsion\n\n\nVolodymyr Vovchenko \nInstitute for Nuclear Theory\nUniversity of Washington\nBox 35155098195SeattleWAUSA\n\nGiersch Science Center\nFrankfurt Institute for Advanced Studies\nD-60438Frankfurt am MainGermany\n\nand Nuclear Science Division\nLawrence Berkeley National Laboratory\n1 Cyclotron Road94720BerkeleyCAUSA\n\nCooper-Frye sampling with short-range repulsion\n\nThis work incorporates the effect of short-range repulsion between particles into the Cooper-Frye hadron sampling procedure. This is achieved by means of a rejection sampling step, which prohibits any pair of particles from overlapping in the coordinate space, effectively modeling the effect of hard-core repulsion. The new procedure -called the FIST sampler -is based on the package Thermal-FIST. It is used here to study the effect of excluded volume on cumulants of the (net-)proton number distribution in central collisions of heavy ions in a broad collision energy range in conjunction with exact global conservation of baryon number, electric charge, and strangeness. The results are compared with earlier calculations based on analytical approximations, quantifying the accuracy of the latter at different collision energies. An additional advantage of the new method over the analytic approaches is that it offers the flexibility provided by event generators, making it straightforwardly extendable to other observables.\n\nI. INTRODUCTION\n\nThe description of bulk observables in relativistic heavy-ion collisions is usually performed in the framework of relativistic hydrodynamics (see e.g. [1,2] for an overview). This modeling incorporates the Cooper-Frye particlization stage, where the expanding QCD fluid is transformed into a gas of hadrons and resonances. In most cases, hadron momenta are sampled from local Maxwell-Boltzmann distributions, with the possible inclusion of shear and bulk viscous corrections [3][4][5][6][7]. Multiplicity distributions of the sampled hadrons usually correspond to Poisson statistics, implying independent particle emission. More involved descriptions additionally implement a (micro-)canonical treatment of conservation laws, such as that of energy-momentum and QCD conserved charges [8][9][10][11][12]. Proper treatment of conservation laws is relevant for observables involving event-by-event fluctuations [13][14][15][16][17]. Physically, the system created at particlization corresponds to an ideal hadron resonance gas, possibly with non-equilibrium corrections due to shear and bulk viscosities and (micro-)canonical effects. On the other hand, extensions of the Cooper-Frye procedure are necessary to incorporate any additional physics. One commonly discussed extension of the ideal HRG is short-range repulsive interactions utilizing excluded volume [18][19][20]. For example, by introducing the excluded volume effect into baryon-baryon (and, by symmetry, antibaryonantibaryon) interaction [21], one can effectively model the presence of hard core in nucleon-nucleon scattering and improve the description of several lattice QCD susceptibilities [22][23][24][25]. Another extension would concern the search for the QCD critical point, which would lead to large non-Gaussian fluctuations of the proton number [26] requiring a generalized Cooper-Frye routine [27,28].\n\nPreviously, a method called subensemble sampler was introduced in [16] to perform particlization of an interacting hadron resonance gas and was used to study the influence of baryonic excluded volume on proton and baryon number cumulants in Pb-Pb collisions at LHC energies. Although this method is rather generic, it requires partitioning the Cooper-Frye hypersurfaces into patches which should, on the one hand, be large enough to capture all the relevant correlations but, on the other hand, also be small compared to inhomogeneity scales across the hypersurface. How to perform the partition into patches can be ambiguous. The method has been used to study rapidity acceptance dependent observables at LHC energies, where the approximate longitudinal boost invariance is realized [16]. Its application at lower collision energies, however, is more complex. Instead, in Ref. [17], the proton cumulants at RHIC-BES were analyzed using an analytic approach incorporating baryon repulsion and conservation. However, analytic approaches lack the flexibility of event generators and are typically restricted to specific observables, such as (net-)proton number cumulants in a particular acceptance [17]. Therefore, it is advisable to incorporate the effect of short-range repulsion directly into the sampling procedure.\n\nThis work introduces the effect of short-range repulsion between particles into the Cooper-Frye particlization through a rejection sampling step, which prohibits any pair of particles from overlapping in the coordinate space. This induces negative correlations between particles that are shown to be consistent with analytical expectations based on the excluded volume model. Then, the method is used to study the effect of baryon repulsion on the cumulants of (net-)proton number distribution in central heavy-ion collisions in a broad energy range √ s NN = 2.4−2760 GeV while simultaneously incorporating exact global conservation of baryon number, electric charge, and strangeness.\n\nThe paper is organized as follows. The modeling of arXiv:2208.13693v2 [hep-ph] 9 Dec 2022 the short-range repulsion effect using the rejection sampling step is described in Sec. II. Section III describes how the effect is introduced into a multi-component hadron system at Cooper-Frye particlization. The method's application to heavy-ion collisions is presented in Sec. IV. Conclusions and outlook in Sec. V close the article.\n\n\nII. METHOD\n\nA. Probability of configurations Consider a uniform system of N classical particles in volume V that is in contact with the heat bath characterized by temperature T . Interactions between particles are mediated by a (non-relativistic) pair potential V (r i , r j ). The properties of the system are characterized by the canonical ensemble (T, V, N ). The microscopic configuration of the system can be given by the set {r i , p i } of coordinates and momenta of all particles in the system. The probability of a particular configuration is determined by the Boltzmann factor, i.e.\nP ({r i , p i }) ∝ e − H({r i ,p i }) T ,(1)\nwhere H ({r i , p i }) is the system Hamiltonian comprising the kinetic and potential energy terms:\nH ({r i , p i }) = N i=1 ε(p i ) + 1 2 N i,j=1 V (r i , r j ).(2)\nHere ε(p i ) is the energy-momentum relation for the particle i, for example, ε(p i ) = p 2 i + m 2 for relativistic particles and ε(p i ) = p 2 i /(2m) for non-relativistic particles.\n\nBased on the structure of the Hamiltonian in Eq. (2) it is clear that the probability P ({r i , p i }) factorizes into momentum-and coordinate-dependent parts. The momentum distribution is given by the (non-)relativistic Maxwell-Boltzmann distribution. Thus, the sampling of particle momenta proceeds in the standard way. However, the probability distribution of the coordinates of the particles is affected by the interaction potential V (r i , r j ). The unnormalized distribution density reads\nP coord ({r i }) = N i,j=1 e − V (r i ,r j ) T .\n(\n\nWithout loss of generality, consider now that the system is placed in a cubic volume. A sampling of momenta and coordinates can be performed via rejection sampling. First, the coordinates are sampled uniformly from the cubic volume. Then, the particle momenta are sampled from the Maxwell-Boltzmann distribution. The sampled configuration is accepted with relative weight proportional toP coord ({r i }) in Eq. (3). For purely repulsive potentials, V (r i , r j ) ≥ 0, the maximum weight computed through Eq. (3) does not exceed unity, and thus the application of rejection sampling is straightforward. The method can, in principle, be generalized to arbitrary two-body potentials that include attraction, for instance, by rescaling the maximum weight or oversampling.\n\n\nB. Hard-core repulsion\n\nHere the focus is on a specific example of the interaction potential, namely the hard-core interaction potential given by\nV HC (r i , r j ) = ∞, |r i − r j | < σ , 0, |r i − r j | ≥ σ .(4)\nWith this choice of V HC (r i , r j ), one deals with the system of hard spheres. Here, σ = 2r c is the hard-sphere diameter, and r c is the radius. As discussed in Appendix A, the equation of state of the hard-sphere system reduces to that of the excluded volume model in the dilute limit, provided that the excluded volume parameter b of the latter model is taken as\nb = 16πr 3 c 3 .(5)\nThe hard-sphere and excluded volume model equations of state are very similar at bn 0.1, where n ≡ N/V . Thus the hard-sphere model can mimic the excluded volume effect if this condition is met. As discussed below, this is the case for the particlization stage in heavy-ion collisions.\n\nFor the case of hard-core potential (4) the probability densityP coord ({r i }) in Eq. (3) vanishes if any pair of particles overlap, i.e. if |r i − r j | < σ for any (i, j) pair. In all other cases,P coord ({r i }) = 1.\n\nIt follows that the system configuration can be sampled by the following algorithm involving a rejection sampling step:\n\n1. The coordinates of N particles are sampled uniformly from the given (cubic) volume.\n\n2. If, for any (i, j) pair of particles, they overlap, i.e., |r i − r j | < σ, the configuration is rejected, and one goes back to step 1.\n\n3. The momenta of the particles are sampled from the Maxwell-Boltzmann distribution.\n\nThe procedure can be further sped up by combining steps 1 and 2: If a newly sampled particle overlaps with any previously sampled particles, the configuration can be rejected outright without needing to sample any remaining particles.\n\nC. Testing the sampling method\n\n\nCanonical ensemble\n\nHere, sampling of the hard-sphere gas is performed, and particle number fluctuations in various coordinate space subsystems are analyzed. The sampling is carried out for a fixed value of the hard-core radius r c and particle number density n = N/V , but for different values of the total number of particles N . In particular, the scaled density bn is fixed to bn = 0.03, where b is given by Eq. (5). The value of b is used to set the length scale, e.g. the dimensionless system volume readsṼ = V /b = N/(bn), the cube length isL =Ṽ 1/3 , and the dimensionless hard-core radius isr c =L 3(bn) 16πN 1/3 = 3 16π 1/3 . To minimize the effect of finite system size, periodic boundary conditions with minimum-image convention (periodic box) are applied when checking the particles for overlap. The results are also compared with the case where periodic boundary conditions are not applied (single box).\n\nAfter sampling the configurations, the first and second moments of particle number distributions inside various subvolumes of the coordinate space along the z direction are studied. More specifically, the results are analyzed as a function of the total volume fraction α < 1 covered by the subvolume. A particle belongs to a subvolume α if its scaled coordinatez is in the range 1−α 2 <z L < 1+α 2 . Monte Carlo simulations are performed for N = 20, 40, 80, 160. For the first moments, one observes that the mean number of particles in a subvolume is consistent within statistical errors with the relation N α = αN , that is, the particles are uniformly distributed throughout the volume on average, as they should.\n\nFor the second moment, the behavior of the scaled variance ω α ≡\nN 2 α − N 2 α N α\nis studied. Note that this quantity vanishes in the limit α → 1 since the total particle number is fixed. For this reason, one looks at a scaled quantityω α = ω α /(1 − α). In Ref. [29] it was shown that this quantity is expected to approach the scaled variance in the grand-canonical limit, i.e.,ω α N →∞ → ω gce . In the excluded volume model, one has [30] \nω ev gce = (1 − bn) 2 ≈ 0.941,(6)\nwhile in a more accurate Carnahan-Starling model (see Appendix A) one has ω CS gce ≈ 0.942. The α-dependence ofw α resulting from the described sampling procedure is shown in Fig. 1, both with and without periodic boundary conditions. The sample size is of the order of several million configurations in each case. It is seen that the results show a suppression of scaled variance by magnitude, which is similar to the one given by the grand-canonical limit ω CS gce , except for the α → 0 and α → 1 limits, where this quantity tends to the Poisson limit of unity. Without periodic boundary conditions, the finite-size effects are more significant. \n\n\nGrand-canonical ensemble\n\nSo far, the sampling has been discussed in the context of the canonical ensemble, i.e. for the case where the total number N is fixed. Consider the situation where the total number N itself fluctuates event-by-event. One example of this scenario is the grand-canonical ensemble, where the system can exchange particles with a heat bath. Even in the canonical ensemble, the total number of particles of a given type can fluctuate if the system is multi-component. A relevant example for the present studies is the baryons and antibaryons in the canonical ensemble hadron resonance gas. Even though the net number of baryons is precisely conserved, the individual numbers of baryons and antibaryons fluctuate.\n\nThe generic way to incorporate the fluctuations of N is to fold the canonical ensemble procedure with the sampling of the total number N in each event. For example, the sampling of the grand-canonical excluded volume model is explored here. Thus, simulations are performed not for a fixed total number N , but for a fixed mean number N that fluctuates following the grandcanonical ensemble distribution given by the EV model. The sampling is performed in two steps:\n\n1. The total number N is sampled from the grandcanonical excluded volume model, where the method described in Ref. [16] is used.\n\n2. The coordinates of the N particles are sampled using the method described in Sec. II B. Figure 2 shows the sampling results for w α as functions of α in the grand-canonical ensemble. For α → 1 the results approach the grand-canonical value, as expected. The finite-size effects play a role at finite α, keepingw α slightly above the grand-canonical limiting value. For α → 0,w α approaches the Poisson limit of unity.\n\n\nD. SPR approximation\n\nOne can introduce the following approximation to the sampling procedure to achieve sufficiently fast sampling of an even larger number of particles. Particles are sampled one by one. When the ith particle is sampled, its possible overlap with any of the already sampled particles is checked. If an overlap is detected, the sampled particle is rejected, and the process is repeated. The difference from the exact method is that the already sampled i − 1 particles are not rejected but retained. This procedure is referred to as the single particle rejection (SPR) approximation, and it allows one to significantly improve the speed of the sampling procedure, especially for large systems. The SPR approximation can be expected to be most accurate for moderate values of bn, i.e. for dilute systems. Indeed, in dilute systems, it is improbable for a newly sampled particle to overlap with more than one other particle. Thus, there is no real need to reject all previously sampled particles if only a single overlap is detected.\n\nThe accuracy of the SPR approximation can be tested with simulations. The results forω α obtained using the SPR approximation are depicted in Fig. 3, for N = 80, 160, 640, for the periodic box only, and in the canonical ensemble. The results for N = 80 and 160 are consistent within errors with the results of the unbiased method, shown in Fig. 3 by the symbols. The results for N = 640 indicate the approaching of the grand-canonical value ω ev gce in the limit N → ∞. The method can be tested further by considering higher values of bn. Carlo results consistently approach the grand-canonical values for all values of bn considered. One can conclude that the SPR approximation produces accurate results for the variance of particle number distributions at densities at least up to bn = 0.5. Differences between the SPR approximation and the full sampler can be seen in more subtle observables than the cumulants. An example based on the radial distribution function is discussed in Appendix B.\n\n\nIII. REPULSIVE CORE IN THE COOPER-FRYE HADRON SAMPLING PROCEDURE\n\nThe system created in heavy-ion collisions is not a static box but a dynamically expanding, inhomogeneous fireball. The momentum distribution for hadron species j emerging from hydrodynamics is given by the Cooper-Frye formula [31] \nω p dN j d 3 p = Σ(x) dΣ µ (x) p µ f j [u µ (x)p µ ; T (x), µ j (x)],(7)\nneglecting the shear and bulk viscous corrections. Here Σ(x) is the Cooper-Frye hypersurface, dΣ µ (x) is the hypersurface element, u µ (x) is the collective four-velocity, p µ is the four-momentum and f j is the distribution function corresponding to a local thermodynamic equilibrium at a space-time point x. The total mean number of particles can be obtained by integrating Eq. (7) over the momenta. This results in\nN j = Σ(x) dN j (x) (8) = Σ(x) dΣ µ (x)u µ (x) n j [T (x), µ j (x)] .(9)\nHere n j is the equilibrium local rest frame density of particle species j near the space-time point x. This density may generally contain the effect of hadronic interactions, such as excluded volume. The total effective volume at particlization is\nV eff = Σ(x) dΣ µ (x)u µ (x),(10)\nthus one can introduce the average effective density of particle species j as n j = N j /V eff . The Cooper-Frye sampling procedure that includes the effect of hard-core repulsion proceeds as follows:\n\n1. The total numbers of all the particle species are sampled. The effect of excluded volume on the distribution of the total particle numbers is included, where applicable, following the method described in Refs. [16,32] and using the average effective densities n j as input. The exact global conservation laws are enforced through rejection sampling, where configurations that do not satisfy the exact conservation laws are discarded.\n\n2. The momenta and coordinates of each hadron are sampled one by one. To do that, first, the hypersurface element from which the given hadron is sampled is determined. This is done via the multinomial distribution where each volume element x is weighted by dN j (x) , i.e. by the grand-canonical mean yield for the given hadron species emitted from that element. Then the momenta and coordinates of the hadron emitted from the chosen hypersurface element x are sampled via the standard procedure.\n\n3. All pairs of hadrons with hard-core repulsion between them are checked for overlap. If an overlap is detected, one goes back to the previous step. As in Sec. II, to speed-up the procedure, one can check the overlap of a newly sampled particle with already sampled ones before sampling the remaining ones. As detailed below, each pair of hadron species can have a different value of the minimum distance σ of their closest possible approach.\n\nNote that a numerical Cooper-Frye hypersurface might have negative volume elements with dΣ µ (x)u µ (x) < 0. Here these volume elements are skipped, i.e., the presence of the θ function, θ(dΣ µ (x)u µ (x)), in all Cooper-Frye integrals is implied.\n\nA few relevant details must be specified regarding the treatment of global conservation laws. The total baryon number to be conserved is calculated from the 4π mean hadron yields from Eq. (8). The calculated value is then rounded to the nearest integer. To maintain consistency, all elements of the hypersurface dΣ(x) are then rescaled by a common factor so that the baryon number calculated by Eq. (8) coincides with the rounded integer number. This has only a minor effect, as the rescaling factor is very close to unity in all considered cases. If strangeness is treated canonically, it is required to be equal to zero in each generated event. Finally, the total net charge is constrained to reproduce the charge-to-baryon ratio of Q/B = 0.4. If the total electric charge satisfying the Q/B = 0.4 condition is not an integer, it is rounded to the nearest integer.\n\nNote that the presented algorithm differs from most of the conventional methods used. There, hadrons are first sampled from each cell one by one, and then constraints from global conservation laws are imposed. In this way, one may have to resort to rejecting all sampled particles if the global conservation laws are not satisfied. The present algorithm deals with this problem more efficiently. Here, first, the total multiplicities of all hadrons are sampled, then the conservation laws are checked, and only after that does the sampling of hadrons' momenta and coordinates begin. The drawback is that, in the case of numerical Cooper-Frye hypersurfaces, this method requires storing the pre-computed multinomial probabilities for all hadron species and all Cooper-Frye hypersurface elements, which increases the memory requirements significantly. For example, the method requires about 5-15 Gb of RAM for sampling the central Au-Au collisions at RHIC-BES using Cooper-Frye hypersurfaces from the MUSIC code [33]. The present algorithm may also be less efficient for event-by-event hydrodynamics, as opposed to single-shot hydrodynamics used in the present study.\n\nThe procedure to check the overlap between two particles at the Cooper-Frye particlization has to be modified compared to the box case to account for the presence of collective motion, relativistic effects, and the fact that particles are emitted at different time moments. This is achieved as follows:\n\n• First, both particles are boosted into their centerof-mass frame.\n\n• Then, the particle emitted at the earlier time is propagated along the straight line to the time of the particle which was emitted later.\n\n• Finally, the distance |r i − r j | between the particles is calculated and checked if it is below the threshold value of σ sisj , where s i and s j is the species type of particles i and j, respectively.\n\nThe hadron resonance gas at the Cooper-Frye particlization stage is a multi-component system. Thus, in general, the hard-core repulsion between different hadron species is characterized not by a single excluded volume parameter b, but by a matrix b αβ of excluded volume parameters where each element corresponds to a distinct pair of hadron species. Note that this matrix need not be symmetric [34]. The matrix b αβ can be used to determine the threshold distances σ αβ . As discussed in the framework of the multi-component excluded volume model [34], the coefficients b αβ are related to σ αβ as\nb αβ + b βα 2 = 2πσ 3 αβ 3 ,(11)\nthus σ αβ = 3b αβ 2π\n1/3 ,(12)where b αβ = (b αβ + b βα )/2.\nThe SPR approximation can be employed similarly to the box case in Sec. II. Using the SPR approximation is essential for sampling central collisions of heavy ions, where the system is so large that applying the full (unbiased) method is prohibitively time-consuming.\n\nIt is instructive to summarize the list of approximations and possible limitations of the method: (i) Total particle numbers N j are sampled from an auxiliary excluded volume HRG model characterized by constant volume V eff and mean hadron yields N j , both calculated via the Cooper-Frye formula. The resulting excluded volume effect on the cumulants of N j -distribution may be slightly different from the true result if the distribution of particle number densities across the Cooper-Frye hypersurface is inhomogeneous. The accuracy of this approximation for central Au-Au collisions at various beam energies is verified in the next section by comparing Monte Carlo results with analytic approximations, yielding only minor differences.\n\n(ii) As the N j -distribution is sampled from the excluded volume model rather than from the exact hard-sphere model distribution, for consistency, it is imperative that the excluded volume model provides an accurate approximation for the hardspheres equation of state. As shown in the Appendix A, this is satisfied with high precision for bn 0.10 − 0.15.\n\nBoth approximations (i) and (ii) concerning the N jdistribution become less relevant when the canonical ensemble is applied, as the canonical ensemble effects then dominate the fluctuations of N j .\n\n(iii) As the hard-core repulsion is an inherently nonrelativistic concept, issues may arise when relativistic effects are strong. In particular, this can be an issue when the two overlapping particles have large relative velocities. This may happen, for instance, when the particles are emitted from different regions of the fireball characterized by significantly different collective velocities. However, as the hard-core repulsion is a short-range phenomenon, the issue becomes irrelevant in this case since these particles would not overlap in the coordinate space. Particles that are close by in the coordinate space, on the other hand, would typically correspond to very similar collective velocities, thus their relative velocities are determined by the local temperature T (x), which is usually of order 160 MeV or lower in heavy-ion collisions at particlization. Therefore, the relativistic effects become less relevant for m/T (x) 1, as is the case for baryons (m B ≥ 938 MeV/c 2 ), but could be important if hard-core repulsion is incorporated for lighter particles such as pions.\n\nThe Cooper-Frye sampling with short-range repulsion is implemented within the open source package Thermal-FIST [35] starting from version 1.4.\n\n\nIV. APPLICATION OF THE METHOD TO PROTON CUMULANTS IN HEAVY-ION COLLISIONS\n\nTo illustrate the newly developed FIST sampler and the corresponding effect of hard-core repulsion, the behavior of (net-)proton cumulants in central collisions of heavy ions at various collision energies is studied. The effect of (anti)baryon excluded volume on the cumulants has previously been studied at LHC [36] and RHIC-BES [17] energies using different methods, thus, it is instructive to compare these results with the present method.\n\nIt is assumed here that excluded volume repulsion is present for all baryon-baryon and, by symmetry, all antibaryon-antibaryon pairs, as motivated by recent analyses of lattice QCD data on baryon number susceptibilities [22,24]. This implies b αβ = b > 0 if either α, β ∈ B or α, β ∈B, and b αβ = 0 otherwise. To be consistent with the earlier studies, the value of the baryon excluded volume parameter is chosen to be b = 1 fm 3 , corresponding to a classical hard-core radius of around r c = 0.39 fm.\n\n\nA. LHC\n\nFirst, the LHC energies are studied. More specifically, the 0-5% central Pb-Pb collisions at √ s NN = 2.76 TeV are analyzed, using a longitudinally boostinvariant Cooper-Frye hypersurface based on the blastwave model. The parametrization of the hypersurface is identical to a previous study [16], where the blast-wave model parameters are based on fits to the p T spectra within a single freeze-out scenario [38]. These parameters are uniform across the entire hypersurface. The particlization temperature is T = 160 MeV, and the chemical potentials are vanishing. For this choice of parameters, one has bn B = bnB ≈ 0.029, where n B and nB are the number densities of baryons and antibaryons, respectively.\n\nA total of 28.3 million events is sampled. The events incorporate the baryon hard-core repulsion as described above, as well as the exact global conservation of baryon number. The behavior of cumulants of net proton number distribution is analyzed. To minimize the statistical error, the fireball volume is reduced by factor 10, from dV /dy = 400 fm 3 to dV /dy = 40 fm 3 . This reduction of the fireball volume is achieved through the reduction of its transverse radius from R ⊥ = 9 fm to R ⊥ ≈ 2.84 fm. The analysis is therefore focused on the ratios of cumulants where the trivial dependence on the volume is canceled. For more details on this procedure to minimize the statistical error, see Ref. [16]. Figure 5 depicts the behavior of (a) the net-proton variance normalized over the Skellam distribution baseline, κ 2 [p −p]/ p +p and (b) the net-proton kurtosis ratio,\nκ 4 [p −p]/κ 2 [p −p].\nThe calculations are performed in the experimental acceptance used in the measurements performed by the ALICE Collaboration [39], corresponding to cuts in (anti)proton momentum, 0.6 < p < 1.5 GeV/c, and pseudorapidity, |η| < η max . The results in Fig. 5 are shown as a function of η max by solid black lines with grey bands depicting the statistical uncertainties. The results are compared with the earlier calculations of Ref. [16] obtained in the framework of the so-called subensemble sampler (dash-dotted black lines), as well as the ideal gas baseline that incorporates the effect of baryon conservation but not hard-core repulsion (dotted blue lines). Note that the calculations for κ 4 [p −p]/κ 2 [p −p] do not account for the effect of volume fluctuations which would influence the measurements [16].\n\nThe FIST sampler results agree with the earlier results based on the subensemble sampler. They also show visible deviations from the ideal gas baseline. This indicates that the FIST sampler provides a reasonable de-scription of the excluded volume effect. In contrast to the subensemble sampler, which is mainly restricted to longitudinally boost invariant systems such as those encountered at the LHC, the FIST sampler can also be straightforwardly applied to lower collision energies where boost invariance does not hold. This is explored at the RHIC beam energy scan energies in the next section.\n\n\nB. RHIC-BES\n\nNet proton fluctuations have been measured by the STAR Collaboration in Au-Au collisions in collision energy range √ s NN = 7.7 − 200 GeV [40,41]. In Ref. [17] the fluctuations have been analyzed based on relativistic hydrodynamics simulations with MUSIC [42], incorporating effects of baryon conservation and excluded volume. Calculations in Ref. [17] were performed analytically, in two steps: (i) proton cumulants are calculated in the experimental acceptance using the Cooper-Frye formula in the grand-canonical limit, including the excluded volume effect; (ii) correction for baryon number conservation is performed using a method called SAM-2.0 [43].\n\nHere, the calculations of proton cumulants are performed using the FIST sampler, by sampling 0-5% central Au-Au collisions at energies √ s NN = 7.7, 14.5, 19.6, 27, 39, 62.4 and 200 GeV. The sampling is performed with and without the effect of exact baryon conservation, and it uses the same Cooper-Frye hypersurfaces produced by the MUSIC code that is based on [42] and available at [33]. For each energy, at least several million events have been generated. These calculations are compared to the analytic approximations of Ref. [17] in Fig. 6, for (a) the second proton cumulant ratio κ 2 /κ 1 and (b) the third order net proton cumulant ratio κ 3 /κ 1 . The FIST sampler results are shown by the bands, while the lines show the analytic results.\n\nIn the absence of baryon conservation (baryon repulsion only, green lines and bands in Fig. 6) deviations of the cumulant ratios from the baseline of unity are driven solely by baryon repulsion, which leads to a modest suppression. The FIST sampler results agree with the analytic approximations, validating the method at RHIC-BES energies.\n\nIncorporating exact baryon conservation (baryon repulsion + baryon cons., black lines, and grey bands in Fig. 6) leads to a more potent suppression of the cumulant ratios. In this case, the analytic and Monte Carlo results agree at high collision energies ( √ s NN 40 GeV), but deviations between the two are visible at lower √ s NN .\n\nThese deviations can be attributed to inaccuracies of the SAM-2.0 framework at lower collision energies. As discussed in the original publication [43], it would tend to overestimate the effect of (repulsive) interactions. Comparison with the FIST sampler allows one to quantify the accuracy of this approximation at RHIC-BES conditions, revealing a modest overestimation of excluded volume effect on proton cumulants which gets worse as [17] with and without exact baryon conservation, respectively. The red symbols correspond to the experimental data of the STAR Collaboration [40,41]. collision energy is decreased.\n\nAs a new application of the FIST sampler, one can simultaneously incorporate the effects of baryon repulsion and exact conservation of all three QCD conserved charges: baryon number, electric charge, and strangeness. The electric charge conservation, in particular, can influence the fluctuations of the proton number since the proton carries the electric charge. At high col-lision energies, the dominant electric charge carriers are pions. Thus, the effect of electric charge conservation on protons is expected to be small [36]. As the collision energy decreases, the fraction of charge carried by protons increases, and so does the effect of charge conservation on proton cumulants. This is reflected in the FIST sampler results shown in Fig. 6 by the blue bands, indicating additional suppression of (net-)proton κ 2 /κ 1 and κ 3 /κ 1 due to electric charge conservation, which becomes visible at √ s NN 40 GeV. Interestingly, at all collision energies considered, the FIST sampler results with exact conservation of baryon number, electric charge, and strangeness are in good agreement with the analytical calculations of proton cumulants from Ref. [43] that incorporate baryon conservation but not electric charge and strangeness. Thus, it appears that the additional suppression due to electric charge conservation at all energies is of the same magnitude as the overestimation of the excluded volume effect in the analytical calculation of Ref. [43] with SAM-2.0. Although this coincidence appears to be purely accidental, it indicates that the results of Ref. [43] serve as an accurate baseline for proton cumulants incorporating non-critical effects like baryon repulsion and exact conservation of multiple conserved charges.\n\n\nC. GSI-SIS\n\nThe final application of the new method concerns the Au-Au collisions at √ s NN = 2.4 GeV, as probed by the HADES experiment at GSI-SIS. The experimental data [44] indicate sizable multiproton correlations. These were recently analyzed in the framework of a fireball model [45]. The Cooper-Frye hypersurface was parameterized in the framework of the Siemens-Rasmussen model with a Hubble-like collective flow, which was found to provide a reasonable description of the p T spectra of pions and protons [46], while the temperature T ≈ 70 MeV and baryochemical potential µ B ≈ 875 MeV were extracted from hadron yields [47]. Here this parametrization of the hypersurface is used in FIST sampler to study the behavior of proton cumulants influenced by conservation laws and baryon repulsion. At √ s NN = 2.4 GeV, a significant fraction of protons is bound into light nuclei in the final state. Based on the preliminary HADES data [46], one can estimate that about 37.5% of the final-state protons are bound. The light nuclei are not incorporated into the FIST sampler employed in the present work. Thus, to model this effect, following Ref. [45], each proton emitted from the fireball is assumed to be bound into light nuclei with a probability of q nucl = 0.375 and not contribute to the measured cumulants of the proton number. The corresponding results for proton number cumulant ratios κ 2 /κ 1 , κ 3 /κ 2 , and κ 4 /κ 2 are shown in Fig. 7 as functions of the rapidity cut y cut in the experimental acceptance of the HADES experiment.\n\nThe results show a similar pattern to RHIC-BES energies: both the baryon conservation and repulsion suppress the cumulant ratios, with baryon conservation having a stronger effect. The additional effect of electric charge conservation (blue lines) is even more notable, which suppresses the cumulant ratios even further. The combined effect of baryon and electric charge conservation is more significant than that of baryon repulsion.\n\nNote that both the total baryon and electric charge of the participant matter fluctuates event by event in the experiment. This is in contrast to the FIST sampler where they are both fixed. The experimental data of the HADES collaboration [44], shown in Fig. 7 by black symbols, is corrected for volume (participant) number fluctuations, which justifies the canonical treatment of baryon number for the participant matter. However, the total electric charge of the participants still fluctuates; thus, the FIST sampler results in Fig. 7 should be considered an upper bound on the effect of exact conservation of the electric charge on the proton number cumulants. It is also clear that the model fails to describe the experimental data even qualitatively. For example, the scaled variance κ 2 /κ 1 increases with the rapidity cut y cut in the experiment, as opposed to the suppression predicted due to conservation laws and short-range repulsion. This indicates that the behavior of the proton cumulants in the experimental data is not driven by the conservation laws or short-range repulsion that are incorporated in FIST sampler. For a discussion of the various possibilities, including the QCD critical point, see Ref. [45] The present analysis indicates that the exact conservation of multiple conserved charges, as opposed to only that of the baryon number, is essential for quantitative analysis of proton number cumulants at moderate collision energies, √ s NN 7.7 GeV. The FIST sampler allows one to evaluate the corresponding baselines in this energy regime, which can then be used to analyze future experimental data coming from the fixed-target program at RHIC [48] or the CBM experiment at FAIR [49].\n\nFor comparisons of the FIST sampler results in the SIS-HADES regime with analytic approximations, see Appendix C.\n\n\nV. CONCLUSIONS AND OUTLOOK\n\nThis study introduced short-range repulsive correlations into the Cooper-Frye hadron sampling procedure. This has been achieved through a rejection sampling step that prohibits any two particles with repulsive interactions from overlapping in the coordinate space, effectively modeling the excluded volume phenomenon. The effect introduces negative correlations between particles visible in normalized cumulants of their distributions. The new method -called the FIST sampler -incorporates this effect and simultaneously allows for both the canonical and grand-canonical treatments of the QCD conserved charges.\n\nFIST sampler was validated in a periodic box setup, where the method yields the behavior of the scaled variance of particle number, which is consistent with the equation of state of hard spheres. nally boost-invariant scenario at LHC energies, the new method produces the behavior of net proton cumulants up to the fourth order, which is consistent with an earlier study of [16] that used the so-called subensemble sampler. The advantage of FIST sampler becomes evident at lower collision energies, where boost invariance no longer holds and where the application of the subensemble sampler is challenging. The new method allowed one to incorporate the simultaneous effects of baryon repulsion and baryon, electric charge, and strangeness conservation at RHIC-BES and SIS-GSI energies. In particular, the effect of electric charge conservation on proton number cumulants becomes increasingly important as the collision energy is decreased, √ s NN 40 GeV, due to a smaller number of pions relative to protons at lower energies. The resulting cumulant ratios of the (net) proton distribution are consistent with estimates of Ref. [17] that were obtained by employing analytic approximations for baryon conservation and repulsion and neglecting electric charge conservation. The reason for this coincidence at lower energies appears to be that the analytic method of Ref. [17] slightly overestimates the effect of baryon repulsion in the canonical ensemble, which at lower collision energies is approximately compensated for by the electric charge conservation in FIST sampler. Due to this coincidence, the results of Ref. [17] can be taken as a baseline for (net) proton cumulants at RHIC-BES that incorporates non-critical effects such as excluded volume and multiple global conservation laws. The mechanisms above do not describe the experimental data of the HADES Collaboration (Fig. 7), indicating that proton number cumulants are driven by different mechanisms that still need to be clarified.\n\nFIST sampler has several future potential applications. For instance, one can couple its output with a hadronic afterburner such as UrQMD [50,51] or SMASH [52], which should ideally include the shortrange repulsion for consistency [53], and study the effect of the hadronic phase on proton number cumulants, in particular, that of baryon annihilation [54,55]. Al-though the effects of hadronic afterburner have not been studied in the present work, the FIST sampler code [56] readily supports output tailored for use with UrQMD afterburner [57]. Since the method offers the full flexibility of an event generator, observables other than proton number cumulants, such as, for example, balance functions [58] or distributions of other hadronic species [59] can be studied as well. In particular, one may revisit the Bayesian constraints on transport coefficients of QCD matter and the switching temperature for the transition from hydrodynamics to transport [60,61] by employing the new particlization routine presented here in such an analysis. One can also straightforwardly incorporate the possible flavor dependence of baryon repulsion suggested by recent analyses of lattice QCD data on off-diagonal susceptibilities [24,25,62] by an appropriate choice of the corresponding threshold distances σ αβ in Eq. (12). It can also be interesting to see how the presence of baryonic hard-core repulsion might affect the light nuclei production, such as employed in phase-space coalescence prescriptions [63,64]. These applications will be the subject of future studies.\n\nUsing the single particle rejection (SPR) approximation was essential to sample central collisions of heavy ions in practical applications of the new routine, as applying a direct unbiased method was prohibitively timeconsuming. The implication is that the sampler is not fully unbiased, even though the tests in a box setup indicate that the sampling of particle number cumulants remains accurate. It is thus advisable to consider other techniques, such as oversampling, to produce a fast, unbiased sampler.\n\nThe sampling with short-range repulsion described in this work has been implemented into the Monte Carlo event generator within the open source package Thermal-FIST [35], starting from version 1.4. The code for the sampling of various central collisions of heavy ions discussed in the present work is available at [56].\n\n\nACKNOWLEDGMENTS\n\nThe author is grateful to Mark Gorenstein for reading the manuscript and for useful comments. The author thanks Volker Koch and Chun Shen for fruitful discussions and collaboration on related projects and Dylan Neff for stimulating discussions. The author also acknowledges the hospitality of the staff at the Frankfurt Institute for Advanced Studies where part of this work was done. This work was supported through the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract number DE-FG02-00ER41132. Computational resources were provided by the Frankfurt Center for Scientific Computing (Goethe-HLR).\n\nAppendix A: The equation of state and particle number fluctuations in the system of hard spheres\n\nThe equation of state that describes the system of classical particles that interact through a hard-core potential (4) can be written in terms of the compressibility factor Z ≡ P/(nT ):\nP nT = g(η),(A1)\nwhere η = bn 4 = 4πr 3 c 3 N V is the so-called packing fraction. The explicit expression for the function g(η) is not known, but different approximations are available that model the equation of state accurately when the values of η are not too large. One well-known approximation is the van der Waals excluded volume model where one has P ev = nT 1−bn , thus\ng ev (η) = (1 − 4η) −1 .(A2)\nA more involved analytical approximation of the equation of state of hard spheres is given by the Carnahan-Starling model [65]:\ng CS (η) = 1 + η + η 2 − η 3 (1 − η) 3 .(A3)\nThe accuracy of the excluded volume and Carnahan-Starling models can be verified by comparing the results to numerical Monte Carlo simulations of the hardsphere system. This comparison is depicted in Fig. 8, where the dependence of the compressibility factor Z on bn, calculated in the two models, is compared to the numerical simulations of [66]. The excluded volume model accurately describes the compressibility factor up to bn ≈ 0.10 − 0.15, while it overestimates the numerical data for Z at higher densities. On the other hand, the Carnahan-Starling model accurately describes the Monte Carlo data for all the densities considered, that is, up to at least bn = 0.6. Figure 9. Density dependence of cumulant ratios κ2/κ1 (black, top), κ3/κ1 (blue, middle), κ4/κ2 (red, bottom) of the grand-canonical particle number distribution calculated for the hard-sphere system within the excluded volume (dashed lines) and Carnahan-Starling (solid lines) approximations.\n\nCumulants of the grand-canonical particle number distribution are given by\nκ n = V T n−1 ∂ n P ∂µ n T .(A4)\nFor n = 1 one has κ 1 ≡ N , therefore, for n > 1 the cumulants read\nκ n = T n−1 ∂ n−1 N ∂µ n−1 T . (A5)\nUsing the thermodynamic identity iteratively, provided that the pressure P is known as a function of density n. This is the case for the excluded volume [Eq. (A2)] and Carnahan-Starling [Eq. (A3)] models. Figure 9 depicts the behavior of the grand-canonical cumulant ratios κ 2 /κ 1 , κ 3 /κ 1 , and κ 4 /κ 2 as a function of the scaled density bn. Given the accuracy of the Carnahan-Starling approximation for the compressibility Z (Fig. 8), it is expected that the corresponding calculations of the cumulant ratios accurately reflect the grandcanonical particle number distribution in the hard-sphere system in the considered density range. Deviations of the excluded volume model from the Carnahan-Starling model become more visible in higher-order cumulants. The excluded volume model provides accurate results up to bn ≈ 0.10.\n\n\nAppendix B: Radial distribution function\n\nThe radial distribution function g(r) describes how the density of particles varies around a reference particle at r = 0 relative to the expectation based on the mean particle number density n = N/V . More specifically, g(r) is defined so that the local particle number density at a distance r from the reference particle is equal to n g(r). Deviations of g(r) from unity signal the presence of correlations between particles.\n\nThe radial distribution function for the hard-sphere system has been studied for a long time [67,68]. As the particles cannot overlap, one has g(r) = 0 for r < σ. g(r) has a discontinuity at r = σ and approaches a value larger than unity in the limit r → σ + 0, indicating the presence of effective attraction as a many-body effect. For large r/σ values, g(r) is expected to approach unity, given that the hard-core repulsion is a short-range phenomenon. For moderate densities of bn, accurate expressions for g(r) of the hard-sphere system can be obtained based on the solution of the Percus-Yevick equation [67][68][69]. This can be used to validate the performance of the FIST sampler. Figure 10 depicts g(r) in a range 1 < r/σ < 3 resulting from FIST sampler for a system of particles in a periodic box at density bn = 0.12, which was obtained by binning the relative distances of the sampled particles. Blue circles correspond to the case where the SPR approximation was applied, whereas black squares show the unbiased calculation without approximations. In the former case N = 320 particles were sampled in each configuration, while in the latter case, only N = 80 particles were used due to slower computational performance.\n\nThe results are compared with the analytical expectation for a hard-sphere system based on the Percus-Yevick approximation, which is expected to be highly accurate for density as low as bn = 0.12 [69]. The full calculation is in quantitative agreement with the analytical expectations, indicating that FIST sampler correctly reproduces the spatial correlations between particles that interact through the hard-core potential (4). The SPR approximation underestimates g(r) at 1 < r/σ 1.5, in- Figure 10. Radial distribution function g(r) resulting from FIST sampler for a system of particles in a periodic box at density bn = 0.12 with (blue circles) and without (black squares) the application of the SPR approximation. The black line corresponds to g(r) of a hard-sphere system calculated through the Percus-Yevick equation [69].\n\ndicating that it does not capture the full strength of the many-body effective attraction effect. At the same time, the SPR approximation reproduces the qualitative structure of g(r), and, as discussed in Sec. II D, reproduces accurately the behavior of cumulants of particle number distribution for densities considered in this study. Here a comparison of FIST sampler calculations with the analytic method recently developed in Ref. [45] is performed. This allows one to verify the accuracy of the analytic approximations of Ref. [45]. In particular, Ref. [45] used a method called SAM-2.0 [43] to perform a correction for exact baryon conservation. As discussed in Ref. [43], the method might overestimate the effect of dynamical correlations among particles, in this case, the effect of short-range hard-core repulsion, at low collision energies such as √ s NN = 2.4 GeV at SIS-GSI. By comparing to FIST sampler, which is free of these issues, one can quantify the accuracy of SAM-2.0 in this collision energy regime.\n\nThe corresponding results are shown in Fig. 11. One sees good agreement between Monte Carlo and analytic calculations for the cases when baryon repulsion is incorporated but not conservation (green lines) or when baryon conservation is implemented but not repulsion (red lines). This validates the analytic approach of Ref. [45] in these regimes, in particular, SAM-2.0 used in [45] to correct for baryon conservation is accurate in the limit of the ideal gas, as expected.\n\nWhen both the baryon repulsion and conservation are incorporated, visible differences between the FIST sampler and the analytic method occur. The analytic Figure 11. The scaled variance of the proton number distribution in the experimental acceptance of the HADES Collaboration as a function of the rapidity cut ycut. The green (top) lines correspond to calculations incorporating the hard-core repulsion among baryons but not baryon conservation. The calculations shown by the red (middle) lines implement exact baryon conservation but not baryon repulsion. The black (bottom) lines show calculations with both the baryon repulsion and conservation included. The differently styled lines correspond to the results obtained using FIST sampler while the solid lines correspond to the analytic method from Ref. [45]. method, based on SAM-2.0, overestimates the suppression of κ 2 /κ 1 compared to the Monte Carlo sampling. This is an artifact of the SAM-2.0 approximation, which assumes that there are no correlations between particles inside and outside the acceptance apart from the baryon conservation. As discussed in [43], this leads to an overestimation of the correlations present in the grand-canonical limit, namely the effect of hard-core repulsion in this study. A similar observation applies to the higher-order cumulant ratios κ 3 /κ 2 and κ 4 /κ 2 that are not shown in Fig. 11.\n\nAs follows from the FIST sampler calculations in Fig. 11, the presence of baryon repulsion in addition to baryon conservation leads to only a minor further suppression of proton cumulant ratios, i.e., the effect of hardcore repulsion in the cumulants appears to be essentially washed out by baryon conservation. This appears to be a consequence of the reduced collective flow effect at HADES energies compared to higher energies, which dilutes the space-momentum correlation, meaning that the excluded volume effect in the coordinate space is almost washed out in the momentum space where the measurements are performed. As shown in recent molecular dynamics simulations [70], even critical fluctuations may be washed out when analyzed in momentum space if exact global conservation laws are enforced, and no collective expansion is imposed.\n\nFigure 1 .\n1The dependence of the corrected scaled variancẽ ωα = ωα/(1 − α) on the acceptance fraction α, evaluated in a box with (bands) and without (dashed lines) periodic boundary conditions resulting from the sampling of particles with a hard-core repulsion for different values of total particle number. The dash-dotted horizontal line corresponds to the expected thermodynamic limit of the hard-sphere equation of state calculated using the Carnahan-Starling model.\n\nFigure 2 .\n2Same asFig. 1, but for the uncorrected scaled variance ωα calculated within the grand-canonical ensemble.\n\nFigure 4 Figure 3 .Figure 4 .\n434depicts the results for bn = 0.03, 0.1, 0.25, 0.5, all calculated for N = 320 within the SPR approximation. The dash-dotted horizontal lines correspond to the grand-canonical values expected in the Same as Fig. 1, but calculated either within the SPR approximation (bands) or without approximations (symbols), for the periodic box only. The bottom panel shows the ratio of the approximate and unbiased calculations. The dependence of the corrected scaled variancẽ ωα = ωα/(1 − α) on the acceptance fraction α calculated in a periodic box within the SPR approximation for different values of the scaled density bn. The dash-dotted horizontal lines correspond to the expected thermodynamic limit of the hard-sphere equation of state calculated using the Carnahan-Starling model. thermodynamic limit, as calculated within the Carnahan-Starling approximation (see Appendix A). The Monte\n\nFigure 5 .\n5Cumulant ratios κ2[p −p]/ p +p (left panel) and κ4[p −p]/κ2[p −p] (right panel) in 0-5% central √ sNN = 2.76 TeV Pb-Pb collisions as functions of the pseudorapidity cut ηmax. Calculations are performed within FIST sampler incorporating the effects of short-range baryon repulsion and global baryon conservation and shown by the black lines with bands. The dash-dotted black lines and dashed blue lines depict the calculations of Ref.[16] within the subensemble sampler with and without the effect of baryon excluded volume, respectively. The symbols correspond to the experimental measurements of the ALICE Collaboration[37].\n\nFigure 6 .\n6Beam energy dependence of the proton cumulant ratio κ2/κ1 (left panel) and net proton cumulant ratio κ3/κ1 (right panel) in 0-5% central Au-Au collisions at RHIC-BES. The results depict FIST sampler calculations incorporating the baryon hard-core repulsion (green bands), plus exact baryon conservation (grey bands), plus exact conservation of electric charge and strangeness (blue bands). The dotted green and dash-dotted black lines correspond to analytic calculations of Ref.\n\nFigure 7 .\n7FIST sampler was then used to model the excluded volume effect in (anti)baryon-(anti)baryon interaction at the particlization stage of heavy-ion collisions in a broad collision energy range, √ s NN = 2.4 − 2760 GeV. In a longitudi-Cumulant ratios (a) κ2/κ1, (b) κ3/κ2, and (c) κ4/κ2 of the proton number distribution in 0-5% central √ sNN = 2.4 GeV Au-Au collisions in the experimental acceptance of the HADES Collaboration, as calculated within FIST sampler. The results are shown as a function of the rapidity cut ycut. The calculations incorporate the baryon hard-core repulsion (dotted green lines), plus exact baryon conservation (dash-dotted black lines), plus exact conservation of electric charge and strangeness (solid blue lines). 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C 105, 044903 (2022), arXiv:2201.08486 [hep-ph].\n"},"corpus_id":{"kind":"number","value":251903578,"string":"251,903,578"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2208.13693\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevc.106.064906\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2208.13693v2.pdf\"\n ],\n \"pdf_sha\": \"bb9bd89a5500b58e789ed5a8d02ded13e49cd722\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Cooper-Frye sampling with short-range repulsion"},"authors":{"kind":"list like","value":["Volodymyr Vovchenko \nInstitute for Nuclear Theory\nUniversity of Washington\nBox 35155098195SeattleWAUSA\n\nGiersch Science Center\nFrankfurt Institute for Advanced Studies\nD-60438Frankfurt am MainGermany\n\nand Nuclear Science Division\nLawrence Berkeley National Laboratory\n1 Cyclotron Road94720BerkeleyCAUSA"],"string":"[\n \"Volodymyr Vovchenko \\nInstitute for Nuclear Theory\\nUniversity of Washington\\nBox 35155098195SeattleWAUSA\\n\\nGiersch Science Center\\nFrankfurt Institute for Advanced Studies\\nD-60438Frankfurt am MainGermany\\n\\nand Nuclear Science Division\\nLawrence Berkeley National Laboratory\\n1 Cyclotron Road94720BerkeleyCAUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"This work incorporates the effect of short-range repulsion between particles into the Cooper-Frye hadron sampling procedure. This is achieved by means of a rejection sampling step, which prohibits any pair of particles from overlapping in the coordinate space, effectively modeling the effect of hard-core repulsion. The new procedure -called the FIST sampler -is based on the package Thermal-FIST. It is used here to study the effect of excluded volume on cumulants of the (net-)proton number distribution in central collisions of heavy ions in a broad collision energy range in conjunction with exact global conservation of baryon number, electric charge, and strangeness. The results are compared with earlier calculations based on analytical approximations, quantifying the accuracy of the latter at different collision energies. An additional advantage of the new method over the analytic approaches is that it offers the flexibility provided by event generators, making it straightforwardly extendable to other observables."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"The description of bulk observables in relativistic heavy-ion collisions is usually performed in the framework of relativistic hydrodynamics (see e.g. [1,2] for an overview). This modeling incorporates the Cooper-Frye particlization stage, where the expanding QCD fluid is transformed into a gas of hadrons and resonances. In most cases, hadron momenta are sampled from local Maxwell-Boltzmann distributions, with the possible inclusion of shear and bulk viscous corrections [3][4][5][6][7]. Multiplicity distributions of the sampled hadrons usually correspond to Poisson statistics, implying independent particle emission. More involved descriptions additionally implement a (micro-)canonical treatment of conservation laws, such as that of energy-momentum and QCD conserved charges [8][9][10][11][12]. Proper treatment of conservation laws is relevant for observables involving event-by-event fluctuations [13][14][15][16][17]. Physically, the system created at particlization corresponds to an ideal hadron resonance gas, possibly with non-equilibrium corrections due to shear and bulk viscosities and (micro-)canonical effects. On the other hand, extensions of the Cooper-Frye procedure are necessary to incorporate any additional physics. One commonly discussed extension of the ideal HRG is short-range repulsive interactions utilizing excluded volume [18][19][20]. For example, by introducing the excluded volume effect into baryon-baryon (and, by symmetry, antibaryonantibaryon) interaction [21], one can effectively model the presence of hard core in nucleon-nucleon scattering and improve the description of several lattice QCD susceptibilities [22][23][24][25]. Another extension would concern the search for the QCD critical point, which would lead to large non-Gaussian fluctuations of the proton number [26] requiring a generalized Cooper-Frye routine [27,28].\nPreviously, a method called subensemble sampler was introduced in [16] to perform particlization of an interacting hadron resonance gas and was used to study the influence of baryonic excluded volume on proton and baryon number cumulants in Pb-Pb collisions at LHC energies. Although this method is rather generic, it requires partitioning the Cooper-Frye hypersurfaces into patches which should, on the one hand, be large enough to capture all the relevant correlations but, on the other hand, also be small compared to inhomogeneity scales across the hypersurface. How to perform the partition into patches can be ambiguous. The method has been used to study rapidity acceptance dependent observables at LHC energies, where the approximate longitudinal boost invariance is realized [16]. Its application at lower collision energies, however, is more complex. Instead, in Ref. [17], the proton cumulants at RHIC-BES were analyzed using an analytic approach incorporating baryon repulsion and conservation. However, analytic approaches lack the flexibility of event generators and are typically restricted to specific observables, such as (net-)proton number cumulants in a particular acceptance [17]. Therefore, it is advisable to incorporate the effect of short-range repulsion directly into the sampling procedure.\nThis work introduces the effect of short-range repulsion between particles into the Cooper-Frye particlization through a rejection sampling step, which prohibits any pair of particles from overlapping in the coordinate space. This induces negative correlations between particles that are shown to be consistent with analytical expectations based on the excluded volume model. Then, the method is used to study the effect of baryon repulsion on the cumulants of (net-)proton number distribution in central heavy-ion collisions in a broad energy range √ s NN = 2.4−2760 GeV while simultaneously incorporating exact global conservation of baryon number, electric charge, and strangeness.\nThe paper is organized as follows. The modeling of arXiv:2208.13693v2 [hep-ph] 9 Dec 2022 the short-range repulsion effect using the rejection sampling step is described in Sec. II. Section III describes how the effect is introduced into a multi-component hadron system at Cooper-Frye particlization. The method's application to heavy-ion collisions is presented in Sec. IV. Conclusions and outlook in Sec. V close the article."},{"title":"II. METHOD","content":"A. Probability of configurations Consider a uniform system of N classical particles in volume V that is in contact with the heat bath characterized by temperature T . Interactions between particles are mediated by a (non-relativistic) pair potential V (r i , r j ). The properties of the system are characterized by the canonical ensemble (T, V, N ). The microscopic configuration of the system can be given by the set {r i , p i } of coordinates and momenta of all particles in the system. The probability of a particular configuration is determined by the Boltzmann factor, i.e.\nwhere H ({r i , p i }) is the system Hamiltonian comprising the kinetic and potential energy terms:\nHere ε(p i ) is the energy-momentum relation for the particle i, for example, ε(p i ) = p 2 i + m 2 for relativistic particles and ε(p i ) = p 2 i /(2m) for non-relativistic particles.\nBased on the structure of the Hamiltonian in Eq. (2) it is clear that the probability P ({r i , p i }) factorizes into momentum-and coordinate-dependent parts. The momentum distribution is given by the (non-)relativistic Maxwell-Boltzmann distribution. Thus, the sampling of particle momenta proceeds in the standard way. However, the probability distribution of the coordinates of the particles is affected by the interaction potential V (r i , r j ). The unnormalized distribution density reads\n(\nWithout loss of generality, consider now that the system is placed in a cubic volume. A sampling of momenta and coordinates can be performed via rejection sampling. First, the coordinates are sampled uniformly from the cubic volume. Then, the particle momenta are sampled from the Maxwell-Boltzmann distribution. The sampled configuration is accepted with relative weight proportional toP coord ({r i }) in Eq. (3). For purely repulsive potentials, V (r i , r j ) ≥ 0, the maximum weight computed through Eq. (3) does not exceed unity, and thus the application of rejection sampling is straightforward. The method can, in principle, be generalized to arbitrary two-body potentials that include attraction, for instance, by rescaling the maximum weight or oversampling."},{"title":"B. Hard-core repulsion","content":"Here the focus is on a specific example of the interaction potential, namely the hard-core interaction potential given by\nWith this choice of V HC (r i , r j ), one deals with the system of hard spheres. Here, σ = 2r c is the hard-sphere diameter, and r c is the radius. As discussed in Appendix A, the equation of state of the hard-sphere system reduces to that of the excluded volume model in the dilute limit, provided that the excluded volume parameter b of the latter model is taken as\nThe hard-sphere and excluded volume model equations of state are very similar at bn 0.1, where n ≡ N/V . Thus the hard-sphere model can mimic the excluded volume effect if this condition is met. As discussed below, this is the case for the particlization stage in heavy-ion collisions.\nFor the case of hard-core potential (4) the probability densityP coord ({r i }) in Eq. (3) vanishes if any pair of particles overlap, i.e. if |r i − r j | < σ for any (i, j) pair. In all other cases,P coord ({r i }) = 1.\nIt follows that the system configuration can be sampled by the following algorithm involving a rejection sampling step:\n1. The coordinates of N particles are sampled uniformly from the given (cubic) volume.\n2. If, for any (i, j) pair of particles, they overlap, i.e., |r i − r j | < σ, the configuration is rejected, and one goes back to step 1.\n3. The momenta of the particles are sampled from the Maxwell-Boltzmann distribution.\nThe procedure can be further sped up by combining steps 1 and 2: If a newly sampled particle overlaps with any previously sampled particles, the configuration can be rejected outright without needing to sample any remaining particles.\nC. Testing the sampling method"},{"title":"Canonical ensemble","content":"Here, sampling of the hard-sphere gas is performed, and particle number fluctuations in various coordinate space subsystems are analyzed. The sampling is carried out for a fixed value of the hard-core radius r c and particle number density n = N/V , but for different values of the total number of particles N . In particular, the scaled density bn is fixed to bn = 0.03, where b is given by Eq. (5). The value of b is used to set the length scale, e.g. the dimensionless system volume readsṼ = V /b = N/(bn), the cube length isL =Ṽ 1/3 , and the dimensionless hard-core radius isr c =L 3(bn) 16πN 1/3 = 3 16π 1/3 . To minimize the effect of finite system size, periodic boundary conditions with minimum-image convention (periodic box) are applied when checking the particles for overlap. The results are also compared with the case where periodic boundary conditions are not applied (single box).\nAfter sampling the configurations, the first and second moments of particle number distributions inside various subvolumes of the coordinate space along the z direction are studied. More specifically, the results are analyzed as a function of the total volume fraction α < 1 covered by the subvolume. A particle belongs to a subvolume α if its scaled coordinatez is in the range 1−α 2 <z L < 1+α 2 . Monte Carlo simulations are performed for N = 20, 40, 80, 160. For the first moments, one observes that the mean number of particles in a subvolume is consistent within statistical errors with the relation N α = αN , that is, the particles are uniformly distributed throughout the volume on average, as they should.\nFor the second moment, the behavior of the scaled variance ω α ≡\nis studied. Note that this quantity vanishes in the limit α → 1 since the total particle number is fixed. For this reason, one looks at a scaled quantityω α = ω α /(1 − α). In Ref. [29] it was shown that this quantity is expected to approach the scaled variance in the grand-canonical limit, i.e.,ω α N →∞ → ω gce . In the excluded volume model, one has [30]\nwhile in a more accurate Carnahan-Starling model (see Appendix A) one has ω CS gce ≈ 0.942. The α-dependence ofw α resulting from the described sampling procedure is shown in Fig. 1, both with and without periodic boundary conditions. The sample size is of the order of several million configurations in each case. It is seen that the results show a suppression of scaled variance by magnitude, which is similar to the one given by the grand-canonical limit ω CS gce , except for the α → 0 and α → 1 limits, where this quantity tends to the Poisson limit of unity. Without periodic boundary conditions, the finite-size effects are more significant."},{"title":"Grand-canonical ensemble","content":"So far, the sampling has been discussed in the context of the canonical ensemble, i.e. for the case where the total number N is fixed. Consider the situation where the total number N itself fluctuates event-by-event. One example of this scenario is the grand-canonical ensemble, where the system can exchange particles with a heat bath. Even in the canonical ensemble, the total number of particles of a given type can fluctuate if the system is multi-component. A relevant example for the present studies is the baryons and antibaryons in the canonical ensemble hadron resonance gas. Even though the net number of baryons is precisely conserved, the individual numbers of baryons and antibaryons fluctuate.\nThe generic way to incorporate the fluctuations of N is to fold the canonical ensemble procedure with the sampling of the total number N in each event. For example, the sampling of the grand-canonical excluded volume model is explored here. Thus, simulations are performed not for a fixed total number N , but for a fixed mean number N that fluctuates following the grandcanonical ensemble distribution given by the EV model. The sampling is performed in two steps:\n1. The total number N is sampled from the grandcanonical excluded volume model, where the method described in Ref. [16] is used.\n2. The coordinates of the N particles are sampled using the method described in Sec. II B. Figure 2 shows the sampling results for w α as functions of α in the grand-canonical ensemble. For α → 1 the results approach the grand-canonical value, as expected. The finite-size effects play a role at finite α, keepingw α slightly above the grand-canonical limiting value. For α → 0,w α approaches the Poisson limit of unity."},{"title":"D. SPR approximation","content":"One can introduce the following approximation to the sampling procedure to achieve sufficiently fast sampling of an even larger number of particles. Particles are sampled one by one. When the ith particle is sampled, its possible overlap with any of the already sampled particles is checked. If an overlap is detected, the sampled particle is rejected, and the process is repeated. The difference from the exact method is that the already sampled i − 1 particles are not rejected but retained. This procedure is referred to as the single particle rejection (SPR) approximation, and it allows one to significantly improve the speed of the sampling procedure, especially for large systems. The SPR approximation can be expected to be most accurate for moderate values of bn, i.e. for dilute systems. Indeed, in dilute systems, it is improbable for a newly sampled particle to overlap with more than one other particle. Thus, there is no real need to reject all previously sampled particles if only a single overlap is detected.\nThe accuracy of the SPR approximation can be tested with simulations. The results forω α obtained using the SPR approximation are depicted in Fig. 3, for N = 80, 160, 640, for the periodic box only, and in the canonical ensemble. The results for N = 80 and 160 are consistent within errors with the results of the unbiased method, shown in Fig. 3 by the symbols. The results for N = 640 indicate the approaching of the grand-canonical value ω ev gce in the limit N → ∞. The method can be tested further by considering higher values of bn. Carlo results consistently approach the grand-canonical values for all values of bn considered. One can conclude that the SPR approximation produces accurate results for the variance of particle number distributions at densities at least up to bn = 0.5. Differences between the SPR approximation and the full sampler can be seen in more subtle observables than the cumulants. An example based on the radial distribution function is discussed in Appendix B."},{"title":"III. REPULSIVE CORE IN THE COOPER-FRYE HADRON SAMPLING PROCEDURE","content":"The system created in heavy-ion collisions is not a static box but a dynamically expanding, inhomogeneous fireball. The momentum distribution for hadron species j emerging from hydrodynamics is given by the Cooper-Frye formula [31]\nneglecting the shear and bulk viscous corrections. Here Σ(x) is the Cooper-Frye hypersurface, dΣ µ (x) is the hypersurface element, u µ (x) is the collective four-velocity, p µ is the four-momentum and f j is the distribution function corresponding to a local thermodynamic equilibrium at a space-time point x. The total mean number of particles can be obtained by integrating Eq. (7) over the momenta. This results in\nHere n j is the equilibrium local rest frame density of particle species j near the space-time point x. This density may generally contain the effect of hadronic interactions, such as excluded volume. The total effective volume at particlization is\nthus one can introduce the average effective density of particle species j as n j = N j /V eff . The Cooper-Frye sampling procedure that includes the effect of hard-core repulsion proceeds as follows:\n1. The total numbers of all the particle species are sampled. The effect of excluded volume on the distribution of the total particle numbers is included, where applicable, following the method described in Refs. [16,32] and using the average effective densities n j as input. The exact global conservation laws are enforced through rejection sampling, where configurations that do not satisfy the exact conservation laws are discarded.\n2. The momenta and coordinates of each hadron are sampled one by one. To do that, first, the hypersurface element from which the given hadron is sampled is determined. This is done via the multinomial distribution where each volume element x is weighted by dN j (x) , i.e. by the grand-canonical mean yield for the given hadron species emitted from that element. Then the momenta and coordinates of the hadron emitted from the chosen hypersurface element x are sampled via the standard procedure.\n3. All pairs of hadrons with hard-core repulsion between them are checked for overlap. If an overlap is detected, one goes back to the previous step. As in Sec. II, to speed-up the procedure, one can check the overlap of a newly sampled particle with already sampled ones before sampling the remaining ones. As detailed below, each pair of hadron species can have a different value of the minimum distance σ of their closest possible approach.\nNote that a numerical Cooper-Frye hypersurface might have negative volume elements with dΣ µ (x)u µ (x) < 0. Here these volume elements are skipped, i.e., the presence of the θ function, θ(dΣ µ (x)u µ (x)), in all Cooper-Frye integrals is implied.\nA few relevant details must be specified regarding the treatment of global conservation laws. The total baryon number to be conserved is calculated from the 4π mean hadron yields from Eq. (8). The calculated value is then rounded to the nearest integer. To maintain consistency, all elements of the hypersurface dΣ(x) are then rescaled by a common factor so that the baryon number calculated by Eq. (8) coincides with the rounded integer number. This has only a minor effect, as the rescaling factor is very close to unity in all considered cases. If strangeness is treated canonically, it is required to be equal to zero in each generated event. Finally, the total net charge is constrained to reproduce the charge-to-baryon ratio of Q/B = 0.4. If the total electric charge satisfying the Q/B = 0.4 condition is not an integer, it is rounded to the nearest integer.\nNote that the presented algorithm differs from most of the conventional methods used. There, hadrons are first sampled from each cell one by one, and then constraints from global conservation laws are imposed. In this way, one may have to resort to rejecting all sampled particles if the global conservation laws are not satisfied. The present algorithm deals with this problem more efficiently. Here, first, the total multiplicities of all hadrons are sampled, then the conservation laws are checked, and only after that does the sampling of hadrons' momenta and coordinates begin. The drawback is that, in the case of numerical Cooper-Frye hypersurfaces, this method requires storing the pre-computed multinomial probabilities for all hadron species and all Cooper-Frye hypersurface elements, which increases the memory requirements significantly. For example, the method requires about 5-15 Gb of RAM for sampling the central Au-Au collisions at RHIC-BES using Cooper-Frye hypersurfaces from the MUSIC code [33]. The present algorithm may also be less efficient for event-by-event hydrodynamics, as opposed to single-shot hydrodynamics used in the present study.\nThe procedure to check the overlap between two particles at the Cooper-Frye particlization has to be modified compared to the box case to account for the presence of collective motion, relativistic effects, and the fact that particles are emitted at different time moments. This is achieved as follows:\n• First, both particles are boosted into their centerof-mass frame.\n• Then, the particle emitted at the earlier time is propagated along the straight line to the time of the particle which was emitted later.\n• Finally, the distance |r i − r j | between the particles is calculated and checked if it is below the threshold value of σ sisj , where s i and s j is the species type of particles i and j, respectively.\nThe hadron resonance gas at the Cooper-Frye particlization stage is a multi-component system. Thus, in general, the hard-core repulsion between different hadron species is characterized not by a single excluded volume parameter b, but by a matrix b αβ of excluded volume parameters where each element corresponds to a distinct pair of hadron species. Note that this matrix need not be symmetric [34]. The matrix b αβ can be used to determine the threshold distances σ αβ . As discussed in the framework of the multi-component excluded volume model [34], the coefficients b αβ are related to σ αβ as\nthus σ αβ = 3b αβ 2π\nThe SPR approximation can be employed similarly to the box case in Sec. II. Using the SPR approximation is essential for sampling central collisions of heavy ions, where the system is so large that applying the full (unbiased) method is prohibitively time-consuming.\nIt is instructive to summarize the list of approximations and possible limitations of the method: (i) Total particle numbers N j are sampled from an auxiliary excluded volume HRG model characterized by constant volume V eff and mean hadron yields N j , both calculated via the Cooper-Frye formula. The resulting excluded volume effect on the cumulants of N j -distribution may be slightly different from the true result if the distribution of particle number densities across the Cooper-Frye hypersurface is inhomogeneous. The accuracy of this approximation for central Au-Au collisions at various beam energies is verified in the next section by comparing Monte Carlo results with analytic approximations, yielding only minor differences.\n(ii) As the N j -distribution is sampled from the excluded volume model rather than from the exact hard-sphere model distribution, for consistency, it is imperative that the excluded volume model provides an accurate approximation for the hardspheres equation of state. As shown in the Appendix A, this is satisfied with high precision for bn 0.10 − 0.15.\nBoth approximations (i) and (ii) concerning the N jdistribution become less relevant when the canonical ensemble is applied, as the canonical ensemble effects then dominate the fluctuations of N j .\n(iii) As the hard-core repulsion is an inherently nonrelativistic concept, issues may arise when relativistic effects are strong. In particular, this can be an issue when the two overlapping particles have large relative velocities. This may happen, for instance, when the particles are emitted from different regions of the fireball characterized by significantly different collective velocities. However, as the hard-core repulsion is a short-range phenomenon, the issue becomes irrelevant in this case since these particles would not overlap in the coordinate space. Particles that are close by in the coordinate space, on the other hand, would typically correspond to very similar collective velocities, thus their relative velocities are determined by the local temperature T (x), which is usually of order 160 MeV or lower in heavy-ion collisions at particlization. Therefore, the relativistic effects become less relevant for m/T (x) 1, as is the case for baryons (m B ≥ 938 MeV/c 2 ), but could be important if hard-core repulsion is incorporated for lighter particles such as pions.\nThe Cooper-Frye sampling with short-range repulsion is implemented within the open source package Thermal-FIST [35] starting from version 1.4."},{"title":"IV. APPLICATION OF THE METHOD TO PROTON CUMULANTS IN HEAVY-ION COLLISIONS","content":"To illustrate the newly developed FIST sampler and the corresponding effect of hard-core repulsion, the behavior of (net-)proton cumulants in central collisions of heavy ions at various collision energies is studied. The effect of (anti)baryon excluded volume on the cumulants has previously been studied at LHC [36] and RHIC-BES [17] energies using different methods, thus, it is instructive to compare these results with the present method.\nIt is assumed here that excluded volume repulsion is present for all baryon-baryon and, by symmetry, all antibaryon-antibaryon pairs, as motivated by recent analyses of lattice QCD data on baryon number susceptibilities [22,24]. This implies b αβ = b > 0 if either α, β ∈ B or α, β ∈B, and b αβ = 0 otherwise. To be consistent with the earlier studies, the value of the baryon excluded volume parameter is chosen to be b = 1 fm 3 , corresponding to a classical hard-core radius of around r c = 0.39 fm."},{"title":"A. LHC","content":"First, the LHC energies are studied. More specifically, the 0-5% central Pb-Pb collisions at √ s NN = 2.76 TeV are analyzed, using a longitudinally boostinvariant Cooper-Frye hypersurface based on the blastwave model. The parametrization of the hypersurface is identical to a previous study [16], where the blast-wave model parameters are based on fits to the p T spectra within a single freeze-out scenario [38]. These parameters are uniform across the entire hypersurface. The particlization temperature is T = 160 MeV, and the chemical potentials are vanishing. For this choice of parameters, one has bn B = bnB ≈ 0.029, where n B and nB are the number densities of baryons and antibaryons, respectively.\nA total of 28.3 million events is sampled. The events incorporate the baryon hard-core repulsion as described above, as well as the exact global conservation of baryon number. The behavior of cumulants of net proton number distribution is analyzed. To minimize the statistical error, the fireball volume is reduced by factor 10, from dV /dy = 400 fm 3 to dV /dy = 40 fm 3 . This reduction of the fireball volume is achieved through the reduction of its transverse radius from R ⊥ = 9 fm to R ⊥ ≈ 2.84 fm. The analysis is therefore focused on the ratios of cumulants where the trivial dependence on the volume is canceled. For more details on this procedure to minimize the statistical error, see Ref. [16]. Figure 5 depicts the behavior of (a) the net-proton variance normalized over the Skellam distribution baseline, κ 2 [p −p]/ p +p and (b) the net-proton kurtosis ratio,\nThe calculations are performed in the experimental acceptance used in the measurements performed by the ALICE Collaboration [39], corresponding to cuts in (anti)proton momentum, 0.6 < p < 1.5 GeV/c, and pseudorapidity, |η| < η max . The results in Fig. 5 are shown as a function of η max by solid black lines with grey bands depicting the statistical uncertainties. The results are compared with the earlier calculations of Ref. [16] obtained in the framework of the so-called subensemble sampler (dash-dotted black lines), as well as the ideal gas baseline that incorporates the effect of baryon conservation but not hard-core repulsion (dotted blue lines). Note that the calculations for κ 4 [p −p]/κ 2 [p −p] do not account for the effect of volume fluctuations which would influence the measurements [16].\nThe FIST sampler results agree with the earlier results based on the subensemble sampler. They also show visible deviations from the ideal gas baseline. This indicates that the FIST sampler provides a reasonable de-scription of the excluded volume effect. In contrast to the subensemble sampler, which is mainly restricted to longitudinally boost invariant systems such as those encountered at the LHC, the FIST sampler can also be straightforwardly applied to lower collision energies where boost invariance does not hold. This is explored at the RHIC beam energy scan energies in the next section."},{"title":"B. RHIC-BES","content":"Net proton fluctuations have been measured by the STAR Collaboration in Au-Au collisions in collision energy range √ s NN = 7.7 − 200 GeV [40,41]. In Ref. [17] the fluctuations have been analyzed based on relativistic hydrodynamics simulations with MUSIC [42], incorporating effects of baryon conservation and excluded volume. Calculations in Ref. [17] were performed analytically, in two steps: (i) proton cumulants are calculated in the experimental acceptance using the Cooper-Frye formula in the grand-canonical limit, including the excluded volume effect; (ii) correction for baryon number conservation is performed using a method called SAM-2.0 [43].\nHere, the calculations of proton cumulants are performed using the FIST sampler, by sampling 0-5% central Au-Au collisions at energies √ s NN = 7.7, 14.5, 19.6, 27, 39, 62.4 and 200 GeV. The sampling is performed with and without the effect of exact baryon conservation, and it uses the same Cooper-Frye hypersurfaces produced by the MUSIC code that is based on [42] and available at [33]. For each energy, at least several million events have been generated. These calculations are compared to the analytic approximations of Ref. [17] in Fig. 6, for (a) the second proton cumulant ratio κ 2 /κ 1 and (b) the third order net proton cumulant ratio κ 3 /κ 1 . The FIST sampler results are shown by the bands, while the lines show the analytic results.\nIn the absence of baryon conservation (baryon repulsion only, green lines and bands in Fig. 6) deviations of the cumulant ratios from the baseline of unity are driven solely by baryon repulsion, which leads to a modest suppression. The FIST sampler results agree with the analytic approximations, validating the method at RHIC-BES energies.\nIncorporating exact baryon conservation (baryon repulsion + baryon cons., black lines, and grey bands in Fig. 6) leads to a more potent suppression of the cumulant ratios. In this case, the analytic and Monte Carlo results agree at high collision energies ( √ s NN 40 GeV), but deviations between the two are visible at lower √ s NN .\nThese deviations can be attributed to inaccuracies of the SAM-2.0 framework at lower collision energies. As discussed in the original publication [43], it would tend to overestimate the effect of (repulsive) interactions. Comparison with the FIST sampler allows one to quantify the accuracy of this approximation at RHIC-BES conditions, revealing a modest overestimation of excluded volume effect on proton cumulants which gets worse as [17] with and without exact baryon conservation, respectively. The red symbols correspond to the experimental data of the STAR Collaboration [40,41]. collision energy is decreased.\nAs a new application of the FIST sampler, one can simultaneously incorporate the effects of baryon repulsion and exact conservation of all three QCD conserved charges: baryon number, electric charge, and strangeness. The electric charge conservation, in particular, can influence the fluctuations of the proton number since the proton carries the electric charge. At high col-lision energies, the dominant electric charge carriers are pions. Thus, the effect of electric charge conservation on protons is expected to be small [36]. As the collision energy decreases, the fraction of charge carried by protons increases, and so does the effect of charge conservation on proton cumulants. This is reflected in the FIST sampler results shown in Fig. 6 by the blue bands, indicating additional suppression of (net-)proton κ 2 /κ 1 and κ 3 /κ 1 due to electric charge conservation, which becomes visible at √ s NN 40 GeV. Interestingly, at all collision energies considered, the FIST sampler results with exact conservation of baryon number, electric charge, and strangeness are in good agreement with the analytical calculations of proton cumulants from Ref. [43] that incorporate baryon conservation but not electric charge and strangeness. Thus, it appears that the additional suppression due to electric charge conservation at all energies is of the same magnitude as the overestimation of the excluded volume effect in the analytical calculation of Ref. [43] with SAM-2.0. Although this coincidence appears to be purely accidental, it indicates that the results of Ref. [43] serve as an accurate baseline for proton cumulants incorporating non-critical effects like baryon repulsion and exact conservation of multiple conserved charges."},{"title":"C. GSI-SIS","content":"The final application of the new method concerns the Au-Au collisions at √ s NN = 2.4 GeV, as probed by the HADES experiment at GSI-SIS. The experimental data [44] indicate sizable multiproton correlations. These were recently analyzed in the framework of a fireball model [45]. The Cooper-Frye hypersurface was parameterized in the framework of the Siemens-Rasmussen model with a Hubble-like collective flow, which was found to provide a reasonable description of the p T spectra of pions and protons [46], while the temperature T ≈ 70 MeV and baryochemical potential µ B ≈ 875 MeV were extracted from hadron yields [47]. Here this parametrization of the hypersurface is used in FIST sampler to study the behavior of proton cumulants influenced by conservation laws and baryon repulsion. At √ s NN = 2.4 GeV, a significant fraction of protons is bound into light nuclei in the final state. Based on the preliminary HADES data [46], one can estimate that about 37.5% of the final-state protons are bound. The light nuclei are not incorporated into the FIST sampler employed in the present work. Thus, to model this effect, following Ref. [45], each proton emitted from the fireball is assumed to be bound into light nuclei with a probability of q nucl = 0.375 and not contribute to the measured cumulants of the proton number. The corresponding results for proton number cumulant ratios κ 2 /κ 1 , κ 3 /κ 2 , and κ 4 /κ 2 are shown in Fig. 7 as functions of the rapidity cut y cut in the experimental acceptance of the HADES experiment.\nThe results show a similar pattern to RHIC-BES energies: both the baryon conservation and repulsion suppress the cumulant ratios, with baryon conservation having a stronger effect. The additional effect of electric charge conservation (blue lines) is even more notable, which suppresses the cumulant ratios even further. The combined effect of baryon and electric charge conservation is more significant than that of baryon repulsion.\nNote that both the total baryon and electric charge of the participant matter fluctuates event by event in the experiment. This is in contrast to the FIST sampler where they are both fixed. The experimental data of the HADES collaboration [44], shown in Fig. 7 by black symbols, is corrected for volume (participant) number fluctuations, which justifies the canonical treatment of baryon number for the participant matter. However, the total electric charge of the participants still fluctuates; thus, the FIST sampler results in Fig. 7 should be considered an upper bound on the effect of exact conservation of the electric charge on the proton number cumulants. It is also clear that the model fails to describe the experimental data even qualitatively. For example, the scaled variance κ 2 /κ 1 increases with the rapidity cut y cut in the experiment, as opposed to the suppression predicted due to conservation laws and short-range repulsion. This indicates that the behavior of the proton cumulants in the experimental data is not driven by the conservation laws or short-range repulsion that are incorporated in FIST sampler. For a discussion of the various possibilities, including the QCD critical point, see Ref. [45] The present analysis indicates that the exact conservation of multiple conserved charges, as opposed to only that of the baryon number, is essential for quantitative analysis of proton number cumulants at moderate collision energies, √ s NN 7.7 GeV. The FIST sampler allows one to evaluate the corresponding baselines in this energy regime, which can then be used to analyze future experimental data coming from the fixed-target program at RHIC [48] or the CBM experiment at FAIR [49].\nFor comparisons of the FIST sampler results in the SIS-HADES regime with analytic approximations, see Appendix C."},{"title":"V. CONCLUSIONS AND OUTLOOK","content":"This study introduced short-range repulsive correlations into the Cooper-Frye hadron sampling procedure. This has been achieved through a rejection sampling step that prohibits any two particles with repulsive interactions from overlapping in the coordinate space, effectively modeling the excluded volume phenomenon. The effect introduces negative correlations between particles visible in normalized cumulants of their distributions. The new method -called the FIST sampler -incorporates this effect and simultaneously allows for both the canonical and grand-canonical treatments of the QCD conserved charges.\nFIST sampler was validated in a periodic box setup, where the method yields the behavior of the scaled variance of particle number, which is consistent with the equation of state of hard spheres. nally boost-invariant scenario at LHC energies, the new method produces the behavior of net proton cumulants up to the fourth order, which is consistent with an earlier study of [16] that used the so-called subensemble sampler. The advantage of FIST sampler becomes evident at lower collision energies, where boost invariance no longer holds and where the application of the subensemble sampler is challenging. The new method allowed one to incorporate the simultaneous effects of baryon repulsion and baryon, electric charge, and strangeness conservation at RHIC-BES and SIS-GSI energies. In particular, the effect of electric charge conservation on proton number cumulants becomes increasingly important as the collision energy is decreased, √ s NN 40 GeV, due to a smaller number of pions relative to protons at lower energies. The resulting cumulant ratios of the (net) proton distribution are consistent with estimates of Ref. [17] that were obtained by employing analytic approximations for baryon conservation and repulsion and neglecting electric charge conservation. The reason for this coincidence at lower energies appears to be that the analytic method of Ref. [17] slightly overestimates the effect of baryon repulsion in the canonical ensemble, which at lower collision energies is approximately compensated for by the electric charge conservation in FIST sampler. Due to this coincidence, the results of Ref. [17] can be taken as a baseline for (net) proton cumulants at RHIC-BES that incorporates non-critical effects such as excluded volume and multiple global conservation laws. The mechanisms above do not describe the experimental data of the HADES Collaboration (Fig. 7), indicating that proton number cumulants are driven by different mechanisms that still need to be clarified.\nFIST sampler has several future potential applications. For instance, one can couple its output with a hadronic afterburner such as UrQMD [50,51] or SMASH [52], which should ideally include the shortrange repulsion for consistency [53], and study the effect of the hadronic phase on proton number cumulants, in particular, that of baryon annihilation [54,55]. Al-though the effects of hadronic afterburner have not been studied in the present work, the FIST sampler code [56] readily supports output tailored for use with UrQMD afterburner [57]. Since the method offers the full flexibility of an event generator, observables other than proton number cumulants, such as, for example, balance functions [58] or distributions of other hadronic species [59] can be studied as well. In particular, one may revisit the Bayesian constraints on transport coefficients of QCD matter and the switching temperature for the transition from hydrodynamics to transport [60,61] by employing the new particlization routine presented here in such an analysis. One can also straightforwardly incorporate the possible flavor dependence of baryon repulsion suggested by recent analyses of lattice QCD data on off-diagonal susceptibilities [24,25,62] by an appropriate choice of the corresponding threshold distances σ αβ in Eq. (12). It can also be interesting to see how the presence of baryonic hard-core repulsion might affect the light nuclei production, such as employed in phase-space coalescence prescriptions [63,64]. These applications will be the subject of future studies.\nUsing the single particle rejection (SPR) approximation was essential to sample central collisions of heavy ions in practical applications of the new routine, as applying a direct unbiased method was prohibitively timeconsuming. The implication is that the sampler is not fully unbiased, even though the tests in a box setup indicate that the sampling of particle number cumulants remains accurate. It is thus advisable to consider other techniques, such as oversampling, to produce a fast, unbiased sampler.\nThe sampling with short-range repulsion described in this work has been implemented into the Monte Carlo event generator within the open source package Thermal-FIST [35], starting from version 1.4. The code for the sampling of various central collisions of heavy ions discussed in the present work is available at [56]."},{"title":"ACKNOWLEDGMENTS","content":"The author is grateful to Mark Gorenstein for reading the manuscript and for useful comments. The author thanks Volker Koch and Chun Shen for fruitful discussions and collaboration on related projects and Dylan Neff for stimulating discussions. The author also acknowledges the hospitality of the staff at the Frankfurt Institute for Advanced Studies where part of this work was done. This work was supported through the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract number DE-FG02-00ER41132. Computational resources were provided by the Frankfurt Center for Scientific Computing (Goethe-HLR).\nAppendix A: The equation of state and particle number fluctuations in the system of hard spheres\nThe equation of state that describes the system of classical particles that interact through a hard-core potential (4) can be written in terms of the compressibility factor Z ≡ P/(nT ):\nwhere η = bn 4 = 4πr 3 c 3 N V is the so-called packing fraction. The explicit expression for the function g(η) is not known, but different approximations are available that model the equation of state accurately when the values of η are not too large. One well-known approximation is the van der Waals excluded volume model where one has P ev = nT 1−bn , thus\nA more involved analytical approximation of the equation of state of hard spheres is given by the Carnahan-Starling model [65]:\nThe accuracy of the excluded volume and Carnahan-Starling models can be verified by comparing the results to numerical Monte Carlo simulations of the hardsphere system. This comparison is depicted in Fig. 8, where the dependence of the compressibility factor Z on bn, calculated in the two models, is compared to the numerical simulations of [66]. The excluded volume model accurately describes the compressibility factor up to bn ≈ 0.10 − 0.15, while it overestimates the numerical data for Z at higher densities. On the other hand, the Carnahan-Starling model accurately describes the Monte Carlo data for all the densities considered, that is, up to at least bn = 0.6. Figure 9. Density dependence of cumulant ratios κ2/κ1 (black, top), κ3/κ1 (blue, middle), κ4/κ2 (red, bottom) of the grand-canonical particle number distribution calculated for the hard-sphere system within the excluded volume (dashed lines) and Carnahan-Starling (solid lines) approximations.\nCumulants of the grand-canonical particle number distribution are given by\nFor n = 1 one has κ 1 ≡ N , therefore, for n > 1 the cumulants read\nUsing the thermodynamic identity iteratively, provided that the pressure P is known as a function of density n. This is the case for the excluded volume [Eq. (A2)] and Carnahan-Starling [Eq. (A3)] models. Figure 9 depicts the behavior of the grand-canonical cumulant ratios κ 2 /κ 1 , κ 3 /κ 1 , and κ 4 /κ 2 as a function of the scaled density bn. Given the accuracy of the Carnahan-Starling approximation for the compressibility Z (Fig. 8), it is expected that the corresponding calculations of the cumulant ratios accurately reflect the grandcanonical particle number distribution in the hard-sphere system in the considered density range. Deviations of the excluded volume model from the Carnahan-Starling model become more visible in higher-order cumulants. The excluded volume model provides accurate results up to bn ≈ 0.10."},{"title":"Appendix B: Radial distribution function","content":"The radial distribution function g(r) describes how the density of particles varies around a reference particle at r = 0 relative to the expectation based on the mean particle number density n = N/V . More specifically, g(r) is defined so that the local particle number density at a distance r from the reference particle is equal to n g(r). Deviations of g(r) from unity signal the presence of correlations between particles.\nThe radial distribution function for the hard-sphere system has been studied for a long time [67,68]. As the particles cannot overlap, one has g(r) = 0 for r < σ. g(r) has a discontinuity at r = σ and approaches a value larger than unity in the limit r → σ + 0, indicating the presence of effective attraction as a many-body effect. For large r/σ values, g(r) is expected to approach unity, given that the hard-core repulsion is a short-range phenomenon. For moderate densities of bn, accurate expressions for g(r) of the hard-sphere system can be obtained based on the solution of the Percus-Yevick equation [67][68][69]. This can be used to validate the performance of the FIST sampler. Figure 10 depicts g(r) in a range 1 < r/σ < 3 resulting from FIST sampler for a system of particles in a periodic box at density bn = 0.12, which was obtained by binning the relative distances of the sampled particles. Blue circles correspond to the case where the SPR approximation was applied, whereas black squares show the unbiased calculation without approximations. In the former case N = 320 particles were sampled in each configuration, while in the latter case, only N = 80 particles were used due to slower computational performance.\nThe results are compared with the analytical expectation for a hard-sphere system based on the Percus-Yevick approximation, which is expected to be highly accurate for density as low as bn = 0.12 [69]. The full calculation is in quantitative agreement with the analytical expectations, indicating that FIST sampler correctly reproduces the spatial correlations between particles that interact through the hard-core potential (4). The SPR approximation underestimates g(r) at 1 < r/σ 1.5, in- Figure 10. Radial distribution function g(r) resulting from FIST sampler for a system of particles in a periodic box at density bn = 0.12 with (blue circles) and without (black squares) the application of the SPR approximation. The black line corresponds to g(r) of a hard-sphere system calculated through the Percus-Yevick equation [69].\ndicating that it does not capture the full strength of the many-body effective attraction effect. At the same time, the SPR approximation reproduces the qualitative structure of g(r), and, as discussed in Sec. II D, reproduces accurately the behavior of cumulants of particle number distribution for densities considered in this study. Here a comparison of FIST sampler calculations with the analytic method recently developed in Ref. [45] is performed. This allows one to verify the accuracy of the analytic approximations of Ref. [45]. In particular, Ref. [45] used a method called SAM-2.0 [43] to perform a correction for exact baryon conservation. As discussed in Ref. [43], the method might overestimate the effect of dynamical correlations among particles, in this case, the effect of short-range hard-core repulsion, at low collision energies such as √ s NN = 2.4 GeV at SIS-GSI. By comparing to FIST sampler, which is free of these issues, one can quantify the accuracy of SAM-2.0 in this collision energy regime.\nThe corresponding results are shown in Fig. 11. One sees good agreement between Monte Carlo and analytic calculations for the cases when baryon repulsion is incorporated but not conservation (green lines) or when baryon conservation is implemented but not repulsion (red lines). This validates the analytic approach of Ref. [45] in these regimes, in particular, SAM-2.0 used in [45] to correct for baryon conservation is accurate in the limit of the ideal gas, as expected.\nWhen both the baryon repulsion and conservation are incorporated, visible differences between the FIST sampler and the analytic method occur. The analytic Figure 11. The scaled variance of the proton number distribution in the experimental acceptance of the HADES Collaboration as a function of the rapidity cut ycut. The green (top) lines correspond to calculations incorporating the hard-core repulsion among baryons but not baryon conservation. The calculations shown by the red (middle) lines implement exact baryon conservation but not baryon repulsion. The black (bottom) lines show calculations with both the baryon repulsion and conservation included. The differently styled lines correspond to the results obtained using FIST sampler while the solid lines correspond to the analytic method from Ref. [45]. method, based on SAM-2.0, overestimates the suppression of κ 2 /κ 1 compared to the Monte Carlo sampling. This is an artifact of the SAM-2.0 approximation, which assumes that there are no correlations between particles inside and outside the acceptance apart from the baryon conservation. As discussed in [43], this leads to an overestimation of the correlations present in the grand-canonical limit, namely the effect of hard-core repulsion in this study. A similar observation applies to the higher-order cumulant ratios κ 3 /κ 2 and κ 4 /κ 2 that are not shown in Fig. 11.\nAs follows from the FIST sampler calculations in Fig. 11, the presence of baryon repulsion in addition to baryon conservation leads to only a minor further suppression of proton cumulant ratios, i.e., the effect of hardcore repulsion in the cumulants appears to be essentially washed out by baryon conservation. This appears to be a consequence of the reduced collective flow effect at HADES energies compared to higher energies, which dilutes the space-momentum correlation, meaning that the excluded volume effect in the coordinate space is almost washed out in the momentum space where the measurements are performed. As shown in recent molecular dynamics simulations [70], even critical fluctuations may be washed out when analyzed in momentum space if exact global conservation laws are enforced, and no collective expansion is imposed."},{"title":"Figure 1 .","content":""},{"title":"Figure 2 .","content":""},{"title":"Figure 4 Figure 3 .Figure 4 .","content":""},{"title":"Figure 5 .","content":""},{"title":"Figure 6 .","content":""},{"title":"Figure 7 .","content":""},{"title":"Figure 8 .","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"The description of bulk observables in relativistic heavy-ion collisions is usually performed in the framework of relativistic hydrodynamics (see e.g. [1,2] for an overview). This modeling incorporates the Cooper-Frye particlization stage, where the expanding QCD fluid is transformed into a gas of hadrons and resonances. In most cases, hadron momenta are sampled from local Maxwell-Boltzmann distributions, with the possible inclusion of shear and bulk viscous corrections [3][4][5][6][7]. Multiplicity distributions of the sampled hadrons usually correspond to Poisson statistics, implying independent particle emission. More involved descriptions additionally implement a (micro-)canonical treatment of conservation laws, such as that of energy-momentum and QCD conserved charges [8][9][10][11][12]. Proper treatment of conservation laws is relevant for observables involving event-by-event fluctuations [13][14][15][16][17]. Physically, the system created at particlization corresponds to an ideal hadron resonance gas, possibly with non-equilibrium corrections due to shear and bulk viscosities and (micro-)canonical effects. On the other hand, extensions of the Cooper-Frye procedure are necessary to incorporate any additional physics. One commonly discussed extension of the ideal HRG is short-range repulsive interactions utilizing excluded volume [18][19][20]. For example, by introducing the excluded volume effect into baryon-baryon (and, by symmetry, antibaryonantibaryon) interaction [21], one can effectively model the presence of hard core in nucleon-nucleon scattering and improve the description of several lattice QCD susceptibilities [22][23][24][25]. Another extension would concern the search for the QCD critical point, which would lead to large non-Gaussian fluctuations of the proton number [26] requiring a generalized Cooper-Frye routine [27,28].\\nPreviously, a method called subensemble sampler was introduced in [16] to perform particlization of an interacting hadron resonance gas and was used to study the influence of baryonic excluded volume on proton and baryon number cumulants in Pb-Pb collisions at LHC energies. Although this method is rather generic, it requires partitioning the Cooper-Frye hypersurfaces into patches which should, on the one hand, be large enough to capture all the relevant correlations but, on the other hand, also be small compared to inhomogeneity scales across the hypersurface. How to perform the partition into patches can be ambiguous. The method has been used to study rapidity acceptance dependent observables at LHC energies, where the approximate longitudinal boost invariance is realized [16]. Its application at lower collision energies, however, is more complex. Instead, in Ref. [17], the proton cumulants at RHIC-BES were analyzed using an analytic approach incorporating baryon repulsion and conservation. However, analytic approaches lack the flexibility of event generators and are typically restricted to specific observables, such as (net-)proton number cumulants in a particular acceptance [17]. Therefore, it is advisable to incorporate the effect of short-range repulsion directly into the sampling procedure.\\nThis work introduces the effect of short-range repulsion between particles into the Cooper-Frye particlization through a rejection sampling step, which prohibits any pair of particles from overlapping in the coordinate space. This induces negative correlations between particles that are shown to be consistent with analytical expectations based on the excluded volume model. Then, the method is used to study the effect of baryon repulsion on the cumulants of (net-)proton number distribution in central heavy-ion collisions in a broad energy range √ s NN = 2.4−2760 GeV while simultaneously incorporating exact global conservation of baryon number, electric charge, and strangeness.\\nThe paper is organized as follows. The modeling of arXiv:2208.13693v2 [hep-ph] 9 Dec 2022 the short-range repulsion effect using the rejection sampling step is described in Sec. II. Section III describes how the effect is introduced into a multi-component hadron system at Cooper-Frye particlization. The method's application to heavy-ion collisions is presented in Sec. IV. Conclusions and outlook in Sec. V close the article.\"\n },\n {\n \"title\": \"II. METHOD\",\n \"content\": \"A. Probability of configurations Consider a uniform system of N classical particles in volume V that is in contact with the heat bath characterized by temperature T . Interactions between particles are mediated by a (non-relativistic) pair potential V (r i , r j ). The properties of the system are characterized by the canonical ensemble (T, V, N ). The microscopic configuration of the system can be given by the set {r i , p i } of coordinates and momenta of all particles in the system. The probability of a particular configuration is determined by the Boltzmann factor, i.e.\\nwhere H ({r i , p i }) is the system Hamiltonian comprising the kinetic and potential energy terms:\\nHere ε(p i ) is the energy-momentum relation for the particle i, for example, ε(p i ) = p 2 i + m 2 for relativistic particles and ε(p i ) = p 2 i /(2m) for non-relativistic particles.\\nBased on the structure of the Hamiltonian in Eq. (2) it is clear that the probability P ({r i , p i }) factorizes into momentum-and coordinate-dependent parts. The momentum distribution is given by the (non-)relativistic Maxwell-Boltzmann distribution. Thus, the sampling of particle momenta proceeds in the standard way. However, the probability distribution of the coordinates of the particles is affected by the interaction potential V (r i , r j ). The unnormalized distribution density reads\\n(\\nWithout loss of generality, consider now that the system is placed in a cubic volume. A sampling of momenta and coordinates can be performed via rejection sampling. First, the coordinates are sampled uniformly from the cubic volume. Then, the particle momenta are sampled from the Maxwell-Boltzmann distribution. The sampled configuration is accepted with relative weight proportional toP coord ({r i }) in Eq. (3). For purely repulsive potentials, V (r i , r j ) ≥ 0, the maximum weight computed through Eq. (3) does not exceed unity, and thus the application of rejection sampling is straightforward. The method can, in principle, be generalized to arbitrary two-body potentials that include attraction, for instance, by rescaling the maximum weight or oversampling.\"\n },\n {\n \"title\": \"B. Hard-core repulsion\",\n \"content\": \"Here the focus is on a specific example of the interaction potential, namely the hard-core interaction potential given by\\nWith this choice of V HC (r i , r j ), one deals with the system of hard spheres. Here, σ = 2r c is the hard-sphere diameter, and r c is the radius. As discussed in Appendix A, the equation of state of the hard-sphere system reduces to that of the excluded volume model in the dilute limit, provided that the excluded volume parameter b of the latter model is taken as\\nThe hard-sphere and excluded volume model equations of state are very similar at bn 0.1, where n ≡ N/V . Thus the hard-sphere model can mimic the excluded volume effect if this condition is met. As discussed below, this is the case for the particlization stage in heavy-ion collisions.\\nFor the case of hard-core potential (4) the probability densityP coord ({r i }) in Eq. (3) vanishes if any pair of particles overlap, i.e. if |r i − r j | < σ for any (i, j) pair. In all other cases,P coord ({r i }) = 1.\\nIt follows that the system configuration can be sampled by the following algorithm involving a rejection sampling step:\\n1. The coordinates of N particles are sampled uniformly from the given (cubic) volume.\\n2. If, for any (i, j) pair of particles, they overlap, i.e., |r i − r j | < σ, the configuration is rejected, and one goes back to step 1.\\n3. The momenta of the particles are sampled from the Maxwell-Boltzmann distribution.\\nThe procedure can be further sped up by combining steps 1 and 2: If a newly sampled particle overlaps with any previously sampled particles, the configuration can be rejected outright without needing to sample any remaining particles.\\nC. Testing the sampling method\"\n },\n {\n \"title\": \"Canonical ensemble\",\n \"content\": \"Here, sampling of the hard-sphere gas is performed, and particle number fluctuations in various coordinate space subsystems are analyzed. The sampling is carried out for a fixed value of the hard-core radius r c and particle number density n = N/V , but for different values of the total number of particles N . In particular, the scaled density bn is fixed to bn = 0.03, where b is given by Eq. (5). The value of b is used to set the length scale, e.g. the dimensionless system volume readsṼ = V /b = N/(bn), the cube length isL =Ṽ 1/3 , and the dimensionless hard-core radius isr c =L 3(bn) 16πN 1/3 = 3 16π 1/3 . To minimize the effect of finite system size, periodic boundary conditions with minimum-image convention (periodic box) are applied when checking the particles for overlap. The results are also compared with the case where periodic boundary conditions are not applied (single box).\\nAfter sampling the configurations, the first and second moments of particle number distributions inside various subvolumes of the coordinate space along the z direction are studied. More specifically, the results are analyzed as a function of the total volume fraction α < 1 covered by the subvolume. A particle belongs to a subvolume α if its scaled coordinatez is in the range 1−α 2 <z L < 1+α 2 . Monte Carlo simulations are performed for N = 20, 40, 80, 160. For the first moments, one observes that the mean number of particles in a subvolume is consistent within statistical errors with the relation N α = αN , that is, the particles are uniformly distributed throughout the volume on average, as they should.\\nFor the second moment, the behavior of the scaled variance ω α ≡\\nis studied. Note that this quantity vanishes in the limit α → 1 since the total particle number is fixed. For this reason, one looks at a scaled quantityω α = ω α /(1 − α). In Ref. [29] it was shown that this quantity is expected to approach the scaled variance in the grand-canonical limit, i.e.,ω α N →∞ → ω gce . In the excluded volume model, one has [30]\\nwhile in a more accurate Carnahan-Starling model (see Appendix A) one has ω CS gce ≈ 0.942. The α-dependence ofw α resulting from the described sampling procedure is shown in Fig. 1, both with and without periodic boundary conditions. The sample size is of the order of several million configurations in each case. It is seen that the results show a suppression of scaled variance by magnitude, which is similar to the one given by the grand-canonical limit ω CS gce , except for the α → 0 and α → 1 limits, where this quantity tends to the Poisson limit of unity. Without periodic boundary conditions, the finite-size effects are more significant.\"\n },\n {\n \"title\": \"Grand-canonical ensemble\",\n \"content\": \"So far, the sampling has been discussed in the context of the canonical ensemble, i.e. for the case where the total number N is fixed. Consider the situation where the total number N itself fluctuates event-by-event. One example of this scenario is the grand-canonical ensemble, where the system can exchange particles with a heat bath. Even in the canonical ensemble, the total number of particles of a given type can fluctuate if the system is multi-component. A relevant example for the present studies is the baryons and antibaryons in the canonical ensemble hadron resonance gas. Even though the net number of baryons is precisely conserved, the individual numbers of baryons and antibaryons fluctuate.\\nThe generic way to incorporate the fluctuations of N is to fold the canonical ensemble procedure with the sampling of the total number N in each event. For example, the sampling of the grand-canonical excluded volume model is explored here. Thus, simulations are performed not for a fixed total number N , but for a fixed mean number N that fluctuates following the grandcanonical ensemble distribution given by the EV model. The sampling is performed in two steps:\\n1. The total number N is sampled from the grandcanonical excluded volume model, where the method described in Ref. [16] is used.\\n2. The coordinates of the N particles are sampled using the method described in Sec. II B. Figure 2 shows the sampling results for w α as functions of α in the grand-canonical ensemble. For α → 1 the results approach the grand-canonical value, as expected. The finite-size effects play a role at finite α, keepingw α slightly above the grand-canonical limiting value. For α → 0,w α approaches the Poisson limit of unity.\"\n },\n {\n \"title\": \"D. SPR approximation\",\n \"content\": \"One can introduce the following approximation to the sampling procedure to achieve sufficiently fast sampling of an even larger number of particles. Particles are sampled one by one. When the ith particle is sampled, its possible overlap with any of the already sampled particles is checked. If an overlap is detected, the sampled particle is rejected, and the process is repeated. The difference from the exact method is that the already sampled i − 1 particles are not rejected but retained. This procedure is referred to as the single particle rejection (SPR) approximation, and it allows one to significantly improve the speed of the sampling procedure, especially for large systems. The SPR approximation can be expected to be most accurate for moderate values of bn, i.e. for dilute systems. Indeed, in dilute systems, it is improbable for a newly sampled particle to overlap with more than one other particle. Thus, there is no real need to reject all previously sampled particles if only a single overlap is detected.\\nThe accuracy of the SPR approximation can be tested with simulations. The results forω α obtained using the SPR approximation are depicted in Fig. 3, for N = 80, 160, 640, for the periodic box only, and in the canonical ensemble. The results for N = 80 and 160 are consistent within errors with the results of the unbiased method, shown in Fig. 3 by the symbols. The results for N = 640 indicate the approaching of the grand-canonical value ω ev gce in the limit N → ∞. The method can be tested further by considering higher values of bn. Carlo results consistently approach the grand-canonical values for all values of bn considered. One can conclude that the SPR approximation produces accurate results for the variance of particle number distributions at densities at least up to bn = 0.5. Differences between the SPR approximation and the full sampler can be seen in more subtle observables than the cumulants. An example based on the radial distribution function is discussed in Appendix B.\"\n },\n {\n \"title\": \"III. REPULSIVE CORE IN THE COOPER-FRYE HADRON SAMPLING PROCEDURE\",\n \"content\": \"The system created in heavy-ion collisions is not a static box but a dynamically expanding, inhomogeneous fireball. The momentum distribution for hadron species j emerging from hydrodynamics is given by the Cooper-Frye formula [31]\\nneglecting the shear and bulk viscous corrections. Here Σ(x) is the Cooper-Frye hypersurface, dΣ µ (x) is the hypersurface element, u µ (x) is the collective four-velocity, p µ is the four-momentum and f j is the distribution function corresponding to a local thermodynamic equilibrium at a space-time point x. The total mean number of particles can be obtained by integrating Eq. (7) over the momenta. This results in\\nHere n j is the equilibrium local rest frame density of particle species j near the space-time point x. This density may generally contain the effect of hadronic interactions, such as excluded volume. The total effective volume at particlization is\\nthus one can introduce the average effective density of particle species j as n j = N j /V eff . The Cooper-Frye sampling procedure that includes the effect of hard-core repulsion proceeds as follows:\\n1. The total numbers of all the particle species are sampled. The effect of excluded volume on the distribution of the total particle numbers is included, where applicable, following the method described in Refs. [16,32] and using the average effective densities n j as input. The exact global conservation laws are enforced through rejection sampling, where configurations that do not satisfy the exact conservation laws are discarded.\\n2. The momenta and coordinates of each hadron are sampled one by one. To do that, first, the hypersurface element from which the given hadron is sampled is determined. This is done via the multinomial distribution where each volume element x is weighted by dN j (x) , i.e. by the grand-canonical mean yield for the given hadron species emitted from that element. Then the momenta and coordinates of the hadron emitted from the chosen hypersurface element x are sampled via the standard procedure.\\n3. All pairs of hadrons with hard-core repulsion between them are checked for overlap. If an overlap is detected, one goes back to the previous step. As in Sec. II, to speed-up the procedure, one can check the overlap of a newly sampled particle with already sampled ones before sampling the remaining ones. As detailed below, each pair of hadron species can have a different value of the minimum distance σ of their closest possible approach.\\nNote that a numerical Cooper-Frye hypersurface might have negative volume elements with dΣ µ (x)u µ (x) < 0. Here these volume elements are skipped, i.e., the presence of the θ function, θ(dΣ µ (x)u µ (x)), in all Cooper-Frye integrals is implied.\\nA few relevant details must be specified regarding the treatment of global conservation laws. The total baryon number to be conserved is calculated from the 4π mean hadron yields from Eq. (8). The calculated value is then rounded to the nearest integer. To maintain consistency, all elements of the hypersurface dΣ(x) are then rescaled by a common factor so that the baryon number calculated by Eq. (8) coincides with the rounded integer number. This has only a minor effect, as the rescaling factor is very close to unity in all considered cases. If strangeness is treated canonically, it is required to be equal to zero in each generated event. Finally, the total net charge is constrained to reproduce the charge-to-baryon ratio of Q/B = 0.4. If the total electric charge satisfying the Q/B = 0.4 condition is not an integer, it is rounded to the nearest integer.\\nNote that the presented algorithm differs from most of the conventional methods used. There, hadrons are first sampled from each cell one by one, and then constraints from global conservation laws are imposed. In this way, one may have to resort to rejecting all sampled particles if the global conservation laws are not satisfied. The present algorithm deals with this problem more efficiently. Here, first, the total multiplicities of all hadrons are sampled, then the conservation laws are checked, and only after that does the sampling of hadrons' momenta and coordinates begin. The drawback is that, in the case of numerical Cooper-Frye hypersurfaces, this method requires storing the pre-computed multinomial probabilities for all hadron species and all Cooper-Frye hypersurface elements, which increases the memory requirements significantly. For example, the method requires about 5-15 Gb of RAM for sampling the central Au-Au collisions at RHIC-BES using Cooper-Frye hypersurfaces from the MUSIC code [33]. The present algorithm may also be less efficient for event-by-event hydrodynamics, as opposed to single-shot hydrodynamics used in the present study.\\nThe procedure to check the overlap between two particles at the Cooper-Frye particlization has to be modified compared to the box case to account for the presence of collective motion, relativistic effects, and the fact that particles are emitted at different time moments. This is achieved as follows:\\n• First, both particles are boosted into their centerof-mass frame.\\n• Then, the particle emitted at the earlier time is propagated along the straight line to the time of the particle which was emitted later.\\n• Finally, the distance |r i − r j | between the particles is calculated and checked if it is below the threshold value of σ sisj , where s i and s j is the species type of particles i and j, respectively.\\nThe hadron resonance gas at the Cooper-Frye particlization stage is a multi-component system. Thus, in general, the hard-core repulsion between different hadron species is characterized not by a single excluded volume parameter b, but by a matrix b αβ of excluded volume parameters where each element corresponds to a distinct pair of hadron species. Note that this matrix need not be symmetric [34]. The matrix b αβ can be used to determine the threshold distances σ αβ . As discussed in the framework of the multi-component excluded volume model [34], the coefficients b αβ are related to σ αβ as\\nthus σ αβ = 3b αβ 2π\\nThe SPR approximation can be employed similarly to the box case in Sec. II. Using the SPR approximation is essential for sampling central collisions of heavy ions, where the system is so large that applying the full (unbiased) method is prohibitively time-consuming.\\nIt is instructive to summarize the list of approximations and possible limitations of the method: (i) Total particle numbers N j are sampled from an auxiliary excluded volume HRG model characterized by constant volume V eff and mean hadron yields N j , both calculated via the Cooper-Frye formula. The resulting excluded volume effect on the cumulants of N j -distribution may be slightly different from the true result if the distribution of particle number densities across the Cooper-Frye hypersurface is inhomogeneous. The accuracy of this approximation for central Au-Au collisions at various beam energies is verified in the next section by comparing Monte Carlo results with analytic approximations, yielding only minor differences.\\n(ii) As the N j -distribution is sampled from the excluded volume model rather than from the exact hard-sphere model distribution, for consistency, it is imperative that the excluded volume model provides an accurate approximation for the hardspheres equation of state. As shown in the Appendix A, this is satisfied with high precision for bn 0.10 − 0.15.\\nBoth approximations (i) and (ii) concerning the N jdistribution become less relevant when the canonical ensemble is applied, as the canonical ensemble effects then dominate the fluctuations of N j .\\n(iii) As the hard-core repulsion is an inherently nonrelativistic concept, issues may arise when relativistic effects are strong. In particular, this can be an issue when the two overlapping particles have large relative velocities. This may happen, for instance, when the particles are emitted from different regions of the fireball characterized by significantly different collective velocities. However, as the hard-core repulsion is a short-range phenomenon, the issue becomes irrelevant in this case since these particles would not overlap in the coordinate space. Particles that are close by in the coordinate space, on the other hand, would typically correspond to very similar collective velocities, thus their relative velocities are determined by the local temperature T (x), which is usually of order 160 MeV or lower in heavy-ion collisions at particlization. Therefore, the relativistic effects become less relevant for m/T (x) 1, as is the case for baryons (m B ≥ 938 MeV/c 2 ), but could be important if hard-core repulsion is incorporated for lighter particles such as pions.\\nThe Cooper-Frye sampling with short-range repulsion is implemented within the open source package Thermal-FIST [35] starting from version 1.4.\"\n },\n {\n \"title\": \"IV. APPLICATION OF THE METHOD TO PROTON CUMULANTS IN HEAVY-ION COLLISIONS\",\n \"content\": \"To illustrate the newly developed FIST sampler and the corresponding effect of hard-core repulsion, the behavior of (net-)proton cumulants in central collisions of heavy ions at various collision energies is studied. The effect of (anti)baryon excluded volume on the cumulants has previously been studied at LHC [36] and RHIC-BES [17] energies using different methods, thus, it is instructive to compare these results with the present method.\\nIt is assumed here that excluded volume repulsion is present for all baryon-baryon and, by symmetry, all antibaryon-antibaryon pairs, as motivated by recent analyses of lattice QCD data on baryon number susceptibilities [22,24]. This implies b αβ = b > 0 if either α, β ∈ B or α, β ∈B, and b αβ = 0 otherwise. To be consistent with the earlier studies, the value of the baryon excluded volume parameter is chosen to be b = 1 fm 3 , corresponding to a classical hard-core radius of around r c = 0.39 fm.\"\n },\n {\n \"title\": \"A. LHC\",\n \"content\": \"First, the LHC energies are studied. More specifically, the 0-5% central Pb-Pb collisions at √ s NN = 2.76 TeV are analyzed, using a longitudinally boostinvariant Cooper-Frye hypersurface based on the blastwave model. The parametrization of the hypersurface is identical to a previous study [16], where the blast-wave model parameters are based on fits to the p T spectra within a single freeze-out scenario [38]. These parameters are uniform across the entire hypersurface. The particlization temperature is T = 160 MeV, and the chemical potentials are vanishing. For this choice of parameters, one has bn B = bnB ≈ 0.029, where n B and nB are the number densities of baryons and antibaryons, respectively.\\nA total of 28.3 million events is sampled. The events incorporate the baryon hard-core repulsion as described above, as well as the exact global conservation of baryon number. The behavior of cumulants of net proton number distribution is analyzed. To minimize the statistical error, the fireball volume is reduced by factor 10, from dV /dy = 400 fm 3 to dV /dy = 40 fm 3 . This reduction of the fireball volume is achieved through the reduction of its transverse radius from R ⊥ = 9 fm to R ⊥ ≈ 2.84 fm. The analysis is therefore focused on the ratios of cumulants where the trivial dependence on the volume is canceled. For more details on this procedure to minimize the statistical error, see Ref. [16]. Figure 5 depicts the behavior of (a) the net-proton variance normalized over the Skellam distribution baseline, κ 2 [p −p]/ p +p and (b) the net-proton kurtosis ratio,\\nThe calculations are performed in the experimental acceptance used in the measurements performed by the ALICE Collaboration [39], corresponding to cuts in (anti)proton momentum, 0.6 < p < 1.5 GeV/c, and pseudorapidity, |η| < η max . The results in Fig. 5 are shown as a function of η max by solid black lines with grey bands depicting the statistical uncertainties. The results are compared with the earlier calculations of Ref. [16] obtained in the framework of the so-called subensemble sampler (dash-dotted black lines), as well as the ideal gas baseline that incorporates the effect of baryon conservation but not hard-core repulsion (dotted blue lines). Note that the calculations for κ 4 [p −p]/κ 2 [p −p] do not account for the effect of volume fluctuations which would influence the measurements [16].\\nThe FIST sampler results agree with the earlier results based on the subensemble sampler. They also show visible deviations from the ideal gas baseline. This indicates that the FIST sampler provides a reasonable de-scription of the excluded volume effect. In contrast to the subensemble sampler, which is mainly restricted to longitudinally boost invariant systems such as those encountered at the LHC, the FIST sampler can also be straightforwardly applied to lower collision energies where boost invariance does not hold. This is explored at the RHIC beam energy scan energies in the next section.\"\n },\n {\n \"title\": \"B. RHIC-BES\",\n \"content\": \"Net proton fluctuations have been measured by the STAR Collaboration in Au-Au collisions in collision energy range √ s NN = 7.7 − 200 GeV [40,41]. In Ref. [17] the fluctuations have been analyzed based on relativistic hydrodynamics simulations with MUSIC [42], incorporating effects of baryon conservation and excluded volume. Calculations in Ref. [17] were performed analytically, in two steps: (i) proton cumulants are calculated in the experimental acceptance using the Cooper-Frye formula in the grand-canonical limit, including the excluded volume effect; (ii) correction for baryon number conservation is performed using a method called SAM-2.0 [43].\\nHere, the calculations of proton cumulants are performed using the FIST sampler, by sampling 0-5% central Au-Au collisions at energies √ s NN = 7.7, 14.5, 19.6, 27, 39, 62.4 and 200 GeV. The sampling is performed with and without the effect of exact baryon conservation, and it uses the same Cooper-Frye hypersurfaces produced by the MUSIC code that is based on [42] and available at [33]. For each energy, at least several million events have been generated. These calculations are compared to the analytic approximations of Ref. [17] in Fig. 6, for (a) the second proton cumulant ratio κ 2 /κ 1 and (b) the third order net proton cumulant ratio κ 3 /κ 1 . The FIST sampler results are shown by the bands, while the lines show the analytic results.\\nIn the absence of baryon conservation (baryon repulsion only, green lines and bands in Fig. 6) deviations of the cumulant ratios from the baseline of unity are driven solely by baryon repulsion, which leads to a modest suppression. The FIST sampler results agree with the analytic approximations, validating the method at RHIC-BES energies.\\nIncorporating exact baryon conservation (baryon repulsion + baryon cons., black lines, and grey bands in Fig. 6) leads to a more potent suppression of the cumulant ratios. In this case, the analytic and Monte Carlo results agree at high collision energies ( √ s NN 40 GeV), but deviations between the two are visible at lower √ s NN .\\nThese deviations can be attributed to inaccuracies of the SAM-2.0 framework at lower collision energies. As discussed in the original publication [43], it would tend to overestimate the effect of (repulsive) interactions. Comparison with the FIST sampler allows one to quantify the accuracy of this approximation at RHIC-BES conditions, revealing a modest overestimation of excluded volume effect on proton cumulants which gets worse as [17] with and without exact baryon conservation, respectively. The red symbols correspond to the experimental data of the STAR Collaboration [40,41]. collision energy is decreased.\\nAs a new application of the FIST sampler, one can simultaneously incorporate the effects of baryon repulsion and exact conservation of all three QCD conserved charges: baryon number, electric charge, and strangeness. The electric charge conservation, in particular, can influence the fluctuations of the proton number since the proton carries the electric charge. At high col-lision energies, the dominant electric charge carriers are pions. Thus, the effect of electric charge conservation on protons is expected to be small [36]. As the collision energy decreases, the fraction of charge carried by protons increases, and so does the effect of charge conservation on proton cumulants. This is reflected in the FIST sampler results shown in Fig. 6 by the blue bands, indicating additional suppression of (net-)proton κ 2 /κ 1 and κ 3 /κ 1 due to electric charge conservation, which becomes visible at √ s NN 40 GeV. Interestingly, at all collision energies considered, the FIST sampler results with exact conservation of baryon number, electric charge, and strangeness are in good agreement with the analytical calculations of proton cumulants from Ref. [43] that incorporate baryon conservation but not electric charge and strangeness. Thus, it appears that the additional suppression due to electric charge conservation at all energies is of the same magnitude as the overestimation of the excluded volume effect in the analytical calculation of Ref. [43] with SAM-2.0. Although this coincidence appears to be purely accidental, it indicates that the results of Ref. [43] serve as an accurate baseline for proton cumulants incorporating non-critical effects like baryon repulsion and exact conservation of multiple conserved charges.\"\n },\n {\n \"title\": \"C. GSI-SIS\",\n \"content\": \"The final application of the new method concerns the Au-Au collisions at √ s NN = 2.4 GeV, as probed by the HADES experiment at GSI-SIS. The experimental data [44] indicate sizable multiproton correlations. These were recently analyzed in the framework of a fireball model [45]. The Cooper-Frye hypersurface was parameterized in the framework of the Siemens-Rasmussen model with a Hubble-like collective flow, which was found to provide a reasonable description of the p T spectra of pions and protons [46], while the temperature T ≈ 70 MeV and baryochemical potential µ B ≈ 875 MeV were extracted from hadron yields [47]. Here this parametrization of the hypersurface is used in FIST sampler to study the behavior of proton cumulants influenced by conservation laws and baryon repulsion. At √ s NN = 2.4 GeV, a significant fraction of protons is bound into light nuclei in the final state. Based on the preliminary HADES data [46], one can estimate that about 37.5% of the final-state protons are bound. The light nuclei are not incorporated into the FIST sampler employed in the present work. Thus, to model this effect, following Ref. [45], each proton emitted from the fireball is assumed to be bound into light nuclei with a probability of q nucl = 0.375 and not contribute to the measured cumulants of the proton number. The corresponding results for proton number cumulant ratios κ 2 /κ 1 , κ 3 /κ 2 , and κ 4 /κ 2 are shown in Fig. 7 as functions of the rapidity cut y cut in the experimental acceptance of the HADES experiment.\\nThe results show a similar pattern to RHIC-BES energies: both the baryon conservation and repulsion suppress the cumulant ratios, with baryon conservation having a stronger effect. The additional effect of electric charge conservation (blue lines) is even more notable, which suppresses the cumulant ratios even further. The combined effect of baryon and electric charge conservation is more significant than that of baryon repulsion.\\nNote that both the total baryon and electric charge of the participant matter fluctuates event by event in the experiment. This is in contrast to the FIST sampler where they are both fixed. The experimental data of the HADES collaboration [44], shown in Fig. 7 by black symbols, is corrected for volume (participant) number fluctuations, which justifies the canonical treatment of baryon number for the participant matter. However, the total electric charge of the participants still fluctuates; thus, the FIST sampler results in Fig. 7 should be considered an upper bound on the effect of exact conservation of the electric charge on the proton number cumulants. It is also clear that the model fails to describe the experimental data even qualitatively. For example, the scaled variance κ 2 /κ 1 increases with the rapidity cut y cut in the experiment, as opposed to the suppression predicted due to conservation laws and short-range repulsion. This indicates that the behavior of the proton cumulants in the experimental data is not driven by the conservation laws or short-range repulsion that are incorporated in FIST sampler. For a discussion of the various possibilities, including the QCD critical point, see Ref. [45] The present analysis indicates that the exact conservation of multiple conserved charges, as opposed to only that of the baryon number, is essential for quantitative analysis of proton number cumulants at moderate collision energies, √ s NN 7.7 GeV. The FIST sampler allows one to evaluate the corresponding baselines in this energy regime, which can then be used to analyze future experimental data coming from the fixed-target program at RHIC [48] or the CBM experiment at FAIR [49].\\nFor comparisons of the FIST sampler results in the SIS-HADES regime with analytic approximations, see Appendix C.\"\n },\n {\n \"title\": \"V. CONCLUSIONS AND OUTLOOK\",\n \"content\": \"This study introduced short-range repulsive correlations into the Cooper-Frye hadron sampling procedure. This has been achieved through a rejection sampling step that prohibits any two particles with repulsive interactions from overlapping in the coordinate space, effectively modeling the excluded volume phenomenon. The effect introduces negative correlations between particles visible in normalized cumulants of their distributions. The new method -called the FIST sampler -incorporates this effect and simultaneously allows for both the canonical and grand-canonical treatments of the QCD conserved charges.\\nFIST sampler was validated in a periodic box setup, where the method yields the behavior of the scaled variance of particle number, which is consistent with the equation of state of hard spheres. nally boost-invariant scenario at LHC energies, the new method produces the behavior of net proton cumulants up to the fourth order, which is consistent with an earlier study of [16] that used the so-called subensemble sampler. The advantage of FIST sampler becomes evident at lower collision energies, where boost invariance no longer holds and where the application of the subensemble sampler is challenging. The new method allowed one to incorporate the simultaneous effects of baryon repulsion and baryon, electric charge, and strangeness conservation at RHIC-BES and SIS-GSI energies. In particular, the effect of electric charge conservation on proton number cumulants becomes increasingly important as the collision energy is decreased, √ s NN 40 GeV, due to a smaller number of pions relative to protons at lower energies. The resulting cumulant ratios of the (net) proton distribution are consistent with estimates of Ref. [17] that were obtained by employing analytic approximations for baryon conservation and repulsion and neglecting electric charge conservation. The reason for this coincidence at lower energies appears to be that the analytic method of Ref. [17] slightly overestimates the effect of baryon repulsion in the canonical ensemble, which at lower collision energies is approximately compensated for by the electric charge conservation in FIST sampler. Due to this coincidence, the results of Ref. [17] can be taken as a baseline for (net) proton cumulants at RHIC-BES that incorporates non-critical effects such as excluded volume and multiple global conservation laws. The mechanisms above do not describe the experimental data of the HADES Collaboration (Fig. 7), indicating that proton number cumulants are driven by different mechanisms that still need to be clarified.\\nFIST sampler has several future potential applications. For instance, one can couple its output with a hadronic afterburner such as UrQMD [50,51] or SMASH [52], which should ideally include the shortrange repulsion for consistency [53], and study the effect of the hadronic phase on proton number cumulants, in particular, that of baryon annihilation [54,55]. Al-though the effects of hadronic afterburner have not been studied in the present work, the FIST sampler code [56] readily supports output tailored for use with UrQMD afterburner [57]. Since the method offers the full flexibility of an event generator, observables other than proton number cumulants, such as, for example, balance functions [58] or distributions of other hadronic species [59] can be studied as well. In particular, one may revisit the Bayesian constraints on transport coefficients of QCD matter and the switching temperature for the transition from hydrodynamics to transport [60,61] by employing the new particlization routine presented here in such an analysis. One can also straightforwardly incorporate the possible flavor dependence of baryon repulsion suggested by recent analyses of lattice QCD data on off-diagonal susceptibilities [24,25,62] by an appropriate choice of the corresponding threshold distances σ αβ in Eq. (12). It can also be interesting to see how the presence of baryonic hard-core repulsion might affect the light nuclei production, such as employed in phase-space coalescence prescriptions [63,64]. These applications will be the subject of future studies.\\nUsing the single particle rejection (SPR) approximation was essential to sample central collisions of heavy ions in practical applications of the new routine, as applying a direct unbiased method was prohibitively timeconsuming. The implication is that the sampler is not fully unbiased, even though the tests in a box setup indicate that the sampling of particle number cumulants remains accurate. It is thus advisable to consider other techniques, such as oversampling, to produce a fast, unbiased sampler.\\nThe sampling with short-range repulsion described in this work has been implemented into the Monte Carlo event generator within the open source package Thermal-FIST [35], starting from version 1.4. The code for the sampling of various central collisions of heavy ions discussed in the present work is available at [56].\"\n },\n {\n \"title\": \"ACKNOWLEDGMENTS\",\n \"content\": \"The author is grateful to Mark Gorenstein for reading the manuscript and for useful comments. The author thanks Volker Koch and Chun Shen for fruitful discussions and collaboration on related projects and Dylan Neff for stimulating discussions. The author also acknowledges the hospitality of the staff at the Frankfurt Institute for Advanced Studies where part of this work was done. This work was supported through the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract number DE-FG02-00ER41132. Computational resources were provided by the Frankfurt Center for Scientific Computing (Goethe-HLR).\\nAppendix A: The equation of state and particle number fluctuations in the system of hard spheres\\nThe equation of state that describes the system of classical particles that interact through a hard-core potential (4) can be written in terms of the compressibility factor Z ≡ P/(nT ):\\nwhere η = bn 4 = 4πr 3 c 3 N V is the so-called packing fraction. The explicit expression for the function g(η) is not known, but different approximations are available that model the equation of state accurately when the values of η are not too large. One well-known approximation is the van der Waals excluded volume model where one has P ev = nT 1−bn , thus\\nA more involved analytical approximation of the equation of state of hard spheres is given by the Carnahan-Starling model [65]:\\nThe accuracy of the excluded volume and Carnahan-Starling models can be verified by comparing the results to numerical Monte Carlo simulations of the hardsphere system. This comparison is depicted in Fig. 8, where the dependence of the compressibility factor Z on bn, calculated in the two models, is compared to the numerical simulations of [66]. The excluded volume model accurately describes the compressibility factor up to bn ≈ 0.10 − 0.15, while it overestimates the numerical data for Z at higher densities. On the other hand, the Carnahan-Starling model accurately describes the Monte Carlo data for all the densities considered, that is, up to at least bn = 0.6. Figure 9. Density dependence of cumulant ratios κ2/κ1 (black, top), κ3/κ1 (blue, middle), κ4/κ2 (red, bottom) of the grand-canonical particle number distribution calculated for the hard-sphere system within the excluded volume (dashed lines) and Carnahan-Starling (solid lines) approximations.\\nCumulants of the grand-canonical particle number distribution are given by\\nFor n = 1 one has κ 1 ≡ N , therefore, for n > 1 the cumulants read\\nUsing the thermodynamic identity iteratively, provided that the pressure P is known as a function of density n. This is the case for the excluded volume [Eq. (A2)] and Carnahan-Starling [Eq. (A3)] models. Figure 9 depicts the behavior of the grand-canonical cumulant ratios κ 2 /κ 1 , κ 3 /κ 1 , and κ 4 /κ 2 as a function of the scaled density bn. Given the accuracy of the Carnahan-Starling approximation for the compressibility Z (Fig. 8), it is expected that the corresponding calculations of the cumulant ratios accurately reflect the grandcanonical particle number distribution in the hard-sphere system in the considered density range. Deviations of the excluded volume model from the Carnahan-Starling model become more visible in higher-order cumulants. The excluded volume model provides accurate results up to bn ≈ 0.10.\"\n },\n {\n \"title\": \"Appendix B: Radial distribution function\",\n \"content\": \"The radial distribution function g(r) describes how the density of particles varies around a reference particle at r = 0 relative to the expectation based on the mean particle number density n = N/V . More specifically, g(r) is defined so that the local particle number density at a distance r from the reference particle is equal to n g(r). Deviations of g(r) from unity signal the presence of correlations between particles.\\nThe radial distribution function for the hard-sphere system has been studied for a long time [67,68]. As the particles cannot overlap, one has g(r) = 0 for r < σ. g(r) has a discontinuity at r = σ and approaches a value larger than unity in the limit r → σ + 0, indicating the presence of effective attraction as a many-body effect. For large r/σ values, g(r) is expected to approach unity, given that the hard-core repulsion is a short-range phenomenon. For moderate densities of bn, accurate expressions for g(r) of the hard-sphere system can be obtained based on the solution of the Percus-Yevick equation [67][68][69]. This can be used to validate the performance of the FIST sampler. Figure 10 depicts g(r) in a range 1 < r/σ < 3 resulting from FIST sampler for a system of particles in a periodic box at density bn = 0.12, which was obtained by binning the relative distances of the sampled particles. Blue circles correspond to the case where the SPR approximation was applied, whereas black squares show the unbiased calculation without approximations. In the former case N = 320 particles were sampled in each configuration, while in the latter case, only N = 80 particles were used due to slower computational performance.\\nThe results are compared with the analytical expectation for a hard-sphere system based on the Percus-Yevick approximation, which is expected to be highly accurate for density as low as bn = 0.12 [69]. The full calculation is in quantitative agreement with the analytical expectations, indicating that FIST sampler correctly reproduces the spatial correlations between particles that interact through the hard-core potential (4). The SPR approximation underestimates g(r) at 1 < r/σ 1.5, in- Figure 10. Radial distribution function g(r) resulting from FIST sampler for a system of particles in a periodic box at density bn = 0.12 with (blue circles) and without (black squares) the application of the SPR approximation. The black line corresponds to g(r) of a hard-sphere system calculated through the Percus-Yevick equation [69].\\ndicating that it does not capture the full strength of the many-body effective attraction effect. At the same time, the SPR approximation reproduces the qualitative structure of g(r), and, as discussed in Sec. II D, reproduces accurately the behavior of cumulants of particle number distribution for densities considered in this study. Here a comparison of FIST sampler calculations with the analytic method recently developed in Ref. [45] is performed. This allows one to verify the accuracy of the analytic approximations of Ref. [45]. In particular, Ref. [45] used a method called SAM-2.0 [43] to perform a correction for exact baryon conservation. As discussed in Ref. [43], the method might overestimate the effect of dynamical correlations among particles, in this case, the effect of short-range hard-core repulsion, at low collision energies such as √ s NN = 2.4 GeV at SIS-GSI. By comparing to FIST sampler, which is free of these issues, one can quantify the accuracy of SAM-2.0 in this collision energy regime.\\nThe corresponding results are shown in Fig. 11. One sees good agreement between Monte Carlo and analytic calculations for the cases when baryon repulsion is incorporated but not conservation (green lines) or when baryon conservation is implemented but not repulsion (red lines). This validates the analytic approach of Ref. [45] in these regimes, in particular, SAM-2.0 used in [45] to correct for baryon conservation is accurate in the limit of the ideal gas, as expected.\\nWhen both the baryon repulsion and conservation are incorporated, visible differences between the FIST sampler and the analytic method occur. The analytic Figure 11. The scaled variance of the proton number distribution in the experimental acceptance of the HADES Collaboration as a function of the rapidity cut ycut. The green (top) lines correspond to calculations incorporating the hard-core repulsion among baryons but not baryon conservation. The calculations shown by the red (middle) lines implement exact baryon conservation but not baryon repulsion. The black (bottom) lines show calculations with both the baryon repulsion and conservation included. The differently styled lines correspond to the results obtained using FIST sampler while the solid lines correspond to the analytic method from Ref. [45]. method, based on SAM-2.0, overestimates the suppression of κ 2 /κ 1 compared to the Monte Carlo sampling. This is an artifact of the SAM-2.0 approximation, which assumes that there are no correlations between particles inside and outside the acceptance apart from the baryon conservation. As discussed in [43], this leads to an overestimation of the correlations present in the grand-canonical limit, namely the effect of hard-core repulsion in this study. A similar observation applies to the higher-order cumulant ratios κ 3 /κ 2 and κ 4 /κ 2 that are not shown in Fig. 11.\\nAs follows from the FIST sampler calculations in Fig. 11, the presence of baryon repulsion in addition to baryon conservation leads to only a minor further suppression of proton cumulant ratios, i.e., the effect of hardcore repulsion in the cumulants appears to be essentially washed out by baryon conservation. This appears to be a consequence of the reduced collective flow effect at HADES energies compared to higher energies, which dilutes the space-momentum correlation, meaning that the excluded volume effect in the coordinate space is almost washed out in the momentum space where the measurements are performed. As shown in recent molecular dynamics simulations [70], even critical fluctuations may be washed out when analyzed in momentum space if exact global conservation laws are enforced, and no collective expansion is imposed.\"\n },\n {\n \"title\": \"Figure 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 4 Figure 3 .Figure 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 5 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 8 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"The dependence of the corrected scaled variancẽ ωα = ωα/(1 − α) on the acceptance fraction α, evaluated in a box with (bands) and without (dashed lines) periodic boundary conditions resulting from the sampling of particles with a hard-core repulsion for different values of total particle number. The dash-dotted horizontal line corresponds to the expected thermodynamic limit of the hard-sphere equation of state calculated using the Carnahan-Starling model.","content":"Figure 1 .\n1The dependence of the corrected scaled variancẽ ωα = ωα/(1 − α) on the acceptance fraction α, evaluated in a box with (bands) and without (dashed lines) periodic boundary conditions resulting from the sampling of particles with a hard-core repulsion for different values of total particle number. The dash-dotted horizontal line corresponds to the expected thermodynamic limit of the hard-sphere equation of state calculated using the Carnahan-Starling model."},{"id":"","caption":"Same asFig. 1, but for the uncorrected scaled variance ωα calculated within the grand-canonical ensemble.","content":"Figure 2 .\n2Same asFig. 1, but for the uncorrected scaled variance ωα calculated within the grand-canonical ensemble."},{"id":"","caption":"depicts the results for bn = 0.03, 0.1, 0.25, 0.5, all calculated for N = 320 within the SPR approximation. The dash-dotted horizontal lines correspond to the grand-canonical values expected in the Same as Fig. 1, but calculated either within the SPR approximation (bands) or without approximations (symbols), for the periodic box only. The bottom panel shows the ratio of the approximate and unbiased calculations. The dependence of the corrected scaled variancẽ ωα = ωα/(1 − α) on the acceptance fraction α calculated in a periodic box within the SPR approximation for different values of the scaled density bn. The dash-dotted horizontal lines correspond to the expected thermodynamic limit of the hard-sphere equation of state calculated using the Carnahan-Starling model. thermodynamic limit, as calculated within the Carnahan-Starling approximation (see Appendix A). The Monte","content":"Figure 4 Figure 3 .Figure 4 .\n434depicts the results for bn = 0.03, 0.1, 0.25, 0.5, all calculated for N = 320 within the SPR approximation. The dash-dotted horizontal lines correspond to the grand-canonical values expected in the Same as Fig. 1, but calculated either within the SPR approximation (bands) or without approximations (symbols), for the periodic box only. The bottom panel shows the ratio of the approximate and unbiased calculations. The dependence of the corrected scaled variancẽ ωα = ωα/(1 − α) on the acceptance fraction α calculated in a periodic box within the SPR approximation for different values of the scaled density bn. The dash-dotted horizontal lines correspond to the expected thermodynamic limit of the hard-sphere equation of state calculated using the Carnahan-Starling model. thermodynamic limit, as calculated within the Carnahan-Starling approximation (see Appendix A). The Monte"},{"id":"","caption":"Cumulant ratios κ2[p −p]/ p +p (left panel) and κ4[p −p]/κ2[p −p] (right panel) in 0-5% central √ sNN = 2.76 TeV Pb-Pb collisions as functions of the pseudorapidity cut ηmax. Calculations are performed within FIST sampler incorporating the effects of short-range baryon repulsion and global baryon conservation and shown by the black lines with bands. The dash-dotted black lines and dashed blue lines depict the calculations of Ref.[16] within the subensemble sampler with and without the effect of baryon excluded volume, respectively. The symbols correspond to the experimental measurements of the ALICE Collaboration[37].","content":"Figure 5 .\n5Cumulant ratios κ2[p −p]/ p +p (left panel) and κ4[p −p]/κ2[p −p] (right panel) in 0-5% central √ sNN = 2.76 TeV Pb-Pb collisions as functions of the pseudorapidity cut ηmax. Calculations are performed within FIST sampler incorporating the effects of short-range baryon repulsion and global baryon conservation and shown by the black lines with bands. The dash-dotted black lines and dashed blue lines depict the calculations of Ref.[16] within the subensemble sampler with and without the effect of baryon excluded volume, respectively. The symbols correspond to the experimental measurements of the ALICE Collaboration[37]."},{"id":"","caption":"Beam energy dependence of the proton cumulant ratio κ2/κ1 (left panel) and net proton cumulant ratio κ3/κ1 (right panel) in 0-5% central Au-Au collisions at RHIC-BES. The results depict FIST sampler calculations incorporating the baryon hard-core repulsion (green bands), plus exact baryon conservation (grey bands), plus exact conservation of electric charge and strangeness (blue bands). The dotted green and dash-dotted black lines correspond to analytic calculations of Ref.","content":"Figure 6 .\n6Beam energy dependence of the proton cumulant ratio κ2/κ1 (left panel) and net proton cumulant ratio κ3/κ1 (right panel) in 0-5% central Au-Au collisions at RHIC-BES. The results depict FIST sampler calculations incorporating the baryon hard-core repulsion (green bands), plus exact baryon conservation (grey bands), plus exact conservation of electric charge and strangeness (blue bands). The dotted green and dash-dotted black lines correspond to analytic calculations of Ref."},{"id":"","caption":"FIST sampler was then used to model the excluded volume effect in (anti)baryon-(anti)baryon interaction at the particlization stage of heavy-ion collisions in a broad collision energy range, √ s NN = 2.4 − 2760 GeV. In a longitudi-Cumulant ratios (a) κ2/κ1, (b) κ3/κ2, and (c) κ4/κ2 of the proton number distribution in 0-5% central √ sNN = 2.4 GeV Au-Au collisions in the experimental acceptance of the HADES Collaboration, as calculated within FIST sampler. The results are shown as a function of the rapidity cut ycut. The calculations incorporate the baryon hard-core repulsion (dotted green lines), plus exact baryon conservation (dash-dotted black lines), plus exact conservation of electric charge and strangeness (solid blue lines). The black symbols correspond to the experimental data of the HADES Collaboration [44].","content":"Figure 7 .\n7FIST sampler was then used to model the excluded volume effect in (anti)baryon-(anti)baryon interaction at the particlization stage of heavy-ion collisions in a broad collision energy range, √ s NN = 2.4 − 2760 GeV. In a longitudi-Cumulant ratios (a) κ2/κ1, (b) κ3/κ2, and (c) κ4/κ2 of the proton number distribution in 0-5% central √ sNN = 2.4 GeV Au-Au collisions in the experimental acceptance of the HADES Collaboration, as calculated within FIST sampler. The results are shown as a function of the rapidity cut ycut. The calculations incorporate the baryon hard-core repulsion (dotted green lines), plus exact baryon conservation (dash-dotted black lines), plus exact conservation of electric charge and strangeness (solid blue lines). The black symbols correspond to the experimental data of the HADES Collaboration [44]."},{"id":"","caption":"Density dependence of the compressibility factor Z = P/(nT ) in the system of hard spheres, as calculated in the excluded volume (dashed black line) and Carnahan-Starling (solid red line) approximations. The symbols show the numerical Monte Carlo results from Ref.[66].","content":"Figure 8 .\n8Density dependence of the compressibility factor Z = P/(nT ) in the system of hard spheres, as calculated in the excluded volume (dashed black line) and Carnahan-Starling (solid red line) approximations. The symbols show the numerical Monte Carlo results from Ref.[66]."},{"id":"","caption":"∂P ∂n T , one can evaluate the derivatives ∂ n−1 N ∂µ n−1 T","content":"∂P ∂n T , one can evaluate the derivatives ∂ n−1 N ∂µ n−1 T"},{"id":"","caption":"Appendix C: Comparison of FIST sampler with analytic calculations at √ sNN = 2.4 GeV","content":"Appendix C: Comparison of FIST sampler with analytic calculations at √ sNN = 2.4 GeV"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"The dependence of the corrected scaled variancẽ ωα = ωα/(1 − α) on the acceptance fraction α, evaluated in a box with (bands) and without (dashed lines) periodic boundary conditions resulting from the sampling of particles with a hard-core repulsion for different values of total particle number. The dash-dotted horizontal line corresponds to the expected thermodynamic limit of the hard-sphere equation of state calculated using the Carnahan-Starling model.\",\n \"content\": \"Figure 1 .\\n1The dependence of the corrected scaled variancẽ ωα = ωα/(1 − α) on the acceptance fraction α, evaluated in a box with (bands) and without (dashed lines) periodic boundary conditions resulting from the sampling of particles with a hard-core repulsion for different values of total particle number. The dash-dotted horizontal line corresponds to the expected thermodynamic limit of the hard-sphere equation of state calculated using the Carnahan-Starling model.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Same asFig. 1, but for the uncorrected scaled variance ωα calculated within the grand-canonical ensemble.\",\n \"content\": \"Figure 2 .\\n2Same asFig. 1, but for the uncorrected scaled variance ωα calculated within the grand-canonical ensemble.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"depicts the results for bn = 0.03, 0.1, 0.25, 0.5, all calculated for N = 320 within the SPR approximation. The dash-dotted horizontal lines correspond to the grand-canonical values expected in the Same as Fig. 1, but calculated either within the SPR approximation (bands) or without approximations (symbols), for the periodic box only. The bottom panel shows the ratio of the approximate and unbiased calculations. The dependence of the corrected scaled variancẽ ωα = ωα/(1 − α) on the acceptance fraction α calculated in a periodic box within the SPR approximation for different values of the scaled density bn. The dash-dotted horizontal lines correspond to the expected thermodynamic limit of the hard-sphere equation of state calculated using the Carnahan-Starling model. thermodynamic limit, as calculated within the Carnahan-Starling approximation (see Appendix A). The Monte\",\n \"content\": \"Figure 4 Figure 3 .Figure 4 .\\n434depicts the results for bn = 0.03, 0.1, 0.25, 0.5, all calculated for N = 320 within the SPR approximation. The dash-dotted horizontal lines correspond to the grand-canonical values expected in the Same as Fig. 1, but calculated either within the SPR approximation (bands) or without approximations (symbols), for the periodic box only. The bottom panel shows the ratio of the approximate and unbiased calculations. The dependence of the corrected scaled variancẽ ωα = ωα/(1 − α) on the acceptance fraction α calculated in a periodic box within the SPR approximation for different values of the scaled density bn. The dash-dotted horizontal lines correspond to the expected thermodynamic limit of the hard-sphere equation of state calculated using the Carnahan-Starling model. thermodynamic limit, as calculated within the Carnahan-Starling approximation (see Appendix A). The Monte\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Cumulant ratios κ2[p −p]/ p +p (left panel) and κ4[p −p]/κ2[p −p] (right panel) in 0-5% central √ sNN = 2.76 TeV Pb-Pb collisions as functions of the pseudorapidity cut ηmax. Calculations are performed within FIST sampler incorporating the effects of short-range baryon repulsion and global baryon conservation and shown by the black lines with bands. The dash-dotted black lines and dashed blue lines depict the calculations of Ref.[16] within the subensemble sampler with and without the effect of baryon excluded volume, respectively. The symbols correspond to the experimental measurements of the ALICE Collaboration[37].\",\n \"content\": \"Figure 5 .\\n5Cumulant ratios κ2[p −p]/ p +p (left panel) and κ4[p −p]/κ2[p −p] (right panel) in 0-5% central √ sNN = 2.76 TeV Pb-Pb collisions as functions of the pseudorapidity cut ηmax. Calculations are performed within FIST sampler incorporating the effects of short-range baryon repulsion and global baryon conservation and shown by the black lines with bands. The dash-dotted black lines and dashed blue lines depict the calculations of Ref.[16] within the subensemble sampler with and without the effect of baryon excluded volume, respectively. The symbols correspond to the experimental measurements of the ALICE Collaboration[37].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Beam energy dependence of the proton cumulant ratio κ2/κ1 (left panel) and net proton cumulant ratio κ3/κ1 (right panel) in 0-5% central Au-Au collisions at RHIC-BES. The results depict FIST sampler calculations incorporating the baryon hard-core repulsion (green bands), plus exact baryon conservation (grey bands), plus exact conservation of electric charge and strangeness (blue bands). The dotted green and dash-dotted black lines correspond to analytic calculations of Ref.\",\n \"content\": \"Figure 6 .\\n6Beam energy dependence of the proton cumulant ratio κ2/κ1 (left panel) and net proton cumulant ratio κ3/κ1 (right panel) in 0-5% central Au-Au collisions at RHIC-BES. The results depict FIST sampler calculations incorporating the baryon hard-core repulsion (green bands), plus exact baryon conservation (grey bands), plus exact conservation of electric charge and strangeness (blue bands). The dotted green and dash-dotted black lines correspond to analytic calculations of Ref.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIST sampler was then used to model the excluded volume effect in (anti)baryon-(anti)baryon interaction at the particlization stage of heavy-ion collisions in a broad collision energy range, √ s NN = 2.4 − 2760 GeV. In a longitudi-Cumulant ratios (a) κ2/κ1, (b) κ3/κ2, and (c) κ4/κ2 of the proton number distribution in 0-5% central √ sNN = 2.4 GeV Au-Au collisions in the experimental acceptance of the HADES Collaboration, as calculated within FIST sampler. The results are shown as a function of the rapidity cut ycut. The calculations incorporate the baryon hard-core repulsion (dotted green lines), plus exact baryon conservation (dash-dotted black lines), plus exact conservation of electric charge and strangeness (solid blue lines). The black symbols correspond to the experimental data of the HADES Collaboration [44].\",\n \"content\": \"Figure 7 .\\n7FIST sampler was then used to model the excluded volume effect in (anti)baryon-(anti)baryon interaction at the particlization stage of heavy-ion collisions in a broad collision energy range, √ s NN = 2.4 − 2760 GeV. In a longitudi-Cumulant ratios (a) κ2/κ1, (b) κ3/κ2, and (c) κ4/κ2 of the proton number distribution in 0-5% central √ sNN = 2.4 GeV Au-Au collisions in the experimental acceptance of the HADES Collaboration, as calculated within FIST sampler. The results are shown as a function of the rapidity cut ycut. The calculations incorporate the baryon hard-core repulsion (dotted green lines), plus exact baryon conservation (dash-dotted black lines), plus exact conservation of electric charge and strangeness (solid blue lines). The black symbols correspond to the experimental data of the HADES Collaboration [44].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Density dependence of the compressibility factor Z = P/(nT ) in the system of hard spheres, as calculated in the excluded volume (dashed black line) and Carnahan-Starling (solid red line) approximations. The symbols show the numerical Monte Carlo results from Ref.[66].\",\n \"content\": \"Figure 8 .\\n8Density dependence of the compressibility factor Z = P/(nT ) in the system of hard spheres, as calculated in the excluded volume (dashed black line) and Carnahan-Starling (solid red line) approximations. The symbols show the numerical Monte Carlo results from Ref.[66].\"\n },\n {\n \"id\": \"\",\n \"caption\": \"∂P ∂n T , one can evaluate the derivatives ∂ n−1 N ∂µ n−1 T\",\n \"content\": \"∂P ∂n T , one can evaluate the derivatives ∂ n−1 N ∂µ n−1 T\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Appendix C: Comparison of FIST sampler with analytic calculations at √ sNN = 2.4 GeV\",\n \"content\": \"Appendix C: Comparison of FIST sampler with analytic calculations at √ sNN = 2.4 GeV\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". A Bzdak, S Esumi, V Koch, J Liao, M Stephanov, N Xu, 10.1016/j.physrep.2020.01.005arXiv:1906.00936Phys. Rept. 853nucl-thA. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov, and N. Xu, Phys. Rept. 853, 1 (2020), arXiv:1906.00936 [nucl-th].","doi":""},{"text":". C Shen, L Yan, 10.1007/s41365-020-00829-zarXiv:2010.12377Nucl. Sci. Tech. 31nucl-thC. Shen and L. Yan, Nucl. Sci. 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C 105, 044903 (2022), arXiv:2201.08486 [hep-ph].\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":576,"cells":{"text":{"kind":"string","value":"\nFLSH -Friendly Library for the Simulation of Humans\n\n\nPablo Ramón \nUniversidad Rey Juan Carlos\nMadridSpain\n\nCristian Romero \nUniversidad Rey Juan Carlos\nMadridSpain\n\nJavier Tapia \nUniversidad Rey Juan Carlos\nMadridSpain\n\nMiguel A Otaduy \nUniversidad Rey Juan Carlos\nMadridSpain\n\nFLSH -Friendly Library for the Simulation of Humans\nFBFA14F0F1DE2033F55EFEFCEAADADB53d avatar model, physics simulation\nComputer models of humans are ubiquitous throughout computer animation and computer vision.However, these models rarely represent the dynamics of human motion, as this requires adding a complex layer that solves body motion in response to external interactions and according to the laws of physics.FLSH is a library that facilitates this task for researchers and developers who are not interested in the nuisances of physics simulation, but want to easily integrate dynamic humans in their applications.FLSH provides easy access to three flavors of body physics, with different features and computational complexity: skeletal dynamics, full soft-tissue dynamics, and reduced-order modeling of soft-tissue dynamics.In all three cases, the simulation models are built on top of the pseudo-standard SMPL parametric body model.\n\nIntroduction\n\nParametric models of humans [2] have greatly simplified the modeling, animation, reconstruction and tracking of humans across computer vision and computer graphics applications.However, to date these parametric models are intrinsically geometric, and their shape, pose and motion must be either designed or inferred through optimization methods.Parametric human models lack the ability to move according to the laws of physics.We believe that endowing parametric human models with physicsbased dynamics can open up novel application possibilities, across animation, virtual reality, or computer vision.\n\nThe images in Fig. 1 highlight the relevance of body physics in two examples: (left) color-coding of the amount of soft-tissue dynamics in 5 different motions; (right) extreme differences in avatar interaction without or with contact mechanics.Deformation, dynamics and contact are obtained in a natural way by solving the laws of physics on top of parametric human representations.To simplify the adoption of this technology across human modeling, animation, reconstruction and tracking, we provide FLSH, a simple software library with dynamic simulation capabilities for the popular SMPL parametric human model [2].In FLSH, the core library runs in C++, but it offers both C++ and Python APIs to allow for flexible integration.\n\nIn FLSH, the simulation of human bodies is presented in three different flavors:\n\n1. Articulated skeleton: This is an articulated skeleton dynamics model with geometric skinning.\n\nBodies react to contact, and they can be controlled according to target trajectories or forces.This is the simplest and fastest representation, when tissue deformation due to contact or dynamics is not necessary.\n\n2. FEM soft skeleton: This is a full finite-element model (FEM) of soft-tissue dynamics on top of the articulated skeleton.Bodies contain both the bone structure and deformable soft-tissue, simulated with state-of-the-art finite-element non-linear elasticity, for high-accuracy applications.3. ROM soft skeleton: This is a reduced-order model (ROM) of the FEM soft skeleton.Bodies solve a real-time approximation of soft-tissue deformation, including contact, following a research solution [6].This representation offers an optimal trade-off between accuracy and performance.\n\nThe rest of the document is structured as follows.First, we describe the different simulation models supported in the library, covering a range of features and allowing maximal performance for each feature set.Second, we outline the high-level API of the simulation library.Third, we show and discuss some simulation results.\n\n\nSimulation Methods\n\nAll three simulation models in FLSH are solved using a common methodology.This simplifies the transition between simulation models based on user needs.The simulation methodology is also easy to extend.In the rest of this section, we present the overall algorithm, the main characteristics of all three simulation models, and some performance comparisons.Fig. 2 shows a comparison of avatar deformation under the three simulation models.\n\n\nOverall algorithm\n\nAll simulation models are defined on top of the parametric body model SMPL [2].SMPL is characterized by a pose and shape .Given a template body ̅ 0 in reference pose, it is mapped to a person-specific body ̅ = ℎ (̅ 0 , ) based on the shape parameters.Then, a skinning transformation = (̅ , ) based on defines the posed body shape .SMPL's skinning transformation is linear blend skinning augmented with pose and shape blend-shapes.Even though the original SMPL is defined only for the body surface, FLSH is built on top of the extension to a volumetric body by [5].\n\nLet us consider a set of generic dynamic degrees of freedom (DoFs) , which depend on the specific simulation model of choice.The DoFs alter the standard SMPL definition of the deformed body , as discussed below for each simulation model.\n\nFLSH simulates dynamics using backward Euler numerical integration.This can be formulated as the following optimization problem [1]:\n𝑞 = arg min 1 2 ℎ 2 (𝑞 − 𝑞 * ) 𝑇 𝑀 (𝑞 − 𝑞 * ) + 𝑉(𝑞)\nIn this optimization, ℎ is the time step, * the tentative state resulting from explicit integration, is the mass matrix, and collects all potential energy terms.\n\nIn essence, the API of FLSH provides the evaluation of the objective, gradient and Hessian of the optimization.This allows extensibility, as users can program their own DoFs and energy terms, which are added to those of FLSH.Furthermore, FLSH offers separate access to the inertial and potentialenergy terms in the optimization.In this way, simply by dropping the inertial term, FLSH can also simulate static deformations.The library is complemented with a solver example.\n\nFLSH offers some default contact handling capabilities, as well as tools for generalizing contact handling.First, it allows the internal definition of some simple parametric colliders (spheres, cylinders).Second, it provides access to the Jacobian\n\n\nArticulated skeleton\n\nFor each body bone in the SMPL skeleton, we define a rigid transformation .Then, the DoFs of the skeletal dynamics model correspond to the collection of rigid bone transformations, = { }.We can obtain the SMPL pose by extracting the skeleton root transformation and the joint angles from , and the deformed body is directly obtained through the skinning transformation defined above.In the articulated skeleton model, the mass matrix is derived from rigid-body inertia terms associated with each bone.The potential energy terms included in the model are: joint constraints to maintain the skeletal structure, and tracking constraints between the skeleton and target configurations (e.g.coming from mocap) implemented as joint rotation springs.\n\n\nFEM soft skeleton\n\nIn the FEM soft skeleton model, we define a skin displacement field (̅ ).We apply the skin deformation in unposed state, hence transforming the reference body shape as ̅ + .By adding SMPL's skinning transformation, the deformed body can be described as = (̅ , ).We use an FEM discretization with nodal displacements { }.Grouping all nodal displacements together with the rigid bone transformations yields the degrees of freedom = ({ }, { }).In FLSH, all body shapes use the same discretization, independent of the shape coefficients .This allows easy transition between body shapes if needed by the user's application.\n\nWe model skin elasticity using the approach in [5], which implies:\n\n1.The definition of deformation gradient = + ̅ ̅\n\n, which accounts for contact and inertial deformations, but respects SMPL's data-driven detail under pose changes.\n\n2. A strain energy density Ψ() corresponding to an orthotropic Saint Venant-Kirchhoff hyperelastic model with Fung-type saturation.The soft-tissue parameters are estimated from the DYNA dataset [3] following the multi-person optimization approach of [4].\n\nIn the FEM soft skeleton model, the energies of the articulated skeleton are extended in the following way.The mass matrix is augmented with terms corresponding to the skin displacements, as well as cross-terms with the bone transformations.The new terms are derived from the kinetic energy of worldspace skin velocities .The potential energy now also includes the integral of the strain energy density Ψ over the body's volume.\n\n\nROM soft skeleton\n\nThe reduced model retains much of the structure of the FEM soft skeleton, but it represents the skin displacement field in a linear reduced basis, (̅ ) = (̅ ) , with the vector of reduced DoFs and the matrix of basis coefficients.We use as reduced model the bounded generalized biharmonic coordinates of [7], following the model architecture of [6].This allows seamless integration in the basis of handle points in the skin together with the skeletal bones.The DoFs of the ROM soft skeleton model are = ({ }, ).The deformed body is defined with the same expression as in the FEM soft skeleton, with the only difference that skin displacements are represented in the reduced basis.\n\nUsing a reduced basis instead of a full FEM discretization dramatically reduces the solver cost, thanks to the reduced size of the problem gradient and Hessian.We also add cubature, following again the data-oblivious approach of [6], to approximate the skin inertia and elastic energy.\n\n\nPerformance\n\nThe performance of the various simulation models is highly dependent on the number of degrees of freedom.The SMPL model, which serves as basis for all models, consists of 24 bones, 13,776 triangles and 6,890 vertices.Pose is parameterized with 144 coefficients and shape with 10 coefficients.Our tetrahedral volume discretization of the body for the FEM soft skeleton consists of around 43500 tetrahedra and 10400 nodes.Furthermore, the ROM skeleton model contains 87 point handles, and we use 500 cubature samples.\n\nIn our performance tests, we have measured both the runtime cost and the generation of the avatars.Note that the soft models require more costly generation process which includes tetrahedral meshing and, in the case of the ROM soft skeleton, cubature sampling.The current version of FLSH implements avatar generation every time a new avatar is created, but future versions will support caching of avatar model data.Table 1 shows the performance measurements on a PC with a 3.4 GHz Intel Core i7-4770 CPU with 32GB of memory.\n\n\nLibrary API\n\nThe FLSH library contains two main objects that are accessible through its API: the SoftAvatar class, which stores the different data structures to represent the avatar body, and the Simulable class, which provides access to simulation state and optimization terms.\n\n\nInitialization and settings\n\nThe creation of an avatar must follow these steps:\n\n1.Call the constructor of the SoftAvatar class.The constructor receives as arguments an SMPL avatar object and a vector of shape coefficients.The creation of the simulation object must follow these steps:\n\n1.Call the constructor of a Simulable class.The constructor receives the following arguments: a SoftAvatar object, the SMPL object, and possibly a sequence of mocap frames to be tracked.\n\n\nCall Simulable::initialize().\n\nIf the user wants to include multiple avatars in the scene, then this is as simple as creating multiple SoftAvatar objects and Simulable interfaces.Each Simulable will read/write data referred to its state size, and this state can be merged into larger vectors and/or matrices corresponding to the full scene.\n\n\nSimulation Results\n\nNext, we discuss some simulation examples that highlight the features of the avatar models in FLSH, as well as differences between the various simulation models.\n\nFig. 2 compares body deformations with all three simulation models while tracking a mocap sequence.The images also highlight the differences in soft-tissue deformation.As expected, the articulated skeleton's deformation is due simply to SMPL's skinning transformation, but there is no additional softtissue deformation.In the soft skeleton models, on the other hand, there is soft-tissue deformation induced by the body's own inertia.The ROM soft skeleton provides a good approximation to the FEM soft skeleton, and therefore offers a good tradeoff between speed and accuracy.\n\nAs mentioned earlier, in the soft skeleton models in FLSH, soft-tissue properties are adapted based on body shape.Fig. 3 shows an example of this shape-based soft-tissue parameterization, for 5 different body shapes.As expected, the differences in soft-tissue parameters produce noticeable differences in the soft-tissue deformation of the various body shapes.\n\nFinally, we highlight the benefit of physics-based simulation to resolve contact-based interactions.Fig. 4 shows an example of articulated skeleton where the lack of physics-based simulation leads to disturbing avatar interpenetration.As shown in the images, the FLSH avatars are endowed with a hierarchy of analytical capsules for fast collision detection.The ROM soft skeleton approximates well contact for large objects like this sphere, but the FEM soft skeleton becomes necessary to resolve contact with small objects and/or sharp features.\n\n\nDiscussion\n\nFLSH is conceived as a living project to offer physics-based avatar simulation to the research community.Its first version includes the three simulation models described in this paper, and we hope that adoption is facilitated by the versatile cross-platform API and the examples provided.\n\nWe understand that in many computational problems involving humans, the shape, pose or motion of the human are variables that are optimized.To provide stronger support to such applications, we would like to augment FLSH with differentiability capabilities in the future.\n\n.\n\nIn this way, users can define externally arbitrary energy terms (), and obtain gradients\n\n\nFig. 2 .\n2\nFig. 2. Comparison of deformations produced by the simulation modes in FLSH.From left to right: articulated skeleton, FEM soft skeleton, and ROM soft skeleton.The colormaps indicate the amount of soft tissue deformation.As shown in the images, the ROM soft skeleton provides a good approximation to the FEM soft skeleton, and therefore offers a good tradeoff between speed and accuracy.\n\n\n2 . 3 .\n23\nEnable settings that define the type of simulation model.There are 4 different settings: enableSoftTissue() If disabled, the simulation model is the default articulated skeleton.enableReducedModel() It allows selecting FEM or ROM soft skeleton.Call SoftAvatar::initVolumetricData().\n\n\nFig. 5\n5\ncompares the accuracy of contact resolution with the different simulation model in FLSH.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2)In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3)\n\n\nFig. 3 .\n3\nFig. 3.In the soft skeleton models in FLSH, soft-tissue properties are adapted based on body shape.This image shows 5 different body shapes (first row), which lead to different distribution of Young modulus (second row) as well as skin thickness (third row).\n\n\nFig. 4 .\n4\nFig. 4. Without physics-based simulation, avatars may interpenetrate each other in disturbing ways, as shown on the left.With the physics-based simulation in FLSH, on the other hand, interactions are much more accurate, as shown on the right.This example uses the articulated skeleton model.For fast collision detection, each avatar is represented with proxy geometry formed by a hierarchy of analytical capsules (shown in the top images).\n\n\nFig. 5 .\n5\nFig. 5. Comparison of contact resolution with the different simulation model in FLSH.To highlight the differences, we display semi-transparent an avatar with no physics-based simulation, and we indicate the position of the chin in the avatars with and without simulation using colored dots.From left to right: avatar with no simulation, articulated skeleton model, FEM soft skeleton model, and ROM soft skeleton model.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2) In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3) The ROM soft skeleton approximates well contact for large objects like this sphere, but the FEM soft skeleton becomes necessary to resolve contact with small objects and/or sharp features.\n\n\n\n\n\n\n\nTable 1 .\n1\nBenchmarking.\nAvatar generationSimulationArticulated skeleton<1 s45 fpsFEM soft skeleton15 s0.2 fpsROM soft skeleton25 s20 fps\n\nThis can be used e.g. in line-search solvers. popState: resets the previous state from the stack. getStateJacobian: it gets the Jacobian of the skeleton pose, for external skeleton-based energies. getNodesStateJacobian: it gets the Jacobian of mesh nodes, for external mesh-based energies. Optimization evaluation: getDynamicEnergyScalar: get the inertial part of the optimization function. getPotentialEnergyScalar: get the potential-energy part of the optimization function. getDynamicGradientVector: get the inertial part of the optimization gradient. getPotentialGradientVector: get the potential-energy part of the gradient optimization gradient. fixVectorKinematic: it receives a full gradient and applies. State management: getState: it gets the current state of the DoFs. updateState: it sets a new state computed by the external solver. pushState: adds the current state to a stack. Dirichlet boundary conditions. getDynamicHessianTripletVector: get the inertial part of the optimization Hessian. getPotentialHessianTripletVector: get the potential-energy part of the optimization Hessian. fixMatrixKinematic: it receives a full Hessian and applies Dirichlet boundary conditions. References\n\nOptimization integrator for large time steps. T Gast, C Schroeder, A Stomakhin, C Jiang, J M Teran, IEEE Trans. on Visualization and Computer Graphics. 21102015\n\nSmpl: A skinned multi-person linear model. M Loper, N Mahmood, J Romero, Pons-Moll , M J Black, ACM Trans. on Graphics. 34616October 2015\n\nDYNA: A model of dynamic human shape in motion. G Pons-Moll, J Romero, N Mahmood, M J Black, in ACM Trans. on Graphics. 3442015\n\nSFLSH: Shape-dependent soft-flesh avatars. P Ramón, C Romero, J Tapia, M A Otaduy, Proc. of the ACM SIGGRAPH Asia Conference. of the ACM SIGGRAPH Asia Conference2023to appear in the\n\nModeling and estimation of nonlinear skin mechanics for animated avatars. C Romero, M A Otaduy, D Casas, J Perez, Computer Graphics Forum. 3922020\n\nParametric skeletons with reduced soft-tissue deformations. J Tapia, C Romero, J Pérez, M A Otaduy, Computer Graphics Forum. 2021\n\nLinear subspace design for real-time shape deformation. Y Wang, A Jacobson, J Barbic, L Kavan, ACM Trans. on Graphics. 3442015\n"},"corpus_id":{"kind":"number","value":263684742,"string":"263,684,742"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2310.18206\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/corr/abs-2310-18206\",\n \"doi\": \"10.15221/23.20\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2310.18206v1.pdf\"\n ],\n \"pdf_sha\": \"b1df2310ea93d691633c683df4974d18a0a166a2\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"FLSH -Friendly Library for the Simulation of Humans"},"authors":{"kind":"list like","value":["Pablo Ramón \nUniversidad Rey Juan Carlos\nMadridSpain","Cristian Romero \nUniversidad Rey Juan Carlos\nMadridSpain","Javier Tapia \nUniversidad Rey Juan Carlos\nMadridSpain","Miguel A Otaduy \nUniversidad Rey Juan Carlos\nMadridSpain"],"string":"[\n \"Pablo Ramón \\nUniversidad Rey Juan Carlos\\nMadridSpain\",\n \"Cristian Romero \\nUniversidad Rey Juan Carlos\\nMadridSpain\",\n \"Javier Tapia \\nUniversidad Rey Juan Carlos\\nMadridSpain\",\n \"Miguel A Otaduy \\nUniversidad Rey Juan Carlos\\nMadridSpain\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Computer models of humans are ubiquitous throughout computer animation and computer vision.However, these models rarely represent the dynamics of human motion, as this requires adding a complex layer that solves body motion in response to external interactions and according to the laws of physics.FLSH is a library that facilitates this task for researchers and developers who are not interested in the nuisances of physics simulation, but want to easily integrate dynamic humans in their applications.FLSH provides easy access to three flavors of body physics, with different features and computational complexity: skeletal dynamics, full soft-tissue dynamics, and reduced-order modeling of soft-tissue dynamics.In all three cases, the simulation models are built on top of the pseudo-standard SMPL parametric body model."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Parametric models of humans [2] have greatly simplified the modeling, animation, reconstruction and tracking of humans across computer vision and computer graphics applications.However, to date these parametric models are intrinsically geometric, and their shape, pose and motion must be either designed or inferred through optimization methods.Parametric human models lack the ability to move according to the laws of physics.We believe that endowing parametric human models with physicsbased dynamics can open up novel application possibilities, across animation, virtual reality, or computer vision.\nThe images in Fig. 1 highlight the relevance of body physics in two examples: (left) color-coding of the amount of soft-tissue dynamics in 5 different motions; (right) extreme differences in avatar interaction without or with contact mechanics.Deformation, dynamics and contact are obtained in a natural way by solving the laws of physics on top of parametric human representations.To simplify the adoption of this technology across human modeling, animation, reconstruction and tracking, we provide FLSH, a simple software library with dynamic simulation capabilities for the popular SMPL parametric human model [2].In FLSH, the core library runs in C++, but it offers both C++ and Python APIs to allow for flexible integration.\nIn FLSH, the simulation of human bodies is presented in three different flavors:\n1. Articulated skeleton: This is an articulated skeleton dynamics model with geometric skinning.\nBodies react to contact, and they can be controlled according to target trajectories or forces.This is the simplest and fastest representation, when tissue deformation due to contact or dynamics is not necessary.\n2. FEM soft skeleton: This is a full finite-element model (FEM) of soft-tissue dynamics on top of the articulated skeleton.Bodies contain both the bone structure and deformable soft-tissue, simulated with state-of-the-art finite-element non-linear elasticity, for high-accuracy applications.3. ROM soft skeleton: This is a reduced-order model (ROM) of the FEM soft skeleton.Bodies solve a real-time approximation of soft-tissue deformation, including contact, following a research solution [6].This representation offers an optimal trade-off between accuracy and performance.\nThe rest of the document is structured as follows.First, we describe the different simulation models supported in the library, covering a range of features and allowing maximal performance for each feature set.Second, we outline the high-level API of the simulation library.Third, we show and discuss some simulation results."},{"title":"Simulation Methods","content":"All three simulation models in FLSH are solved using a common methodology.This simplifies the transition between simulation models based on user needs.The simulation methodology is also easy to extend.In the rest of this section, we present the overall algorithm, the main characteristics of all three simulation models, and some performance comparisons.Fig. 2 shows a comparison of avatar deformation under the three simulation models."},{"title":"Overall algorithm","content":"All simulation models are defined on top of the parametric body model SMPL [2].SMPL is characterized by a pose and shape .Given a template body ̅ 0 in reference pose, it is mapped to a person-specific body ̅ = ℎ (̅ 0 , ) based on the shape parameters.Then, a skinning transformation = (̅ , ) based on defines the posed body shape .SMPL's skinning transformation is linear blend skinning augmented with pose and shape blend-shapes.Even though the original SMPL is defined only for the body surface, FLSH is built on top of the extension to a volumetric body by [5].\nLet us consider a set of generic dynamic degrees of freedom (DoFs) , which depend on the specific simulation model of choice.The DoFs alter the standard SMPL definition of the deformed body , as discussed below for each simulation model.\nFLSH simulates dynamics using backward Euler numerical integration.This can be formulated as the following optimization problem [1]:\nIn this optimization, ℎ is the time step, * the tentative state resulting from explicit integration, is the mass matrix, and collects all potential energy terms.\nIn essence, the API of FLSH provides the evaluation of the objective, gradient and Hessian of the optimization.This allows extensibility, as users can program their own DoFs and energy terms, which are added to those of FLSH.Furthermore, FLSH offers separate access to the inertial and potentialenergy terms in the optimization.In this way, simply by dropping the inertial term, FLSH can also simulate static deformations.The library is complemented with a solver example.\nFLSH offers some default contact handling capabilities, as well as tools for generalizing contact handling.First, it allows the internal definition of some simple parametric colliders (spheres, cylinders).Second, it provides access to the Jacobian"},{"title":"Articulated skeleton","content":"For each body bone in the SMPL skeleton, we define a rigid transformation .Then, the DoFs of the skeletal dynamics model correspond to the collection of rigid bone transformations, = { }.We can obtain the SMPL pose by extracting the skeleton root transformation and the joint angles from , and the deformed body is directly obtained through the skinning transformation defined above.In the articulated skeleton model, the mass matrix is derived from rigid-body inertia terms associated with each bone.The potential energy terms included in the model are: joint constraints to maintain the skeletal structure, and tracking constraints between the skeleton and target configurations (e.g.coming from mocap) implemented as joint rotation springs."},{"title":"FEM soft skeleton","content":"In the FEM soft skeleton model, we define a skin displacement field (̅ ).We apply the skin deformation in unposed state, hence transforming the reference body shape as ̅ + .By adding SMPL's skinning transformation, the deformed body can be described as = (̅ , ).We use an FEM discretization with nodal displacements { }.Grouping all nodal displacements together with the rigid bone transformations yields the degrees of freedom = ({ }, { }).In FLSH, all body shapes use the same discretization, independent of the shape coefficients .This allows easy transition between body shapes if needed by the user's application.\nWe model skin elasticity using the approach in [5], which implies:\n1.The definition of deformation gradient = + ̅ ̅\n, which accounts for contact and inertial deformations, but respects SMPL's data-driven detail under pose changes.\n2. A strain energy density Ψ() corresponding to an orthotropic Saint Venant-Kirchhoff hyperelastic model with Fung-type saturation.The soft-tissue parameters are estimated from the DYNA dataset [3] following the multi-person optimization approach of [4].\nIn the FEM soft skeleton model, the energies of the articulated skeleton are extended in the following way.The mass matrix is augmented with terms corresponding to the skin displacements, as well as cross-terms with the bone transformations.The new terms are derived from the kinetic energy of worldspace skin velocities .The potential energy now also includes the integral of the strain energy density Ψ over the body's volume."},{"title":"ROM soft skeleton","content":"The reduced model retains much of the structure of the FEM soft skeleton, but it represents the skin displacement field in a linear reduced basis, (̅ ) = (̅ ) , with the vector of reduced DoFs and the matrix of basis coefficients.We use as reduced model the bounded generalized biharmonic coordinates of [7], following the model architecture of [6].This allows seamless integration in the basis of handle points in the skin together with the skeletal bones.The DoFs of the ROM soft skeleton model are = ({ }, ).The deformed body is defined with the same expression as in the FEM soft skeleton, with the only difference that skin displacements are represented in the reduced basis.\nUsing a reduced basis instead of a full FEM discretization dramatically reduces the solver cost, thanks to the reduced size of the problem gradient and Hessian.We also add cubature, following again the data-oblivious approach of [6], to approximate the skin inertia and elastic energy."},{"title":"Performance","content":"The performance of the various simulation models is highly dependent on the number of degrees of freedom.The SMPL model, which serves as basis for all models, consists of 24 bones, 13,776 triangles and 6,890 vertices.Pose is parameterized with 144 coefficients and shape with 10 coefficients.Our tetrahedral volume discretization of the body for the FEM soft skeleton consists of around 43500 tetrahedra and 10400 nodes.Furthermore, the ROM skeleton model contains 87 point handles, and we use 500 cubature samples.\nIn our performance tests, we have measured both the runtime cost and the generation of the avatars.Note that the soft models require more costly generation process which includes tetrahedral meshing and, in the case of the ROM soft skeleton, cubature sampling.The current version of FLSH implements avatar generation every time a new avatar is created, but future versions will support caching of avatar model data.Table 1 shows the performance measurements on a PC with a 3.4 GHz Intel Core i7-4770 CPU with 32GB of memory."},{"title":"Library API","content":"The FLSH library contains two main objects that are accessible through its API: the SoftAvatar class, which stores the different data structures to represent the avatar body, and the Simulable class, which provides access to simulation state and optimization terms."},{"title":"Initialization and settings","content":"The creation of an avatar must follow these steps:\n1.Call the constructor of the SoftAvatar class.The constructor receives as arguments an SMPL avatar object and a vector of shape coefficients.The creation of the simulation object must follow these steps:\n1.Call the constructor of a Simulable class.The constructor receives the following arguments: a SoftAvatar object, the SMPL object, and possibly a sequence of mocap frames to be tracked."},{"title":"Call Simulable::initialize().","content":"If the user wants to include multiple avatars in the scene, then this is as simple as creating multiple SoftAvatar objects and Simulable interfaces.Each Simulable will read/write data referred to its state size, and this state can be merged into larger vectors and/or matrices corresponding to the full scene."},{"title":"Simulation Results","content":"Next, we discuss some simulation examples that highlight the features of the avatar models in FLSH, as well as differences between the various simulation models.\nFig. 2 compares body deformations with all three simulation models while tracking a mocap sequence.The images also highlight the differences in soft-tissue deformation.As expected, the articulated skeleton's deformation is due simply to SMPL's skinning transformation, but there is no additional softtissue deformation.In the soft skeleton models, on the other hand, there is soft-tissue deformation induced by the body's own inertia.The ROM soft skeleton provides a good approximation to the FEM soft skeleton, and therefore offers a good tradeoff between speed and accuracy.\nAs mentioned earlier, in the soft skeleton models in FLSH, soft-tissue properties are adapted based on body shape.Fig. 3 shows an example of this shape-based soft-tissue parameterization, for 5 different body shapes.As expected, the differences in soft-tissue parameters produce noticeable differences in the soft-tissue deformation of the various body shapes.\nFinally, we highlight the benefit of physics-based simulation to resolve contact-based interactions.Fig. 4 shows an example of articulated skeleton where the lack of physics-based simulation leads to disturbing avatar interpenetration.As shown in the images, the FLSH avatars are endowed with a hierarchy of analytical capsules for fast collision detection.The ROM soft skeleton approximates well contact for large objects like this sphere, but the FEM soft skeleton becomes necessary to resolve contact with small objects and/or sharp features."},{"title":"Discussion","content":"FLSH is conceived as a living project to offer physics-based avatar simulation to the research community.Its first version includes the three simulation models described in this paper, and we hope that adoption is facilitated by the versatile cross-platform API and the examples provided.\nWe understand that in many computational problems involving humans, the shape, pose or motion of the human are variables that are optimized.To provide stronger support to such applications, we would like to augment FLSH with differentiability capabilities in the future."},{"title":".","content":"In this way, users can define externally arbitrary energy terms (), and obtain gradients"},{"title":"Fig. 2 .","content":"Fig. 2. Comparison of deformations produced by the simulation modes in FLSH.From left to right: articulated skeleton, FEM soft skeleton, and ROM soft skeleton.The colormaps indicate the amount of soft tissue deformation.As shown in the images, the ROM soft skeleton provides a good approximation to the FEM soft skeleton, and therefore offers a good tradeoff between speed and accuracy."},{"title":"2 . 3 .","content":"Enable settings that define the type of simulation model.There are 4 different settings: enableSoftTissue() If disabled, the simulation model is the default articulated skeleton.enableReducedModel() It allows selecting FEM or ROM soft skeleton.Call SoftAvatar::initVolumetricData()."},{"title":"Fig. 5","content":"compares the accuracy of contact resolution with the different simulation model in FLSH.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2)In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3)"},{"title":"Fig. 3 .","content":"Fig. 3.In the soft skeleton models in FLSH, soft-tissue properties are adapted based on body shape.This image shows 5 different body shapes (first row), which lead to different distribution of Young modulus (second row) as well as skin thickness (third row)."},{"title":"Fig. 4 .","content":"Fig. 4. Without physics-based simulation, avatars may interpenetrate each other in disturbing ways, as shown on the left.With the physics-based simulation in FLSH, on the other hand, interactions are much more accurate, as shown on the right.This example uses the articulated skeleton model.For fast collision detection, each avatar is represented with proxy geometry formed by a hierarchy of analytical capsules (shown in the top images)."},{"title":"Fig. 5 .","content":"Fig. 5. Comparison of contact resolution with the different simulation model in FLSH.To highlight the differences, we display semi-transparent an avatar with no physics-based simulation, and we indicate the position of the chin in the avatars with and without simulation using colored dots.From left to right: avatar with no simulation, articulated skeleton model, FEM soft skeleton model, and ROM soft skeleton model.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2) In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3) The ROM soft skeleton approximates well contact for large objects like this sphere, but the FEM soft skeleton becomes necessary to resolve contact with small objects and/or sharp features."},{"title":"Table 1 .","content":"Benchmarking."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Parametric models of humans [2] have greatly simplified the modeling, animation, reconstruction and tracking of humans across computer vision and computer graphics applications.However, to date these parametric models are intrinsically geometric, and their shape, pose and motion must be either designed or inferred through optimization methods.Parametric human models lack the ability to move according to the laws of physics.We believe that endowing parametric human models with physicsbased dynamics can open up novel application possibilities, across animation, virtual reality, or computer vision.\\nThe images in Fig. 1 highlight the relevance of body physics in two examples: (left) color-coding of the amount of soft-tissue dynamics in 5 different motions; (right) extreme differences in avatar interaction without or with contact mechanics.Deformation, dynamics and contact are obtained in a natural way by solving the laws of physics on top of parametric human representations.To simplify the adoption of this technology across human modeling, animation, reconstruction and tracking, we provide FLSH, a simple software library with dynamic simulation capabilities for the popular SMPL parametric human model [2].In FLSH, the core library runs in C++, but it offers both C++ and Python APIs to allow for flexible integration.\\nIn FLSH, the simulation of human bodies is presented in three different flavors:\\n1. Articulated skeleton: This is an articulated skeleton dynamics model with geometric skinning.\\nBodies react to contact, and they can be controlled according to target trajectories or forces.This is the simplest and fastest representation, when tissue deformation due to contact or dynamics is not necessary.\\n2. FEM soft skeleton: This is a full finite-element model (FEM) of soft-tissue dynamics on top of the articulated skeleton.Bodies contain both the bone structure and deformable soft-tissue, simulated with state-of-the-art finite-element non-linear elasticity, for high-accuracy applications.3. ROM soft skeleton: This is a reduced-order model (ROM) of the FEM soft skeleton.Bodies solve a real-time approximation of soft-tissue deformation, including contact, following a research solution [6].This representation offers an optimal trade-off between accuracy and performance.\\nThe rest of the document is structured as follows.First, we describe the different simulation models supported in the library, covering a range of features and allowing maximal performance for each feature set.Second, we outline the high-level API of the simulation library.Third, we show and discuss some simulation results.\"\n },\n {\n \"title\": \"Simulation Methods\",\n \"content\": \"All three simulation models in FLSH are solved using a common methodology.This simplifies the transition between simulation models based on user needs.The simulation methodology is also easy to extend.In the rest of this section, we present the overall algorithm, the main characteristics of all three simulation models, and some performance comparisons.Fig. 2 shows a comparison of avatar deformation under the three simulation models.\"\n },\n {\n \"title\": \"Overall algorithm\",\n \"content\": \"All simulation models are defined on top of the parametric body model SMPL [2].SMPL is characterized by a pose and shape .Given a template body ̅ 0 in reference pose, it is mapped to a person-specific body ̅ = ℎ (̅ 0 , ) based on the shape parameters.Then, a skinning transformation = (̅ , ) based on defines the posed body shape .SMPL's skinning transformation is linear blend skinning augmented with pose and shape blend-shapes.Even though the original SMPL is defined only for the body surface, FLSH is built on top of the extension to a volumetric body by [5].\\nLet us consider a set of generic dynamic degrees of freedom (DoFs) , which depend on the specific simulation model of choice.The DoFs alter the standard SMPL definition of the deformed body , as discussed below for each simulation model.\\nFLSH simulates dynamics using backward Euler numerical integration.This can be formulated as the following optimization problem [1]:\\nIn this optimization, ℎ is the time step, * the tentative state resulting from explicit integration, is the mass matrix, and collects all potential energy terms.\\nIn essence, the API of FLSH provides the evaluation of the objective, gradient and Hessian of the optimization.This allows extensibility, as users can program their own DoFs and energy terms, which are added to those of FLSH.Furthermore, FLSH offers separate access to the inertial and potentialenergy terms in the optimization.In this way, simply by dropping the inertial term, FLSH can also simulate static deformations.The library is complemented with a solver example.\\nFLSH offers some default contact handling capabilities, as well as tools for generalizing contact handling.First, it allows the internal definition of some simple parametric colliders (spheres, cylinders).Second, it provides access to the Jacobian\"\n },\n {\n \"title\": \"Articulated skeleton\",\n \"content\": \"For each body bone in the SMPL skeleton, we define a rigid transformation .Then, the DoFs of the skeletal dynamics model correspond to the collection of rigid bone transformations, = { }.We can obtain the SMPL pose by extracting the skeleton root transformation and the joint angles from , and the deformed body is directly obtained through the skinning transformation defined above.In the articulated skeleton model, the mass matrix is derived from rigid-body inertia terms associated with each bone.The potential energy terms included in the model are: joint constraints to maintain the skeletal structure, and tracking constraints between the skeleton and target configurations (e.g.coming from mocap) implemented as joint rotation springs.\"\n },\n {\n \"title\": \"FEM soft skeleton\",\n \"content\": \"In the FEM soft skeleton model, we define a skin displacement field (̅ ).We apply the skin deformation in unposed state, hence transforming the reference body shape as ̅ + .By adding SMPL's skinning transformation, the deformed body can be described as = (̅ , ).We use an FEM discretization with nodal displacements { }.Grouping all nodal displacements together with the rigid bone transformations yields the degrees of freedom = ({ }, { }).In FLSH, all body shapes use the same discretization, independent of the shape coefficients .This allows easy transition between body shapes if needed by the user's application.\\nWe model skin elasticity using the approach in [5], which implies:\\n1.The definition of deformation gradient = + ̅ ̅\\n, which accounts for contact and inertial deformations, but respects SMPL's data-driven detail under pose changes.\\n2. A strain energy density Ψ() corresponding to an orthotropic Saint Venant-Kirchhoff hyperelastic model with Fung-type saturation.The soft-tissue parameters are estimated from the DYNA dataset [3] following the multi-person optimization approach of [4].\\nIn the FEM soft skeleton model, the energies of the articulated skeleton are extended in the following way.The mass matrix is augmented with terms corresponding to the skin displacements, as well as cross-terms with the bone transformations.The new terms are derived from the kinetic energy of worldspace skin velocities .The potential energy now also includes the integral of the strain energy density Ψ over the body's volume.\"\n },\n {\n \"title\": \"ROM soft skeleton\",\n \"content\": \"The reduced model retains much of the structure of the FEM soft skeleton, but it represents the skin displacement field in a linear reduced basis, (̅ ) = (̅ ) , with the vector of reduced DoFs and the matrix of basis coefficients.We use as reduced model the bounded generalized biharmonic coordinates of [7], following the model architecture of [6].This allows seamless integration in the basis of handle points in the skin together with the skeletal bones.The DoFs of the ROM soft skeleton model are = ({ }, ).The deformed body is defined with the same expression as in the FEM soft skeleton, with the only difference that skin displacements are represented in the reduced basis.\\nUsing a reduced basis instead of a full FEM discretization dramatically reduces the solver cost, thanks to the reduced size of the problem gradient and Hessian.We also add cubature, following again the data-oblivious approach of [6], to approximate the skin inertia and elastic energy.\"\n },\n {\n \"title\": \"Performance\",\n \"content\": \"The performance of the various simulation models is highly dependent on the number of degrees of freedom.The SMPL model, which serves as basis for all models, consists of 24 bones, 13,776 triangles and 6,890 vertices.Pose is parameterized with 144 coefficients and shape with 10 coefficients.Our tetrahedral volume discretization of the body for the FEM soft skeleton consists of around 43500 tetrahedra and 10400 nodes.Furthermore, the ROM skeleton model contains 87 point handles, and we use 500 cubature samples.\\nIn our performance tests, we have measured both the runtime cost and the generation of the avatars.Note that the soft models require more costly generation process which includes tetrahedral meshing and, in the case of the ROM soft skeleton, cubature sampling.The current version of FLSH implements avatar generation every time a new avatar is created, but future versions will support caching of avatar model data.Table 1 shows the performance measurements on a PC with a 3.4 GHz Intel Core i7-4770 CPU with 32GB of memory.\"\n },\n {\n \"title\": \"Library API\",\n \"content\": \"The FLSH library contains two main objects that are accessible through its API: the SoftAvatar class, which stores the different data structures to represent the avatar body, and the Simulable class, which provides access to simulation state and optimization terms.\"\n },\n {\n \"title\": \"Initialization and settings\",\n \"content\": \"The creation of an avatar must follow these steps:\\n1.Call the constructor of the SoftAvatar class.The constructor receives as arguments an SMPL avatar object and a vector of shape coefficients.The creation of the simulation object must follow these steps:\\n1.Call the constructor of a Simulable class.The constructor receives the following arguments: a SoftAvatar object, the SMPL object, and possibly a sequence of mocap frames to be tracked.\"\n },\n {\n \"title\": \"Call Simulable::initialize().\",\n \"content\": \"If the user wants to include multiple avatars in the scene, then this is as simple as creating multiple SoftAvatar objects and Simulable interfaces.Each Simulable will read/write data referred to its state size, and this state can be merged into larger vectors and/or matrices corresponding to the full scene.\"\n },\n {\n \"title\": \"Simulation Results\",\n \"content\": \"Next, we discuss some simulation examples that highlight the features of the avatar models in FLSH, as well as differences between the various simulation models.\\nFig. 2 compares body deformations with all three simulation models while tracking a mocap sequence.The images also highlight the differences in soft-tissue deformation.As expected, the articulated skeleton's deformation is due simply to SMPL's skinning transformation, but there is no additional softtissue deformation.In the soft skeleton models, on the other hand, there is soft-tissue deformation induced by the body's own inertia.The ROM soft skeleton provides a good approximation to the FEM soft skeleton, and therefore offers a good tradeoff between speed and accuracy.\\nAs mentioned earlier, in the soft skeleton models in FLSH, soft-tissue properties are adapted based on body shape.Fig. 3 shows an example of this shape-based soft-tissue parameterization, for 5 different body shapes.As expected, the differences in soft-tissue parameters produce noticeable differences in the soft-tissue deformation of the various body shapes.\\nFinally, we highlight the benefit of physics-based simulation to resolve contact-based interactions.Fig. 4 shows an example of articulated skeleton where the lack of physics-based simulation leads to disturbing avatar interpenetration.As shown in the images, the FLSH avatars are endowed with a hierarchy of analytical capsules for fast collision detection.The ROM soft skeleton approximates well contact for large objects like this sphere, but the FEM soft skeleton becomes necessary to resolve contact with small objects and/or sharp features.\"\n },\n {\n \"title\": \"Discussion\",\n \"content\": \"FLSH is conceived as a living project to offer physics-based avatar simulation to the research community.Its first version includes the three simulation models described in this paper, and we hope that adoption is facilitated by the versatile cross-platform API and the examples provided.\\nWe understand that in many computational problems involving humans, the shape, pose or motion of the human are variables that are optimized.To provide stronger support to such applications, we would like to augment FLSH with differentiability capabilities in the future.\"\n },\n {\n \"title\": \".\",\n \"content\": \"In this way, users can define externally arbitrary energy terms (), and obtain gradients\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"Fig. 2. Comparison of deformations produced by the simulation modes in FLSH.From left to right: articulated skeleton, FEM soft skeleton, and ROM soft skeleton.The colormaps indicate the amount of soft tissue deformation.As shown in the images, the ROM soft skeleton provides a good approximation to the FEM soft skeleton, and therefore offers a good tradeoff between speed and accuracy.\"\n },\n {\n \"title\": \"2 . 3 .\",\n \"content\": \"Enable settings that define the type of simulation model.There are 4 different settings: enableSoftTissue() If disabled, the simulation model is the default articulated skeleton.enableReducedModel() It allows selecting FEM or ROM soft skeleton.Call SoftAvatar::initVolumetricData().\"\n },\n {\n \"title\": \"Fig. 5\",\n \"content\": \"compares the accuracy of contact resolution with the different simulation model in FLSH.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2)In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3)\"\n },\n {\n \"title\": \"Fig. 3 .\",\n \"content\": \"Fig. 3.In the soft skeleton models in FLSH, soft-tissue properties are adapted based on body shape.This image shows 5 different body shapes (first row), which lead to different distribution of Young modulus (second row) as well as skin thickness (third row).\"\n },\n {\n \"title\": \"Fig. 4 .\",\n \"content\": \"Fig. 4. Without physics-based simulation, avatars may interpenetrate each other in disturbing ways, as shown on the left.With the physics-based simulation in FLSH, on the other hand, interactions are much more accurate, as shown on the right.This example uses the articulated skeleton model.For fast collision detection, each avatar is represented with proxy geometry formed by a hierarchy of analytical capsules (shown in the top images).\"\n },\n {\n \"title\": \"Fig. 5 .\",\n \"content\": \"Fig. 5. Comparison of contact resolution with the different simulation model in FLSH.To highlight the differences, we display semi-transparent an avatar with no physics-based simulation, and we indicate the position of the chin in the avatars with and without simulation using colored dots.From left to right: avatar with no simulation, articulated skeleton model, FEM soft skeleton model, and ROM soft skeleton model.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2) In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3) The ROM soft skeleton approximates well contact for large objects like this sphere, but the FEM soft skeleton becomes necessary to resolve contact with small objects and/or sharp features.\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"Benchmarking.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"In this way, users can define externally arbitrary energy terms (), and obtain gradients","content":".\n\nIn this way, users can define externally arbitrary energy terms (), and obtain gradients"},{"id":"","caption":"Fig. 2. Comparison of deformations produced by the simulation modes in FLSH.From left to right: articulated skeleton, FEM soft skeleton, and ROM soft skeleton.The colormaps indicate the amount of soft tissue deformation.As shown in the images, the ROM soft skeleton provides a good approximation to the FEM soft skeleton, and therefore offers a good tradeoff between speed and accuracy.","content":"Fig. 2 .\n2\nFig. 2. Comparison of deformations produced by the simulation modes in FLSH.From left to right: articulated skeleton, FEM soft skeleton, and ROM soft skeleton.The colormaps indicate the amount of soft tissue deformation.As shown in the images, the ROM soft skeleton provides a good approximation to the FEM soft skeleton, and therefore offers a good tradeoff between speed and accuracy."},{"id":"","caption":"Enable settings that define the type of simulation model.There are 4 different settings: enableSoftTissue() If disabled, the simulation model is the default articulated skeleton.enableReducedModel() It allows selecting FEM or ROM soft skeleton.Call SoftAvatar::initVolumetricData().","content":"2 . 3 .\n23\nEnable settings that define the type of simulation model.There are 4 different settings: enableSoftTissue() If disabled, the simulation model is the default articulated skeleton.enableReducedModel() It allows selecting FEM or ROM soft skeleton.Call SoftAvatar::initVolumetricData()."},{"id":"","caption":"compares the accuracy of contact resolution with the different simulation model in FLSH.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2)In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3)","content":"Fig. 5\n5\ncompares the accuracy of contact resolution with the different simulation model in FLSH.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2)In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3)"},{"id":"","caption":"Fig. 3.In the soft skeleton models in FLSH, soft-tissue properties are adapted based on body shape.This image shows 5 different body shapes (first row), which lead to different distribution of Young modulus (second row) as well as skin thickness (third row).","content":"Fig. 3 .\n3\nFig. 3.In the soft skeleton models in FLSH, soft-tissue properties are adapted based on body shape.This image shows 5 different body shapes (first row), which lead to different distribution of Young modulus (second row) as well as skin thickness (third row)."},{"id":"","caption":"Fig. 4. Without physics-based simulation, avatars may interpenetrate each other in disturbing ways, as shown on the left.With the physics-based simulation in FLSH, on the other hand, interactions are much more accurate, as shown on the right.This example uses the articulated skeleton model.For fast collision detection, each avatar is represented with proxy geometry formed by a hierarchy of analytical capsules (shown in the top images).","content":"Fig. 4 .\n4\nFig. 4. Without physics-based simulation, avatars may interpenetrate each other in disturbing ways, as shown on the left.With the physics-based simulation in FLSH, on the other hand, interactions are much more accurate, as shown on the right.This example uses the articulated skeleton model.For fast collision detection, each avatar is represented with proxy geometry formed by a hierarchy of analytical capsules (shown in the top images)."},{"id":"","caption":"Fig. 5. Comparison of contact resolution with the different simulation model in FLSH.To highlight the differences, we display semi-transparent an avatar with no physics-based simulation, and we indicate the position of the chin in the avatars with and without simulation using colored dots.From left to right: avatar with no simulation, articulated skeleton model, FEM soft skeleton model, and ROM soft skeleton model.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2) In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3) The ROM soft skeleton approximates well contact for large objects like this sphere, but the FEM soft skeleton becomes necessary to resolve contact with small objects and/or sharp features.","content":"Fig. 5 .\n5\nFig. 5. Comparison of contact resolution with the different simulation model in FLSH.To highlight the differences, we display semi-transparent an avatar with no physics-based simulation, and we indicate the position of the chin in the avatars with and without simulation using colored dots.From left to right: avatar with no simulation, articulated skeleton model, FEM soft skeleton model, and ROM soft skeleton model.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2) In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3) The ROM soft skeleton approximates well contact for large objects like this sphere, but the FEM soft skeleton becomes necessary to resolve contact with small objects and/or sharp features."},{"id":"","caption":"","content":""},{"id":"","caption":"Benchmarking.","content":"Table 1 .\n1\nBenchmarking.\nAvatar generationSimulationArticulated skeleton<1 s45 fpsFEM soft skeleton15 s0.2 fpsROM soft skeleton25 s20 fps"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"In this way, users can define externally arbitrary energy terms (), and obtain gradients\",\n \"content\": \".\\n\\nIn this way, users can define externally arbitrary energy terms (), and obtain gradients\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 2. Comparison of deformations produced by the simulation modes in FLSH.From left to right: articulated skeleton, FEM soft skeleton, and ROM soft skeleton.The colormaps indicate the amount of soft tissue deformation.As shown in the images, the ROM soft skeleton provides a good approximation to the FEM soft skeleton, and therefore offers a good tradeoff between speed and accuracy.\",\n \"content\": \"Fig. 2 .\\n2\\nFig. 2. Comparison of deformations produced by the simulation modes in FLSH.From left to right: articulated skeleton, FEM soft skeleton, and ROM soft skeleton.The colormaps indicate the amount of soft tissue deformation.As shown in the images, the ROM soft skeleton provides a good approximation to the FEM soft skeleton, and therefore offers a good tradeoff between speed and accuracy.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Enable settings that define the type of simulation model.There are 4 different settings: enableSoftTissue() If disabled, the simulation model is the default articulated skeleton.enableReducedModel() It allows selecting FEM or ROM soft skeleton.Call SoftAvatar::initVolumetricData().\",\n \"content\": \"2 . 3 .\\n23\\nEnable settings that define the type of simulation model.There are 4 different settings: enableSoftTissue() If disabled, the simulation model is the default articulated skeleton.enableReducedModel() It allows selecting FEM or ROM soft skeleton.Call SoftAvatar::initVolumetricData().\"\n },\n {\n \"id\": \"\",\n \"caption\": \"compares the accuracy of contact resolution with the different simulation model in FLSH.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2)In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3)\",\n \"content\": \"Fig. 5\\n5\\ncompares the accuracy of contact resolution with the different simulation model in FLSH.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2)In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 3.In the soft skeleton models in FLSH, soft-tissue properties are adapted based on body shape.This image shows 5 different body shapes (first row), which lead to different distribution of Young modulus (second row) as well as skin thickness (third row).\",\n \"content\": \"Fig. 3 .\\n3\\nFig. 3.In the soft skeleton models in FLSH, soft-tissue properties are adapted based on body shape.This image shows 5 different body shapes (first row), which lead to different distribution of Young modulus (second row) as well as skin thickness (third row).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 4. Without physics-based simulation, avatars may interpenetrate each other in disturbing ways, as shown on the left.With the physics-based simulation in FLSH, on the other hand, interactions are much more accurate, as shown on the right.This example uses the articulated skeleton model.For fast collision detection, each avatar is represented with proxy geometry formed by a hierarchy of analytical capsules (shown in the top images).\",\n \"content\": \"Fig. 4 .\\n4\\nFig. 4. Without physics-based simulation, avatars may interpenetrate each other in disturbing ways, as shown on the left.With the physics-based simulation in FLSH, on the other hand, interactions are much more accurate, as shown on the right.This example uses the articulated skeleton model.For fast collision detection, each avatar is represented with proxy geometry formed by a hierarchy of analytical capsules (shown in the top images).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 5. Comparison of contact resolution with the different simulation model in FLSH.To highlight the differences, we display semi-transparent an avatar with no physics-based simulation, and we indicate the position of the chin in the avatars with and without simulation using colored dots.From left to right: avatar with no simulation, articulated skeleton model, FEM soft skeleton model, and ROM soft skeleton model.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2) In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3) The ROM soft skeleton approximates well contact for large objects like this sphere, but the FEM soft skeleton becomes necessary to resolve contact with small objects and/or sharp features.\",\n \"content\": \"Fig. 5 .\\n5\\nFig. 5. Comparison of contact resolution with the different simulation model in FLSH.To highlight the differences, we display semi-transparent an avatar with no physics-based simulation, and we indicate the position of the chin in the avatars with and without simulation using colored dots.From left to right: avatar with no simulation, articulated skeleton model, FEM soft skeleton model, and ROM soft skeleton model.From the images, we draw three main conclusions: (1) Lack of simulation produces a very unrealistic result.(2) In the articulated skeleton model, the full reaction to contact is absorbed by the skeleton, while in the soft skeleton models part is absorbed by the skeleton and part by the soft tissue, as evidenced by the displacement of the chin.(3) The ROM soft skeleton approximates well contact for large objects like this sphere, but the FEM soft skeleton becomes necessary to resolve contact with small objects and/or sharp features.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"\",\n \"content\": \"\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Benchmarking.\",\n \"content\": \"Table 1 .\\n1\\nBenchmarking.\\nAvatar generationSimulationArticulated skeleton<1 s45 fpsFEM soft skeleton15 s0.2 fpsROM soft skeleton25 s20 fps\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"This can be used e.g. in line-search solvers. popState: resets the previous state from the stack. getStateJacobian: it gets the Jacobian of the skeleton pose, for external skeleton-based energies. getNodesStateJacobian: it gets the Jacobian of mesh nodes, for external mesh-based energies. Optimization evaluation: getDynamicEnergyScalar: get the inertial part of the optimization function. getPotentialEnergyScalar: get the potential-energy part of the optimization function. getDynamicGradientVector: get the inertial part of the optimization gradient. getPotentialGradientVector: get the potential-energy part of the gradient optimization gradient. fixVectorKinematic: it receives a full gradient and applies. State management: getState: it gets the current state of the DoFs. updateState: it sets a new state computed by the external solver. pushState: adds the current state to a stack. Dirichlet boundary conditions. getDynamicHessianTripletVector: get the inertial part of the optimization Hessian. getPotentialHessianTripletVector: get the potential-energy part of the optimization Hessian. fixMatrixKinematic: it receives a full Hessian and applies Dirichlet boundary conditions. References","doi":""},{"text":"Optimization integrator for large time steps. T Gast, C Schroeder, A Stomakhin, C Jiang, J M Teran, IEEE Trans. on Visualization and Computer Graphics. 21102015","doi":""},{"text":"Smpl: A skinned multi-person linear model. M Loper, N Mahmood, J Romero, Pons-Moll , M J Black, ACM Trans. on Graphics. 34616October 2015","doi":""},{"text":"DYNA: A model of dynamic human shape in motion. G Pons-Moll, J Romero, N Mahmood, M J Black, in ACM Trans. on Graphics. 3442015","doi":""},{"text":"SFLSH: Shape-dependent soft-flesh avatars. P Ramón, C Romero, J Tapia, M A Otaduy, Proc. of the ACM SIGGRAPH Asia Conference. of the ACM SIGGRAPH Asia Conference2023to appear in the","doi":""},{"text":"Modeling and estimation of nonlinear skin mechanics for animated avatars. C Romero, M A Otaduy, D Casas, J Perez, Computer Graphics Forum. 3922020","doi":""},{"text":"Parametric skeletons with reduced soft-tissue deformations. J Tapia, C Romero, J Pérez, M A Otaduy, Computer Graphics Forum. 2021","doi":""},{"text":"Linear subspace design for real-time shape deformation. Y Wang, A Jacobson, J Barbic, L Kavan, ACM Trans. on Graphics. 3442015","doi":""}],"string":"[\n {\n \"text\": \"This can be used e.g. in line-search solvers. popState: resets the previous state from the stack. getStateJacobian: it gets the Jacobian of the skeleton pose, for external skeleton-based energies. getNodesStateJacobian: it gets the Jacobian of mesh nodes, for external mesh-based energies. Optimization evaluation: getDynamicEnergyScalar: get the inertial part of the optimization function. getPotentialEnergyScalar: get the potential-energy part of the optimization function. getDynamicGradientVector: get the inertial part of the optimization gradient. getPotentialGradientVector: get the potential-energy part of the gradient optimization gradient. fixVectorKinematic: it receives a full gradient and applies. State management: getState: it gets the current state of the DoFs. updateState: it sets a new state computed by the external solver. pushState: adds the current state to a stack. Dirichlet boundary conditions. getDynamicHessianTripletVector: get the inertial part of the optimization Hessian. getPotentialHessianTripletVector: get the potential-energy part of the optimization Hessian. fixMatrixKinematic: it receives a full Hessian and applies Dirichlet boundary conditions. References\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Optimization integrator for large time steps. T Gast, C Schroeder, A Stomakhin, C Jiang, J M Teran, IEEE Trans. on Visualization and Computer Graphics. 21102015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Smpl: A skinned multi-person linear model. M Loper, N Mahmood, J Romero, Pons-Moll , M J Black, ACM Trans. on Graphics. 34616October 2015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"DYNA: A model of dynamic human shape in motion. G Pons-Moll, J Romero, N Mahmood, M J Black, in ACM Trans. on Graphics. 3442015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"SFLSH: Shape-dependent soft-flesh avatars. P Ramón, C Romero, J Tapia, M A Otaduy, Proc. of the ACM SIGGRAPH Asia Conference. of the ACM SIGGRAPH Asia Conference2023to appear in the\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Modeling and estimation of nonlinear skin mechanics for animated avatars. C Romero, M A Otaduy, D Casas, J Perez, Computer Graphics Forum. 3922020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Parametric skeletons with reduced soft-tissue deformations. J Tapia, C Romero, J Pérez, M A Otaduy, Computer Graphics Forum. 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Linear subspace design for real-time shape deformation. Y Wang, A Jacobson, J Barbic, L Kavan, ACM Trans. on Graphics. 3442015\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":577,"cells":{"text":{"kind":"string","value":"\nFluid Antenna Multiple Access with Simultaneous Non-unique Decoding in Strong Interference Channel\n28 Oct 2024\n\nMember, IEEEFarshad Rostami Ghadi f.rostamighadi@ucl.ac.uk \nFellow, IEEEKai-Kit K Wong kai-kit.wong@ucl.ac.uk \nMasoud Kaveh masoud.kaveh@aalto.fi \nMember, IEEEH Xu \nMember, IEEEW K New \nSenior Member, IEEEF Javier López-Martínez \nFellow, IEEEHyundong Shin hshin@khu.ac.kr \nDe- \nF J López-Martínez \n\npartment of Electronic and Electrical Engineering\nUniversity College London\nLondonUK\n\n\nDepartment of Electronic Engineering\nKyung Hee University\n17104Yongin-si, GyeonggiKorea\n\n\nDepartment of Information and Communication Engi-neering\nAalto University\nEspooFinland\n\n\nDepartment of Signal Theory\nResearch Centre for Information and Communication Technologies (CITIC-UGR)\nNetworking and Communications\nUniversity of Granada\n18071GranadaSpain\n\n\nDepartment of Electronic Engineering\nKyung Hee Uni-versity\n17104Yongin-si, GyeonggiKorea\n\nFluid Antenna Multiple Access with Simultaneous Non-unique Decoding in Strong Interference Channel\n28 Oct 20245993FB6EF60B05CF33341496FF1373F210.13039/50110001103arXiv:2410.20930v1[cs.IT]Delay outage ratefluid antenna multiple accessfluid antenna systeminterference channeloutage probabilitysimultaneous non-unique decoding\nFluid antenna system (FAS) is gaining attention as an innovative technology for boosting diversity and multiplexing gains.As a key innovation, it presents the possibility to overcome interference by position reconfigurability on one radio frequency (RF) chain, giving rise to the concept of fluid antenna multiple access (FAMA).While FAMA is originally designed to deal with interference mainly by position change and treat interference as noise, this is not rate optimal, especially when suffering from a strong interference channel (IC) where all positions have strong interference.To tackle this, this paper considers a two-user strong IC where FAMA is used in conjunction with simultaneous nonunique decoding (SND).Specifically, we analyze the key statistics for the signal-to-noise ratio (SNR) and interference-to-noise ratio (INR) for a canonical two-user IC setup, and subsequently derive the delay outage rate (DOR), outage probability (OP) and ergodic capacity (EC) of the FAMA-IC.Our numerical results illustrate huge benefits of FAMA with SND over traditional fixed-position antenna systems (TAS) with SND in the fading IC.\n\nI. INTRODUCTION A. Context and Literature Review\n\nI N THE RAPIDLY-EVOLVING landscape of wireless com- munication, the emergence of fluid antenna systems (FAS) represents a transformative leap forward [1], [2], [3].As a new form of reconfigurable antennas, FAS emphasizes on position and shape flexibility in antennas for wireless communications.FAS can be implemented by liquid-based antennas [4], radiofrequency reconfigurable pixels [5], [6], metamaterials [7], [8], mechanically movable antennas [9] and etc.An easy way to understand the benefits of FAS is that it presents a new degree of freedom (DoF) to the physical layer that can be translated into diversity [10] and multiplexing gains [11].In recent years, efforts on FAS have been spent onto improving the accuracy of channel model [12], [13], and studying the diversity order in different fading channels such as Rayleigh [14], Nakagami [15], α-µ distributed [16] and even arbitrarily distributed [17].\n\nCopulas were shown to be useful in the performance analysis of FAS [18].Multiple fluid antennas at both ends were recently considered in [19], revealing extraordinary diversity gains.\n\nOne interesting application for FAS is multiuser communications.In particular, a unique way of using FAS is to mitigate inter-user interference by seeking the antenna position where the interference suffers from deep fades, leading to the concept of fluid antenna multiple access (FAMA).By doing so, each receiver handles its interference by position flexibility of FAS without the requirement of precoding at the base station (BS) transmitter.The main advantage is a reduced burden at the BS for scalability in terms of both channel state information (CSI) estimation and optimization.Depending on how fast the FAS position (a.k.a.'port') changes, FAMA can be classified into being fast [20] and slow [21], [22].Opportunistic scheduling can also combine with FAMA for improved performance [23], [24].In [25], a mean-field game was formulated to empower slow FAMA for energy-efficiency maximization.\n\nEvidently, FAS can also be adopted as an additional DoF to elevate traditional beamforming approaches.For instance, [26] considered an uplink scenario where the BS used traditional beamforming but the users employed multiple fluid antennas for joint port selection and beamforming in order to maximize the capacity.Moreover, FAS has been considered for integrated sensing and communications in [27], [28], [29].Dirty multiple access channels with FAS were also studied in [30] when side information was assumed at the transmitters.The application of FAS has also extended to backscatter communications [31] and wireless power transfer [32].Clearly, CSI is crucial to the performance of FAS and hence recent efforts have addressed the channel estimation problem for FAS in [33], [34].Besides, machine learning has also been demonstrated to be effective in reducing the complexity for establishing the channels at the FAS ports [35], [36].Finally, it is worth mentioning the recent experimental results in [37], [38], validating the capability of FAS in wireless communications systems.\n\n\nB. Motivation and Contributions\n\nWhile FAMA is attractive in simplifying how interference can be handled, it is not rate optimal or capacity achieving.In [39], a two-user interference channel (IC) with FAS was studied and the Han-Kobayashi's achievable rate was investigated.It was revealed that FAMA without power splitting nor rate splitting could achieve a similar sum-rate to that with, if the size and resolution of FAS were sufficient.In other words, if FAS can find a port that has weak interference, then it will be optimal to treat interference as noise [40].However, this is not the case when facing a strong IC in which FAS would be unable to find any port that has weak interference and therefore treating interference as noise is far from optimal.\n\nMotivated by the above, in this paper, we aim to investigate the performance of a canonical IC where two transmitters wish to communicate to their respective receivers over a shared IC, and each of the receivers is equipped with FAS.Specifically, our interest is on the strong IC scenarios where joint decoding of both messages at both receivers is capacity optimal [40].For this reason, we consider the use of simultaneous non-unique decoding (SND) [41].This scheme enables receivers to decode multiple signals simultaneously, even when there is no clear separation between the desired signal and the interference. 1By integrating SND into the FAMA framework, we can enhance the system's ability to overcome interference, enabling each receiver to retrieve both its own message and those from the interfering transmitters without additional rate constraints.\n\nOur key contributions are summarized as follows:\n\n• We formalize a definition of the two-user FAMA setup in the strong interference regime, and define its instantaneous capacity region.In this scenario, we derive the probability density function (PDF) and cumulative distribution function (CDF) of the signal-to-noise ratio (SNR) and interference-to-noise ratio (INR) at the receivers.• Additionally, we derive closed-form expressions for the probabilistic measures such as outage probability (OP) and delay outage rate (DOR), in terms of the CDF of the multivariate normal distribution.We also provide asymptotic expressions in the high SNR regime, considering the SND interference management technique.• We then derive an analytical expression for the ergodic capacity (EC) of the FAMA-IC with strong interference.This is obtained by approximating the expected value of the equivalent SNRs, i.e., the maximum of N correlated random variables (RVs), using a heuristic approach.• Finally, we conduct numerical evaluations to assess how the implementation of FAS influences the performance of FAMA-IC under strong interference scenarios, highlighting its performance gain over the traditional fixedposition antenna system (TAS) counterpart.\n\n\nC. Organization\n\nThe remainder of this manuscript is structured as follows: Section II introduces the system model of FAMA-IC.The statistical characteristics of the equivalent channel, including the CDF and PDF, are then derived in Section III.Section IV presents the analytical results of key performance metrics such as OP, DOR, and EC.Section V provides the numerical results, and finally, Section VI concludes the paper.\n\n\nII. SYSTEM MODEL\n\nConsider a setup as shown in Fig. 1, where each transmitter t i , i ∈ {1, 2}, sends inputs x i with transmit power P i to its respective receiver r i over a shared IC.It is assumed that the transmitters t i are equipped with a traditional fixed antenna while receivers r i take advantage of a 2D planar fluid antenna that includes N ri preset ports which are uniformly distributed over an area of W ri .To be more precise, we assume a grid structure for the fluid antenna so that N ri,l ports are uniformly distributed along a linear space of length λW ri,l for l ∈ {1, 2}, where λ denotes the wavelength of the carrier frequency, i.e., N ri = N ri,1 ×N ri,2 and W ri = λW ri,1 ×λW ri,2 .Moreover, in order to convert the 2D indices to the 1D index for each port (i.e., the (n ri,1 , n ri,2 )-th port), we introduce an appropriate one-to-one mapping function as F (n ri ) = (n ri,1 , n ri,2 ) and F −1 (n ri,1 , n ri,2 ) = n ri in which n ri ∈ {1, . . ., N ri } and n ri,l ∈ {1, . . ., N ri,l }.Also, given that the fluid antenna ports can be arbitrarily close to each other, the corresponding channels are spatially correlated.Therefore, in a 3D isotropic scattering environment, the spatial correlation between the n rith port and the ñri -th port is given by [19]\nϱ ri nr i ,ñr i = cov {n ri , ñri } = J 0 2π |nr i ,1−ñr i ,1| Nr i ,1−1 W ri,1 2 + |nr i ,2−ñr i ,2| Nr i ,2−1 W ri,2 2 ,(1)\nwhere J 0 (•) is the spherical Bessel function of the first kind, and F (ñ ri ) = (ñ ri,1 , ñri,2 ) and F −1 (ñ ri,1 , ñri,2 ) = ñri where ñri ∈ {1, . . ., N ri } and ñri,l ∈ {1, . . ., N ri,l }.Hence, the spatial correlation matrix R ri is defined as\nR ri ≜ ϱ ri 1,1 ϱ ri 1,2 . . . ϱ ri 1,Nr i ϱ ri 2,1 ϱ ri 2,2 . . . ϱ ri 2,Nr i . . . . . . . . . . . . ϱ ri Nr i ,1 ϱ ri Nr i ,2 . . . ϱ ri Nr i ,Nr i . (2)\nUnlike TAS, the radiating element of each FAS receiver r i can switch its location among N ri ports.To facilitate the notations, we define ι ∈ {1, 2} as the complement of subscript i, so that if i = 1, then ι = 2 and vice versa.Therefore, the received signal at the n ri -th port of receiver r i is given by\ny nr i ri = h nr i ti,ri L i,i x i + h nr i tι,ri L ι,i x ι + z nr i ri ,(3)\nwhere z nr i ri denotes the additive white Gaussian noise (AWGN) with zero mean and variance of σ2 i at the n ri -th port of r i , and L j,k is the path loss experienced by the message from transmitter j arriving at receiver k, for j, k ∈ {i, ι}.The term h nr i ti,ri defines the corresponding normalized Rayleigh fading channel from transmitter t i to receiver r i , i.e., E |h\nnr i ti,ri | 2 = 1, where E {•} represents the expectation operator.\nSpecifically, we consider the scenario where each receiver r i within the FAMA-IC is physically closer to the interfering transmitter t ι than to its own transmitter t i .This can be the case, for instance, where users are located in the expanded region of a pico-cell in the presence of a macro-cell [42].As a result, the signal received from the interfering transmitter t ι surpasses that from its respective transmitter t i in signal strength.Under such strong interference conditions, employing SND yields superior results compared to treating interference as noise. 2 By decoding multiple signals simultaneously using approaches such as multiuser detection or joint detection, SND can achieve higher spectral efficiency and offer a larger capacity region [40].Consequently, each receiver can effectively retrieve the message of the interfering transmitter without requiring an additional constraint on its rate.With all the above definitions, the capacity region for a general two-user IC is expressed as [40, Theorem 6.2]\nR 1 < log 2 (1 + SNR 1 ) , (4a) R 2 < log 2 (1 + SNR 2 ) , (4b) R 1 + R 2 < min {log 2 (1 + SNR 1 + INR 1 ) , log 2 (1 + SNR 2 + INR 2 )} ,(4c)\nin which SNR i and INR i represent the SNR for the desired and interfering messages at user i, respectively.From these, the instantaneous capacity region for the considered strong interference scenario in the FAMA-IC can be defined as\nR 1 < log 2 1 + max nr 1 ∈{1,...,Nr 1 } γ nr 1 r1 , (5a) R 2 < log 2 1 + max nr 2 ∈{1,...,Nr 2 } γ nr 2 r2 ,(5b)R 1 + R 2 < min log 2 1 + max nr 1 ∈{1,...,Nr 1 } κ nr 1 r1 , log 2 1 + max nr 2 ∈{1,...,Nr 2 } κ nr 2 r2 ,(5c)\nwhere\nκ nr i ri = γ nr i ri + ζ nr i ri , and γ ňr i ri = P i L i,i h ňr i tiri 2 σ 2 i = γ ri h ňr i tiri 2 , ζ ňr i ri = P ι L ι,i h ňr i tιri 2 σ 2 i = ζ ri h ňr i tιri 2 , (6)provided that P ι L ι,i |h tι,ri | 2 > P i L i,i |h ti,ri | 2 (strong interfer- ence condition).\nThe terms γ ri and ζ ri denote the average SNR and the average INR at receiver r i , respectively.Note that ň ∈ {n, n}, F (n ri ) = (n ri,1 , nri,2 ) and F −1 (n ri,1 , nri,2 ) = nri , where nri ∈ {1, . . ., N ri } and nri,l ∈ {1, . . ., N ri,l }.That is to say, we consider that only the port which maximizes the received SNR at the receiver is activated.\n\n\nIII. STATISTICAL CHARACTERIZATION\n\nTo mathematically characterize the capacity region of the FAMA-IC in (5) under Rayleigh fading, we find it necessary to determine the distribution of the maximum amongst the N ri correlated SNRs, INRs, and the sum of SNRs and INRs.In this regard, we first note that all channels experience Rayleigh fading, so the corresponding channel gains are exponentially distributed, i.e., h\nf δ ňr i r i δ ňr i ri = 1 δ ri exp − δ ňr i ri δ ri(7)\nand\nF δ ňr i r i δ ňr i ri = 1 − exp − δ ňr i ri δ ri ,(8)\nin which δ ∈ {γ, ζ}.Following a similar approach as in [18], the joint multivariate distribution for the maximum of d correlated arbitrary RVs can be accurately generated by using Sklar's theorem.More specifically, the joint CDF of d RVs S 1 , . . ., S d with the corresponding marginal CDF F S l (s l ), l ∈ {1, . . ., d}, in the extended real line domain R, is given by\nF S1,...,S d (s 1 , . . . , s d ) = C (F S1 (s 1 ) , . . . , F S d (s d ) ; ϑ) , (9)\nwhere\nC : [0, 1] d → [0, 1] is the d-dimensional copula, which\nis defined as a joint CDF of d RVs on the unit cube [0, 1] d with uniform marginal distributions.Besides, ϑ is the copula dependence parameter that measures the linear and/or nonlinear correlation between the underling RVs.In this regard, Gaussian copula is one of the practical models to describe the inherent spatial correlation between fluid antenna ports due to its appealing properties [18].Thus, by defining the maximum of γ\nnr i ri and ζ nr i ri as δ fama ri = max δ 1 ri , . . . , δ Nr i ri , δ ∈ {γ, ζ} ,(10)\nthe CDF of δ fama ri can be found as [18]\nF δ fama r i (r) = Φ Rr i φ −1 F δ 1 r i (r) , . . . , φ −1 F δ Nr i r i (r) ; ϖ ri ,(11)in which F δ nr i r i (r) is defined in (8), Φ R (•)\nis the joint CDF of the multivariate normal distribution with zero mean vector and correlation matrix\nR ri , φ −1 (x) = √ 2erf −1 (2x − 1) is the quantile function of the standard normal distribution, where erf −1 (•) is the inverse of the error function erf (z) = 2 √ π z 0 e −t 2 dt.\nIn addition, ϖ ri defines the dependence parameter of the Gaussian copula.It is noted that by exploiting the Cholesky decomposition, we have ϖ ri nr i ,ñr i ≈ ϱ ri nr i ,ñr i [18].By applying the chain rule to (11), the PDF of δ fama ri is given by\nf δ fama r i (r) = Kr i nr i =1 f nr i δr i (r) exp − 1 2 φ −1 δr i T R −1 ri − I φ −1 δr i det (R ri ) ,(12)\nwhere f δ nr i r i (r) is given in (7), det (R ri ) is the determinant of the correlation matrix R ri , I is an identity matrix, and\nφ −1 δr i = φ −1 F δ 1 r i (r) , . . . , φ −1 F δ Nr i r i (r) T .(13)\nNow, we need to determine the distribution corresponding to the maximum of the sum of the SNR and INR, i.e., κ\nnr i ri = γ nr i ri + ζ nr i\nri , and we define\nκ fama ri = max κ 1 ri , . . . , κ Nr i ri . (14)\nGiven that κ nr i ri includes the sum of two independent exponential RVs with different scale parameters γ ri and ζ ri , it has the hypoexponential distribution with the following PDF and CDF [43]:\nf κ nr i r i κ nr i ri = 1 ∆ ri e − κ nr i r i γ r i − e − κ nr i r i ζ r i (15)\nand\nF κ nr i r i κ nr i ri = 1 − 1 ∆ ri γ ri e − κ nr i r i γ r i − ζ ri e − κ nr i r i ζ r i ,(16)\nrespectively, where ∆ ri = γ ri − ζ ri .As such, using Sklar's theorem in (9) and the Gaussian copula definition, the CDF and PDF of κ fama ri are found as\nF κ fama r i (r) = Φ Rr i φ −1 F κ 1 r i (r) , . . . , φ −1 F κ Nr i r i (r) ; ϖ ri ,(17)\nand\nf κ fama r i (r) = Nr i nr i =1 f nr i κr i (r) exp − 1 2 φ −1 κr i T R −1 ri − I φ −1 κr i det (R ri ) ,(18)\nin which F\nκ nr i r i (r) and f κ nr i r i\n(r) are defined in ( 16) and ( 15), respectively.Moreover, φ −1 κr i is denoted as\nφ −1 κr i ≜ φ −1 F κ 1 r i (r) , . . . , φ −1 F κ Nr i r i (r) T .(19)\nNote that the copula approach allows for a tractable characterization for the joint distribution of κ fama ri , from their underlying marginal distributions.Having characterized the key statistics of the equivalent channels under consideration, these will be used to analyze a number of performance metrics.\n\n\nIV. PERFORMANCE ANALYSIS A. OP\n\nOP is an important performance metric in evaluating wireless communication systems, and is defined as the probability that the capacity falls below a given rate threshold R th , i.e., P out = Pr (C ≤ R th ), where C = log 2 (1 + δ).Hence, for the considered FAMA-IC, an outage occurs if at least one of the following events happens:\nE 1 : log 2 1 + γ fama r1 ≤ R th 1 ,(20a)E 2 : log 2 1 + γ fama r2 ≤ R th 2 ,(20b)E 3 : min log 2 1 + κ fama r1 , log 2 1 + κ fama r2 ≤ R th sum ,(20c)\nwhere R th i and R th sum = R th 1 + R th 2 are the rate threshold and the sum-rate threshold, respectively.Therefore, the OP for the considered FAMA-IC is defined as\nP out = Pr (E 1 ∪ E 2 ∪ E 3 ) . (21)\nTheorem 1.The OP for the FAMA-IC system under the strong interference scenario is given by\nP out = 1 − F κ fama r 1 Rth sum × F κ fama r 2 Rth sum F γ fama r 1 ,γ fama r 2 Rth 1 , Rth 2 ,(22, Rth i = 2 R th i − 1, and Rth sum = 2 R th sum − 1.\nProof.The OP definition in (21) can be extended as\nP out = Pr (E 1 ) + Pr (E 2 ) + Pr (E 3 ) − Pr (E 1 ∩ E 2 ) − Pr (E 1 ∩ E 3 ) − Pr (E 2 ∩ E 3 ) + Pr (E 1 ∩ E 2 ∩ E 3 ) ,(23)\nwhere Pr (E 1 ) can be determined as\nPr (E 1 ) = Pr log 2 1 + γ fama r1 ≤ R th 1 (24a) = Pr γ fama r1 ≤ Rth 1 (24b) = F γ fama r 1 Rth 1 ,(24c)\nwhere Rth\n1 = 2 R th 1 − 1. Similarly, Pr (E 2 ) = F γ fama r 2\nRth 2 .As for the event E 3 , we have\nPr (E 3 ) = Pr min log 2 1 + κ fama r1 , log 2 1 + κ fama r2 ≤ R th sum (25a) = 1 − Pr log 2 1 + κ fama r1 > R th sum × Pr log 2 1 + κ fama r1 > R th sum (25b) = 1 − Pr κ fama r1 > Rth sum Pr κ fama r2 > Rth sum (25c) = 1 − F κ fama r 1 Rth sum F κ fama r 2 Rth sum ,(25d)\nwhere (25b) is obtained from the independence of the events,\nRth sum = 2 R th sum − 1, and F κ fama r i (r) = 1 − F κ fama r i (r) is the CCDF of κ fama ri\n. Since all events E j for j ∈ {1, 2, 3} are independent of each other, the other probabilities can be obtained according to Pr (E j ).Therefore, by considering the CDF of γ fama ri , ζ fama ri , and κ fama ri , and after some mathematical simplification, the proof is accomplished.\n\nFrom the derived closed-form expression for OP, we now aim to understand the asymptotic behavior of the OP at high SNR, i.e., γ ri → ∞ while meeting the strong interference condition.As we will later see, this asymptotic expression provides more insights into the system's reliability.\n\nCorollary 1.The asymptotic OP in the high SNR regime for the FAMA-IC system under the strong interference scenario is given by\nP ∞ out = 1 − F ∞ κ fama r 1 Rth sum × F ∞ κ fama r 2 Rth sum F ∞ γ fama r 1 ,γ fama r 2 Rth 1 , Rth 2 ,(26)\nwhere\nF ∞ κ fama r i Rth sum and F ∞ γ fama r 1\n,γ fama Proof.Given that γ nr i ri and κ nr i ri follow the exponential and hypoexponential distributions, respectively, their corresponding CDFs in the high SNR regime can be derived with the help of the Taylor series expansion.Therefore, as γ ri → ∞, the term e − Rth ν /γ r i for ν ∈ {1, 2, sum} becomes very small such that we have\nexp − Rth ν γ ri ≈ 1 − Rth ν γ ri , ν ∈ {1, 2, sum} .(29)\nBy substituting ( 29) into ( 8) and ( 16), the marginal CDFs of γ nr i ri and κ nr i ri at high SNR are, respectively, given by\nF ∞ κ nr i r i Rth sum = 1 ∆ ri Rth sum + ζ ri exp − Rth sum ζ ri ,(30)\nand\nF ∞ γ ňr i r i Rth i = Rth i γ ri .(31)\nNow, by inserting ( 31) into (11) and plugging ( 30) into (17),\nF ∞ κ fama r i Rth sum and F ∞ γ fama r 1 ,γ fama r 2\nRth 1 , Rth 2 are obtained and the proof becomes complete.\n\n\nB. DOR\n\nThe DOR is a crucial metric in the design of various wireless communication scenarios, particularly in ultra-reliable and low-latency communication systems.It is defined as the probability that the time required to transmit a specific amount of data R over a wireless channel with bandwidth B exceeds a predefined threshold time T th [44].Mathematically, this is expressed as Pr (T d > T th ), where T d = R BC indicates the delivery time such that C denotes the channel capacity [31].This expression captures the essential relationship between data transmission time, channel bandwidth, and signal quality.\n\nThus, for the considered FAMA-IC, the DOR is achieved if at least one of the following events occurs:\nT 1 : R 1 B 1 C 1 > T th 1 ,(32a)T 2 : R 2 B 2 C 2 > T th 2 ,(32b)T 3 : R 1 + R 2 B sum C sum > T th sum ,(32c)\nwhere C ν for ν ∈ {1, 2, sum} is defined in (5), and B ν and R i denote the corresponding bandwidth and rates, respectively.Therefore, the DOR for FAMA-IC is defined as\nP dor = Pr (T 1 ∪ T 2 ∪ T 3 ) . (33)\nTheorem 2. The OP for the FAMA-IC system under the strong interference scenario is given by\nP dor = 1 − F κ fama r 1 T th sum × F κ fama r 2 T th sum F γ fama r 1 ,γ fama r 2 T th 1 , T th 2 ,(34)\nin which Ti = e R i ln 2\nBT th i − 1 and Tsum = e (R 1 +R 2 ) ln 2 Bsum T th sum .\nProof.By extending the definition of DOR in (33), we have\nP dor = Pr (T 1 ) + Pr (T 2 ) + Pr (T 3 ) − Pr (T 1 ∩ T 2 ) − Pr (T 1 ∩ T 3 ) − Pr (T 2 ∩ T 3 ) + Pr (T 1 ∩ T 2 ∩ T 3 ) , (35)\nwhere Pr (T 1 ) can be mathematically obtained as\nPr (T 1 ) = Pr R 1 B 1 log 2 1 + γ fama r1 > T th 1 (36a) = Pr γ fama r1 ≤ exp R 1 ln 2 B 1 T th 1 − 1 (36b) = F γ fama r 1 T1 (36c) F ∞ κ fama r i Rth sum = 1 − Φ Ri φ −1 1 ∆ ri Rth sum + ζ ri e − Rth sum ζ r i , . . . , φ −1 1 ∆ ri Rth sum + ζ ri e − Rth sum ζ r i ; ϖ ri (27) F ∞ γ fama r 1 ,γ fama r 2 Rth 1 , Rth 2 = 1 − Φ Rr 1 φ −1 Rth 1 γ r1 , . . . , φ −1 Rth 1 γ r1 ; ϖ r1 1 − Φ Rr 2 φ −1 Rth 2 γ r2 , . . . , φ −1 Rth 2 γ r2 ; ϖ r2(28)\nwhere\nT1 = e R 1 ln 2 B 1 T th 1 − 1. Similarly, Pr (T 2 ) = F γ fama r 2 T2 in which T2 = e R 2 ln 2 B 1 T th 2 − 1.\nMoreover, Pr (T 3 ) is derived as\nPr (T 3 ) = Pr (R1+R2) Bsum min{log 2 (1+κ fama r 1\n),log 2 (1+κ fama r 2\n)} >T th sum (37a) = 1 − Pr min log 2 1 + κ fama r1 , log 2 1 + κ fama r2 < (R 1 + R 2 ) ln 2 B sum T th sum (37b) = 1 − Pr κ r1 > exp (R 1 + R 2 ) ln 2 B sum T th sum − 1 × Pr κ r2 > exp (R 1 + R 2 ) ln 2 B sum T th sum − 1 (37c) = 1 − F κ fama r 1 Tsum F κ fama r 2\nTsum ,\n\nwhere Tsum = e 2(R 1 +R 2 ) ln 2 Bsum T th sum .Now, given the independence of all events T j for j ∈ {1, 2, 3}, other probabilities can be obtained according to Pr (T j ).Hence, by considering the CDFs of γ fama ri , ζ fama ri , and κ fama ri , and after some mathematical simplification, the proof is completed.\n\nCorollary 2. The asymptotic DOR in the high SNR regime for the FAMA-IC under the strong interference scenario is given by\nP ∞ dor = 1 − F ∞ κ fama r 1 T th sum × F ∞ κ fama r 2 T th sum F ∞ γ fama r 1 ,γ fama r 2 T th 1 , T th 2 . (38)\nProof.Following the same approach used to prove Corollary 1, the proof is completed.\n\n\nC. EC\n\nThe EC (per bandwidth unit) for the system model under consideration is defined as\nC [bps/Hz] ≜ E [C] = ∞ 0 log 2 (1 + η) f η (η)dη,(39)\nwhere η ∈ γ fama ri , κ fama ri and f η (η) denotes the corresponding PDF.Therefore, the EC for the considered system model is derived and presented in the following theorem.Theorem 3. The EC for the FAMA-IC system under the strong interference scenario is given by\nC 1 ≈ log 2 1 + γ r1 H 1 1 − ϖ r1 H 1 2N r1 ,(40)C 2 ≈ log 2 1 + γ r2 H 2 1 − ϖ r2 H 2 2N r2 ,(41)\nand\nC sum ≈ min log 2 1 + κ r1 H 1 1 − ϖ r1 H 1 2N r1 , log 2 1 + κ r2 H 2 1 − ϖ r2 H 2 2N r2 ,(42)\nin which\nH 1 = Nr 1 nr 1 =1 1 n r1\n, and\nH 2 = Nr 2 nr 2 =1 1 n r2 ,(43)\nwhere κ ri = E [κ ri ] = γ ri + ζ ri defines the average of κ ri .\n\nProof.In order to derive the exact EC, we need to insert the PDFs from ( 12) and ( 18) into (39) and evaluate the integral.However, due to the complexity of f γ fama r i (r) and f κ fama r i (r) that include the joint PDF of multivariate normal distribution, it is mathematically intractable to solve the provided integrals in closed form.For this purpose, we apply Jensen's inequality into (12) to derive a tight approximation of the EC, i.e.,\nC ≈ log 2 (1 + E [η]) .(44)\nTherefore, we need to determine the expectation of γ fama ri and κ fama ri , which involves the expectation of the maximum of N ri correlated RVs.To do so, we introduce a heuristic approach such that we first derive the expectation of the maximum of N ri independent RVs and then it is extended to the correlated scenario by adding a heuristic term based on the Gaussian copula [45], [46].As for γ ind ri , which includes N ri exponential RVs, the expected value for the independent case is given by\nE γ ind ri = γ ri H i = γ ri Nr i nr i =1 1 n ri ,(45)\nwhere H i is the N ri -th harmonic number.Now, when the exponential RVs become correlated, the expected maximum decreases because the dependence among the RVs reduces the variability of the maximum value.Therefore, this reduction in the expected maximum needs to be accounted for the correlation case.In our correlation model, for the Gaussian copula with the correlation matrix R ri , the parameter ϖ ri influences how the RVs are related.Therefore, given that the average correlation, e.g., ϖ ri , among the RVs has a diminishing effect on the maximum values, a heuristic approximation of the expectation γ fama ri for the correlated case can be given by\nE γ fama ri ≈ γ ri H i 1 − ϖ ri H i 2N ri .(46)\nThe term 1 − ϖ r i Hi 2Nr i is based on the statistical principle of the Gaussian copula and heuristic arguments that adjust the expected value of the maximum to account for the correlation among the RVs.In other words, the term ϖ r i Hi 2Nr i is an approximation that adjusts the expectation for the reduction caused by the average pairwise correlation ϖ ri among the RVs.\n\nFollowing the same approach, we first derive the expected value of κ fama ri that involves N ri independent hypoexponential RVs as\nE κ ind ri = κ ri H i = κ ri Nr i nr i =1 1 n ri ,(47)\nwhere κ ri = γ ri + ζ ri is the average of κ ri .Next, we adjust the harmonic number H i by the term ϖ r i Hi 2Nr i to account for the correlation among the correlated RVs.Thus, we have\nE κ fama ri ≈ κ ri H i 1 − ϖ ri H i 2N ri . (48)\nBy substituting the derived expectations from ( 46) and ( 48) to (44), we have ( 40)-( 42) and the proof is completed.\n\nRemark 1.It is noteworthy that the approximation provided for the maximum of correlated exponential and hypoexponential RVs using the Gaussian copula is a heuristic approach, which attempts to incorporate the effect of correlation in a manner analogous to methods used for normal distributions.Nevertheless, the accuracy of the proposed analytical approach needs to be evaluated.In this regard, the best way to validate this accuracy is through empirical simulations.\n\nBy generating correlated exponential and hypoexponential RVs using the Gaussian copula and comparing the simulated expectations of the maximum to the approximated values, we can gauge the accuracy of the analytical derivations.\n\nRemark 2. In Fig. 2, we compare the approximation and the empirical results of the expected value for γ fama ri and κ fama ri , respectively.It can be seen that the proposed approximations are perfectly matched with the empirical expectations.Furthermore, we can observe that the expected value of the maximum value of N ri exponential and hypoexponential RVs, i.e., E γ fama ri and E κ fama ri , respectively, increases with N ri when considering the harmonic number\nH i = Nr i nr i =1 1 nr i\n. This is because as N ri increases, the harmonic number H i grows, leading to a higher expected maximum.Corollary 3. The asymptotic EC in the high SNR regime for the FAMA-IC system under the strong interference scenario is given by\nC ∞ 1 ≈ log 2 γ r1 H 1 1 − ϖ r1 H 1 2N r1 , (49) C ∞ 2 ≈ log 2 γ r2 H 2 1 − ϖ r2 H 2 2N r2 ,(50)\nand where H i has been defined in (43).\nC ∞ sum ≈ min log 2 κ r1 H 1 1 − ϖ r1 H 1 2N r1 , log 2 κ r2 H 2 1 − ϖ r2 H 2 2N r2 ,(51)\nProof.In the high SNR regime, i.e., η ≫ 1, in (39), we have log 2 (1 + η) ≈ log 2 (η).Now, by directly applying this approximation into (40)-( 42), the proof is accomplished.\n\n\nV. NUMERICAL RESULTS\n\nIn this section, we evaluate the analytical derivations, which are double-checked in all instances by Monte Carlo simulation.Figs. 3 and 4 show the performance of OP and DOR against the average SNR γ ri for different values of fluid antenna size, W ri , and number of fluid antenna ports, N ri .First, we see that the asymptotic results closely match the numerical results at high SNRs.Moreover, assuming a strong interference scenario, i.e., ζ ri > γ rι , as γ ri grows, the OP and DOR for both FAS and TAS decrease which is expected since the channel condition and the quality of the desired signal improve.It is also evident that the performance of OP and DOR enhances as W ri and N ri increase.This improvement is mainly because increasing the fluid antenna size increases the spatial separation between the ports, and therefore, reduces the spatial correlation.Although increasing the number of ports leads to stronger spatial correlation, it can potentially enhance the channel capacity, diversity gain, and spatial multiplexing.In other words, increasing N ri while keeping W ri fixed causes the ports to become too close to one another, which leads to dominating spatial correlation effects.As a result, the diversity gain diminishes beyond a certain point, and the reduction in OP and DOR slows down, eventually reaching saturation.Thus, by simultaneously increasing W ri and N ri , the spatial correlation between the fluid antenna ports becomes balanced; consequently, a lower OP and DOR is provided.Furthermore, it can be seen that deploying a fluid antenna with only one activated port at users obtains significantly lower OP and DOR compared to TAS in the considered FAMA-IC with strong interference scenario.For instance, for a fixed average SNR γ r1 = 10 dB, the OP for user r i under FAMA is in the order of 10 −4 , while it is around 10 −1 under TAS.Moreover, by setting R i = 1 Kbits, B i = 1 MHz, T th i = 1 ms, and γ ri , we can observe that the DOR for user r i under FAMA is around 10 −2 , whereas it is near to 1 for the TAS counterpart.\n\nAdditionally, a comparison of the results for FAMA-IC and the benchmark FAMA reveals that FAMA-IC outperforms FAMA in terms of both OP and DOR under conditions of strong interference because it leverages fluid antenna diversity to optimize the SINR while employing advanced decoding techniques, such as SND, to mitigate interference.This approach enables the system to maintain reliable data rates by utilizing the structure of the strong interference instead of being fully degraded by it, which would occur if interference were treated as noise.As a result, the increased channel capacity achievable under strong interference reduces the likelihood of falling below the threshold rate, making strong interference advantageous for minimizing OP in FAMA systems.\n\nAssuming R i = R and B i = B, Fig. 5 investigates the impact of increasing the data rate R on the DOR performance under FAMA and TAS deployments in the strong IC.While transmitting a larger amount of data within a fixed bandwidth B is challenging for both FAMA and TAS, it becomes much more feasible when using FAS.For example, sending R = 1.5 Kbits over a strong IC with B = 1 MHz is nearly impossible with TAS, but it can be achieved with an acceptable DOR under FAMA.The impact of bandwidth B on the DOR for selected values of W ri and N ri is presented in Fig. 6.It can be seen that increasing B improves the DOR performance for both FAMA and TAS in strong IC, which means that the same amount of data can be transmitted with less delay as the IC bandwidth increases.However, it can be observed that this improvement is much more remarkable under the FAMA deployment than the TAS, thanks to the position reconfigurability, enhanced spatial diversity, and precise interference management provided by FAS.For instance, with a predefined bandwidth of B = 1 MHz, sending 1 Kbits of data results in a DOR close to 10 −5 for FAMA and nearly 1 for TAS.Fig. 7 studies the performance of the sum EC as a function of γ ri for selected values of W ri and N ri .It is evident that the asymptotic results closely align with the analytical derivations at high SNR, confirming the reliability of the theoretical model to predict the system performance.As expected, increasing W ri and N ri leads to a significant improvement in the sum EC.Even under the challenging strong IC, FAMA outperforms TAS, with the latter showing lower capacity across all SNR values.This highlights the advantage of using FAS with only one activated port and even with smaller sizes, over a fixed single-antenna in interference-dominated environments.However, in strong IC, the EC is constrained by the interference power relative to the signal power.As W ri and N ri increase, the system can initially mitigate interference more effectively by leveraging spatial diversity.Nevertheless, beyond a certain point, the interference remains dominant, and the system capacity cannot improve further, leading to saturation.\n\n\nVI. CONCLUSION\n\nThis paper has provided a comprehensive study of FAMA for two-user strong IC, with each mobile user equipped with a FAS.By applying the realistic SND interference management technique, analytical expressions for the CDF and PDF of the SNR and several key performance metrics, including the OP, DOR, and EC were derived.The results demonstrated significant advantages of FAS, showcasing its ability to improve system performance even with only one activated port under strong interference conditions.These findings contribute to more realistic interference management models in future wireless communication systems and offer valuable insights for optimizing FAMA in practical deployments.\n\nFig. 1 .\n1\nFig. 1.System model for a two-user FAMA-IC.\n\n\nnr i tiri 2 ∼\n2\nexp(1).Hence, the PDF and CDF for γ ňr i ri and ζ ňr i ri are, respectively, expressed as\n\n\n)\n\nin which F κ fama r iRthsum represents the complementary CDF (CCDF) of κ fama ri ,\n\n\n\n\nregime, provided in (27) and (28), respectively.\n\n\nFig. 2 .\n2\nFig. 2. The expected values of RVs (a) γ fama r i and (b) κ fama r i .\n\n\nFig. 3 .\n3\nFig. 3. OP versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, and ζ r 1 = ζ r 2 = γ rι + 20 dB.\n\n\nFig. 4 .\n4\nFig. 4. DOR versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R 1 = R 2 = 1 Kbits, B 1 = B 2 = 1 MHz, T th 1 = T th 2 = 1 ms, and ζ r 1 = ζ r 2 = γ rι + 20 dB.\n\n\nFig. 5 .Fig. 6 .\n56\nFig. 5. DOR versus rate R for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when B 1 = B 2 = 1 MHz, γ r i = 10 dB, and T th 1 = T th 2 = 1 ms.\n\n\nFig. 7 .\n7\nFig. 7. EC versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, ζ r 1 = ζ r 2 = γ rι + 20 dB.\n\nThis is particularly useful in scenarios in which users experience similar channel strengths or where interference is not suited to be effectively canceled by conventional methods such as successive interference cancellation (SIC). Unlike SIC, which decodes the strongest signal first and then cancels it out before decoding weaker ones, SND decodes multiple signals concurrently. This approach allows for more efficient use of the spectrum and expands the capacity region, especially in high-interference scenarios.\nConventional sequential decoding schemes such as SIC typically aim to find a single best solution for decoding the desired signal at the receiver, especially when the users have different channel strengths. 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P Hennig, arXiv:0910.01152009arXiv preprint\n"},"corpus_id":{"kind":"number","value":273654288,"string":"273,654,288"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2410.20930\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/corr/abs-2410-20930\",\n \"doi\": \"10.48550/arxiv.2410.20930\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2410.20930v1.pdf\"\n ],\n \"pdf_sha\": \"35efa64f1c804b92e5c64e7c1ce3fe9b1c4b814a\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Fluid Antenna Multiple Access with Simultaneous Non-unique Decoding in Strong Interference Channel"},"authors":{"kind":"list like","value":["Member, IEEEFarshad Rostami Ghadi f.rostamighadi@ucl.ac.uk","Fellow, IEEEKai-Kit K Wong kai-kit.wong@ucl.ac.uk","Masoud Kaveh masoud.kaveh@aalto.fi","Member, IEEEH Xu","Member, IEEEW K New","Senior Member, IEEEF Javier López-Martínez","Fellow, IEEEHyundong Shin hshin@khu.ac.kr","De-","F J López-Martínez","partment of Electronic and Electrical Engineering\nUniversity College London\nLondonUK","Department of Electronic Engineering\nKyung Hee University\n17104Yongin-si, GyeonggiKorea","Department of Information and Communication Engi-neering\nAalto University\nEspooFinland","Department of Signal Theory\nResearch Centre for Information and Communication Technologies (CITIC-UGR)\nNetworking and Communications\nUniversity of Granada\n18071GranadaSpain","Department of Electronic Engineering\nKyung Hee Uni-versity\n17104Yongin-si, GyeonggiKorea"],"string":"[\n \"Member, IEEEFarshad Rostami Ghadi f.rostamighadi@ucl.ac.uk\",\n \"Fellow, IEEEKai-Kit K Wong kai-kit.wong@ucl.ac.uk\",\n \"Masoud Kaveh masoud.kaveh@aalto.fi\",\n \"Member, IEEEH Xu\",\n \"Member, IEEEW K New\",\n \"Senior Member, IEEEF Javier López-Martínez\",\n \"Fellow, IEEEHyundong Shin hshin@khu.ac.kr\",\n \"De-\",\n \"F J López-Martínez\",\n \"partment of Electronic and Electrical Engineering\\nUniversity College London\\nLondonUK\",\n \"Department of Electronic Engineering\\nKyung Hee University\\n17104Yongin-si, GyeonggiKorea\",\n \"Department of Information and Communication Engi-neering\\nAalto University\\nEspooFinland\",\n \"Department of Signal Theory\\nResearch Centre for Information and Communication Technologies (CITIC-UGR)\\nNetworking and Communications\\nUniversity of Granada\\n18071GranadaSpain\",\n \"Department of Electronic Engineering\\nKyung Hee Uni-versity\\n17104Yongin-si, GyeonggiKorea\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Fluid antenna system (FAS) is gaining attention as an innovative technology for boosting diversity and multiplexing gains.As a key innovation, it presents the possibility to overcome interference by position reconfigurability on one radio frequency (RF) chain, giving rise to the concept of fluid antenna multiple access (FAMA).While FAMA is originally designed to deal with interference mainly by position change and treat interference as noise, this is not rate optimal, especially when suffering from a strong interference channel (IC) where all positions have strong interference.To tackle this, this paper considers a two-user strong IC where FAMA is used in conjunction with simultaneous nonunique decoding (SND).Specifically, we analyze the key statistics for the signal-to-noise ratio (SNR) and interference-to-noise ratio (INR) for a canonical two-user IC setup, and subsequently derive the delay outage rate (DOR), outage probability (OP) and ergodic capacity (EC) of the FAMA-IC.Our numerical results illustrate huge benefits of FAMA with SND over traditional fixed-position antenna systems (TAS) with SND in the fading IC."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION A. Context and Literature Review","content":"I N THE RAPIDLY-EVOLVING landscape of wireless com- munication, the emergence of fluid antenna systems (FAS) represents a transformative leap forward [1], [2], [3].As a new form of reconfigurable antennas, FAS emphasizes on position and shape flexibility in antennas for wireless communications.FAS can be implemented by liquid-based antennas [4], radiofrequency reconfigurable pixels [5], [6], metamaterials [7], [8], mechanically movable antennas [9] and etc.An easy way to understand the benefits of FAS is that it presents a new degree of freedom (DoF) to the physical layer that can be translated into diversity [10] and multiplexing gains [11].In recent years, efforts on FAS have been spent onto improving the accuracy of channel model [12], [13], and studying the diversity order in different fading channels such as Rayleigh [14], Nakagami [15], α-µ distributed [16] and even arbitrarily distributed [17].\nCopulas were shown to be useful in the performance analysis of FAS [18].Multiple fluid antennas at both ends were recently considered in [19], revealing extraordinary diversity gains.\nOne interesting application for FAS is multiuser communications.In particular, a unique way of using FAS is to mitigate inter-user interference by seeking the antenna position where the interference suffers from deep fades, leading to the concept of fluid antenna multiple access (FAMA).By doing so, each receiver handles its interference by position flexibility of FAS without the requirement of precoding at the base station (BS) transmitter.The main advantage is a reduced burden at the BS for scalability in terms of both channel state information (CSI) estimation and optimization.Depending on how fast the FAS position (a.k.a.'port') changes, FAMA can be classified into being fast [20] and slow [21], [22].Opportunistic scheduling can also combine with FAMA for improved performance [23], [24].In [25], a mean-field game was formulated to empower slow FAMA for energy-efficiency maximization.\nEvidently, FAS can also be adopted as an additional DoF to elevate traditional beamforming approaches.For instance, [26] considered an uplink scenario where the BS used traditional beamforming but the users employed multiple fluid antennas for joint port selection and beamforming in order to maximize the capacity.Moreover, FAS has been considered for integrated sensing and communications in [27], [28], [29].Dirty multiple access channels with FAS were also studied in [30] when side information was assumed at the transmitters.The application of FAS has also extended to backscatter communications [31] and wireless power transfer [32].Clearly, CSI is crucial to the performance of FAS and hence recent efforts have addressed the channel estimation problem for FAS in [33], [34].Besides, machine learning has also been demonstrated to be effective in reducing the complexity for establishing the channels at the FAS ports [35], [36].Finally, it is worth mentioning the recent experimental results in [37], [38], validating the capability of FAS in wireless communications systems."},{"title":"B. Motivation and Contributions","content":"While FAMA is attractive in simplifying how interference can be handled, it is not rate optimal or capacity achieving.In [39], a two-user interference channel (IC) with FAS was studied and the Han-Kobayashi's achievable rate was investigated.It was revealed that FAMA without power splitting nor rate splitting could achieve a similar sum-rate to that with, if the size and resolution of FAS were sufficient.In other words, if FAS can find a port that has weak interference, then it will be optimal to treat interference as noise [40].However, this is not the case when facing a strong IC in which FAS would be unable to find any port that has weak interference and therefore treating interference as noise is far from optimal.\nMotivated by the above, in this paper, we aim to investigate the performance of a canonical IC where two transmitters wish to communicate to their respective receivers over a shared IC, and each of the receivers is equipped with FAS.Specifically, our interest is on the strong IC scenarios where joint decoding of both messages at both receivers is capacity optimal [40].For this reason, we consider the use of simultaneous non-unique decoding (SND) [41].This scheme enables receivers to decode multiple signals simultaneously, even when there is no clear separation between the desired signal and the interference. 1By integrating SND into the FAMA framework, we can enhance the system's ability to overcome interference, enabling each receiver to retrieve both its own message and those from the interfering transmitters without additional rate constraints.\nOur key contributions are summarized as follows:\n• We formalize a definition of the two-user FAMA setup in the strong interference regime, and define its instantaneous capacity region.In this scenario, we derive the probability density function (PDF) and cumulative distribution function (CDF) of the signal-to-noise ratio (SNR) and interference-to-noise ratio (INR) at the receivers.• Additionally, we derive closed-form expressions for the probabilistic measures such as outage probability (OP) and delay outage rate (DOR), in terms of the CDF of the multivariate normal distribution.We also provide asymptotic expressions in the high SNR regime, considering the SND interference management technique.• We then derive an analytical expression for the ergodic capacity (EC) of the FAMA-IC with strong interference.This is obtained by approximating the expected value of the equivalent SNRs, i.e., the maximum of N correlated random variables (RVs), using a heuristic approach.• Finally, we conduct numerical evaluations to assess how the implementation of FAS influences the performance of FAMA-IC under strong interference scenarios, highlighting its performance gain over the traditional fixedposition antenna system (TAS) counterpart."},{"title":"C. Organization","content":"The remainder of this manuscript is structured as follows: Section II introduces the system model of FAMA-IC.The statistical characteristics of the equivalent channel, including the CDF and PDF, are then derived in Section III.Section IV presents the analytical results of key performance metrics such as OP, DOR, and EC.Section V provides the numerical results, and finally, Section VI concludes the paper."},{"title":"II. SYSTEM MODEL","content":"Consider a setup as shown in Fig. 1, where each transmitter t i , i ∈ {1, 2}, sends inputs x i with transmit power P i to its respective receiver r i over a shared IC.It is assumed that the transmitters t i are equipped with a traditional fixed antenna while receivers r i take advantage of a 2D planar fluid antenna that includes N ri preset ports which are uniformly distributed over an area of W ri .To be more precise, we assume a grid structure for the fluid antenna so that N ri,l ports are uniformly distributed along a linear space of length λW ri,l for l ∈ {1, 2}, where λ denotes the wavelength of the carrier frequency, i.e., N ri = N ri,1 ×N ri,2 and W ri = λW ri,1 ×λW ri,2 .Moreover, in order to convert the 2D indices to the 1D index for each port (i.e., the (n ri,1 , n ri,2 )-th port), we introduce an appropriate one-to-one mapping function as F (n ri ) = (n ri,1 , n ri,2 ) and F −1 (n ri,1 , n ri,2 ) = n ri in which n ri ∈ {1, . . ., N ri } and n ri,l ∈ {1, . . ., N ri,l }.Also, given that the fluid antenna ports can be arbitrarily close to each other, the corresponding channels are spatially correlated.Therefore, in a 3D isotropic scattering environment, the spatial correlation between the n rith port and the ñri -th port is given by [19]\nwhere J 0 (•) is the spherical Bessel function of the first kind, and F (ñ ri ) = (ñ ri,1 , ñri,2 ) and F −1 (ñ ri,1 , ñri,2 ) = ñri where ñri ∈ {1, . . ., N ri } and ñri,l ∈ {1, . . ., N ri,l }.Hence, the spatial correlation matrix R ri is defined as\nUnlike TAS, the radiating element of each FAS receiver r i can switch its location among N ri ports.To facilitate the notations, we define ι ∈ {1, 2} as the complement of subscript i, so that if i = 1, then ι = 2 and vice versa.Therefore, the received signal at the n ri -th port of receiver r i is given by\nwhere z nr i ri denotes the additive white Gaussian noise (AWGN) with zero mean and variance of σ2 i at the n ri -th port of r i , and L j,k is the path loss experienced by the message from transmitter j arriving at receiver k, for j, k ∈ {i, ι}.The term h nr i ti,ri defines the corresponding normalized Rayleigh fading channel from transmitter t i to receiver r i , i.e., E |h\nSpecifically, we consider the scenario where each receiver r i within the FAMA-IC is physically closer to the interfering transmitter t ι than to its own transmitter t i .This can be the case, for instance, where users are located in the expanded region of a pico-cell in the presence of a macro-cell [42].As a result, the signal received from the interfering transmitter t ι surpasses that from its respective transmitter t i in signal strength.Under such strong interference conditions, employing SND yields superior results compared to treating interference as noise. 2 By decoding multiple signals simultaneously using approaches such as multiuser detection or joint detection, SND can achieve higher spectral efficiency and offer a larger capacity region [40].Consequently, each receiver can effectively retrieve the message of the interfering transmitter without requiring an additional constraint on its rate.With all the above definitions, the capacity region for a general two-user IC is expressed as [40, Theorem 6.2]\nin which SNR i and INR i represent the SNR for the desired and interfering messages at user i, respectively.From these, the instantaneous capacity region for the considered strong interference scenario in the FAMA-IC can be defined as\nwhere\nThe terms γ ri and ζ ri denote the average SNR and the average INR at receiver r i , respectively.Note that ň ∈ {n, n}, F (n ri ) = (n ri,1 , nri,2 ) and F −1 (n ri,1 , nri,2 ) = nri , where nri ∈ {1, . . ., N ri } and nri,l ∈ {1, . . ., N ri,l }.That is to say, we consider that only the port which maximizes the received SNR at the receiver is activated."},{"title":"III. STATISTICAL CHARACTERIZATION","content":"To mathematically characterize the capacity region of the FAMA-IC in (5) under Rayleigh fading, we find it necessary to determine the distribution of the maximum amongst the N ri correlated SNRs, INRs, and the sum of SNRs and INRs.In this regard, we first note that all channels experience Rayleigh fading, so the corresponding channel gains are exponentially distributed, i.e., h\nand\nin which δ ∈ {γ, ζ}.Following a similar approach as in [18], the joint multivariate distribution for the maximum of d correlated arbitrary RVs can be accurately generated by using Sklar's theorem.More specifically, the joint CDF of d RVs S 1 , . . ., S d with the corresponding marginal CDF F S l (s l ), l ∈ {1, . . ., d}, in the extended real line domain R, is given by\nwhere\nis defined as a joint CDF of d RVs on the unit cube [0, 1] d with uniform marginal distributions.Besides, ϑ is the copula dependence parameter that measures the linear and/or nonlinear correlation between the underling RVs.In this regard, Gaussian copula is one of the practical models to describe the inherent spatial correlation between fluid antenna ports due to its appealing properties [18].Thus, by defining the maximum of γ\nthe CDF of δ fama ri can be found as [18]\nis the joint CDF of the multivariate normal distribution with zero mean vector and correlation matrix\nIn addition, ϖ ri defines the dependence parameter of the Gaussian copula.It is noted that by exploiting the Cholesky decomposition, we have ϖ ri nr i ,ñr i ≈ ϱ ri nr i ,ñr i [18].By applying the chain rule to (11), the PDF of δ fama ri is given by\nwhere f δ nr i r i (r) is given in (7), det (R ri ) is the determinant of the correlation matrix R ri , I is an identity matrix, and\nNow, we need to determine the distribution corresponding to the maximum of the sum of the SNR and INR, i.e., κ\nri , and we define\nGiven that κ nr i ri includes the sum of two independent exponential RVs with different scale parameters γ ri and ζ ri , it has the hypoexponential distribution with the following PDF and CDF [43]:\nand\nrespectively, where ∆ ri = γ ri − ζ ri .As such, using Sklar's theorem in (9) and the Gaussian copula definition, the CDF and PDF of κ fama ri are found as\nand\nin which F\n(r) are defined in ( 16) and ( 15), respectively.Moreover, φ −1 κr i is denoted as\nNote that the copula approach allows for a tractable characterization for the joint distribution of κ fama ri , from their underlying marginal distributions.Having characterized the key statistics of the equivalent channels under consideration, these will be used to analyze a number of performance metrics."},{"title":"IV. PERFORMANCE ANALYSIS A. OP","content":"OP is an important performance metric in evaluating wireless communication systems, and is defined as the probability that the capacity falls below a given rate threshold R th , i.e., P out = Pr (C ≤ R th ), where C = log 2 (1 + δ).Hence, for the considered FAMA-IC, an outage occurs if at least one of the following events happens:\nwhere R th i and R th sum = R th 1 + R th 2 are the rate threshold and the sum-rate threshold, respectively.Therefore, the OP for the considered FAMA-IC is defined as\nTheorem 1.The OP for the FAMA-IC system under the strong interference scenario is given by\nProof.The OP definition in (21) can be extended as\nwhere Pr (E 1 ) can be determined as\nwhere Rth\nRth 2 .As for the event E 3 , we have\nwhere (25b) is obtained from the independence of the events,\n. Since all events E j for j ∈ {1, 2, 3} are independent of each other, the other probabilities can be obtained according to Pr (E j ).Therefore, by considering the CDF of γ fama ri , ζ fama ri , and κ fama ri , and after some mathematical simplification, the proof is accomplished.\nFrom the derived closed-form expression for OP, we now aim to understand the asymptotic behavior of the OP at high SNR, i.e., γ ri → ∞ while meeting the strong interference condition.As we will later see, this asymptotic expression provides more insights into the system's reliability.\nCorollary 1.The asymptotic OP in the high SNR regime for the FAMA-IC system under the strong interference scenario is given by\nwhere\n,γ fama Proof.Given that γ nr i ri and κ nr i ri follow the exponential and hypoexponential distributions, respectively, their corresponding CDFs in the high SNR regime can be derived with the help of the Taylor series expansion.Therefore, as γ ri → ∞, the term e − Rth ν /γ r i for ν ∈ {1, 2, sum} becomes very small such that we have\nBy substituting ( 29) into ( 8) and ( 16), the marginal CDFs of γ nr i ri and κ nr i ri at high SNR are, respectively, given by\nand\nNow, by inserting ( 31) into (11) and plugging ( 30) into (17),\nRth 1 , Rth 2 are obtained and the proof becomes complete."},{"title":"B. DOR","content":"The DOR is a crucial metric in the design of various wireless communication scenarios, particularly in ultra-reliable and low-latency communication systems.It is defined as the probability that the time required to transmit a specific amount of data R over a wireless channel with bandwidth B exceeds a predefined threshold time T th [44].Mathematically, this is expressed as Pr (T d > T th ), where T d = R BC indicates the delivery time such that C denotes the channel capacity [31].This expression captures the essential relationship between data transmission time, channel bandwidth, and signal quality.\nThus, for the considered FAMA-IC, the DOR is achieved if at least one of the following events occurs:\nwhere C ν for ν ∈ {1, 2, sum} is defined in (5), and B ν and R i denote the corresponding bandwidth and rates, respectively.Therefore, the DOR for FAMA-IC is defined as\nTheorem 2. The OP for the FAMA-IC system under the strong interference scenario is given by\nin which Ti = e R i ln 2\nProof.By extending the definition of DOR in (33), we have\nwhere Pr (T 1 ) can be mathematically obtained as\nwhere\nMoreover, Pr (T 3 ) is derived as\n),log 2 (1+κ fama r 2\nTsum ,\nwhere Tsum = e 2(R 1 +R 2 ) ln 2 Bsum T th sum .Now, given the independence of all events T j for j ∈ {1, 2, 3}, other probabilities can be obtained according to Pr (T j ).Hence, by considering the CDFs of γ fama ri , ζ fama ri , and κ fama ri , and after some mathematical simplification, the proof is completed.\nCorollary 2. The asymptotic DOR in the high SNR regime for the FAMA-IC under the strong interference scenario is given by\nProof.Following the same approach used to prove Corollary 1, the proof is completed."},{"title":"C. EC","content":"The EC (per bandwidth unit) for the system model under consideration is defined as\nwhere η ∈ γ fama ri , κ fama ri and f η (η) denotes the corresponding PDF.Therefore, the EC for the considered system model is derived and presented in the following theorem.Theorem 3. The EC for the FAMA-IC system under the strong interference scenario is given by\nand\nin which\n, and\nwhere κ ri = E [κ ri ] = γ ri + ζ ri defines the average of κ ri .\nProof.In order to derive the exact EC, we need to insert the PDFs from ( 12) and ( 18) into (39) and evaluate the integral.However, due to the complexity of f γ fama r i (r) and f κ fama r i (r) that include the joint PDF of multivariate normal distribution, it is mathematically intractable to solve the provided integrals in closed form.For this purpose, we apply Jensen's inequality into (12) to derive a tight approximation of the EC, i.e.,\nTherefore, we need to determine the expectation of γ fama ri and κ fama ri , which involves the expectation of the maximum of N ri correlated RVs.To do so, we introduce a heuristic approach such that we first derive the expectation of the maximum of N ri independent RVs and then it is extended to the correlated scenario by adding a heuristic term based on the Gaussian copula [45], [46].As for γ ind ri , which includes N ri exponential RVs, the expected value for the independent case is given by\nwhere H i is the N ri -th harmonic number.Now, when the exponential RVs become correlated, the expected maximum decreases because the dependence among the RVs reduces the variability of the maximum value.Therefore, this reduction in the expected maximum needs to be accounted for the correlation case.In our correlation model, for the Gaussian copula with the correlation matrix R ri , the parameter ϖ ri influences how the RVs are related.Therefore, given that the average correlation, e.g., ϖ ri , among the RVs has a diminishing effect on the maximum values, a heuristic approximation of the expectation γ fama ri for the correlated case can be given by\nThe term 1 − ϖ r i Hi 2Nr i is based on the statistical principle of the Gaussian copula and heuristic arguments that adjust the expected value of the maximum to account for the correlation among the RVs.In other words, the term ϖ r i Hi 2Nr i is an approximation that adjusts the expectation for the reduction caused by the average pairwise correlation ϖ ri among the RVs.\nFollowing the same approach, we first derive the expected value of κ fama ri that involves N ri independent hypoexponential RVs as\nwhere κ ri = γ ri + ζ ri is the average of κ ri .Next, we adjust the harmonic number H i by the term ϖ r i Hi 2Nr i to account for the correlation among the correlated RVs.Thus, we have\nBy substituting the derived expectations from ( 46) and ( 48) to (44), we have ( 40)-( 42) and the proof is completed.\nRemark 1.It is noteworthy that the approximation provided for the maximum of correlated exponential and hypoexponential RVs using the Gaussian copula is a heuristic approach, which attempts to incorporate the effect of correlation in a manner analogous to methods used for normal distributions.Nevertheless, the accuracy of the proposed analytical approach needs to be evaluated.In this regard, the best way to validate this accuracy is through empirical simulations.\nBy generating correlated exponential and hypoexponential RVs using the Gaussian copula and comparing the simulated expectations of the maximum to the approximated values, we can gauge the accuracy of the analytical derivations.\nRemark 2. In Fig. 2, we compare the approximation and the empirical results of the expected value for γ fama ri and κ fama ri , respectively.It can be seen that the proposed approximations are perfectly matched with the empirical expectations.Furthermore, we can observe that the expected value of the maximum value of N ri exponential and hypoexponential RVs, i.e., E γ fama ri and E κ fama ri , respectively, increases with N ri when considering the harmonic number\n. This is because as N ri increases, the harmonic number H i grows, leading to a higher expected maximum.Corollary 3. The asymptotic EC in the high SNR regime for the FAMA-IC system under the strong interference scenario is given by\nand where H i has been defined in (43).\nProof.In the high SNR regime, i.e., η ≫ 1, in (39), we have log 2 (1 + η) ≈ log 2 (η).Now, by directly applying this approximation into (40)-( 42), the proof is accomplished."},{"title":"V. NUMERICAL RESULTS","content":"In this section, we evaluate the analytical derivations, which are double-checked in all instances by Monte Carlo simulation.Figs. 3 and 4 show the performance of OP and DOR against the average SNR γ ri for different values of fluid antenna size, W ri , and number of fluid antenna ports, N ri .First, we see that the asymptotic results closely match the numerical results at high SNRs.Moreover, assuming a strong interference scenario, i.e., ζ ri > γ rι , as γ ri grows, the OP and DOR for both FAS and TAS decrease which is expected since the channel condition and the quality of the desired signal improve.It is also evident that the performance of OP and DOR enhances as W ri and N ri increase.This improvement is mainly because increasing the fluid antenna size increases the spatial separation between the ports, and therefore, reduces the spatial correlation.Although increasing the number of ports leads to stronger spatial correlation, it can potentially enhance the channel capacity, diversity gain, and spatial multiplexing.In other words, increasing N ri while keeping W ri fixed causes the ports to become too close to one another, which leads to dominating spatial correlation effects.As a result, the diversity gain diminishes beyond a certain point, and the reduction in OP and DOR slows down, eventually reaching saturation.Thus, by simultaneously increasing W ri and N ri , the spatial correlation between the fluid antenna ports becomes balanced; consequently, a lower OP and DOR is provided.Furthermore, it can be seen that deploying a fluid antenna with only one activated port at users obtains significantly lower OP and DOR compared to TAS in the considered FAMA-IC with strong interference scenario.For instance, for a fixed average SNR γ r1 = 10 dB, the OP for user r i under FAMA is in the order of 10 −4 , while it is around 10 −1 under TAS.Moreover, by setting R i = 1 Kbits, B i = 1 MHz, T th i = 1 ms, and γ ri , we can observe that the DOR for user r i under FAMA is around 10 −2 , whereas it is near to 1 for the TAS counterpart.\nAdditionally, a comparison of the results for FAMA-IC and the benchmark FAMA reveals that FAMA-IC outperforms FAMA in terms of both OP and DOR under conditions of strong interference because it leverages fluid antenna diversity to optimize the SINR while employing advanced decoding techniques, such as SND, to mitigate interference.This approach enables the system to maintain reliable data rates by utilizing the structure of the strong interference instead of being fully degraded by it, which would occur if interference were treated as noise.As a result, the increased channel capacity achievable under strong interference reduces the likelihood of falling below the threshold rate, making strong interference advantageous for minimizing OP in FAMA systems.\nAssuming R i = R and B i = B, Fig. 5 investigates the impact of increasing the data rate R on the DOR performance under FAMA and TAS deployments in the strong IC.While transmitting a larger amount of data within a fixed bandwidth B is challenging for both FAMA and TAS, it becomes much more feasible when using FAS.For example, sending R = 1.5 Kbits over a strong IC with B = 1 MHz is nearly impossible with TAS, but it can be achieved with an acceptable DOR under FAMA.The impact of bandwidth B on the DOR for selected values of W ri and N ri is presented in Fig. 6.It can be seen that increasing B improves the DOR performance for both FAMA and TAS in strong IC, which means that the same amount of data can be transmitted with less delay as the IC bandwidth increases.However, it can be observed that this improvement is much more remarkable under the FAMA deployment than the TAS, thanks to the position reconfigurability, enhanced spatial diversity, and precise interference management provided by FAS.For instance, with a predefined bandwidth of B = 1 MHz, sending 1 Kbits of data results in a DOR close to 10 −5 for FAMA and nearly 1 for TAS.Fig. 7 studies the performance of the sum EC as a function of γ ri for selected values of W ri and N ri .It is evident that the asymptotic results closely align with the analytical derivations at high SNR, confirming the reliability of the theoretical model to predict the system performance.As expected, increasing W ri and N ri leads to a significant improvement in the sum EC.Even under the challenging strong IC, FAMA outperforms TAS, with the latter showing lower capacity across all SNR values.This highlights the advantage of using FAS with only one activated port and even with smaller sizes, over a fixed single-antenna in interference-dominated environments.However, in strong IC, the EC is constrained by the interference power relative to the signal power.As W ri and N ri increase, the system can initially mitigate interference more effectively by leveraging spatial diversity.Nevertheless, beyond a certain point, the interference remains dominant, and the system capacity cannot improve further, leading to saturation."},{"title":"VI. CONCLUSION","content":"This paper has provided a comprehensive study of FAMA for two-user strong IC, with each mobile user equipped with a FAS.By applying the realistic SND interference management technique, analytical expressions for the CDF and PDF of the SNR and several key performance metrics, including the OP, DOR, and EC were derived.The results demonstrated significant advantages of FAS, showcasing its ability to improve system performance even with only one activated port under strong interference conditions.These findings contribute to more realistic interference management models in future wireless communication systems and offer valuable insights for optimizing FAMA in practical deployments."},{"title":"Fig. 1 .","content":"Fig. 1.System model for a two-user FAMA-IC."},{"title":"nr i tiri 2 ∼","content":"exp(1).Hence, the PDF and CDF for γ ňr i ri and ζ ňr i ri are, respectively, expressed as"},{"title":")","content":"in which F κ fama r iRthsum represents the complementary CDF (CCDF) of κ fama ri ,\nregime, provided in (27) and (28), respectively."},{"title":"Fig. 2 .","content":"Fig. 2. The expected values of RVs (a) γ fama r i and (b) κ fama r i ."},{"title":"Fig. 3 .","content":"Fig. 3. OP versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, and ζ r 1 = ζ r 2 = γ rι + 20 dB."},{"title":"Fig. 4 .","content":"Fig. 4. DOR versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R 1 = R 2 = 1 Kbits, B 1 = B 2 = 1 MHz, T th 1 = T th 2 = 1 ms, and ζ r 1 = ζ r 2 = γ rι + 20 dB."},{"title":"Fig. 5 .Fig. 6 .","content":"Fig. 5. DOR versus rate R for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when B 1 = B 2 = 1 MHz, γ r i = 10 dB, and T th 1 = T th 2 = 1 ms."},{"title":"Fig. 7 .","content":"Fig. 7. EC versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, ζ r 1 = ζ r 2 = γ rι + 20 dB."}],"string":"[\n {\n \"title\": \"I. INTRODUCTION A. Context and Literature Review\",\n \"content\": \"I N THE RAPIDLY-EVOLVING landscape of wireless com- munication, the emergence of fluid antenna systems (FAS) represents a transformative leap forward [1], [2], [3].As a new form of reconfigurable antennas, FAS emphasizes on position and shape flexibility in antennas for wireless communications.FAS can be implemented by liquid-based antennas [4], radiofrequency reconfigurable pixels [5], [6], metamaterials [7], [8], mechanically movable antennas [9] and etc.An easy way to understand the benefits of FAS is that it presents a new degree of freedom (DoF) to the physical layer that can be translated into diversity [10] and multiplexing gains [11].In recent years, efforts on FAS have been spent onto improving the accuracy of channel model [12], [13], and studying the diversity order in different fading channels such as Rayleigh [14], Nakagami [15], α-µ distributed [16] and even arbitrarily distributed [17].\\nCopulas were shown to be useful in the performance analysis of FAS [18].Multiple fluid antennas at both ends were recently considered in [19], revealing extraordinary diversity gains.\\nOne interesting application for FAS is multiuser communications.In particular, a unique way of using FAS is to mitigate inter-user interference by seeking the antenna position where the interference suffers from deep fades, leading to the concept of fluid antenna multiple access (FAMA).By doing so, each receiver handles its interference by position flexibility of FAS without the requirement of precoding at the base station (BS) transmitter.The main advantage is a reduced burden at the BS for scalability in terms of both channel state information (CSI) estimation and optimization.Depending on how fast the FAS position (a.k.a.'port') changes, FAMA can be classified into being fast [20] and slow [21], [22].Opportunistic scheduling can also combine with FAMA for improved performance [23], [24].In [25], a mean-field game was formulated to empower slow FAMA for energy-efficiency maximization.\\nEvidently, FAS can also be adopted as an additional DoF to elevate traditional beamforming approaches.For instance, [26] considered an uplink scenario where the BS used traditional beamforming but the users employed multiple fluid antennas for joint port selection and beamforming in order to maximize the capacity.Moreover, FAS has been considered for integrated sensing and communications in [27], [28], [29].Dirty multiple access channels with FAS were also studied in [30] when side information was assumed at the transmitters.The application of FAS has also extended to backscatter communications [31] and wireless power transfer [32].Clearly, CSI is crucial to the performance of FAS and hence recent efforts have addressed the channel estimation problem for FAS in [33], [34].Besides, machine learning has also been demonstrated to be effective in reducing the complexity for establishing the channels at the FAS ports [35], [36].Finally, it is worth mentioning the recent experimental results in [37], [38], validating the capability of FAS in wireless communications systems.\"\n },\n {\n \"title\": \"B. Motivation and Contributions\",\n \"content\": \"While FAMA is attractive in simplifying how interference can be handled, it is not rate optimal or capacity achieving.In [39], a two-user interference channel (IC) with FAS was studied and the Han-Kobayashi's achievable rate was investigated.It was revealed that FAMA without power splitting nor rate splitting could achieve a similar sum-rate to that with, if the size and resolution of FAS were sufficient.In other words, if FAS can find a port that has weak interference, then it will be optimal to treat interference as noise [40].However, this is not the case when facing a strong IC in which FAS would be unable to find any port that has weak interference and therefore treating interference as noise is far from optimal.\\nMotivated by the above, in this paper, we aim to investigate the performance of a canonical IC where two transmitters wish to communicate to their respective receivers over a shared IC, and each of the receivers is equipped with FAS.Specifically, our interest is on the strong IC scenarios where joint decoding of both messages at both receivers is capacity optimal [40].For this reason, we consider the use of simultaneous non-unique decoding (SND) [41].This scheme enables receivers to decode multiple signals simultaneously, even when there is no clear separation between the desired signal and the interference. 1By integrating SND into the FAMA framework, we can enhance the system's ability to overcome interference, enabling each receiver to retrieve both its own message and those from the interfering transmitters without additional rate constraints.\\nOur key contributions are summarized as follows:\\n• We formalize a definition of the two-user FAMA setup in the strong interference regime, and define its instantaneous capacity region.In this scenario, we derive the probability density function (PDF) and cumulative distribution function (CDF) of the signal-to-noise ratio (SNR) and interference-to-noise ratio (INR) at the receivers.• Additionally, we derive closed-form expressions for the probabilistic measures such as outage probability (OP) and delay outage rate (DOR), in terms of the CDF of the multivariate normal distribution.We also provide asymptotic expressions in the high SNR regime, considering the SND interference management technique.• We then derive an analytical expression for the ergodic capacity (EC) of the FAMA-IC with strong interference.This is obtained by approximating the expected value of the equivalent SNRs, i.e., the maximum of N correlated random variables (RVs), using a heuristic approach.• Finally, we conduct numerical evaluations to assess how the implementation of FAS influences the performance of FAMA-IC under strong interference scenarios, highlighting its performance gain over the traditional fixedposition antenna system (TAS) counterpart.\"\n },\n {\n \"title\": \"C. Organization\",\n \"content\": \"The remainder of this manuscript is structured as follows: Section II introduces the system model of FAMA-IC.The statistical characteristics of the equivalent channel, including the CDF and PDF, are then derived in Section III.Section IV presents the analytical results of key performance metrics such as OP, DOR, and EC.Section V provides the numerical results, and finally, Section VI concludes the paper.\"\n },\n {\n \"title\": \"II. SYSTEM MODEL\",\n \"content\": \"Consider a setup as shown in Fig. 1, where each transmitter t i , i ∈ {1, 2}, sends inputs x i with transmit power P i to its respective receiver r i over a shared IC.It is assumed that the transmitters t i are equipped with a traditional fixed antenna while receivers r i take advantage of a 2D planar fluid antenna that includes N ri preset ports which are uniformly distributed over an area of W ri .To be more precise, we assume a grid structure for the fluid antenna so that N ri,l ports are uniformly distributed along a linear space of length λW ri,l for l ∈ {1, 2}, where λ denotes the wavelength of the carrier frequency, i.e., N ri = N ri,1 ×N ri,2 and W ri = λW ri,1 ×λW ri,2 .Moreover, in order to convert the 2D indices to the 1D index for each port (i.e., the (n ri,1 , n ri,2 )-th port), we introduce an appropriate one-to-one mapping function as F (n ri ) = (n ri,1 , n ri,2 ) and F −1 (n ri,1 , n ri,2 ) = n ri in which n ri ∈ {1, . . ., N ri } and n ri,l ∈ {1, . . ., N ri,l }.Also, given that the fluid antenna ports can be arbitrarily close to each other, the corresponding channels are spatially correlated.Therefore, in a 3D isotropic scattering environment, the spatial correlation between the n rith port and the ñri -th port is given by [19]\\nwhere J 0 (•) is the spherical Bessel function of the first kind, and F (ñ ri ) = (ñ ri,1 , ñri,2 ) and F −1 (ñ ri,1 , ñri,2 ) = ñri where ñri ∈ {1, . . ., N ri } and ñri,l ∈ {1, . . ., N ri,l }.Hence, the spatial correlation matrix R ri is defined as\\nUnlike TAS, the radiating element of each FAS receiver r i can switch its location among N ri ports.To facilitate the notations, we define ι ∈ {1, 2} as the complement of subscript i, so that if i = 1, then ι = 2 and vice versa.Therefore, the received signal at the n ri -th port of receiver r i is given by\\nwhere z nr i ri denotes the additive white Gaussian noise (AWGN) with zero mean and variance of σ2 i at the n ri -th port of r i , and L j,k is the path loss experienced by the message from transmitter j arriving at receiver k, for j, k ∈ {i, ι}.The term h nr i ti,ri defines the corresponding normalized Rayleigh fading channel from transmitter t i to receiver r i , i.e., E |h\\nSpecifically, we consider the scenario where each receiver r i within the FAMA-IC is physically closer to the interfering transmitter t ι than to its own transmitter t i .This can be the case, for instance, where users are located in the expanded region of a pico-cell in the presence of a macro-cell [42].As a result, the signal received from the interfering transmitter t ι surpasses that from its respective transmitter t i in signal strength.Under such strong interference conditions, employing SND yields superior results compared to treating interference as noise. 2 By decoding multiple signals simultaneously using approaches such as multiuser detection or joint detection, SND can achieve higher spectral efficiency and offer a larger capacity region [40].Consequently, each receiver can effectively retrieve the message of the interfering transmitter without requiring an additional constraint on its rate.With all the above definitions, the capacity region for a general two-user IC is expressed as [40, Theorem 6.2]\\nin which SNR i and INR i represent the SNR for the desired and interfering messages at user i, respectively.From these, the instantaneous capacity region for the considered strong interference scenario in the FAMA-IC can be defined as\\nwhere\\nThe terms γ ri and ζ ri denote the average SNR and the average INR at receiver r i , respectively.Note that ň ∈ {n, n}, F (n ri ) = (n ri,1 , nri,2 ) and F −1 (n ri,1 , nri,2 ) = nri , where nri ∈ {1, . . ., N ri } and nri,l ∈ {1, . . ., N ri,l }.That is to say, we consider that only the port which maximizes the received SNR at the receiver is activated.\"\n },\n {\n \"title\": \"III. STATISTICAL CHARACTERIZATION\",\n \"content\": \"To mathematically characterize the capacity region of the FAMA-IC in (5) under Rayleigh fading, we find it necessary to determine the distribution of the maximum amongst the N ri correlated SNRs, INRs, and the sum of SNRs and INRs.In this regard, we first note that all channels experience Rayleigh fading, so the corresponding channel gains are exponentially distributed, i.e., h\\nand\\nin which δ ∈ {γ, ζ}.Following a similar approach as in [18], the joint multivariate distribution for the maximum of d correlated arbitrary RVs can be accurately generated by using Sklar's theorem.More specifically, the joint CDF of d RVs S 1 , . . ., S d with the corresponding marginal CDF F S l (s l ), l ∈ {1, . . ., d}, in the extended real line domain R, is given by\\nwhere\\nis defined as a joint CDF of d RVs on the unit cube [0, 1] d with uniform marginal distributions.Besides, ϑ is the copula dependence parameter that measures the linear and/or nonlinear correlation between the underling RVs.In this regard, Gaussian copula is one of the practical models to describe the inherent spatial correlation between fluid antenna ports due to its appealing properties [18].Thus, by defining the maximum of γ\\nthe CDF of δ fama ri can be found as [18]\\nis the joint CDF of the multivariate normal distribution with zero mean vector and correlation matrix\\nIn addition, ϖ ri defines the dependence parameter of the Gaussian copula.It is noted that by exploiting the Cholesky decomposition, we have ϖ ri nr i ,ñr i ≈ ϱ ri nr i ,ñr i [18].By applying the chain rule to (11), the PDF of δ fama ri is given by\\nwhere f δ nr i r i (r) is given in (7), det (R ri ) is the determinant of the correlation matrix R ri , I is an identity matrix, and\\nNow, we need to determine the distribution corresponding to the maximum of the sum of the SNR and INR, i.e., κ\\nri , and we define\\nGiven that κ nr i ri includes the sum of two independent exponential RVs with different scale parameters γ ri and ζ ri , it has the hypoexponential distribution with the following PDF and CDF [43]:\\nand\\nrespectively, where ∆ ri = γ ri − ζ ri .As such, using Sklar's theorem in (9) and the Gaussian copula definition, the CDF and PDF of κ fama ri are found as\\nand\\nin which F\\n(r) are defined in ( 16) and ( 15), respectively.Moreover, φ −1 κr i is denoted as\\nNote that the copula approach allows for a tractable characterization for the joint distribution of κ fama ri , from their underlying marginal distributions.Having characterized the key statistics of the equivalent channels under consideration, these will be used to analyze a number of performance metrics.\"\n },\n {\n \"title\": \"IV. PERFORMANCE ANALYSIS A. OP\",\n \"content\": \"OP is an important performance metric in evaluating wireless communication systems, and is defined as the probability that the capacity falls below a given rate threshold R th , i.e., P out = Pr (C ≤ R th ), where C = log 2 (1 + δ).Hence, for the considered FAMA-IC, an outage occurs if at least one of the following events happens:\\nwhere R th i and R th sum = R th 1 + R th 2 are the rate threshold and the sum-rate threshold, respectively.Therefore, the OP for the considered FAMA-IC is defined as\\nTheorem 1.The OP for the FAMA-IC system under the strong interference scenario is given by\\nProof.The OP definition in (21) can be extended as\\nwhere Pr (E 1 ) can be determined as\\nwhere Rth\\nRth 2 .As for the event E 3 , we have\\nwhere (25b) is obtained from the independence of the events,\\n. Since all events E j for j ∈ {1, 2, 3} are independent of each other, the other probabilities can be obtained according to Pr (E j ).Therefore, by considering the CDF of γ fama ri , ζ fama ri , and κ fama ri , and after some mathematical simplification, the proof is accomplished.\\nFrom the derived closed-form expression for OP, we now aim to understand the asymptotic behavior of the OP at high SNR, i.e., γ ri → ∞ while meeting the strong interference condition.As we will later see, this asymptotic expression provides more insights into the system's reliability.\\nCorollary 1.The asymptotic OP in the high SNR regime for the FAMA-IC system under the strong interference scenario is given by\\nwhere\\n,γ fama Proof.Given that γ nr i ri and κ nr i ri follow the exponential and hypoexponential distributions, respectively, their corresponding CDFs in the high SNR regime can be derived with the help of the Taylor series expansion.Therefore, as γ ri → ∞, the term e − Rth ν /γ r i for ν ∈ {1, 2, sum} becomes very small such that we have\\nBy substituting ( 29) into ( 8) and ( 16), the marginal CDFs of γ nr i ri and κ nr i ri at high SNR are, respectively, given by\\nand\\nNow, by inserting ( 31) into (11) and plugging ( 30) into (17),\\nRth 1 , Rth 2 are obtained and the proof becomes complete.\"\n },\n {\n \"title\": \"B. DOR\",\n \"content\": \"The DOR is a crucial metric in the design of various wireless communication scenarios, particularly in ultra-reliable and low-latency communication systems.It is defined as the probability that the time required to transmit a specific amount of data R over a wireless channel with bandwidth B exceeds a predefined threshold time T th [44].Mathematically, this is expressed as Pr (T d > T th ), where T d = R BC indicates the delivery time such that C denotes the channel capacity [31].This expression captures the essential relationship between data transmission time, channel bandwidth, and signal quality.\\nThus, for the considered FAMA-IC, the DOR is achieved if at least one of the following events occurs:\\nwhere C ν for ν ∈ {1, 2, sum} is defined in (5), and B ν and R i denote the corresponding bandwidth and rates, respectively.Therefore, the DOR for FAMA-IC is defined as\\nTheorem 2. The OP for the FAMA-IC system under the strong interference scenario is given by\\nin which Ti = e R i ln 2\\nProof.By extending the definition of DOR in (33), we have\\nwhere Pr (T 1 ) can be mathematically obtained as\\nwhere\\nMoreover, Pr (T 3 ) is derived as\\n),log 2 (1+κ fama r 2\\nTsum ,\\nwhere Tsum = e 2(R 1 +R 2 ) ln 2 Bsum T th sum .Now, given the independence of all events T j for j ∈ {1, 2, 3}, other probabilities can be obtained according to Pr (T j ).Hence, by considering the CDFs of γ fama ri , ζ fama ri , and κ fama ri , and after some mathematical simplification, the proof is completed.\\nCorollary 2. The asymptotic DOR in the high SNR regime for the FAMA-IC under the strong interference scenario is given by\\nProof.Following the same approach used to prove Corollary 1, the proof is completed.\"\n },\n {\n \"title\": \"C. EC\",\n \"content\": \"The EC (per bandwidth unit) for the system model under consideration is defined as\\nwhere η ∈ γ fama ri , κ fama ri and f η (η) denotes the corresponding PDF.Therefore, the EC for the considered system model is derived and presented in the following theorem.Theorem 3. The EC for the FAMA-IC system under the strong interference scenario is given by\\nand\\nin which\\n, and\\nwhere κ ri = E [κ ri ] = γ ri + ζ ri defines the average of κ ri .\\nProof.In order to derive the exact EC, we need to insert the PDFs from ( 12) and ( 18) into (39) and evaluate the integral.However, due to the complexity of f γ fama r i (r) and f κ fama r i (r) that include the joint PDF of multivariate normal distribution, it is mathematically intractable to solve the provided integrals in closed form.For this purpose, we apply Jensen's inequality into (12) to derive a tight approximation of the EC, i.e.,\\nTherefore, we need to determine the expectation of γ fama ri and κ fama ri , which involves the expectation of the maximum of N ri correlated RVs.To do so, we introduce a heuristic approach such that we first derive the expectation of the maximum of N ri independent RVs and then it is extended to the correlated scenario by adding a heuristic term based on the Gaussian copula [45], [46].As for γ ind ri , which includes N ri exponential RVs, the expected value for the independent case is given by\\nwhere H i is the N ri -th harmonic number.Now, when the exponential RVs become correlated, the expected maximum decreases because the dependence among the RVs reduces the variability of the maximum value.Therefore, this reduction in the expected maximum needs to be accounted for the correlation case.In our correlation model, for the Gaussian copula with the correlation matrix R ri , the parameter ϖ ri influences how the RVs are related.Therefore, given that the average correlation, e.g., ϖ ri , among the RVs has a diminishing effect on the maximum values, a heuristic approximation of the expectation γ fama ri for the correlated case can be given by\\nThe term 1 − ϖ r i Hi 2Nr i is based on the statistical principle of the Gaussian copula and heuristic arguments that adjust the expected value of the maximum to account for the correlation among the RVs.In other words, the term ϖ r i Hi 2Nr i is an approximation that adjusts the expectation for the reduction caused by the average pairwise correlation ϖ ri among the RVs.\\nFollowing the same approach, we first derive the expected value of κ fama ri that involves N ri independent hypoexponential RVs as\\nwhere κ ri = γ ri + ζ ri is the average of κ ri .Next, we adjust the harmonic number H i by the term ϖ r i Hi 2Nr i to account for the correlation among the correlated RVs.Thus, we have\\nBy substituting the derived expectations from ( 46) and ( 48) to (44), we have ( 40)-( 42) and the proof is completed.\\nRemark 1.It is noteworthy that the approximation provided for the maximum of correlated exponential and hypoexponential RVs using the Gaussian copula is a heuristic approach, which attempts to incorporate the effect of correlation in a manner analogous to methods used for normal distributions.Nevertheless, the accuracy of the proposed analytical approach needs to be evaluated.In this regard, the best way to validate this accuracy is through empirical simulations.\\nBy generating correlated exponential and hypoexponential RVs using the Gaussian copula and comparing the simulated expectations of the maximum to the approximated values, we can gauge the accuracy of the analytical derivations.\\nRemark 2. In Fig. 2, we compare the approximation and the empirical results of the expected value for γ fama ri and κ fama ri , respectively.It can be seen that the proposed approximations are perfectly matched with the empirical expectations.Furthermore, we can observe that the expected value of the maximum value of N ri exponential and hypoexponential RVs, i.e., E γ fama ri and E κ fama ri , respectively, increases with N ri when considering the harmonic number\\n. This is because as N ri increases, the harmonic number H i grows, leading to a higher expected maximum.Corollary 3. The asymptotic EC in the high SNR regime for the FAMA-IC system under the strong interference scenario is given by\\nand where H i has been defined in (43).\\nProof.In the high SNR regime, i.e., η ≫ 1, in (39), we have log 2 (1 + η) ≈ log 2 (η).Now, by directly applying this approximation into (40)-( 42), the proof is accomplished.\"\n },\n {\n \"title\": \"V. NUMERICAL RESULTS\",\n \"content\": \"In this section, we evaluate the analytical derivations, which are double-checked in all instances by Monte Carlo simulation.Figs. 3 and 4 show the performance of OP and DOR against the average SNR γ ri for different values of fluid antenna size, W ri , and number of fluid antenna ports, N ri .First, we see that the asymptotic results closely match the numerical results at high SNRs.Moreover, assuming a strong interference scenario, i.e., ζ ri > γ rι , as γ ri grows, the OP and DOR for both FAS and TAS decrease which is expected since the channel condition and the quality of the desired signal improve.It is also evident that the performance of OP and DOR enhances as W ri and N ri increase.This improvement is mainly because increasing the fluid antenna size increases the spatial separation between the ports, and therefore, reduces the spatial correlation.Although increasing the number of ports leads to stronger spatial correlation, it can potentially enhance the channel capacity, diversity gain, and spatial multiplexing.In other words, increasing N ri while keeping W ri fixed causes the ports to become too close to one another, which leads to dominating spatial correlation effects.As a result, the diversity gain diminishes beyond a certain point, and the reduction in OP and DOR slows down, eventually reaching saturation.Thus, by simultaneously increasing W ri and N ri , the spatial correlation between the fluid antenna ports becomes balanced; consequently, a lower OP and DOR is provided.Furthermore, it can be seen that deploying a fluid antenna with only one activated port at users obtains significantly lower OP and DOR compared to TAS in the considered FAMA-IC with strong interference scenario.For instance, for a fixed average SNR γ r1 = 10 dB, the OP for user r i under FAMA is in the order of 10 −4 , while it is around 10 −1 under TAS.Moreover, by setting R i = 1 Kbits, B i = 1 MHz, T th i = 1 ms, and γ ri , we can observe that the DOR for user r i under FAMA is around 10 −2 , whereas it is near to 1 for the TAS counterpart.\\nAdditionally, a comparison of the results for FAMA-IC and the benchmark FAMA reveals that FAMA-IC outperforms FAMA in terms of both OP and DOR under conditions of strong interference because it leverages fluid antenna diversity to optimize the SINR while employing advanced decoding techniques, such as SND, to mitigate interference.This approach enables the system to maintain reliable data rates by utilizing the structure of the strong interference instead of being fully degraded by it, which would occur if interference were treated as noise.As a result, the increased channel capacity achievable under strong interference reduces the likelihood of falling below the threshold rate, making strong interference advantageous for minimizing OP in FAMA systems.\\nAssuming R i = R and B i = B, Fig. 5 investigates the impact of increasing the data rate R on the DOR performance under FAMA and TAS deployments in the strong IC.While transmitting a larger amount of data within a fixed bandwidth B is challenging for both FAMA and TAS, it becomes much more feasible when using FAS.For example, sending R = 1.5 Kbits over a strong IC with B = 1 MHz is nearly impossible with TAS, but it can be achieved with an acceptable DOR under FAMA.The impact of bandwidth B on the DOR for selected values of W ri and N ri is presented in Fig. 6.It can be seen that increasing B improves the DOR performance for both FAMA and TAS in strong IC, which means that the same amount of data can be transmitted with less delay as the IC bandwidth increases.However, it can be observed that this improvement is much more remarkable under the FAMA deployment than the TAS, thanks to the position reconfigurability, enhanced spatial diversity, and precise interference management provided by FAS.For instance, with a predefined bandwidth of B = 1 MHz, sending 1 Kbits of data results in a DOR close to 10 −5 for FAMA and nearly 1 for TAS.Fig. 7 studies the performance of the sum EC as a function of γ ri for selected values of W ri and N ri .It is evident that the asymptotic results closely align with the analytical derivations at high SNR, confirming the reliability of the theoretical model to predict the system performance.As expected, increasing W ri and N ri leads to a significant improvement in the sum EC.Even under the challenging strong IC, FAMA outperforms TAS, with the latter showing lower capacity across all SNR values.This highlights the advantage of using FAS with only one activated port and even with smaller sizes, over a fixed single-antenna in interference-dominated environments.However, in strong IC, the EC is constrained by the interference power relative to the signal power.As W ri and N ri increase, the system can initially mitigate interference more effectively by leveraging spatial diversity.Nevertheless, beyond a certain point, the interference remains dominant, and the system capacity cannot improve further, leading to saturation.\"\n },\n {\n \"title\": \"VI. CONCLUSION\",\n \"content\": \"This paper has provided a comprehensive study of FAMA for two-user strong IC, with each mobile user equipped with a FAS.By applying the realistic SND interference management technique, analytical expressions for the CDF and PDF of the SNR and several key performance metrics, including the OP, DOR, and EC were derived.The results demonstrated significant advantages of FAS, showcasing its ability to improve system performance even with only one activated port under strong interference conditions.These findings contribute to more realistic interference management models in future wireless communication systems and offer valuable insights for optimizing FAMA in practical deployments.\"\n },\n {\n \"title\": \"Fig. 1 .\",\n \"content\": \"Fig. 1.System model for a two-user FAMA-IC.\"\n },\n {\n \"title\": \"nr i tiri 2 ∼\",\n \"content\": \"exp(1).Hence, the PDF and CDF for γ ňr i ri and ζ ňr i ri are, respectively, expressed as\"\n },\n {\n \"title\": \")\",\n \"content\": \"in which F κ fama r iRthsum represents the complementary CDF (CCDF) of κ fama ri ,\\nregime, provided in (27) and (28), respectively.\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"Fig. 2. The expected values of RVs (a) γ fama r i and (b) κ fama r i .\"\n },\n {\n \"title\": \"Fig. 3 .\",\n \"content\": \"Fig. 3. OP versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, and ζ r 1 = ζ r 2 = γ rι + 20 dB.\"\n },\n {\n \"title\": \"Fig. 4 .\",\n \"content\": \"Fig. 4. DOR versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R 1 = R 2 = 1 Kbits, B 1 = B 2 = 1 MHz, T th 1 = T th 2 = 1 ms, and ζ r 1 = ζ r 2 = γ rι + 20 dB.\"\n },\n {\n \"title\": \"Fig. 5 .Fig. 6 .\",\n \"content\": \"Fig. 5. DOR versus rate R for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when B 1 = B 2 = 1 MHz, γ r i = 10 dB, and T th 1 = T th 2 = 1 ms.\"\n },\n {\n \"title\": \"Fig. 7 .\",\n \"content\": \"Fig. 7. EC versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, ζ r 1 = ζ r 2 = γ rι + 20 dB.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Fig. 1.System model for a two-user FAMA-IC.","content":"Fig. 1 .\n1\nFig. 1.System model for a two-user FAMA-IC."},{"id":"","caption":"exp(1).Hence, the PDF and CDF for γ ňr i ri and ζ ňr i ri are, respectively, expressed as","content":"nr i tiri 2 ∼\n2\nexp(1).Hence, the PDF and CDF for γ ňr i ri and ζ ňr i ri are, respectively, expressed as"},{"id":"","caption":"in which F κ fama r iRthsum represents the complementary CDF (CCDF) of κ fama ri ,","content":")\n\nin which F κ fama r iRthsum represents the complementary CDF (CCDF) of κ fama ri ,"},{"id":"","caption":"regime, provided in (27) and (28), respectively.","content":"regime, provided in (27) and (28), respectively."},{"id":"","caption":"Fig. 2. The expected values of RVs (a) γ fama r i and (b) κ fama r i .","content":"Fig. 2 .\n2\nFig. 2. The expected values of RVs (a) γ fama r i and (b) κ fama r i ."},{"id":"","caption":"Fig. 3. OP versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, and ζ r 1 = ζ r 2 = γ rι + 20 dB.","content":"Fig. 3 .\n3\nFig. 3. OP versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, and ζ r 1 = ζ r 2 = γ rι + 20 dB."},{"id":"","caption":"Fig. 4. DOR versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R 1 = R 2 = 1 Kbits, B 1 = B 2 = 1 MHz, T th 1 = T th 2 = 1 ms, and ζ r 1 = ζ r 2 = γ rι + 20 dB.","content":"Fig. 4 .\n4\nFig. 4. DOR versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R 1 = R 2 = 1 Kbits, B 1 = B 2 = 1 MHz, T th 1 = T th 2 = 1 ms, and ζ r 1 = ζ r 2 = γ rι + 20 dB."},{"id":"","caption":"Fig. 5. DOR versus rate R for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when B 1 = B 2 = 1 MHz, γ r i = 10 dB, and T th 1 = T th 2 = 1 ms.","content":"Fig. 5 .Fig. 6 .\n56\nFig. 5. DOR versus rate R for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when B 1 = B 2 = 1 MHz, γ r i = 10 dB, and T th 1 = T th 2 = 1 ms."},{"id":"","caption":"Fig. 7. EC versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, ζ r 1 = ζ r 2 = γ rι + 20 dB.","content":"Fig. 7 .\n7\nFig. 7. EC versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, ζ r 1 = ζ r 2 = γ rι + 20 dB."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Fig. 1.System model for a two-user FAMA-IC.\",\n \"content\": \"Fig. 1 .\\n1\\nFig. 1.System model for a two-user FAMA-IC.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"exp(1).Hence, the PDF and CDF for γ ňr i ri and ζ ňr i ri are, respectively, expressed as\",\n \"content\": \"nr i tiri 2 ∼\\n2\\nexp(1).Hence, the PDF and CDF for γ ňr i ri and ζ ňr i ri are, respectively, expressed as\"\n },\n {\n \"id\": \"\",\n \"caption\": \"in which F κ fama r iRthsum represents the complementary CDF (CCDF) of κ fama ri ,\",\n \"content\": \")\\n\\nin which F κ fama r iRthsum represents the complementary CDF (CCDF) of κ fama ri ,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"regime, provided in (27) and (28), respectively.\",\n \"content\": \"regime, provided in (27) and (28), respectively.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 2. The expected values of RVs (a) γ fama r i and (b) κ fama r i .\",\n \"content\": \"Fig. 2 .\\n2\\nFig. 2. The expected values of RVs (a) γ fama r i and (b) κ fama r i .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 3. OP versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, and ζ r 1 = ζ r 2 = γ rι + 20 dB.\",\n \"content\": \"Fig. 3 .\\n3\\nFig. 3. OP versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, and ζ r 1 = ζ r 2 = γ rι + 20 dB.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 4. DOR versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R 1 = R 2 = 1 Kbits, B 1 = B 2 = 1 MHz, T th 1 = T th 2 = 1 ms, and ζ r 1 = ζ r 2 = γ rι + 20 dB.\",\n \"content\": \"Fig. 4 .\\n4\\nFig. 4. DOR versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R 1 = R 2 = 1 Kbits, B 1 = B 2 = 1 MHz, T th 1 = T th 2 = 1 ms, and ζ r 1 = ζ r 2 = γ rι + 20 dB.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 5. DOR versus rate R for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when B 1 = B 2 = 1 MHz, γ r i = 10 dB, and T th 1 = T th 2 = 1 ms.\",\n \"content\": \"Fig. 5 .Fig. 6 .\\n56\\nFig. 5. DOR versus rate R for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when B 1 = B 2 = 1 MHz, γ r i = 10 dB, and T th 1 = T th 2 = 1 ms.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Fig. 7. EC versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, ζ r 1 = ζ r 2 = γ rι + 20 dB.\",\n \"content\": \"Fig. 7 .\\n7\\nFig. 7. EC versus average SNR γ r i for different number of fluid antenna ports, Nr i , and fluid antenna size, Wr i , when R th 1 = R th 2 = 0.5 bits, ζ r 1 = ζ r 2 = γ rι + 20 dB.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"W K New, arXiv:2407.03449A tutorial on fluid antenna system for 6G networks: Encompassing communication theory, optimization methods and hardware designs. Jul. 2024arXiv preprint","doi":""},{"text":"Bruce Leeinspired fluid antenna system: Six research topics and the potentials for 6G. K K Wong, K F Tong, Y Shen, Y Chen, Y Zhang, Frontiers Commun. and Netw., section Wireless Commun. 3853416Mar. 2022","doi":""},{"text":"AI-empowered fluid antenna systems: Opportunities, challenges and future directions. C Wang, IEEE Wireless Commun. 315October 2024","doi":""},{"text":"Liquid antennas: Past, present and future. Y Huang, L Xing, C Song, S Wang, F Elhouni, IEEE Open J. Antennas & Propag. 22021","doi":""},{"text":"Frequency, radiation pattern and polarization reconfigurable antenna using a parasitic pixel layer. D Rodrigo, B A Cetiner, L Jofre, IEEE Trans. Antennas & Propag. 626Jun. 2014","doi":""},{"text":"Compact pattern reconfigurable pixel antenna with diagonal pixel connections. L Jing, M Li, R Murch, IEEE Trans. Antennas & Propag. 7010Oct. 2022","doi":""},{"text":"Computational polarimetric imaging using two-dimensional dynamic metasurface apertures. T V Hoang, V Fusco, T Fromenteze, O Yurduseven, IEEE Open J. Antennas & Propag. 22021","doi":""},{"text":"Reconfigurable holographic surfaces for ultra-massive MIMO in 6G: Practical design, optimization and implementation. R Deng, IEEE J. Select. Areas Commun. 418Aug. 2023","doi":""},{"text":"Historical review of fluid antennas and movable antennas. L Zhu, K K Wong, arXiv:2401.02362v2Jan. 2024arXiv preprint","doi":""},{"text":"Fluid antenna systems. 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Lett. 132Feb. 2024","doi":""},{"text":"On the performance of fluid antennas systems under α-µ fading channels. P D Alvim, IEEE Wireless Commun. Lett. 131Jan. 2024","doi":""},{"text":"Copula-based performance analysis for fluid antenna systems under arbitrary fading channels. F R Ghadi, K.-K Wong, F J López-Martínez, K.-F Tong, IEEE Commun. Lett. 2711Nov. 2023","doi":""},{"text":"A Gaussian copula approach to the performance analysis of fluid antenna systems. F R Ghadi, 10.1109/TWC.2024.3454558IEEE Trans. Wireless Commun., early access. 2024","doi":""},{"text":"An information-theoretic characterization of MIMO-FAS: Optimization, diversity-multiplexing tradeoff and q-outage capacity. W K New, K K Wong, H Xu, K F Tong, C.-B Chae, IEEE Trans. Wireless Commun. 236Jun. 2024","doi":""},{"text":"Fast fluid antenna multiple access enabling massive connectivity. K K Wong, K F Tong, Y Chen, Y Zhang, IEEE Commun. Lett. 272Feb. 2023","doi":""},{"text":"Slow fluid antenna multiple access. 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C Wang, IEEE Wireless Commun. 315October 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Liquid antennas: Past, present and future. Y Huang, L Xing, C Song, S Wang, F Elhouni, IEEE Open J. Antennas & Propag. 22021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Frequency, radiation pattern and polarization reconfigurable antenna using a parasitic pixel layer. D Rodrigo, B A Cetiner, L Jofre, IEEE Trans. Antennas & Propag. 626Jun. 2014\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Compact pattern reconfigurable pixel antenna with diagonal pixel connections. L Jing, M Li, R Murch, IEEE Trans. Antennas & Propag. 7010Oct. 2022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Computational polarimetric imaging using two-dimensional dynamic metasurface apertures. T V Hoang, V Fusco, T Fromenteze, O Yurduseven, IEEE Open J. Antennas & Propag. 22021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Reconfigurable holographic surfaces for ultra-massive MIMO in 6G: Practical design, optimization and implementation. R Deng, IEEE J. Select. Areas Commun. 418Aug. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Historical review of fluid antennas and movable antennas. L Zhu, K K Wong, arXiv:2401.02362v2Jan. 2024arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluid antenna systems. K K Wong, A Shojaeifard, K F Tong, Y Zhang, IEEE Trans. Wireless Commun. 203Mar. 2021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluid antenna multiple access. K K Wong, K F Tong, IEEE Trans. Wireless Commun. 217Jul. 2022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A new analytical approximation of the fluid antenna system channel. M Khammassi, A Kammoun, M.-S Alouini, IEEE Trans. Wireless Commun. 2212Dec. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A new spatial block-correlation model for fluid antenna systems. P Ramirez-Espinosa, D Morales-Jimenez, K.-K Wong, 10.1109/TWC.2024.3434509IEEE Trans. Wireless Commun., early access. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluid antenna system: New insights on outage probability and diversity gain. W K New, K K Wong, H Xu, K F Tong, C.-B Chae, IEEE Trans. Wireless Commun. 231Jan. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A simple method for the performance analysis of fluid antenna systems under correlated Nakagami-m fading. J D Vega-Sánchez, L Urquiza-Aguiar, M C P Paredes, D P M Osorio, IEEE Wireless Commun. Lett. 132Feb. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the performance of fluid antennas systems under α-µ fading channels. P D Alvim, IEEE Wireless Commun. Lett. 131Jan. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Copula-based performance analysis for fluid antenna systems under arbitrary fading channels. F R Ghadi, K.-K Wong, F J López-Martínez, K.-F Tong, IEEE Commun. Lett. 2711Nov. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Gaussian copula approach to the performance analysis of fluid antenna systems. F R Ghadi, 10.1109/TWC.2024.3454558IEEE Trans. Wireless Commun., early access. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An information-theoretic characterization of MIMO-FAS: Optimization, diversity-multiplexing tradeoff and q-outage capacity. W K New, K K Wong, H Xu, K F Tong, C.-B Chae, IEEE Trans. Wireless Commun. 236Jun. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fast fluid antenna multiple access enabling massive connectivity. K K Wong, K F Tong, Y Chen, Y Zhang, IEEE Commun. Lett. 272Feb. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Slow fluid antenna multiple access. K K Wong, D Morales-Jimenez, K F Tong, C B Chae, IEEE Trans. Commun. 715May 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Revisiting outage probability analysis for two-user fluid antenna multiple access system. H Xu, IEEE Trans. Wireless Commun. 238Aug. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Opportunistic fluid antenna multiple access. K.-K Wong, K.-F Tong, Y Chen, Y Zhang, C.-B Chae, IEEE Trans. Wireless Commun. 2211Nov. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Opportunistic fluid antenna multiple access via teaminspired reinforcement learning. N Waqar, IEEE Trans. Wireless Commun. 239Sept. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Energy-efficiency optimization for slow fluid antenna multiple access using mean-field game. Y Chen, S Li, Y Hou, X Tao, IEEE Wireless Commun. Lett. 134Apr. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Capacity maximization for FAS-assisted multiple access channels. H Xu, arXiv:2311.110372023arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluid antenna system liberating multiuser MIMO for ISAC via deep reinforcement learning. C Wang, IEEE Trans. Wireless Commun. 239Sept. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluid antenna-assisted ISAC systems. L Zhou, J Yao, M Jin, T Wu, K K Wong, 10.1109/LWC.2024.3476148IEEE Wireless Commun. Lett., early access. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Shifting the ISAC trade-off with fluid antenna systems. J Zou, 10.1109/LWC.2024.3473991IEEE Wireless Commun. Lett., early access. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluid antenna-assisted dirty multiple access channels over composite fading. F R Ghadi, IEEE Commun. Lett. 282Feb. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Performance analysis of FAS-aided backscatter communications. F R Ghadi, M Kaveh, K.-K Wong, Y Zhang, IEEE Wireless Commun. Lett. 139Sept. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Fluid antenna multiple access assisted integrated data and energy transfer: Outage and multiplexing gain analysis. X Lin, Y Zhao, H Yang, J Hu, K K Wong, arXiv:2407.105482024arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Channel estimation for FAS-assisted multiuser mmWave systems. H Xu, IEEE Commun. Lett. 283Mar. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Successive Bayesian reconstructor for channel estimation in fluid antenna systems. Z Zhang, J Zhu, L Dai, R W HeathJr, arXiv:2312.06551v32024arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Deep learning enabled slow fluid antenna multiple access. N Waqar, K K Wong, K F Tong, A Sharples, Y Zhang, IEEE Commun. Lett. 273Mar. 2023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Learning-induced channel extrapolation for fluid antenna systems using asymmetric graph masked autoencoder. H Zhang, IEEE Wireless Commun. Lett. 136Jun. 2024\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Design and implementation of mmWave surface wave enabled fluid antennas and experimental results for fluid antenna multiple access. 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P Hennig, arXiv:0910.01152009arXiv preprint\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":578,"cells":{"text":{"kind":"string","value":"\nHexagonal-Close-Packed Lattice: Phase Transition and Spin Transport\n24 Dec 2011\n\nDanh-Tai Hoang \nLaboratoire de Physique Théorique et Modélisation\nUniversité de Cergy-Pontoise\nCNRS\nAvenue Adolphe Chauvin8089 2, 95302Cergy-Pontoise CedexUMRFrance\n\nH T Diep \nLaboratoire de Physique Théorique et Modélisation\nUniversité de Cergy-Pontoise\nCNRS\nAvenue Adolphe Chauvin8089 2, 95302Cergy-Pontoise CedexUMRFrance\n\nHexagonal-Close-Packed Lattice: Phase Transition and Spin Transport\n24 Dec 2011(Dated: December 30, 2011)numbers: 0570Fh7510Hk7540Mg7547-m :\nWe study the ground state (GS) and the phase transition in a hexagonal-close-packed lattice with both XY and Ising models by using extensive Monte Carlo simulation. We suppose the in-plane interaction J1 and inter-plane interaction J2, both antiferromagnetic. The system is frustrated with two kinds of GS configuration below and above a critical value of η = J1/J2 (ηc). For the Ising case, one has ηc = 0.5 which separates in-plane ferromagnetic and antiferromagnetic states, while for the XY case ηc = 1/3 separates the collinear and non collinear spin configurations. The phase transition is shown to be of first (second) order for η > (<)ηc. The spin resistivity is calculated for the Ising case. It shows a rounded maximum at the magnetic transition in the second-order region, and a discontinuity in the first-order region of η. PACS numbers: 05.70.Fh ; 75.10.Hk ; 75.40.Mg ;\n\nI. INTRODUCTION\n\nFrustrated spin systems have been subject of intensive investigations during the last 30 years [1]. These systems are very unstable due to the competition between antagonist interactions or due to a geometrical frustration such as in the antiferromagnetic (AF) triangular lattice. A spin is said frustrated when it cannot find an orientation which \"fully\" satisfies all the interactions with its neighbors [2,3]. As a consequence of the frustration, the ground state (GS) is very highly degenerate. In the Ising case the GS degeneracy is often infinite as in the triangular lattice, the face-centered cubic (FCC) and hexagonalclose-packed (HCP) lattices, with AF interaction. In the case of vector spins, the GS is non collinear such as the 120-degree configuration in the XY and Heisenberg AF stacked triangular lattice (STL). In two dimensions (2D), several frustrated systems with Ising spin model have been exactly solved [4,5]. Among the most interesting models one can mention the frustrated generalized Kagome lattice [6] and the honeycomb lattice [7] where exotic features such as the existence of several phase transitions, the reentrance and the disorder lines have been exactly found by mapping these systems into vertex models [8]. In three dimensions (3D), the situation is complicated. The renormalization group (RG) [9,10], which provided a good understanding of the nature of the phase transition in non frustrated systems, encounters much of difficulties in application to frustrated systems. Among the most studied subjects during the last 20 years, one can mention the nature of the phase transition in the XY and Heisenberg STL. After a long debate [11] on whether it is a second-, or a first-order transition or it belongs to a * danh-tai.hoang@u-cergy.fr † diep@u-cergy.fr, corresponding author new universality class, the controversy has recently ended with the conclusion of a first-order transition [12,13].\n\nIn this paper, we are interested in the HCP antiferromagnet with Ising and XY spin models. Our purpose is to study its properties such as the ground state, the phase transition and the spin transport, in the case of anisotropic exchange interactions. The isotropic nearestneighbor (NN) AF interaction has been studied for Ising [14] and XY and Heisenberg spins [15]. These isotropic cases have been shown to undergo a phase transition of first order, and the infinite GS degeneracy is reduced to 6 at low temperatures [14,15]. The effect of anisotropic interaction and anisotropy on the GS in the case of vector spins has been studied [16].\n\nIn section II, we show our model and analyze the ground state. Results of Monte Carlo (MC) simulations are shown and discussed in section III. Concluding remarks are given in section V.\n\n\nII. MODEL AND GROUND-STATE ANALYSIS\n\nWe consider the HCP lattice shown in Fig. 1. The stacking direction is z. The Hamiltonian is given by\nH = − (i,j) J i,j S i · S j(1)\nwhere S i is the spin at lattice site i and J ij is the AF interaction between nearest-neighbors (NN). We suppose that J ij = J 1 if the NN are on the xy triangular plane, and J ij = J 2 if the NN are on two adjacent planes (see Fig. 1). The GS can be determined by the steepestdescent method. This method is very simple [17] (i) we generate an initial configuration at random (ii) we calculate the local field created at a site by its neighbors using (1) (iii) we align the spin of that site along the calculated local field to minimize its energy (iv) we go to another site and repeat until all sites are visited: we say we make one sweep (v) we do a large number of sweeps per site until a good convergence is reached. However, one can also minimize the interaction energy as shown below to calculate the GS configuration. Since both interactions are AF (negative), for simplicity, we fix J 2 = −1 and vary J 1 . The unit of energy is taken as |J 2 | and the temperature T is in the unit of |J 2 |/k B where k B is the Boltzmann constant.\n\nLet us recall the GS in the case of isotropic interaction, namely J 1 = J 2 [15]. For the HCP lattice, each spin is shared by eight tetrahedra (four in the upper half-space and four in the lower half-space along the z axis) and a NN bond is shared by two tetrahedra. The GS spin configuration of the system is formed by stacking neighboring tetrahedra. In the GS, one has two pairs of antiparallel spins on each tetrahedron. Their axes form an arbitrary angle α. The degeneracy is thus infinite (see Fig. 2a of Ref. [15]). Of course, the periodic boundary conditions will reduce a number of configurations, but the degeneracy is still infinite. Of these GS's, one particular family of configurations of interest for both XY and Heisenberg cases is when α = 0. The GS is then collinear with two spins up and the other two down. The stacking sequence is then simplest: there are three equivalent configurations since there are three ways to choose the parallel spin pair in the original tetrahedron.\n\nWe now examine the case where J 1 = J 2 .\n\n• Ising case:\n\nThe steepest descent method with varying J 1 (J 2 = −1 gives two kinds of GS spin configuration: the first consists of xy ferromagnetic planes stacked antiferromagnetically along the z direction, while the second one is the stacking of xy AF planes such that each tetrahedron has two up and two down spins. The transition between the two configurations is determined as follows: one simply writes down the respective energies of a tetrahedron and compares them\nE 1 = 3(−J 1 + J 2 ) (2) E 2 = J 1 + J 2(3)\nOne sees that\nE 1 < E 2 when J 1 > 0.5J 2 , i.e. |J 1 | < 0.5|J 2 |.\nThus the first configuration is more stable when |J 1 | < 0.5|J 2 |.\n\n• XY case:\n\nConsider one tetrahedron of the HCP lattice. The Hamiltonian for this cell is given by\nH c = −J 1 (S 1 · S 2 + S 2 · S 3 + S 3 · S 1 ) −J 2 (S 1 · S 4 + S 2 · S 4 + S 3 · S 4 ) (4) Suppose that |S i | = 1, one has H c = −J 1 [cos α + cos β + cos(α + β)] −J 2 [cos γ + cos(γ − α) + cos(γ − α − β)] (5)\nwhere the angles are defined in Fig. 2. The steepest descent method shows that while β and α have unique values for a given J 1 /J 2 , α is arbitrary, just as in the case of isotropic interaction [15] discussed above. To simplify the formulae, we take α = 0 in the following. The energy of the cell is written as\nH c = −J 1 [1 + 2cosβ] − J 2 [2 cos γ + cos(γ − β)] (6)\nThe critical values of β and γ are determined from the relations\n∂H c ∂β = 2J 1 sin β − 2J 2 sin(γ − β) = 0 (7) ∂H c ∂γ = 2J 2 sin γ + J 2 sin(γ − β) = 0(8)\nWe find the following solutions:\n(i) β = 0, γ = 0, (ii)β = 0, γ = π, (iii) β = π, γ = 0, (iv) β = π, γ = π, and (v) cos β = 1 4(J1/J2) 2 − 5 4 , cos γ = − 1+3(J1/J2) 2 4(J1/J2)\n. By comparing the energy values at these solutions, we obtain the minimum energy with the last solution: one has where cos β and cos γ are given above.\nH c = − 1 + 3(J 1 /J 2 ) 2 2(J 1 /J 2 )(9)\nBecause −1 ≤ cos β ≤ 1 and −1 ≤ cos γ ≤ 1, the above solution is valid for 1 3 ≤ J 1 /J 2 ≤ 1. We plot cos β, cos γ, β and γ in Fig. 3 where we observe that the non collinear GS configuration occurs in the interval 1/3 ≤ η = J 1 /J 2 ≤ 1.\n\n.\n\n\nIII. PHASE TRANSITION: RESULTS\n\nWe consider a sample size of L×L×L z where L and L z vary from 12 to 36 but L z can be different from L in order to detect the dependence of the GS on J 1 . The exchange interaction |J 2 | = 1 is used as the unit of energy. We use here MC simulations with a histogram technique to detect first-order transition. We equilibrate the system and average physical quantities over several millions of MC steps per spin. The averaged energy U and the heat capacity C V are calculated by\nU = H (10) C V = U 2 − U 2 k B T 2(11)\nwhere ... indicates the thermal average taken over microscopic states at T . The order parameter M is defined from the sublattice magnetization by\nM = K | i∈K S i |(12)\nwhere S i belongs to the sublattice K. Note that there are at least four sublattices which define the ordering of the spins on the tetrahedra if a long-range order is observed. The susceptibility is defined by\nχ = M 2 − M 2 k B T(13)\nIn MC simulations, we work at finite sizes, so for each size we have to determine the \"pseudo\" transition which corresponds in general to the maximum of the specific heat or of the susceptibility. The maxima of these quantities need not to be at the same temperature. Only at the infinite size, they should coincide. The theory of finite-size scaling [18][19][20] permits to deduce properties of a system at its thermodynamic limit. In order to check the first-order nature of the transition, we used the histogram technique which is very efficient in detecting weak first-order transitions and in calculating critical exponents of second-order transitions. [19,20] The main idea of this technique is to make an energy histogram at a temperature T 0 as close as possible to the transition temperature. Often, one has to try at several temperatures in the transition region. Using this histogram in the formulae of statistical physics for canonical distribution, one obtains energy histograms in a range of temperature around T 0 . In second-order transitions, these histograms are gaussian. They allows us to calculate averages of physical quantities as well as critical exponents using the finite-size scaling. In first-order transitions, the energy histogram shows a double-peak structure.\n\n\nA. Ising case\n\nIn the following, the results in different figures are shown with L = 18 and L z = 8 (16 atomic planes along z). Since the GS changes at η c = 0.5, we show here examples on both sides of this value. Figure 4 shows the energy per spin E, the specific heat per spin C V , the order parameter M and the susceptibility χ, for η = 0.3. The transition is of second order. On the other side, we show in Fig. 5 the energy per spin and the order parameter versus T , for η = 0.85 and 1. We find a strong first-order transition in both cases. The discontinuity of E and M at the transition is very large. We show in Fig. 6 the energy histogram taken at the transition temperature for three values η = 0.3 (black), 0.85 (blue) and 1 (red). As seen, the first case is a gaussian distribution indicating a second-order transition, while the last two cases show a double-peak structure indicating a first-order transition.\n\nWe have calculated the critical temperature T C as a function of η. The phase diagram is shown in Fig. 7 where I and II indicate the ordering of the first, and second kinds, respectively. P indicates the paramagnetic phase. Note that the transition line between I and P is a second-order line, while that between II and P is a first-order line. \n\n\nB. XY case\n\nIn the XY case, the change of the GS takes place at η = 1/3. Let us show in Fig. 8 the result for η = 0.3 where the GS is composed of ferromagnetic planes antiferromagnetically stacked in the z direction. The transi- tion is of second order.\n\nWe show now the result for the non collinear GS region in Fig. 9. The energy and the order parameter show clearly a discontinuity at the transition for η = 0.58 and 1. Using the histogram method described above, we have calculated the histogram shown in Fig. 10 for η = 0.3, 0.58 and 1. For η = 0.3 which is in the collinear region of the GS, the histogram is gaussian, confirming the secondorder transition observed in the data shown above. For η = 0.58 and 1 belonging to the non collinear region, the histogram shows a two-peak structure which confirms the first-order character of the transitions in this region.\n\nThe two peaks are very well separated with the dip going down to zero, indicating an energy discontinuity. The distance between the two peaks is the latent heat ∆E.\n\nWe show in Fig. 11 the transition temperature versus η where I and II indicate the collinear and non collinear phases, respectively. P denotes the paramagnetic state. The line separating I and P is a second-order transition line, while that separating II and P is the first-order one.\n\nTo close this section, we emphasize that all 3D frustrated systems known so far undergo a first-order transition: let us mention the AF STL [12,13], the FCC antiferromagnets [21], the simple cubic fully frustrated lattices [22][23][24][25], helimagnets [26] and the HCP lattice studied here.\n\n\nIV. SPIN TRANSPORT IN THE ISING CASE\n\nWe have recently studied the spin transport in magnetically ordered materials by using an efficient MC simulation method [27,28]. The tour-de-force of the method was realized when we studied the semiconducting MnTe where the agreement with experiment is excellent [29]. In ferromagnets, the spin resistivity shows a sharp peak at the magnetic phase transition. The origin of this anomaly comes from the spin-spin correlation [30][31][32]. In antiferromagnets, such a peak is not sharp and often a broad maximum [33]. We have also studied the spin resistivity in two frustrated cases: the FCC Ising case [34] and the J 1 − J 2 simple cubic lattice [35]. The results show that the discontinuous transition in these systems gives rise to a discontinuity of the spin resistivity.\n\nUsing the method which has been described in detail elsewhere [27,34], we calculate the current of itinerant spins moving across the crystal under an electric field ǫ applied in the x direction. We suppose that each itinerant spin σ i interacts with the lattice spins around it within a sphere of radius D 1 at any point on its trajectory. The interaction Hamiltonian reads\nH l = − j I ij σ i · S j(14)\nwhere the sum is performed over all lattice spins in the sphere around the itinerant spin σ i and I ij is the interaction which depends on the distance between σ i and S j . We suppose also the following interaction between an itinerant spin with its neighboring itinerant spins within a sphere of radius D 2\nH i = − j K ij σ i · σ j(15)\nFor simplicity, we suppose\nI ij = I 0 exp(−Br ij ) (16) K ij = K 0 exp(−Cr ij )(17)\nwhere I 0 , B, K 0 and C are constants chosen in such a way that the energy of an itinerant spin is smaller than that of a lattice spin. This choice is made to avoid the influence of itinerant spins on the ordering of the lattice spins. Note that in the almost-free electron model, I 0 ≃ K 0 ≃ 0, and in semiconductors they are larger but still weak with respect to J 1 and J 2 . A discussion on the choice of different parameters has been given for example in Ref. [27]. We suppose here a concentration of one itinerant per two lattice cells. At such a low concentration, the averaged distance between itinerant spins is much larger than the cutoff distance D 2 . In spite of this, due to the attractive nature of their interaction, Eq. (17), we have to use a chemical potential term to insure that itinerant spins do not form clusters. This potential is taken of the form D[n( r) − n 0 ] where D is a positive constant, n( r) the concentration of itinerant spins in the sphere of cutoff radius D 2 centered at the position r of the itinerant spin under consideration, and n 0 the averaged concentration. We perform the simulation by taking into account the relaxation time of the lattice spins [18,28,29] which is given by\nτ L = A |1 − T /T C | zν(18)\nwhere A is a constant, ν the correlation critical exponent, and z the dynamic exponent. We see that as T tends to T C , τ L diverges. This phenomenon is known as the critical slowing-down. For the Ising spin model, ν = 0.638 (3D Ising universality) and z = 2.02 [36]. We have previously shown that τ L strongly affects the shape of ρ at T C [28]. By choosing A = 1, we fix τ L = 1 at T = 2T C deep inside the paramagnetic phase far above T C . This value is what we expect for thermal fluctuations in the disordered phase.\n\nThe spin resistivity ρ versus T for the two typical cases η = 0.3 and 1 is shown in Fig. 12 where distances D 1 and D 2 are in unit of the NN distance, energy constants I 0 , K 0 and D are in unit of |J 2 | = 1. We observe here that in the second-order region ρ has a rounded maximum and in the first-order region it undergoes an almost discontinuous jump at the transition.\n\nWe have varied the radius D 1 to see its effect on the resistivity ρ at the transition. We observe the same effect which has been seen in some other antiferromagnets [34,35]: at a given T , ρ oscillates slightly with distance. We have found that this oscillation comes from the oscillatory behavior of the difference between the numbers of up and down spins in the sphere as D 1 varies.\n\n\nV. CONCLUSION\n\nWe have studied in this paper some properties of the HCP antiferromagnet with Ising and XY spin models. The in-plane J 1 and inter-plane J 2 interactions are supposed to be different. As a result, the GS spin configuration depends on the ratio η = J 1 /J 2 . We show that there exists a critical value η c where the GS changes. For the Ising case, we find η c = 0.5 below (above) which the spins in the xy planes are ferromagnetic (antiferromagnetic). For the XY case, the GS is collinear below η c = 1/3, and is non collinear above that value. The nature of the transition changes from a second order below η c to a first order above η c for both Ising and XY cases. We have also studied the spin resistivity in the Ising case. We found that the shape of ρ depends on the nature of the transition: in the second-order region, a rounded maximum is observed at the transition while in the first-order region, ρ undergoes a discontinuity at the transition as we have observed in other frustrated cases. These findings may help to understand the transition nature and the spin transport in different compounds with HCP structure.\n\nFIG. 1 .\n1HCP lattice. The in-plane and inter-plane interactions are indicated by J1 and J2.\n\nFIG. 2 .FIG. 3 .\n23Ground state in the XY case. The tetrahedron is projected on the xy plane. The spins are numbered from 1 to 4. See text for comments. Ground state in the XY case (color online). The angles β (black circles) and γ (blue void circles) and their cosinus are shown as functions of η = J1/J2. Non collinear GS configurations occur in the region 1/3 ≤ η ≤ 1 . See text for comments.\n\nFIG. 5 .FIG. 6 .\n56Ising case (color online): Energy E and order parameter M for η = J1/J2 = 0.85 (blue void circles) and 1 (red triangles). See text for comments. Ising case (color online): Energy histogram P (E) versus E for η = 1( red triangles), 0.85 (blue void circles) and 0.3 (black circles). See text for comments.\n\nFIG. 7 .\n7Ising case: TC versus η. I, II and P denote the first, second and paramagnetic phases, respectively. See text for comments.\n\nFIG. 8 .\n8XY case: Energy E, specific heat CV , order parameter M and susceptibility χ versus T for η = 0.3. See text for comments.\n\nFIG. 9 .FIG. 10 .\n910XY case (color online): E and M versus T for η = 0.58 (blue void circles) and 1 (red triangles). XY case (color online): Energy histogram P versus E for η = 0.3 (black circles), 0.58 (blue void circles), 1 (red triangles) at the respective transition temperatures. See text for comments.\n\n\n. 11. XY case: TC versus η. I, II and P denote the collinear, non collinear and paramagnetic phases, respectively. See text for comments.\n\nFIG. 12 .\n12Ising case: Spin resistivity ρ versus T for η = 0.3 (upper) and 1 (lower). L = 18, Lz = 8 (16 planes in the z direction), D1 = D2 = 1, ǫ = 1, I0 = 2, K0 = 0.5, B = C = 1, A = 1, D = 0.5. All distances are in unit of the NN distance, energy constants are in unit of |J2| = 1. See text for comments.\n\n\nFIG. 4. Ising case: Energy E, specific heat CV , order parameter M and susceptibility χ versus T for η = J1/J2 = 0.3. See text for comments.-2 \n\n-1.8 \n\n-1.6 \n\n-1.4 \n\n-1.2 \n\n-1 \n\n-0.8 \n\n1.8 \n2 \n2.2 \n2.4 \n2.6 \n2.8 \n3 \n\nT \n\nE \n\n0 \n\n0.5 \n\n1 \n\n1.5 \n\n2 \n\n2.5 \n\n3 \n\n1.8 \n2 \n2.2 \n2.4 \n2.6 \n2.8 \n3 \n\nT \n\nC V \n\n0 \n\n0.1 \n\n0.2 \n\n0.3 \n\n0.4 \n\n0.5 \n\n0.6 \n\n0.7 \n\n0.8 \n\n0.9 \n\n1 \n\n1.8 \n2 \n2.2 \n2.4 \n2.6 \n2.8 \n3 \n\nT \n\nM \n\n0 \n\n10 \n\n20 \n\n30 \n\n40 \n\n50 \n\n60 \n\n70 \n\n1.8 \n2 \n2.2 \n2.4 \n2.6 \n2.8 \n3 \n\nT \n\nχ \n\n\n\nFrustrated Spin Systems. World Scientific. H. T. DiepH. T. Diep (ed.), Frustrated Spin Systems, World Scien- tific, Singapor (2005).\n\n. G Toulouse, Commun. Phys. 2115G. Toulouse, Commun. Phys. 2, 115 (1977).\n\n. J Villain, J. Phys. C. 101717J. Villain, J. Phys. C 10, 1717 (1977).\n\nStatistical Mechanics of Periodic Frustrated Ising Systems. 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Auerbach, E. Domany, and J. E. Gubernatis, Phys. Rev. B 37, 1719 (1988), and references therein.\n\n. H T Diep, Phys. Rev. B. 452863H. T. Diep, Phys. Rev. B 45, 2863 (1992).\n\n. B E Larson, J. Appl. Phys. 675240B.E. Larson, J. Appl. Phys. 67, 5240 (1990).\n\n. V T Ngo, H T Diep, Phys. Rev. B. 7535412V. T. Ngo and H. T. Diep, Phys. Rev. B 75, 035412 (2007).\n\n. P C Hohenberg, B I Halperin, Rev. Mod. Phys. 49435P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49 435 (1977).\n\n. A M Ferrenberg, R H Swendsen, Phys. Rev. Lett. 611195A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988) ; ibid 63, 1195(1989) .\n\n. A M Ferrenberg, D P Landau, Phys. Rev. B. 445081A. M. Ferrenberg and D. P. Landau, Phys. Rev. B 44, 5081 (1991).\n\n. H T Diep, H Kawamura, Phys. Rev. B. 407019H. T. Diep and H. Kawamura, Phys. Rev. B 40, 7019 (1989).\n\n. D Blankschtein, M Ma, A N Berker, Phys. Rev. B. 301362D. Blankschtein, M.Ma and A. N. Berker, Phys. Rev. B 30 (1984) 1362.\n\n. V T Ngo, D T Hoang, H T Diep, Phys. Rev. E. 8241123V. T. Ngo, D. T. Hoang and H. T. Diep, Phys. Rev. E 82 , 041123 (2010).\n\n. V Thanh Ngo, D Tien Hoang, H T Diep, Mod. Phys. Lett. B. 25929V. Thanh Ngo, D. Tien Hoang and H. T. Diep, Mod. Phys. Lett. B 25, 929 (2011).\n\n. V Thanh Ngo, D Tien Hoang, H T Diep, J. Phys.-Cond. Matt. 23226002V. Thanh Ngo, D. Tien Hoang and H. T. Diep, J. Phys.- Cond. Matt. 23, 226002 (2011).\n\n. H T Diep, Phys. Rev. B. 39397H. T. Diep, Phys. Rev. B 39, 397 (1989).\n\n. Y Magnin, K Akabli, H T Diep, I Harada, Computational Materials Science. 49Y. Magnin, K. Akabli, H. T. Diep and I. Harada, Com- putational Materials Science 49, S204-S209 (2010).\n\n. Y Magnin, Danh-Tai Hoang, H T Diep, Mod. Phys. Lett. B. 251029Y. Magnin, Danh-Tai Hoang and H. T. Diep, Mod. Phys. Lett. B 25, 1029 (2011).\n\n. Y Magnin, H T Diep, arXiv:1111.1507Y. Magnin and H. T. Diep, arXiv:1111.1507 (2011).\n\n. P.-G De Gennes, J Friedel, J. Phys. Chem. Solids. 471P.-G. de Gennes and J. Friedel, J. Phys. Chem. Solids 4, 71 (1958).\n\n. M E Fisher, J S Langer, Phys. Rev. Lett. 20665M. E. Fisher and J.S. Langer, Phys. Rev. Lett. 20 , 665 (1968).\n\n. M Kataoka, Phys. Rev. B. 63134435M. Kataoka, Phys. Rev. B 63, 134435 (2001).\n\n. C Haas, Phys. Rev. 168531C. Haas, Phys. Rev. 168, 531 (1968).\n\n. Y Magnin, K Akabli, H T Diep, Phys. Rev. B. 83144406Y. Magnin, K. Akabli and H. T. Diep, Phys. Rev. B 83, 144406(2011).\n\n. Danh-Tai Hoang, Y Magnin, H T Diep, Mod. Phys. Lett. B. 25937Danh-Tai Hoang, Y. Magnin and H. T. Diep, Mod. Phys. Lett. B 25, 937 (2011).\n\n. Vladimir V Prudnikov, Pavel V Prudnikov, Aleksandr S Krinitsyn, Andrei N Vakilov, Evgenii A Pospelov, Mikhail V Rychkov, Phys. Rev. E. 8111130and references thereinVladimir V. Prudnikov, Pavel V. Prudnikov, Aleksandr S. Krinitsyn, Andrei N. Vakilov, Evgenii A. Pospelov, and Mikhail V. Rychkov, Phys. Rev. E 81 011130 (2010), and references therein.\n"},"corpus_id":{"kind":"number","value":85520964,"string":"85,520,964"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1112.5724\",\n \"mag\": 1540957907,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1112.5724v1.pdf\"\n ],\n \"pdf_sha\": \"50c317a952dc3ee4b8e83d63b07756a7561df400\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Hexagonal-Close-Packed Lattice: Phase Transition and Spin Transport"},"authors":{"kind":"list like","value":["Danh-Tai Hoang \nLaboratoire de Physique Théorique et Modélisation\nUniversité de Cergy-Pontoise\nCNRS\nAvenue Adolphe Chauvin8089 2, 95302Cergy-Pontoise CedexUMRFrance","H T Diep \nLaboratoire de Physique Théorique et Modélisation\nUniversité de Cergy-Pontoise\nCNRS\nAvenue Adolphe Chauvin8089 2, 95302Cergy-Pontoise CedexUMRFrance"],"string":"[\n \"Danh-Tai Hoang \\nLaboratoire de Physique Théorique et Modélisation\\nUniversité de Cergy-Pontoise\\nCNRS\\nAvenue Adolphe Chauvin8089 2, 95302Cergy-Pontoise CedexUMRFrance\",\n \"H T Diep \\nLaboratoire de Physique Théorique et Modélisation\\nUniversité de Cergy-Pontoise\\nCNRS\\nAvenue Adolphe Chauvin8089 2, 95302Cergy-Pontoise CedexUMRFrance\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We study the ground state (GS) and the phase transition in a hexagonal-close-packed lattice with both XY and Ising models by using extensive Monte Carlo simulation. We suppose the in-plane interaction J1 and inter-plane interaction J2, both antiferromagnetic. The system is frustrated with two kinds of GS configuration below and above a critical value of η = J1/J2 (ηc). For the Ising case, one has ηc = 0.5 which separates in-plane ferromagnetic and antiferromagnetic states, while for the XY case ηc = 1/3 separates the collinear and non collinear spin configurations. The phase transition is shown to be of first (second) order for η > (<)ηc. The spin resistivity is calculated for the Ising case. It shows a rounded maximum at the magnetic transition in the second-order region, and a discontinuity in the first-order region of η. PACS numbers: 05.70.Fh ; 75.10.Hk ; 75.40.Mg ;"},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"Frustrated spin systems have been subject of intensive investigations during the last 30 years [1]. These systems are very unstable due to the competition between antagonist interactions or due to a geometrical frustration such as in the antiferromagnetic (AF) triangular lattice. A spin is said frustrated when it cannot find an orientation which \"fully\" satisfies all the interactions with its neighbors [2,3]. As a consequence of the frustration, the ground state (GS) is very highly degenerate. In the Ising case the GS degeneracy is often infinite as in the triangular lattice, the face-centered cubic (FCC) and hexagonalclose-packed (HCP) lattices, with AF interaction. In the case of vector spins, the GS is non collinear such as the 120-degree configuration in the XY and Heisenberg AF stacked triangular lattice (STL). In two dimensions (2D), several frustrated systems with Ising spin model have been exactly solved [4,5]. Among the most interesting models one can mention the frustrated generalized Kagome lattice [6] and the honeycomb lattice [7] where exotic features such as the existence of several phase transitions, the reentrance and the disorder lines have been exactly found by mapping these systems into vertex models [8]. In three dimensions (3D), the situation is complicated. The renormalization group (RG) [9,10], which provided a good understanding of the nature of the phase transition in non frustrated systems, encounters much of difficulties in application to frustrated systems. Among the most studied subjects during the last 20 years, one can mention the nature of the phase transition in the XY and Heisenberg STL. After a long debate [11] on whether it is a second-, or a first-order transition or it belongs to a * danh-tai.hoang@u-cergy.fr † diep@u-cergy.fr, corresponding author new universality class, the controversy has recently ended with the conclusion of a first-order transition [12,13].\nIn this paper, we are interested in the HCP antiferromagnet with Ising and XY spin models. Our purpose is to study its properties such as the ground state, the phase transition and the spin transport, in the case of anisotropic exchange interactions. The isotropic nearestneighbor (NN) AF interaction has been studied for Ising [14] and XY and Heisenberg spins [15]. These isotropic cases have been shown to undergo a phase transition of first order, and the infinite GS degeneracy is reduced to 6 at low temperatures [14,15]. The effect of anisotropic interaction and anisotropy on the GS in the case of vector spins has been studied [16].\nIn section II, we show our model and analyze the ground state. Results of Monte Carlo (MC) simulations are shown and discussed in section III. Concluding remarks are given in section V."},{"title":"II. MODEL AND GROUND-STATE ANALYSIS","content":"We consider the HCP lattice shown in Fig. 1. The stacking direction is z. The Hamiltonian is given by\nwhere S i is the spin at lattice site i and J ij is the AF interaction between nearest-neighbors (NN). We suppose that J ij = J 1 if the NN are on the xy triangular plane, and J ij = J 2 if the NN are on two adjacent planes (see Fig. 1). The GS can be determined by the steepestdescent method. This method is very simple [17] (i) we generate an initial configuration at random (ii) we calculate the local field created at a site by its neighbors using (1) (iii) we align the spin of that site along the calculated local field to minimize its energy (iv) we go to another site and repeat until all sites are visited: we say we make one sweep (v) we do a large number of sweeps per site until a good convergence is reached. However, one can also minimize the interaction energy as shown below to calculate the GS configuration. Since both interactions are AF (negative), for simplicity, we fix J 2 = −1 and vary J 1 . The unit of energy is taken as |J 2 | and the temperature T is in the unit of |J 2 |/k B where k B is the Boltzmann constant.\nLet us recall the GS in the case of isotropic interaction, namely J 1 = J 2 [15]. For the HCP lattice, each spin is shared by eight tetrahedra (four in the upper half-space and four in the lower half-space along the z axis) and a NN bond is shared by two tetrahedra. The GS spin configuration of the system is formed by stacking neighboring tetrahedra. In the GS, one has two pairs of antiparallel spins on each tetrahedron. Their axes form an arbitrary angle α. The degeneracy is thus infinite (see Fig. 2a of Ref. [15]). Of course, the periodic boundary conditions will reduce a number of configurations, but the degeneracy is still infinite. Of these GS's, one particular family of configurations of interest for both XY and Heisenberg cases is when α = 0. The GS is then collinear with two spins up and the other two down. The stacking sequence is then simplest: there are three equivalent configurations since there are three ways to choose the parallel spin pair in the original tetrahedron.\nWe now examine the case where J 1 = J 2 .\n• Ising case:\nThe steepest descent method with varying J 1 (J 2 = −1 gives two kinds of GS spin configuration: the first consists of xy ferromagnetic planes stacked antiferromagnetically along the z direction, while the second one is the stacking of xy AF planes such that each tetrahedron has two up and two down spins. The transition between the two configurations is determined as follows: one simply writes down the respective energies of a tetrahedron and compares them\nOne sees that\nThus the first configuration is more stable when |J 1 | < 0.5|J 2 |.\n• XY case:\nConsider one tetrahedron of the HCP lattice. The Hamiltonian for this cell is given by\nwhere the angles are defined in Fig. 2. The steepest descent method shows that while β and α have unique values for a given J 1 /J 2 , α is arbitrary, just as in the case of isotropic interaction [15] discussed above. To simplify the formulae, we take α = 0 in the following. The energy of the cell is written as\nThe critical values of β and γ are determined from the relations\nWe find the following solutions:\n. By comparing the energy values at these solutions, we obtain the minimum energy with the last solution: one has where cos β and cos γ are given above.\nBecause −1 ≤ cos β ≤ 1 and −1 ≤ cos γ ≤ 1, the above solution is valid for 1 3 ≤ J 1 /J 2 ≤ 1. We plot cos β, cos γ, β and γ in Fig. 3 where we observe that the non collinear GS configuration occurs in the interval 1/3 ≤ η = J 1 /J 2 ≤ 1.\n."},{"title":"III. PHASE TRANSITION: RESULTS","content":"We consider a sample size of L×L×L z where L and L z vary from 12 to 36 but L z can be different from L in order to detect the dependence of the GS on J 1 . The exchange interaction |J 2 | = 1 is used as the unit of energy. We use here MC simulations with a histogram technique to detect first-order transition. We equilibrate the system and average physical quantities over several millions of MC steps per spin. The averaged energy U and the heat capacity C V are calculated by\nwhere ... indicates the thermal average taken over microscopic states at T . The order parameter M is defined from the sublattice magnetization by\nwhere S i belongs to the sublattice K. Note that there are at least four sublattices which define the ordering of the spins on the tetrahedra if a long-range order is observed. The susceptibility is defined by\nIn MC simulations, we work at finite sizes, so for each size we have to determine the \"pseudo\" transition which corresponds in general to the maximum of the specific heat or of the susceptibility. The maxima of these quantities need not to be at the same temperature. Only at the infinite size, they should coincide. The theory of finite-size scaling [18][19][20] permits to deduce properties of a system at its thermodynamic limit. In order to check the first-order nature of the transition, we used the histogram technique which is very efficient in detecting weak first-order transitions and in calculating critical exponents of second-order transitions. [19,20] The main idea of this technique is to make an energy histogram at a temperature T 0 as close as possible to the transition temperature. Often, one has to try at several temperatures in the transition region. Using this histogram in the formulae of statistical physics for canonical distribution, one obtains energy histograms in a range of temperature around T 0 . In second-order transitions, these histograms are gaussian. They allows us to calculate averages of physical quantities as well as critical exponents using the finite-size scaling. In first-order transitions, the energy histogram shows a double-peak structure."},{"title":"A. Ising case","content":"In the following, the results in different figures are shown with L = 18 and L z = 8 (16 atomic planes along z). Since the GS changes at η c = 0.5, we show here examples on both sides of this value. Figure 4 shows the energy per spin E, the specific heat per spin C V , the order parameter M and the susceptibility χ, for η = 0.3. The transition is of second order. On the other side, we show in Fig. 5 the energy per spin and the order parameter versus T , for η = 0.85 and 1. We find a strong first-order transition in both cases. The discontinuity of E and M at the transition is very large. We show in Fig. 6 the energy histogram taken at the transition temperature for three values η = 0.3 (black), 0.85 (blue) and 1 (red). As seen, the first case is a gaussian distribution indicating a second-order transition, while the last two cases show a double-peak structure indicating a first-order transition.\nWe have calculated the critical temperature T C as a function of η. The phase diagram is shown in Fig. 7 where I and II indicate the ordering of the first, and second kinds, respectively. P indicates the paramagnetic phase. Note that the transition line between I and P is a second-order line, while that between II and P is a first-order line."},{"title":"B. XY case","content":"In the XY case, the change of the GS takes place at η = 1/3. Let us show in Fig. 8 the result for η = 0.3 where the GS is composed of ferromagnetic planes antiferromagnetically stacked in the z direction. The transi- tion is of second order.\nWe show now the result for the non collinear GS region in Fig. 9. The energy and the order parameter show clearly a discontinuity at the transition for η = 0.58 and 1. Using the histogram method described above, we have calculated the histogram shown in Fig. 10 for η = 0.3, 0.58 and 1. For η = 0.3 which is in the collinear region of the GS, the histogram is gaussian, confirming the secondorder transition observed in the data shown above. For η = 0.58 and 1 belonging to the non collinear region, the histogram shows a two-peak structure which confirms the first-order character of the transitions in this region.\nThe two peaks are very well separated with the dip going down to zero, indicating an energy discontinuity. The distance between the two peaks is the latent heat ∆E.\nWe show in Fig. 11 the transition temperature versus η where I and II indicate the collinear and non collinear phases, respectively. P denotes the paramagnetic state. The line separating I and P is a second-order transition line, while that separating II and P is the first-order one.\nTo close this section, we emphasize that all 3D frustrated systems known so far undergo a first-order transition: let us mention the AF STL [12,13], the FCC antiferromagnets [21], the simple cubic fully frustrated lattices [22][23][24][25], helimagnets [26] and the HCP lattice studied here."},{"title":"IV. SPIN TRANSPORT IN THE ISING CASE","content":"We have recently studied the spin transport in magnetically ordered materials by using an efficient MC simulation method [27,28]. The tour-de-force of the method was realized when we studied the semiconducting MnTe where the agreement with experiment is excellent [29]. In ferromagnets, the spin resistivity shows a sharp peak at the magnetic phase transition. The origin of this anomaly comes from the spin-spin correlation [30][31][32]. In antiferromagnets, such a peak is not sharp and often a broad maximum [33]. We have also studied the spin resistivity in two frustrated cases: the FCC Ising case [34] and the J 1 − J 2 simple cubic lattice [35]. The results show that the discontinuous transition in these systems gives rise to a discontinuity of the spin resistivity.\nUsing the method which has been described in detail elsewhere [27,34], we calculate the current of itinerant spins moving across the crystal under an electric field ǫ applied in the x direction. We suppose that each itinerant spin σ i interacts with the lattice spins around it within a sphere of radius D 1 at any point on its trajectory. The interaction Hamiltonian reads\nwhere the sum is performed over all lattice spins in the sphere around the itinerant spin σ i and I ij is the interaction which depends on the distance between σ i and S j . We suppose also the following interaction between an itinerant spin with its neighboring itinerant spins within a sphere of radius D 2\nFor simplicity, we suppose\nwhere I 0 , B, K 0 and C are constants chosen in such a way that the energy of an itinerant spin is smaller than that of a lattice spin. This choice is made to avoid the influence of itinerant spins on the ordering of the lattice spins. Note that in the almost-free electron model, I 0 ≃ K 0 ≃ 0, and in semiconductors they are larger but still weak with respect to J 1 and J 2 . A discussion on the choice of different parameters has been given for example in Ref. [27]. We suppose here a concentration of one itinerant per two lattice cells. At such a low concentration, the averaged distance between itinerant spins is much larger than the cutoff distance D 2 . In spite of this, due to the attractive nature of their interaction, Eq. (17), we have to use a chemical potential term to insure that itinerant spins do not form clusters. This potential is taken of the form D[n( r) − n 0 ] where D is a positive constant, n( r) the concentration of itinerant spins in the sphere of cutoff radius D 2 centered at the position r of the itinerant spin under consideration, and n 0 the averaged concentration. We perform the simulation by taking into account the relaxation time of the lattice spins [18,28,29] which is given by\nwhere A is a constant, ν the correlation critical exponent, and z the dynamic exponent. We see that as T tends to T C , τ L diverges. This phenomenon is known as the critical slowing-down. For the Ising spin model, ν = 0.638 (3D Ising universality) and z = 2.02 [36]. We have previously shown that τ L strongly affects the shape of ρ at T C [28]. By choosing A = 1, we fix τ L = 1 at T = 2T C deep inside the paramagnetic phase far above T C . This value is what we expect for thermal fluctuations in the disordered phase.\nThe spin resistivity ρ versus T for the two typical cases η = 0.3 and 1 is shown in Fig. 12 where distances D 1 and D 2 are in unit of the NN distance, energy constants I 0 , K 0 and D are in unit of |J 2 | = 1. We observe here that in the second-order region ρ has a rounded maximum and in the first-order region it undergoes an almost discontinuous jump at the transition.\nWe have varied the radius D 1 to see its effect on the resistivity ρ at the transition. We observe the same effect which has been seen in some other antiferromagnets [34,35]: at a given T , ρ oscillates slightly with distance. We have found that this oscillation comes from the oscillatory behavior of the difference between the numbers of up and down spins in the sphere as D 1 varies."},{"title":"V. CONCLUSION","content":"We have studied in this paper some properties of the HCP antiferromagnet with Ising and XY spin models. The in-plane J 1 and inter-plane J 2 interactions are supposed to be different. As a result, the GS spin configuration depends on the ratio η = J 1 /J 2 . We show that there exists a critical value η c where the GS changes. For the Ising case, we find η c = 0.5 below (above) which the spins in the xy planes are ferromagnetic (antiferromagnetic). For the XY case, the GS is collinear below η c = 1/3, and is non collinear above that value. The nature of the transition changes from a second order below η c to a first order above η c for both Ising and XY cases. We have also studied the spin resistivity in the Ising case. We found that the shape of ρ depends on the nature of the transition: in the second-order region, a rounded maximum is observed at the transition while in the first-order region, ρ undergoes a discontinuity at the transition as we have observed in other frustrated cases. These findings may help to understand the transition nature and the spin transport in different compounds with HCP structure."},{"title":"FIG. 1 .","content":""},{"title":"FIG. 2 .FIG. 3 .","content":""},{"title":"FIG. 5 .FIG. 6 .","content":""},{"title":"FIG. 7 .","content":""},{"title":"FIG. 8 .","content":""},{"title":"FIG. 9 .FIG. 10 .","content":""},{"title":"FIG. 12 .","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"Frustrated spin systems have been subject of intensive investigations during the last 30 years [1]. These systems are very unstable due to the competition between antagonist interactions or due to a geometrical frustration such as in the antiferromagnetic (AF) triangular lattice. A spin is said frustrated when it cannot find an orientation which \\\"fully\\\" satisfies all the interactions with its neighbors [2,3]. As a consequence of the frustration, the ground state (GS) is very highly degenerate. In the Ising case the GS degeneracy is often infinite as in the triangular lattice, the face-centered cubic (FCC) and hexagonalclose-packed (HCP) lattices, with AF interaction. In the case of vector spins, the GS is non collinear such as the 120-degree configuration in the XY and Heisenberg AF stacked triangular lattice (STL). In two dimensions (2D), several frustrated systems with Ising spin model have been exactly solved [4,5]. Among the most interesting models one can mention the frustrated generalized Kagome lattice [6] and the honeycomb lattice [7] where exotic features such as the existence of several phase transitions, the reentrance and the disorder lines have been exactly found by mapping these systems into vertex models [8]. In three dimensions (3D), the situation is complicated. The renormalization group (RG) [9,10], which provided a good understanding of the nature of the phase transition in non frustrated systems, encounters much of difficulties in application to frustrated systems. Among the most studied subjects during the last 20 years, one can mention the nature of the phase transition in the XY and Heisenberg STL. After a long debate [11] on whether it is a second-, or a first-order transition or it belongs to a * danh-tai.hoang@u-cergy.fr † diep@u-cergy.fr, corresponding author new universality class, the controversy has recently ended with the conclusion of a first-order transition [12,13].\\nIn this paper, we are interested in the HCP antiferromagnet with Ising and XY spin models. Our purpose is to study its properties such as the ground state, the phase transition and the spin transport, in the case of anisotropic exchange interactions. The isotropic nearestneighbor (NN) AF interaction has been studied for Ising [14] and XY and Heisenberg spins [15]. These isotropic cases have been shown to undergo a phase transition of first order, and the infinite GS degeneracy is reduced to 6 at low temperatures [14,15]. The effect of anisotropic interaction and anisotropy on the GS in the case of vector spins has been studied [16].\\nIn section II, we show our model and analyze the ground state. Results of Monte Carlo (MC) simulations are shown and discussed in section III. Concluding remarks are given in section V.\"\n },\n {\n \"title\": \"II. MODEL AND GROUND-STATE ANALYSIS\",\n \"content\": \"We consider the HCP lattice shown in Fig. 1. The stacking direction is z. The Hamiltonian is given by\\nwhere S i is the spin at lattice site i and J ij is the AF interaction between nearest-neighbors (NN). We suppose that J ij = J 1 if the NN are on the xy triangular plane, and J ij = J 2 if the NN are on two adjacent planes (see Fig. 1). The GS can be determined by the steepestdescent method. This method is very simple [17] (i) we generate an initial configuration at random (ii) we calculate the local field created at a site by its neighbors using (1) (iii) we align the spin of that site along the calculated local field to minimize its energy (iv) we go to another site and repeat until all sites are visited: we say we make one sweep (v) we do a large number of sweeps per site until a good convergence is reached. However, one can also minimize the interaction energy as shown below to calculate the GS configuration. Since both interactions are AF (negative), for simplicity, we fix J 2 = −1 and vary J 1 . The unit of energy is taken as |J 2 | and the temperature T is in the unit of |J 2 |/k B where k B is the Boltzmann constant.\\nLet us recall the GS in the case of isotropic interaction, namely J 1 = J 2 [15]. For the HCP lattice, each spin is shared by eight tetrahedra (four in the upper half-space and four in the lower half-space along the z axis) and a NN bond is shared by two tetrahedra. The GS spin configuration of the system is formed by stacking neighboring tetrahedra. In the GS, one has two pairs of antiparallel spins on each tetrahedron. Their axes form an arbitrary angle α. The degeneracy is thus infinite (see Fig. 2a of Ref. [15]). Of course, the periodic boundary conditions will reduce a number of configurations, but the degeneracy is still infinite. Of these GS's, one particular family of configurations of interest for both XY and Heisenberg cases is when α = 0. The GS is then collinear with two spins up and the other two down. The stacking sequence is then simplest: there are three equivalent configurations since there are three ways to choose the parallel spin pair in the original tetrahedron.\\nWe now examine the case where J 1 = J 2 .\\n• Ising case:\\nThe steepest descent method with varying J 1 (J 2 = −1 gives two kinds of GS spin configuration: the first consists of xy ferromagnetic planes stacked antiferromagnetically along the z direction, while the second one is the stacking of xy AF planes such that each tetrahedron has two up and two down spins. The transition between the two configurations is determined as follows: one simply writes down the respective energies of a tetrahedron and compares them\\nOne sees that\\nThus the first configuration is more stable when |J 1 | < 0.5|J 2 |.\\n• XY case:\\nConsider one tetrahedron of the HCP lattice. The Hamiltonian for this cell is given by\\nwhere the angles are defined in Fig. 2. The steepest descent method shows that while β and α have unique values for a given J 1 /J 2 , α is arbitrary, just as in the case of isotropic interaction [15] discussed above. To simplify the formulae, we take α = 0 in the following. The energy of the cell is written as\\nThe critical values of β and γ are determined from the relations\\nWe find the following solutions:\\n. By comparing the energy values at these solutions, we obtain the minimum energy with the last solution: one has where cos β and cos γ are given above.\\nBecause −1 ≤ cos β ≤ 1 and −1 ≤ cos γ ≤ 1, the above solution is valid for 1 3 ≤ J 1 /J 2 ≤ 1. We plot cos β, cos γ, β and γ in Fig. 3 where we observe that the non collinear GS configuration occurs in the interval 1/3 ≤ η = J 1 /J 2 ≤ 1.\\n.\"\n },\n {\n \"title\": \"III. PHASE TRANSITION: RESULTS\",\n \"content\": \"We consider a sample size of L×L×L z where L and L z vary from 12 to 36 but L z can be different from L in order to detect the dependence of the GS on J 1 . The exchange interaction |J 2 | = 1 is used as the unit of energy. We use here MC simulations with a histogram technique to detect first-order transition. We equilibrate the system and average physical quantities over several millions of MC steps per spin. The averaged energy U and the heat capacity C V are calculated by\\nwhere ... indicates the thermal average taken over microscopic states at T . The order parameter M is defined from the sublattice magnetization by\\nwhere S i belongs to the sublattice K. Note that there are at least four sublattices which define the ordering of the spins on the tetrahedra if a long-range order is observed. The susceptibility is defined by\\nIn MC simulations, we work at finite sizes, so for each size we have to determine the \\\"pseudo\\\" transition which corresponds in general to the maximum of the specific heat or of the susceptibility. The maxima of these quantities need not to be at the same temperature. Only at the infinite size, they should coincide. The theory of finite-size scaling [18][19][20] permits to deduce properties of a system at its thermodynamic limit. In order to check the first-order nature of the transition, we used the histogram technique which is very efficient in detecting weak first-order transitions and in calculating critical exponents of second-order transitions. [19,20] The main idea of this technique is to make an energy histogram at a temperature T 0 as close as possible to the transition temperature. Often, one has to try at several temperatures in the transition region. Using this histogram in the formulae of statistical physics for canonical distribution, one obtains energy histograms in a range of temperature around T 0 . In second-order transitions, these histograms are gaussian. They allows us to calculate averages of physical quantities as well as critical exponents using the finite-size scaling. In first-order transitions, the energy histogram shows a double-peak structure.\"\n },\n {\n \"title\": \"A. Ising case\",\n \"content\": \"In the following, the results in different figures are shown with L = 18 and L z = 8 (16 atomic planes along z). Since the GS changes at η c = 0.5, we show here examples on both sides of this value. Figure 4 shows the energy per spin E, the specific heat per spin C V , the order parameter M and the susceptibility χ, for η = 0.3. The transition is of second order. On the other side, we show in Fig. 5 the energy per spin and the order parameter versus T , for η = 0.85 and 1. We find a strong first-order transition in both cases. The discontinuity of E and M at the transition is very large. We show in Fig. 6 the energy histogram taken at the transition temperature for three values η = 0.3 (black), 0.85 (blue) and 1 (red). As seen, the first case is a gaussian distribution indicating a second-order transition, while the last two cases show a double-peak structure indicating a first-order transition.\\nWe have calculated the critical temperature T C as a function of η. The phase diagram is shown in Fig. 7 where I and II indicate the ordering of the first, and second kinds, respectively. P indicates the paramagnetic phase. Note that the transition line between I and P is a second-order line, while that between II and P is a first-order line.\"\n },\n {\n \"title\": \"B. XY case\",\n \"content\": \"In the XY case, the change of the GS takes place at η = 1/3. Let us show in Fig. 8 the result for η = 0.3 where the GS is composed of ferromagnetic planes antiferromagnetically stacked in the z direction. The transi- tion is of second order.\\nWe show now the result for the non collinear GS region in Fig. 9. The energy and the order parameter show clearly a discontinuity at the transition for η = 0.58 and 1. Using the histogram method described above, we have calculated the histogram shown in Fig. 10 for η = 0.3, 0.58 and 1. For η = 0.3 which is in the collinear region of the GS, the histogram is gaussian, confirming the secondorder transition observed in the data shown above. For η = 0.58 and 1 belonging to the non collinear region, the histogram shows a two-peak structure which confirms the first-order character of the transitions in this region.\\nThe two peaks are very well separated with the dip going down to zero, indicating an energy discontinuity. The distance between the two peaks is the latent heat ∆E.\\nWe show in Fig. 11 the transition temperature versus η where I and II indicate the collinear and non collinear phases, respectively. P denotes the paramagnetic state. The line separating I and P is a second-order transition line, while that separating II and P is the first-order one.\\nTo close this section, we emphasize that all 3D frustrated systems known so far undergo a first-order transition: let us mention the AF STL [12,13], the FCC antiferromagnets [21], the simple cubic fully frustrated lattices [22][23][24][25], helimagnets [26] and the HCP lattice studied here.\"\n },\n {\n \"title\": \"IV. SPIN TRANSPORT IN THE ISING CASE\",\n \"content\": \"We have recently studied the spin transport in magnetically ordered materials by using an efficient MC simulation method [27,28]. The tour-de-force of the method was realized when we studied the semiconducting MnTe where the agreement with experiment is excellent [29]. In ferromagnets, the spin resistivity shows a sharp peak at the magnetic phase transition. The origin of this anomaly comes from the spin-spin correlation [30][31][32]. In antiferromagnets, such a peak is not sharp and often a broad maximum [33]. We have also studied the spin resistivity in two frustrated cases: the FCC Ising case [34] and the J 1 − J 2 simple cubic lattice [35]. The results show that the discontinuous transition in these systems gives rise to a discontinuity of the spin resistivity.\\nUsing the method which has been described in detail elsewhere [27,34], we calculate the current of itinerant spins moving across the crystal under an electric field ǫ applied in the x direction. We suppose that each itinerant spin σ i interacts with the lattice spins around it within a sphere of radius D 1 at any point on its trajectory. The interaction Hamiltonian reads\\nwhere the sum is performed over all lattice spins in the sphere around the itinerant spin σ i and I ij is the interaction which depends on the distance between σ i and S j . We suppose also the following interaction between an itinerant spin with its neighboring itinerant spins within a sphere of radius D 2\\nFor simplicity, we suppose\\nwhere I 0 , B, K 0 and C are constants chosen in such a way that the energy of an itinerant spin is smaller than that of a lattice spin. This choice is made to avoid the influence of itinerant spins on the ordering of the lattice spins. Note that in the almost-free electron model, I 0 ≃ K 0 ≃ 0, and in semiconductors they are larger but still weak with respect to J 1 and J 2 . A discussion on the choice of different parameters has been given for example in Ref. [27]. We suppose here a concentration of one itinerant per two lattice cells. At such a low concentration, the averaged distance between itinerant spins is much larger than the cutoff distance D 2 . In spite of this, due to the attractive nature of their interaction, Eq. (17), we have to use a chemical potential term to insure that itinerant spins do not form clusters. This potential is taken of the form D[n( r) − n 0 ] where D is a positive constant, n( r) the concentration of itinerant spins in the sphere of cutoff radius D 2 centered at the position r of the itinerant spin under consideration, and n 0 the averaged concentration. We perform the simulation by taking into account the relaxation time of the lattice spins [18,28,29] which is given by\\nwhere A is a constant, ν the correlation critical exponent, and z the dynamic exponent. We see that as T tends to T C , τ L diverges. This phenomenon is known as the critical slowing-down. For the Ising spin model, ν = 0.638 (3D Ising universality) and z = 2.02 [36]. We have previously shown that τ L strongly affects the shape of ρ at T C [28]. By choosing A = 1, we fix τ L = 1 at T = 2T C deep inside the paramagnetic phase far above T C . This value is what we expect for thermal fluctuations in the disordered phase.\\nThe spin resistivity ρ versus T for the two typical cases η = 0.3 and 1 is shown in Fig. 12 where distances D 1 and D 2 are in unit of the NN distance, energy constants I 0 , K 0 and D are in unit of |J 2 | = 1. We observe here that in the second-order region ρ has a rounded maximum and in the first-order region it undergoes an almost discontinuous jump at the transition.\\nWe have varied the radius D 1 to see its effect on the resistivity ρ at the transition. We observe the same effect which has been seen in some other antiferromagnets [34,35]: at a given T , ρ oscillates slightly with distance. We have found that this oscillation comes from the oscillatory behavior of the difference between the numbers of up and down spins in the sphere as D 1 varies.\"\n },\n {\n \"title\": \"V. CONCLUSION\",\n \"content\": \"We have studied in this paper some properties of the HCP antiferromagnet with Ising and XY spin models. The in-plane J 1 and inter-plane J 2 interactions are supposed to be different. As a result, the GS spin configuration depends on the ratio η = J 1 /J 2 . We show that there exists a critical value η c where the GS changes. For the Ising case, we find η c = 0.5 below (above) which the spins in the xy planes are ferromagnetic (antiferromagnetic). For the XY case, the GS is collinear below η c = 1/3, and is non collinear above that value. The nature of the transition changes from a second order below η c to a first order above η c for both Ising and XY cases. We have also studied the spin resistivity in the Ising case. We found that the shape of ρ depends on the nature of the transition: in the second-order region, a rounded maximum is observed at the transition while in the first-order region, ρ undergoes a discontinuity at the transition as we have observed in other frustrated cases. These findings may help to understand the transition nature and the spin transport in different compounds with HCP structure.\"\n },\n {\n \"title\": \"FIG. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 2 .FIG. 3 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 5 .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. 9 .FIG. 10 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 12 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"HCP lattice. The in-plane and inter-plane interactions are indicated by J1 and J2.","content":"FIG. 1 .\n1HCP lattice. The in-plane and inter-plane interactions are indicated by J1 and J2."},{"id":"","caption":"Ground state in the XY case. The tetrahedron is projected on the xy plane. The spins are numbered from 1 to 4. See text for comments. Ground state in the XY case (color online). The angles β (black circles) and γ (blue void circles) and their cosinus are shown as functions of η = J1/J2. Non collinear GS configurations occur in the region 1/3 ≤ η ≤ 1 . See text for comments.","content":"FIG. 2 .FIG. 3 .\n23Ground state in the XY case. The tetrahedron is projected on the xy plane. The spins are numbered from 1 to 4. See text for comments. Ground state in the XY case (color online). The angles β (black circles) and γ (blue void circles) and their cosinus are shown as functions of η = J1/J2. Non collinear GS configurations occur in the region 1/3 ≤ η ≤ 1 . See text for comments."},{"id":"","caption":"Ising case (color online): Energy E and order parameter M for η = J1/J2 = 0.85 (blue void circles) and 1 (red triangles). See text for comments. Ising case (color online): Energy histogram P (E) versus E for η = 1( red triangles), 0.85 (blue void circles) and 0.3 (black circles). See text for comments.","content":"FIG. 5 .FIG. 6 .\n56Ising case (color online): Energy E and order parameter M for η = J1/J2 = 0.85 (blue void circles) and 1 (red triangles). See text for comments. Ising case (color online): Energy histogram P (E) versus E for η = 1( red triangles), 0.85 (blue void circles) and 0.3 (black circles). See text for comments."},{"id":"","caption":"Ising case: TC versus η. I, II and P denote the first, second and paramagnetic phases, respectively. See text for comments.","content":"FIG. 7 .\n7Ising case: TC versus η. I, II and P denote the first, second and paramagnetic phases, respectively. See text for comments."},{"id":"","caption":"XY case: Energy E, specific heat CV , order parameter M and susceptibility χ versus T for η = 0.3. See text for comments.","content":"FIG. 8 .\n8XY case: Energy E, specific heat CV , order parameter M and susceptibility χ versus T for η = 0.3. See text for comments."},{"id":"","caption":"XY case (color online): E and M versus T for η = 0.58 (blue void circles) and 1 (red triangles). XY case (color online): Energy histogram P versus E for η = 0.3 (black circles), 0.58 (blue void circles), 1 (red triangles) at the respective transition temperatures. See text for comments.","content":"FIG. 9 .FIG. 10 .\n910XY case (color online): E and M versus T for η = 0.58 (blue void circles) and 1 (red triangles). XY case (color online): Energy histogram P versus E for η = 0.3 (black circles), 0.58 (blue void circles), 1 (red triangles) at the respective transition temperatures. See text for comments."},{"id":"","caption":". 11. XY case: TC versus η. I, II and P denote the collinear, non collinear and paramagnetic phases, respectively. See text for comments.","content":". 11. XY case: TC versus η. I, II and P denote the collinear, non collinear and paramagnetic phases, respectively. See text for comments."},{"id":"","caption":"Ising case: Spin resistivity ρ versus T for η = 0.3 (upper) and 1 (lower). L = 18, Lz = 8 (16 planes in the z direction), D1 = D2 = 1, ǫ = 1, I0 = 2, K0 = 0.5, B = C = 1, A = 1, D = 0.5. All distances are in unit of the NN distance, energy constants are in unit of |J2| = 1. See text for comments.","content":"FIG. 12 .\n12Ising case: Spin resistivity ρ versus T for η = 0.3 (upper) and 1 (lower). L = 18, Lz = 8 (16 planes in the z direction), D1 = D2 = 1, ǫ = 1, I0 = 2, K0 = 0.5, B = C = 1, A = 1, D = 0.5. All distances are in unit of the NN distance, energy constants are in unit of |J2| = 1. See text for comments."},{"id":"","caption":"FIG. 4. Ising case: Energy E, specific heat CV , order parameter M and susceptibility χ versus T for η = J1/J2 = 0.3. See text for comments.","content":"FIG. 4. Ising case: Energy E, specific heat CV , order parameter M and susceptibility χ versus T for η = J1/J2 = 0.3. See text for comments.-2 \n\n-1.8 \n\n-1.6 \n\n-1.4 \n\n-1.2 \n\n-1 \n\n-0.8 \n\n1.8 \n2 \n2.2 \n2.4 \n2.6 \n2.8 \n3 \n\nT \n\nE \n\n0 \n\n0.5 \n\n1 \n\n1.5 \n\n2 \n\n2.5 \n\n3 \n\n1.8 \n2 \n2.2 \n2.4 \n2.6 \n2.8 \n3 \n\nT \n\nC V \n\n0 \n\n0.1 \n\n0.2 \n\n0.3 \n\n0.4 \n\n0.5 \n\n0.6 \n\n0.7 \n\n0.8 \n\n0.9 \n\n1 \n\n1.8 \n2 \n2.2 \n2.4 \n2.6 \n2.8 \n3 \n\nT \n\nM \n\n0 \n\n10 \n\n20 \n\n30 \n\n40 \n\n50 \n\n60 \n\n70 \n\n1.8 \n2 \n2.2 \n2.4 \n2.6 \n2.8 \n3 \n\nT \n\nχ"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"HCP lattice. The in-plane and inter-plane interactions are indicated by J1 and J2.\",\n \"content\": \"FIG. 1 .\\n1HCP lattice. The in-plane and inter-plane interactions are indicated by J1 and J2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Ground state in the XY case. The tetrahedron is projected on the xy plane. The spins are numbered from 1 to 4. See text for comments. Ground state in the XY case (color online). The angles β (black circles) and γ (blue void circles) and their cosinus are shown as functions of η = J1/J2. Non collinear GS configurations occur in the region 1/3 ≤ η ≤ 1 . See text for comments.\",\n \"content\": \"FIG. 2 .FIG. 3 .\\n23Ground state in the XY case. The tetrahedron is projected on the xy plane. The spins are numbered from 1 to 4. See text for comments. Ground state in the XY case (color online). The angles β (black circles) and γ (blue void circles) and their cosinus are shown as functions of η = J1/J2. Non collinear GS configurations occur in the region 1/3 ≤ η ≤ 1 . See text for comments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Ising case (color online): Energy E and order parameter M for η = J1/J2 = 0.85 (blue void circles) and 1 (red triangles). See text for comments. Ising case (color online): Energy histogram P (E) versus E for η = 1( red triangles), 0.85 (blue void circles) and 0.3 (black circles). See text for comments.\",\n \"content\": \"FIG. 5 .FIG. 6 .\\n56Ising case (color online): Energy E and order parameter M for η = J1/J2 = 0.85 (blue void circles) and 1 (red triangles). See text for comments. Ising case (color online): Energy histogram P (E) versus E for η = 1( red triangles), 0.85 (blue void circles) and 0.3 (black circles). See text for comments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Ising case: TC versus η. I, II and P denote the first, second and paramagnetic phases, respectively. See text for comments.\",\n \"content\": \"FIG. 7 .\\n7Ising case: TC versus η. I, II and P denote the first, second and paramagnetic phases, respectively. See text for comments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"XY case: Energy E, specific heat CV , order parameter M and susceptibility χ versus T for η = 0.3. See text for comments.\",\n \"content\": \"FIG. 8 .\\n8XY case: Energy E, specific heat CV , order parameter M and susceptibility χ versus T for η = 0.3. See text for comments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"XY case (color online): E and M versus T for η = 0.58 (blue void circles) and 1 (red triangles). XY case (color online): Energy histogram P versus E for η = 0.3 (black circles), 0.58 (blue void circles), 1 (red triangles) at the respective transition temperatures. See text for comments.\",\n \"content\": \"FIG. 9 .FIG. 10 .\\n910XY case (color online): E and M versus T for η = 0.58 (blue void circles) and 1 (red triangles). XY case (color online): Energy histogram P versus E for η = 0.3 (black circles), 0.58 (blue void circles), 1 (red triangles) at the respective transition temperatures. See text for comments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \". 11. XY case: TC versus η. I, II and P denote the collinear, non collinear and paramagnetic phases, respectively. See text for comments.\",\n \"content\": \". 11. XY case: TC versus η. I, II and P denote the collinear, non collinear and paramagnetic phases, respectively. See text for comments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Ising case: Spin resistivity ρ versus T for η = 0.3 (upper) and 1 (lower). L = 18, Lz = 8 (16 planes in the z direction), D1 = D2 = 1, ǫ = 1, I0 = 2, K0 = 0.5, B = C = 1, A = 1, D = 0.5. All distances are in unit of the NN distance, energy constants are in unit of |J2| = 1. See text for comments.\",\n \"content\": \"FIG. 12 .\\n12Ising case: Spin resistivity ρ versus T for η = 0.3 (upper) and 1 (lower). L = 18, Lz = 8 (16 planes in the z direction), D1 = D2 = 1, ǫ = 1, I0 = 2, K0 = 0.5, B = C = 1, A = 1, D = 0.5. All distances are in unit of the NN distance, energy constants are in unit of |J2| = 1. See text for comments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"FIG. 4. Ising case: Energy E, specific heat CV , order parameter M and susceptibility χ versus T for η = J1/J2 = 0.3. See text for comments.\",\n \"content\": \"FIG. 4. Ising case: Energy E, specific heat CV , order parameter M and susceptibility χ versus T for η = J1/J2 = 0.3. See text for comments.-2 \\n\\n-1.8 \\n\\n-1.6 \\n\\n-1.4 \\n\\n-1.2 \\n\\n-1 \\n\\n-0.8 \\n\\n1.8 \\n2 \\n2.2 \\n2.4 \\n2.6 \\n2.8 \\n3 \\n\\nT \\n\\nE \\n\\n0 \\n\\n0.5 \\n\\n1 \\n\\n1.5 \\n\\n2 \\n\\n2.5 \\n\\n3 \\n\\n1.8 \\n2 \\n2.2 \\n2.4 \\n2.6 \\n2.8 \\n3 \\n\\nT \\n\\nC V \\n\\n0 \\n\\n0.1 \\n\\n0.2 \\n\\n0.3 \\n\\n0.4 \\n\\n0.5 \\n\\n0.6 \\n\\n0.7 \\n\\n0.8 \\n\\n0.9 \\n\\n1 \\n\\n1.8 \\n2 \\n2.2 \\n2.4 \\n2.6 \\n2.8 \\n3 \\n\\nT \\n\\nM \\n\\n0 \\n\\n10 \\n\\n20 \\n\\n30 \\n\\n40 \\n\\n50 \\n\\n60 \\n\\n70 \\n\\n1.8 \\n2 \\n2.2 \\n2.4 \\n2.6 \\n2.8 \\n3 \\n\\nT \\n\\nχ\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Frustrated Spin Systems. World Scientific. H. T. DiepH. T. Diep (ed.), Frustrated Spin Systems, World Scien- tific, Singapor (2005).","doi":""},{"text":". G Toulouse, Commun. Phys. 2115G. Toulouse, Commun. Phys. 2, 115 (1977).","doi":""},{"text":". J Villain, J. Phys. C. 101717J. Villain, J. Phys. 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Rev. B 39, 397 (1989).","doi":""},{"text":". Y Magnin, K Akabli, H T Diep, I Harada, Computational Materials Science. 49Y. Magnin, K. Akabli, H. T. Diep and I. Harada, Com- putational Materials Science 49, S204-S209 (2010).","doi":""},{"text":". Y Magnin, Danh-Tai Hoang, H T Diep, Mod. Phys. Lett. B. 251029Y. Magnin, Danh-Tai Hoang and H. T. Diep, Mod. Phys. Lett. B 25, 1029 (2011).","doi":""},{"text":". Y Magnin, H T Diep, arXiv:1111.1507Y. Magnin and H. T. Diep, arXiv:1111.1507 (2011).","doi":"arXiv:1111.1507"},{"text":". P.-G De Gennes, J Friedel, J. Phys. Chem. Solids. 471P.-G. de Gennes and J. Friedel, J. Phys. Chem. Solids 4, 71 (1958).","doi":""},{"text":". M E Fisher, J S Langer, Phys. Rev. Lett. 20665M. E. Fisher and J.S. Langer, Phys. Rev. Lett. 20 , 665 (1968).","doi":""},{"text":". M Kataoka, Phys. Rev. B. 63134435M. Kataoka, Phys. Rev. B 63, 134435 (2001).","doi":""},{"text":". C Haas, Phys. Rev. 168531C. Haas, Phys. Rev. 168, 531 (1968).","doi":""},{"text":". Y Magnin, K Akabli, H T Diep, Phys. Rev. B. 83144406Y. Magnin, K. Akabli and H. T. Diep, Phys. Rev. B 83, 144406(2011).","doi":""},{"text":". Danh-Tai Hoang, Y Magnin, H T Diep, Mod. Phys. Lett. B. 25937Danh-Tai Hoang, Y. Magnin and H. T. Diep, Mod. Phys. Lett. B 25, 937 (2011).","doi":""},{"text":". Vladimir V Prudnikov, Pavel V Prudnikov, Aleksandr S Krinitsyn, Andrei N Vakilov, Evgenii A Pospelov, Mikhail V Rychkov, Phys. Rev. E. 8111130and references thereinVladimir V. Prudnikov, Pavel V. Prudnikov, Aleksandr S. Krinitsyn, Andrei N. Vakilov, Evgenii A. Pospelov, and Mikhail V. Rychkov, Phys. Rev. E 81 011130 (2010), and references therein.","doi":""}],"string":"[\n {\n \"text\": \"Frustrated Spin Systems. World Scientific. H. T. DiepH. T. Diep (ed.), Frustrated Spin Systems, World Scien- tific, Singapor (2005).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Toulouse, Commun. Phys. 2115G. Toulouse, Commun. Phys. 2, 115 (1977).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Villain, J. Phys. C. 101717J. Villain, J. Phys. C 10, 1717 (1977).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Statistical Mechanics of Periodic Frustrated Ising Systems. R Liebmann, Lecture Notes in Physics. 251Springer-VerlagR. Liebmann, Statistical Mechanics of Periodic Frus- trated Ising Systems, Lecture Notes in Physics Vol. 251 (Springer-Verlag, Berlin, 1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"chapter Exactly Solved Frustrated Models in Ref. H T Diep, H Giacomini, and references thereinH. T. Diep and H. Giacomini, chapter Exactly Solved Frustrated Models in Ref. [1], and references therein.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Debauche, H T Diep, P Azaria, H Giacomini, Phys. Rev. B. 442369M. Debauche, H. T. Diep, P. Azaria and H. Giacomini, Phys. Rev. B 44, 2369 (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H T Diep, M Debauche, H Giacomini, Phys. Rev. B. 438759H. T. Diep, M. Debauche, and H. Giacomini, Phys. Rev. B 43, R8759 (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Exactly Solved Models in Statistical Physics. R J Baxter, Academic Press IncLondonR. J. Baxter, Exactly Solved Models in Statistical Physics, Academic Press Inc., London (1982).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K G Wilson, Phys. Rev. B. 43174K. G. Wilson, Phys. Rev. B 4, 3174 (1971).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Zinn-Justin, Quantum Field Theory and Critical Phenomena. Oxford Univ. Press4th ed.J. Zinn-Justin, Quantum Field Theory and Critical Phe- nomena, 4th ed., Oxford Univ. Press (2002);\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Field theory, the renormalization group and critical phenomena. D J Amit, World ScientificSingaporD. J. Amit, Field theory, the renormalization group and critical phe- nomena, World Scientific, Singapor (1984).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"See review chapter Renormalization Group Approaches to Frustrated Magnets in D=3 by B. Delamotte, D. Mouhanna and M. Tissier in Ref. See review chapter Renormalization Group Approaches to Frustrated Magnets in D=3 by B. Delamotte, D. Mouhanna and M. Tissier in Ref. [1].\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V , Thanh Ngo, H T Diep, J. Appl. Phys. 103V. Thanh Ngo and H. T. Diep, J. Appl. Phys. 103, 07C712 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V T Ngo, H T Diep, Phys. Rev. E. 7831119V. T. Ngo and H. T. Diep, Phys. Rev. E 78, 031119 (2008).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Auerbach, E Domany, J E Gubernatis, Phys. Rev. B. 371719and references thereinD. Auerbach, E. Domany, and J. E. Gubernatis, Phys. Rev. B 37, 1719 (1988), and references therein.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H T Diep, Phys. Rev. B. 452863H. T. Diep, Phys. Rev. B 45, 2863 (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B E Larson, J. Appl. Phys. 675240B.E. Larson, J. Appl. Phys. 67, 5240 (1990).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V T Ngo, H T Diep, Phys. Rev. B. 7535412V. T. Ngo and H. T. Diep, Phys. Rev. B 75, 035412 (2007).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P C Hohenberg, B I Halperin, Rev. Mod. Phys. 49435P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49 435 (1977).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A M Ferrenberg, R H Swendsen, Phys. Rev. Lett. 611195A. M. Ferrenberg and R. H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988) ; ibid 63, 1195(1989) .\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A M Ferrenberg, D P Landau, Phys. Rev. B. 445081A. M. Ferrenberg and D. P. Landau, Phys. Rev. B 44, 5081 (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H T Diep, H Kawamura, Phys. Rev. B. 407019H. T. Diep and H. Kawamura, Phys. Rev. B 40, 7019 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Blankschtein, M Ma, A N Berker, Phys. Rev. B. 301362D. Blankschtein, M.Ma and A. N. Berker, Phys. Rev. B 30 (1984) 1362.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V T Ngo, D T Hoang, H T Diep, Phys. Rev. E. 8241123V. T. Ngo, D. T. Hoang and H. T. Diep, Phys. Rev. E 82 , 041123 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Thanh Ngo, D Tien Hoang, H T Diep, Mod. Phys. Lett. B. 25929V. Thanh Ngo, D. Tien Hoang and H. T. Diep, Mod. Phys. Lett. B 25, 929 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Thanh Ngo, D Tien Hoang, H T Diep, J. Phys.-Cond. Matt. 23226002V. Thanh Ngo, D. Tien Hoang and H. T. Diep, J. Phys.- Cond. Matt. 23, 226002 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H T Diep, Phys. Rev. B. 39397H. T. Diep, Phys. Rev. B 39, 397 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Magnin, K Akabli, H T Diep, I Harada, Computational Materials Science. 49Y. Magnin, K. Akabli, H. T. Diep and I. Harada, Com- putational Materials Science 49, S204-S209 (2010).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Magnin, Danh-Tai Hoang, H T Diep, Mod. Phys. Lett. B. 251029Y. Magnin, Danh-Tai Hoang and H. T. Diep, Mod. Phys. Lett. B 25, 1029 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Magnin, H T Diep, arXiv:1111.1507Y. Magnin and H. T. Diep, arXiv:1111.1507 (2011).\",\n \"doi\": \"arXiv:1111.1507\"\n },\n {\n \"text\": \". P.-G De Gennes, J Friedel, J. Phys. Chem. Solids. 471P.-G. de Gennes and J. Friedel, J. Phys. Chem. Solids 4, 71 (1958).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M E Fisher, J S Langer, Phys. Rev. Lett. 20665M. E. Fisher and J.S. Langer, Phys. Rev. Lett. 20 , 665 (1968).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Kataoka, Phys. Rev. B. 63134435M. Kataoka, Phys. Rev. B 63, 134435 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C Haas, Phys. Rev. 168531C. Haas, Phys. Rev. 168, 531 (1968).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Magnin, K Akabli, H T Diep, Phys. Rev. B. 83144406Y. Magnin, K. Akabli and H. T. Diep, Phys. Rev. B 83, 144406(2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Danh-Tai Hoang, Y Magnin, H T Diep, Mod. Phys. Lett. B. 25937Danh-Tai Hoang, Y. Magnin and H. T. Diep, Mod. Phys. Lett. B 25, 937 (2011).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Vladimir V Prudnikov, Pavel V Prudnikov, Aleksandr S Krinitsyn, Andrei N Vakilov, Evgenii A Pospelov, Mikhail V Rychkov, Phys. Rev. E. 8111130and references thereinVladimir V. Prudnikov, Pavel V. Prudnikov, Aleksandr S. Krinitsyn, Andrei N. Vakilov, Evgenii A. Pospelov, and Mikhail V. Rychkov, Phys. Rev. E 81 011130 (2010), and references therein.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":579,"cells":{"text":{"kind":"string","value":"\nHardware-In-the-Loop Measurements of the Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) System\n\n\nCarsten Bockelmann bockelmann@ant.uni-bremen.de \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen\n\nFabian Monsees monsees@ant.uni-bremen.de \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen\n\nMatthias Woltering woltering@ant.uni-bremen.de \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen\n\nArmin Dekorsy dekorsy@ant.uni-bremen.de \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen\n\nArbeitsbereich Nachrichtentechnik \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen\n\nHardware-In-the-Loop Measurements of the Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) System\n\nMCSM is a recently proposed novel system concept to solve the massive access problem envisioned in future communication systems like 5G and industry 4.0 systems. This work focuses on the practical verification of the theoretical gains that MCSM provides using a Hardware-In-the-Loop (HIL) measurement setup. We present results in two different scenarios: (i) a LoS lab setup and (ii) a non-LoS machine hall. In both scenarios MCSM shows promising performance in terms of the number of supported users and the achieved reliability.\n\nIntroduction\n\nMassive Machine-to-Machine (M2M) communication is expected to be one of the major drivers for new radio access technologies in the development of the 5th generation mobile networks (5G) as well as industrial radio systems for industry 4.0 (I4.0). For cellular networks forecasts indicate that in 2020 a factor 10 to 100 more machine type devices (MTD) than personal mobile phones will be served by a single base station with the number of MTDs connected in the range from 10.000 to 100.000 [NGMN15]. Similarly, the number of devices in industrial contexts will also grow tremendously given the needs of future I4.0 factories. While most current standards like UMTS/LTE or WLAN were conceived for relatively few devices with high data rates, a single MTD often only generates small amounts of user data, shows very diverse channel access or traffic patterns (triggered, periodically, sporadic or random), and for some applications also needs to be low-cost and very energy efficient to operate for long-lifetimes. As a consequence, new radio access technologies are required that are capable of supporting a variety of requirements. On the one hand massive access requires lean signalling and access structures; on the other hand, strict latency or reliability requirements need to be considered. Thus, PHY and MAC technologies with low signalling overhead and adaptable reliability, e.g. in terms of error protection, need to be designed for handling low data rate sporadic machine type communication (MTC) balancing the payload to overhead ratio as well as the reliability.\n\nTo this end, we recently proposed the so-called multi-carrier compressed sensing multiuser detection (MCSM) system [MWB15a] which specifically addresses massive access, bandwidth efficiency and flexible bandwidth allocation out of the many challenges in MTC design. Theoretical work on MCSM has shown promising performance in terms of error rates and the number of connected devices, but practical verification in different scenarios was still missing. The focus of this work is a measurement campaign to show the performance of the proposed MCSM system by a Hardware-In-the-Loop (HIL) setup. Future solutions for either 5G MTC or I4.0 need to fulfil a number of basic requirements that strongly influence the design. First, coexistence has to be guaranteed. In a 5G context coexistence with other services like broadband access is crucial, whereas in I4.0 coexistence with other wireless systems in the same frequency band has to be ensured. Furthermore, we focus on the basic MTC requirements of low signalling overhead, high reliability and flexible resource use with a special attention on large number of served users. The recently proposed MCSM system [MWB15a] tackles these MTC requirements by combining three different technology components as depicted in Figure 1: (i) multi-carrier concepts, (ii) compressive sensing multi-user detection (CS-MUD) and (iii) non-coherent modulation. Each component will be discussed in more detail in the following section.\n\n\nA multi-carrier system for machine type communication\n\n\nMCSM Components\n\n\nMulti-Carrier-Modulation\n\nFor flexible spectrum allocation multi-carrier systems with carefully designed waveforms have been identified as a potential solution [METI15]. On the one hand, coexistence management is enabled by good spectral containment; on the other hand, spectrally efficient multi-user approaches can be used for low-overhead direct random access communication. (OFDM) as a multi-carrier scheme, but general waveforms providing better spectral containment are equally applicable. Assume that the bandwidth shown in Figure 1 is divided into overall ܰ subcarriers with a subcarrier spacing ∆݂. Then a subset of ܰ ௌ subcarriers of is allocated to one particular MCSM system. MCSM systems are narrowband systems serving up to ܭ nodes per system and several systems can coexist within a certain bandwidth by simply allocating non-overlapping subcarrier blocks. This also enables blanking of sub-bands used by other communication systems. We restrict the following descriptions to a single MCSM system to ease notation. The most important design criterion in choosing ܰ ௌ determines the bandwidth of the MCSM system: in order to enable non-coherent detection, i.e. the third MCSM component, the bandwidth ܰ ௌ • Δ has to be smaller than or equal to the coherence bandwidth of the wireless channel ܤ ≈1/߬ , where ߬ denotes the delay spread of the channel.\n\n\nCompressive Sensing Multiuser Detection (CS-MUD)\n\nWhile multi-carrier modulation provides flexible spectrum allocation the physical channel access for each MCSM system has to be designed with very low signalling overhead. To this end, direct random access, i.e. the transmission of payload without a multi-transmission handshake to reserve resources beforehand, has been identified as a crucial component to enable massive access [MWB15a,SBD13a]. The basis for MCSM is CS-MUD, which has been proposed in literature to achieve direct random access with good collision resolution [BSDe13]. CS-MUD is a multi-user detection scheme that exploits certain structures in the multiuser signal. In massive MTC it can be assumed that only a subset of all nodes is active at any time leading to so-called sporadic access of the nodes. This sporadic nature can be exploited at the receiver by modelling inactive nodes as transmitting zeros instead of modulation symbols, yielding an efficient joint detection of activity and user data. The main advantage is a highly reduced signalling overhead compared to known techniques. In this context, sparse multi-user detection facilitating such a joint data detection and signal acquisition is a promising candidate for the application in 5G cellular networks [BSDe13, SBDe13, JSBD14].\n\n\nNon-coherent Modulation\n\nIn addition to CS-MUD a third component is required to lower the required signalling overhead even further. Many of today's communication systems employ coherent communication technologies to achieve high data rates at the cost of signalling, i.e. pilots are required for high quality channel estimation. However, differential modulation concepts like Differential M-Phase Shift Keying (D-MPSK) allow for a robust transmission without need for channel estimation as long as the channel can be assumed flat.\n\n\nMCSM Node Processing and Detection Model\n\nIn the following receiver and transmitter side processing are shortly summarized. For a complete theoretical description please refer to [MWB15b].\n\n\nTransmitter (Node)\n\nFigure 2 -Transmitter side processing at every MCSM node. Figure 2 shows the block diagram of a single node out of ܭ nodes summarizing one MCSM system. Each node in a MCSM system can either be active or inactive. In the following, we describe the transmitter processing of an active node.\n\nFirst, the ܰ data bits ∈ ሼ0,1ሽ ே ೆ of a single node ݇ are protected by a forward error correcting code and are modulated according to a D-MPSK modulation. In particular, the encoded information sequence ∈ ሼ0,1ሽ ே will be modulated to an -ܯPhase Shift Keying (M-PSK) alphabet resulting in symbols ∈ ℂ , where ܮ denotes the frame length. These symbols are differentially modulated to a so called D-MPSK signal through ܾ ሺ݅ሻ = ܽ ሺ݅ሻܾ ሺ݅ − 1ሻ, where ݅ denotes the symbol clock and a known starting phase such as ܾ ሺ0ሻ = 1 is assumed. In the following we restrict ourselves to the case that a MCSM node ݇ spreads each differentially modulated symbol ܾ (݅) to one OFDM symbol consisting of ܰ ௌ subcarriers, Figure 1 by using a unique node-specific spreading sequence ܛ ∈ ℂ ே ೄ that is known at the base station [VER98]. In this case we have ܰ ௌ = ܰ ௌ and the symbol clock ݅ remains unchanged for the OFDM symbols. Note that other constellations are possible.\n\nThe spread frequency domain symbol is then converted to a time domain symbol via a partial IDFT matrix ۴ ு ∈ ℂ ே×ே ೄ such that the node occupies only the ܰ ௌ out of ܰ subcarriers allocated to that particular MCSM system. Subsequently, a cyclic prefix is added in time domain by multiplication with a matrix P to counteract inter-symbol interference. Thus, the transmit signal vector ࢠ (݅) of node ݇ in time domain is formally ࢠ ሺ݅ሻ = ு ܾ ሺ݅ሻ.\n\n(1)\n\n\nReceiver Processing (Access Point)\n\nThe MCSM detection model employed at the base station contains the symbols of all ܭ nodes in the system that are potentially active and superimposed in time domain. Active as well as inactive users are included in the detection model due to the unknown user activity. Assuming a frequency flat fading channel over the allocated ܰ ௌ subcarriers (coherence bandwidth), the received signal at the base station for the ݅-th symbol is\nܡሺ݅ሻ = ۴ h ۷۴ ு ܛ b ሺ݅ሻ ୀ .(2)\nHere, h ۷ ∈ ℂ ே ೄ ×ே ೄ denotes a frequency flat fading channel with a single node specific channel tap. Furthermore, the user symbols ܾ (݅) can be either D-MPSK or zero\nb k u k {0, 1} OFDM Spread D-MPSK FEC × e −j2πf cfo,k s k b k c k PF H P s k b k\nwhich models the inactivity. Clearly, we can reformulate such that ۴ ۴ ு = ۷, and b ෨ (݅) = b (݅)h is the differentially modulated symbol weighted with the node specific channel coefficient. Hence, we can write the received signal in frequency domain in matrix form as\n)݅(ܡ = ෩ (݅),(3)\nwhere ෩ (݅) ∈ ℂ contains the differentially modulated symbols from all ܭ nodes weighted with the individual channel taps. The matrix ܁ ∈ ℂ ே ೄ × contains the spreading sequences of all ܭ nodes as column vectors. For a whole transmit frame, of ܮ symbols, we can write\n܇ = ෩ ,(4)\nwhere ܇ ∈ ℂ ே ೄ × and ෩ ∈ ℂ × summarize the received and transmitted frame, respectively. The block diagram of the Base-Station processing is exemplary shown in Figure 3. After converting the time domain symbols to frequency domain leading to (4), a Group Orthogonal Matching Pursuit (GOMP) is employed as Compressed Sensing algorithm to detect the activity of the nodes. The GOMP performs two steps consecutively: first it estimates the activity of one node and, subsequently, estimates the data of that node. This procedure is repeated until a certain stopping criterion is met. For a more detailed description of the GOMP the reader is referred to [SBD13b]. At the output of the GOMP, we have estimates ܾ ሺ݅ሻ for the modulated symbols of the active nodes. The receiver recovers the modulation symbols by performing differential demodulation\nܽ ො ሺ݅ሻ = ܾ ሺ݅ሻ ܾ ሺ݅ − 1ሻ = ܾ ሺ݅ሻܾ ሺ݅ − 1ሻ * ,(5)\nwhich are subsequently demapped to the code word estimate ො . Finally, the estimated information bit sequence ෝ is obtained by decoding the code word. Please note that this is the simplest form of a D-MPSK demodulator and more advanced techniques exist but are out of the scope of this paper. \nu 1 (i) u 2 (i) u K (i)\n\nMeasurement Setup\n\nA practical measurement with a simplified transmission scenario is implemented to evaluate the proposed MCSM system. Two development hardware platforms serve as measurement environment. First, the platforms, the frame structure and the modelling of the massive access will be introduced. Then, we shortly describe the two different measurement scenarios: 1) A Line of Sight (LoS) setup within our laboratory and 2) a non-Line of Sight (non-LoS) setup in a machine hall, where different interference factors like metal switch cabinets, machines and material affect the transmission. Furthermore, both platforms contain a Windows PC that controls the hardware environment and facilitates the offline processing of the HIL setup using MATLAB. A direct Ethernet connection between both platforms simulates an ideal feedback control channel to enable automated measurements and error rate calculations.\n\n\nHardware Platform and Setup\n\n\nFrame Structure and Emulation Massive Machine Access\n\nThe practical evaluation of massive access faces the challenge to realize a massive number of nodes that access a common base station. As illustrated in Figure 4 the two available transceivers are setup as HIL-devices: one serve as base station (RX), the other servers as a \"user emulator\" (TX). The base station processing is relatively straightforward along the lines described in section 2.2 with additional steps to ensure synchronization, CFO estimation and so on. The \"user emulation\", however, requires the generation of many virtual users and their mapping to the available hardware, i.e. primarily the four available antennas to emulate channel variations. The overall frame structure of our MCSM system is depicted in Figure 5. Multiple MCSM systems occupy the available 20 MHz bandwidth at a carrier frequency of 2.484 GHz. As a starting point to show the massive access capabilities of MCSM we target an LTE like implementation. The left part of Table 1 summarizes the resulting system parameters in comparison with LTE parameters. Additionally, Figure 5 (left) shows the principle timing of one OFDM symbol: the bandwidth is subdivided by an ܰ ൌ 2048 point-IFFT leading to a core symbol timing is ܶ ைிெ ൌ ݖܪܯ62/8402 ൌ 78.77 μݏ with a sub-carrier spacing Δ݂ ൌ 1/ܶ ைிெ ൌ 12.695 .ݖܪ݇ Additionally, the cyclic prefix (CP) of length 144 adds ܶ ൌ 5.538 μݏ to the overall OFDM symbol. ܰ ௌ Δ݂ = 253 ݖܪ݇ and consists of 150 OFDM symbols. This gives a total frame length of ܰ ெௌெ = ܰ * ሺܰ + ܰ ሻ = 328800 sample points and a total frame duration of ܶ ி,ெௌெ = ܰ ெௌெ * ܶ ௦ = 12.65 ,ݏ݉ which is slightly longer than the LTE frame of ܶ ி,்ா = 10 .ݏ݉ Additionally, a regular frequency hopping of the MCSM systems is applied every 10 OFDM symbol to exploit frequency diversity if the channel is not frequency flat or changing over time. Note that more general frequency hopping strategies can also be applied. To evaluate the start of one frame an LTE related synchronization sequence of 128 samples or ܶ ௦௬ = 4.92 μݏ is added to the frame. The virtual massive access of all nodes is realized at the transmitter (TX) seen in Figure 5 (right). Here, all signals of every active node (highlighted) are generated like described in (2). Each active node is randomly mapped to one antenna to emulate different user channels. The inactive nodes are modelled to be zero and therefore are not mapped to an antenna.\n\n\nFrame Structure and Node Parameters\n\n\nMeasurement Environments\n\nSetup 1: Line of Sight (LoS) Figure 6 shows the setup of the LoS scenario. The transmitter simulating the massive access (TX) is in the upper left, the receiver (RX) is located roughly 5 m away close to the door. The Line of Sight is unobstructed and mostly wooden furniture (some tables, cabinets, etc.) may act as scatters. Hence, the scenario is expected to show a mostly frequency flat behaviour which is reflected by the channel measurements depicted on the right. \n\n\nSetup 2: Non-Line of Sight (non-LoS)\n\nAs a second setup we choose a more realistic industrial environment, i.e. a machine hall used for automation laboratories and research. The left part of Figure 7 shows a picture of the scenario. The TX and the RX are located as far away as possible, which leads to a larger distance of roughly 14.5 m. Furthermore, the line of sight is now obstructed by metal switch cabinets, metal machines and other electrical components that are located within this hall. In addition to the setup 1 with only wooden cabinets and tables a high degree of reflections and stronger dampening of the signal is to be expected. Hence, the measured channel seen on the right of Figure 7 is much more frequency selective. Figure 8 shows the results of our measurements. The left plot depicts the frame error percentage for different node activities at a fixed transmit power in the LoS case (setup 1). We see that MCSM performs highly reliable if up to 10 nodes are simultaneously active. Increasing the number of active nodes gracefully increases the likelihood for frame errors due to the higher system load. In setup 1 MCSM is mainly limited by the node activity due to the frequency flat channel and sufficiently high received power. To support higher number of active nodes, the MCSM parametrization can be adapted towards longer spreading codes and higher bandwidth allocation\n\n\nsub-carrier index\n\n\nMeasurement Results\n\nIn contrast, the right plot of Figure 8 shows the average frame error rate (FER) performance of MCSM in a non-LoS scenario (setup 2) with random node activity. The nodes are active with probability ൌ 0.1, which means we have 0.1 * 60 ൌ 6 simultaneously active nodes on average using 273 kHz in a single MCSM system. In setup 2 the influence of different transmit powers can be observed due to the multipath sub-carrier index channel and the additional path loss compared to the LoS setup. For low transmit powers the FER is severely degraded but for high transmit powers a comparable performance to the LoS setup is achievable. For maximum transmit power of 26 dBm no errors could be observed during our measurement. Naturally, the coverage of any MCSM system given a minimum required reliability is strongly coupled with the propagation environment and allowed transmit powers.\n\n\nConclusion\n\nIn this work we have presented a practical evaluation of the MCSM system concept in a lab environment as well as a more realistic machine hall setup. Our measurements validate the theoretical findings of previous works with respect to the robustness and flexibility of the scheme. The presented results are only one exemplary MCSM parametrization which is specifically tuned to support a high number of potential users in massive M2M situations. Furthermore, other parametrizations can be tuned towards latency reduction (frame length) and higher reliability, e.g. by longer spreading sequences or by using stronger codes. In conclusion we have shown that the recently introduced MCSM system can be considered as a candidate technology for novel MTC applications in 5G and I4.0 applications.\n\nFigure 1 -\n1Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) concept.\n\nFigure 3 -\n3Base Station processing.\n\nFigure 4 -\n4Hardware-in-the-Loop (HIL) setup.\n\nFigure 5 -\n5Frame design and symbol timing (left) and virtual massive access (right)\n\nFigure 6 -\n6LoS scenario: (left) Schematic of the laboratory setup; (right) exemplary channel measurement with subcarrier 200 up to 700 and the bandwidth of the MCSM system considered.\n\nFigure 7 -\n7Non-LoS scenario -(left) Schematic of the machine hall setup; (right) exemplary channel measurement.\n\nFigure 8 -\n8Measured frame error percentage vs. the number of active users in LoS setup (left). Frame Error Rate vs. transmit power in non-LoS setup with random node activity (right).\n\n\nThus, the first technology component of MCSM is a suitable multi-carrier concept to flexibly allocate spectrum for multiple MCSM systems in one frequency band. In this paper we restrict to Orthogonal Frequency Division MultiplexingMCSM \nMCSM \nMCSM \n\nexisting wireless system \n\nexisting wireless system \nChannel in frequency domain \n\ncoherence \nbandwidth \n\nCompresssive Sensing Multi-\nUser Detection (CS-MUD) \n\nCollision resolution \nExploit sporadic activity \n\nMulti-Carrier Concepts \n\nFlexible spectrum use \nFrequency flat spectrum \n(coherence bandwidth) \n\nNon-Coherent Modulation \n\nAvoid channel estimation \nRobust against offsets \n\nfrequency \n\ntime \n\nTechnology Components \n\n\n\n\nBoth Nutaq hardware development platforms used in our measurement setup are equipped with an 8 channel 14 Bit AD converter with up to 125 MSPS as well as an 8 channel 14 Bit DA converter with up to 500 MSPS. The maximum sampling rate is 104 MHz, which is decreased to 26 MHz for this measurement due to memory bandwidth limitations. Hence, a sample timing of ܶ ௦ ൌ 1/26 ݖܪܯ ൌ 38.462 ݏ݊ results. Hz up to a maximum bandwidth of 20 MHz pass the frontend. The typical CFO is specified as 100ppm of the actual carrier frequency. Assuming an ISM Band with a carrier frequency ݂ ൌ 2.4 ݖܪܩ a CFO of approx. ݂ ிை, ൎ േ 2.4 ݖܪ݇ per transceiver is expected. During operation CFOs of up to ݂ ிை ൌ ሺ݂ ,் ݂ ிை,் ሻ െ ሺ݂ ,ோ ݂ ிை,ோ ሻ ൎ 5 ݖܪ݇ can be observed, which is well within specifications.Furthermore, both provide transceivers with four RF Frontends with 26 dBm maximum \ntransmission power for MIMO transmission within the 2.4 GHz and 5 GHz ISM Bands. \nThe RF frontend is implemented as a non-DC-coupled highpass, where all frequencies \nlarger 100 \n\nCompressive Sensing based Multi-User Detection for Machine-to-Machine. C Bockelmann, H Schepker, A Dekorsy, Transactions on Emerging Telecommunications Technologies: Special Issue on Machine-to-Machine: An emerging communication paradigm. 24Bockelmann, C.; Schepker, H.; Dekorsy, A.: Compressive Sensing based Multi-User Detection for Machine-to-Machine. In Transactions on Emerging Telecommunications Technologies: Special Issue on Machine-to-Machine: An emerging communication paradigm, Vol. 24, No. 4, pp. 389-400, June 2013.\n\nExploiting Sparsity in Channel and Data Estimation for Sporadic Multi-User Communication. H Schepker, C Bockelmann, A Dekorsy, Proc. 10th International Symposium on Wireless Communication Systems (ISWCS 13). 10th International Symposium on Wireless Communication Systems (ISWCS 13)Ilmenau, GermanySchepker, H.; Bockelmann, C.; Dekorsy, A.: Exploiting Sparsity in Channel and Data Estimation for Sporadic Multi-User Communication. In: Proc. 10th International Symposium on Wireless Communication Systems (ISWCS 13), Ilmenau, Germany, 27. -30. August 2013\n\nImproving Group Orthogonal Matching Pursuit with Iterative Feedback. H Schepker, C Bockelmann, A Dekorsy, Proc. IEEE 78 th Vehicular Technology Conference (VTC 2013-Fall). IEEE 78 th Vehicular Technology Conference (VTC 2013-Fall)Las-Vegas, USA, 2Schepker, H.; Bockelmann, C.; Dekorsy, A.: Improving Group Orthogonal Matching Pursuit with Iterative Feedback. In: Proc. IEEE 78 th Vehicular Technology Conference (VTC 2013-Fall), Las-Vegas, USA, 2. -5. September 2013\n\nCharacterization of Coded Random Access with Compressive Sensing based Multi-User Detection. Y Ji, Č Stefanović, C Bockelmann, A Dekorsy, P Popovski, Proc. IEEE Globecom. IEEE GlobecomAustin, TX, USA, 8Ji, Y.; Stefanović, Č.; Bockelmann, C.; Dekorsy, A.; Popovski, P.: Characterization of Coded Random Access with Compressive Sensing based Multi-User Detection. In: Proc. IEEE Globecom 2014, Austin, TX, USA, 8. -11. December 2014\n\nNGMN: 5G White Paper. NGMN: 5G White Paper, 2015\n\nMETIS D6.6: Final Report on the METIS System Concept and Technology Roadmap. METIS D6.6: Final Report on the METIS System Concept and Technology Roadmap, 2015.\n\nCompressive Sensing Multi-User Detection for Multi-Carrier Systems in Sporadic Machine Type Communication. F Monsees, M Woltering, C Bockelmann, A Dekorsy, Proc. IEEE 81th Vehicular Technology Conference (VTC2015-Spring). IEEE 81th Vehicular Technology Conference (VTC2015-Spring)Glasgow, Great BritainMonsees, F.; Woltering, M.; Bockelmann, C.; Dekorsy, A.: Compressive Sensing Multi-User Detection for Multi-Carrier Systems in Sporadic Machine Type Communication. In: Proc. IEEE 81th Vehicular Technology Conference (VTC2015- Spring), Glasgow, Great Britain, May 2015.\n\nMultiuser Detection. S Verdu, Cambridge University PressS. Verdu, Multiuser Detection, Cambridge University Press, 1998\n\nA Potential Solution for MTC: Multi-Carrier Compressive Sensing Multi-User Detection. F Monsees, M Woltering, C Bockelmann, A Dekorsy, Proc. IEEE The Asilomar Conference on Signals, Systems, and Computers, Asilomar Hotel and Conference Grounds. IEEE The Asilomar Conference on Signals, Systems, and Computers, Asilomar Hotel and Conference GroundsUSAMonsees, F.; Woltering, M.; Bockelmann, C.; Dekorsy, A.: A Potential Solution for MTC: Multi-Carrier Compressive Sensing Multi-User Detection. In: Proc. IEEE The Asilomar Conference on Signals, Systems, and Computers, Asilomar Hotel and Conference Grounds, USA, November 2015.\n"},"corpus_id":{"kind":"number","value":10927775,"string":"10,927,775"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1511.09278\",\n \"mag\": 2277299231,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/corr/BockelmannMWD15\",\n \"doi\": null\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1511.09278v1.pdf\"\n ],\n \"pdf_sha\": \"13b7000def6f40dc5da4107012539265b7ee0796\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Hardware-In-the-Loop Measurements of the Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) System"},"authors":{"kind":"list like","value":["Carsten Bockelmann bockelmann@ant.uni-bremen.de \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen","Fabian Monsees monsees@ant.uni-bremen.de \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen","Matthias Woltering woltering@ant.uni-bremen.de \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen","Armin Dekorsy dekorsy@ant.uni-bremen.de \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen","Arbeitsbereich Nachrichtentechnik \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen"],"string":"[\n \"Carsten Bockelmann bockelmann@ant.uni-bremen.de \\nUniversität Bremen Otto-Hahn-Alle\\nNW1 28359Bremen\",\n \"Fabian Monsees monsees@ant.uni-bremen.de \\nUniversität Bremen Otto-Hahn-Alle\\nNW1 28359Bremen\",\n \"Matthias Woltering woltering@ant.uni-bremen.de \\nUniversität Bremen Otto-Hahn-Alle\\nNW1 28359Bremen\",\n \"Armin Dekorsy dekorsy@ant.uni-bremen.de \\nUniversität Bremen Otto-Hahn-Alle\\nNW1 28359Bremen\",\n \"Arbeitsbereich Nachrichtentechnik \\nUniversität Bremen Otto-Hahn-Alle\\nNW1 28359Bremen\"\n]","isConversational":false},"abstract":{"kind":"string","value":"MCSM is a recently proposed novel system concept to solve the massive access problem envisioned in future communication systems like 5G and industry 4.0 systems. This work focuses on the practical verification of the theoretical gains that MCSM provides using a Hardware-In-the-Loop (HIL) measurement setup. We present results in two different scenarios: (i) a LoS lab setup and (ii) a non-LoS machine hall. In both scenarios MCSM shows promising performance in terms of the number of supported users and the achieved reliability."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Massive Machine-to-Machine (M2M) communication is expected to be one of the major drivers for new radio access technologies in the development of the 5th generation mobile networks (5G) as well as industrial radio systems for industry 4.0 (I4.0). For cellular networks forecasts indicate that in 2020 a factor 10 to 100 more machine type devices (MTD) than personal mobile phones will be served by a single base station with the number of MTDs connected in the range from 10.000 to 100.000 [NGMN15]. Similarly, the number of devices in industrial contexts will also grow tremendously given the needs of future I4.0 factories. While most current standards like UMTS/LTE or WLAN were conceived for relatively few devices with high data rates, a single MTD often only generates small amounts of user data, shows very diverse channel access or traffic patterns (triggered, periodically, sporadic or random), and for some applications also needs to be low-cost and very energy efficient to operate for long-lifetimes. As a consequence, new radio access technologies are required that are capable of supporting a variety of requirements. On the one hand massive access requires lean signalling and access structures; on the other hand, strict latency or reliability requirements need to be considered. Thus, PHY and MAC technologies with low signalling overhead and adaptable reliability, e.g. in terms of error protection, need to be designed for handling low data rate sporadic machine type communication (MTC) balancing the payload to overhead ratio as well as the reliability.\nTo this end, we recently proposed the so-called multi-carrier compressed sensing multiuser detection (MCSM) system [MWB15a] which specifically addresses massive access, bandwidth efficiency and flexible bandwidth allocation out of the many challenges in MTC design. Theoretical work on MCSM has shown promising performance in terms of error rates and the number of connected devices, but practical verification in different scenarios was still missing. The focus of this work is a measurement campaign to show the performance of the proposed MCSM system by a Hardware-In-the-Loop (HIL) setup. Future solutions for either 5G MTC or I4.0 need to fulfil a number of basic requirements that strongly influence the design. First, coexistence has to be guaranteed. In a 5G context coexistence with other services like broadband access is crucial, whereas in I4.0 coexistence with other wireless systems in the same frequency band has to be ensured. Furthermore, we focus on the basic MTC requirements of low signalling overhead, high reliability and flexible resource use with a special attention on large number of served users. The recently proposed MCSM system [MWB15a] tackles these MTC requirements by combining three different technology components as depicted in Figure 1: (i) multi-carrier concepts, (ii) compressive sensing multi-user detection (CS-MUD) and (iii) non-coherent modulation. Each component will be discussed in more detail in the following section."},{"title":"A multi-carrier system for machine type communication","content":""},{"title":"MCSM Components","content":""},{"title":"Multi-Carrier-Modulation","content":"For flexible spectrum allocation multi-carrier systems with carefully designed waveforms have been identified as a potential solution [METI15]. On the one hand, coexistence management is enabled by good spectral containment; on the other hand, spectrally efficient multi-user approaches can be used for low-overhead direct random access communication. (OFDM) as a multi-carrier scheme, but general waveforms providing better spectral containment are equally applicable. Assume that the bandwidth shown in Figure 1 is divided into overall ܰ subcarriers with a subcarrier spacing ∆݂. Then a subset of ܰ ௌ subcarriers of is allocated to one particular MCSM system. MCSM systems are narrowband systems serving up to ܭ nodes per system and several systems can coexist within a certain bandwidth by simply allocating non-overlapping subcarrier blocks. This also enables blanking of sub-bands used by other communication systems. We restrict the following descriptions to a single MCSM system to ease notation. The most important design criterion in choosing ܰ ௌ determines the bandwidth of the MCSM system: in order to enable non-coherent detection, i.e. the third MCSM component, the bandwidth ܰ ௌ • Δ has to be smaller than or equal to the coherence bandwidth of the wireless channel ܤ ≈1/߬ , where ߬ denotes the delay spread of the channel."},{"title":"Compressive Sensing Multiuser Detection (CS-MUD)","content":"While multi-carrier modulation provides flexible spectrum allocation the physical channel access for each MCSM system has to be designed with very low signalling overhead. To this end, direct random access, i.e. the transmission of payload without a multi-transmission handshake to reserve resources beforehand, has been identified as a crucial component to enable massive access [MWB15a,SBD13a]. The basis for MCSM is CS-MUD, which has been proposed in literature to achieve direct random access with good collision resolution [BSDe13]. CS-MUD is a multi-user detection scheme that exploits certain structures in the multiuser signal. In massive MTC it can be assumed that only a subset of all nodes is active at any time leading to so-called sporadic access of the nodes. This sporadic nature can be exploited at the receiver by modelling inactive nodes as transmitting zeros instead of modulation symbols, yielding an efficient joint detection of activity and user data. The main advantage is a highly reduced signalling overhead compared to known techniques. In this context, sparse multi-user detection facilitating such a joint data detection and signal acquisition is a promising candidate for the application in 5G cellular networks [BSDe13, SBDe13, JSBD14]."},{"title":"Non-coherent Modulation","content":"In addition to CS-MUD a third component is required to lower the required signalling overhead even further. Many of today's communication systems employ coherent communication technologies to achieve high data rates at the cost of signalling, i.e. pilots are required for high quality channel estimation. However, differential modulation concepts like Differential M-Phase Shift Keying (D-MPSK) allow for a robust transmission without need for channel estimation as long as the channel can be assumed flat."},{"title":"MCSM Node Processing and Detection Model","content":"In the following receiver and transmitter side processing are shortly summarized. For a complete theoretical description please refer to [MWB15b]."},{"title":"Transmitter (Node)","content":"Figure 2 -Transmitter side processing at every MCSM node. Figure 2 shows the block diagram of a single node out of ܭ nodes summarizing one MCSM system. Each node in a MCSM system can either be active or inactive. In the following, we describe the transmitter processing of an active node.\nFirst, the ܰ data bits ∈ ሼ0,1ሽ ே ೆ of a single node ݇ are protected by a forward error correcting code and are modulated according to a D-MPSK modulation. In particular, the encoded information sequence ∈ ሼ0,1ሽ ே will be modulated to an -ܯPhase Shift Keying (M-PSK) alphabet resulting in symbols ∈ ℂ , where ܮ denotes the frame length. These symbols are differentially modulated to a so called D-MPSK signal through ܾ ሺ݅ሻ = ܽ ሺ݅ሻܾ ሺ݅ − 1ሻ, where ݅ denotes the symbol clock and a known starting phase such as ܾ ሺ0ሻ = 1 is assumed. In the following we restrict ourselves to the case that a MCSM node ݇ spreads each differentially modulated symbol ܾ (݅) to one OFDM symbol consisting of ܰ ௌ subcarriers, Figure 1 by using a unique node-specific spreading sequence ܛ ∈ ℂ ே ೄ that is known at the base station [VER98]. In this case we have ܰ ௌ = ܰ ௌ and the symbol clock ݅ remains unchanged for the OFDM symbols. Note that other constellations are possible.\nThe spread frequency domain symbol is then converted to a time domain symbol via a partial IDFT matrix ۴ ு ∈ ℂ ே×ே ೄ such that the node occupies only the ܰ ௌ out of ܰ subcarriers allocated to that particular MCSM system. Subsequently, a cyclic prefix is added in time domain by multiplication with a matrix P to counteract inter-symbol interference. Thus, the transmit signal vector ࢠ (݅) of node ݇ in time domain is formally ࢠ ሺ݅ሻ = ு ܾ ሺ݅ሻ.\n(1)"},{"title":"Receiver Processing (Access Point)","content":"The MCSM detection model employed at the base station contains the symbols of all ܭ nodes in the system that are potentially active and superimposed in time domain. Active as well as inactive users are included in the detection model due to the unknown user activity. Assuming a frequency flat fading channel over the allocated ܰ ௌ subcarriers (coherence bandwidth), the received signal at the base station for the ݅-th symbol is\nHere, h ۷ ∈ ℂ ே ೄ ×ே ೄ denotes a frequency flat fading channel with a single node specific channel tap. Furthermore, the user symbols ܾ (݅) can be either D-MPSK or zero\nwhich models the inactivity. Clearly, we can reformulate such that ۴ ۴ ு = ۷, and b ෨ (݅) = b (݅)h is the differentially modulated symbol weighted with the node specific channel coefficient. Hence, we can write the received signal in frequency domain in matrix form as\nwhere ෩ (݅) ∈ ℂ contains the differentially modulated symbols from all ܭ nodes weighted with the individual channel taps. The matrix ܁ ∈ ℂ ே ೄ × contains the spreading sequences of all ܭ nodes as column vectors. For a whole transmit frame, of ܮ symbols, we can write\nwhere ܇ ∈ ℂ ே ೄ × and ෩ ∈ ℂ × summarize the received and transmitted frame, respectively. The block diagram of the Base-Station processing is exemplary shown in Figure 3. After converting the time domain symbols to frequency domain leading to (4), a Group Orthogonal Matching Pursuit (GOMP) is employed as Compressed Sensing algorithm to detect the activity of the nodes. The GOMP performs two steps consecutively: first it estimates the activity of one node and, subsequently, estimates the data of that node. This procedure is repeated until a certain stopping criterion is met. For a more detailed description of the GOMP the reader is referred to [SBD13b]. At the output of the GOMP, we have estimates ܾ ሺ݅ሻ for the modulated symbols of the active nodes. The receiver recovers the modulation symbols by performing differential demodulation\nwhich are subsequently demapped to the code word estimate ො . Finally, the estimated information bit sequence ෝ is obtained by decoding the code word. Please note that this is the simplest form of a D-MPSK demodulator and more advanced techniques exist but are out of the scope of this paper."},{"title":"Measurement Setup","content":"A practical measurement with a simplified transmission scenario is implemented to evaluate the proposed MCSM system. Two development hardware platforms serve as measurement environment. First, the platforms, the frame structure and the modelling of the massive access will be introduced. Then, we shortly describe the two different measurement scenarios: 1) A Line of Sight (LoS) setup within our laboratory and 2) a non-Line of Sight (non-LoS) setup in a machine hall, where different interference factors like metal switch cabinets, machines and material affect the transmission. Furthermore, both platforms contain a Windows PC that controls the hardware environment and facilitates the offline processing of the HIL setup using MATLAB. A direct Ethernet connection between both platforms simulates an ideal feedback control channel to enable automated measurements and error rate calculations."},{"title":"Hardware Platform and Setup","content":""},{"title":"Frame Structure and Emulation Massive Machine Access","content":"The practical evaluation of massive access faces the challenge to realize a massive number of nodes that access a common base station. As illustrated in Figure 4 the two available transceivers are setup as HIL-devices: one serve as base station (RX), the other servers as a \"user emulator\" (TX). The base station processing is relatively straightforward along the lines described in section 2.2 with additional steps to ensure synchronization, CFO estimation and so on. The \"user emulation\", however, requires the generation of many virtual users and their mapping to the available hardware, i.e. primarily the four available antennas to emulate channel variations. The overall frame structure of our MCSM system is depicted in Figure 5. Multiple MCSM systems occupy the available 20 MHz bandwidth at a carrier frequency of 2.484 GHz. As a starting point to show the massive access capabilities of MCSM we target an LTE like implementation. The left part of Table 1 summarizes the resulting system parameters in comparison with LTE parameters. Additionally, Figure 5 (left) shows the principle timing of one OFDM symbol: the bandwidth is subdivided by an ܰ ൌ 2048 point-IFFT leading to a core symbol timing is ܶ ைிெ ൌ ݖܪܯ62/8402 ൌ 78.77 μݏ with a sub-carrier spacing Δ݂ ൌ 1/ܶ ைிெ ൌ 12.695 .ݖܪ݇ Additionally, the cyclic prefix (CP) of length 144 adds ܶ ൌ 5.538 μݏ to the overall OFDM symbol. ܰ ௌ Δ݂ = 253 ݖܪ݇ and consists of 150 OFDM symbols. This gives a total frame length of ܰ ெௌெ = ܰ * ሺܰ + ܰ ሻ = 328800 sample points and a total frame duration of ܶ ி,ெௌெ = ܰ ெௌெ * ܶ ௦ = 12.65 ,ݏ݉ which is slightly longer than the LTE frame of ܶ ி,்ா = 10 .ݏ݉ Additionally, a regular frequency hopping of the MCSM systems is applied every 10 OFDM symbol to exploit frequency diversity if the channel is not frequency flat or changing over time. Note that more general frequency hopping strategies can also be applied. To evaluate the start of one frame an LTE related synchronization sequence of 128 samples or ܶ ௦௬ = 4.92 μݏ is added to the frame. The virtual massive access of all nodes is realized at the transmitter (TX) seen in Figure 5 (right). Here, all signals of every active node (highlighted) are generated like described in (2). Each active node is randomly mapped to one antenna to emulate different user channels. The inactive nodes are modelled to be zero and therefore are not mapped to an antenna."},{"title":"Frame Structure and Node Parameters","content":""},{"title":"Measurement Environments","content":"Setup 1: Line of Sight (LoS) Figure 6 shows the setup of the LoS scenario. The transmitter simulating the massive access (TX) is in the upper left, the receiver (RX) is located roughly 5 m away close to the door. The Line of Sight is unobstructed and mostly wooden furniture (some tables, cabinets, etc.) may act as scatters. Hence, the scenario is expected to show a mostly frequency flat behaviour which is reflected by the channel measurements depicted on the right."},{"title":"Setup 2: Non-Line of Sight (non-LoS)","content":"As a second setup we choose a more realistic industrial environment, i.e. a machine hall used for automation laboratories and research. The left part of Figure 7 shows a picture of the scenario. The TX and the RX are located as far away as possible, which leads to a larger distance of roughly 14.5 m. Furthermore, the line of sight is now obstructed by metal switch cabinets, metal machines and other electrical components that are located within this hall. In addition to the setup 1 with only wooden cabinets and tables a high degree of reflections and stronger dampening of the signal is to be expected. Hence, the measured channel seen on the right of Figure 7 is much more frequency selective. Figure 8 shows the results of our measurements. The left plot depicts the frame error percentage for different node activities at a fixed transmit power in the LoS case (setup 1). We see that MCSM performs highly reliable if up to 10 nodes are simultaneously active. Increasing the number of active nodes gracefully increases the likelihood for frame errors due to the higher system load. In setup 1 MCSM is mainly limited by the node activity due to the frequency flat channel and sufficiently high received power. To support higher number of active nodes, the MCSM parametrization can be adapted towards longer spreading codes and higher bandwidth allocation"},{"title":"sub-carrier index","content":""},{"title":"Measurement Results","content":"In contrast, the right plot of Figure 8 shows the average frame error rate (FER) performance of MCSM in a non-LoS scenario (setup 2) with random node activity. The nodes are active with probability ൌ 0.1, which means we have 0.1 * 60 ൌ 6 simultaneously active nodes on average using 273 kHz in a single MCSM system. In setup 2 the influence of different transmit powers can be observed due to the multipath sub-carrier index channel and the additional path loss compared to the LoS setup. For low transmit powers the FER is severely degraded but for high transmit powers a comparable performance to the LoS setup is achievable. For maximum transmit power of 26 dBm no errors could be observed during our measurement. Naturally, the coverage of any MCSM system given a minimum required reliability is strongly coupled with the propagation environment and allowed transmit powers."},{"title":"Conclusion","content":"In this work we have presented a practical evaluation of the MCSM system concept in a lab environment as well as a more realistic machine hall setup. Our measurements validate the theoretical findings of previous works with respect to the robustness and flexibility of the scheme. The presented results are only one exemplary MCSM parametrization which is specifically tuned to support a high number of potential users in massive M2M situations. Furthermore, other parametrizations can be tuned towards latency reduction (frame length) and higher reliability, e.g. by longer spreading sequences or by using stronger codes. In conclusion we have shown that the recently introduced MCSM system can be considered as a candidate technology for novel MTC applications in 5G and I4.0 applications."},{"title":"Figure 1 -","content":""},{"title":"Figure 3 -","content":""},{"title":"Figure 4 -","content":""},{"title":"Figure 5 -","content":""},{"title":"Figure 6 -","content":""},{"title":"Figure 7 -","content":""},{"title":"Figure 8 -","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Massive Machine-to-Machine (M2M) communication is expected to be one of the major drivers for new radio access technologies in the development of the 5th generation mobile networks (5G) as well as industrial radio systems for industry 4.0 (I4.0). For cellular networks forecasts indicate that in 2020 a factor 10 to 100 more machine type devices (MTD) than personal mobile phones will be served by a single base station with the number of MTDs connected in the range from 10.000 to 100.000 [NGMN15]. Similarly, the number of devices in industrial contexts will also grow tremendously given the needs of future I4.0 factories. While most current standards like UMTS/LTE or WLAN were conceived for relatively few devices with high data rates, a single MTD often only generates small amounts of user data, shows very diverse channel access or traffic patterns (triggered, periodically, sporadic or random), and for some applications also needs to be low-cost and very energy efficient to operate for long-lifetimes. As a consequence, new radio access technologies are required that are capable of supporting a variety of requirements. On the one hand massive access requires lean signalling and access structures; on the other hand, strict latency or reliability requirements need to be considered. Thus, PHY and MAC technologies with low signalling overhead and adaptable reliability, e.g. in terms of error protection, need to be designed for handling low data rate sporadic machine type communication (MTC) balancing the payload to overhead ratio as well as the reliability.\\nTo this end, we recently proposed the so-called multi-carrier compressed sensing multiuser detection (MCSM) system [MWB15a] which specifically addresses massive access, bandwidth efficiency and flexible bandwidth allocation out of the many challenges in MTC design. Theoretical work on MCSM has shown promising performance in terms of error rates and the number of connected devices, but practical verification in different scenarios was still missing. The focus of this work is a measurement campaign to show the performance of the proposed MCSM system by a Hardware-In-the-Loop (HIL) setup. Future solutions for either 5G MTC or I4.0 need to fulfil a number of basic requirements that strongly influence the design. First, coexistence has to be guaranteed. In a 5G context coexistence with other services like broadband access is crucial, whereas in I4.0 coexistence with other wireless systems in the same frequency band has to be ensured. Furthermore, we focus on the basic MTC requirements of low signalling overhead, high reliability and flexible resource use with a special attention on large number of served users. The recently proposed MCSM system [MWB15a] tackles these MTC requirements by combining three different technology components as depicted in Figure 1: (i) multi-carrier concepts, (ii) compressive sensing multi-user detection (CS-MUD) and (iii) non-coherent modulation. Each component will be discussed in more detail in the following section.\"\n },\n {\n \"title\": \"A multi-carrier system for machine type communication\",\n \"content\": \"\"\n },\n {\n \"title\": \"MCSM Components\",\n \"content\": \"\"\n },\n {\n \"title\": \"Multi-Carrier-Modulation\",\n \"content\": \"For flexible spectrum allocation multi-carrier systems with carefully designed waveforms have been identified as a potential solution [METI15]. On the one hand, coexistence management is enabled by good spectral containment; on the other hand, spectrally efficient multi-user approaches can be used for low-overhead direct random access communication. (OFDM) as a multi-carrier scheme, but general waveforms providing better spectral containment are equally applicable. Assume that the bandwidth shown in Figure 1 is divided into overall ܰ subcarriers with a subcarrier spacing ∆݂. Then a subset of ܰ ௌ subcarriers of is allocated to one particular MCSM system. MCSM systems are narrowband systems serving up to ܭ nodes per system and several systems can coexist within a certain bandwidth by simply allocating non-overlapping subcarrier blocks. This also enables blanking of sub-bands used by other communication systems. We restrict the following descriptions to a single MCSM system to ease notation. The most important design criterion in choosing ܰ ௌ determines the bandwidth of the MCSM system: in order to enable non-coherent detection, i.e. the third MCSM component, the bandwidth ܰ ௌ • Δ has to be smaller than or equal to the coherence bandwidth of the wireless channel ܤ ≈1/߬ , where ߬ denotes the delay spread of the channel.\"\n },\n {\n \"title\": \"Compressive Sensing Multiuser Detection (CS-MUD)\",\n \"content\": \"While multi-carrier modulation provides flexible spectrum allocation the physical channel access for each MCSM system has to be designed with very low signalling overhead. To this end, direct random access, i.e. the transmission of payload without a multi-transmission handshake to reserve resources beforehand, has been identified as a crucial component to enable massive access [MWB15a,SBD13a]. The basis for MCSM is CS-MUD, which has been proposed in literature to achieve direct random access with good collision resolution [BSDe13]. CS-MUD is a multi-user detection scheme that exploits certain structures in the multiuser signal. In massive MTC it can be assumed that only a subset of all nodes is active at any time leading to so-called sporadic access of the nodes. This sporadic nature can be exploited at the receiver by modelling inactive nodes as transmitting zeros instead of modulation symbols, yielding an efficient joint detection of activity and user data. The main advantage is a highly reduced signalling overhead compared to known techniques. In this context, sparse multi-user detection facilitating such a joint data detection and signal acquisition is a promising candidate for the application in 5G cellular networks [BSDe13, SBDe13, JSBD14].\"\n },\n {\n \"title\": \"Non-coherent Modulation\",\n \"content\": \"In addition to CS-MUD a third component is required to lower the required signalling overhead even further. Many of today's communication systems employ coherent communication technologies to achieve high data rates at the cost of signalling, i.e. pilots are required for high quality channel estimation. However, differential modulation concepts like Differential M-Phase Shift Keying (D-MPSK) allow for a robust transmission without need for channel estimation as long as the channel can be assumed flat.\"\n },\n {\n \"title\": \"MCSM Node Processing and Detection Model\",\n \"content\": \"In the following receiver and transmitter side processing are shortly summarized. For a complete theoretical description please refer to [MWB15b].\"\n },\n {\n \"title\": \"Transmitter (Node)\",\n \"content\": \"Figure 2 -Transmitter side processing at every MCSM node. Figure 2 shows the block diagram of a single node out of ܭ nodes summarizing one MCSM system. Each node in a MCSM system can either be active or inactive. In the following, we describe the transmitter processing of an active node.\\nFirst, the ܰ data bits ∈ ሼ0,1ሽ ே ೆ of a single node ݇ are protected by a forward error correcting code and are modulated according to a D-MPSK modulation. In particular, the encoded information sequence ∈ ሼ0,1ሽ ே will be modulated to an -ܯPhase Shift Keying (M-PSK) alphabet resulting in symbols ∈ ℂ , where ܮ denotes the frame length. These symbols are differentially modulated to a so called D-MPSK signal through ܾ ሺ݅ሻ = ܽ ሺ݅ሻܾ ሺ݅ − 1ሻ, where ݅ denotes the symbol clock and a known starting phase such as ܾ ሺ0ሻ = 1 is assumed. In the following we restrict ourselves to the case that a MCSM node ݇ spreads each differentially modulated symbol ܾ (݅) to one OFDM symbol consisting of ܰ ௌ subcarriers, Figure 1 by using a unique node-specific spreading sequence ܛ ∈ ℂ ே ೄ that is known at the base station [VER98]. In this case we have ܰ ௌ = ܰ ௌ and the symbol clock ݅ remains unchanged for the OFDM symbols. Note that other constellations are possible.\\nThe spread frequency domain symbol is then converted to a time domain symbol via a partial IDFT matrix ۴ ு ∈ ℂ ே×ே ೄ such that the node occupies only the ܰ ௌ out of ܰ subcarriers allocated to that particular MCSM system. Subsequently, a cyclic prefix is added in time domain by multiplication with a matrix P to counteract inter-symbol interference. Thus, the transmit signal vector ࢠ (݅) of node ݇ in time domain is formally ࢠ ሺ݅ሻ = ு ܾ ሺ݅ሻ.\\n(1)\"\n },\n {\n \"title\": \"Receiver Processing (Access Point)\",\n \"content\": \"The MCSM detection model employed at the base station contains the symbols of all ܭ nodes in the system that are potentially active and superimposed in time domain. Active as well as inactive users are included in the detection model due to the unknown user activity. Assuming a frequency flat fading channel over the allocated ܰ ௌ subcarriers (coherence bandwidth), the received signal at the base station for the ݅-th symbol is\\nHere, h ۷ ∈ ℂ ே ೄ ×ே ೄ denotes a frequency flat fading channel with a single node specific channel tap. Furthermore, the user symbols ܾ (݅) can be either D-MPSK or zero\\nwhich models the inactivity. Clearly, we can reformulate such that ۴ ۴ ு = ۷, and b ෨ (݅) = b (݅)h is the differentially modulated symbol weighted with the node specific channel coefficient. Hence, we can write the received signal in frequency domain in matrix form as\\nwhere ෩ (݅) ∈ ℂ contains the differentially modulated symbols from all ܭ nodes weighted with the individual channel taps. The matrix ܁ ∈ ℂ ே ೄ × contains the spreading sequences of all ܭ nodes as column vectors. For a whole transmit frame, of ܮ symbols, we can write\\nwhere ܇ ∈ ℂ ே ೄ × and ෩ ∈ ℂ × summarize the received and transmitted frame, respectively. The block diagram of the Base-Station processing is exemplary shown in Figure 3. After converting the time domain symbols to frequency domain leading to (4), a Group Orthogonal Matching Pursuit (GOMP) is employed as Compressed Sensing algorithm to detect the activity of the nodes. The GOMP performs two steps consecutively: first it estimates the activity of one node and, subsequently, estimates the data of that node. This procedure is repeated until a certain stopping criterion is met. For a more detailed description of the GOMP the reader is referred to [SBD13b]. At the output of the GOMP, we have estimates ܾ ሺ݅ሻ for the modulated symbols of the active nodes. The receiver recovers the modulation symbols by performing differential demodulation\\nwhich are subsequently demapped to the code word estimate ො . Finally, the estimated information bit sequence ෝ is obtained by decoding the code word. Please note that this is the simplest form of a D-MPSK demodulator and more advanced techniques exist but are out of the scope of this paper.\"\n },\n {\n \"title\": \"Measurement Setup\",\n \"content\": \"A practical measurement with a simplified transmission scenario is implemented to evaluate the proposed MCSM system. Two development hardware platforms serve as measurement environment. First, the platforms, the frame structure and the modelling of the massive access will be introduced. Then, we shortly describe the two different measurement scenarios: 1) A Line of Sight (LoS) setup within our laboratory and 2) a non-Line of Sight (non-LoS) setup in a machine hall, where different interference factors like metal switch cabinets, machines and material affect the transmission. Furthermore, both platforms contain a Windows PC that controls the hardware environment and facilitates the offline processing of the HIL setup using MATLAB. A direct Ethernet connection between both platforms simulates an ideal feedback control channel to enable automated measurements and error rate calculations.\"\n },\n {\n \"title\": \"Hardware Platform and Setup\",\n \"content\": \"\"\n },\n {\n \"title\": \"Frame Structure and Emulation Massive Machine Access\",\n \"content\": \"The practical evaluation of massive access faces the challenge to realize a massive number of nodes that access a common base station. As illustrated in Figure 4 the two available transceivers are setup as HIL-devices: one serve as base station (RX), the other servers as a \\\"user emulator\\\" (TX). The base station processing is relatively straightforward along the lines described in section 2.2 with additional steps to ensure synchronization, CFO estimation and so on. The \\\"user emulation\\\", however, requires the generation of many virtual users and their mapping to the available hardware, i.e. primarily the four available antennas to emulate channel variations. The overall frame structure of our MCSM system is depicted in Figure 5. Multiple MCSM systems occupy the available 20 MHz bandwidth at a carrier frequency of 2.484 GHz. As a starting point to show the massive access capabilities of MCSM we target an LTE like implementation. The left part of Table 1 summarizes the resulting system parameters in comparison with LTE parameters. Additionally, Figure 5 (left) shows the principle timing of one OFDM symbol: the bandwidth is subdivided by an ܰ ൌ 2048 point-IFFT leading to a core symbol timing is ܶ ைிெ ൌ ݖܪܯ62/8402 ൌ 78.77 μݏ with a sub-carrier spacing Δ݂ ൌ 1/ܶ ைிெ ൌ 12.695 .ݖܪ݇ Additionally, the cyclic prefix (CP) of length 144 adds ܶ ൌ 5.538 μݏ to the overall OFDM symbol. ܰ ௌ Δ݂ = 253 ݖܪ݇ and consists of 150 OFDM symbols. This gives a total frame length of ܰ ெௌெ = ܰ * ሺܰ + ܰ ሻ = 328800 sample points and a total frame duration of ܶ ி,ெௌெ = ܰ ெௌெ * ܶ ௦ = 12.65 ,ݏ݉ which is slightly longer than the LTE frame of ܶ ி,்ா = 10 .ݏ݉ Additionally, a regular frequency hopping of the MCSM systems is applied every 10 OFDM symbol to exploit frequency diversity if the channel is not frequency flat or changing over time. Note that more general frequency hopping strategies can also be applied. To evaluate the start of one frame an LTE related synchronization sequence of 128 samples or ܶ ௦௬ = 4.92 μݏ is added to the frame. The virtual massive access of all nodes is realized at the transmitter (TX) seen in Figure 5 (right). Here, all signals of every active node (highlighted) are generated like described in (2). Each active node is randomly mapped to one antenna to emulate different user channels. The inactive nodes are modelled to be zero and therefore are not mapped to an antenna.\"\n },\n {\n \"title\": \"Frame Structure and Node Parameters\",\n \"content\": \"\"\n },\n {\n \"title\": \"Measurement Environments\",\n \"content\": \"Setup 1: Line of Sight (LoS) Figure 6 shows the setup of the LoS scenario. The transmitter simulating the massive access (TX) is in the upper left, the receiver (RX) is located roughly 5 m away close to the door. The Line of Sight is unobstructed and mostly wooden furniture (some tables, cabinets, etc.) may act as scatters. Hence, the scenario is expected to show a mostly frequency flat behaviour which is reflected by the channel measurements depicted on the right.\"\n },\n {\n \"title\": \"Setup 2: Non-Line of Sight (non-LoS)\",\n \"content\": \"As a second setup we choose a more realistic industrial environment, i.e. a machine hall used for automation laboratories and research. The left part of Figure 7 shows a picture of the scenario. The TX and the RX are located as far away as possible, which leads to a larger distance of roughly 14.5 m. Furthermore, the line of sight is now obstructed by metal switch cabinets, metal machines and other electrical components that are located within this hall. In addition to the setup 1 with only wooden cabinets and tables a high degree of reflections and stronger dampening of the signal is to be expected. Hence, the measured channel seen on the right of Figure 7 is much more frequency selective. Figure 8 shows the results of our measurements. The left plot depicts the frame error percentage for different node activities at a fixed transmit power in the LoS case (setup 1). We see that MCSM performs highly reliable if up to 10 nodes are simultaneously active. Increasing the number of active nodes gracefully increases the likelihood for frame errors due to the higher system load. In setup 1 MCSM is mainly limited by the node activity due to the frequency flat channel and sufficiently high received power. To support higher number of active nodes, the MCSM parametrization can be adapted towards longer spreading codes and higher bandwidth allocation\"\n },\n {\n \"title\": \"sub-carrier index\",\n \"content\": \"\"\n },\n {\n \"title\": \"Measurement Results\",\n \"content\": \"In contrast, the right plot of Figure 8 shows the average frame error rate (FER) performance of MCSM in a non-LoS scenario (setup 2) with random node activity. The nodes are active with probability ൌ 0.1, which means we have 0.1 * 60 ൌ 6 simultaneously active nodes on average using 273 kHz in a single MCSM system. In setup 2 the influence of different transmit powers can be observed due to the multipath sub-carrier index channel and the additional path loss compared to the LoS setup. For low transmit powers the FER is severely degraded but for high transmit powers a comparable performance to the LoS setup is achievable. For maximum transmit power of 26 dBm no errors could be observed during our measurement. Naturally, the coverage of any MCSM system given a minimum required reliability is strongly coupled with the propagation environment and allowed transmit powers.\"\n },\n {\n \"title\": \"Conclusion\",\n \"content\": \"In this work we have presented a practical evaluation of the MCSM system concept in a lab environment as well as a more realistic machine hall setup. Our measurements validate the theoretical findings of previous works with respect to the robustness and flexibility of the scheme. The presented results are only one exemplary MCSM parametrization which is specifically tuned to support a high number of potential users in massive M2M situations. Furthermore, other parametrizations can be tuned towards latency reduction (frame length) and higher reliability, e.g. by longer spreading sequences or by using stronger codes. In conclusion we have shown that the recently introduced MCSM system can be considered as a candidate technology for novel MTC applications in 5G and I4.0 applications.\"\n },\n {\n \"title\": \"Figure 1 -\",\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 {\n \"title\": \"Figure 7 -\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 8 -\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) concept.","content":"Figure 1 -\n1Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) concept."},{"id":"","caption":"Base Station processing.","content":"Figure 3 -\n3Base Station processing."},{"id":"","caption":"Hardware-in-the-Loop (HIL) setup.","content":"Figure 4 -\n4Hardware-in-the-Loop (HIL) setup."},{"id":"","caption":"Frame design and symbol timing (left) and virtual massive access (right)","content":"Figure 5 -\n5Frame design and symbol timing (left) and virtual massive access (right)"},{"id":"","caption":"LoS scenario: (left) Schematic of the laboratory setup; (right) exemplary channel measurement with subcarrier 200 up to 700 and the bandwidth of the MCSM system considered.","content":"Figure 6 -\n6LoS scenario: (left) Schematic of the laboratory setup; (right) exemplary channel measurement with subcarrier 200 up to 700 and the bandwidth of the MCSM system considered."},{"id":"","caption":"Non-LoS scenario -(left) Schematic of the machine hall setup; (right) exemplary channel measurement.","content":"Figure 7 -\n7Non-LoS scenario -(left) Schematic of the machine hall setup; (right) exemplary channel measurement."},{"id":"","caption":"Measured frame error percentage vs. the number of active users in LoS setup (left). Frame Error Rate vs. transmit power in non-LoS setup with random node activity (right).","content":"Figure 8 -\n8Measured frame error percentage vs. the number of active users in LoS setup (left). Frame Error Rate vs. transmit power in non-LoS setup with random node activity (right)."},{"id":"","caption":"Thus, the first technology component of MCSM is a suitable multi-carrier concept to flexibly allocate spectrum for multiple MCSM systems in one frequency band. In this paper we restrict to Orthogonal Frequency Division Multiplexing","content":"Thus, the first technology component of MCSM is a suitable multi-carrier concept to flexibly allocate spectrum for multiple MCSM systems in one frequency band. In this paper we restrict to Orthogonal Frequency Division MultiplexingMCSM \nMCSM \nMCSM \n\nexisting wireless system \n\nexisting wireless system \nChannel in frequency domain \n\ncoherence \nbandwidth \n\nCompresssive Sensing Multi-\nUser Detection (CS-MUD) \n\nCollision resolution \nExploit sporadic activity \n\nMulti-Carrier Concepts \n\nFlexible spectrum use \nFrequency flat spectrum \n(coherence bandwidth) \n\nNon-Coherent Modulation \n\nAvoid channel estimation \nRobust against offsets \n\nfrequency \n\ntime \n\nTechnology Components"},{"id":"","caption":"Both Nutaq hardware development platforms used in our measurement setup are equipped with an 8 channel 14 Bit AD converter with up to 125 MSPS as well as an 8 channel 14 Bit DA converter with up to 500 MSPS. The maximum sampling rate is 104 MHz, which is decreased to 26 MHz for this measurement due to memory bandwidth limitations. Hence, a sample timing of ܶ ௦ ൌ 1/26 ݖܪܯ ൌ 38.462 ݏ݊ results. Hz up to a maximum bandwidth of 20 MHz pass the frontend. The typical CFO is specified as 100ppm of the actual carrier frequency. Assuming an ISM Band with a carrier frequency ݂ ൌ 2.4 ݖܪܩ a CFO of approx. ݂ ிை, ൎ േ 2.4 ݖܪ݇ per transceiver is expected. During operation CFOs of up to ݂ ிை ൌ ሺ݂ ,் ݂ ிை,் ሻ െ ሺ݂ ,ோ ݂ ிை,ோ ሻ ൎ 5 ݖܪ݇ can be observed, which is well within specifications.","content":"Both Nutaq hardware development platforms used in our measurement setup are equipped with an 8 channel 14 Bit AD converter with up to 125 MSPS as well as an 8 channel 14 Bit DA converter with up to 500 MSPS. The maximum sampling rate is 104 MHz, which is decreased to 26 MHz for this measurement due to memory bandwidth limitations. Hence, a sample timing of ܶ ௦ ൌ 1/26 ݖܪܯ ൌ 38.462 ݏ݊ results. Hz up to a maximum bandwidth of 20 MHz pass the frontend. The typical CFO is specified as 100ppm of the actual carrier frequency. Assuming an ISM Band with a carrier frequency ݂ ൌ 2.4 ݖܪܩ a CFO of approx. ݂ ிை, ൎ േ 2.4 ݖܪ݇ per transceiver is expected. During operation CFOs of up to ݂ ிை ൌ ሺ݂ ,் ݂ ிை,் ሻ െ ሺ݂ ,ோ ݂ ிை,ோ ሻ ൎ 5 ݖܪ݇ can be observed, which is well within specifications.Furthermore, both provide transceivers with four RF Frontends with 26 dBm maximum \ntransmission power for MIMO transmission within the 2.4 GHz and 5 GHz ISM Bands. \nThe RF frontend is implemented as a non-DC-coupled highpass, where all frequencies \nlarger 100"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) concept.\",\n \"content\": \"Figure 1 -\\n1Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) concept.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Base Station processing.\",\n \"content\": \"Figure 3 -\\n3Base Station processing.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Hardware-in-the-Loop (HIL) setup.\",\n \"content\": \"Figure 4 -\\n4Hardware-in-the-Loop (HIL) setup.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Frame design and symbol timing (left) and virtual massive access (right)\",\n \"content\": \"Figure 5 -\\n5Frame design and symbol timing (left) and virtual massive access (right)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"LoS scenario: (left) Schematic of the laboratory setup; (right) exemplary channel measurement with subcarrier 200 up to 700 and the bandwidth of the MCSM system considered.\",\n \"content\": \"Figure 6 -\\n6LoS scenario: (left) Schematic of the laboratory setup; (right) exemplary channel measurement with subcarrier 200 up to 700 and the bandwidth of the MCSM system considered.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Non-LoS scenario -(left) Schematic of the machine hall setup; (right) exemplary channel measurement.\",\n \"content\": \"Figure 7 -\\n7Non-LoS scenario -(left) Schematic of the machine hall setup; (right) exemplary channel measurement.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Measured frame error percentage vs. the number of active users in LoS setup (left). Frame Error Rate vs. transmit power in non-LoS setup with random node activity (right).\",\n \"content\": \"Figure 8 -\\n8Measured frame error percentage vs. the number of active users in LoS setup (left). Frame Error Rate vs. transmit power in non-LoS setup with random node activity (right).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Thus, the first technology component of MCSM is a suitable multi-carrier concept to flexibly allocate spectrum for multiple MCSM systems in one frequency band. In this paper we restrict to Orthogonal Frequency Division Multiplexing\",\n \"content\": \"Thus, the first technology component of MCSM is a suitable multi-carrier concept to flexibly allocate spectrum for multiple MCSM systems in one frequency band. In this paper we restrict to Orthogonal Frequency Division MultiplexingMCSM \\nMCSM \\nMCSM \\n\\nexisting wireless system \\n\\nexisting wireless system \\nChannel in frequency domain \\n\\ncoherence \\nbandwidth \\n\\nCompresssive Sensing Multi-\\nUser Detection (CS-MUD) \\n\\nCollision resolution \\nExploit sporadic activity \\n\\nMulti-Carrier Concepts \\n\\nFlexible spectrum use \\nFrequency flat spectrum \\n(coherence bandwidth) \\n\\nNon-Coherent Modulation \\n\\nAvoid channel estimation \\nRobust against offsets \\n\\nfrequency \\n\\ntime \\n\\nTechnology Components\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Both Nutaq hardware development platforms used in our measurement setup are equipped with an 8 channel 14 Bit AD converter with up to 125 MSPS as well as an 8 channel 14 Bit DA converter with up to 500 MSPS. The maximum sampling rate is 104 MHz, which is decreased to 26 MHz for this measurement due to memory bandwidth limitations. Hence, a sample timing of ܶ ௦ ൌ 1/26 ݖܪܯ ൌ 38.462 ݏ݊ results. Hz up to a maximum bandwidth of 20 MHz pass the frontend. The typical CFO is specified as 100ppm of the actual carrier frequency. Assuming an ISM Band with a carrier frequency ݂ ൌ 2.4 ݖܪܩ a CFO of approx. ݂ ிை, ൎ േ 2.4 ݖܪ݇ per transceiver is expected. During operation CFOs of up to ݂ ிை ൌ ሺ݂ ,் ݂ ிை,் ሻ െ ሺ݂ ,ோ ݂ ிை,ோ ሻ ൎ 5 ݖܪ݇ can be observed, which is well within specifications.\",\n \"content\": \"Both Nutaq hardware development platforms used in our measurement setup are equipped with an 8 channel 14 Bit AD converter with up to 125 MSPS as well as an 8 channel 14 Bit DA converter with up to 500 MSPS. The maximum sampling rate is 104 MHz, which is decreased to 26 MHz for this measurement due to memory bandwidth limitations. Hence, a sample timing of ܶ ௦ ൌ 1/26 ݖܪܯ ൌ 38.462 ݏ݊ results. Hz up to a maximum bandwidth of 20 MHz pass the frontend. The typical CFO is specified as 100ppm of the actual carrier frequency. Assuming an ISM Band with a carrier frequency ݂ ൌ 2.4 ݖܪܩ a CFO of approx. ݂ ிை, ൎ േ 2.4 ݖܪ݇ per transceiver is expected. During operation CFOs of up to ݂ ிை ൌ ሺ݂ ,் ݂ ிை,் ሻ െ ሺ݂ ,ோ ݂ ிை,ோ ሻ ൎ 5 ݖܪ݇ can be observed, which is well within specifications.Furthermore, both provide transceivers with four RF Frontends with 26 dBm maximum \\ntransmission power for MIMO transmission within the 2.4 GHz and 5 GHz ISM Bands. \\nThe RF frontend is implemented as a non-DC-coupled highpass, where all frequencies \\nlarger 100\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Compressive Sensing based Multi-User Detection for Machine-to-Machine. C Bockelmann, H Schepker, A Dekorsy, Transactions on Emerging Telecommunications Technologies: Special Issue on Machine-to-Machine: An emerging communication paradigm. 24Bockelmann, C.; Schepker, H.; Dekorsy, A.: Compressive Sensing based Multi-User Detection for Machine-to-Machine. In Transactions on Emerging Telecommunications Technologies: Special Issue on Machine-to-Machine: An emerging communication paradigm, Vol. 24, No. 4, pp. 389-400, June 2013.","doi":""},{"text":"Exploiting Sparsity in Channel and Data Estimation for Sporadic Multi-User Communication. H Schepker, C Bockelmann, A Dekorsy, Proc. 10th International Symposium on Wireless Communication Systems (ISWCS 13). 10th International Symposium on Wireless Communication Systems (ISWCS 13)Ilmenau, GermanySchepker, H.; Bockelmann, C.; Dekorsy, A.: Exploiting Sparsity in Channel and Data Estimation for Sporadic Multi-User Communication. In: Proc. 10th International Symposium on Wireless Communication Systems (ISWCS 13), Ilmenau, Germany, 27. -30. August 2013","doi":""},{"text":"Improving Group Orthogonal Matching Pursuit with Iterative Feedback. H Schepker, C Bockelmann, A Dekorsy, Proc. IEEE 78 th Vehicular Technology Conference (VTC 2013-Fall). IEEE 78 th Vehicular Technology Conference (VTC 2013-Fall)Las-Vegas, USA, 2Schepker, H.; Bockelmann, C.; Dekorsy, A.: Improving Group Orthogonal Matching Pursuit with Iterative Feedback. In: Proc. IEEE 78 th Vehicular Technology Conference (VTC 2013-Fall), Las-Vegas, USA, 2. -5. September 2013","doi":""},{"text":"Characterization of Coded Random Access with Compressive Sensing based Multi-User Detection. Y Ji, Č Stefanović, C Bockelmann, A Dekorsy, P Popovski, Proc. IEEE Globecom. IEEE GlobecomAustin, TX, USA, 8Ji, Y.; Stefanović, Č.; Bockelmann, C.; Dekorsy, A.; Popovski, P.: Characterization of Coded Random Access with Compressive Sensing based Multi-User Detection. In: Proc. IEEE Globecom 2014, Austin, TX, USA, 8. -11. December 2014","doi":""},{"text":"NGMN: 5G White Paper. NGMN: 5G White Paper, 2015","doi":""},{"text":"METIS D6.6: Final Report on the METIS System Concept and Technology Roadmap. METIS D6.6: Final Report on the METIS System Concept and Technology Roadmap, 2015.","doi":""},{"text":"Compressive Sensing Multi-User Detection for Multi-Carrier Systems in Sporadic Machine Type Communication. F Monsees, M Woltering, C Bockelmann, A Dekorsy, Proc. IEEE 81th Vehicular Technology Conference (VTC2015-Spring). IEEE 81th Vehicular Technology Conference (VTC2015-Spring)Glasgow, Great BritainMonsees, F.; Woltering, M.; Bockelmann, C.; Dekorsy, A.: Compressive Sensing Multi-User Detection for Multi-Carrier Systems in Sporadic Machine Type Communication. In: Proc. IEEE 81th Vehicular Technology Conference (VTC2015- Spring), Glasgow, Great Britain, May 2015.","doi":""},{"text":"Multiuser Detection. S Verdu, Cambridge University PressS. Verdu, Multiuser Detection, Cambridge University Press, 1998","doi":""},{"text":"A Potential Solution for MTC: Multi-Carrier Compressive Sensing Multi-User Detection. F Monsees, M Woltering, C Bockelmann, A Dekorsy, Proc. IEEE The Asilomar Conference on Signals, Systems, and Computers, Asilomar Hotel and Conference Grounds. IEEE The Asilomar Conference on Signals, Systems, and Computers, Asilomar Hotel and Conference GroundsUSAMonsees, F.; Woltering, M.; Bockelmann, C.; Dekorsy, A.: A Potential Solution for MTC: Multi-Carrier Compressive Sensing Multi-User Detection. In: Proc. IEEE The Asilomar Conference on Signals, Systems, and Computers, Asilomar Hotel and Conference Grounds, USA, November 2015.","doi":""}],"string":"[\n {\n \"text\": \"Compressive Sensing based Multi-User Detection for Machine-to-Machine. C Bockelmann, H Schepker, A Dekorsy, Transactions on Emerging Telecommunications Technologies: Special Issue on Machine-to-Machine: An emerging communication paradigm. 24Bockelmann, C.; Schepker, H.; Dekorsy, A.: Compressive Sensing based Multi-User Detection for Machine-to-Machine. In Transactions on Emerging Telecommunications Technologies: Special Issue on Machine-to-Machine: An emerging communication paradigm, Vol. 24, No. 4, pp. 389-400, June 2013.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Exploiting Sparsity in Channel and Data Estimation for Sporadic Multi-User Communication. H Schepker, C Bockelmann, A Dekorsy, Proc. 10th International Symposium on Wireless Communication Systems (ISWCS 13). 10th International Symposium on Wireless Communication Systems (ISWCS 13)Ilmenau, GermanySchepker, H.; Bockelmann, C.; Dekorsy, A.: Exploiting Sparsity in Channel and Data Estimation for Sporadic Multi-User Communication. In: Proc. 10th International Symposium on Wireless Communication Systems (ISWCS 13), Ilmenau, Germany, 27. -30. August 2013\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Improving Group Orthogonal Matching Pursuit with Iterative Feedback. H Schepker, C Bockelmann, A Dekorsy, Proc. IEEE 78 th Vehicular Technology Conference (VTC 2013-Fall). IEEE 78 th Vehicular Technology Conference (VTC 2013-Fall)Las-Vegas, USA, 2Schepker, H.; Bockelmann, C.; Dekorsy, A.: Improving Group Orthogonal Matching Pursuit with Iterative Feedback. In: Proc. IEEE 78 th Vehicular Technology Conference (VTC 2013-Fall), Las-Vegas, USA, 2. -5. September 2013\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Characterization of Coded Random Access with Compressive Sensing based Multi-User Detection. Y Ji, Č Stefanović, C Bockelmann, A Dekorsy, P Popovski, Proc. IEEE Globecom. IEEE GlobecomAustin, TX, USA, 8Ji, Y.; Stefanović, Č.; Bockelmann, C.; Dekorsy, A.; Popovski, P.: Characterization of Coded Random Access with Compressive Sensing based Multi-User Detection. In: Proc. IEEE Globecom 2014, Austin, TX, USA, 8. -11. December 2014\",\n \"doi\": \"\"\n },\n {\n \"text\": \"NGMN: 5G White Paper. NGMN: 5G White Paper, 2015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"METIS D6.6: Final Report on the METIS System Concept and Technology Roadmap. METIS D6.6: Final Report on the METIS System Concept and Technology Roadmap, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Compressive Sensing Multi-User Detection for Multi-Carrier Systems in Sporadic Machine Type Communication. F Monsees, M Woltering, C Bockelmann, A Dekorsy, Proc. IEEE 81th Vehicular Technology Conference (VTC2015-Spring). IEEE 81th Vehicular Technology Conference (VTC2015-Spring)Glasgow, Great BritainMonsees, F.; Woltering, M.; Bockelmann, C.; Dekorsy, A.: Compressive Sensing Multi-User Detection for Multi-Carrier Systems in Sporadic Machine Type Communication. In: Proc. IEEE 81th Vehicular Technology Conference (VTC2015- Spring), Glasgow, Great Britain, May 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multiuser Detection. S Verdu, Cambridge University PressS. Verdu, Multiuser Detection, Cambridge University Press, 1998\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Potential Solution for MTC: Multi-Carrier Compressive Sensing Multi-User Detection. F Monsees, M Woltering, C Bockelmann, A Dekorsy, Proc. IEEE The Asilomar Conference on Signals, Systems, and Computers, Asilomar Hotel and Conference Grounds. IEEE The Asilomar Conference on Signals, Systems, and Computers, Asilomar Hotel and Conference GroundsUSAMonsees, F.; Woltering, M.; Bockelmann, C.; Dekorsy, A.: A Potential Solution for MTC: Multi-Carrier Compressive Sensing Multi-User Detection. In: Proc. IEEE The Asilomar Conference on Signals, Systems, and Computers, Asilomar Hotel and Conference Grounds, USA, November 2015.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":580,"cells":{"text":{"kind":"string","value":"\nANALYSIS OF THE TRANSPORT MODEL THAT APPROXIMATES DECISION TAKER'S PREFERENCES\n\n\nV Ya \nUniversity of Technology\nKorolevMoscow RegionRussia\n\nVilisov \nUniversity of Technology\nKorolevMoscow RegionRussia\n\nANALYSIS OF THE TRANSPORT MODEL THAT APPROXIMATES DECISION TAKER'S PREFERENCES\n1transport problemreverse problemdecision-takingdecision-makingadaptation\nPaper provides a method for solving the reverse Monge-Kantorovich transport problem (TP). It allows to accumulate positive decision-taking experience made by decision-taker in situations that can be presented in the form of TP. The initial data for the solution of the inverse TP is the information on orders, inventories and effective decisions take by decision-taker. The result of solving the inverse TP contains evaluations of the TP's payoff matrix elements. It can be used in new situations to select the solution corresponding to the preferences of the decision-taker. The method allows to gain decision taker's experience, so it can be used by others. The method allows to build the model of decision taker's preferences in a specific application area. The model can be updated regularly to ensure its relevance and adequacy to the decision taker's system of preferences. This model is adaptive to the current preferences of the decision taker.\n\nIntroduction\n\nModern information technologies provide new possibilities for managing complicated social and economic objects. However, computers' computational abilities very often outpace technological capabilities of the existing management procedures. Thus, the tendency of \"burdening\" the computers with a larger number of managerial functions, which was predicted by the fathers of the management science [3,5,11], is well preserved and gradually turns into a common practice. Modern management includes different tools such as elements of artificial intellect (expert systems, genetic algorithms, neural nets etc.), simulation modelling tools, adaptive management ideas and so forth.\n\nThis work is devoted to some issues of the enterprise adaptive management reviewed on the basis of the transport-related business processes. Ideas of adaptive management related to the social and economic objects are developing for quite a while already [1,2,3]. Some authors [1,9] think that the ideal economics (and its all-level objects) should possess a high degree of adaptability like living creatures in nature. Here it means that social and economic objects should be able to adapt to the unfavorable external disturbances by rebuilding their structure or by changing parameters. At that it is the social component that plays the adaptability role in relation to the environment (it consists of managers, operators etc., i.e. the people that take the decisions -Decision Takers). Currently, the algorithmic component is not yet fully developed, only playing a role of the computational support. Within the framework of this adaptation technology, it is only the decision takers that accumulate the adaptability experience, so that when they are changed or are absent in some form or way, the whole system's experience is lost thus decreasing the effectiveness of its function.\n\nThe work considers another adaptation aspect where the adaptability experience stays within the system even when the decision taker is taken away or when he/she is absent temporarily. This approach is considered within the context of the transport system management problems. The experience is kept by the economic and mathematical models, parameters (sometimes it could be even the structure itself), which are set according to the decisions taken by the decision taker in the specific situations. Thus, these models approximate the preferences of the decision taker for the real-time situation by taking into account the uncertainty and instability of the environment. It is possible to say that they preserve the experience of decision taker. Still, like every preservation, the experience of decision taker has a limited validity period due to the instability of the internal characteristics of the system and the environment. Such preservations can be used by the system either without direct participation of the experience donor (the decision taker), with his/her minimal participation or with the participation of other decision takers that manage the same objects. Such accumulators of the decision taker's positive experience show his/her preferences as criteria and objective functions. Apart from the property that separates the experience (objective preferences) of the decision taker from its bearer, the technology under review plays another important role being the convolution of many real-time objective indices into the scalar objective function that approximates the objective indices vector. It is necessary to mention that part of these objective indices can be considered by the decision taker only on the subconscious level. Some scientists had already mentioned the necessity of formal characterization of the decision taker's experience and its further usage in the management procedures [7,8]. And according to Herbert Simon [11] all situations, involving decision-taking when managing social and economic objects can be divided into the structured and non-structured ones. All procedures should gradually become structured, being performed only by the computations means of the Corporate Information Systems (CIS).\n\nThere also exist other approaches for the formal characterization of the decision taker's experience, such as expert systems and neural nets. Still, their practical application technique does not yet provide effective ways for managing the business processes under review.\n\nAs a rule, modern enterprise's CIS [11] includes some elements of ERP-, APS-and MESsystems. The algorithms that are considered in this work are directed at developing and amending the functions of APS-systems.\n\nWithin the framework of the suggested technology the object management is performed in the two-circuit configuration. In the first circuit we perform the adjustment of the model parameters basing on the decisions taken by the manager (decision taker), while in the second one we perform direct object management basing on the model. In this scheme the first circuit works according to the manager's natural tempo while the second one works in the rhythm of the managing processes. Thus, the high intensity of the data flow that is present during the working management of the processes does not reduce the quality of approved managerial decisions. However, here (according to H. Simon [11]), we see the reduction of the manager's bounded rationality impact upon the management quality.\n\nThe work investigates usage of transport models (its peculiarities and properties) as the preservatives of the positive management experience within the transport systems. Materials of this work develop the author's earlier research conducted in the field of the adaptive management within the social and economic and technical systems [4,13,14]. The performed analysis is based on the simulation experiment, using the MS Excel's Data Analysis add-in. For the purposes of better result visualization, but without losing the communality, the research deals with the minimum dimensionality models.\n\n\nDirect and Reverse Formulation of the Transport Problem\n\nTransport problem model (TPM) is the special case of the linear programming model [12].\n\nTo solve the transport problem means to find the number of goods ( ) that are sent from the point of departure to the destination points. Here the plan's optimality criterion is the minimum amount of the total transportation cost. Usually we know such source data as the stock quantity ( ) in the point of departure and the goods demand rate ( ) per each destination point. We also know the commodity unit transportation cost matrix ( ij c ) from i-th departure point to j-th destination point.\n\nHistorically, these linear programming problems were included into a specific group due to their special structure that allows solving them more effectively using specially developed handcalculation methods. However, today, using the existing software of modern and powerful computers, TPM can be solved as a usual direct linear programming problem (LPP). At that the problem's characteristics will be reflected only in its source setting. Further we shall show how we can represent the TPM source setting as a standard LPP. In this case, for solving the reverse LPP (RLPP) we shall use one of its solution algorithms [4,12], so that after performing a reverse transition we could obtain the estimates of TPM coefficients.\n\nUsually, when compiling the transportation plan, the total transportation cost is minimized. But in real time the problem not only contains many criteria, but is also flexible in time (unstable). Therefore it is difficult to state a priori whether it is the total cost or the time that is the dominating factor, or maybe it is some other non-formalized indices, about which only the decision taker knows. This is why we think that it would be highly useful to find some common convolution that would include decision taker's integral preferences in relation to the multitude of alternatives.\n\nIn comparison with the standard LPP, the TPM's peculiarity lies in the bigger dimensionality even with the fewer number of the departure and destination points, at the same time possessing a higher sparsity of the solution matrix (many zero cells in the solution matrix). Standard form of the transport model is usually used when describing situations involving homogeneous goods, which can be delivered from any point to a random point.\n\nStill, the situations involving heterogeneous goods happen more frequently when dealing with practical transport decisions. Here the stocks and inquiries change frequently and the coefficient matrix is unstable (it depends on season, daily and random fluctuations of the transport network). Moreover, in the real time (where the optimization models are used), the transport problem is usually solved together with optimizing the workload of the existing car park.\n\nDespite the differences between the classic set-up of the transport problem and the big number of real-time situations, there are some cases where we can use TPM. For the purposes of TPM's higher adequacy, let us agree that the transport cost of one commodity unit from a i-th point to a j-th point shall stand not only for a money equivalent, but for some general expenses that are taken into consideration by the decision taker when choosing the transport plan. In this case the solution of the reverse transport problem shall be the following: basing on the situation monitoring (stocks and inquiries) as well as the transport plans that are defined as positive according to their operational results, we shall build ̂ estimates, according to which it will be possible to make transport plans for the new situations, thus solving a new transport problem. This will allow to automate the planning process either by completely replacing the decision taker (within the stable environment) or by significantly decreasing the decision taker's workload in the part of the transport plan compilation. However, we shall preserve the compliance of the transport plans with the decision taker's preferences and his/her practical experience. When using the adjusted model for planning within the flexible environment, prior to its sending for execution, the model-type plan variant can be given to the decision taker for approval or corrections.\n\nFor the purposes of discussion, the decision taker can be defined as a \"black box\" that transforms the stocks and inquiries vectors (that define the decision making situations -DMS) into the transport plan (see Picture 1).\n\nSolution of the reverse transport problem (RTP) is the following: by monitoring the situations and actions of the decision taker, we shall discover the decision taker's preference system (see Picture 2). Basing on this we shall solve the direct transport problem using one of the methods. \n( ̅ ) = ∑ ∑ =1 =1 ,(4)\nwhich can be defined as general transport expenses when transporting a commodity unit from i-th departure point to j-th destination point. When solving a direct TP, we shall provide ( ̅ ) → min.\n\n\nTransformation of the Transport Problem into the Linear Programming Problem\n\nFor the purposes of solving the RTP, let us maximally convert TPM into the LPP using inequality constraints. For this let us transform equality constraints into inequality constraints, minimum OF into the maximum OF, also performing other transformations.\n\nThese transformations are necessary because the reverse problem solution algorithm exists [4] for the mentioned LPP, not existing for the TPM, which is the specific type of problem.\n\nGiven that within the set of constraints of ( + ) equations only ( + − 1) equations are linearly independent (one equation is redundant, as the sum of equations derived according to the lines of the payment matrix equals to the sum of equations that are derived according to the columns -order-inquiry balancing property), let us express ( + − 1) of variables (basic ones) via the rest (free ones). For the purposes of certainty, these variables shall be 1 , 1 ; = 1, ̅̅̅̅̅̅ ; = 2, ̅̅̅̅̅ . Let us express 1 , 1 ( = 1, ̅̅̅̅̅̅ ; = 2, ̅̅̅̅̅ ) variables via the rest:\n̅ ̅ ̅ Direct Transport Problem's Solution Algorithm (with coefficients of matrix) ̅ , ̅ ̅ ̂ Reverse Transport Problem's Solution Algorithm 11 = 1 − ∑ =2 + ∑ ∑ =2 ; (5) =2 1 = − ∑ =2 , = 2, ̅̅̅̅̅̅ ; (6) 1 = − ∑ =2 , = 2, ̅̅̅̅̅ ;(7)\nHaving also transformed the objective function and by placing the values of basic variables into the source OF (4) using free variables (5)-(7), we shall obtain OF and corresponding delimitations represented as maximum LPP. Here all OF coefficients should be multiplied by (-1), which, with new OF maximization shall correspond to the minimization of the source OF.\n( ̅ ) = ∑ ∑= 2 =2 → max,(8)\nwhere ̃= −( 11 − 1 − 1 + )\n∑ =2 − 1 − ∑ ∑ =2 ≤ 0; (9) =2 ∑ =2 − ≤ 0 , = 2, ̅̅̅̅̅̅ ; (10) ∑ =2 − ≤ 0 , = 2, ̅̅̅̅̅ ;(11)\n≤ 0; = 2, ̅̅̅̅̅̅ ; = 2, ̅̅̅̅̅ .\n\n(12) ̃ coefficients should be estimated according to the solution algorithm of the reverse linear programming problem (RLPP). For the purposes of solving the direct transport problem on the basis of the built model, it is necessary, using the appearing DMS (i.e. ̅, ̅ vector population) to solve the linear programming problem (8)- (12). As a result of this, we shall find variables ( − 1) × ( − 1), while the rest ( + ( − 1)) should be calculated according to the formulae (5)- (7).\n\nThis research uses the point-like algorithm, which is one of the RLPP solving algorithms. Its essence will be shown below using one of the examples.\n\n\nResearching Properties of the Reverse Transport Problem\n\nFor the purposes of simplicity and result interpretation, let us consider a TP possessing two points of departure (m=2) and three points of destination (n=3). All data for this problem is provided in Table 1 and Table 2. The total number of variables is 6, but only 2 variables ( 22 , 23 ) stay independent (free), which provides us an opportunity to clearly demonstrate the solutions of direct and reverse problems at the two-dimensional subspace. Table 1 Transport Table (2 × 3) 11 12 13 21 22 23 Table 2 Variable Problems (2 × 3) 11 12 13\n\n\n22 23\n\nLet us express basic variables of the first column and the first line 1 , 1 , ( = 1, 2; = 2, 3) using the rest (free) variables , ( = 1, 2; = 2, 3):\n{ 11 = 1 − 2 − 3 + 22 + 23 21 = 2 − 22 − 23 12 = 2 − 22 13 = 3 − 23(13)\nAs all variables should be non-negative, this property should be fulfilled for the basic variables. Thus in (13) both left and right parts should be nonnegative. Let us write inequality-delimitations using a standard way (accepted for LPP with a maximum). For this we should multiply parts of both inequalities by (-1):\n{ 2 + 3 − 1 − 22 − 23 ≤ 0 22 + 23 − 2 ≤ 0 22 − 2 ≤ 0 23 − 3 ≤ 0(14)\nMoreover, here the non-negativity conditions should also apply: 22 ≥ 0; 23 ≥ 0.\n\nLet us put basic variables into the source objective function for expressing it through two free variables. At that, for the purposes of changing the TP optimization operator from min to max (for LPP) let us change the sign of the objective function by multiplying OF of TP by (-1 In the end, maximum LPP possessing 2 variables, which was received from the TP (2 × 3), shall look like below.\n\nLet us simplify the combination of OF coefficients by making a replacement: ̃2 2 = − 11 + 12 + 21 − 22 ; ̃2 3 = − 11 + 13 + 21 − 23 . In this case the OF (16) shall look like:\n( ̅ ) =̃2 2 22 +̃2 3 23 → max,(17)\nAfter transforming the inequalitydelimitations into the standard form, we shall receive:\n{ − 22 − 23 ≤ 1 − 2 − 3 22 + 23 ≤ 2 22 ≤ 2 23 ≤ 3 − 22 ≤ 0 − 23 ≤ 0(18)\nModelling Source Data Quite often, simulation modelling is the only possible way to research the properties of the economic object management algorithms [6]. Below we shall provide the source data and the simulation modelling results pertaining to the appearing planning situations. We will choose the best transportation plan (supposedly chosen by the decision taker) and will monitor (using the equivalent coefficients) the table of expenditures, which, in new situations, can be used for making a new plan without participation of the decision taker. As it is shown above, the TP can be transformed into LPP, therefore they are equivalent and can be transformed into each other using a oneone principle.\n\nAs is known, left part delimitation coefficients and the OF coefficients represent vector coordinates that are normal in relation to the corresponding lines (hyperplanes). Generally speaking, lengths of these normal vectors can be random, being defined by the values of the left parts. However, as is known, the inequality (or OF) will not change should both its parts be divided with the same positive number. Should the vector's original length (specific for each delimitation and the OF) be such a number, then all lines (hyperplanes) of the delimitations and the OF become comparable, i.e. they correspond to the unit length normal vectors (ULNV), thus becoming the normalized ones. It is necessary to mention that when solving direct LPP (DLPP) it does not matter whether the delimitations and/or OF are normalized. All parameters can be in their original form, or some of the parameters can be normalized while others are not. Making non-normalized delimitations and OF normalized is important for solving reverse LPP (RLPP) [4].\n\nNormalized coefficients ̃2 2 and ̃2 3 shall represent the coordinates of the vector that is normal in relation to the line (hyperplane) of the objective function (17), i.e. UNLV ̅ = [ 1 2 ] . The formulae for calculating the UNLV coordinates are:\n1 =̃2 2 √̃2 2 2 +̃2 3 2 ; 2 =̃2 3 √̃2 2 2 +̃2 3 2 . (19)\nThey represent the transportation expenditures, i.e. should the expenditure table of the example under review contains free variables, the coefficients shall look like: ̃2 2 = −0.225 and ̃2 3 = 0.974. When TP contains similar situations of choice, it is usually implied that transportation costs stay the same, therefore their normalized images also stay unchanged from one stage to another. It is possible to suggest that when solving RLPP, the estimate vector ̂̅ = [̂1̂2] should result in its actual value, which is ̅ = [ 1 2 ] vector. Therefore, as soon as the estimates are close enough to the true values that correspond to the ones of the decision taker, we can use these estimates as adequate approximation of the decision taker's criterion preferences (for the purposes of solving the direct problem).\n\nNormalized OF of direct LPP shall look like: ( ̅ ) = −0.225 22 + 0.974 23 → max.\n\n(20)\n\nFor the purposes of further representation and analysis let us write these delimitations as a coefficient table of left and rights parts (Table 4). Let us also add there the coefficients of the normalized OF. \n1 -1 -1 − − 2 1 1 3 1 0 4 0 1 5 -1 0 0 6 0 -1 0 OF -0.225 0.974 max\nThe reverse transport problem is solved on the basis of several stages (steps) of the research. Every new research is the situation (DMS) that consists of recurrent values (vectors) of supply and demand as well as the decision that is made by the decision taker in this situation.\n\nCoefficients of the left-part delimitations stay the same at each stage of the research (see Table 4) with only right parts changing as they represent values of supply and demand that appear at a stage. It is necessary to mention that only 4 out of 5 elements of supply and demand vectors take part in the right parts ( 1 does not participate). This is explained by the fact that these elements follow the equilibrium criterion, that's why we are using only one degree of freedom. Moreover, last two inequalities contained in Table 4 or in delimitations (18) stay unchanged for all observations as they represent the non-negativity property of the desired solution or it may mean that the problem's tolerance region (TR) always lies within the first quadrant. Given that the TR's current spectrum vectors play an important role in the solution of reverse TP, we shall briefly explain the meaning of the spectrum (according to [4]). TR spectrum is a cluster of vectors (UNLV) where each of them is orthogonal to one (its own) hyperplane that is included into the number of the hyperplanes that create the TR.\n\nTherefore, the TP being transformed into LLP is a problem with the fixed (discrete) spectrum. Here TP is similar to the production problems [4]. Still, there are some differences: coefficients of the left and right parts of the delimitations as well as the OF coefficients are not always positive. These differences result in the fact that the number of active delimitations (those that create the TR) does not always include two latter ones (see (18)), which means that TR can be \"hanging\" in the first quadrant without touching the coordinate axis. But the UNLV of OF is turned at any other side. Besides, all this diversity of TR and OF's UNLV is defined only by the values of ̅ = [ 1 2 ] and ̅ = [ 1 2 3 ] vector elements. Here we should mention that the spectrum has its own characteristics in the TP. Thus, the discrete spectrum for the given dimensionality ( × ) can be random in the production problems (it can vary from one problem to another, being defined only by the matrix of the left-part delimitations). But in the TP with the concrete dimensionality (for example, the one that is reviewed here: 2 × 3) the spectrum is defined only by dimensionality, being independent from the coefficient values of the expenditure table. Only the OF of LPP that is built according to TP, depends on them. Table 5 we can see the data that is obtained with the help of the random number generator (Data Analysis add-in within MS Excel environment). We generated the numbers within [1:100] interval per each column of supply and demand (excluding the last ones -2 and 3 ), after which, in order to provide the equilibrium, we computed the rest 2 columns and/or corrected original random numbers should it be necessary. Let us decide that the decision taker (i.e. the OF (20) that simulates his/her choice) chose values of variables per each DMS. These step-by-step decisions are shown in Table 6. There (for the purposes of further analysis) we also provide OF values (in the normalized form -until (20)), giving as well the numbers of two delimitations (numbering is made according to Table 4) that create the extreme point, chosen by the decision taker as the optimal one. These delimitations are called \"active delimitations\" because they participate in forming the optimal point, i.e. the solution for the given DMS.\n\nIn the last line of Table 5 and Table 6 we provide a polygon that corresponds to the specific DMS that was built according to the problem's spectrum (left parts of delimitations). Further we shall use this DMS for checking quality of the objective function.\n\n\nPeculiarity Analysis of LPP Built on the Basis of TP\n\nIn spite of a seemingly big and possible variety of TR variants, TP-caused LLP, unlike other types of linear programming models (e.g. production-type), possess very special characteristics.\n\nLet us consider special characteristics pertaining to this type of problems.\n\n\nPossible TR-Configurations for the Problem under Review.\n\nPictures 1 to 20 show all possible TR configurations of LPP for TP (2 × 3). The figures represent numbers of delimitations (numbering is made according to Table 4). Last five DMS (Pic. 16 -Pic. 20) are different from the rest because of the special value combination of the transport table, i.e. the 1 st delimitation is placed beyond the first quadrant, thus not participating in the TR creation. Generally speaking, there can be TRs that generate into intervals, i.e. when some pair of parallel delimitations coincides, e.g. 1-2, 3-5 or 4-6. However, such situations are very rare. Therefore we shall consider them generated and will exclude them from the further consideration. We shall also not consider the situations where the extreme point is created by three and not by two lines. Such cases are rather rare and should they happen, it is always possible to choose the pair of lines that are directly adjacent to TR. Pic. 14. \n\n\nPolygon.\n\nThe polygon is [4] such a TR (see Pic. 21), which possesses many important and special properties that allow it to use a corresponding DMS as a control situation to check the quality of model's settings in relation to the observations and other types of research.\n\n\nPic. 21. TR-Polygon\n\nThe polygon includes the following important properties:\n\n All delimitations of the polygon are active, i.e. they participate in the TR border formation. All TR alternatives are maximally informative. In the case of 2-dimensional situation it stands for the maximum possible obtuse angles at the extreme TR points. The alternatives are evenly (maximally) contrast, i.e. ideally the paired distances between the alternatives (at the TR border) are the same. Such variant is not always easy to perform technically, but we should strive to it. Quite often the compromise is the polygon where separate lines (hyperplanes) of delimitations are the tangents to some circumference (hypersphere). This variant is shown on Pic. 21. The polygon is used when researching the model parameter setting process, playing the role of a \"litmus test paper\". It checks whether the decision made on the basis of the set-up model corresponds to the decision made by the decision taker (or his/her simulation).\n\nPeculiarity of TP lies in the fact that type of the polygon does not depend on data, depending only on the problem's dimensionality. Therefore, for all values of supply and demand vectors the polygon shall look like Pic. 21.\n\n\nSpectra of the Problem (Delimitations, OF) and Observations.\n\nOn Picture 22 we can see the problem's spectrum, i.e. the UNLV population of six lines of delimitations and one line of OF level.\n\n\nPic. 22. TR-Polygon\n\nIt is necessary to mention that TP's UNLV of OF can be directed anywhere (unlike productiontype LPP where UNLV of OF can lie only within the first quadrant).\n\n\nPossible Types of Decisions (for Random OF). Pairs of Spectral Vectors.\n\nFor the TR that corresponds to the situation in place (DMS), the decision taker (or OF that simulates him/her) chooses one of the extreme points as a solution. One of UNLV pairs corresponds to the pair of delimitation lines that form the chosen extreme point (see Pic. 22). If to consider the potentially possible variants of UNLV pairs that can participate in the formation of extreme points, we shall see 6 two-figure combinations. Here we do not include three pairs of UNLV that are parallel to each other (1-2, 3-5 and 4-6). This multitude consists of 12 variants: 1-3, 1-4, 1-5, 1-6, 2-3, 2-4, 2-5, 2-6, 3-4, 3-6, 4-5, 5-6. The decision taker chooses one of the TR's extreme points formed by the corresponding UNLV pair as an optimal one. Variants of these pairs (arrows in bold) are shown in Table 7. Here we can also see sum vectors (double arrows) for each UNLV pair. All 12 pairs are divided into three groups that are distinguished by the angle between the paired vectors and consequently, by the length of the sum vector. Looking ahead, we can notice that the length of the sum vector reflects the informativeness of the solution, i.e. of that extreme TR point, to which the given UNLV pair corresponds. The sum vector length is used as the weights of the corresponding observations when solving the reverse problem. Thus, all possible observations can belong to one of three groups: to the 1st group -the least informative, to the 3rd group -the most informative and to the 2nd group -medium informative. Table 7 UNLV Pairs Groups UNLV Pairs in Groups 1 2 3 Apart from DMS, the observation includes the decision that was made and which corresponds to one of TR's extreme points, i.e. to the sum vector, shown as a double arrow at the pictures of Table 7. Eventually, the result of each observation is the only vector (observation vector), which is used in the solution algorithm of the reverse problem for estimating the decision taker's OF vector. Only direction of the desired decision taker's OF vector is important, its length is not of any importance. As far as observation vectors are concerned, their direction and length are of a big interest because the length shows the informative value of the given observation, i.e. its contribution into the evaluation process of OF performed by the decision taker. On Picture 23 we can see all observation vectors (sums of vector pairs) that are available for the TP example under review. Thus, in each observation the DMS is represented as some population (from three to six) of the active delimitation vectors. The decision that was made by the decision taker is shown in the following manner: one of these vectors is marked by a dot like on Pic. 23 where out of 25 observations of the example under review we chose only three as the optimal ones: 1-4, 2-5, 4-5. It is necessary to mention though that pairs 1-4 and 2-5 have minimum weights (lengths), while pair 4-5 has an average weight. On Pic. 24 we can see UNLVs both of observations and the decision taker simulated OF. In Table 8 we can see their combined characteristics. In Table 8 the alternative's rank is a whole number (r=1,2,3) that can have one value out of three: = 1 with = 0.076; = 2 with = 0.293; = 3 with = 0.617, where is the observational weight. General rank is the total sum of ranks of all alternatives related to this observation. Average rank is the rank that was averaged according to the multitude of alternatives related to this observation. Average weight is the averaged weight value related to the multitude of observations.\n\nAverage rank or average weight define the informative value of this observation, i.e. its contribution into the information gain about the evaluated OF made by the decision taker who can actually perform this observation. It is clear that the TR, whose configuration is similar to the polygon (Pic. 21) possesses a larger informative value (Pic. 7). In Table 8 we can see the groups of one-type TRs, inside of which the situation changes probably on the basis of the region's turn. These groups are numbered according to the increase of their average weight (or average rank).\n\nIt is necessary to pay attention to the 5 th observation. All its characteristics are outstanding: it possesses maximal (in comparison with other observational variants) average weight and average rank together with the maximum number of active delimitations, i.e. the delimitations that form the TR. In this case all existing delimitations take place in the process. Polygon also looks like this delimitation.\n\n\nSolution of Reverse Problem (Restoration of OF Parameters on the basis of Observations)\n\nThe main computational formula of a singlepoint step-by-step algorithm used for the estimation of model's parameters within the observations looks like [4]:\n̂= 1 √(∑ 1 =1 ) 2 + (∑ 2 =1 ) 2 ∑ =1 , (21)\nwhere = 1; 2 is the coordinate number;\n\nis the weight coefficient of j-th observation.\n\nThere is no discounting in this algorithm, its role is played indirectly by the coordinate normalization of the estimate vector (reduction to a single length) where the numerator uses an accumulated coordinate. Whereas the sum gradually increases within the accumulated coordinates, the relative contribution of each new observation shall decrease in proportion to this accumulation.\n\nShould we introduce a sliding summing-up interval (e.g. using a K length), the summing-up limits within the sums of formula (21) shall look like: ∑ ⋯ = − +1 for > . Evaluation Results. As a continuation of the above-mentioned observations (Table 5 and Table 6) let us provide in Table 9 the calculation results received according to the single-point step-by-step algorithm.\n k j i j j e 1 i k с i=1 i=2 i=1 i=2 i=1 i=2 i=1 i=2 i=1\nEvaluation algorithm (21) is actually an averaging procedure performed with the spectral observation vectors that are taken with the corresponding weights and which are related to the multitude of the observational steps. Considering frequencies of three spectral vectors that are observed during 25 steps, it is clear that their average value should be formed within the neighborhood of 4-5 spectral vector.\n\nThe calculations, provided in Table 9, confirmed this conclusion. It is necessary to note that the estimate is getting close to the Polygon's observation vector (see Polygon line in Table 9).\n\nExplanation of the Obtained Results Algorithm (21) is built on the basis of averaging the weighted observation vectors. Thus, if the data was formed randomly, the appearances of any observational vector (see Pic. 23) are equally possible. As it can be seen on Pictures 23 and 24, not all directions possess equal informativeness. Therefore, the wide pattern of the observation spectrum leads to the fact that the final setting-up vector is displaced in relation to the actual (modelled) OF vector of the decision taker. However, the wide pattern of the observation spectrum can also play a positive role: thus, if the representative spectral line is \"taken\", it will provide high quality of the decisions made in the future.\n\nThe estimate convergence is shown graphically on Pic. 25.\n\n\nPic. 25. Convergence of Decision Taker's OF Estimates\n\nHere it can be seen that the estimates of OF coefficients (its UNLV) are converged to the values of the solution (its UNLV) at the Polygon and not to the actual (modelled) values. Still, according to the modelling results, we can see that within any newly appearing DNS (see Pic. 3 -Pic. 20) the solutions, which are obtained with the adjusted (estimated) OF do not lead to errors (they correspond to the solutions obtained in relation to the modelled OF). All model-type solutions that are accepted in any DMS shall correspond to the solutions, accepted by the decision taker (that simulate his/her OF). Thus, the approximation of OF, which was not estimate-effective, turned out to be a solution-effective one.\n\nPic. 26. Convergence of Vectors' Differences Here we do not provide the solution convergence because right after the first step of settings the model's estimates turned out to be accurate enough for the solutions accepted for all other appearing DMS to completely correspond between each other according to the OF of the decision taker and according to the set-up model (its approximation).\n\n\nOn Adequacy Logic of the Reestablished Model\n\nPresence of fast solution-convergence and bad estimate-convergence is explained by the fact that the decision taker's OF becomes apparent only through DMS (TR). Out of all possible DMS only the DMS-polygon is the most representative and the most informative (see Table 8) representative of the environment, where the decision taker works.\n\nExternal observer sees objective preferences (OF-shaped) of the decision taker through DMS, therefore the OF of the decision taker should look like one of DMS elements. Problem spectrum or polygon spectrum vectors are such DMS elements.\n\nIn the process of reestablishment (estimation) of the decision taker's OF we can find the UNLV of OF (as an image of OF), approximating it with one of the polygon spectrum vectors. Thus we can talk about approximating OF of the decision taker using one of the polygon observation vectors (see vectors 2-4, 2-3, 3-6, 1-6, 1-5 and 4-5 on Pic. 24). Thus, OF of the decision taker, represented by the continuous UNLV (Pic. 24) and being projected at DMS is discretized by the problem's spectrum, whose full informational representative is the polygon's spectrum. This is why the search for the estimate of the decision taker's OF that approximates his/her preferences can be performed only at the discrete spectrum of the problem (polygon). This explains the fact that the estimates (UNLV of OF) converge to one of the polygon's spectrum vectors and not to the continuous and real UNLV of decision taker's OF.\n\nIt is also necessary to note that quality of approximation depends on the representativeness degree of DMS-multitude at the estimation stage, i.e. how completely it reflects the variety of all possible situations.\n\nIf DMS-multitude is representative (adequate to the environment), we can talk about approximation that is adequate to any potentially possible DMS. If DMS-multitude reflects only a part of possible situations, then here we see local approximation, where we use only a part of the problem's spectrum or the polygon's spectrum (i.e. only a local spectrum is used) within the setting-up procedure and in the course of the further solution of the direct TP. In this case, the solutions obtained on the basis of the adjusted model shall be reliable only for the new DMS, which appears within the same local region of the spectrum. It is possible to say that this search will be performed \"with the light\" in that part of the problem's spectrum, which was already \"lit up\" by the previous set-up steps. If there appears a DMS that extends beyond the borders of the local one, it is necessary to test decision taker again for the knowledge of this new area, afterwards correcting the OF estimates.\n\n\nConclusions\n\n\n1.\n\nModelling approximation process of the decision taker preferences within the transport system using general transport table shows high speed of the solution convergence, thus providing grounds for application of similar approximations in the transportation planning systems as well as in other applications, described by the scheme of the transport problem.\n\n2. Research of both the approximation algorithm and properties of the constructed model of the transport problem showed that solutions obtained with the help of the adjusted model can possess local effectiveness, i.e. solutions of the direct transport problem, obtained with the help of the adjusted model can be as good as the solutions, obtained by the decision taker in the same situations.\n\n3. Stopping rules of the model's set-up process can be based on the statistic characteristics of the variations pertaining to the estimate vector of decision taker's OF such as average value and average quadratic deviation.\n\n\nFor the TPM, ̅ solution per each DMS can be represented as the R-values matrix ̅ = ‖ ‖ . In this case, the approximation of the decision taker's preferences performed by the transport problem model lies in the evaluation of coefficients, pertaining to the target function:\n\n\n23. Variants of Observational Vectors Pic. 24. UNLV of Observations and OF Comparative Characteristics of the Observation Variants. Different TR configurations, shown on Pictures 3-20 differ from each other on the basis of some properties' values (number of alternatives, their informativeness etc.).\n\n\napproximating OF in relation to the actual OF of UNLV (upper one) and in relation to the closest spectral vector of the Polygon are shown on Picture 26.\n\n\nAfter removing the constant component that does not influence the solution, we shall obtain a working variant of the OF:). In this \ncase, within the new (free) coordinates ( 22 and \n23 ) the complete form of LPP's OF shall look like: \n( ̅ ) = −( 11 ( 1 − 2 − 3 ) + 12 2 + 13 3 \n+ 21 2 ) \n+ (− 11 + 12 + 21 − 22 ) 22 \n+ (− 11 + 13 + 21 − 23 ) 23 \n→ max. \n(15) \n\n( ̅ ) = (− 11 + 12 + 21 − 22 ) 22 \n+ (− 11 + 13 + 21 − 23 ) 23 \n→ max. \n(16) \n\n\n\nTable 3\n3contains expenditures expressed in absolute units (e.g. in Rubles). Right table column contains one of the variants of the supply (stocks) vector value ̅ = [ 1 2 ] , while the bottom line contains the variant of the demand vector value ̅ = [ 1 2 3 ] . When the line and the column intersect we can see the supply and demand balance value.\n\nTable 3 Modelling\n3Data \n\n10 2 20 10 \n\n12 7 \n9 25 \n\n5 15 15 35 \n\n\n\nTable 4\n4Left and Right Delimitation Parts of LPP That Is Equivalent to the Original TP DelimitationNumberVariables \nCondition \nRight Parts \n\n22 \n23 \n\n\n\nTable 5 Observation\n5Sample (DMS) \nObservation \nStep \n\nSupply \nDemand \nEquilibrium \n\n1 \n10 \n25 \n5 \n15 \n15 \n35 \n2 \n13 \n52 \n26 \n19 \n20 \n65 \n3 \n71 \n79 \n17 \n87 \n46 \n150 \n4 \n2 \n29 \n12 \n13 \n6 \n31 \n5 \n5 \n4 \n2 \n5 \n2 \n9 \n6 \n65 \n70 \n56 \n43 \n36 \n135 \n7 \n107 \n23 \n55 \n19 \n56 \n130 \n8 \n96 \n6 \n24 \n5 \n73 \n102 \n9 \n32 \n54 \n27 \n54 \n5 \n86 \n10 \n31 \n79 \n32 \n47 \n31 \n110 \n11 \n92 \n4 \n25 \n41 \n30 \n96 \n12 \n44 \n50 \n47 \n45 \n2 \n94 \n13 \n24 \n74 \n9 \n36 \n53 \n98 \n14 \n64 \n81 \n83 \n56 \n6 \n145 \n15 \n97 \n22 \n35 \n54 \n30 \n119 \n16 \n14 \n6 \n9 \n8 \n3 \n20 \n17 \n90 \n4 \n12 \n51 \n31 \n94 \n18 \n27 \n56 \n45 \n13 \n25 \n83 \n19 \n78 \n66 \n52 \n48 \n44 \n144 \n20 \n75 \n99 \n65 \n52 \n57 \n174 \n21 \n12 \n1 \n6 \n4 \n3 \n13 \n22 \n31 \n69 \n24 \n44 \n32 \n100 \n23 \n64 \n39 \n38 \n34 \n31 \n103 \n24 \n83 \n36 \n28 \n51 \n40 \n119 \n25 \n15 \n12 \n16 \n1 \n10 \n27 \nPolygon \n5 \n3 \n4 \n2 \n2 \n8 \n\n\nTable 6\n6Decisions Made by Decision Taker (Simulation) in relation to the Observation SampleObservation \nStep \n\nSolution \nOF \nActive \nDelimitations \n\nL norm_2 \nDel. 1 Del. 2 \n1 \n0 \n10 \n0 \n5 \n5 \n15 \n8.963 \n1 \n4 \n2 \n0 \n13 \n0 \n26 \n6 \n20 \n20.077 \n1 \n4 \n3 \n0 \n71 \n0 \n17 \n16 \n46 \n31.263 \n1 \n4 \n4 \n0 \n2 \n0 \n12 \n11 \n6 \n10.003 \n1 \n4 \n5 \n0 \n5 \n0 \n2 \n0 \n2 \n1.864 \n1 \n4 \n6 \n22 \n43 \n0 \n34 \n0 \n36 \n37.214 \n4 \n5 \n7 \n55 \n19 \n33 \n0 \n0 \n23 \n52.164 \n2 \n5 \n8 \n24 \n5 \n67 \n0 \n0 \n6 \n58.940 \n2 \n5 \n9 \n0 \n32 \n0 \n27 \n22 \n5 \n21.045 \n1 \n4 \n10 \n0 \n31 \n0 \n32 \n16 \n31 \n30.008 \n1 \n4 \n11 \n25 \n41 \n26 \n0 \n0 \n4 \n31.836 \n2 \n5 \n12 \n0 \n44 \n0 \n47 \n1 \n2 \n24.272 \n1 \n4 \n13 \n0 \n24 \n0 \n9 \n12 \n53 \n25.706 \n1 \n4 \n14 \n8 \n56 \n0 \n75 \n0 \n6 \n41.086 \n4 \n5 \n15 \n35 \n54 \n8 \n0 \n0 \n22 \n29.255 \n2 \n5 \n16 \n6 \n8 \n0 \n3 \n0 \n3 \n4.983 \n1 \n4 \n17 \n12 \n51 \n27 \n0 \n0 \n4 \n28.610 \n2 \n5 \n18 \n14 \n13 \n0 \n31 \n0 \n25 \n27.355 \n4 \n5 \n19 \n30 \n48 \n0 \n22 \n0 \n44 \n37.859 \n4 \n5 \n20 \n23 \n52 \n0 \n42 \n0 \n57 \n48.436 \n4 \n5 \n21 \n6 \n4 \n2 \n0 \n0 \n1 \n4.195 \n2 \n5 \n22 \n0 \n31 \n0 \n24 \n13 \n32 \n26.136 \n1 \n4 \n23 \n30 \n34 \n0 \n8 \n0 \n31 \n26.638 \n4 \n5 \n24 \n28 \n51 \n4 \n0 \n0 \n36 \n28.179 \n2 \n5 \n25 \n14 \n1 \n0 \n2 \n0 \n10 \n9.178 \n4 \n5 \nPolygon \n3 \n2 \n0 \n1 \n0 \n2 \n4 \n5 \n\nObservations: DMS and Decisions Made by \nDecision Taker \nIn each observation, DMS are represented by \nthe values of two vectors, which are ̅ = \n[ 1 \n2 ] and ̅ = [ 1 \n\n2 \n\n3 ] . Let the \n\nobservation sample for the simulation experiment \nconsist of 25 situations (DMS) where it is \nnecessary to build the transportation plan. In \n\nTable 8\n8Properties of Observational VariantsNo. Type of TRActive Delimitations \nInformativeness Indices \nGroups \nof One-\nType TR \nQuantity \nNumbers \nAlternative's \nRanks \n\nGeneral \nRank \n\nAverage \nRank \n\nAverage \nWeight \n1 \nPic. 3. \n4 \n1,2,5,6 \n1, 1, 3, 3 \n8 \n2 \n0.347 \n5 \n2 \nPic. 4. \n5 \n1,2,4,5,6 \n1, 2, 3, 3, 3 \n12 \n2.4 \n0.444 \n7 \n3 \nPic. 5. \n4 \n1,2,4,6 \n1, 1, 3, 3 \n8 \n2 \n0.347 \n4 \n4 \nPic. 6. \n5 \n1,2,3,5,6 \n1, 2, 3, 3, 3 \n12 \n2.4 \n0.444 \n7 \n\n5 \nPic. 7. \n6 \n1,2,3,4,5,6 \n2, 2, 3, 3, 3, 3 \n16 \n2.7 \n0.509 \n9 \n\n6 \nPic. 8. \n5 \n1,3,4,5,6 \n2, 2, 2, 3, 3 \n12 \n2.4 \n0.423 \n8 \n7 \nPic. 9. \n4 \n1,3,4,6 \n1, 2, 2, 3 \n8 \n2 \n0.320 \n3 \n8 \nPic. 10. \n5 \n1,2,3,4,6 \n1, 2, 3, 3, 3 \n12 \n2.4 \n0.444 \n6 \n9 \nPic. 11. \n4 \n1,2,3,5 \n1, 1, 3, 3 \n6 \n2 \n0.347 \n4 \n10 \nPic. 12. \n4 \n1,3,4,5 \n1, 2, 2, 3 \n8 \n2 \n0.320 \n3 \n11 \nPic. 13. \n5 \n1,2,3,4,5 \n1, 2, 3, 3, 3 \n12 \n2.4 \n0.444 \n6 \n12 \nPic. 14. \n3 \n1,3,4 \n1, 1, 2 \n4 \n1.3 \n0.148 \n1 \n13 \nPic. 15. \n4 \n1,2,3,4 \n1, 1, 3, 3 \n8 \n2 \n0.347 \n5 \n14 \nPic. 16. \n3 \n2,5,6 \n1, 1, 2 \n4 \n1.3 \n0.148 \n1 \n15 \nPic. 17. \n4 \n2,4,5,6 \n1, 2, 2, 3 \n8 \n2 \n0.320 \n3 \n16 \nPic. 18. \n5 \n2,3,4,5,6 \n2, 2, 2, 3, 3 \n12 \n2.4 \n0.423 \n8 \n17 \nPic. 19. \n4 \n2,3,5,6 \n1, 2, 2, 3 \n8 \n2 \n0.320 \n3 \n18 \nPic. 20. \n4 \n3,4,5,6 \n2, 2, 2, 2 \n8 \n2 \n0.293 \n2 \n\n\n\nTable 9\n9Decisions Made by Decision Taker in relation to the Observation SampleObservation \nStep \n\nDelimitation \nPair \n\nUNLV 1 \nUNLV 2 \nUNLV of \nObservation \nObservational \nWeight \n\n\n\nA concept of Corporate Planning. 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Introduction into MIS. -Kiev: Tekhnika Publishing House, 1974.\n\nSimulation Modelling of Economic Processes. A A Yemelyanov, Ye A Vlasova, R V Duma, A.A. Yemelyanov. MoscowFinances and Statistics Publishing HouseA.A. Yemelyanov, Ye.A. Vlasova, R.V. Duma. Simulation Modelling of Economic Processes / Edited by A.A. Yemelyanov. Moscow: Finances and Statistics Publishing House, 2009.\n\nMathematics Makes an Experiment. N N Moiseyev, Nauka Publishing HouseMoscowN.N. Moiseyev. Mathematics Makes an Experiment. -Moscow: Nauka Publishing House, 1979\n\nManagement Science: a Bayesian Introduction. W T Morris, Prentice HallW.T. Morris. Management Science: a Bayesian Introduction, Prentice Hall, 1968.\n\nToyota Production System: Beyond Large-Scale Production. Taiichi Ohno, Productivity PressTaiichi Ohno. Toyota Production System: Beyond Large-Scale Production. - Productivity Press, 1988.\n\nRight on Time for Russia: ERP-Systems Application. S V Piterkin, Practice / S.V. Piterkin, I.A. Oladov, D.V. IsayevAlpina Business BooksMoscowS.V. Piterkin. Right on Time for Russia: ERP- Systems Application Practice / S.V. Piterkin, I.A. Oladov, D.V. Isayev. -Moscow: Alpina Business Books, 2006.\n\nTheories of Decision-Making in Economics and Behavioral Science. H A Simon, The American Economic Review. 493H.A. Simon. Theories of Decision-Making in Economics and Behavioral Science. -The American Economic Review, Vol. 49, No. 3 (Jun., 1959), pp. 253-283.\n\nOperation research: An Introduction. 8-th ed. H A Taha, Pearson Prentice Hall813New JerseyH.A. Taha. Operation research: An Introduction. 8-th ed., Pearson Prentice Hall, New Jersey, 2007, 813 p.\n\nRobot Training Under Conditions of Incomplete Information. V Ya, Vilisov, International Scientific and Technological Conference EXTREME ROBOTICS, 1.10. St.PetersburgV.Ya. Vilisov. Robot Training Under Conditions of Incomplete Information. - International Scientific and Technological Conference EXTREME ROBOTICS, 1.10.2013, St.Petersburg. - URL: http://arxiv.org/abs/1402.2996\n\nResearch of the Robot's Learning Effectiveness in the Changing Environment. -International Scientific and Technological Conference EXTREME ROBOTICS, 1.10. V Ya, Vilisov, 10.13140/RG.2.1.3343.7926St.PetersburgV.Ya. Vilisov. Research of the Robot's Learning Effectiveness in the Changing Environment. -International Scientific and Technological Conference EXTREME ROBOTICS, 1.10.2014, St.Petersburg. - http://dx.doi.org/10.13140/RG.2.1.3343.7926\n"},"corpus_id":{"kind":"number","value":13275516,"string":"13,275,516"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1509.01815\",\n \"mag\": 2229806270,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/corr/Vilisov15b\",\n \"doi\": \"10.13140/rg.2.1.5085.6166\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1509.01815v1.pdf\"\n ],\n \"pdf_sha\": \"c04f17b1c5f02e86e859b451171159a6b0980cd8\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"ANALYSIS OF THE TRANSPORT MODEL THAT APPROXIMATES DECISION TAKER'S PREFERENCES"},"authors":{"kind":"list like","value":["V Ya \nUniversity of Technology\nKorolevMoscow RegionRussia","Vilisov \nUniversity of Technology\nKorolevMoscow RegionRussia"],"string":"[\n \"V Ya \\nUniversity of Technology\\nKorolevMoscow RegionRussia\",\n \"Vilisov \\nUniversity of Technology\\nKorolevMoscow RegionRussia\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Paper provides a method for solving the reverse Monge-Kantorovich transport problem (TP). It allows to accumulate positive decision-taking experience made by decision-taker in situations that can be presented in the form of TP. The initial data for the solution of the inverse TP is the information on orders, inventories and effective decisions take by decision-taker. The result of solving the inverse TP contains evaluations of the TP's payoff matrix elements. It can be used in new situations to select the solution corresponding to the preferences of the decision-taker. The method allows to gain decision taker's experience, so it can be used by others. The method allows to build the model of decision taker's preferences in a specific application area. The model can be updated regularly to ensure its relevance and adequacy to the decision taker's system of preferences. This model is adaptive to the current preferences of the decision taker."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Modern information technologies provide new possibilities for managing complicated social and economic objects. However, computers' computational abilities very often outpace technological capabilities of the existing management procedures. Thus, the tendency of \"burdening\" the computers with a larger number of managerial functions, which was predicted by the fathers of the management science [3,5,11], is well preserved and gradually turns into a common practice. Modern management includes different tools such as elements of artificial intellect (expert systems, genetic algorithms, neural nets etc.), simulation modelling tools, adaptive management ideas and so forth.\nThis work is devoted to some issues of the enterprise adaptive management reviewed on the basis of the transport-related business processes. Ideas of adaptive management related to the social and economic objects are developing for quite a while already [1,2,3]. Some authors [1,9] think that the ideal economics (and its all-level objects) should possess a high degree of adaptability like living creatures in nature. Here it means that social and economic objects should be able to adapt to the unfavorable external disturbances by rebuilding their structure or by changing parameters. At that it is the social component that plays the adaptability role in relation to the environment (it consists of managers, operators etc., i.e. the people that take the decisions -Decision Takers). Currently, the algorithmic component is not yet fully developed, only playing a role of the computational support. Within the framework of this adaptation technology, it is only the decision takers that accumulate the adaptability experience, so that when they are changed or are absent in some form or way, the whole system's experience is lost thus decreasing the effectiveness of its function.\nThe work considers another adaptation aspect where the adaptability experience stays within the system even when the decision taker is taken away or when he/she is absent temporarily. This approach is considered within the context of the transport system management problems. The experience is kept by the economic and mathematical models, parameters (sometimes it could be even the structure itself), which are set according to the decisions taken by the decision taker in the specific situations. Thus, these models approximate the preferences of the decision taker for the real-time situation by taking into account the uncertainty and instability of the environment. It is possible to say that they preserve the experience of decision taker. Still, like every preservation, the experience of decision taker has a limited validity period due to the instability of the internal characteristics of the system and the environment. Such preservations can be used by the system either without direct participation of the experience donor (the decision taker), with his/her minimal participation or with the participation of other decision takers that manage the same objects. Such accumulators of the decision taker's positive experience show his/her preferences as criteria and objective functions. Apart from the property that separates the experience (objective preferences) of the decision taker from its bearer, the technology under review plays another important role being the convolution of many real-time objective indices into the scalar objective function that approximates the objective indices vector. It is necessary to mention that part of these objective indices can be considered by the decision taker only on the subconscious level. Some scientists had already mentioned the necessity of formal characterization of the decision taker's experience and its further usage in the management procedures [7,8]. And according to Herbert Simon [11] all situations, involving decision-taking when managing social and economic objects can be divided into the structured and non-structured ones. All procedures should gradually become structured, being performed only by the computations means of the Corporate Information Systems (CIS).\nThere also exist other approaches for the formal characterization of the decision taker's experience, such as expert systems and neural nets. Still, their practical application technique does not yet provide effective ways for managing the business processes under review.\nAs a rule, modern enterprise's CIS [11] includes some elements of ERP-, APS-and MESsystems. The algorithms that are considered in this work are directed at developing and amending the functions of APS-systems.\nWithin the framework of the suggested technology the object management is performed in the two-circuit configuration. In the first circuit we perform the adjustment of the model parameters basing on the decisions taken by the manager (decision taker), while in the second one we perform direct object management basing on the model. In this scheme the first circuit works according to the manager's natural tempo while the second one works in the rhythm of the managing processes. Thus, the high intensity of the data flow that is present during the working management of the processes does not reduce the quality of approved managerial decisions. However, here (according to H. Simon [11]), we see the reduction of the manager's bounded rationality impact upon the management quality.\nThe work investigates usage of transport models (its peculiarities and properties) as the preservatives of the positive management experience within the transport systems. Materials of this work develop the author's earlier research conducted in the field of the adaptive management within the social and economic and technical systems [4,13,14]. The performed analysis is based on the simulation experiment, using the MS Excel's Data Analysis add-in. For the purposes of better result visualization, but without losing the communality, the research deals with the minimum dimensionality models."},{"title":"Direct and Reverse Formulation of the Transport Problem","content":"Transport problem model (TPM) is the special case of the linear programming model [12].\nTo solve the transport problem means to find the number of goods ( ) that are sent from the point of departure to the destination points. Here the plan's optimality criterion is the minimum amount of the total transportation cost. Usually we know such source data as the stock quantity ( ) in the point of departure and the goods demand rate ( ) per each destination point. We also know the commodity unit transportation cost matrix ( ij c ) from i-th departure point to j-th destination point.\nHistorically, these linear programming problems were included into a specific group due to their special structure that allows solving them more effectively using specially developed handcalculation methods. However, today, using the existing software of modern and powerful computers, TPM can be solved as a usual direct linear programming problem (LPP). At that the problem's characteristics will be reflected only in its source setting. Further we shall show how we can represent the TPM source setting as a standard LPP. In this case, for solving the reverse LPP (RLPP) we shall use one of its solution algorithms [4,12], so that after performing a reverse transition we could obtain the estimates of TPM coefficients.\nUsually, when compiling the transportation plan, the total transportation cost is minimized. But in real time the problem not only contains many criteria, but is also flexible in time (unstable). Therefore it is difficult to state a priori whether it is the total cost or the time that is the dominating factor, or maybe it is some other non-formalized indices, about which only the decision taker knows. This is why we think that it would be highly useful to find some common convolution that would include decision taker's integral preferences in relation to the multitude of alternatives.\nIn comparison with the standard LPP, the TPM's peculiarity lies in the bigger dimensionality even with the fewer number of the departure and destination points, at the same time possessing a higher sparsity of the solution matrix (many zero cells in the solution matrix). Standard form of the transport model is usually used when describing situations involving homogeneous goods, which can be delivered from any point to a random point.\nStill, the situations involving heterogeneous goods happen more frequently when dealing with practical transport decisions. Here the stocks and inquiries change frequently and the coefficient matrix is unstable (it depends on season, daily and random fluctuations of the transport network). Moreover, in the real time (where the optimization models are used), the transport problem is usually solved together with optimizing the workload of the existing car park.\nDespite the differences between the classic set-up of the transport problem and the big number of real-time situations, there are some cases where we can use TPM. For the purposes of TPM's higher adequacy, let us agree that the transport cost of one commodity unit from a i-th point to a j-th point shall stand not only for a money equivalent, but for some general expenses that are taken into consideration by the decision taker when choosing the transport plan. In this case the solution of the reverse transport problem shall be the following: basing on the situation monitoring (stocks and inquiries) as well as the transport plans that are defined as positive according to their operational results, we shall build ̂ estimates, according to which it will be possible to make transport plans for the new situations, thus solving a new transport problem. This will allow to automate the planning process either by completely replacing the decision taker (within the stable environment) or by significantly decreasing the decision taker's workload in the part of the transport plan compilation. However, we shall preserve the compliance of the transport plans with the decision taker's preferences and his/her practical experience. When using the adjusted model for planning within the flexible environment, prior to its sending for execution, the model-type plan variant can be given to the decision taker for approval or corrections.\nFor the purposes of discussion, the decision taker can be defined as a \"black box\" that transforms the stocks and inquiries vectors (that define the decision making situations -DMS) into the transport plan (see Picture 1).\nSolution of the reverse transport problem (RTP) is the following: by monitoring the situations and actions of the decision taker, we shall discover the decision taker's preference system (see Picture 2). Basing on this we shall solve the direct transport problem using one of the methods.\nwhich can be defined as general transport expenses when transporting a commodity unit from i-th departure point to j-th destination point. When solving a direct TP, we shall provide ( ̅ ) → min."},{"title":"Transformation of the Transport Problem into the Linear Programming Problem","content":"For the purposes of solving the RTP, let us maximally convert TPM into the LPP using inequality constraints. For this let us transform equality constraints into inequality constraints, minimum OF into the maximum OF, also performing other transformations.\nThese transformations are necessary because the reverse problem solution algorithm exists [4] for the mentioned LPP, not existing for the TPM, which is the specific type of problem.\nGiven that within the set of constraints of ( + ) equations only ( + − 1) equations are linearly independent (one equation is redundant, as the sum of equations derived according to the lines of the payment matrix equals to the sum of equations that are derived according to the columns -order-inquiry balancing property), let us express ( + − 1) of variables (basic ones) via the rest (free ones). For the purposes of certainty, these variables shall be 1 , 1 ; = 1, ̅̅̅̅̅̅ ; = 2, ̅̅̅̅̅ . Let us express 1 , 1 ( = 1, ̅̅̅̅̅̅ ; = 2, ̅̅̅̅̅ ) variables via the rest:\nHaving also transformed the objective function and by placing the values of basic variables into the source OF (4) using free variables (5)-(7), we shall obtain OF and corresponding delimitations represented as maximum LPP. Here all OF coefficients should be multiplied by (-1), which, with new OF maximization shall correspond to the minimization of the source OF.\nwhere ̃= −( 11 − 1 − 1 + )\n≤ 0; = 2, ̅̅̅̅̅̅ ; = 2, ̅̅̅̅̅ .\n(12) ̃ coefficients should be estimated according to the solution algorithm of the reverse linear programming problem (RLPP). For the purposes of solving the direct transport problem on the basis of the built model, it is necessary, using the appearing DMS (i.e. ̅, ̅ vector population) to solve the linear programming problem (8)- (12). As a result of this, we shall find variables ( − 1) × ( − 1), while the rest ( + ( − 1)) should be calculated according to the formulae (5)- (7).\nThis research uses the point-like algorithm, which is one of the RLPP solving algorithms. Its essence will be shown below using one of the examples."},{"title":"Researching Properties of the Reverse Transport Problem","content":"For the purposes of simplicity and result interpretation, let us consider a TP possessing two points of departure (m=2) and three points of destination (n=3). All data for this problem is provided in Table 1 and Table 2. The total number of variables is 6, but only 2 variables ( 22 , 23 ) stay independent (free), which provides us an opportunity to clearly demonstrate the solutions of direct and reverse problems at the two-dimensional subspace. Table 1 Transport Table (2 × 3) 11 12 13 21 22 23 Table 2 Variable Problems (2 × 3) 11 12 13"},{"title":"22 23","content":"Let us express basic variables of the first column and the first line 1 , 1 , ( = 1, 2; = 2, 3) using the rest (free) variables , ( = 1, 2; = 2, 3):\nAs all variables should be non-negative, this property should be fulfilled for the basic variables. Thus in (13) both left and right parts should be nonnegative. Let us write inequality-delimitations using a standard way (accepted for LPP with a maximum). For this we should multiply parts of both inequalities by (-1):\nMoreover, here the non-negativity conditions should also apply: 22 ≥ 0; 23 ≥ 0.\nLet us put basic variables into the source objective function for expressing it through two free variables. At that, for the purposes of changing the TP optimization operator from min to max (for LPP) let us change the sign of the objective function by multiplying OF of TP by (-1 In the end, maximum LPP possessing 2 variables, which was received from the TP (2 × 3), shall look like below.\nLet us simplify the combination of OF coefficients by making a replacement: ̃2 2 = − 11 + 12 + 21 − 22 ; ̃2 3 = − 11 + 13 + 21 − 23 . In this case the OF (16) shall look like:\nAfter transforming the inequalitydelimitations into the standard form, we shall receive:\nModelling Source Data Quite often, simulation modelling is the only possible way to research the properties of the economic object management algorithms [6]. Below we shall provide the source data and the simulation modelling results pertaining to the appearing planning situations. We will choose the best transportation plan (supposedly chosen by the decision taker) and will monitor (using the equivalent coefficients) the table of expenditures, which, in new situations, can be used for making a new plan without participation of the decision taker. As it is shown above, the TP can be transformed into LPP, therefore they are equivalent and can be transformed into each other using a oneone principle.\nAs is known, left part delimitation coefficients and the OF coefficients represent vector coordinates that are normal in relation to the corresponding lines (hyperplanes). Generally speaking, lengths of these normal vectors can be random, being defined by the values of the left parts. However, as is known, the inequality (or OF) will not change should both its parts be divided with the same positive number. Should the vector's original length (specific for each delimitation and the OF) be such a number, then all lines (hyperplanes) of the delimitations and the OF become comparable, i.e. they correspond to the unit length normal vectors (ULNV), thus becoming the normalized ones. It is necessary to mention that when solving direct LPP (DLPP) it does not matter whether the delimitations and/or OF are normalized. All parameters can be in their original form, or some of the parameters can be normalized while others are not. Making non-normalized delimitations and OF normalized is important for solving reverse LPP (RLPP) [4].\nNormalized coefficients ̃2 2 and ̃2 3 shall represent the coordinates of the vector that is normal in relation to the line (hyperplane) of the objective function (17), i.e. UNLV ̅ = [ 1 2 ] . The formulae for calculating the UNLV coordinates are:\nThey represent the transportation expenditures, i.e. should the expenditure table of the example under review contains free variables, the coefficients shall look like: ̃2 2 = −0.225 and ̃2 3 = 0.974. When TP contains similar situations of choice, it is usually implied that transportation costs stay the same, therefore their normalized images also stay unchanged from one stage to another. It is possible to suggest that when solving RLPP, the estimate vector ̂̅ = [̂1̂2] should result in its actual value, which is ̅ = [ 1 2 ] vector. Therefore, as soon as the estimates are close enough to the true values that correspond to the ones of the decision taker, we can use these estimates as adequate approximation of the decision taker's criterion preferences (for the purposes of solving the direct problem).\nNormalized OF of direct LPP shall look like: ( ̅ ) = −0.225 22 + 0.974 23 → max.\n(20)\nFor the purposes of further representation and analysis let us write these delimitations as a coefficient table of left and rights parts (Table 4). Let us also add there the coefficients of the normalized OF.\nThe reverse transport problem is solved on the basis of several stages (steps) of the research. Every new research is the situation (DMS) that consists of recurrent values (vectors) of supply and demand as well as the decision that is made by the decision taker in this situation.\nCoefficients of the left-part delimitations stay the same at each stage of the research (see Table 4) with only right parts changing as they represent values of supply and demand that appear at a stage. It is necessary to mention that only 4 out of 5 elements of supply and demand vectors take part in the right parts ( 1 does not participate). This is explained by the fact that these elements follow the equilibrium criterion, that's why we are using only one degree of freedom. Moreover, last two inequalities contained in Table 4 or in delimitations (18) stay unchanged for all observations as they represent the non-negativity property of the desired solution or it may mean that the problem's tolerance region (TR) always lies within the first quadrant. Given that the TR's current spectrum vectors play an important role in the solution of reverse TP, we shall briefly explain the meaning of the spectrum (according to [4]). TR spectrum is a cluster of vectors (UNLV) where each of them is orthogonal to one (its own) hyperplane that is included into the number of the hyperplanes that create the TR.\nTherefore, the TP being transformed into LLP is a problem with the fixed (discrete) spectrum. Here TP is similar to the production problems [4]. Still, there are some differences: coefficients of the left and right parts of the delimitations as well as the OF coefficients are not always positive. These differences result in the fact that the number of active delimitations (those that create the TR) does not always include two latter ones (see (18)), which means that TR can be \"hanging\" in the first quadrant without touching the coordinate axis. But the UNLV of OF is turned at any other side. Besides, all this diversity of TR and OF's UNLV is defined only by the values of ̅ = [ 1 2 ] and ̅ = [ 1 2 3 ] vector elements. Here we should mention that the spectrum has its own characteristics in the TP. Thus, the discrete spectrum for the given dimensionality ( × ) can be random in the production problems (it can vary from one problem to another, being defined only by the matrix of the left-part delimitations). But in the TP with the concrete dimensionality (for example, the one that is reviewed here: 2 × 3) the spectrum is defined only by dimensionality, being independent from the coefficient values of the expenditure table. Only the OF of LPP that is built according to TP, depends on them. Table 5 we can see the data that is obtained with the help of the random number generator (Data Analysis add-in within MS Excel environment). We generated the numbers within [1:100] interval per each column of supply and demand (excluding the last ones -2 and 3 ), after which, in order to provide the equilibrium, we computed the rest 2 columns and/or corrected original random numbers should it be necessary. Let us decide that the decision taker (i.e. the OF (20) that simulates his/her choice) chose values of variables per each DMS. These step-by-step decisions are shown in Table 6. There (for the purposes of further analysis) we also provide OF values (in the normalized form -until (20)), giving as well the numbers of two delimitations (numbering is made according to Table 4) that create the extreme point, chosen by the decision taker as the optimal one. These delimitations are called \"active delimitations\" because they participate in forming the optimal point, i.e. the solution for the given DMS.\nIn the last line of Table 5 and Table 6 we provide a polygon that corresponds to the specific DMS that was built according to the problem's spectrum (left parts of delimitations). Further we shall use this DMS for checking quality of the objective function."},{"title":"Peculiarity Analysis of LPP Built on the Basis of TP","content":"In spite of a seemingly big and possible variety of TR variants, TP-caused LLP, unlike other types of linear programming models (e.g. production-type), possess very special characteristics.\nLet us consider special characteristics pertaining to this type of problems."},{"title":"Possible TR-Configurations for the Problem under Review.","content":"Pictures 1 to 20 show all possible TR configurations of LPP for TP (2 × 3). The figures represent numbers of delimitations (numbering is made according to Table 4). Last five DMS (Pic. 16 -Pic. 20) are different from the rest because of the special value combination of the transport table, i.e. the 1 st delimitation is placed beyond the first quadrant, thus not participating in the TR creation. Generally speaking, there can be TRs that generate into intervals, i.e. when some pair of parallel delimitations coincides, e.g. 1-2, 3-5 or 4-6. However, such situations are very rare. Therefore we shall consider them generated and will exclude them from the further consideration. We shall also not consider the situations where the extreme point is created by three and not by two lines. Such cases are rather rare and should they happen, it is always possible to choose the pair of lines that are directly adjacent to TR. Pic. 14."},{"title":"Polygon.","content":"The polygon is [4] such a TR (see Pic. 21), which possesses many important and special properties that allow it to use a corresponding DMS as a control situation to check the quality of model's settings in relation to the observations and other types of research."},{"title":"Pic. 21. TR-Polygon","content":"The polygon includes the following important properties:\n All delimitations of the polygon are active, i.e. they participate in the TR border formation. All TR alternatives are maximally informative. In the case of 2-dimensional situation it stands for the maximum possible obtuse angles at the extreme TR points. The alternatives are evenly (maximally) contrast, i.e. ideally the paired distances between the alternatives (at the TR border) are the same. Such variant is not always easy to perform technically, but we should strive to it. Quite often the compromise is the polygon where separate lines (hyperplanes) of delimitations are the tangents to some circumference (hypersphere). This variant is shown on Pic. 21. The polygon is used when researching the model parameter setting process, playing the role of a \"litmus test paper\". It checks whether the decision made on the basis of the set-up model corresponds to the decision made by the decision taker (or his/her simulation).\nPeculiarity of TP lies in the fact that type of the polygon does not depend on data, depending only on the problem's dimensionality. Therefore, for all values of supply and demand vectors the polygon shall look like Pic. 21."},{"title":"Spectra of the Problem (Delimitations, OF) and Observations.","content":"On Picture 22 we can see the problem's spectrum, i.e. the UNLV population of six lines of delimitations and one line of OF level."},{"title":"Pic. 22. TR-Polygon","content":"It is necessary to mention that TP's UNLV of OF can be directed anywhere (unlike productiontype LPP where UNLV of OF can lie only within the first quadrant)."},{"title":"Possible Types of Decisions (for Random OF). Pairs of Spectral Vectors.","content":"For the TR that corresponds to the situation in place (DMS), the decision taker (or OF that simulates him/her) chooses one of the extreme points as a solution. One of UNLV pairs corresponds to the pair of delimitation lines that form the chosen extreme point (see Pic. 22). If to consider the potentially possible variants of UNLV pairs that can participate in the formation of extreme points, we shall see 6 two-figure combinations. Here we do not include three pairs of UNLV that are parallel to each other (1-2, 3-5 and 4-6). This multitude consists of 12 variants: 1-3, 1-4, 1-5, 1-6, 2-3, 2-4, 2-5, 2-6, 3-4, 3-6, 4-5, 5-6. The decision taker chooses one of the TR's extreme points formed by the corresponding UNLV pair as an optimal one. Variants of these pairs (arrows in bold) are shown in Table 7. Here we can also see sum vectors (double arrows) for each UNLV pair. All 12 pairs are divided into three groups that are distinguished by the angle between the paired vectors and consequently, by the length of the sum vector. Looking ahead, we can notice that the length of the sum vector reflects the informativeness of the solution, i.e. of that extreme TR point, to which the given UNLV pair corresponds. The sum vector length is used as the weights of the corresponding observations when solving the reverse problem. Thus, all possible observations can belong to one of three groups: to the 1st group -the least informative, to the 3rd group -the most informative and to the 2nd group -medium informative. Table 7 UNLV Pairs Groups UNLV Pairs in Groups 1 2 3 Apart from DMS, the observation includes the decision that was made and which corresponds to one of TR's extreme points, i.e. to the sum vector, shown as a double arrow at the pictures of Table 7. Eventually, the result of each observation is the only vector (observation vector), which is used in the solution algorithm of the reverse problem for estimating the decision taker's OF vector. Only direction of the desired decision taker's OF vector is important, its length is not of any importance. As far as observation vectors are concerned, their direction and length are of a big interest because the length shows the informative value of the given observation, i.e. its contribution into the evaluation process of OF performed by the decision taker. On Picture 23 we can see all observation vectors (sums of vector pairs) that are available for the TP example under review. Thus, in each observation the DMS is represented as some population (from three to six) of the active delimitation vectors. The decision that was made by the decision taker is shown in the following manner: one of these vectors is marked by a dot like on Pic. 23 where out of 25 observations of the example under review we chose only three as the optimal ones: 1-4, 2-5, 4-5. It is necessary to mention though that pairs 1-4 and 2-5 have minimum weights (lengths), while pair 4-5 has an average weight. On Pic. 24 we can see UNLVs both of observations and the decision taker simulated OF. In Table 8 we can see their combined characteristics. In Table 8 the alternative's rank is a whole number (r=1,2,3) that can have one value out of three: = 1 with = 0.076; = 2 with = 0.293; = 3 with = 0.617, where is the observational weight. General rank is the total sum of ranks of all alternatives related to this observation. Average rank is the rank that was averaged according to the multitude of alternatives related to this observation. Average weight is the averaged weight value related to the multitude of observations.\nAverage rank or average weight define the informative value of this observation, i.e. its contribution into the information gain about the evaluated OF made by the decision taker who can actually perform this observation. It is clear that the TR, whose configuration is similar to the polygon (Pic. 21) possesses a larger informative value (Pic. 7). In Table 8 we can see the groups of one-type TRs, inside of which the situation changes probably on the basis of the region's turn. These groups are numbered according to the increase of their average weight (or average rank).\nIt is necessary to pay attention to the 5 th observation. All its characteristics are outstanding: it possesses maximal (in comparison with other observational variants) average weight and average rank together with the maximum number of active delimitations, i.e. the delimitations that form the TR. In this case all existing delimitations take place in the process. Polygon also looks like this delimitation."},{"title":"Solution of Reverse Problem (Restoration of OF Parameters on the basis of Observations)","content":"The main computational formula of a singlepoint step-by-step algorithm used for the estimation of model's parameters within the observations looks like [4]:\nwhere = 1; 2 is the coordinate number;\nis the weight coefficient of j-th observation.\nThere is no discounting in this algorithm, its role is played indirectly by the coordinate normalization of the estimate vector (reduction to a single length) where the numerator uses an accumulated coordinate. Whereas the sum gradually increases within the accumulated coordinates, the relative contribution of each new observation shall decrease in proportion to this accumulation.\nShould we introduce a sliding summing-up interval (e.g. using a K length), the summing-up limits within the sums of formula (21) shall look like: ∑ ⋯ = − +1 for > . Evaluation Results. As a continuation of the above-mentioned observations (Table 5 and Table 6) let us provide in Table 9 the calculation results received according to the single-point step-by-step algorithm.\nEvaluation algorithm (21) is actually an averaging procedure performed with the spectral observation vectors that are taken with the corresponding weights and which are related to the multitude of the observational steps. Considering frequencies of three spectral vectors that are observed during 25 steps, it is clear that their average value should be formed within the neighborhood of 4-5 spectral vector.\nThe calculations, provided in Table 9, confirmed this conclusion. It is necessary to note that the estimate is getting close to the Polygon's observation vector (see Polygon line in Table 9).\nExplanation of the Obtained Results Algorithm (21) is built on the basis of averaging the weighted observation vectors. Thus, if the data was formed randomly, the appearances of any observational vector (see Pic. 23) are equally possible. As it can be seen on Pictures 23 and 24, not all directions possess equal informativeness. Therefore, the wide pattern of the observation spectrum leads to the fact that the final setting-up vector is displaced in relation to the actual (modelled) OF vector of the decision taker. However, the wide pattern of the observation spectrum can also play a positive role: thus, if the representative spectral line is \"taken\", it will provide high quality of the decisions made in the future.\nThe estimate convergence is shown graphically on Pic. 25."},{"title":"Pic. 25. Convergence of Decision Taker's OF Estimates","content":"Here it can be seen that the estimates of OF coefficients (its UNLV) are converged to the values of the solution (its UNLV) at the Polygon and not to the actual (modelled) values. Still, according to the modelling results, we can see that within any newly appearing DNS (see Pic. 3 -Pic. 20) the solutions, which are obtained with the adjusted (estimated) OF do not lead to errors (they correspond to the solutions obtained in relation to the modelled OF). All model-type solutions that are accepted in any DMS shall correspond to the solutions, accepted by the decision taker (that simulate his/her OF). Thus, the approximation of OF, which was not estimate-effective, turned out to be a solution-effective one.\nPic. 26. Convergence of Vectors' Differences Here we do not provide the solution convergence because right after the first step of settings the model's estimates turned out to be accurate enough for the solutions accepted for all other appearing DMS to completely correspond between each other according to the OF of the decision taker and according to the set-up model (its approximation)."},{"title":"On Adequacy Logic of the Reestablished Model","content":"Presence of fast solution-convergence and bad estimate-convergence is explained by the fact that the decision taker's OF becomes apparent only through DMS (TR). Out of all possible DMS only the DMS-polygon is the most representative and the most informative (see Table 8) representative of the environment, where the decision taker works.\nExternal observer sees objective preferences (OF-shaped) of the decision taker through DMS, therefore the OF of the decision taker should look like one of DMS elements. Problem spectrum or polygon spectrum vectors are such DMS elements.\nIn the process of reestablishment (estimation) of the decision taker's OF we can find the UNLV of OF (as an image of OF), approximating it with one of the polygon spectrum vectors. Thus we can talk about approximating OF of the decision taker using one of the polygon observation vectors (see vectors 2-4, 2-3, 3-6, 1-6, 1-5 and 4-5 on Pic. 24). Thus, OF of the decision taker, represented by the continuous UNLV (Pic. 24) and being projected at DMS is discretized by the problem's spectrum, whose full informational representative is the polygon's spectrum. This is why the search for the estimate of the decision taker's OF that approximates his/her preferences can be performed only at the discrete spectrum of the problem (polygon). This explains the fact that the estimates (UNLV of OF) converge to one of the polygon's spectrum vectors and not to the continuous and real UNLV of decision taker's OF.\nIt is also necessary to note that quality of approximation depends on the representativeness degree of DMS-multitude at the estimation stage, i.e. how completely it reflects the variety of all possible situations.\nIf DMS-multitude is representative (adequate to the environment), we can talk about approximation that is adequate to any potentially possible DMS. If DMS-multitude reflects only a part of possible situations, then here we see local approximation, where we use only a part of the problem's spectrum or the polygon's spectrum (i.e. only a local spectrum is used) within the setting-up procedure and in the course of the further solution of the direct TP. In this case, the solutions obtained on the basis of the adjusted model shall be reliable only for the new DMS, which appears within the same local region of the spectrum. It is possible to say that this search will be performed \"with the light\" in that part of the problem's spectrum, which was already \"lit up\" by the previous set-up steps. If there appears a DMS that extends beyond the borders of the local one, it is necessary to test decision taker again for the knowledge of this new area, afterwards correcting the OF estimates."},{"title":"Conclusions","content":""},{"title":"1.","content":"Modelling approximation process of the decision taker preferences within the transport system using general transport table shows high speed of the solution convergence, thus providing grounds for application of similar approximations in the transportation planning systems as well as in other applications, described by the scheme of the transport problem.\n2. Research of both the approximation algorithm and properties of the constructed model of the transport problem showed that solutions obtained with the help of the adjusted model can possess local effectiveness, i.e. solutions of the direct transport problem, obtained with the help of the adjusted model can be as good as the solutions, obtained by the decision taker in the same situations.\n3. Stopping rules of the model's set-up process can be based on the statistic characteristics of the variations pertaining to the estimate vector of decision taker's OF such as average value and average quadratic deviation."},{"title":"Table 3","content":""},{"title":"Table 3 Modelling","content":""},{"title":"Table 4","content":""},{"title":"Table 5 Observation","content":""},{"title":"Table 6","content":""},{"title":"Table 8","content":""},{"title":"Table 9","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Modern information technologies provide new possibilities for managing complicated social and economic objects. However, computers' computational abilities very often outpace technological capabilities of the existing management procedures. Thus, the tendency of \\\"burdening\\\" the computers with a larger number of managerial functions, which was predicted by the fathers of the management science [3,5,11], is well preserved and gradually turns into a common practice. Modern management includes different tools such as elements of artificial intellect (expert systems, genetic algorithms, neural nets etc.), simulation modelling tools, adaptive management ideas and so forth.\\nThis work is devoted to some issues of the enterprise adaptive management reviewed on the basis of the transport-related business processes. Ideas of adaptive management related to the social and economic objects are developing for quite a while already [1,2,3]. Some authors [1,9] think that the ideal economics (and its all-level objects) should possess a high degree of adaptability like living creatures in nature. Here it means that social and economic objects should be able to adapt to the unfavorable external disturbances by rebuilding their structure or by changing parameters. At that it is the social component that plays the adaptability role in relation to the environment (it consists of managers, operators etc., i.e. the people that take the decisions -Decision Takers). Currently, the algorithmic component is not yet fully developed, only playing a role of the computational support. Within the framework of this adaptation technology, it is only the decision takers that accumulate the adaptability experience, so that when they are changed or are absent in some form or way, the whole system's experience is lost thus decreasing the effectiveness of its function.\\nThe work considers another adaptation aspect where the adaptability experience stays within the system even when the decision taker is taken away or when he/she is absent temporarily. This approach is considered within the context of the transport system management problems. The experience is kept by the economic and mathematical models, parameters (sometimes it could be even the structure itself), which are set according to the decisions taken by the decision taker in the specific situations. Thus, these models approximate the preferences of the decision taker for the real-time situation by taking into account the uncertainty and instability of the environment. It is possible to say that they preserve the experience of decision taker. Still, like every preservation, the experience of decision taker has a limited validity period due to the instability of the internal characteristics of the system and the environment. Such preservations can be used by the system either without direct participation of the experience donor (the decision taker), with his/her minimal participation or with the participation of other decision takers that manage the same objects. Such accumulators of the decision taker's positive experience show his/her preferences as criteria and objective functions. Apart from the property that separates the experience (objective preferences) of the decision taker from its bearer, the technology under review plays another important role being the convolution of many real-time objective indices into the scalar objective function that approximates the objective indices vector. It is necessary to mention that part of these objective indices can be considered by the decision taker only on the subconscious level. Some scientists had already mentioned the necessity of formal characterization of the decision taker's experience and its further usage in the management procedures [7,8]. And according to Herbert Simon [11] all situations, involving decision-taking when managing social and economic objects can be divided into the structured and non-structured ones. All procedures should gradually become structured, being performed only by the computations means of the Corporate Information Systems (CIS).\\nThere also exist other approaches for the formal characterization of the decision taker's experience, such as expert systems and neural nets. Still, their practical application technique does not yet provide effective ways for managing the business processes under review.\\nAs a rule, modern enterprise's CIS [11] includes some elements of ERP-, APS-and MESsystems. The algorithms that are considered in this work are directed at developing and amending the functions of APS-systems.\\nWithin the framework of the suggested technology the object management is performed in the two-circuit configuration. In the first circuit we perform the adjustment of the model parameters basing on the decisions taken by the manager (decision taker), while in the second one we perform direct object management basing on the model. In this scheme the first circuit works according to the manager's natural tempo while the second one works in the rhythm of the managing processes. Thus, the high intensity of the data flow that is present during the working management of the processes does not reduce the quality of approved managerial decisions. However, here (according to H. Simon [11]), we see the reduction of the manager's bounded rationality impact upon the management quality.\\nThe work investigates usage of transport models (its peculiarities and properties) as the preservatives of the positive management experience within the transport systems. Materials of this work develop the author's earlier research conducted in the field of the adaptive management within the social and economic and technical systems [4,13,14]. The performed analysis is based on the simulation experiment, using the MS Excel's Data Analysis add-in. For the purposes of better result visualization, but without losing the communality, the research deals with the minimum dimensionality models.\"\n },\n {\n \"title\": \"Direct and Reverse Formulation of the Transport Problem\",\n \"content\": \"Transport problem model (TPM) is the special case of the linear programming model [12].\\nTo solve the transport problem means to find the number of goods ( ) that are sent from the point of departure to the destination points. Here the plan's optimality criterion is the minimum amount of the total transportation cost. Usually we know such source data as the stock quantity ( ) in the point of departure and the goods demand rate ( ) per each destination point. We also know the commodity unit transportation cost matrix ( ij c ) from i-th departure point to j-th destination point.\\nHistorically, these linear programming problems were included into a specific group due to their special structure that allows solving them more effectively using specially developed handcalculation methods. However, today, using the existing software of modern and powerful computers, TPM can be solved as a usual direct linear programming problem (LPP). At that the problem's characteristics will be reflected only in its source setting. Further we shall show how we can represent the TPM source setting as a standard LPP. In this case, for solving the reverse LPP (RLPP) we shall use one of its solution algorithms [4,12], so that after performing a reverse transition we could obtain the estimates of TPM coefficients.\\nUsually, when compiling the transportation plan, the total transportation cost is minimized. But in real time the problem not only contains many criteria, but is also flexible in time (unstable). Therefore it is difficult to state a priori whether it is the total cost or the time that is the dominating factor, or maybe it is some other non-formalized indices, about which only the decision taker knows. This is why we think that it would be highly useful to find some common convolution that would include decision taker's integral preferences in relation to the multitude of alternatives.\\nIn comparison with the standard LPP, the TPM's peculiarity lies in the bigger dimensionality even with the fewer number of the departure and destination points, at the same time possessing a higher sparsity of the solution matrix (many zero cells in the solution matrix). Standard form of the transport model is usually used when describing situations involving homogeneous goods, which can be delivered from any point to a random point.\\nStill, the situations involving heterogeneous goods happen more frequently when dealing with practical transport decisions. Here the stocks and inquiries change frequently and the coefficient matrix is unstable (it depends on season, daily and random fluctuations of the transport network). Moreover, in the real time (where the optimization models are used), the transport problem is usually solved together with optimizing the workload of the existing car park.\\nDespite the differences between the classic set-up of the transport problem and the big number of real-time situations, there are some cases where we can use TPM. For the purposes of TPM's higher adequacy, let us agree that the transport cost of one commodity unit from a i-th point to a j-th point shall stand not only for a money equivalent, but for some general expenses that are taken into consideration by the decision taker when choosing the transport plan. In this case the solution of the reverse transport problem shall be the following: basing on the situation monitoring (stocks and inquiries) as well as the transport plans that are defined as positive according to their operational results, we shall build ̂ estimates, according to which it will be possible to make transport plans for the new situations, thus solving a new transport problem. This will allow to automate the planning process either by completely replacing the decision taker (within the stable environment) or by significantly decreasing the decision taker's workload in the part of the transport plan compilation. However, we shall preserve the compliance of the transport plans with the decision taker's preferences and his/her practical experience. When using the adjusted model for planning within the flexible environment, prior to its sending for execution, the model-type plan variant can be given to the decision taker for approval or corrections.\\nFor the purposes of discussion, the decision taker can be defined as a \\\"black box\\\" that transforms the stocks and inquiries vectors (that define the decision making situations -DMS) into the transport plan (see Picture 1).\\nSolution of the reverse transport problem (RTP) is the following: by monitoring the situations and actions of the decision taker, we shall discover the decision taker's preference system (see Picture 2). Basing on this we shall solve the direct transport problem using one of the methods.\\nwhich can be defined as general transport expenses when transporting a commodity unit from i-th departure point to j-th destination point. When solving a direct TP, we shall provide ( ̅ ) → min.\"\n },\n {\n \"title\": \"Transformation of the Transport Problem into the Linear Programming Problem\",\n \"content\": \"For the purposes of solving the RTP, let us maximally convert TPM into the LPP using inequality constraints. For this let us transform equality constraints into inequality constraints, minimum OF into the maximum OF, also performing other transformations.\\nThese transformations are necessary because the reverse problem solution algorithm exists [4] for the mentioned LPP, not existing for the TPM, which is the specific type of problem.\\nGiven that within the set of constraints of ( + ) equations only ( + − 1) equations are linearly independent (one equation is redundant, as the sum of equations derived according to the lines of the payment matrix equals to the sum of equations that are derived according to the columns -order-inquiry balancing property), let us express ( + − 1) of variables (basic ones) via the rest (free ones). For the purposes of certainty, these variables shall be 1 , 1 ; = 1, ̅̅̅̅̅̅ ; = 2, ̅̅̅̅̅ . Let us express 1 , 1 ( = 1, ̅̅̅̅̅̅ ; = 2, ̅̅̅̅̅ ) variables via the rest:\\nHaving also transformed the objective function and by placing the values of basic variables into the source OF (4) using free variables (5)-(7), we shall obtain OF and corresponding delimitations represented as maximum LPP. Here all OF coefficients should be multiplied by (-1), which, with new OF maximization shall correspond to the minimization of the source OF.\\nwhere ̃= −( 11 − 1 − 1 + )\\n≤ 0; = 2, ̅̅̅̅̅̅ ; = 2, ̅̅̅̅̅ .\\n(12) ̃ coefficients should be estimated according to the solution algorithm of the reverse linear programming problem (RLPP). For the purposes of solving the direct transport problem on the basis of the built model, it is necessary, using the appearing DMS (i.e. ̅, ̅ vector population) to solve the linear programming problem (8)- (12). As a result of this, we shall find variables ( − 1) × ( − 1), while the rest ( + ( − 1)) should be calculated according to the formulae (5)- (7).\\nThis research uses the point-like algorithm, which is one of the RLPP solving algorithms. Its essence will be shown below using one of the examples.\"\n },\n {\n \"title\": \"Researching Properties of the Reverse Transport Problem\",\n \"content\": \"For the purposes of simplicity and result interpretation, let us consider a TP possessing two points of departure (m=2) and three points of destination (n=3). All data for this problem is provided in Table 1 and Table 2. The total number of variables is 6, but only 2 variables ( 22 , 23 ) stay independent (free), which provides us an opportunity to clearly demonstrate the solutions of direct and reverse problems at the two-dimensional subspace. Table 1 Transport Table (2 × 3) 11 12 13 21 22 23 Table 2 Variable Problems (2 × 3) 11 12 13\"\n },\n {\n \"title\": \"22 23\",\n \"content\": \"Let us express basic variables of the first column and the first line 1 , 1 , ( = 1, 2; = 2, 3) using the rest (free) variables , ( = 1, 2; = 2, 3):\\nAs all variables should be non-negative, this property should be fulfilled for the basic variables. Thus in (13) both left and right parts should be nonnegative. Let us write inequality-delimitations using a standard way (accepted for LPP with a maximum). For this we should multiply parts of both inequalities by (-1):\\nMoreover, here the non-negativity conditions should also apply: 22 ≥ 0; 23 ≥ 0.\\nLet us put basic variables into the source objective function for expressing it through two free variables. At that, for the purposes of changing the TP optimization operator from min to max (for LPP) let us change the sign of the objective function by multiplying OF of TP by (-1 In the end, maximum LPP possessing 2 variables, which was received from the TP (2 × 3), shall look like below.\\nLet us simplify the combination of OF coefficients by making a replacement: ̃2 2 = − 11 + 12 + 21 − 22 ; ̃2 3 = − 11 + 13 + 21 − 23 . In this case the OF (16) shall look like:\\nAfter transforming the inequalitydelimitations into the standard form, we shall receive:\\nModelling Source Data Quite often, simulation modelling is the only possible way to research the properties of the economic object management algorithms [6]. Below we shall provide the source data and the simulation modelling results pertaining to the appearing planning situations. We will choose the best transportation plan (supposedly chosen by the decision taker) and will monitor (using the equivalent coefficients) the table of expenditures, which, in new situations, can be used for making a new plan without participation of the decision taker. As it is shown above, the TP can be transformed into LPP, therefore they are equivalent and can be transformed into each other using a oneone principle.\\nAs is known, left part delimitation coefficients and the OF coefficients represent vector coordinates that are normal in relation to the corresponding lines (hyperplanes). Generally speaking, lengths of these normal vectors can be random, being defined by the values of the left parts. However, as is known, the inequality (or OF) will not change should both its parts be divided with the same positive number. Should the vector's original length (specific for each delimitation and the OF) be such a number, then all lines (hyperplanes) of the delimitations and the OF become comparable, i.e. they correspond to the unit length normal vectors (ULNV), thus becoming the normalized ones. It is necessary to mention that when solving direct LPP (DLPP) it does not matter whether the delimitations and/or OF are normalized. All parameters can be in their original form, or some of the parameters can be normalized while others are not. Making non-normalized delimitations and OF normalized is important for solving reverse LPP (RLPP) [4].\\nNormalized coefficients ̃2 2 and ̃2 3 shall represent the coordinates of the vector that is normal in relation to the line (hyperplane) of the objective function (17), i.e. UNLV ̅ = [ 1 2 ] . The formulae for calculating the UNLV coordinates are:\\nThey represent the transportation expenditures, i.e. should the expenditure table of the example under review contains free variables, the coefficients shall look like: ̃2 2 = −0.225 and ̃2 3 = 0.974. When TP contains similar situations of choice, it is usually implied that transportation costs stay the same, therefore their normalized images also stay unchanged from one stage to another. It is possible to suggest that when solving RLPP, the estimate vector ̂̅ = [̂1̂2] should result in its actual value, which is ̅ = [ 1 2 ] vector. Therefore, as soon as the estimates are close enough to the true values that correspond to the ones of the decision taker, we can use these estimates as adequate approximation of the decision taker's criterion preferences (for the purposes of solving the direct problem).\\nNormalized OF of direct LPP shall look like: ( ̅ ) = −0.225 22 + 0.974 23 → max.\\n(20)\\nFor the purposes of further representation and analysis let us write these delimitations as a coefficient table of left and rights parts (Table 4). Let us also add there the coefficients of the normalized OF.\\nThe reverse transport problem is solved on the basis of several stages (steps) of the research. Every new research is the situation (DMS) that consists of recurrent values (vectors) of supply and demand as well as the decision that is made by the decision taker in this situation.\\nCoefficients of the left-part delimitations stay the same at each stage of the research (see Table 4) with only right parts changing as they represent values of supply and demand that appear at a stage. It is necessary to mention that only 4 out of 5 elements of supply and demand vectors take part in the right parts ( 1 does not participate). This is explained by the fact that these elements follow the equilibrium criterion, that's why we are using only one degree of freedom. Moreover, last two inequalities contained in Table 4 or in delimitations (18) stay unchanged for all observations as they represent the non-negativity property of the desired solution or it may mean that the problem's tolerance region (TR) always lies within the first quadrant. Given that the TR's current spectrum vectors play an important role in the solution of reverse TP, we shall briefly explain the meaning of the spectrum (according to [4]). TR spectrum is a cluster of vectors (UNLV) where each of them is orthogonal to one (its own) hyperplane that is included into the number of the hyperplanes that create the TR.\\nTherefore, the TP being transformed into LLP is a problem with the fixed (discrete) spectrum. Here TP is similar to the production problems [4]. Still, there are some differences: coefficients of the left and right parts of the delimitations as well as the OF coefficients are not always positive. These differences result in the fact that the number of active delimitations (those that create the TR) does not always include two latter ones (see (18)), which means that TR can be \\\"hanging\\\" in the first quadrant without touching the coordinate axis. But the UNLV of OF is turned at any other side. Besides, all this diversity of TR and OF's UNLV is defined only by the values of ̅ = [ 1 2 ] and ̅ = [ 1 2 3 ] vector elements. Here we should mention that the spectrum has its own characteristics in the TP. Thus, the discrete spectrum for the given dimensionality ( × ) can be random in the production problems (it can vary from one problem to another, being defined only by the matrix of the left-part delimitations). But in the TP with the concrete dimensionality (for example, the one that is reviewed here: 2 × 3) the spectrum is defined only by dimensionality, being independent from the coefficient values of the expenditure table. Only the OF of LPP that is built according to TP, depends on them. Table 5 we can see the data that is obtained with the help of the random number generator (Data Analysis add-in within MS Excel environment). We generated the numbers within [1:100] interval per each column of supply and demand (excluding the last ones -2 and 3 ), after which, in order to provide the equilibrium, we computed the rest 2 columns and/or corrected original random numbers should it be necessary. Let us decide that the decision taker (i.e. the OF (20) that simulates his/her choice) chose values of variables per each DMS. These step-by-step decisions are shown in Table 6. There (for the purposes of further analysis) we also provide OF values (in the normalized form -until (20)), giving as well the numbers of two delimitations (numbering is made according to Table 4) that create the extreme point, chosen by the decision taker as the optimal one. These delimitations are called \\\"active delimitations\\\" because they participate in forming the optimal point, i.e. the solution for the given DMS.\\nIn the last line of Table 5 and Table 6 we provide a polygon that corresponds to the specific DMS that was built according to the problem's spectrum (left parts of delimitations). Further we shall use this DMS for checking quality of the objective function.\"\n },\n {\n \"title\": \"Peculiarity Analysis of LPP Built on the Basis of TP\",\n \"content\": \"In spite of a seemingly big and possible variety of TR variants, TP-caused LLP, unlike other types of linear programming models (e.g. production-type), possess very special characteristics.\\nLet us consider special characteristics pertaining to this type of problems.\"\n },\n {\n \"title\": \"Possible TR-Configurations for the Problem under Review.\",\n \"content\": \"Pictures 1 to 20 show all possible TR configurations of LPP for TP (2 × 3). The figures represent numbers of delimitations (numbering is made according to Table 4). Last five DMS (Pic. 16 -Pic. 20) are different from the rest because of the special value combination of the transport table, i.e. the 1 st delimitation is placed beyond the first quadrant, thus not participating in the TR creation. Generally speaking, there can be TRs that generate into intervals, i.e. when some pair of parallel delimitations coincides, e.g. 1-2, 3-5 or 4-6. However, such situations are very rare. Therefore we shall consider them generated and will exclude them from the further consideration. We shall also not consider the situations where the extreme point is created by three and not by two lines. Such cases are rather rare and should they happen, it is always possible to choose the pair of lines that are directly adjacent to TR. Pic. 14.\"\n },\n {\n \"title\": \"Polygon.\",\n \"content\": \"The polygon is [4] such a TR (see Pic. 21), which possesses many important and special properties that allow it to use a corresponding DMS as a control situation to check the quality of model's settings in relation to the observations and other types of research.\"\n },\n {\n \"title\": \"Pic. 21. TR-Polygon\",\n \"content\": \"The polygon includes the following important properties:\\n All delimitations of the polygon are active, i.e. they participate in the TR border formation. All TR alternatives are maximally informative. In the case of 2-dimensional situation it stands for the maximum possible obtuse angles at the extreme TR points. The alternatives are evenly (maximally) contrast, i.e. ideally the paired distances between the alternatives (at the TR border) are the same. Such variant is not always easy to perform technically, but we should strive to it. Quite often the compromise is the polygon where separate lines (hyperplanes) of delimitations are the tangents to some circumference (hypersphere). This variant is shown on Pic. 21. The polygon is used when researching the model parameter setting process, playing the role of a \\\"litmus test paper\\\". It checks whether the decision made on the basis of the set-up model corresponds to the decision made by the decision taker (or his/her simulation).\\nPeculiarity of TP lies in the fact that type of the polygon does not depend on data, depending only on the problem's dimensionality. Therefore, for all values of supply and demand vectors the polygon shall look like Pic. 21.\"\n },\n {\n \"title\": \"Spectra of the Problem (Delimitations, OF) and Observations.\",\n \"content\": \"On Picture 22 we can see the problem's spectrum, i.e. the UNLV population of six lines of delimitations and one line of OF level.\"\n },\n {\n \"title\": \"Pic. 22. TR-Polygon\",\n \"content\": \"It is necessary to mention that TP's UNLV of OF can be directed anywhere (unlike productiontype LPP where UNLV of OF can lie only within the first quadrant).\"\n },\n {\n \"title\": \"Possible Types of Decisions (for Random OF). Pairs of Spectral Vectors.\",\n \"content\": \"For the TR that corresponds to the situation in place (DMS), the decision taker (or OF that simulates him/her) chooses one of the extreme points as a solution. One of UNLV pairs corresponds to the pair of delimitation lines that form the chosen extreme point (see Pic. 22). If to consider the potentially possible variants of UNLV pairs that can participate in the formation of extreme points, we shall see 6 two-figure combinations. Here we do not include three pairs of UNLV that are parallel to each other (1-2, 3-5 and 4-6). This multitude consists of 12 variants: 1-3, 1-4, 1-5, 1-6, 2-3, 2-4, 2-5, 2-6, 3-4, 3-6, 4-5, 5-6. The decision taker chooses one of the TR's extreme points formed by the corresponding UNLV pair as an optimal one. Variants of these pairs (arrows in bold) are shown in Table 7. Here we can also see sum vectors (double arrows) for each UNLV pair. All 12 pairs are divided into three groups that are distinguished by the angle between the paired vectors and consequently, by the length of the sum vector. Looking ahead, we can notice that the length of the sum vector reflects the informativeness of the solution, i.e. of that extreme TR point, to which the given UNLV pair corresponds. The sum vector length is used as the weights of the corresponding observations when solving the reverse problem. Thus, all possible observations can belong to one of three groups: to the 1st group -the least informative, to the 3rd group -the most informative and to the 2nd group -medium informative. Table 7 UNLV Pairs Groups UNLV Pairs in Groups 1 2 3 Apart from DMS, the observation includes the decision that was made and which corresponds to one of TR's extreme points, i.e. to the sum vector, shown as a double arrow at the pictures of Table 7. Eventually, the result of each observation is the only vector (observation vector), which is used in the solution algorithm of the reverse problem for estimating the decision taker's OF vector. Only direction of the desired decision taker's OF vector is important, its length is not of any importance. As far as observation vectors are concerned, their direction and length are of a big interest because the length shows the informative value of the given observation, i.e. its contribution into the evaluation process of OF performed by the decision taker. On Picture 23 we can see all observation vectors (sums of vector pairs) that are available for the TP example under review. Thus, in each observation the DMS is represented as some population (from three to six) of the active delimitation vectors. The decision that was made by the decision taker is shown in the following manner: one of these vectors is marked by a dot like on Pic. 23 where out of 25 observations of the example under review we chose only three as the optimal ones: 1-4, 2-5, 4-5. It is necessary to mention though that pairs 1-4 and 2-5 have minimum weights (lengths), while pair 4-5 has an average weight. On Pic. 24 we can see UNLVs both of observations and the decision taker simulated OF. In Table 8 we can see their combined characteristics. In Table 8 the alternative's rank is a whole number (r=1,2,3) that can have one value out of three: = 1 with = 0.076; = 2 with = 0.293; = 3 with = 0.617, where is the observational weight. General rank is the total sum of ranks of all alternatives related to this observation. Average rank is the rank that was averaged according to the multitude of alternatives related to this observation. Average weight is the averaged weight value related to the multitude of observations.\\nAverage rank or average weight define the informative value of this observation, i.e. its contribution into the information gain about the evaluated OF made by the decision taker who can actually perform this observation. It is clear that the TR, whose configuration is similar to the polygon (Pic. 21) possesses a larger informative value (Pic. 7). In Table 8 we can see the groups of one-type TRs, inside of which the situation changes probably on the basis of the region's turn. These groups are numbered according to the increase of their average weight (or average rank).\\nIt is necessary to pay attention to the 5 th observation. All its characteristics are outstanding: it possesses maximal (in comparison with other observational variants) average weight and average rank together with the maximum number of active delimitations, i.e. the delimitations that form the TR. In this case all existing delimitations take place in the process. Polygon also looks like this delimitation.\"\n },\n {\n \"title\": \"Solution of Reverse Problem (Restoration of OF Parameters on the basis of Observations)\",\n \"content\": \"The main computational formula of a singlepoint step-by-step algorithm used for the estimation of model's parameters within the observations looks like [4]:\\nwhere = 1; 2 is the coordinate number;\\nis the weight coefficient of j-th observation.\\nThere is no discounting in this algorithm, its role is played indirectly by the coordinate normalization of the estimate vector (reduction to a single length) where the numerator uses an accumulated coordinate. Whereas the sum gradually increases within the accumulated coordinates, the relative contribution of each new observation shall decrease in proportion to this accumulation.\\nShould we introduce a sliding summing-up interval (e.g. using a K length), the summing-up limits within the sums of formula (21) shall look like: ∑ ⋯ = − +1 for > . Evaluation Results. As a continuation of the above-mentioned observations (Table 5 and Table 6) let us provide in Table 9 the calculation results received according to the single-point step-by-step algorithm.\\nEvaluation algorithm (21) is actually an averaging procedure performed with the spectral observation vectors that are taken with the corresponding weights and which are related to the multitude of the observational steps. Considering frequencies of three spectral vectors that are observed during 25 steps, it is clear that their average value should be formed within the neighborhood of 4-5 spectral vector.\\nThe calculations, provided in Table 9, confirmed this conclusion. It is necessary to note that the estimate is getting close to the Polygon's observation vector (see Polygon line in Table 9).\\nExplanation of the Obtained Results Algorithm (21) is built on the basis of averaging the weighted observation vectors. Thus, if the data was formed randomly, the appearances of any observational vector (see Pic. 23) are equally possible. As it can be seen on Pictures 23 and 24, not all directions possess equal informativeness. Therefore, the wide pattern of the observation spectrum leads to the fact that the final setting-up vector is displaced in relation to the actual (modelled) OF vector of the decision taker. However, the wide pattern of the observation spectrum can also play a positive role: thus, if the representative spectral line is \\\"taken\\\", it will provide high quality of the decisions made in the future.\\nThe estimate convergence is shown graphically on Pic. 25.\"\n },\n {\n \"title\": \"Pic. 25. Convergence of Decision Taker's OF Estimates\",\n \"content\": \"Here it can be seen that the estimates of OF coefficients (its UNLV) are converged to the values of the solution (its UNLV) at the Polygon and not to the actual (modelled) values. Still, according to the modelling results, we can see that within any newly appearing DNS (see Pic. 3 -Pic. 20) the solutions, which are obtained with the adjusted (estimated) OF do not lead to errors (they correspond to the solutions obtained in relation to the modelled OF). All model-type solutions that are accepted in any DMS shall correspond to the solutions, accepted by the decision taker (that simulate his/her OF). Thus, the approximation of OF, which was not estimate-effective, turned out to be a solution-effective one.\\nPic. 26. Convergence of Vectors' Differences Here we do not provide the solution convergence because right after the first step of settings the model's estimates turned out to be accurate enough for the solutions accepted for all other appearing DMS to completely correspond between each other according to the OF of the decision taker and according to the set-up model (its approximation).\"\n },\n {\n \"title\": \"On Adequacy Logic of the Reestablished Model\",\n \"content\": \"Presence of fast solution-convergence and bad estimate-convergence is explained by the fact that the decision taker's OF becomes apparent only through DMS (TR). Out of all possible DMS only the DMS-polygon is the most representative and the most informative (see Table 8) representative of the environment, where the decision taker works.\\nExternal observer sees objective preferences (OF-shaped) of the decision taker through DMS, therefore the OF of the decision taker should look like one of DMS elements. Problem spectrum or polygon spectrum vectors are such DMS elements.\\nIn the process of reestablishment (estimation) of the decision taker's OF we can find the UNLV of OF (as an image of OF), approximating it with one of the polygon spectrum vectors. Thus we can talk about approximating OF of the decision taker using one of the polygon observation vectors (see vectors 2-4, 2-3, 3-6, 1-6, 1-5 and 4-5 on Pic. 24). Thus, OF of the decision taker, represented by the continuous UNLV (Pic. 24) and being projected at DMS is discretized by the problem's spectrum, whose full informational representative is the polygon's spectrum. This is why the search for the estimate of the decision taker's OF that approximates his/her preferences can be performed only at the discrete spectrum of the problem (polygon). This explains the fact that the estimates (UNLV of OF) converge to one of the polygon's spectrum vectors and not to the continuous and real UNLV of decision taker's OF.\\nIt is also necessary to note that quality of approximation depends on the representativeness degree of DMS-multitude at the estimation stage, i.e. how completely it reflects the variety of all possible situations.\\nIf DMS-multitude is representative (adequate to the environment), we can talk about approximation that is adequate to any potentially possible DMS. If DMS-multitude reflects only a part of possible situations, then here we see local approximation, where we use only a part of the problem's spectrum or the polygon's spectrum (i.e. only a local spectrum is used) within the setting-up procedure and in the course of the further solution of the direct TP. In this case, the solutions obtained on the basis of the adjusted model shall be reliable only for the new DMS, which appears within the same local region of the spectrum. It is possible to say that this search will be performed \\\"with the light\\\" in that part of the problem's spectrum, which was already \\\"lit up\\\" by the previous set-up steps. If there appears a DMS that extends beyond the borders of the local one, it is necessary to test decision taker again for the knowledge of this new area, afterwards correcting the OF estimates.\"\n },\n {\n \"title\": \"Conclusions\",\n \"content\": \"\"\n },\n {\n \"title\": \"1.\",\n \"content\": \"Modelling approximation process of the decision taker preferences within the transport system using general transport table shows high speed of the solution convergence, thus providing grounds for application of similar approximations in the transportation planning systems as well as in other applications, described by the scheme of the transport problem.\\n2. Research of both the approximation algorithm and properties of the constructed model of the transport problem showed that solutions obtained with the help of the adjusted model can possess local effectiveness, i.e. solutions of the direct transport problem, obtained with the help of the adjusted model can be as good as the solutions, obtained by the decision taker in the same situations.\\n3. Stopping rules of the model's set-up process can be based on the statistic characteristics of the variations pertaining to the estimate vector of decision taker's OF such as average value and average quadratic deviation.\"\n },\n {\n \"title\": \"Table 3\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3 Modelling\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 4\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 5 Observation\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 6\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 8\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 9\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"For the TPM, ̅ solution per each DMS can be represented as the R-values matrix ̅ = ‖ ‖ . In this case, the approximation of the decision taker's preferences performed by the transport problem model lies in the evaluation of coefficients, pertaining to the target function:","content":"For the TPM, ̅ solution per each DMS can be represented as the R-values matrix ̅ = ‖ ‖ . In this case, the approximation of the decision taker's preferences performed by the transport problem model lies in the evaluation of coefficients, pertaining to the target function:"},{"id":"","caption":"23. Variants of Observational Vectors Pic. 24. UNLV of Observations and OF Comparative Characteristics of the Observation Variants. Different TR configurations, shown on Pictures 3-20 differ from each other on the basis of some properties' values (number of alternatives, their informativeness etc.).","content":"23. Variants of Observational Vectors Pic. 24. UNLV of Observations and OF Comparative Characteristics of the Observation Variants. Different TR configurations, shown on Pictures 3-20 differ from each other on the basis of some properties' values (number of alternatives, their informativeness etc.)."},{"id":"","caption":"approximating OF in relation to the actual OF of UNLV (upper one) and in relation to the closest spectral vector of the Polygon are shown on Picture 26.","content":"approximating OF in relation to the actual OF of UNLV (upper one) and in relation to the closest spectral vector of the Polygon are shown on Picture 26."},{"id":"","caption":"After removing the constant component that does not influence the solution, we shall obtain a working variant of the OF:","content":"After removing the constant component that does not influence the solution, we shall obtain a working variant of the OF:). In this \ncase, within the new (free) coordinates ( 22 and \n23 ) the complete form of LPP's OF shall look like: \n( ̅ ) = −( 11 ( 1 − 2 − 3 ) + 12 2 + 13 3 \n+ 21 2 ) \n+ (− 11 + 12 + 21 − 22 ) 22 \n+ (− 11 + 13 + 21 − 23 ) 23 \n→ max. \n(15) \n\n( ̅ ) = (− 11 + 12 + 21 − 22 ) 22 \n+ (− 11 + 13 + 21 − 23 ) 23 \n→ max. \n(16)"},{"id":"","caption":"contains expenditures expressed in absolute units (e.g. in Rubles). Right table column contains one of the variants of the supply (stocks) vector value ̅ = [ 1 2 ] , while the bottom line contains the variant of the demand vector value ̅ = [ 1 2 3 ] . When the line and the column intersect we can see the supply and demand balance value.","content":"Table 3\n3contains expenditures expressed in absolute units (e.g. in Rubles). Right table column contains one of the variants of the supply (stocks) vector value ̅ = [ 1 2 ] , while the bottom line contains the variant of the demand vector value ̅ = [ 1 2 3 ] . When the line and the column intersect we can see the supply and demand balance value."},{"id":"","caption":"Left and Right Delimitation Parts of LPP That Is Equivalent to the Original TP DelimitationNumber","content":"Table 3 Modelling\n3Data \n\n10 2 20 10 \n\n12 7 \n9 25 \n\n5 15 15 35"},{"id":"","caption":"Decisions Made by Decision Taker (Simulation) in relation to the Observation Sample","content":"Table 4\n4Left and Right Delimitation Parts of LPP That Is Equivalent to the Original TP DelimitationNumberVariables \nCondition \nRight Parts \n\n22 \n23"},{"id":"","caption":"Properties of Observational VariantsNo. Type of TR","content":"Table 5 Observation\n5Sample (DMS) \nObservation \nStep \n\nSupply \nDemand \nEquilibrium \n\n1 \n10 \n25 \n5 \n15 \n15 \n35 \n2 \n13 \n52 \n26 \n19 \n20 \n65 \n3 \n71 \n79 \n17 \n87 \n46 \n150 \n4 \n2 \n29 \n12 \n13 \n6 \n31 \n5 \n5 \n4 \n2 \n5 \n2 \n9 \n6 \n65 \n70 \n56 \n43 \n36 \n135 \n7 \n107 \n23 \n55 \n19 \n56 \n130 \n8 \n96 \n6 \n24 \n5 \n73 \n102 \n9 \n32 \n54 \n27 \n54 \n5 \n86 \n10 \n31 \n79 \n32 \n47 \n31 \n110 \n11 \n92 \n4 \n25 \n41 \n30 \n96 \n12 \n44 \n50 \n47 \n45 \n2 \n94 \n13 \n24 \n74 \n9 \n36 \n53 \n98 \n14 \n64 \n81 \n83 \n56 \n6 \n145 \n15 \n97 \n22 \n35 \n54 \n30 \n119 \n16 \n14 \n6 \n9 \n8 \n3 \n20 \n17 \n90 \n4 \n12 \n51 \n31 \n94 \n18 \n27 \n56 \n45 \n13 \n25 \n83 \n19 \n78 \n66 \n52 \n48 \n44 \n144 \n20 \n75 \n99 \n65 \n52 \n57 \n174 \n21 \n12 \n1 \n6 \n4 \n3 \n13 \n22 \n31 \n69 \n24 \n44 \n32 \n100 \n23 \n64 \n39 \n38 \n34 \n31 \n103 \n24 \n83 \n36 \n28 \n51 \n40 \n119 \n25 \n15 \n12 \n16 \n1 \n10 \n27 \nPolygon \n5 \n3 \n4 \n2 \n2 \n8"},{"id":"","caption":"Decisions Made by Decision Taker in relation to the Observation Sample","content":"Table 6\n6Decisions Made by Decision Taker (Simulation) in relation to the Observation SampleObservation \nStep \n\nSolution \nOF \nActive \nDelimitations \n\nL norm_2 \nDel. 1 Del. 2 \n1 \n0 \n10 \n0 \n5 \n5 \n15 \n8.963 \n1 \n4 \n2 \n0 \n13 \n0 \n26 \n6 \n20 \n20.077 \n1 \n4 \n3 \n0 \n71 \n0 \n17 \n16 \n46 \n31.263 \n1 \n4 \n4 \n0 \n2 \n0 \n12 \n11 \n6 \n10.003 \n1 \n4 \n5 \n0 \n5 \n0 \n2 \n0 \n2 \n1.864 \n1 \n4 \n6 \n22 \n43 \n0 \n34 \n0 \n36 \n37.214 \n4 \n5 \n7 \n55 \n19 \n33 \n0 \n0 \n23 \n52.164 \n2 \n5 \n8 \n24 \n5 \n67 \n0 \n0 \n6 \n58.940 \n2 \n5 \n9 \n0 \n32 \n0 \n27 \n22 \n5 \n21.045 \n1 \n4 \n10 \n0 \n31 \n0 \n32 \n16 \n31 \n30.008 \n1 \n4 \n11 \n25 \n41 \n26 \n0 \n0 \n4 \n31.836 \n2 \n5 \n12 \n0 \n44 \n0 \n47 \n1 \n2 \n24.272 \n1 \n4 \n13 \n0 \n24 \n0 \n9 \n12 \n53 \n25.706 \n1 \n4 \n14 \n8 \n56 \n0 \n75 \n0 \n6 \n41.086 \n4 \n5 \n15 \n35 \n54 \n8 \n0 \n0 \n22 \n29.255 \n2 \n5 \n16 \n6 \n8 \n0 \n3 \n0 \n3 \n4.983 \n1 \n4 \n17 \n12 \n51 \n27 \n0 \n0 \n4 \n28.610 \n2 \n5 \n18 \n14 \n13 \n0 \n31 \n0 \n25 \n27.355 \n4 \n5 \n19 \n30 \n48 \n0 \n22 \n0 \n44 \n37.859 \n4 \n5 \n20 \n23 \n52 \n0 \n42 \n0 \n57 \n48.436 \n4 \n5 \n21 \n6 \n4 \n2 \n0 \n0 \n1 \n4.195 \n2 \n5 \n22 \n0 \n31 \n0 \n24 \n13 \n32 \n26.136 \n1 \n4 \n23 \n30 \n34 \n0 \n8 \n0 \n31 \n26.638 \n4 \n5 \n24 \n28 \n51 \n4 \n0 \n0 \n36 \n28.179 \n2 \n5 \n25 \n14 \n1 \n0 \n2 \n0 \n10 \n9.178 \n4 \n5 \nPolygon \n3 \n2 \n0 \n1 \n0 \n2 \n4 \n5 \n\nObservations: DMS and Decisions Made by \nDecision Taker \nIn each observation, DMS are represented by \nthe values of two vectors, which are ̅ = \n[ 1 \n2 ] and ̅ = [ 1 \n\n2 \n\n3 ] . Let the \n\nobservation sample for the simulation experiment \nconsist of 25 situations (DMS) where it is \nnecessary to build the transportation plan. In"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"For the TPM, ̅ solution per each DMS can be represented as the R-values matrix ̅ = ‖ ‖ . In this case, the approximation of the decision taker's preferences performed by the transport problem model lies in the evaluation of coefficients, pertaining to the target function:\",\n \"content\": \"For the TPM, ̅ solution per each DMS can be represented as the R-values matrix ̅ = ‖ ‖ . In this case, the approximation of the decision taker's preferences performed by the transport problem model lies in the evaluation of coefficients, pertaining to the target function:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"23. Variants of Observational Vectors Pic. 24. UNLV of Observations and OF Comparative Characteristics of the Observation Variants. Different TR configurations, shown on Pictures 3-20 differ from each other on the basis of some properties' values (number of alternatives, their informativeness etc.).\",\n \"content\": \"23. Variants of Observational Vectors Pic. 24. UNLV of Observations and OF Comparative Characteristics of the Observation Variants. Different TR configurations, shown on Pictures 3-20 differ from each other on the basis of some properties' values (number of alternatives, their informativeness etc.).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"approximating OF in relation to the actual OF of UNLV (upper one) and in relation to the closest spectral vector of the Polygon are shown on Picture 26.\",\n \"content\": \"approximating OF in relation to the actual OF of UNLV (upper one) and in relation to the closest spectral vector of the Polygon are shown on Picture 26.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"After removing the constant component that does not influence the solution, we shall obtain a working variant of the OF:\",\n \"content\": \"After removing the constant component that does not influence the solution, we shall obtain a working variant of the OF:). In this \\ncase, within the new (free) coordinates ( 22 and \\n23 ) the complete form of LPP's OF shall look like: \\n( ̅ ) = −( 11 ( 1 − 2 − 3 ) + 12 2 + 13 3 \\n+ 21 2 ) \\n+ (− 11 + 12 + 21 − 22 ) 22 \\n+ (− 11 + 13 + 21 − 23 ) 23 \\n→ max. \\n(15) \\n\\n( ̅ ) = (− 11 + 12 + 21 − 22 ) 22 \\n+ (− 11 + 13 + 21 − 23 ) 23 \\n→ max. \\n(16)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"contains expenditures expressed in absolute units (e.g. in Rubles). Right table column contains one of the variants of the supply (stocks) vector value ̅ = [ 1 2 ] , while the bottom line contains the variant of the demand vector value ̅ = [ 1 2 3 ] . When the line and the column intersect we can see the supply and demand balance value.\",\n \"content\": \"Table 3\\n3contains expenditures expressed in absolute units (e.g. in Rubles). Right table column contains one of the variants of the supply (stocks) vector value ̅ = [ 1 2 ] , while the bottom line contains the variant of the demand vector value ̅ = [ 1 2 3 ] . When the line and the column intersect we can see the supply and demand balance value.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Left and Right Delimitation Parts of LPP That Is Equivalent to the Original TP DelimitationNumber\",\n \"content\": \"Table 3 Modelling\\n3Data \\n\\n10 2 20 10 \\n\\n12 7 \\n9 25 \\n\\n5 15 15 35\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Decisions Made by Decision Taker (Simulation) in relation to the Observation Sample\",\n \"content\": \"Table 4\\n4Left and Right Delimitation Parts of LPP That Is Equivalent to the Original TP DelimitationNumberVariables \\nCondition \\nRight Parts \\n\\n22 \\n23\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Properties of Observational VariantsNo. Type of TR\",\n \"content\": \"Table 5 Observation\\n5Sample (DMS) \\nObservation \\nStep \\n\\nSupply \\nDemand \\nEquilibrium \\n\\n1 \\n10 \\n25 \\n5 \\n15 \\n15 \\n35 \\n2 \\n13 \\n52 \\n26 \\n19 \\n20 \\n65 \\n3 \\n71 \\n79 \\n17 \\n87 \\n46 \\n150 \\n4 \\n2 \\n29 \\n12 \\n13 \\n6 \\n31 \\n5 \\n5 \\n4 \\n2 \\n5 \\n2 \\n9 \\n6 \\n65 \\n70 \\n56 \\n43 \\n36 \\n135 \\n7 \\n107 \\n23 \\n55 \\n19 \\n56 \\n130 \\n8 \\n96 \\n6 \\n24 \\n5 \\n73 \\n102 \\n9 \\n32 \\n54 \\n27 \\n54 \\n5 \\n86 \\n10 \\n31 \\n79 \\n32 \\n47 \\n31 \\n110 \\n11 \\n92 \\n4 \\n25 \\n41 \\n30 \\n96 \\n12 \\n44 \\n50 \\n47 \\n45 \\n2 \\n94 \\n13 \\n24 \\n74 \\n9 \\n36 \\n53 \\n98 \\n14 \\n64 \\n81 \\n83 \\n56 \\n6 \\n145 \\n15 \\n97 \\n22 \\n35 \\n54 \\n30 \\n119 \\n16 \\n14 \\n6 \\n9 \\n8 \\n3 \\n20 \\n17 \\n90 \\n4 \\n12 \\n51 \\n31 \\n94 \\n18 \\n27 \\n56 \\n45 \\n13 \\n25 \\n83 \\n19 \\n78 \\n66 \\n52 \\n48 \\n44 \\n144 \\n20 \\n75 \\n99 \\n65 \\n52 \\n57 \\n174 \\n21 \\n12 \\n1 \\n6 \\n4 \\n3 \\n13 \\n22 \\n31 \\n69 \\n24 \\n44 \\n32 \\n100 \\n23 \\n64 \\n39 \\n38 \\n34 \\n31 \\n103 \\n24 \\n83 \\n36 \\n28 \\n51 \\n40 \\n119 \\n25 \\n15 \\n12 \\n16 \\n1 \\n10 \\n27 \\nPolygon \\n5 \\n3 \\n4 \\n2 \\n2 \\n8\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Decisions Made by Decision Taker in relation to the Observation Sample\",\n \"content\": \"Table 6\\n6Decisions Made by Decision Taker (Simulation) in relation to the Observation SampleObservation \\nStep \\n\\nSolution \\nOF \\nActive \\nDelimitations \\n\\nL norm_2 \\nDel. 1 Del. 2 \\n1 \\n0 \\n10 \\n0 \\n5 \\n5 \\n15 \\n8.963 \\n1 \\n4 \\n2 \\n0 \\n13 \\n0 \\n26 \\n6 \\n20 \\n20.077 \\n1 \\n4 \\n3 \\n0 \\n71 \\n0 \\n17 \\n16 \\n46 \\n31.263 \\n1 \\n4 \\n4 \\n0 \\n2 \\n0 \\n12 \\n11 \\n6 \\n10.003 \\n1 \\n4 \\n5 \\n0 \\n5 \\n0 \\n2 \\n0 \\n2 \\n1.864 \\n1 \\n4 \\n6 \\n22 \\n43 \\n0 \\n34 \\n0 \\n36 \\n37.214 \\n4 \\n5 \\n7 \\n55 \\n19 \\n33 \\n0 \\n0 \\n23 \\n52.164 \\n2 \\n5 \\n8 \\n24 \\n5 \\n67 \\n0 \\n0 \\n6 \\n58.940 \\n2 \\n5 \\n9 \\n0 \\n32 \\n0 \\n27 \\n22 \\n5 \\n21.045 \\n1 \\n4 \\n10 \\n0 \\n31 \\n0 \\n32 \\n16 \\n31 \\n30.008 \\n1 \\n4 \\n11 \\n25 \\n41 \\n26 \\n0 \\n0 \\n4 \\n31.836 \\n2 \\n5 \\n12 \\n0 \\n44 \\n0 \\n47 \\n1 \\n2 \\n24.272 \\n1 \\n4 \\n13 \\n0 \\n24 \\n0 \\n9 \\n12 \\n53 \\n25.706 \\n1 \\n4 \\n14 \\n8 \\n56 \\n0 \\n75 \\n0 \\n6 \\n41.086 \\n4 \\n5 \\n15 \\n35 \\n54 \\n8 \\n0 \\n0 \\n22 \\n29.255 \\n2 \\n5 \\n16 \\n6 \\n8 \\n0 \\n3 \\n0 \\n3 \\n4.983 \\n1 \\n4 \\n17 \\n12 \\n51 \\n27 \\n0 \\n0 \\n4 \\n28.610 \\n2 \\n5 \\n18 \\n14 \\n13 \\n0 \\n31 \\n0 \\n25 \\n27.355 \\n4 \\n5 \\n19 \\n30 \\n48 \\n0 \\n22 \\n0 \\n44 \\n37.859 \\n4 \\n5 \\n20 \\n23 \\n52 \\n0 \\n42 \\n0 \\n57 \\n48.436 \\n4 \\n5 \\n21 \\n6 \\n4 \\n2 \\n0 \\n0 \\n1 \\n4.195 \\n2 \\n5 \\n22 \\n0 \\n31 \\n0 \\n24 \\n13 \\n32 \\n26.136 \\n1 \\n4 \\n23 \\n30 \\n34 \\n0 \\n8 \\n0 \\n31 \\n26.638 \\n4 \\n5 \\n24 \\n28 \\n51 \\n4 \\n0 \\n0 \\n36 \\n28.179 \\n2 \\n5 \\n25 \\n14 \\n1 \\n0 \\n2 \\n0 \\n10 \\n9.178 \\n4 \\n5 \\nPolygon \\n3 \\n2 \\n0 \\n1 \\n0 \\n2 \\n4 \\n5 \\n\\nObservations: DMS and Decisions Made by \\nDecision Taker \\nIn each observation, DMS are represented by \\nthe values of two vectors, which are ̅ = \\n[ 1 \\n2 ] and ̅ = [ 1 \\n\\n2 \\n\\n3 ] . Let the \\n\\nobservation sample for the simulation experiment \\nconsist of 25 situations (DMS) where it is \\nnecessary to build the transportation plan. In\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"A concept of Corporate Planning. R L Ackoff, John WileyNew YorkR.L. Ackoff. A concept of Corporate Planning. -New York: John Wiley, 1970.","doi":""},{"text":"On Adaptive Management Methods in Transitional Economy // Economic Science of Modern Russia. K A Bagrinovskii, CEMI RSSI Publishing House2K.A. Bagrinovskii. On Adaptive Management Methods in Transitional Economy // Economic Science of Modern Russia, CEMI RSSI Publishing House, 2, 1999.","doi":""},{"text":"Brain of the Firm. S Beer, John WileyNew YorkS. Beer. Brain of the Firm. -New York: John Wiley, 1988.","doi":""},{"text":"Adaptive choice of control solutions. Models of operation research as the method for storage of knowledge of the decision-maker. V Ya, Vilisov, LAP LAMBERT Academic PublishingSaarbruckenV.Ya. Vilisov. Adaptive choice of control solutions. Models of operation research as the method for storage of knowledge of the decision-maker. -Saarbrucken: LAP LAMBERT Academic Publishing, 2011.","doi":""},{"text":"Introduction into MIS. V M Glushkov, Tekhnika Publishing HouseKievV.M. Glushkov. Introduction into MIS. -Kiev: Tekhnika Publishing House, 1974.","doi":""},{"text":"Simulation Modelling of Economic Processes. A A Yemelyanov, Ye A Vlasova, R V Duma, A.A. Yemelyanov. MoscowFinances and Statistics Publishing HouseA.A. Yemelyanov, Ye.A. Vlasova, R.V. Duma. Simulation Modelling of Economic Processes / Edited by A.A. Yemelyanov. Moscow: Finances and Statistics Publishing House, 2009.","doi":""},{"text":"Mathematics Makes an Experiment. N N Moiseyev, Nauka Publishing HouseMoscowN.N. Moiseyev. Mathematics Makes an Experiment. -Moscow: Nauka Publishing House, 1979","doi":""},{"text":"Management Science: a Bayesian Introduction. W T Morris, Prentice HallW.T. Morris. Management Science: a Bayesian Introduction, Prentice Hall, 1968.","doi":""},{"text":"Toyota Production System: Beyond Large-Scale Production. Taiichi Ohno, Productivity PressTaiichi Ohno. Toyota Production System: Beyond Large-Scale Production. - Productivity Press, 1988.","doi":""},{"text":"Right on Time for Russia: ERP-Systems Application. S V Piterkin, Practice / S.V. Piterkin, I.A. Oladov, D.V. IsayevAlpina Business BooksMoscowS.V. Piterkin. Right on Time for Russia: ERP- Systems Application Practice / S.V. Piterkin, I.A. Oladov, D.V. Isayev. -Moscow: Alpina Business Books, 2006.","doi":""},{"text":"Theories of Decision-Making in Economics and Behavioral Science. H A Simon, The American Economic Review. 493H.A. Simon. Theories of Decision-Making in Economics and Behavioral Science. -The American Economic Review, Vol. 49, No. 3 (Jun., 1959), pp. 253-283.","doi":""},{"text":"Operation research: An Introduction. 8-th ed. H A Taha, Pearson Prentice Hall813New JerseyH.A. Taha. Operation research: An Introduction. 8-th ed., Pearson Prentice Hall, New Jersey, 2007, 813 p.","doi":""},{"text":"Robot Training Under Conditions of Incomplete Information. V Ya, Vilisov, International Scientific and Technological Conference EXTREME ROBOTICS, 1.10. St.PetersburgV.Ya. Vilisov. Robot Training Under Conditions of Incomplete Information. - International Scientific and Technological Conference EXTREME ROBOTICS, 1.10.2013, St.Petersburg. - URL: http://arxiv.org/abs/1402.2996","doi":""},{"text":"Research of the Robot's Learning Effectiveness in the Changing Environment. -International Scientific and Technological Conference EXTREME ROBOTICS, 1.10. V Ya, Vilisov, 10.13140/RG.2.1.3343.7926St.PetersburgV.Ya. Vilisov. Research of the Robot's Learning Effectiveness in the Changing Environment. -International Scientific and Technological Conference EXTREME ROBOTICS, 1.10.2014, St.Petersburg. - http://dx.doi.org/10.13140/RG.2.1.3343.7926","doi":"10.13140/RG.2.1.3343.7926"}],"string":"[\n {\n \"text\": \"A concept of Corporate Planning. R L Ackoff, John WileyNew YorkR.L. Ackoff. A concept of Corporate Planning. -New York: John Wiley, 1970.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On Adaptive Management Methods in Transitional Economy // Economic Science of Modern Russia. K A Bagrinovskii, CEMI RSSI Publishing House2K.A. Bagrinovskii. On Adaptive Management Methods in Transitional Economy // Economic Science of Modern Russia, CEMI RSSI Publishing House, 2, 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Brain of the Firm. S Beer, John WileyNew YorkS. Beer. Brain of the Firm. -New York: John Wiley, 1988.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Adaptive choice of control solutions. Models of operation research as the method for storage of knowledge of the decision-maker. V Ya, Vilisov, LAP LAMBERT Academic PublishingSaarbruckenV.Ya. Vilisov. Adaptive choice of control solutions. Models of operation research as the method for storage of knowledge of the decision-maker. -Saarbrucken: LAP LAMBERT Academic Publishing, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Introduction into MIS. V M Glushkov, Tekhnika Publishing HouseKievV.M. Glushkov. Introduction into MIS. -Kiev: Tekhnika Publishing House, 1974.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Simulation Modelling of Economic Processes. A A Yemelyanov, Ye A Vlasova, R V Duma, A.A. Yemelyanov. MoscowFinances and Statistics Publishing HouseA.A. Yemelyanov, Ye.A. Vlasova, R.V. Duma. Simulation Modelling of Economic Processes / Edited by A.A. Yemelyanov. Moscow: Finances and Statistics Publishing House, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Mathematics Makes an Experiment. N N Moiseyev, Nauka Publishing HouseMoscowN.N. Moiseyev. Mathematics Makes an Experiment. -Moscow: Nauka Publishing House, 1979\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Management Science: a Bayesian Introduction. W T Morris, Prentice HallW.T. Morris. Management Science: a Bayesian Introduction, Prentice Hall, 1968.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Toyota Production System: Beyond Large-Scale Production. Taiichi Ohno, Productivity PressTaiichi Ohno. Toyota Production System: Beyond Large-Scale Production. - Productivity Press, 1988.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Right on Time for Russia: ERP-Systems Application. S V Piterkin, Practice / S.V. Piterkin, I.A. Oladov, D.V. IsayevAlpina Business BooksMoscowS.V. Piterkin. Right on Time for Russia: ERP- Systems Application Practice / S.V. Piterkin, I.A. Oladov, D.V. Isayev. -Moscow: Alpina Business Books, 2006.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Theories of Decision-Making in Economics and Behavioral Science. H A Simon, The American Economic Review. 493H.A. Simon. Theories of Decision-Making in Economics and Behavioral Science. -The American Economic Review, Vol. 49, No. 3 (Jun., 1959), pp. 253-283.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Operation research: An Introduction. 8-th ed. H A Taha, Pearson Prentice Hall813New JerseyH.A. Taha. Operation research: An Introduction. 8-th ed., Pearson Prentice Hall, New Jersey, 2007, 813 p.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Robot Training Under Conditions of Incomplete Information. V Ya, Vilisov, International Scientific and Technological Conference EXTREME ROBOTICS, 1.10. St.PetersburgV.Ya. Vilisov. Robot Training Under Conditions of Incomplete Information. - International Scientific and Technological Conference EXTREME ROBOTICS, 1.10.2013, St.Petersburg. - URL: http://arxiv.org/abs/1402.2996\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Research of the Robot's Learning Effectiveness in the Changing Environment. -International Scientific and Technological Conference EXTREME ROBOTICS, 1.10. V Ya, Vilisov, 10.13140/RG.2.1.3343.7926St.PetersburgV.Ya. Vilisov. Research of the Robot's Learning Effectiveness in the Changing Environment. -International Scientific and Technological Conference EXTREME ROBOTICS, 1.10.2014, St.Petersburg. - http://dx.doi.org/10.13140/RG.2.1.3343.7926\",\n \"doi\": \"10.13140/RG.2.1.3343.7926\"\n }\n]","isConversational":false}}},{"rowIdx":581,"cells":{"text":{"kind":"string","value":"\nProperties of Laughlin states on fractal lattices\n\n\nMani Chandra \nMax-Planck-Institut für Physik komplexer Systeme\nD-01187DresdenGermany\n\nJha \nAnne E B Nielsen \nDepartment of Physics and Astronomy\nAarhus University\nDK-8000Aarhus CDenmark\n\nProperties of Laughlin states on fractal lattices\n\nLaughlin states have recently been constructed on fractal lattices and have been shown to be topological in such systems. Some of their properties are, however, quite different from the two-dimensional case. On the Sierpinski triangle, for instance, the entanglement entropy shows oscillations as a function of particle number and does not obey the area law despite being topologically ordered, and the particle density is non-uniform in the bulk. Here, we investigate these deviant properties in greater detail on the Sierpinski triangle, and we also study the properties on the Sierpinski carpet and the T-fractal. We find that the density variations across the fractal are present for all the considered fractal lattices and for most choices of the number of particles. The size of anyons inserted into the lattice Laughlin state also varies with position on the fractal lattice. We observe that quasiholes and quasiparticles have similar sizes and that the size of the anyons typically increases with decreasing Hausdorff dimension. As opposed to periodic lattices in two dimensions, the Sierpinski triangle and carpet have inner edges. We construct trial states for both inner and outer edge states. We find that oscillations of the entropy as a function of particle number are present for the T-fractal, but not for the Sierpinski carpet. Finally, we observe deviations from the area law for several different bipartitions on the Sierpinski triangle.\n\nIntroduction\n\nWhen a two-dimensional electron gas is placed in a strong magnetic field and at very low temperatures, the fractional quantum Hall (FQH) effect can appear [1]. The Hamiltonian describing this effect, which is basically the Coulomb interaction projected to the lowest Landau level, is macroscopically degenerate and extremely hard to diagonalize, except for a small number of particles. In 1983, R. B. Laughlin proposed an ansatz [2] for the ground state wavefunction of the FQH effect at odd-denominator filling factors. The so-called Laughlin wavefunction predicted the existence of quasihole excitations with fractional electric charge. It was shown numerically to have more than 99 percent overlap with the exact ground state of the Coulomb repulsion for small systems [2].\n\nIn 1987, Kalmeyer and Laughlin showed that the ground state of a frustrated Heisenberg antiferromagnet on a triangular lattice is well described by a Laughlin wave function for bosons in which the particle-coordinates were restricted to be the lattice coordinates [3]. This so-called Kalmeyer-Laughlin wave function showed that FQH physics can be found in settings different from where it was originally discovered. Later, several models describing FQH physics in lattices were proposed [4,5].\n\nThe work on FQH lattice models is, in part, motivated by the developments in quantum engineering, e.g. in ultracold atoms. Such systems also allow for going beyond periodic lattices, which opens several possibilities. With Rydberg atoms in optical tweezers arbitrary atomic arrangements can be obtained [6,7], and there are also ideas for implementing artificial gauge fields in these systems [8][9][10], which is a starting point for obtaining quantum Hall physics.\n\nFractals are an interesting example of nonperiodic systems, because their Hausdorff dimension can be noninteger, and hence one can use the spatial dimension as a parameter to change the properties of the physical system. The quantum Hall effect was originally studied in two dimensions, but it has turned out that it can also appear in fractal dimensions. Examples of integer quantum Hall models in fractals [11][12][13][14][15][16][17] already show features that are not present for periodic lattices in two dimensions, such as inner edge states. FQH physics has also been investigated on fractal lattices [18][19][20][21], and one of the interesting observations here is that the entropy can scale differently with subsystem size than it does in two dimensions.\n\nTrial states, such as the Laughlin state, are important tools to gain insight into the FQH effect. Moore and Read showed that the Laughlin state can be expressed as a correlation function in conformal field theory [22], and this construction was later modified to obtain lattice versions of the Laughlin state that share the same topological features as the continuum Laughlin state for several different lattices and parameters [23,24]. The construction can also be applied to obtain Laughlin states on fractal lattices as long as the fractal lattices can be embedded in the two-dimensional plane. This again leads to topological systems in many cases, allowing quasiholes to be added and investigated [18]. The states were studied further on the Sierpinski triangle in [19], where it was shown numerically that the entanglement entropy of the states does not follow an area law. It was also found that the density of particles on the Sierpinski triangle is different from the two-dimensional case.\n\nIn this paper, we take a deeper look into the properties of Laughlin states on different fractal lattices. In addition to the Sierpinski triangle, we also consider the Sierpinski carpet and the T-fractal. These fractals differ both with respect to Hausdorff dimension and ramification number. The considered fractal lattices and the lattice Laughlin wavefunction are described in section 2. Since fractal lattices do not have a periodic bulk, the density of particles generally varies across the fractal lattice as observed for the Sierpinski triangle in [19]. In section 3, we study how the density variations across the fractal lattice depend on the number of particles for the three fractals. We also find that the variations go to zero at half-filling of the lattice, which is due to particle-hole symmetry at half-filling. In section 4, we introduce both quasiholes and quasiparticles into the lattice Laughlin state. We investigate how the presence of the anyons modify the density, and we investigate how the size of the anyons depend on their position in the fractal lattice. In section 5, we construct trial states for edge states on both the inner and outer edges of the Sierpinski triangle and the Sierpinski carpet. In section 6, we compute the behaviour of the entropy as a function of particle number. It was found previously [19] that the entropy shows oscillations with a fixed period as a function of particle number for the Sierpinksi triangle, but not for the square lattice. Here, we find that oscillations also occur for the T-fractal, but not for the Sierpinski carpet. This suggests that the ramification number could play a role for the presence or absence of the oscillations. In section 7, we investigate how the entropy scales with subsystem size for different bipartitions of the Sierpinski triangle. We find that different bipartitions give rise to different behaviors violating the area law. In section 8, we summarize the conclusions. In Appendix A, we describe analytically how the density and entanglement entropy transform under the particle-hole transformation.\n\n\nThe lattice Laughlin wavefunction\n\nThe lattice Laughlin states considered in [23,24] have the following form\n|ψ = n 1 ,...,n N ψ(n 1 , . . . , n N )|n 1 , . . . , n N ,(1)ψ(n 1 , . . . , n N ) = C M δ n,M i<j (z i − z j ) qn i n j k =l (z k − z l ) −ηn l ,\nwhere the z i are the coordinates of the lattice sites written as complex numbers and n i are their occupation numbers, which are either 0 or 1, since the particles are fermions for odd q and hardcore bosons for even q. The positive integer q = 1/ν is the flux per particle. The Kronecker delta\nδ n,M ≡ δ i n i ,M = 1 for i n i = M 0 otherwise(2)\nensures that i n i = M satisfies qM = N η for all nonzero terms of the wavefunction, where M is the total number of particles and N is the total number of lattice sites.\n\nSince qM is the total number of fluxes, η can be interpreted as the flux per lattice site. The constant of proportionality C M is chosen such that the wavefunction |ψ is normalized. The last factor in (1) differs from the Gaussian factor that appears in the original Laughlin state, because the background charge is here placed on the lattice sites only. This point turns out to be important to obtain the desired physics on fractal lattices [18]. An exact parent Hamiltonian for (1) can be found in [18,24], and [21] made a systematic search for simpler, approximate parent Hamiltonians. In the following sections, we study the properties of the wavefunction (1) when the z i are given by the coordinates of a finite generation fractal lattice or a square lattice. Specifically, we consider the Sierpinski triangle depicted in figure 1(a) with N = 3 g sites, the Sierpinski carpet in figure 1(d) with N = 8 g sites, and the T-fractal in figure 1(g) with N = 5 g sites, where g is the generation. The infinite version of these fractals have Hausdorff dimensions ln(3)/ ln(2) ≈ 1.58, ln(8)/ ln(3) ≈ 1.89, and ln(5)/ ln(3) ≈ 1.46, respectively. They also differ in ramification number [25]. If we imagine all the nearestneighbor sites being linked by a bond, an arbitrarily large piece of the T-fractal can be isolated by cutting one bond, while that number is two for the Sierpinski triangle and infinite for the Sierpinski carpet.\n\n\nParticle density\n\nWe first consider the particle density on the lattice sites, i.e. ψ|n i |ψ , wheren i =ĉ † iĉi is the number operator andĉ i is the operator that annihilates a particle on the ith lattice site. This quantity can be computed via Monte Carlo simulations of the expression\nn i = n 1 ,...,n N n i |ψ(n 1 , . . . , n N )| 2 .(3)\nFor Laughlin states on two-dimensional, periodic lattices, we expect the expectation value of the number operator to be uniform in the bulk of the system (see, e.g., [19] for computations on square and triangular lattices). For fractal structures, however, there is not a distinct edge or bulk. The density pattern is, hence, not obvious a priori. In [19], the particle density of the Laughlin state on the g = 5 Sierpinski triangle for a fixed particle number (M = 27) was computed and compared to the density on a triangular lattice. While the density in the bulk of the triangular lattice is constant, there are fluctuations around the mean value near the edge. On the fractal lattice, the density fluctuates from site to site, roughly imitating the edge density pattern of the triangular lattice. It was also found that the density on the corner sites of the fractal differs significantly from the density on the other sites of the fractal lattice.\n\nHere, we take this analysis further by studying the densities on the Sierpinski triangle as a function of M , as opposed to studying the spatial pattern for a fixed value of M . Moreover, we study the density using this approach also for the Sierpinski carpet and the T-fractal. At zero and complete filling the densities are trivially constant across the lattice, so we will not discuss those cases in the following. For almost all other values of M , we find that there is a spatial variation, but at exactly half filling, the variance goes to zero. The plots also have a symmetry around the half-filling point, which appears in all the studied fractals at all values of q. Both of these observations can be explained by studying the particle-hole transformation of the lattice Laughlin state as noted in [26] and discussed in detail in Appendix A below. We also see that the corner densities behave differently from the rest of the lattice, except near and at half-filling. We propose that the density structure can be used as a tool to define and distinguish a bulk and edge for a fractal where, a priori, there is no clear distinction between them. There is a significant dispersion within the central band at lower values of M . At M = 27, however, which corresponds to the case of η = qM/N = 1, the dispersion decreases somewhat. After this the dispersion rises slightly and briefly until M = 30, at which point it starts falling monotonically. At M = 40 and M = 41 (corresponding to almost half filling) it is close to zero. As a whole, the densities reflect the particle-hole symmetry. This means that the plot of n i r as a function of M is unchanged by a rotation by π around the point ( n i r = 0, M = N/2). In other words, n i r,M = − n i r,N −M , where n i r,M is n i r for the state with M particles. This behaviour of the densities is derived analytically in Appendix A.\n(a) (b) (c) (d) (e) (f) (g) (h) (i)\nIn figure 1(c), the densities are plotted for the next generation of the fractal with 243 sites and q = 3. We see that the structure of a central band surrounded by deviant lines corresponding to corner sites and their immediate neighbours becomes even more apparent. Again we see a pinching of the densities at η = 1, i.e. M = 81, as well as close to half-filling (M = 120, 121). At η = 1, only the central band seems to pinch while the corner (and next to corner) lines remain separated, but at half-filling, even the densities at the corner sites go to zero.\n\nWe next consider the Sierpinski carpet in figure 1(d). The densities are shown in figure 1(e-f). The values for larger fillings can be obtained utilizing the symmetry discussed above. We see a central band which is initially separated from a set of lines below it. The points below correspond to the four corner sites. At around M = 150, they merge with the central band. For the case of q = 2, this happens at around M = 175. For both values of q, n i r goes to zero at half filling (i.e. M = 256) for all lattice sites. This is not immediately visible in the figure, however, due to the large number of points plotted.\n\nIn figure 1(h-i), we plot the particle densities for the T-fractal. Compared to the Sierpinski triangle and carpet, there are larger variations in the densities with several sites having | n i r | > 0.05. At half-filling, the densities converge to zero as for the other fractal lattices.\n\n\nQuasiholes and quasiparticles\n\nIn this section, we investigate quasiholes and quasiparticle in the Laughlin states. We first give examples of density profiles of the anyons, and we then study the size of the anyons as a function of position on the fractal lattices. Building on [2], it was shown in [27] that states describing quasiholes and quasiparticles on periodic lattices with open boundary conditions can be obtained by modifying the lattice Laughlin state (1) into\n|ψ p, w = n 1 ,...,n N ψ p, w (n 1 , . . . , n N )|n 1 , . . . , n N ,(4)ψ p, w (n 1 , . . . , n N ) ∝ δ n,M i,j (w i − z j ) p i n j i<j (z i − z j ) qn i n j k =l (z k − z l ) −ηn l ,\nwhere w i is the coordinate of the ith anyon, and p i = +1 (p i = −1) if the ith anyon is a quasihole (quasiparticle). Also,\nη = qM + i p i /N.(5)\nThe anyons are extended objects that appear as local variations in the density. The w i can be anywhere in the complex plane, and the density variations appear on the lattice sites close to w i , i.e. the anyon lives on the lattice sites even if w i does not coincide with lattice sites. The sum of the density variations over a region that is large enough to contain the anyon is −p i /q, so for q = 2, for instance, a quasihole gives rise to half a particle missing on average in a local region, while a quasiparticle gives rise to half a particle extra on average in a local region. One can show [27] that the braiding properties are as expected for anyons in systems with Laughlin type topology, as long as (4) produces the correct local density variations and the local density variations do not overlap during the braiding process. It is hence sufficient to study the density variations to see the topology.\n\nThis construction also applies to fractal lattices, but for each lattice considered one needs to check numerically whether the state still describes anyons, i.e. whether (4) produces the correct changes in density compared to (1). This question was studied for quasiholes in [18]. Here, we extend that analysis by also considering quasiparticles, by considering further fractals (specifically the Sierpinski carpet and the T-fractal), and by quantifying the size of the anyons as a function of position on the fractal lattice.\n\nWe shall here consider states with one quasihole and one quasiparticle, such that i p i = 0, and we shall take q = 2 throughout. To limit the number of possible positions of the anyons, we shall take all w i to be on lattice sites, but the case where the w i are not on lattice sites can be studied with the same approach. To quantify the density variations, we define\nρ i = ψ p, w |n i |ψ p, w − ψ|n i |ψ .(6)\nWe shall also consider\nD r = i ρ i θ(r − |z i − w k |)(7)\nfor the kth anyon, i.e. the sum of ρ i over a circular region around w k . The Heaviside step function θ selects the i for which the distance between z i and w k is less than r. For a system with properly screened anyons, D r for the kth anyon approaches −p k /q when r is large, but still small compared to the distance to other anyons in the system. We compute ρ i and D r through Monte Carlo simulations of ψ p, w |n i |ψ p, w and ψ|n i |ψ . Results for the square lattice, the Sierpinski carpet, the Sierpinski triangle, and the T-fractal are given in figure 2. For all cases, we take M/N as close to 1/9 as possible to be able to compare the different lattices. For the square lattice, the Sierpinski carpet, and the Sierpinski triangle, well-separated quasiholes and quasiparticles are seen, and D r is close to ±1/2 for r larger than about 5 measured in units of the lattice spacing. It is also seen that the quasihole and the quasiparticle have approximately the same size. For the T-fractal with 125 sites the separation is less clear, and D r has larger variations for r around 5 to 10 than for the other lattices. By increasing the system size to 625 sites, we can increase the separation between w 1 and w 2 , and we observe that D r is closer to ±1/2 for large r for this larger lattice. This suggests that we can have well-separated anyons in large enough T-fractals. It seems plausible that the larger size of the anyons in the T-fractal appears because several sites in the T-fractal only have two neighbors, which produces a less good screening.\n\nWe next make a more systematic study of the size of the quasiholes and the quasiparticles as a function of position on the lattice. It is not straightforward to precisely define the size of an anyon, as the ρ i are only defined on lattice sites. Here, we quantify the size of an anyon as the number of lattice sites within a disk shaped region that have |ρ i | > 0.002. Note that this number is large compared to the Monte Carlo errors. The disk shaped region is centered at w k and its radius is half the distance to the nearest w j with j = k. We are, of course, interested in the situation, where the quasihole and the quasiparticle are well-separated, and we hence put the other anyon at (6)). To make finer details visible, the color bars have been saturated at ±0.01, so values outside this range are shown as dark blue or dark red. The number of particles is chosen such that the lattice filling M/N is as close to 1/9 as possible for all lattices, and we take q = 2. D r (see (7)) is shown as a function of the radius r for (e) the quasihole and (f) the quasiparticle. The radius is measured in units of the distance between nearest neighbor sites. For the square lattice, the Sierpinski carpet, and the Sierpinski triangle, D r is seen to be close to −1/2 and 1/2, respectively, for r larger than about 5, while there are larger fluctuations for the T-fractal. The fluctuations decrease in size, however, when the number of sites is increased from 125 (T-125) to 625 (T-625). The lines are terminated at half the distance between the anyons, except for the T-fractal with 625 sites, for which half the distance is outside the interval shown.\n\nthe lattice site furthest away from w k . The computed sizes are shown for the quasiholes in figure 3 and for the quasiparticles in figure 4. We also show the sum of ρ i over the sites inside the disk shaped region with |ρ i | > 0.002. This number should be close to −1/2 for quasiholes and close the +1/2 for quasiparticles in order for the size estimates to be reasonable. Comparing the two figures, the results are seen to be quite similar for quasiholes and quasiparticles. For the square lattice, we observe that the anyon size is constant in the bulk of the system due to the periodicity, while the sizes vary on the edges. For the fractal lattices, the sizes vary more with position and depend on the lattice structure. For the Sierpinski carpet, the largest sizes appear for sites that are not on the outer edges and have only two nearest neighbors. For the Sierpinski triangle all sites except the three corner sites have three nearest neighbors, and here the largest sizes appear for positions that are not close to one of the corners and also not close to the central Figure 3. Size of quasiholes. We consider q = 2 and place the w-coordinate of a quasihole on lattice site i. The w-coordinate of a quasiparticle is placed on the lattice site, which is the longest distance away from site i measured in the two-dimensional plane. We consider a disk around the quasihole with a radius that is half the distance between the anyons. To estimate the size of the quasihole, we show the number of sites within this disk for which |ρ i | > 0.002 in (a), (b), (c), and (g), and we show the sum of ρ i over the sites within the disk with |ρ i | > 0.002 in (d), (e), (f), and (h).\n\npart of the fractal lattice. For the T-fractal on the other hand, the largest sizes appear for positions furthest to the left, right, or bottom of the fractal lattice. We also observe from the figure that the anyon sizes increase with decreasing Hausdorff dimension of the fractals. For a two-dimensional system, the linear size of a region scales as the square root of the number of sites in the region. For a one-dimensional system linear sizes instead scale linearly with the number of sites. For the considered fractals, the scaling is in between these two cases. The linear size of the anyons hence increase even faster with decreasing Hausdorff dimension than the sizes considered here, meaning that the lattice should be bigger to avoid overlap between anyons. \n\n\nTrial states for inner and outer edge states\n\nIn studies of chiral topological phases in noninteracting systems on fractal lattices, it has been observed that some of the states in the spectra are localized on inner edges of the lattices [11,15]. One may therefore ask, if such inner edge states can also appear in interacting systems. The Laughlin state can be expressed as a correlator in conformal field theory, and for periodic systems with open boundaries, it has been found that trial edge states can be obtained by adding further operators into the correlator [28][29][30][31]. A special case of this is to add fluxes. Here, we add fluxes to the lattice Laughlin state on the Sierpinski triangle and carpet and show that this gives rise to density modifications along the (inner) edges surrounding the added flux. Due to the screening properties of the states, the properties of the states away from the edges are not expected to be affected, and hence the states with added fluxes can be interpreted as (inner) edge states. Our starting point is again the state (4). In this state, anyons are added to the lattice Laughlin state by adding extra fluxes at the positions w i , which lead to the density modifications that are the anyons. Here, we instead add the fluxes in regions, where there are no lattice sites, as we expect this produces density variations on the edges closest to were the flux is added. We start by putting one flux at infinity and one flux at the center of the fractal lattice. We choose the fluxes to have opposite signs, i.e. p 1 = −p 2 . The resulting density modifications are shown in figure 5(a,b,g) for the Sierpinski triangle and carpet. Alternatively, one can put both fluxes at the center of holes in the fractal lattice as illustrated in figure 5(c). The figure shows that the fluxes indeed lead to density variations on the sites closest to the fluxes. When the fluxes are placed at the center of holes, inner edge states are formed on the sites surrounding the holes, and the fluxes at infinity produce density modifications on the outer edges of the fractal lattices. It is also seen in the figure that the produced density modifications are not uniform along the edges. This is particular clear for the outer edges, where the fluxes at infinity primarily lead to density modifications of the sites at the corners of the fractal lattices. Density variations that are more uniform along the (inner) edges can be obtained by splitting the fluxes into several pieces and placing them next to the sites forming the (inner) edges as illustrated in figures 5(d,e,f,h,i). Computationally this is done by splitting each w i into K positions w i,k and replacing the factors i (w i − z j ) p i n j in (4) by i,k (w i,k − z j ) p i n j /K . The figures also display the sum of ρ i over the sites at the (inner) edges. Deviations of these numbers from ±1/2 is a measure of to what extent the (inner) edges penetrate deeper into the lattice than the outermost layer of sites.\n; i ; i ; i ; i ; i ; i ; i ; i ; i\n\nEntropy as a function of particle number\n\nThe entanglement entropy is a measure of the amount of entanglement between two parts of a quantum mechanical system. In [19], it was found that the entanglement entropy shows oscillations as a function of the particle number M on the Sierpinski triangle, while such oscillations are absent for the square lattice. Here, we find that such oscillations are also present for T-fractals, but not for the Sierpinski carpet. The origin of the oscillations is an open problem, but the results presented here suggest that the ramification number could play a role, as for the studied fractals, oscillations are seen only for the fractals with finite ramification number.\n\nWe consider the Renyi entropy S A , which is defined as follows. We start from a pure state |ψ and subdivide the system into two parts A and B. The reduced density matrix of subsystem A reads ρ A = Tr B (|ψ ψ|). Here, all the degrees of freedom of subsystem B are traced over. Then the Renyi entanglement entropy of subsystem A is defined as\nS A = −ln[Tr(ρ 2 A )].(8)\nWe consider the Renyi entropy rather than the von Neumann entropy, since the former can be computed efficiently with Monte Carlo simulations using the so-called replica trick [32,33]. The trick is to simulate\ne −S A = n,m,n ,m δ (n ,m),M δ (n,m ),Mψ (n, m )ψ(n , m) ψ(n, m)ψ(n , m ) |ψ(n, m)| 2 |ψ(n , m )| 2 ,(9)\nwhich is an average of a quantity over the probability distribution |ψ(n, m)| 2 |ψ(n , m )| 2 .\n\nHere, n = {n 1 , n 2 , . . . , n N A } is a basis for subsystem A, m = {n N A +1 , n N A +2 , . . . , n N } is a basis for subsystem B, and n and m are independent copies of n and m. We choose the numbering of the sites such that the sites in A are number 1 to N A , and the sites in B are number N A + 1 to N . The delta functions are defined as The computations for the Sierpinski triangle were also done in [19], and we include the results here for comparison. We observe that the entropy versus particle number shows periodic oscillations for the Sierpinski triangle and the T-fractal, but not for the square lattice and the Sierpinski carpet. For the T-fractal, the local minima lie at 5n, where n is a natural number.\nδ (n ,m),M = 1 for N A i=1 n i + N i=N A +1 n i = M 0 otherwise ,(10)andψ (n 1 , . . . , n N ) ≡ i<j (z i − z j ) qn i n j k =l (z k − z l ) −ηn l . (11) (a) (b) (c) (d) (e)\nPlots of S A versus M for the state (1) have a mirror symmetry around M = N/2. This symmetry was observed and explained in [26], and a detailed derivation is given in Appendix A below. The entanglement entropy for an 8 × 8 square lattice is shown in figure 6(a). A similar computation was done for the case of the cylinder in [26]. The entropy increases initially and then becomes constant and decreases. The second half of the curve is the mirror image of the first half. Initially, more particles means more entanglement and hence the increase makes sense.\n\nWhen the same state is hosted on a fractal lattice, however, the behaviour can be rather different. In [19], the entropy as a function of M was computed for the case of the Sierpinski triangle. We also recompute and plot the result for one specific case in figure 6(b). Instead of varying smoothly with M , the entropy shows oscillations with a precise period. The origin of these oscillations is, as of now, unclear.\n\nThe entropy plots for the T-fractal in figure 6(d-e) also show oscillations. In (d), part A includes 25 sites in the upper central part. The minima occur at M = 5n, where n is a natural number. This seems to follow the pattern observed in [19], where the minima in the oscillations were found to be at M = nN/N A for the cases studied. In (e), however, we show the results for N A = 50. For this case, N/N A = 5/2 is not an integer. We nevertheless still find oscillations with the minima located at M = 5n. This is also the case for the T-fractal with 25 sites. Figure 6(c) shows results for the Sierpinski carpet. There are a few points that are lower than the others, but the oscillations, present in both the Sierpinski triangle and the T-fractal, are surprisingly absent for the Sierpinski carpet.\n\n\nEntropy scaling\n\nFor systems with local interactions and a gapped energy spectrum, the entanglement entropy of the ground state is expected to follow an area law. If such a system has topological order, the intercept of the linear scaling with system size is a universal constant and is called the topological entanglement entropy [34,35]. In [19], it was found that the lattice Laughlin state on a fractal does not obey an area law. In fact, for a specific bipartition, an approximately logarithmic scaling was found. Here, we investigate the scaling further for different bipartitions in order to gain deeper insights into the entanglement of such a strongly correlated state on the Sierpinski triangle. The scaling is done by increasing the fractal generation and the number of particles, while keeping M/N , N A /N , and the number of nearest neighbour bonds crossing the border of the bipartition constant. With this scaling, the entropy should be independent of system size if the system follows the area law. We instead find that the entropy generally increases with system size. We consider only the Sierpinski triangle here, because the number of sites increases faster with the generation for the Sierpinski carpet and the T-fractal, which limits the number of data points we can obtain.\n\nOur results are shown in figure 7. The chosen bipartitions are illustrated in (a-c) and (g-i). When the generation is increased by one, each lattice site turns into three lattice sites, which are assigned to the same bipartition as the original site. In this process the number of nearest neighbor bonds crossing the border of the bipartition remains constant, e.g. 4 for the bipartition in (a). In other words, the area remains constant. We also increase the number of particles such that the average particle density M/N remains constant. The behavior of the entropy with system size as well as linear fits are shown in (d-f) and (j-l). While the entropy increases with system size for all cases, showing that the area law in not fulfilled, the scaling behavior depends on the choice of bipartition. In some cases, the data points follow a scaling that is close to logarithmic, while in other cases the increase of the entropy with subsystem size is faster, and in some cases the entropy is closer to scaling with the volume of the subsystem, at least for the system sizes considered. The results hence reveal a quite complicated behavior of the entropy. \n(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)\n\nConclusion\n\nMotivated by our findings in [19], we studied the properties of the lattice Laughlin state on different kinds of fractal lattices with different Hausdorff dimensions and different ramification numbers, and we found a variety of non-trivial and interesting features. First, we studied the particle densities on the fractal lattice as a function of the particle number M . We observed that for most values of M , the densities at different sites vary with position, but at half-filling, the particle density is one-half for all sites due to particle-hole symmetry. For the Sierpinski triangle, we also observed that the variation of the particle density with position is slightly reduced at η = 1. For most values of the particle number, the densities of the Sierpinski triangle and Sierpinski carpet fall in two distinct groups corresponding to the sites in the corners and sites in the 'bulk'. The densities at the 'bulk' sites form a thick central band around the mean value, while the corner sites deviate from this band. Thus, even though a fractal typically does not offer a distinct concept of 'bulk' and 'edge', one sees a clear distinction in the density patterns, but only for large enough system sizes.\n\nWe inserted both quasiholes and quasiparticles into the Laughlin state on fractal lattices and studied how their presence affect the particle density. We found that the quasiparticles are about the same size as the quasiholes. We also showed how the size of the anyons varies with position in the fractal lattices. For the considered fractal lattices, we observed that the size of the anyons tends to increase when the Hausdorff dimension decreases.\n\nAn important difference between periodic lattices in two dimensions and the Sierpinski triangle and carpet is that the Sierpinski triangle and carpet have inner edges in addition to the outer edge. We constructed trial states describing edge states on both the outer and the inner edges of the Sierpinski triangle and carpet.\n\nIt has been found previously [19] that the entropy shows oscillations as a function of the particle number for the Sierpinski triangle, but not for the square lattice. Here, we showed that oscillations are also present for the T-fractal, but not for the Sierpinski carpet. This suggests that the ramification number could play a role for the oscillations, as the ramification number is infinite for the square lattice and the Sierpinski carpet and finite for the Sierpinski triangle and the T-fractal.\n\nWe also investigated how the entanglement entropy scales with subsystem size for the Sierpinski triangle. We took a sequence of subsystems in such a way that the generation of the fractal increased without changing the number of nearest neighbour bonds crossing the border of the bipartition. We observed for several different bipartitions that the entropy increases with subsystem size, and in some cases the increase is faster than logarithmic. This is starkly different from the two-dimensional case, where the area law applies.\n\nOur results show that topologically ordered systems, when hosted on fractal lattices, can show a variety of non-trivial properties which have no counterpart in periodic lattices in two dimensions.\n\nFinally, we consider the normalization factor. Considering a system with M particles, we must have 1 |C M | 2 = In this section, we show analytically that the Renyi entanglement entropy for the system with M particles is the same as the Renyi entanglement entropy for the system with N − M particles. We shall here denote the Renyi entanglement entropy for the system with M particles by S A,M .\n\nWe first observe that \n\nFigure 1 (\n1b) shows n i r = n i − M/N of a generation g = 4 Sierpinski triangle for all values of the number of particles M in the range [1,80]. The data roughly form a thick central band and two sets of lines (blue and reddish) that are quite far from the central band for most values of M . The blue line is composed of the three sites at the corners of the fractal, which all have the same density because of the C 3 symmetry of the Sierpinski triangle. The reddish line is composed of the six sites that are neighbors of the three corner sites.\n\nFigure 1 .\n1(a) The g = 4 Sierpinski triangle with N = 81 sites. (b) The particle densities of all 81 sites, plotted against particle number M , for the q = 3 lattice Laughlin state. The densities n i r are obtained by subtracting the mean value M/N , i.e., n i r = n i − M/N . (c) The particle densities of all 243 sites of a g = 5 Sierpinski triangle for the q = 3 Laughlin state. (d) The g = 3 Sierpinski carpet with N = 512 sites. The particle densities of all 512 sites are shown for (e) q = 2 and (f) q = 3. (g) The g = 3 T-fractal with N = 125 sites. The particle densities for all 125 sites are shown for (h) q = 2 and (i) q = 3\n\nFigure 2 .\n2A quasihole and a quasiparticle with the w-coordinates placed at the sites marked by the green arrows on (a) a square lattice with 256 sites, (b) a Sierpinski carpet with 512 sites, (c) a Sierpinski triangle with 243 sites, and (d) a T-fractal with 125 sites. The color of the lattice sites shows ρ i (see\n\nFigure 4 .\n4Size of quasiparticles. The plotted quantities are the same as infigure 3, except that the w-coordinate of the quasiparticle is now placed on site i, while the w-coordinate of the quasihole is placed on the site furthest from site i.\n\nFigure 5 .\n5Inner and outer edge states on the Sierpinski triangle and carpet produced by inserting fluxes in the Laughlin state at q = 2. The color shows the density with the fluxes inserted minus the density when fluxes are not inserted. Color primarily on the edges hence signifies edge states. The blue and red dots are the positions of the positive and negative fluxes (in (a), (b), and (g) one flux is placed at infinity).\n\nFigure 6 .\n6The entanglement entropy as a function of particle number M for (a) an 8 × 8 square lattice, (b) the Sierpinski triangle with 81 sites, (c) the Sierpinski carpet with 64 sites, and (d-e) the T-fractal with 125 sites. For all the plots, q = 3. Part A is marked in red in the insets. The plots are symmetric around half-filling.\n\nFigure 7 .\n7Scaling of the Renyi entropy with subsytem size for various bipartitions on a Sierpinski triangle. For each bipartition, we increase the system size by increasing the generation. In doing so, we keep N A /N and the shape of the subsystem unchanged. The average density of particles M/N is also unchanged. The figure above each fit shows the chosen subsystem (marked red) for the fractal with N = 81 sites, and the other two data points are for N = 243 and N = 729. The lines are linear fits with the slope and intersection given in the plots.\n\n\nn 1 ,...,n N δ n,M |ψ(n 1 , . . . , n N )| 2 (A.12) By changing summation variables and using (A.11) we get n 1 ,...,n N δ n,M |ψ(n 1 , . . . , n N )| 2 = n 1 ,...,n N δ n,N −M |ψ(1 − n 1 , . . . , 1 − n N )| 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N ) i<j (z i − z j ) (2η−q) i<j (−1) (−q+η+qn i ) | 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N ) i<j (z i − z j ) (2η−q) i<j (−1) (−q+η) | 2 . (A.13)Considering a system with N − M particles, however, we must also have 1|C N −M | 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N )| 2 .(A.14)Combining the last three equations, we conclude|C N −M | = |C M | i<j (z i − z j ) (2η−q) i<j (−1) (−q+η) .(A.15) Appendix A.1. Density We will here denote the expectation value of the number operator in the state with M particles by n i M . Consider 1 − n i M = |C M | 2 n 1 ,...,n N (1 − n i )δ n,M |ψ(n 1 , . . . , n N )| 2 = |C M | 2 n 1 ,...,n N n i δ n,N −M |ψ(1 − n 1 , . . . , 1 − n N )| 2 = |C N −M | 2 n 1 ,...,n N n i δ n,N −M |ψ(n 1 , . . . , n N )| 2 = n i N −M . (A.16) At the second equality sign, we changed summation variables, and at the third equality sign, we used (A.11) and (A.15). Subtracting (N − M )/N on both sides of this equation gives n i N −M − N − M N = − n i M − M N (A.17) This expresses the symmetry observed in the plots in section 3. For the special case of M = N − M = N/2, (A.16) yields n i N/2 = 1/2 for all i. Appendix A.2. Entanglement entropy\n\n2 =\n2the factors (−1) qn k (N −k) in the numerator with and without primes cancel corresponding factors in the denominator. We can hence write e −S A,M = |C M | 4 n,m,n ,m δ (n,m),M δ (n ,m ),M δ (n ,m),M δ (n,m ),Mψ (n, m )ψ(n n,m),N −M δ (n ,m ),N −M δ (n ,m),N −M δ (n,m ),N −M ×ψ (n, m )ψ(n , m) ψ(n, m)ψ(n , m ) |ψ(1 − n, 1 − m)| 2 |ψ(1 − n , 1 − m )| |C N −M | 4 n,m,n ,m δ (n,m),N −M δ (n ,m ),N −M δ (n ,m),N −M δ (n,m ),N −M ×ψ (n, m )ψ(n , m) ψ(n, m)ψ(n , m ) |ψ(n, m)| 2 |ψ(n , m )| 2 = e −S A,N −M . (A.19) From this we conclude that S A,M = S A,N −M .\nAcknowledgmentsThis work has been supported by the Independent Research Fund Denmark under grant number 8049-00074B.Appendix A. Particle-hole transformation of the lattice Laughlin stateIn this section, we show explicitly how the lattice Laughlin state (1) transforms when particles and holes are exchanged, i.e., n i → 1 − n i , ∀i ∈ {1, 2, . . . , N }. When doing this operation, we should note that η = q i n i /N , so η → q − η.We first note that the delta functionpicks out terms with N − M particles. Next we consider the factorsThey have the propertỹWe relabel the indices k → i and l → j =ψ(n 1 , . . . , n N )Now we switch i and j in the second product only =ψ(n 1 , . . . , n N )Note that the factor i<j (−1) (−q+η) i<j (z i − z j ) (2η−q) is independent of n i . Note also that for the special case of M = N/2, the Kronecker delta remains the same, and the wavefunction only changes by the sign factors in (A.11) as q = 2η.\n. D C Tsui, H Stormer, A C Gossard, Phys. Rev. Lett. 481559Tsui D C, Stormer H L and Gossard A C 1982 Phys. Rev. Lett. 48 1559\n\n. R Laughlin, Phys. Rev. Lett. 501395Laughlin R B 1983 Phys. Rev. Lett. 50 1395\n\n. V Kalmeyer, R Laughlin, Phys. Rev. Lett. 592095Kalmeyer V and Laughlin R 1987 Phys. Rev. Lett. 59 2095\n\n. E J Bergholtz, Z Liu, International Journal of Modern Physics B. 271330017Bergholtz E J and Liu Z 2013 International Journal of Modern Physics B 27 1330017\n\n. 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A Kitaev, J Preskill, Phys. Rev. Lett. 9611110404Kitaev A and Preskill J 2006 Phys. Rev. Lett. 96(11) 110404\n\n. Levin M Wen, X G , Phys. Rev. Lett. 9611110405Levin M and Wen X G 2006 Phys. Rev. Lett. 96(11) 110405\n"},"corpus_id":{"kind":"number","value":257038438,"string":"257,038,438"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2302.10010\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/1742-5468/acd104\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2302.10010v1.pdf\"\n ],\n \"pdf_sha\": \"5a27cc07258fc1957a87ec0b476e2e3ea50d0bcc\",\n \"open_access\": {\n \"license\": null,\n \"url\": null,\n \"status\": null\n }\n}"},"title":{"kind":"string","value":"Properties of Laughlin states on fractal lattices"},"authors":{"kind":"list like","value":["Mani Chandra \nMax-Planck-Institut für Physik komplexer Systeme\nD-01187DresdenGermany","Jha","Anne E B Nielsen \nDepartment of Physics and Astronomy\nAarhus University\nDK-8000Aarhus CDenmark"],"string":"[\n \"Mani Chandra \\nMax-Planck-Institut für Physik komplexer Systeme\\nD-01187DresdenGermany\",\n \"Jha\",\n \"Anne E B Nielsen \\nDepartment of Physics and Astronomy\\nAarhus University\\nDK-8000Aarhus CDenmark\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Laughlin states have recently been constructed on fractal lattices and have been shown to be topological in such systems. Some of their properties are, however, quite different from the two-dimensional case. On the Sierpinski triangle, for instance, the entanglement entropy shows oscillations as a function of particle number and does not obey the area law despite being topologically ordered, and the particle density is non-uniform in the bulk. Here, we investigate these deviant properties in greater detail on the Sierpinski triangle, and we also study the properties on the Sierpinski carpet and the T-fractal. We find that the density variations across the fractal are present for all the considered fractal lattices and for most choices of the number of particles. The size of anyons inserted into the lattice Laughlin state also varies with position on the fractal lattice. We observe that quasiholes and quasiparticles have similar sizes and that the size of the anyons typically increases with decreasing Hausdorff dimension. As opposed to periodic lattices in two dimensions, the Sierpinski triangle and carpet have inner edges. We construct trial states for both inner and outer edge states. We find that oscillations of the entropy as a function of particle number are present for the T-fractal, but not for the Sierpinski carpet. Finally, we observe deviations from the area law for several different bipartitions on the Sierpinski triangle."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"When a two-dimensional electron gas is placed in a strong magnetic field and at very low temperatures, the fractional quantum Hall (FQH) effect can appear [1]. The Hamiltonian describing this effect, which is basically the Coulomb interaction projected to the lowest Landau level, is macroscopically degenerate and extremely hard to diagonalize, except for a small number of particles. In 1983, R. B. Laughlin proposed an ansatz [2] for the ground state wavefunction of the FQH effect at odd-denominator filling factors. The so-called Laughlin wavefunction predicted the existence of quasihole excitations with fractional electric charge. It was shown numerically to have more than 99 percent overlap with the exact ground state of the Coulomb repulsion for small systems [2].\nIn 1987, Kalmeyer and Laughlin showed that the ground state of a frustrated Heisenberg antiferromagnet on a triangular lattice is well described by a Laughlin wave function for bosons in which the particle-coordinates were restricted to be the lattice coordinates [3]. This so-called Kalmeyer-Laughlin wave function showed that FQH physics can be found in settings different from where it was originally discovered. Later, several models describing FQH physics in lattices were proposed [4,5].\nThe work on FQH lattice models is, in part, motivated by the developments in quantum engineering, e.g. in ultracold atoms. Such systems also allow for going beyond periodic lattices, which opens several possibilities. With Rydberg atoms in optical tweezers arbitrary atomic arrangements can be obtained [6,7], and there are also ideas for implementing artificial gauge fields in these systems [8][9][10], which is a starting point for obtaining quantum Hall physics.\nFractals are an interesting example of nonperiodic systems, because their Hausdorff dimension can be noninteger, and hence one can use the spatial dimension as a parameter to change the properties of the physical system. The quantum Hall effect was originally studied in two dimensions, but it has turned out that it can also appear in fractal dimensions. Examples of integer quantum Hall models in fractals [11][12][13][14][15][16][17] already show features that are not present for periodic lattices in two dimensions, such as inner edge states. FQH physics has also been investigated on fractal lattices [18][19][20][21], and one of the interesting observations here is that the entropy can scale differently with subsystem size than it does in two dimensions.\nTrial states, such as the Laughlin state, are important tools to gain insight into the FQH effect. Moore and Read showed that the Laughlin state can be expressed as a correlation function in conformal field theory [22], and this construction was later modified to obtain lattice versions of the Laughlin state that share the same topological features as the continuum Laughlin state for several different lattices and parameters [23,24]. The construction can also be applied to obtain Laughlin states on fractal lattices as long as the fractal lattices can be embedded in the two-dimensional plane. This again leads to topological systems in many cases, allowing quasiholes to be added and investigated [18]. The states were studied further on the Sierpinski triangle in [19], where it was shown numerically that the entanglement entropy of the states does not follow an area law. It was also found that the density of particles on the Sierpinski triangle is different from the two-dimensional case.\nIn this paper, we take a deeper look into the properties of Laughlin states on different fractal lattices. In addition to the Sierpinski triangle, we also consider the Sierpinski carpet and the T-fractal. These fractals differ both with respect to Hausdorff dimension and ramification number. The considered fractal lattices and the lattice Laughlin wavefunction are described in section 2. Since fractal lattices do not have a periodic bulk, the density of particles generally varies across the fractal lattice as observed for the Sierpinski triangle in [19]. In section 3, we study how the density variations across the fractal lattice depend on the number of particles for the three fractals. We also find that the variations go to zero at half-filling of the lattice, which is due to particle-hole symmetry at half-filling. In section 4, we introduce both quasiholes and quasiparticles into the lattice Laughlin state. We investigate how the presence of the anyons modify the density, and we investigate how the size of the anyons depend on their position in the fractal lattice. In section 5, we construct trial states for edge states on both the inner and outer edges of the Sierpinski triangle and the Sierpinski carpet. In section 6, we compute the behaviour of the entropy as a function of particle number. It was found previously [19] that the entropy shows oscillations with a fixed period as a function of particle number for the Sierpinksi triangle, but not for the square lattice. Here, we find that oscillations also occur for the T-fractal, but not for the Sierpinski carpet. This suggests that the ramification number could play a role for the presence or absence of the oscillations. In section 7, we investigate how the entropy scales with subsystem size for different bipartitions of the Sierpinski triangle. We find that different bipartitions give rise to different behaviors violating the area law. In section 8, we summarize the conclusions. In Appendix A, we describe analytically how the density and entanglement entropy transform under the particle-hole transformation."},{"title":"The lattice Laughlin wavefunction","content":"The lattice Laughlin states considered in [23,24] have the following form\nwhere the z i are the coordinates of the lattice sites written as complex numbers and n i are their occupation numbers, which are either 0 or 1, since the particles are fermions for odd q and hardcore bosons for even q. The positive integer q = 1/ν is the flux per particle. The Kronecker delta\nensures that i n i = M satisfies qM = N η for all nonzero terms of the wavefunction, where M is the total number of particles and N is the total number of lattice sites.\nSince qM is the total number of fluxes, η can be interpreted as the flux per lattice site. The constant of proportionality C M is chosen such that the wavefunction |ψ is normalized. The last factor in (1) differs from the Gaussian factor that appears in the original Laughlin state, because the background charge is here placed on the lattice sites only. This point turns out to be important to obtain the desired physics on fractal lattices [18]. An exact parent Hamiltonian for (1) can be found in [18,24], and [21] made a systematic search for simpler, approximate parent Hamiltonians. In the following sections, we study the properties of the wavefunction (1) when the z i are given by the coordinates of a finite generation fractal lattice or a square lattice. Specifically, we consider the Sierpinski triangle depicted in figure 1(a) with N = 3 g sites, the Sierpinski carpet in figure 1(d) with N = 8 g sites, and the T-fractal in figure 1(g) with N = 5 g sites, where g is the generation. The infinite version of these fractals have Hausdorff dimensions ln(3)/ ln(2) ≈ 1.58, ln(8)/ ln(3) ≈ 1.89, and ln(5)/ ln(3) ≈ 1.46, respectively. They also differ in ramification number [25]. If we imagine all the nearestneighbor sites being linked by a bond, an arbitrarily large piece of the T-fractal can be isolated by cutting one bond, while that number is two for the Sierpinski triangle and infinite for the Sierpinski carpet."},{"title":"Particle density","content":"We first consider the particle density on the lattice sites, i.e. ψ|n i |ψ , wheren i =ĉ † iĉi is the number operator andĉ i is the operator that annihilates a particle on the ith lattice site. This quantity can be computed via Monte Carlo simulations of the expression\nFor Laughlin states on two-dimensional, periodic lattices, we expect the expectation value of the number operator to be uniform in the bulk of the system (see, e.g., [19] for computations on square and triangular lattices). For fractal structures, however, there is not a distinct edge or bulk. The density pattern is, hence, not obvious a priori. In [19], the particle density of the Laughlin state on the g = 5 Sierpinski triangle for a fixed particle number (M = 27) was computed and compared to the density on a triangular lattice. While the density in the bulk of the triangular lattice is constant, there are fluctuations around the mean value near the edge. On the fractal lattice, the density fluctuates from site to site, roughly imitating the edge density pattern of the triangular lattice. It was also found that the density on the corner sites of the fractal differs significantly from the density on the other sites of the fractal lattice.\nHere, we take this analysis further by studying the densities on the Sierpinski triangle as a function of M , as opposed to studying the spatial pattern for a fixed value of M . Moreover, we study the density using this approach also for the Sierpinski carpet and the T-fractal. At zero and complete filling the densities are trivially constant across the lattice, so we will not discuss those cases in the following. For almost all other values of M , we find that there is a spatial variation, but at exactly half filling, the variance goes to zero. The plots also have a symmetry around the half-filling point, which appears in all the studied fractals at all values of q. Both of these observations can be explained by studying the particle-hole transformation of the lattice Laughlin state as noted in [26] and discussed in detail in Appendix A below. We also see that the corner densities behave differently from the rest of the lattice, except near and at half-filling. We propose that the density structure can be used as a tool to define and distinguish a bulk and edge for a fractal where, a priori, there is no clear distinction between them. There is a significant dispersion within the central band at lower values of M . At M = 27, however, which corresponds to the case of η = qM/N = 1, the dispersion decreases somewhat. After this the dispersion rises slightly and briefly until M = 30, at which point it starts falling monotonically. At M = 40 and M = 41 (corresponding to almost half filling) it is close to zero. As a whole, the densities reflect the particle-hole symmetry. This means that the plot of n i r as a function of M is unchanged by a rotation by π around the point ( n i r = 0, M = N/2). In other words, n i r,M = − n i r,N −M , where n i r,M is n i r for the state with M particles. This behaviour of the densities is derived analytically in Appendix A.\nIn figure 1(c), the densities are plotted for the next generation of the fractal with 243 sites and q = 3. We see that the structure of a central band surrounded by deviant lines corresponding to corner sites and their immediate neighbours becomes even more apparent. Again we see a pinching of the densities at η = 1, i.e. M = 81, as well as close to half-filling (M = 120, 121). At η = 1, only the central band seems to pinch while the corner (and next to corner) lines remain separated, but at half-filling, even the densities at the corner sites go to zero.\nWe next consider the Sierpinski carpet in figure 1(d). The densities are shown in figure 1(e-f). The values for larger fillings can be obtained utilizing the symmetry discussed above. We see a central band which is initially separated from a set of lines below it. The points below correspond to the four corner sites. At around M = 150, they merge with the central band. For the case of q = 2, this happens at around M = 175. For both values of q, n i r goes to zero at half filling (i.e. M = 256) for all lattice sites. This is not immediately visible in the figure, however, due to the large number of points plotted.\nIn figure 1(h-i), we plot the particle densities for the T-fractal. Compared to the Sierpinski triangle and carpet, there are larger variations in the densities with several sites having | n i r | > 0.05. At half-filling, the densities converge to zero as for the other fractal lattices."},{"title":"Quasiholes and quasiparticles","content":"In this section, we investigate quasiholes and quasiparticle in the Laughlin states. We first give examples of density profiles of the anyons, and we then study the size of the anyons as a function of position on the fractal lattices. Building on [2], it was shown in [27] that states describing quasiholes and quasiparticles on periodic lattices with open boundary conditions can be obtained by modifying the lattice Laughlin state (1) into\nwhere w i is the coordinate of the ith anyon, and p i = +1 (p i = −1) if the ith anyon is a quasihole (quasiparticle). Also,\nThe anyons are extended objects that appear as local variations in the density. The w i can be anywhere in the complex plane, and the density variations appear on the lattice sites close to w i , i.e. the anyon lives on the lattice sites even if w i does not coincide with lattice sites. The sum of the density variations over a region that is large enough to contain the anyon is −p i /q, so for q = 2, for instance, a quasihole gives rise to half a particle missing on average in a local region, while a quasiparticle gives rise to half a particle extra on average in a local region. One can show [27] that the braiding properties are as expected for anyons in systems with Laughlin type topology, as long as (4) produces the correct local density variations and the local density variations do not overlap during the braiding process. It is hence sufficient to study the density variations to see the topology.\nThis construction also applies to fractal lattices, but for each lattice considered one needs to check numerically whether the state still describes anyons, i.e. whether (4) produces the correct changes in density compared to (1). This question was studied for quasiholes in [18]. Here, we extend that analysis by also considering quasiparticles, by considering further fractals (specifically the Sierpinski carpet and the T-fractal), and by quantifying the size of the anyons as a function of position on the fractal lattice.\nWe shall here consider states with one quasihole and one quasiparticle, such that i p i = 0, and we shall take q = 2 throughout. To limit the number of possible positions of the anyons, we shall take all w i to be on lattice sites, but the case where the w i are not on lattice sites can be studied with the same approach. To quantify the density variations, we define\nWe shall also consider\nfor the kth anyon, i.e. the sum of ρ i over a circular region around w k . The Heaviside step function θ selects the i for which the distance between z i and w k is less than r. For a system with properly screened anyons, D r for the kth anyon approaches −p k /q when r is large, but still small compared to the distance to other anyons in the system. We compute ρ i and D r through Monte Carlo simulations of ψ p, w |n i |ψ p, w and ψ|n i |ψ . Results for the square lattice, the Sierpinski carpet, the Sierpinski triangle, and the T-fractal are given in figure 2. For all cases, we take M/N as close to 1/9 as possible to be able to compare the different lattices. For the square lattice, the Sierpinski carpet, and the Sierpinski triangle, well-separated quasiholes and quasiparticles are seen, and D r is close to ±1/2 for r larger than about 5 measured in units of the lattice spacing. It is also seen that the quasihole and the quasiparticle have approximately the same size. For the T-fractal with 125 sites the separation is less clear, and D r has larger variations for r around 5 to 10 than for the other lattices. By increasing the system size to 625 sites, we can increase the separation between w 1 and w 2 , and we observe that D r is closer to ±1/2 for large r for this larger lattice. This suggests that we can have well-separated anyons in large enough T-fractals. It seems plausible that the larger size of the anyons in the T-fractal appears because several sites in the T-fractal only have two neighbors, which produces a less good screening.\nWe next make a more systematic study of the size of the quasiholes and the quasiparticles as a function of position on the lattice. It is not straightforward to precisely define the size of an anyon, as the ρ i are only defined on lattice sites. Here, we quantify the size of an anyon as the number of lattice sites within a disk shaped region that have |ρ i | > 0.002. Note that this number is large compared to the Monte Carlo errors. The disk shaped region is centered at w k and its radius is half the distance to the nearest w j with j = k. We are, of course, interested in the situation, where the quasihole and the quasiparticle are well-separated, and we hence put the other anyon at (6)). To make finer details visible, the color bars have been saturated at ±0.01, so values outside this range are shown as dark blue or dark red. The number of particles is chosen such that the lattice filling M/N is as close to 1/9 as possible for all lattices, and we take q = 2. D r (see (7)) is shown as a function of the radius r for (e) the quasihole and (f) the quasiparticle. The radius is measured in units of the distance between nearest neighbor sites. For the square lattice, the Sierpinski carpet, and the Sierpinski triangle, D r is seen to be close to −1/2 and 1/2, respectively, for r larger than about 5, while there are larger fluctuations for the T-fractal. The fluctuations decrease in size, however, when the number of sites is increased from 125 (T-125) to 625 (T-625). The lines are terminated at half the distance between the anyons, except for the T-fractal with 625 sites, for which half the distance is outside the interval shown.\nthe lattice site furthest away from w k . The computed sizes are shown for the quasiholes in figure 3 and for the quasiparticles in figure 4. We also show the sum of ρ i over the sites inside the disk shaped region with |ρ i | > 0.002. This number should be close to −1/2 for quasiholes and close the +1/2 for quasiparticles in order for the size estimates to be reasonable. Comparing the two figures, the results are seen to be quite similar for quasiholes and quasiparticles. For the square lattice, we observe that the anyon size is constant in the bulk of the system due to the periodicity, while the sizes vary on the edges. For the fractal lattices, the sizes vary more with position and depend on the lattice structure. For the Sierpinski carpet, the largest sizes appear for sites that are not on the outer edges and have only two nearest neighbors. For the Sierpinski triangle all sites except the three corner sites have three nearest neighbors, and here the largest sizes appear for positions that are not close to one of the corners and also not close to the central Figure 3. Size of quasiholes. We consider q = 2 and place the w-coordinate of a quasihole on lattice site i. The w-coordinate of a quasiparticle is placed on the lattice site, which is the longest distance away from site i measured in the two-dimensional plane. We consider a disk around the quasihole with a radius that is half the distance between the anyons. To estimate the size of the quasihole, we show the number of sites within this disk for which |ρ i | > 0.002 in (a), (b), (c), and (g), and we show the sum of ρ i over the sites within the disk with |ρ i | > 0.002 in (d), (e), (f), and (h).\npart of the fractal lattice. For the T-fractal on the other hand, the largest sizes appear for positions furthest to the left, right, or bottom of the fractal lattice. We also observe from the figure that the anyon sizes increase with decreasing Hausdorff dimension of the fractals. For a two-dimensional system, the linear size of a region scales as the square root of the number of sites in the region. For a one-dimensional system linear sizes instead scale linearly with the number of sites. For the considered fractals, the scaling is in between these two cases. The linear size of the anyons hence increase even faster with decreasing Hausdorff dimension than the sizes considered here, meaning that the lattice should be bigger to avoid overlap between anyons."},{"title":"Trial states for inner and outer edge states","content":"In studies of chiral topological phases in noninteracting systems on fractal lattices, it has been observed that some of the states in the spectra are localized on inner edges of the lattices [11,15]. One may therefore ask, if such inner edge states can also appear in interacting systems. The Laughlin state can be expressed as a correlator in conformal field theory, and for periodic systems with open boundaries, it has been found that trial edge states can be obtained by adding further operators into the correlator [28][29][30][31]. A special case of this is to add fluxes. Here, we add fluxes to the lattice Laughlin state on the Sierpinski triangle and carpet and show that this gives rise to density modifications along the (inner) edges surrounding the added flux. Due to the screening properties of the states, the properties of the states away from the edges are not expected to be affected, and hence the states with added fluxes can be interpreted as (inner) edge states. Our starting point is again the state (4). In this state, anyons are added to the lattice Laughlin state by adding extra fluxes at the positions w i , which lead to the density modifications that are the anyons. Here, we instead add the fluxes in regions, where there are no lattice sites, as we expect this produces density variations on the edges closest to were the flux is added. We start by putting one flux at infinity and one flux at the center of the fractal lattice. We choose the fluxes to have opposite signs, i.e. p 1 = −p 2 . The resulting density modifications are shown in figure 5(a,b,g) for the Sierpinski triangle and carpet. Alternatively, one can put both fluxes at the center of holes in the fractal lattice as illustrated in figure 5(c). The figure shows that the fluxes indeed lead to density variations on the sites closest to the fluxes. When the fluxes are placed at the center of holes, inner edge states are formed on the sites surrounding the holes, and the fluxes at infinity produce density modifications on the outer edges of the fractal lattices. It is also seen in the figure that the produced density modifications are not uniform along the edges. This is particular clear for the outer edges, where the fluxes at infinity primarily lead to density modifications of the sites at the corners of the fractal lattices. Density variations that are more uniform along the (inner) edges can be obtained by splitting the fluxes into several pieces and placing them next to the sites forming the (inner) edges as illustrated in figures 5(d,e,f,h,i). Computationally this is done by splitting each w i into K positions w i,k and replacing the factors i (w i − z j ) p i n j in (4) by i,k (w i,k − z j ) p i n j /K . The figures also display the sum of ρ i over the sites at the (inner) edges. Deviations of these numbers from ±1/2 is a measure of to what extent the (inner) edges penetrate deeper into the lattice than the outermost layer of sites."},{"title":"Entropy as a function of particle number","content":"The entanglement entropy is a measure of the amount of entanglement between two parts of a quantum mechanical system. In [19], it was found that the entanglement entropy shows oscillations as a function of the particle number M on the Sierpinski triangle, while such oscillations are absent for the square lattice. Here, we find that such oscillations are also present for T-fractals, but not for the Sierpinski carpet. The origin of the oscillations is an open problem, but the results presented here suggest that the ramification number could play a role, as for the studied fractals, oscillations are seen only for the fractals with finite ramification number.\nWe consider the Renyi entropy S A , which is defined as follows. We start from a pure state |ψ and subdivide the system into two parts A and B. The reduced density matrix of subsystem A reads ρ A = Tr B (|ψ ψ|). Here, all the degrees of freedom of subsystem B are traced over. Then the Renyi entanglement entropy of subsystem A is defined as\nWe consider the Renyi entropy rather than the von Neumann entropy, since the former can be computed efficiently with Monte Carlo simulations using the so-called replica trick [32,33]. The trick is to simulate\nwhich is an average of a quantity over the probability distribution |ψ(n, m)| 2 |ψ(n , m )| 2 .\nHere, n = {n 1 , n 2 , . . . , n N A } is a basis for subsystem A, m = {n N A +1 , n N A +2 , . . . , n N } is a basis for subsystem B, and n and m are independent copies of n and m. We choose the numbering of the sites such that the sites in A are number 1 to N A , and the sites in B are number N A + 1 to N . The delta functions are defined as The computations for the Sierpinski triangle were also done in [19], and we include the results here for comparison. We observe that the entropy versus particle number shows periodic oscillations for the Sierpinski triangle and the T-fractal, but not for the square lattice and the Sierpinski carpet. For the T-fractal, the local minima lie at 5n, where n is a natural number.\nPlots of S A versus M for the state (1) have a mirror symmetry around M = N/2. This symmetry was observed and explained in [26], and a detailed derivation is given in Appendix A below. The entanglement entropy for an 8 × 8 square lattice is shown in figure 6(a). A similar computation was done for the case of the cylinder in [26]. The entropy increases initially and then becomes constant and decreases. The second half of the curve is the mirror image of the first half. Initially, more particles means more entanglement and hence the increase makes sense.\nWhen the same state is hosted on a fractal lattice, however, the behaviour can be rather different. In [19], the entropy as a function of M was computed for the case of the Sierpinski triangle. We also recompute and plot the result for one specific case in figure 6(b). Instead of varying smoothly with M , the entropy shows oscillations with a precise period. The origin of these oscillations is, as of now, unclear.\nThe entropy plots for the T-fractal in figure 6(d-e) also show oscillations. In (d), part A includes 25 sites in the upper central part. The minima occur at M = 5n, where n is a natural number. This seems to follow the pattern observed in [19], where the minima in the oscillations were found to be at M = nN/N A for the cases studied. In (e), however, we show the results for N A = 50. For this case, N/N A = 5/2 is not an integer. We nevertheless still find oscillations with the minima located at M = 5n. This is also the case for the T-fractal with 25 sites. Figure 6(c) shows results for the Sierpinski carpet. There are a few points that are lower than the others, but the oscillations, present in both the Sierpinski triangle and the T-fractal, are surprisingly absent for the Sierpinski carpet."},{"title":"Entropy scaling","content":"For systems with local interactions and a gapped energy spectrum, the entanglement entropy of the ground state is expected to follow an area law. If such a system has topological order, the intercept of the linear scaling with system size is a universal constant and is called the topological entanglement entropy [34,35]. In [19], it was found that the lattice Laughlin state on a fractal does not obey an area law. In fact, for a specific bipartition, an approximately logarithmic scaling was found. Here, we investigate the scaling further for different bipartitions in order to gain deeper insights into the entanglement of such a strongly correlated state on the Sierpinski triangle. The scaling is done by increasing the fractal generation and the number of particles, while keeping M/N , N A /N , and the number of nearest neighbour bonds crossing the border of the bipartition constant. With this scaling, the entropy should be independent of system size if the system follows the area law. We instead find that the entropy generally increases with system size. We consider only the Sierpinski triangle here, because the number of sites increases faster with the generation for the Sierpinski carpet and the T-fractal, which limits the number of data points we can obtain.\nOur results are shown in figure 7. The chosen bipartitions are illustrated in (a-c) and (g-i). When the generation is increased by one, each lattice site turns into three lattice sites, which are assigned to the same bipartition as the original site. In this process the number of nearest neighbor bonds crossing the border of the bipartition remains constant, e.g. 4 for the bipartition in (a). In other words, the area remains constant. We also increase the number of particles such that the average particle density M/N remains constant. The behavior of the entropy with system size as well as linear fits are shown in (d-f) and (j-l). While the entropy increases with system size for all cases, showing that the area law in not fulfilled, the scaling behavior depends on the choice of bipartition. In some cases, the data points follow a scaling that is close to logarithmic, while in other cases the increase of the entropy with subsystem size is faster, and in some cases the entropy is closer to scaling with the volume of the subsystem, at least for the system sizes considered. The results hence reveal a quite complicated behavior of the entropy."},{"title":"Conclusion","content":"Motivated by our findings in [19], we studied the properties of the lattice Laughlin state on different kinds of fractal lattices with different Hausdorff dimensions and different ramification numbers, and we found a variety of non-trivial and interesting features. First, we studied the particle densities on the fractal lattice as a function of the particle number M . We observed that for most values of M , the densities at different sites vary with position, but at half-filling, the particle density is one-half for all sites due to particle-hole symmetry. For the Sierpinski triangle, we also observed that the variation of the particle density with position is slightly reduced at η = 1. For most values of the particle number, the densities of the Sierpinski triangle and Sierpinski carpet fall in two distinct groups corresponding to the sites in the corners and sites in the 'bulk'. The densities at the 'bulk' sites form a thick central band around the mean value, while the corner sites deviate from this band. Thus, even though a fractal typically does not offer a distinct concept of 'bulk' and 'edge', one sees a clear distinction in the density patterns, but only for large enough system sizes.\nWe inserted both quasiholes and quasiparticles into the Laughlin state on fractal lattices and studied how their presence affect the particle density. We found that the quasiparticles are about the same size as the quasiholes. We also showed how the size of the anyons varies with position in the fractal lattices. For the considered fractal lattices, we observed that the size of the anyons tends to increase when the Hausdorff dimension decreases.\nAn important difference between periodic lattices in two dimensions and the Sierpinski triangle and carpet is that the Sierpinski triangle and carpet have inner edges in addition to the outer edge. We constructed trial states describing edge states on both the outer and the inner edges of the Sierpinski triangle and carpet.\nIt has been found previously [19] that the entropy shows oscillations as a function of the particle number for the Sierpinski triangle, but not for the square lattice. Here, we showed that oscillations are also present for the T-fractal, but not for the Sierpinski carpet. This suggests that the ramification number could play a role for the oscillations, as the ramification number is infinite for the square lattice and the Sierpinski carpet and finite for the Sierpinski triangle and the T-fractal.\nWe also investigated how the entanglement entropy scales with subsystem size for the Sierpinski triangle. We took a sequence of subsystems in such a way that the generation of the fractal increased without changing the number of nearest neighbour bonds crossing the border of the bipartition. We observed for several different bipartitions that the entropy increases with subsystem size, and in some cases the increase is faster than logarithmic. This is starkly different from the two-dimensional case, where the area law applies.\nOur results show that topologically ordered systems, when hosted on fractal lattices, can show a variety of non-trivial properties which have no counterpart in periodic lattices in two dimensions.\nFinally, we consider the normalization factor. Considering a system with M particles, we must have 1 |C M | 2 = In this section, we show analytically that the Renyi entanglement entropy for the system with M particles is the same as the Renyi entanglement entropy for the system with N − M particles. We shall here denote the Renyi entanglement entropy for the system with M particles by S A,M .\nWe first observe that"},{"title":"Figure 1 (","content":""},{"title":"Figure 1 .","content":""},{"title":"Figure 2 .","content":""},{"title":"Figure 4 .","content":""},{"title":"Figure 5 .","content":""},{"title":"Figure 6 .","content":""},{"title":"Figure 7 .","content":""},{"title":"2 =","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"When a two-dimensional electron gas is placed in a strong magnetic field and at very low temperatures, the fractional quantum Hall (FQH) effect can appear [1]. The Hamiltonian describing this effect, which is basically the Coulomb interaction projected to the lowest Landau level, is macroscopically degenerate and extremely hard to diagonalize, except for a small number of particles. In 1983, R. B. Laughlin proposed an ansatz [2] for the ground state wavefunction of the FQH effect at odd-denominator filling factors. The so-called Laughlin wavefunction predicted the existence of quasihole excitations with fractional electric charge. It was shown numerically to have more than 99 percent overlap with the exact ground state of the Coulomb repulsion for small systems [2].\\nIn 1987, Kalmeyer and Laughlin showed that the ground state of a frustrated Heisenberg antiferromagnet on a triangular lattice is well described by a Laughlin wave function for bosons in which the particle-coordinates were restricted to be the lattice coordinates [3]. This so-called Kalmeyer-Laughlin wave function showed that FQH physics can be found in settings different from where it was originally discovered. Later, several models describing FQH physics in lattices were proposed [4,5].\\nThe work on FQH lattice models is, in part, motivated by the developments in quantum engineering, e.g. in ultracold atoms. Such systems also allow for going beyond periodic lattices, which opens several possibilities. With Rydberg atoms in optical tweezers arbitrary atomic arrangements can be obtained [6,7], and there are also ideas for implementing artificial gauge fields in these systems [8][9][10], which is a starting point for obtaining quantum Hall physics.\\nFractals are an interesting example of nonperiodic systems, because their Hausdorff dimension can be noninteger, and hence one can use the spatial dimension as a parameter to change the properties of the physical system. The quantum Hall effect was originally studied in two dimensions, but it has turned out that it can also appear in fractal dimensions. Examples of integer quantum Hall models in fractals [11][12][13][14][15][16][17] already show features that are not present for periodic lattices in two dimensions, such as inner edge states. FQH physics has also been investigated on fractal lattices [18][19][20][21], and one of the interesting observations here is that the entropy can scale differently with subsystem size than it does in two dimensions.\\nTrial states, such as the Laughlin state, are important tools to gain insight into the FQH effect. Moore and Read showed that the Laughlin state can be expressed as a correlation function in conformal field theory [22], and this construction was later modified to obtain lattice versions of the Laughlin state that share the same topological features as the continuum Laughlin state for several different lattices and parameters [23,24]. The construction can also be applied to obtain Laughlin states on fractal lattices as long as the fractal lattices can be embedded in the two-dimensional plane. This again leads to topological systems in many cases, allowing quasiholes to be added and investigated [18]. The states were studied further on the Sierpinski triangle in [19], where it was shown numerically that the entanglement entropy of the states does not follow an area law. It was also found that the density of particles on the Sierpinski triangle is different from the two-dimensional case.\\nIn this paper, we take a deeper look into the properties of Laughlin states on different fractal lattices. In addition to the Sierpinski triangle, we also consider the Sierpinski carpet and the T-fractal. These fractals differ both with respect to Hausdorff dimension and ramification number. The considered fractal lattices and the lattice Laughlin wavefunction are described in section 2. Since fractal lattices do not have a periodic bulk, the density of particles generally varies across the fractal lattice as observed for the Sierpinski triangle in [19]. In section 3, we study how the density variations across the fractal lattice depend on the number of particles for the three fractals. We also find that the variations go to zero at half-filling of the lattice, which is due to particle-hole symmetry at half-filling. In section 4, we introduce both quasiholes and quasiparticles into the lattice Laughlin state. We investigate how the presence of the anyons modify the density, and we investigate how the size of the anyons depend on their position in the fractal lattice. In section 5, we construct trial states for edge states on both the inner and outer edges of the Sierpinski triangle and the Sierpinski carpet. In section 6, we compute the behaviour of the entropy as a function of particle number. It was found previously [19] that the entropy shows oscillations with a fixed period as a function of particle number for the Sierpinksi triangle, but not for the square lattice. Here, we find that oscillations also occur for the T-fractal, but not for the Sierpinski carpet. This suggests that the ramification number could play a role for the presence or absence of the oscillations. In section 7, we investigate how the entropy scales with subsystem size for different bipartitions of the Sierpinski triangle. We find that different bipartitions give rise to different behaviors violating the area law. In section 8, we summarize the conclusions. In Appendix A, we describe analytically how the density and entanglement entropy transform under the particle-hole transformation.\"\n },\n {\n \"title\": \"The lattice Laughlin wavefunction\",\n \"content\": \"The lattice Laughlin states considered in [23,24] have the following form\\nwhere the z i are the coordinates of the lattice sites written as complex numbers and n i are their occupation numbers, which are either 0 or 1, since the particles are fermions for odd q and hardcore bosons for even q. The positive integer q = 1/ν is the flux per particle. The Kronecker delta\\nensures that i n i = M satisfies qM = N η for all nonzero terms of the wavefunction, where M is the total number of particles and N is the total number of lattice sites.\\nSince qM is the total number of fluxes, η can be interpreted as the flux per lattice site. The constant of proportionality C M is chosen such that the wavefunction |ψ is normalized. The last factor in (1) differs from the Gaussian factor that appears in the original Laughlin state, because the background charge is here placed on the lattice sites only. This point turns out to be important to obtain the desired physics on fractal lattices [18]. An exact parent Hamiltonian for (1) can be found in [18,24], and [21] made a systematic search for simpler, approximate parent Hamiltonians. In the following sections, we study the properties of the wavefunction (1) when the z i are given by the coordinates of a finite generation fractal lattice or a square lattice. Specifically, we consider the Sierpinski triangle depicted in figure 1(a) with N = 3 g sites, the Sierpinski carpet in figure 1(d) with N = 8 g sites, and the T-fractal in figure 1(g) with N = 5 g sites, where g is the generation. The infinite version of these fractals have Hausdorff dimensions ln(3)/ ln(2) ≈ 1.58, ln(8)/ ln(3) ≈ 1.89, and ln(5)/ ln(3) ≈ 1.46, respectively. They also differ in ramification number [25]. If we imagine all the nearestneighbor sites being linked by a bond, an arbitrarily large piece of the T-fractal can be isolated by cutting one bond, while that number is two for the Sierpinski triangle and infinite for the Sierpinski carpet.\"\n },\n {\n \"title\": \"Particle density\",\n \"content\": \"We first consider the particle density on the lattice sites, i.e. ψ|n i |ψ , wheren i =ĉ † iĉi is the number operator andĉ i is the operator that annihilates a particle on the ith lattice site. This quantity can be computed via Monte Carlo simulations of the expression\\nFor Laughlin states on two-dimensional, periodic lattices, we expect the expectation value of the number operator to be uniform in the bulk of the system (see, e.g., [19] for computations on square and triangular lattices). For fractal structures, however, there is not a distinct edge or bulk. The density pattern is, hence, not obvious a priori. In [19], the particle density of the Laughlin state on the g = 5 Sierpinski triangle for a fixed particle number (M = 27) was computed and compared to the density on a triangular lattice. While the density in the bulk of the triangular lattice is constant, there are fluctuations around the mean value near the edge. On the fractal lattice, the density fluctuates from site to site, roughly imitating the edge density pattern of the triangular lattice. It was also found that the density on the corner sites of the fractal differs significantly from the density on the other sites of the fractal lattice.\\nHere, we take this analysis further by studying the densities on the Sierpinski triangle as a function of M , as opposed to studying the spatial pattern for a fixed value of M . Moreover, we study the density using this approach also for the Sierpinski carpet and the T-fractal. At zero and complete filling the densities are trivially constant across the lattice, so we will not discuss those cases in the following. For almost all other values of M , we find that there is a spatial variation, but at exactly half filling, the variance goes to zero. The plots also have a symmetry around the half-filling point, which appears in all the studied fractals at all values of q. Both of these observations can be explained by studying the particle-hole transformation of the lattice Laughlin state as noted in [26] and discussed in detail in Appendix A below. We also see that the corner densities behave differently from the rest of the lattice, except near and at half-filling. We propose that the density structure can be used as a tool to define and distinguish a bulk and edge for a fractal where, a priori, there is no clear distinction between them. There is a significant dispersion within the central band at lower values of M . At M = 27, however, which corresponds to the case of η = qM/N = 1, the dispersion decreases somewhat. After this the dispersion rises slightly and briefly until M = 30, at which point it starts falling monotonically. At M = 40 and M = 41 (corresponding to almost half filling) it is close to zero. As a whole, the densities reflect the particle-hole symmetry. This means that the plot of n i r as a function of M is unchanged by a rotation by π around the point ( n i r = 0, M = N/2). In other words, n i r,M = − n i r,N −M , where n i r,M is n i r for the state with M particles. This behaviour of the densities is derived analytically in Appendix A.\\nIn figure 1(c), the densities are plotted for the next generation of the fractal with 243 sites and q = 3. We see that the structure of a central band surrounded by deviant lines corresponding to corner sites and their immediate neighbours becomes even more apparent. Again we see a pinching of the densities at η = 1, i.e. M = 81, as well as close to half-filling (M = 120, 121). At η = 1, only the central band seems to pinch while the corner (and next to corner) lines remain separated, but at half-filling, even the densities at the corner sites go to zero.\\nWe next consider the Sierpinski carpet in figure 1(d). The densities are shown in figure 1(e-f). The values for larger fillings can be obtained utilizing the symmetry discussed above. We see a central band which is initially separated from a set of lines below it. The points below correspond to the four corner sites. At around M = 150, they merge with the central band. For the case of q = 2, this happens at around M = 175. For both values of q, n i r goes to zero at half filling (i.e. M = 256) for all lattice sites. This is not immediately visible in the figure, however, due to the large number of points plotted.\\nIn figure 1(h-i), we plot the particle densities for the T-fractal. Compared to the Sierpinski triangle and carpet, there are larger variations in the densities with several sites having | n i r | > 0.05. At half-filling, the densities converge to zero as for the other fractal lattices.\"\n },\n {\n \"title\": \"Quasiholes and quasiparticles\",\n \"content\": \"In this section, we investigate quasiholes and quasiparticle in the Laughlin states. We first give examples of density profiles of the anyons, and we then study the size of the anyons as a function of position on the fractal lattices. Building on [2], it was shown in [27] that states describing quasiholes and quasiparticles on periodic lattices with open boundary conditions can be obtained by modifying the lattice Laughlin state (1) into\\nwhere w i is the coordinate of the ith anyon, and p i = +1 (p i = −1) if the ith anyon is a quasihole (quasiparticle). Also,\\nThe anyons are extended objects that appear as local variations in the density. The w i can be anywhere in the complex plane, and the density variations appear on the lattice sites close to w i , i.e. the anyon lives on the lattice sites even if w i does not coincide with lattice sites. The sum of the density variations over a region that is large enough to contain the anyon is −p i /q, so for q = 2, for instance, a quasihole gives rise to half a particle missing on average in a local region, while a quasiparticle gives rise to half a particle extra on average in a local region. One can show [27] that the braiding properties are as expected for anyons in systems with Laughlin type topology, as long as (4) produces the correct local density variations and the local density variations do not overlap during the braiding process. It is hence sufficient to study the density variations to see the topology.\\nThis construction also applies to fractal lattices, but for each lattice considered one needs to check numerically whether the state still describes anyons, i.e. whether (4) produces the correct changes in density compared to (1). This question was studied for quasiholes in [18]. Here, we extend that analysis by also considering quasiparticles, by considering further fractals (specifically the Sierpinski carpet and the T-fractal), and by quantifying the size of the anyons as a function of position on the fractal lattice.\\nWe shall here consider states with one quasihole and one quasiparticle, such that i p i = 0, and we shall take q = 2 throughout. To limit the number of possible positions of the anyons, we shall take all w i to be on lattice sites, but the case where the w i are not on lattice sites can be studied with the same approach. To quantify the density variations, we define\\nWe shall also consider\\nfor the kth anyon, i.e. the sum of ρ i over a circular region around w k . The Heaviside step function θ selects the i for which the distance between z i and w k is less than r. For a system with properly screened anyons, D r for the kth anyon approaches −p k /q when r is large, but still small compared to the distance to other anyons in the system. We compute ρ i and D r through Monte Carlo simulations of ψ p, w |n i |ψ p, w and ψ|n i |ψ . Results for the square lattice, the Sierpinski carpet, the Sierpinski triangle, and the T-fractal are given in figure 2. For all cases, we take M/N as close to 1/9 as possible to be able to compare the different lattices. For the square lattice, the Sierpinski carpet, and the Sierpinski triangle, well-separated quasiholes and quasiparticles are seen, and D r is close to ±1/2 for r larger than about 5 measured in units of the lattice spacing. It is also seen that the quasihole and the quasiparticle have approximately the same size. For the T-fractal with 125 sites the separation is less clear, and D r has larger variations for r around 5 to 10 than for the other lattices. By increasing the system size to 625 sites, we can increase the separation between w 1 and w 2 , and we observe that D r is closer to ±1/2 for large r for this larger lattice. This suggests that we can have well-separated anyons in large enough T-fractals. It seems plausible that the larger size of the anyons in the T-fractal appears because several sites in the T-fractal only have two neighbors, which produces a less good screening.\\nWe next make a more systematic study of the size of the quasiholes and the quasiparticles as a function of position on the lattice. It is not straightforward to precisely define the size of an anyon, as the ρ i are only defined on lattice sites. Here, we quantify the size of an anyon as the number of lattice sites within a disk shaped region that have |ρ i | > 0.002. Note that this number is large compared to the Monte Carlo errors. The disk shaped region is centered at w k and its radius is half the distance to the nearest w j with j = k. We are, of course, interested in the situation, where the quasihole and the quasiparticle are well-separated, and we hence put the other anyon at (6)). To make finer details visible, the color bars have been saturated at ±0.01, so values outside this range are shown as dark blue or dark red. The number of particles is chosen such that the lattice filling M/N is as close to 1/9 as possible for all lattices, and we take q = 2. D r (see (7)) is shown as a function of the radius r for (e) the quasihole and (f) the quasiparticle. The radius is measured in units of the distance between nearest neighbor sites. For the square lattice, the Sierpinski carpet, and the Sierpinski triangle, D r is seen to be close to −1/2 and 1/2, respectively, for r larger than about 5, while there are larger fluctuations for the T-fractal. The fluctuations decrease in size, however, when the number of sites is increased from 125 (T-125) to 625 (T-625). The lines are terminated at half the distance between the anyons, except for the T-fractal with 625 sites, for which half the distance is outside the interval shown.\\nthe lattice site furthest away from w k . The computed sizes are shown for the quasiholes in figure 3 and for the quasiparticles in figure 4. We also show the sum of ρ i over the sites inside the disk shaped region with |ρ i | > 0.002. This number should be close to −1/2 for quasiholes and close the +1/2 for quasiparticles in order for the size estimates to be reasonable. Comparing the two figures, the results are seen to be quite similar for quasiholes and quasiparticles. For the square lattice, we observe that the anyon size is constant in the bulk of the system due to the periodicity, while the sizes vary on the edges. For the fractal lattices, the sizes vary more with position and depend on the lattice structure. For the Sierpinski carpet, the largest sizes appear for sites that are not on the outer edges and have only two nearest neighbors. For the Sierpinski triangle all sites except the three corner sites have three nearest neighbors, and here the largest sizes appear for positions that are not close to one of the corners and also not close to the central Figure 3. Size of quasiholes. We consider q = 2 and place the w-coordinate of a quasihole on lattice site i. The w-coordinate of a quasiparticle is placed on the lattice site, which is the longest distance away from site i measured in the two-dimensional plane. We consider a disk around the quasihole with a radius that is half the distance between the anyons. To estimate the size of the quasihole, we show the number of sites within this disk for which |ρ i | > 0.002 in (a), (b), (c), and (g), and we show the sum of ρ i over the sites within the disk with |ρ i | > 0.002 in (d), (e), (f), and (h).\\npart of the fractal lattice. For the T-fractal on the other hand, the largest sizes appear for positions furthest to the left, right, or bottom of the fractal lattice. We also observe from the figure that the anyon sizes increase with decreasing Hausdorff dimension of the fractals. For a two-dimensional system, the linear size of a region scales as the square root of the number of sites in the region. For a one-dimensional system linear sizes instead scale linearly with the number of sites. For the considered fractals, the scaling is in between these two cases. The linear size of the anyons hence increase even faster with decreasing Hausdorff dimension than the sizes considered here, meaning that the lattice should be bigger to avoid overlap between anyons.\"\n },\n {\n \"title\": \"Trial states for inner and outer edge states\",\n \"content\": \"In studies of chiral topological phases in noninteracting systems on fractal lattices, it has been observed that some of the states in the spectra are localized on inner edges of the lattices [11,15]. One may therefore ask, if such inner edge states can also appear in interacting systems. The Laughlin state can be expressed as a correlator in conformal field theory, and for periodic systems with open boundaries, it has been found that trial edge states can be obtained by adding further operators into the correlator [28][29][30][31]. A special case of this is to add fluxes. Here, we add fluxes to the lattice Laughlin state on the Sierpinski triangle and carpet and show that this gives rise to density modifications along the (inner) edges surrounding the added flux. Due to the screening properties of the states, the properties of the states away from the edges are not expected to be affected, and hence the states with added fluxes can be interpreted as (inner) edge states. Our starting point is again the state (4). In this state, anyons are added to the lattice Laughlin state by adding extra fluxes at the positions w i , which lead to the density modifications that are the anyons. Here, we instead add the fluxes in regions, where there are no lattice sites, as we expect this produces density variations on the edges closest to were the flux is added. We start by putting one flux at infinity and one flux at the center of the fractal lattice. We choose the fluxes to have opposite signs, i.e. p 1 = −p 2 . The resulting density modifications are shown in figure 5(a,b,g) for the Sierpinski triangle and carpet. Alternatively, one can put both fluxes at the center of holes in the fractal lattice as illustrated in figure 5(c). The figure shows that the fluxes indeed lead to density variations on the sites closest to the fluxes. When the fluxes are placed at the center of holes, inner edge states are formed on the sites surrounding the holes, and the fluxes at infinity produce density modifications on the outer edges of the fractal lattices. It is also seen in the figure that the produced density modifications are not uniform along the edges. This is particular clear for the outer edges, where the fluxes at infinity primarily lead to density modifications of the sites at the corners of the fractal lattices. Density variations that are more uniform along the (inner) edges can be obtained by splitting the fluxes into several pieces and placing them next to the sites forming the (inner) edges as illustrated in figures 5(d,e,f,h,i). Computationally this is done by splitting each w i into K positions w i,k and replacing the factors i (w i − z j ) p i n j in (4) by i,k (w i,k − z j ) p i n j /K . The figures also display the sum of ρ i over the sites at the (inner) edges. Deviations of these numbers from ±1/2 is a measure of to what extent the (inner) edges penetrate deeper into the lattice than the outermost layer of sites.\"\n },\n {\n \"title\": \"Entropy as a function of particle number\",\n \"content\": \"The entanglement entropy is a measure of the amount of entanglement between two parts of a quantum mechanical system. In [19], it was found that the entanglement entropy shows oscillations as a function of the particle number M on the Sierpinski triangle, while such oscillations are absent for the square lattice. Here, we find that such oscillations are also present for T-fractals, but not for the Sierpinski carpet. The origin of the oscillations is an open problem, but the results presented here suggest that the ramification number could play a role, as for the studied fractals, oscillations are seen only for the fractals with finite ramification number.\\nWe consider the Renyi entropy S A , which is defined as follows. We start from a pure state |ψ and subdivide the system into two parts A and B. The reduced density matrix of subsystem A reads ρ A = Tr B (|ψ ψ|). Here, all the degrees of freedom of subsystem B are traced over. Then the Renyi entanglement entropy of subsystem A is defined as\\nWe consider the Renyi entropy rather than the von Neumann entropy, since the former can be computed efficiently with Monte Carlo simulations using the so-called replica trick [32,33]. The trick is to simulate\\nwhich is an average of a quantity over the probability distribution |ψ(n, m)| 2 |ψ(n , m )| 2 .\\nHere, n = {n 1 , n 2 , . . . , n N A } is a basis for subsystem A, m = {n N A +1 , n N A +2 , . . . , n N } is a basis for subsystem B, and n and m are independent copies of n and m. We choose the numbering of the sites such that the sites in A are number 1 to N A , and the sites in B are number N A + 1 to N . The delta functions are defined as The computations for the Sierpinski triangle were also done in [19], and we include the results here for comparison. We observe that the entropy versus particle number shows periodic oscillations for the Sierpinski triangle and the T-fractal, but not for the square lattice and the Sierpinski carpet. For the T-fractal, the local minima lie at 5n, where n is a natural number.\\nPlots of S A versus M for the state (1) have a mirror symmetry around M = N/2. This symmetry was observed and explained in [26], and a detailed derivation is given in Appendix A below. The entanglement entropy for an 8 × 8 square lattice is shown in figure 6(a). A similar computation was done for the case of the cylinder in [26]. The entropy increases initially and then becomes constant and decreases. The second half of the curve is the mirror image of the first half. Initially, more particles means more entanglement and hence the increase makes sense.\\nWhen the same state is hosted on a fractal lattice, however, the behaviour can be rather different. In [19], the entropy as a function of M was computed for the case of the Sierpinski triangle. We also recompute and plot the result for one specific case in figure 6(b). Instead of varying smoothly with M , the entropy shows oscillations with a precise period. The origin of these oscillations is, as of now, unclear.\\nThe entropy plots for the T-fractal in figure 6(d-e) also show oscillations. In (d), part A includes 25 sites in the upper central part. The minima occur at M = 5n, where n is a natural number. This seems to follow the pattern observed in [19], where the minima in the oscillations were found to be at M = nN/N A for the cases studied. In (e), however, we show the results for N A = 50. For this case, N/N A = 5/2 is not an integer. We nevertheless still find oscillations with the minima located at M = 5n. This is also the case for the T-fractal with 25 sites. Figure 6(c) shows results for the Sierpinski carpet. There are a few points that are lower than the others, but the oscillations, present in both the Sierpinski triangle and the T-fractal, are surprisingly absent for the Sierpinski carpet.\"\n },\n {\n \"title\": \"Entropy scaling\",\n \"content\": \"For systems with local interactions and a gapped energy spectrum, the entanglement entropy of the ground state is expected to follow an area law. If such a system has topological order, the intercept of the linear scaling with system size is a universal constant and is called the topological entanglement entropy [34,35]. In [19], it was found that the lattice Laughlin state on a fractal does not obey an area law. In fact, for a specific bipartition, an approximately logarithmic scaling was found. Here, we investigate the scaling further for different bipartitions in order to gain deeper insights into the entanglement of such a strongly correlated state on the Sierpinski triangle. The scaling is done by increasing the fractal generation and the number of particles, while keeping M/N , N A /N , and the number of nearest neighbour bonds crossing the border of the bipartition constant. With this scaling, the entropy should be independent of system size if the system follows the area law. We instead find that the entropy generally increases with system size. We consider only the Sierpinski triangle here, because the number of sites increases faster with the generation for the Sierpinski carpet and the T-fractal, which limits the number of data points we can obtain.\\nOur results are shown in figure 7. The chosen bipartitions are illustrated in (a-c) and (g-i). When the generation is increased by one, each lattice site turns into three lattice sites, which are assigned to the same bipartition as the original site. In this process the number of nearest neighbor bonds crossing the border of the bipartition remains constant, e.g. 4 for the bipartition in (a). In other words, the area remains constant. We also increase the number of particles such that the average particle density M/N remains constant. The behavior of the entropy with system size as well as linear fits are shown in (d-f) and (j-l). While the entropy increases with system size for all cases, showing that the area law in not fulfilled, the scaling behavior depends on the choice of bipartition. In some cases, the data points follow a scaling that is close to logarithmic, while in other cases the increase of the entropy with subsystem size is faster, and in some cases the entropy is closer to scaling with the volume of the subsystem, at least for the system sizes considered. The results hence reveal a quite complicated behavior of the entropy.\"\n },\n {\n \"title\": \"Conclusion\",\n \"content\": \"Motivated by our findings in [19], we studied the properties of the lattice Laughlin state on different kinds of fractal lattices with different Hausdorff dimensions and different ramification numbers, and we found a variety of non-trivial and interesting features. First, we studied the particle densities on the fractal lattice as a function of the particle number M . We observed that for most values of M , the densities at different sites vary with position, but at half-filling, the particle density is one-half for all sites due to particle-hole symmetry. For the Sierpinski triangle, we also observed that the variation of the particle density with position is slightly reduced at η = 1. For most values of the particle number, the densities of the Sierpinski triangle and Sierpinski carpet fall in two distinct groups corresponding to the sites in the corners and sites in the 'bulk'. The densities at the 'bulk' sites form a thick central band around the mean value, while the corner sites deviate from this band. Thus, even though a fractal typically does not offer a distinct concept of 'bulk' and 'edge', one sees a clear distinction in the density patterns, but only for large enough system sizes.\\nWe inserted both quasiholes and quasiparticles into the Laughlin state on fractal lattices and studied how their presence affect the particle density. We found that the quasiparticles are about the same size as the quasiholes. We also showed how the size of the anyons varies with position in the fractal lattices. For the considered fractal lattices, we observed that the size of the anyons tends to increase when the Hausdorff dimension decreases.\\nAn important difference between periodic lattices in two dimensions and the Sierpinski triangle and carpet is that the Sierpinski triangle and carpet have inner edges in addition to the outer edge. We constructed trial states describing edge states on both the outer and the inner edges of the Sierpinski triangle and carpet.\\nIt has been found previously [19] that the entropy shows oscillations as a function of the particle number for the Sierpinski triangle, but not for the square lattice. Here, we showed that oscillations are also present for the T-fractal, but not for the Sierpinski carpet. This suggests that the ramification number could play a role for the oscillations, as the ramification number is infinite for the square lattice and the Sierpinski carpet and finite for the Sierpinski triangle and the T-fractal.\\nWe also investigated how the entanglement entropy scales with subsystem size for the Sierpinski triangle. We took a sequence of subsystems in such a way that the generation of the fractal increased without changing the number of nearest neighbour bonds crossing the border of the bipartition. We observed for several different bipartitions that the entropy increases with subsystem size, and in some cases the increase is faster than logarithmic. This is starkly different from the two-dimensional case, where the area law applies.\\nOur results show that topologically ordered systems, when hosted on fractal lattices, can show a variety of non-trivial properties which have no counterpart in periodic lattices in two dimensions.\\nFinally, we consider the normalization factor. Considering a system with M particles, we must have 1 |C M | 2 = In this section, we show analytically that the Renyi entanglement entropy for the system with M particles is the same as the Renyi entanglement entropy for the system with N − M particles. We shall here denote the Renyi entanglement entropy for the system with M particles by S A,M .\\nWe first observe that\"\n },\n {\n \"title\": \"Figure 1 (\",\n \"content\": \"\"\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\": \"Figure 7 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"2 =\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"b) shows n i r = n i − M/N of a generation g = 4 Sierpinski triangle for all values of the number of particles M in the range [1,80]. The data roughly form a thick central band and two sets of lines (blue and reddish) that are quite far from the central band for most values of M . The blue line is composed of the three sites at the corners of the fractal, which all have the same density because of the C 3 symmetry of the Sierpinski triangle. The reddish line is composed of the six sites that are neighbors of the three corner sites.","content":"Figure 1 (\n1b) shows n i r = n i − M/N of a generation g = 4 Sierpinski triangle for all values of the number of particles M in the range [1,80]. The data roughly form a thick central band and two sets of lines (blue and reddish) that are quite far from the central band for most values of M . The blue line is composed of the three sites at the corners of the fractal, which all have the same density because of the C 3 symmetry of the Sierpinski triangle. The reddish line is composed of the six sites that are neighbors of the three corner sites."},{"id":"","caption":"(a) The g = 4 Sierpinski triangle with N = 81 sites. (b) The particle densities of all 81 sites, plotted against particle number M , for the q = 3 lattice Laughlin state. The densities n i r are obtained by subtracting the mean value M/N , i.e., n i r = n i − M/N . (c) The particle densities of all 243 sites of a g = 5 Sierpinski triangle for the q = 3 Laughlin state. (d) The g = 3 Sierpinski carpet with N = 512 sites. The particle densities of all 512 sites are shown for (e) q = 2 and (f) q = 3. (g) The g = 3 T-fractal with N = 125 sites. The particle densities for all 125 sites are shown for (h) q = 2 and (i) q = 3","content":"Figure 1 .\n1(a) The g = 4 Sierpinski triangle with N = 81 sites. (b) The particle densities of all 81 sites, plotted against particle number M , for the q = 3 lattice Laughlin state. The densities n i r are obtained by subtracting the mean value M/N , i.e., n i r = n i − M/N . (c) The particle densities of all 243 sites of a g = 5 Sierpinski triangle for the q = 3 Laughlin state. (d) The g = 3 Sierpinski carpet with N = 512 sites. The particle densities of all 512 sites are shown for (e) q = 2 and (f) q = 3. (g) The g = 3 T-fractal with N = 125 sites. The particle densities for all 125 sites are shown for (h) q = 2 and (i) q = 3"},{"id":"","caption":"A quasihole and a quasiparticle with the w-coordinates placed at the sites marked by the green arrows on (a) a square lattice with 256 sites, (b) a Sierpinski carpet with 512 sites, (c) a Sierpinski triangle with 243 sites, and (d) a T-fractal with 125 sites. The color of the lattice sites shows ρ i (see","content":"Figure 2 .\n2A quasihole and a quasiparticle with the w-coordinates placed at the sites marked by the green arrows on (a) a square lattice with 256 sites, (b) a Sierpinski carpet with 512 sites, (c) a Sierpinski triangle with 243 sites, and (d) a T-fractal with 125 sites. The color of the lattice sites shows ρ i (see"},{"id":"","caption":"Size of quasiparticles. The plotted quantities are the same as infigure 3, except that the w-coordinate of the quasiparticle is now placed on site i, while the w-coordinate of the quasihole is placed on the site furthest from site i.","content":"Figure 4 .\n4Size of quasiparticles. The plotted quantities are the same as infigure 3, except that the w-coordinate of the quasiparticle is now placed on site i, while the w-coordinate of the quasihole is placed on the site furthest from site i."},{"id":"","caption":"Inner and outer edge states on the Sierpinski triangle and carpet produced by inserting fluxes in the Laughlin state at q = 2. The color shows the density with the fluxes inserted minus the density when fluxes are not inserted. Color primarily on the edges hence signifies edge states. The blue and red dots are the positions of the positive and negative fluxes (in (a), (b), and (g) one flux is placed at infinity).","content":"Figure 5 .\n5Inner and outer edge states on the Sierpinski triangle and carpet produced by inserting fluxes in the Laughlin state at q = 2. The color shows the density with the fluxes inserted minus the density when fluxes are not inserted. Color primarily on the edges hence signifies edge states. The blue and red dots are the positions of the positive and negative fluxes (in (a), (b), and (g) one flux is placed at infinity)."},{"id":"","caption":"The entanglement entropy as a function of particle number M for (a) an 8 × 8 square lattice, (b) the Sierpinski triangle with 81 sites, (c) the Sierpinski carpet with 64 sites, and (d-e) the T-fractal with 125 sites. For all the plots, q = 3. Part A is marked in red in the insets. The plots are symmetric around half-filling.","content":"Figure 6 .\n6The entanglement entropy as a function of particle number M for (a) an 8 × 8 square lattice, (b) the Sierpinski triangle with 81 sites, (c) the Sierpinski carpet with 64 sites, and (d-e) the T-fractal with 125 sites. For all the plots, q = 3. Part A is marked in red in the insets. The plots are symmetric around half-filling."},{"id":"","caption":"Scaling of the Renyi entropy with subsytem size for various bipartitions on a Sierpinski triangle. For each bipartition, we increase the system size by increasing the generation. In doing so, we keep N A /N and the shape of the subsystem unchanged. The average density of particles M/N is also unchanged. The figure above each fit shows the chosen subsystem (marked red) for the fractal with N = 81 sites, and the other two data points are for N = 243 and N = 729. The lines are linear fits with the slope and intersection given in the plots.","content":"Figure 7 .\n7Scaling of the Renyi entropy with subsytem size for various bipartitions on a Sierpinski triangle. For each bipartition, we increase the system size by increasing the generation. In doing so, we keep N A /N and the shape of the subsystem unchanged. The average density of particles M/N is also unchanged. The figure above each fit shows the chosen subsystem (marked red) for the fractal with N = 81 sites, and the other two data points are for N = 243 and N = 729. The lines are linear fits with the slope and intersection given in the plots."},{"id":"","caption":"n 1 ,...,n N δ n,M |ψ(n 1 , . . . , n N )| 2 (A.12) By changing summation variables and using (A.11) we get n 1 ,...,n N δ n,M |ψ(n 1 , . . . , n N )| 2 = n 1 ,...,n N δ n,N −M |ψ(1 − n 1 , . . . , 1 − n N )| 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N ) i<j (z i − z j ) (2η−q) i<j (−1) (−q+η+qn i ) | 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N ) i<j (z i − z j ) (2η−q) i<j (−1) (−q+η) | 2 . (A.13)Considering a system with N − M particles, however, we must also have 1|C N −M | 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N )| 2 .(A.14)Combining the last three equations, we conclude|C N −M | = |C M | i<j (z i − z j ) (2η−q) i<j (−1) (−q+η) .(A.15) Appendix A.1. Density We will here denote the expectation value of the number operator in the state with M particles by n i M . Consider 1 − n i M = |C M | 2 n 1 ,...,n N (1 − n i )δ n,M |ψ(n 1 , . . . , n N )| 2 = |C M | 2 n 1 ,...,n N n i δ n,N −M |ψ(1 − n 1 , . . . , 1 − n N )| 2 = |C N −M | 2 n 1 ,...,n N n i δ n,N −M |ψ(n 1 , . . . , n N )| 2 = n i N −M . (A.16) At the second equality sign, we changed summation variables, and at the third equality sign, we used (A.11) and (A.15). Subtracting (N − M )/N on both sides of this equation gives n i N −M − N − M N = − n i M − M N (A.17) This expresses the symmetry observed in the plots in section 3. For the special case of M = N − M = N/2, (A.16) yields n i N/2 = 1/2 for all i. Appendix A.2. Entanglement entropy","content":"n 1 ,...,n N δ n,M |ψ(n 1 , . . . , n N )| 2 (A.12) By changing summation variables and using (A.11) we get n 1 ,...,n N δ n,M |ψ(n 1 , . . . , n N )| 2 = n 1 ,...,n N δ n,N −M |ψ(1 − n 1 , . . . , 1 − n N )| 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N ) i<j (z i − z j ) (2η−q) i<j (−1) (−q+η+qn i ) | 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N ) i<j (z i − z j ) (2η−q) i<j (−1) (−q+η) | 2 . (A.13)Considering a system with N − M particles, however, we must also have 1|C N −M | 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N )| 2 .(A.14)Combining the last three equations, we conclude|C N −M | = |C M | i<j (z i − z j ) (2η−q) i<j (−1) (−q+η) .(A.15) Appendix A.1. Density We will here denote the expectation value of the number operator in the state with M particles by n i M . Consider 1 − n i M = |C M | 2 n 1 ,...,n N (1 − n i )δ n,M |ψ(n 1 , . . . , n N )| 2 = |C M | 2 n 1 ,...,n N n i δ n,N −M |ψ(1 − n 1 , . . . , 1 − n N )| 2 = |C N −M | 2 n 1 ,...,n N n i δ n,N −M |ψ(n 1 , . . . , n N )| 2 = n i N −M . (A.16) At the second equality sign, we changed summation variables, and at the third equality sign, we used (A.11) and (A.15). Subtracting (N − M )/N on both sides of this equation gives n i N −M − N − M N = − n i M − M N (A.17) This expresses the symmetry observed in the plots in section 3. For the special case of M = N − M = N/2, (A.16) yields n i N/2 = 1/2 for all i. Appendix A.2. Entanglement entropy"},{"id":"","caption":"the factors (−1) qn k (N −k) in the numerator with and without primes cancel corresponding factors in the denominator. We can hence write e −S A,M = |C M | 4 n,m,n ,m δ (n,m),M δ (n ,m ),M δ (n ,m),M δ (n,m ),Mψ (n, m )ψ(n n,m),N −M δ (n ,m ),N −M δ (n ,m),N −M δ (n,m ),N −M ×ψ (n, m )ψ(n , m) ψ(n, m)ψ(n , m ) |ψ(1 − n, 1 − m)| 2 |ψ(1 − n , 1 − m )| |C N −M | 4 n,m,n ,m δ (n,m),N −M δ (n ,m ),N −M δ (n ,m),N −M δ (n,m ),N −M ×ψ (n, m )ψ(n , m) ψ(n, m)ψ(n , m ) |ψ(n, m)| 2 |ψ(n , m )| 2 = e −S A,N −M . (A.19) From this we conclude that S A,M = S A,N −M .","content":"2 =\n2the factors (−1) qn k (N −k) in the numerator with and without primes cancel corresponding factors in the denominator. We can hence write e −S A,M = |C M | 4 n,m,n ,m δ (n,m),M δ (n ,m ),M δ (n ,m),M δ (n,m ),Mψ (n, m )ψ(n n,m),N −M δ (n ,m ),N −M δ (n ,m),N −M δ (n,m ),N −M ×ψ (n, m )ψ(n , m) ψ(n, m)ψ(n , m ) |ψ(1 − n, 1 − m)| 2 |ψ(1 − n , 1 − m )| |C N −M | 4 n,m,n ,m δ (n,m),N −M δ (n ,m ),N −M δ (n ,m),N −M δ (n,m ),N −M ×ψ (n, m )ψ(n , m) ψ(n, m)ψ(n , m ) |ψ(n, m)| 2 |ψ(n , m )| 2 = e −S A,N −M . (A.19) From this we conclude that S A,M = S A,N −M ."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"b) shows n i r = n i − M/N of a generation g = 4 Sierpinski triangle for all values of the number of particles M in the range [1,80]. The data roughly form a thick central band and two sets of lines (blue and reddish) that are quite far from the central band for most values of M . The blue line is composed of the three sites at the corners of the fractal, which all have the same density because of the C 3 symmetry of the Sierpinski triangle. The reddish line is composed of the six sites that are neighbors of the three corner sites.\",\n \"content\": \"Figure 1 (\\n1b) shows n i r = n i − M/N of a generation g = 4 Sierpinski triangle for all values of the number of particles M in the range [1,80]. The data roughly form a thick central band and two sets of lines (blue and reddish) that are quite far from the central band for most values of M . The blue line is composed of the three sites at the corners of the fractal, which all have the same density because of the C 3 symmetry of the Sierpinski triangle. The reddish line is composed of the six sites that are neighbors of the three corner sites.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) The g = 4 Sierpinski triangle with N = 81 sites. (b) The particle densities of all 81 sites, plotted against particle number M , for the q = 3 lattice Laughlin state. The densities n i r are obtained by subtracting the mean value M/N , i.e., n i r = n i − M/N . (c) The particle densities of all 243 sites of a g = 5 Sierpinski triangle for the q = 3 Laughlin state. (d) The g = 3 Sierpinski carpet with N = 512 sites. The particle densities of all 512 sites are shown for (e) q = 2 and (f) q = 3. (g) The g = 3 T-fractal with N = 125 sites. The particle densities for all 125 sites are shown for (h) q = 2 and (i) q = 3\",\n \"content\": \"Figure 1 .\\n1(a) The g = 4 Sierpinski triangle with N = 81 sites. (b) The particle densities of all 81 sites, plotted against particle number M , for the q = 3 lattice Laughlin state. The densities n i r are obtained by subtracting the mean value M/N , i.e., n i r = n i − M/N . (c) The particle densities of all 243 sites of a g = 5 Sierpinski triangle for the q = 3 Laughlin state. (d) The g = 3 Sierpinski carpet with N = 512 sites. The particle densities of all 512 sites are shown for (e) q = 2 and (f) q = 3. (g) The g = 3 T-fractal with N = 125 sites. The particle densities for all 125 sites are shown for (h) q = 2 and (i) q = 3\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A quasihole and a quasiparticle with the w-coordinates placed at the sites marked by the green arrows on (a) a square lattice with 256 sites, (b) a Sierpinski carpet with 512 sites, (c) a Sierpinski triangle with 243 sites, and (d) a T-fractal with 125 sites. The color of the lattice sites shows ρ i (see\",\n \"content\": \"Figure 2 .\\n2A quasihole and a quasiparticle with the w-coordinates placed at the sites marked by the green arrows on (a) a square lattice with 256 sites, (b) a Sierpinski carpet with 512 sites, (c) a Sierpinski triangle with 243 sites, and (d) a T-fractal with 125 sites. The color of the lattice sites shows ρ i (see\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Size of quasiparticles. The plotted quantities are the same as infigure 3, except that the w-coordinate of the quasiparticle is now placed on site i, while the w-coordinate of the quasihole is placed on the site furthest from site i.\",\n \"content\": \"Figure 4 .\\n4Size of quasiparticles. The plotted quantities are the same as infigure 3, except that the w-coordinate of the quasiparticle is now placed on site i, while the w-coordinate of the quasihole is placed on the site furthest from site i.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Inner and outer edge states on the Sierpinski triangle and carpet produced by inserting fluxes in the Laughlin state at q = 2. The color shows the density with the fluxes inserted minus the density when fluxes are not inserted. Color primarily on the edges hence signifies edge states. The blue and red dots are the positions of the positive and negative fluxes (in (a), (b), and (g) one flux is placed at infinity).\",\n \"content\": \"Figure 5 .\\n5Inner and outer edge states on the Sierpinski triangle and carpet produced by inserting fluxes in the Laughlin state at q = 2. The color shows the density with the fluxes inserted minus the density when fluxes are not inserted. Color primarily on the edges hence signifies edge states. The blue and red dots are the positions of the positive and negative fluxes (in (a), (b), and (g) one flux is placed at infinity).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The entanglement entropy as a function of particle number M for (a) an 8 × 8 square lattice, (b) the Sierpinski triangle with 81 sites, (c) the Sierpinski carpet with 64 sites, and (d-e) the T-fractal with 125 sites. For all the plots, q = 3. Part A is marked in red in the insets. The plots are symmetric around half-filling.\",\n \"content\": \"Figure 6 .\\n6The entanglement entropy as a function of particle number M for (a) an 8 × 8 square lattice, (b) the Sierpinski triangle with 81 sites, (c) the Sierpinski carpet with 64 sites, and (d-e) the T-fractal with 125 sites. For all the plots, q = 3. Part A is marked in red in the insets. The plots are symmetric around half-filling.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Scaling of the Renyi entropy with subsytem size for various bipartitions on a Sierpinski triangle. For each bipartition, we increase the system size by increasing the generation. In doing so, we keep N A /N and the shape of the subsystem unchanged. The average density of particles M/N is also unchanged. The figure above each fit shows the chosen subsystem (marked red) for the fractal with N = 81 sites, and the other two data points are for N = 243 and N = 729. The lines are linear fits with the slope and intersection given in the plots.\",\n \"content\": \"Figure 7 .\\n7Scaling of the Renyi entropy with subsytem size for various bipartitions on a Sierpinski triangle. For each bipartition, we increase the system size by increasing the generation. In doing so, we keep N A /N and the shape of the subsystem unchanged. The average density of particles M/N is also unchanged. The figure above each fit shows the chosen subsystem (marked red) for the fractal with N = 81 sites, and the other two data points are for N = 243 and N = 729. The lines are linear fits with the slope and intersection given in the plots.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"n 1 ,...,n N δ n,M |ψ(n 1 , . . . , n N )| 2 (A.12) By changing summation variables and using (A.11) we get n 1 ,...,n N δ n,M |ψ(n 1 , . . . , n N )| 2 = n 1 ,...,n N δ n,N −M |ψ(1 − n 1 , . . . , 1 − n N )| 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N ) i<j (z i − z j ) (2η−q) i<j (−1) (−q+η+qn i ) | 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N ) i<j (z i − z j ) (2η−q) i<j (−1) (−q+η) | 2 . (A.13)Considering a system with N − M particles, however, we must also have 1|C N −M | 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N )| 2 .(A.14)Combining the last three equations, we conclude|C N −M | = |C M | i<j (z i − z j ) (2η−q) i<j (−1) (−q+η) .(A.15) Appendix A.1. Density We will here denote the expectation value of the number operator in the state with M particles by n i M . Consider 1 − n i M = |C M | 2 n 1 ,...,n N (1 − n i )δ n,M |ψ(n 1 , . . . , n N )| 2 = |C M | 2 n 1 ,...,n N n i δ n,N −M |ψ(1 − n 1 , . . . , 1 − n N )| 2 = |C N −M | 2 n 1 ,...,n N n i δ n,N −M |ψ(n 1 , . . . , n N )| 2 = n i N −M . (A.16) At the second equality sign, we changed summation variables, and at the third equality sign, we used (A.11) and (A.15). Subtracting (N − M )/N on both sides of this equation gives n i N −M − N − M N = − n i M − M N (A.17) This expresses the symmetry observed in the plots in section 3. For the special case of M = N − M = N/2, (A.16) yields n i N/2 = 1/2 for all i. Appendix A.2. Entanglement entropy\",\n \"content\": \"n 1 ,...,n N δ n,M |ψ(n 1 , . . . , n N )| 2 (A.12) By changing summation variables and using (A.11) we get n 1 ,...,n N δ n,M |ψ(n 1 , . . . , n N )| 2 = n 1 ,...,n N δ n,N −M |ψ(1 − n 1 , . . . , 1 − n N )| 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N ) i<j (z i − z j ) (2η−q) i<j (−1) (−q+η+qn i ) | 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N ) i<j (z i − z j ) (2η−q) i<j (−1) (−q+η) | 2 . (A.13)Considering a system with N − M particles, however, we must also have 1|C N −M | 2 = n 1 ,...,n N δ n,N −M |ψ(n 1 , . . . , n N )| 2 .(A.14)Combining the last three equations, we conclude|C N −M | = |C M | i<j (z i − z j ) (2η−q) i<j (−1) (−q+η) .(A.15) Appendix A.1. Density We will here denote the expectation value of the number operator in the state with M particles by n i M . Consider 1 − n i M = |C M | 2 n 1 ,...,n N (1 − n i )δ n,M |ψ(n 1 , . . . , n N )| 2 = |C M | 2 n 1 ,...,n N n i δ n,N −M |ψ(1 − n 1 , . . . , 1 − n N )| 2 = |C N −M | 2 n 1 ,...,n N n i δ n,N −M |ψ(n 1 , . . . , n N )| 2 = n i N −M . (A.16) At the second equality sign, we changed summation variables, and at the third equality sign, we used (A.11) and (A.15). Subtracting (N − M )/N on both sides of this equation gives n i N −M − N − M N = − n i M − M N (A.17) This expresses the symmetry observed in the plots in section 3. For the special case of M = N − M = N/2, (A.16) yields n i N/2 = 1/2 for all i. Appendix A.2. Entanglement entropy\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the factors (−1) qn k (N −k) in the numerator with and without primes cancel corresponding factors in the denominator. We can hence write e −S A,M = |C M | 4 n,m,n ,m δ (n,m),M δ (n ,m ),M δ (n ,m),M δ (n,m ),Mψ (n, m )ψ(n n,m),N −M δ (n ,m ),N −M δ (n ,m),N −M δ (n,m ),N −M ×ψ (n, m )ψ(n , m) ψ(n, m)ψ(n , m ) |ψ(1 − n, 1 − m)| 2 |ψ(1 − n , 1 − m )| |C N −M | 4 n,m,n ,m δ (n,m),N −M δ (n ,m ),N −M δ (n ,m),N −M δ (n,m ),N −M ×ψ (n, m )ψ(n , m) ψ(n, m)ψ(n , m ) |ψ(n, m)| 2 |ψ(n , m )| 2 = e −S A,N −M . (A.19) From this we conclude that S A,M = S A,N −M .\",\n \"content\": \"2 =\\n2the factors (−1) qn k (N −k) in the numerator with and without primes cancel corresponding factors in the denominator. We can hence write e −S A,M = |C M | 4 n,m,n ,m δ (n,m),M δ (n ,m ),M δ (n ,m),M δ (n,m ),Mψ (n, m )ψ(n n,m),N −M δ (n ,m ),N −M δ (n ,m),N −M δ (n,m ),N −M ×ψ (n, m )ψ(n , m) ψ(n, m)ψ(n , m ) |ψ(1 − n, 1 − m)| 2 |ψ(1 − n , 1 − m )| |C N −M | 4 n,m,n ,m δ (n,m),N −M δ (n ,m ),N −M δ (n ,m),N −M δ (n,m ),N −M ×ψ (n, m )ψ(n , m) ψ(n, m)ψ(n , m ) |ψ(n, m)| 2 |ψ(n , m )| 2 = e −S A,N −M . (A.19) From this we conclude that S A,M = S A,N −M .\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". D C Tsui, H Stormer, A C Gossard, Phys. Rev. Lett. 481559Tsui D C, Stormer H L and Gossard A C 1982 Phys. Rev. Lett. 48 1559","doi":""},{"text":". R Laughlin, Phys. Rev. Lett. 501395Laughlin R B 1983 Phys. Rev. Lett. 50 1395","doi":""},{"text":". V Kalmeyer, R Laughlin, Phys. Rev. Lett. 592095Kalmeyer V and Laughlin R 1987 Phys. Rev. Lett. 59 2095","doi":""},{"text":". E J Bergholtz, Z Liu, International Journal of Modern Physics B. 271330017Bergholtz E J and Liu Z 2013 International Journal of Modern Physics B 27 1330017","doi":""},{"text":". S A Parameswaran, Roy R Sondhi, S L , Comptes Rendus Physique. 14Parameswaran S A, Roy R and Sondhi S L 2013 Comptes Rendus Physique 14 816-839","doi":""},{"text":". D Barredo, De Léséleuc, S , Lienhard V Lahaye, T Browaeys, A , Science. 354Barredo D and De Léséleuc S and Lienhard V and Lahaye T and Browaeys A 2016 Science 354 1021-1023","doi":""},{"text":". D Barredo, V Lienhard, De Leseleuc, S Lahaye, T Browaeys, A , Nature. 561Barredo D, Lienhard V, De Leseleuc S, Lahaye T and Browaeys A 2018 Nature 561 79-82","doi":""},{"text":". V Lienhard, P Scholl, S Weber, D Barredo, S De Léséleuc, R Bai, N Lang, M Fleischhauer, H P Büchler, T Lahaye, A Browaeys, Phys. Rev. X. 10221031Lienhard V, Scholl P, Weber S, Barredo D, de Léséleuc S, Bai R, Lang N, Fleischhauer M, Büchler H P, Lahaye T and Browaeys A 2020 Phys. Rev. X 10(2) 021031","doi":""},{"text":". S Weber, R Bai, N Makki, J Mögerle, T Lahaye, A Browaeys, M Daghofer, Lang N Büchler, H P , PRX Quantum. 3330302Weber S, Bai R, Makki N, Mögerle J, Lahaye T, Browaeys A, Daghofer M, Lang N and Büchler H P 2022 PRX Quantum 3(3) 030302","doi":""},{"text":". X Wu, F Yang, S Yang, K Mølmer, T Pohl, M Tey, L You, Phys. Rev. Res. 4332046Wu X, Yang F, Yang S, Mølmer K, Pohl T, Tey M K and You L 2022 Phys. Rev. Res. 4(3) L032046","doi":""},{"text":". M Brzezińska, A Cook, Neupert , Phys. Rev. B. 98205116Brzezińska M, Cook A M and Neupert T 2018 Phys. Rev. B 98(20) 205116","doi":""},{"text":". S Pai, A Prem, Phys. Rev. B. 10015155135Pai S and Prem A 2019 Phys. Rev. B 100(15) 155135","doi":""},{"text":". A A Iliasov, M Katsnelson, S Yuan, Phys. Rev. B. 101445413Iliasov A A, Katsnelson M I and Yuan S 2020 Phys. Rev. B 101(4) 045413","doi":""},{"text":". M Fremling, M Van Hooft, C Smith, L Fritz, Phys. Rev. Res. 2113044Fremling M, van Hooft M, Smith C M and Fritz L 2020 Phys. Rev. Res. 2(1) 013044","doi":""},{"text":". S Fischer, M Van Hooft, T Van Der Meijden, C M Smith, Fritz L Fremling, M , Phys. Rev. Res. 3443103Fischer S, van Hooft M, van der Meijden T, Smith C M, Fritz L and Fremling M 2021 Phys. Rev. Res. 3(4) 043103","doi":""},{"text":". S Manna, S Nandy, B Roy, Phys. Rev. B. 105201301Manna S, Nandy S and Roy B 2022 Phys. Rev. B 105(20) L201301","doi":""},{"text":". M N Ivaki, I Sahlberg, K Pöyhönen, T Ojanen, Communications Physics. 5327Ivaki M N, Sahlberg I, Pöyhönen K and Ojanen T 2022 Communications Physics 5 327","doi":""},{"text":". S Manna, B Pal, Wang W Nielsen, A E B , Phys. Rev. Research. 223401Manna S, Pal B, Wang W and Nielsen A E B 2020 Phys. Rev. Research 2 023401","doi":""},{"text":". X Li, M Jha, A E B Nielsen, Phys. Rev. B. 105885152Li X, Jha M C and Nielsen A E B 2022 Phys. Rev. B 105(8) 085152","doi":""},{"text":". G Zhu, -O' Jochym, T Connor, A Dua, PRX Quantum. 3330338Zhu G, Jochym-O'Connor T and Dua A 2022 PRX Quantum 3(3) 030338","doi":""},{"text":". B Jaworowski, M Iversen, A E Nielsen, arXiv:2301.09971arXiv preprintJaworowski B, Iversen M and Nielsen A E 2023 arXiv preprint arXiv:2301.09971","doi":""},{"text":". Moore G Read, N , Nuclear Physics B. 360Moore G and Read N 1991 Nuclear Physics B 360 362-396","doi":""},{"text":". A E B Nielsen, J Cirac, G Sierra, Phys. Rev. Lett. 108257206Nielsen A E B, Cirac J I and Sierra G 2012 Phys. Rev. Lett. 108 257206","doi":""},{"text":". H H Tu, A E B Nielsen, J Cirac, G Sierra, New Journal of Physics. 1633025Tu H H, Nielsen A E B, Cirac J I and Sierra G 2014 New Journal of Physics 16 033025","doi":""},{"text":". A S Balankin, Martínez-Cruz M A Andálvarez-Jasso , M D , Patiño-Ortiz M , Patiño-Ortiz , J , Physics Letters A. 383Balankin A S and Martínez-Cruz M A andÁlvarez-Jasso M D and Patiño-Ortiz M and Patiño- Ortiz J 2019 Physics Letters A 383 957-966","doi":""},{"text":". I Glasser, J I Cirac, Sierra G Nielsen, A E B , Phys. Rev. B. 9424245104Glasser I, Cirac J I, Sierra G and Nielsen A E B 2016 Phys. Rev. B 94(24) 245104","doi":""},{"text":". A E B Nielsen, I Glasser, I D Rodrãguez, New Journal of Physics. 2033029Nielsen A E B, Glasser I and RodrÃguez I D 2018 New Journal of Physics 20 033029","doi":""},{"text":". X Wen, International journal of modern physics B. 6Wen X G 1992 International journal of modern physics B 6 1711-1762","doi":""},{"text":". J Dubail, N Read, E H Rezayi, Phys. Rev. B. 8624245310Dubail J, Read N and Rezayi E H 2012 Phys. Rev. B 86(24) 245310","doi":""},{"text":". B Estienne, Z Papić, Regnault N Bernevig, B A , Phys. Rev. B. 8716161112Estienne B, Papić Z, Regnault N and Bernevig B A 2013 Phys. Rev. B 87(16) 161112","doi":""},{"text":". B Herwerth, G Sierra, H H Tu, J Cirac, A E B Nielsen, Phys. Rev. B. 9224245111Herwerth B, Sierra G, Tu H H, Cirac J I and Nielsen A E B 2015 Phys. Rev. B 92(24) 245111","doi":""},{"text":". J Cirac, G Sierra, Phys. Rev. B. 81104431Cirac J I and Sierra G 2010 Phys. Rev. B 81 104431","doi":""},{"text":". M B Hastings, I González, A Kallin, R G Melko, Phys. Rev. Lett. 104157201Hastings M B, González I, Kallin A B and Melko R G 2010 Phys. Rev. Lett. 104 157201","doi":""},{"text":". A Kitaev, J Preskill, Phys. Rev. Lett. 9611110404Kitaev A and Preskill J 2006 Phys. Rev. Lett. 96(11) 110404","doi":""},{"text":". Levin M Wen, X G , Phys. Rev. Lett. 9611110405Levin M and Wen X G 2006 Phys. Rev. Lett. 96(11) 110405","doi":""}],"string":"[\n {\n \"text\": \". D C Tsui, H Stormer, A C Gossard, Phys. Rev. Lett. 481559Tsui D C, Stormer H L and Gossard A C 1982 Phys. Rev. Lett. 48 1559\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Laughlin, Phys. Rev. Lett. 501395Laughlin R B 1983 Phys. Rev. Lett. 50 1395\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Kalmeyer, R Laughlin, Phys. Rev. Lett. 592095Kalmeyer V and Laughlin R 1987 Phys. Rev. Lett. 59 2095\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E J Bergholtz, Z Liu, International Journal of Modern Physics B. 271330017Bergholtz E J and Liu Z 2013 International Journal of Modern Physics B 27 1330017\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S A Parameswaran, Roy R Sondhi, S L , Comptes Rendus Physique. 14Parameswaran S A, Roy R and Sondhi S L 2013 Comptes Rendus Physique 14 816-839\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Barredo, De Léséleuc, S , Lienhard V Lahaye, T Browaeys, A , Science. 354Barredo D and De Léséleuc S and Lienhard V and Lahaye T and Browaeys A 2016 Science 354 1021-1023\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Barredo, V Lienhard, De Leseleuc, S Lahaye, T Browaeys, A , Nature. 561Barredo D, Lienhard V, De Leseleuc S, Lahaye T and Browaeys A 2018 Nature 561 79-82\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Lienhard, P Scholl, S Weber, D Barredo, S De Léséleuc, R Bai, N Lang, M Fleischhauer, H P Büchler, T Lahaye, A Browaeys, Phys. Rev. X. 10221031Lienhard V, Scholl P, Weber S, Barredo D, de Léséleuc S, Bai R, Lang N, Fleischhauer M, Büchler H P, Lahaye T and Browaeys A 2020 Phys. Rev. X 10(2) 021031\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Weber, R Bai, N Makki, J Mögerle, T Lahaye, A Browaeys, M Daghofer, Lang N Büchler, H P , PRX Quantum. 3330302Weber S, Bai R, Makki N, Mögerle J, Lahaye T, Browaeys A, Daghofer M, Lang N and Büchler H P 2022 PRX Quantum 3(3) 030302\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Wu, F Yang, S Yang, K Mølmer, T Pohl, M Tey, L You, Phys. Rev. Res. 4332046Wu X, Yang F, Yang S, Mølmer K, Pohl T, Tey M K and You L 2022 Phys. Rev. Res. 4(3) L032046\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Brzezińska, A Cook, Neupert , Phys. Rev. B. 98205116Brzezińska M, Cook A M and Neupert T 2018 Phys. Rev. B 98(20) 205116\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Pai, A Prem, Phys. Rev. B. 10015155135Pai S and Prem A 2019 Phys. Rev. B 100(15) 155135\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A A Iliasov, M Katsnelson, S Yuan, Phys. Rev. B. 101445413Iliasov A A, Katsnelson M I and Yuan S 2020 Phys. Rev. B 101(4) 045413\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Fremling, M Van Hooft, C Smith, L Fritz, Phys. Rev. Res. 2113044Fremling M, van Hooft M, Smith C M and Fritz L 2020 Phys. Rev. Res. 2(1) 013044\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Fischer, M Van Hooft, T Van Der Meijden, C M Smith, Fritz L Fremling, M , Phys. Rev. Res. 3443103Fischer S, van Hooft M, van der Meijden T, Smith C M, Fritz L and Fremling M 2021 Phys. Rev. Res. 3(4) 043103\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Manna, S Nandy, B Roy, Phys. Rev. B. 105201301Manna S, Nandy S and Roy B 2022 Phys. Rev. B 105(20) L201301\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M N Ivaki, I Sahlberg, K Pöyhönen, T Ojanen, Communications Physics. 5327Ivaki M N, Sahlberg I, Pöyhönen K and Ojanen T 2022 Communications Physics 5 327\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Manna, B Pal, Wang W Nielsen, A E B , Phys. Rev. Research. 223401Manna S, Pal B, Wang W and Nielsen A E B 2020 Phys. Rev. Research 2 023401\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Li, M Jha, A E B Nielsen, Phys. Rev. B. 105885152Li X, Jha M C and Nielsen A E B 2022 Phys. Rev. B 105(8) 085152\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Zhu, -O' Jochym, T Connor, A Dua, PRX Quantum. 3330338Zhu G, Jochym-O'Connor T and Dua A 2022 PRX Quantum 3(3) 030338\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Jaworowski, M Iversen, A E Nielsen, arXiv:2301.09971arXiv preprintJaworowski B, Iversen M and Nielsen A E 2023 arXiv preprint arXiv:2301.09971\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Moore G Read, N , Nuclear Physics B. 360Moore G and Read N 1991 Nuclear Physics B 360 362-396\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A E B Nielsen, J Cirac, G Sierra, Phys. Rev. Lett. 108257206Nielsen A E B, Cirac J I and Sierra G 2012 Phys. Rev. Lett. 108 257206\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H H Tu, A E B Nielsen, J Cirac, G Sierra, New Journal of Physics. 1633025Tu H H, Nielsen A E B, Cirac J I and Sierra G 2014 New Journal of Physics 16 033025\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A S Balankin, Martínez-Cruz M A Andálvarez-Jasso , M D , Patiño-Ortiz M , Patiño-Ortiz , J , Physics Letters A. 383Balankin A S and Martínez-Cruz M A andÁlvarez-Jasso M D and Patiño-Ortiz M and Patiño- Ortiz J 2019 Physics Letters A 383 957-966\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Glasser, J I Cirac, Sierra G Nielsen, A E B , Phys. Rev. B. 9424245104Glasser I, Cirac J I, Sierra G and Nielsen A E B 2016 Phys. Rev. B 94(24) 245104\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A E B Nielsen, I Glasser, I D Rodrãguez, New Journal of Physics. 2033029Nielsen A E B, Glasser I and RodrÃguez I D 2018 New Journal of Physics 20 033029\",\n \"doi\": \"\"\n },\n {\n \"text\": \". X Wen, International journal of modern physics B. 6Wen X G 1992 International journal of modern physics B 6 1711-1762\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Dubail, N Read, E H Rezayi, Phys. Rev. B. 8624245310Dubail J, Read N and Rezayi E H 2012 Phys. Rev. B 86(24) 245310\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Estienne, Z Papić, Regnault N Bernevig, B A , Phys. Rev. B. 8716161112Estienne B, Papić Z, Regnault N and Bernevig B A 2013 Phys. Rev. B 87(16) 161112\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B Herwerth, G Sierra, H H Tu, J Cirac, A E B Nielsen, Phys. Rev. B. 9224245111Herwerth B, Sierra G, Tu H H, Cirac J I and Nielsen A E B 2015 Phys. Rev. B 92(24) 245111\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J Cirac, G Sierra, Phys. Rev. B. 81104431Cirac J I and Sierra G 2010 Phys. Rev. B 81 104431\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M B Hastings, I González, A Kallin, R G Melko, Phys. Rev. Lett. 104157201Hastings M B, González I, Kallin A B and Melko R G 2010 Phys. Rev. Lett. 104 157201\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Kitaev, J Preskill, Phys. Rev. Lett. 9611110404Kitaev A and Preskill J 2006 Phys. Rev. Lett. 96(11) 110404\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Levin M Wen, X G , Phys. Rev. Lett. 9611110405Levin M and Wen X G 2006 Phys. Rev. Lett. 96(11) 110405\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":582,"cells":{"text":{"kind":"string","value":"\nNon-local non-linear sigma models\nJune 2019 25 Jun 2019\n\nSteven S Gubser ssgubser@princeton.edu \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA\n\nChristian B Jepsen cjepsen@princeton.edu \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA\n\nZiming Ji \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA\n\nBrian Trundy btrundy@princeton.edu \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA\n\nAmos Yarom ayarom@physics.technion.ac.il \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA\n\nDepartment of Physics\n32000Technion, HaifaIsrael\n\nNon-local non-linear sigma models\nJune 2019 25 Jun 2019PUPT-2592\nWe study non-local non-linear sigma models in arbitrary dimension, focusing on the scale invariant limit in which the scalar fields naturally have scaling dimension zero, so that the free propagator is logarithmic. The classical action is a bi-local integral of the square of the arc length between points on the target manifold. One-loop divergences can be canceled by introducing an additional bi-local term in the action, proportional to the target space laplacian of the square of the arc length. The metric renormalization that one encounters in the two-derivative non-linear sigma model is absent in the non-local case. In our analysis, the target space manifold is assumed to be smooth and Archimedean; however, the base space may be either Archimedean or ultrametric. We comment on the relation to higher derivative non-linear sigma models and speculate on a possible application to the dynamics of M2-branes.\n\nIntroduction\n\nScalar field theories over the reals with bi-local kinetic terms were introduced in [1], and the recent work [2] provides a useful point of entry into the extensive literature. Similar field theories over the p-adic numbers were considered in [3] as a continuum description of Dyson's hierarchical model [4]. A unifying point of view on the bi-local O(N ) vector model was provided in [5], showing that the standard large N development can be framed in terms that are largely independent of whether the theory is formulated over the reals or the p-adics. The present work extends the study of bi-local theories to bi-local non-linear sigma models, starting with the action S = µ n−s 2γ V ×V d n xd n y |x − y| n+s d(φ(x), φ(y)) 2 ,\n\nwhere |x − y| is the distance function on the n-dimensional base space V and d(φ(x), φ(y)) is the distance function on the target manifold. In the limit s → n, where the theory (1) becomes classically scale invariant, we find logarithmic divergences in one-loop diagrams which can be canceled by counterterms that can be expressed in terms of the target space laplacian of the square of the distance function, together with field redefinitions. 1 Ricci flatness suppresses the one-loop divergences that we encounter, so in a sense (and with significant caveats) we may claim that we are deriving the vacuum Einstein equations from conformal invariance, as in [6]. Our work was partly motivated by the more recent results of [7], which were derived for a nearest neighbor arc length model on the Bruhat-Tits tree-in other words, on the other side of the p-adic AdS/CFT duality [8,9] from our results for field theories over the p-adic numbers. However, the particular structure of counterterm we find suggests that renormalization of our theories have less to do with renormalization of the local metric as normally understood (i.e. Ricci flow) than with an augmentation of the action (1) to include the target space laplacian of d(φ(x), φ(y)) 2 .\n\nA conservative expectation is that once non-local terms are allowed in a field theory, they proliferate and the theory becomes non-renormalizable. Theories with purely quadratic bi-local kinetic terms, as studied in [1,3] (as well as many subsequent works) avoid such problems through a non-renormalization theorem: If we write\nS = 1 2 V d n kφ(−k)|k| sφ (k) + V d n x U (φ(x)) ,(2)\nthen the claim is that the quadratic bi-local term is never renormalized (at least perturbatively), though the purely local term U (φ(x)) certainly is-and depending on details, derivative terms might be radiatively generated. Non-local interaction terms vitiate this non-renormalization theorem, and one's suspicions could be renewed that there is no sensible theory. We will not be able in this work entirely to allay such concerns, because we do not give a demonstration parallel to the one in [6] that Ward identities based on diffeomorphism invariance guarantee that loop divergences can only modify the original form of the action. Indeed, the counterterms we generate at one loop do modify the bi-local action in an unexpected way, but one which appears to be controlled in a derivative expansion, so that higher derivative terms can be radiatively generated at each new order without spoiling results from lower orders. We will revisit the question of renormalizability in section 13. The organization of the rest of this paper is as follows. In section 2 we present the main results in Fourier analysis that we need, both over the reals and the p-adics. In section 3 we explain how double integrals such as the one in (1) can be regulated if divergences arise as |x − y| → 0. In section 4 we introduce the classical action for the bi-local non-linear sigma model. In section 5 we discuss loop divergences in general terms, including an introductory account of the non-renormalization property of the kinetic term in (2). In sections 6-10 we investigate the simplest one-loop divergences of the bi-local non-linear sigma model, and then in section 11 we argue that all these divergences can be canceled by a laplacian counterterm in place of renormalization of the local metric, together with field redefinitions. As a byproduct of our analysis, we recover in section 12 the usual beta function for the two-derivative theory in two dimensions. We conclude in section 13 with a summary of possible future directions.\n\n\nFourier transforms\n\nIn loop calculations we will often need to go back and forth between momentum space expressions similar to the ones presented in (2) and their real space counterparts, using the Fourier transforms\nφ(x) = V d n k e 2πik·xφ (k)φ(k) = V d n x e −2πik·x φ(x) .(3)\nThe relevant results are fairly similar between real and p-adic cases, so we present them together. When V = R n , the definitions (3) are entirely standard, and k·x can be understood as the ordinary dot product. Likewise, in this case, |x| is understood as the standard L 2 norm on R n . We refer to the real case as Archimedean because the norm | · | has the property that if 0 < |x| < |y|, then there is some n ∈ Z such that |y| < |nx|. The simplest n-dimensional p-adic construction is based on letting V = Q p n be the (unique) unramified n-dimensional extension of Q p . Let N and Tr be the field norm and field trace with respect to the extension Q p n /Q p . Then we define |x| = | N(x)| 1/n p where | · | p is the usual p-adic norm. We will refer to the p-adic case as ultrametric because the norm | · | just defined has the property |x + y| ≤ max{|x|, |y|}. Next we define k · x = 1 n Tr(kx). Note that k · x ∈ Q p , so to give meaning to e 2πik·x we now only need to define e 2πiξ for ξ ∈ Q p . To this end we find the unique p-adic integer ξ such that ξ − ξ ∈ [0, 1) ∩ Q, and we understand that by e 2πiξ we really mean e 2πi(ξ− ξ ) .\n\nWe are particularly interested in the Fourier transform of powers of |k|: V d n k e 2πik·x |k| s = Γ V (n + s) |x| n+s + (contact terms) .\n\nHere Γ V (s) is a meromorphic function of s which can be evaluated as\nΓ V (s) = ζ v (s) ζ v (n − s)(5)\nwhere we set v = ∞ in the Archimedean case and v = p in the ultrametric case, with\nζ ∞ (s) ≡ π −s/2 Γ Euler (s/2) ζ p (s) ≡ 1 1 − p −s .(6)\nIntuitively, Γ V is a variant of the Euler gamma, specific to the choice of V , and constructed so as to be the coefficient of the 1/|x| n+s term in (4). In the remainder of our discussion, integrals are over V unless otherwise indicated. The contact terms in (4) are somewhat delicate and dependent on detail. When −n < s < 0, the integral in (4) is convergent, and no contact terms are needed. One can easily check that Γ V (n + s) → 0 as s → 0, so when s = 0 the power law term goes away and we recover the obvious result d n k e 2πik·x = δ n (x) .\n\nFor s > 0, the integral in (4) diverges, and we need a more careful approach. A good first step is to understand (4) in terms of its action on a test function φ : V → R:\nd n k e 2πik·x |k| sφ (k) = D s φ(x) ,(8)\nwhere D s is some linear map on functions φ(x). A suitable class of test functions are socalled Schwartz-Bruhat functions. When φ : Q p n → R, we require that φ is locally constant with compact support. For example, the characteristic function of the p-adic integers is a Schwartz-Bruhat function on Q p . When φ : R n → R, the test functions are more appropriately called Schwartz functions, and their defining property is that they go to 0 faster than any power of |x|, as do all their derivatives. An example is a Gaussian. Both in the real and ultrametric cases, the Fourier transformφ(k) of a Schwartz-Bruhat function is again a Schwartz-Bruhat function.\n\nWith (8) taken as the definition of D s , our task is to find a representation of D s entirely in position space. In the ultrametric case for arbitrarily positive s, one finds\nD s φ(x) = Γ V (n + s) d n y φ(y) − φ(x) |x − y| n+s .(9)\nThis is the Vladimirov derivative. In the Archimedean case, the same expression (9) is valid for 0 < s < 2. There is one more easy case to dispose of: even positive integer s for Archimedean V . Then Γ V (n + s) = 0, which makes sense in (4) because the right hand should be purely distributional, on account of |k| s = (k 2 ) s/2 being analytic in k 2 . Explicitly,\nD s φ(x) = 1 (2π) s (− ) s/2 φ(x)\nfor positive even s ,\n\nwhere = n i=1 ∂ 2 x i . We are left with the task of defining D s for Archimedean V and for s > 2 but not an even integer. Heuristically, the contact terms in (4) are a sum of terms of the form r δ n (x), where 0 ≤ r < s/2, with divergent coefficients. To state this more precisely, we write\nD s φ(x) = Γ V (n + s) d n y φ(y) − φ(x) |x − y| n+s ,(11)\nwhere a regulated integral\nd n y G(x, y) |x − y| n+s(12)\nis rendered finite (if possible) by allowing the subtraction from G(x, y) of a finite sum of smooth functions of either of the following types:\n\nI. Pure powers: more precisely, any function whose y dependence comes solely through a factor |x − y| α where α is a real number. This is meant to include, through the case α = 0, functions which have no y dependence.\n\nII. Higher partial waves: more precisely, any function of the form Y ( y − x)g(|x−y|) where Y (ẑ) is a spherical harmonic on S n−1 other than the s-wave.\n\nType I functions are never integrable, whereas type II functions may or may not be; so at best there is a unique choice of type I functions that will work, whereas many choices of type II functions are possible. An alternative approach, generalizing the principle value prescription, is to eschew modifications of the integrand and instead carry out y integration in polar coordinates centered around x, as follows. One first performs the angular integrals. Then the radial integral is restricted to run from l to L. One next allows the subtraction of an arbitrary finite sum of negative powers of l and/or positive powers of L, chosen (if possible) so that the limits l → 0 and L → ∞, taken independently, lead to a finite result. Doing the angular integration first obviates the need for type II functions, while the ultraviolet and infrared cutoffs, l and L, obviate the need for type I. 2\n\nWhile the subtractions described can in principle cure either ultraviolet (UV) or infrared (IR) divergences, we will be interested only in applications where UV divergences matter: that is, divergences arising when |x − y| → 0 (with x held fixed). Type II subtractions are relatively innocuous because they follow automatically from performing angular integrations first; therefore we will use the notation d n y . . . to indicate a y integration with type II subtractions which we usually omit to write explicitly.\n\nAlthough we have stated our integration prescriptions in the abstract, it is easy to see how to apply them to (11) when φ is a Schwartz function. Consider the case 2 < s < 3, and set x = 0 for simplicity. Then (11) becomes\nd n y |y| n+s [φ(y) − φ(0)] = d n y |y| n+s φ(y) − φ(0) − y i ∂ i φ(0) − 1 2 y i 1 y i 2 ∂ i 1 ∂ i 2 φ(0) .(13)\nThe extra terms in square brackets on the right hand side of (13) evidently render the integral convergent near y = 0 for 2 < s < 3. The term linear in y is clearly a type II function, and the term quadratic in y is a sum of a type II function proportional to y i 1 y i 2 − y 2 n δ i 1 i 2 (a d-wave term) and a type I function proportional to y 2 . If 3 ≤ s < 4, then we would need one additional term in the Taylor series expansion of φ around y = 0, and this additional term is a type II function. In summary, for 2 < s < 4, and omitting type II subtractions,\nD s φ(0) = d n y |y| n+s φ(y) − φ(0) − y 2 2n φ(0) .(14)\nEvidently, if 0 < s < 2, a simpler subtraction scheme would work, resulting in (14) with the laplacian term omitted, in agreement with (9). For general s > 0 (other than positive even integers) and Archimedean V ,\nD s φ(0) = Γ V (n + s) d n y |y| n+s φ(y) − s/2 r=0 y 2r b r r φ(0) (15) where b r = Γ Euler ( n 2 ) 2 2r Γ Euler (r + n 2 )Γ Euler (r + 1) .(16)\nIn principle, one may derive (15) by subtracting an appropriate number of terms in the Taylor series expansion of φ(y) and then finding appropriate type II subtractions to bring the result into the form (15).\n\ncannot be cured using powers of l and/or L after a cutoff integration. Because we avoid even integer s as well as functions G(x, y) which grow as positive powers of large separation |x − y|, we do not need to specify a resolution to this inequivalence.\n\nA more efficient way to determine the coefficients b r is to start from (15) and Fourier transform:\nd n x e −2πik·x D s φ(x) = Γ V (n + s) d n x e −2πik·x d n y |x − y| n+s φ(y) − s/2 r=0 (x − y) 2r b r r φ(x) = 2Γ V (n + s) ζ ∞ (n − 1) ∞ 0 dỹ y s+1 π 0 dθ (sin θ) n−2φ (k) e 2πi|k|ỹ cos θ − s/2 r=0 b r (2πi) 2r k 2rỹ2r = Γ V (n + s)(2π) n 2 +s |k| sφ (k) ∞ 0 dρ ρ − n 2 −s J n 2 −1 (ρ) − s/2 r=0 a r ρ 2r−s−1 (17)\nIn the second equality of (17), we have partially carried out the y integral in polar coordinates around the point y = x, introducing a radial variableỹ = |x − y|. In the third equality, we have carried out the angular θ integral and introduced a new radial variable, ρ = 2π|k|ỹ. The ρ integral in the last line of (17) converges, provided s is positive but not an even integer, and provided the coefficients a r are coefficients in the Taylor series expansion of the Bessel function around ρ = 0. These coefficients a r are well known, and from them one can recover the expression (16) for the b r .\n\n\nBi-local integrals\n\nWe are particularly interested in double integrals of the form\nV ×V d n xd n y |x − y| n+s G(x, y)(18)\nwhere s > 0 and G(x, y) is piecewise constant if V is ultrametric and smooth if V is Archimedean. Unless otherwise noted, all double integrals over x and y will by taken over all of V × V . In the ultrametric case, for any s > 0, following (9) we define\nd n xd n y |x − y| n+s G(x, y) ≡ d n xd n y |x − y| n+s [G(x, y) − G(x, x)] .(19)\nIn the Archimedean case, we define d n xd n y |x − y| n+s G(x, y)\n\nby performing the y integration first and allowing the subtraction of type I and type II functions to G(x, y) in order to achieve a finite result (if possible). As in the previous section, type II subtractions are deemed relatively inconsequential, so even unprimed integration over x and y means to perform the y integration first, allowing the subtraction of type II functions in order to achieve a finite result (if possible). Explicitly, for s not a positive even integer,\nd n xd n y |x − y| n+s G(x, y) = d n xd n y |x − y| n+s G(x, y) − s/2 r=0 b r r y G(x, y) y=x (y − x) 2r ,(21)\nwhere the coefficients b r are as given in (16). We avoid positive even integer s when V is Archimedean because in this case we expect that our constructions will lead instead to purely local theories; also, precisely in this case, the subtleties pointed out in footnote 2 regarding logarithmic divergences come into play.\n\nOur computational strategy will turn on converting bi-local position space integrals into Fourier space integrals. Let's start with the simplest example of that calculation, valid for ultrametric V and any s > 0, and also for Archimedean V and 0 < s < 2. Let φ : V → R be a Schwartz-Bruhat function. Then\nd n xd n y |x − y| n+s [φ(x) − φ(y)] 2 = d n xd n y |x − y| n+s [φ(x) − φ(y)] 2 + φ(x) 2 − φ(y) 2 = −2 d n x φ(x) d n y |x − y| n+s [φ(y) − φ(x)] = − 2 Γ V (n + s) d n x φ(x)D s φ(x) = − 2 Γ V (n + s) d n kφ(−k)|k| sφ (k) .(22)\nThe first step is actually the trickiest, because it is not clear from the rules of integration set forth following (20) that we are allowed to add a function like φ(x) 2 − φ(y) 2 to the integrand. To justify this step, we denote f (x) = φ(x) 2 , and we argue that\nd n xd n y |x − y| n+s φ(y) 2 − φ(x) 2 = 1 Γ V (n + s) d n x D s f (x) = 0 .(23)\nThe second integral in (23) is the k = 0 component of the Fourier transform of D s f (x). But this Fourier transform is |k| sf (k), and since s > 0 the k = 0 component indeed vanishes.\n\nLet's now pursue the same computation for the Archimedean case with 2 < s < 4. On one hand, using (21),\nd n xd n y |x − y| n+s [φ(x) − φ(y)] 2 = d n xd n y |x − y| n+s [φ(x) − φ(y)] 2 − (y − x) 2 n (∂φ(x)) 2 = d n xd n y |x − y| n+s [φ(x) − φ(y)] 2 − (y − x) 2 2n −2φ(x) φ(x) + φ(x) 2 .(24)\nOn the other hand, using (14),\n− 2 Γ V (n + s) d n kφ(−k)|k| sφ (k) = −2 d n x φ(x) d n z |z| n+s [φ(x + z) − φ(x)] = d n xd n y |x − y| n+s 2φ(x) [φ(x) − φ(y)] + (y − x) 2 n φ(x) φ(x) .(25)\nIn order to conclude\nd n xd n y |x − y| n+s [φ(x) − φ(y)] 2 = − 2 Γ V (n + s) d n kφ(−k)|k| sφ (k) ,(26)\nwe must therefore argue that the final integrals in (24) and (25) agree. Subtracting (25) from (24) and simplifying slightly with the definition f (x) = φ(x) 2 , we arrive at\nd n xd n y |x − y| n+s f (y) − f (x) − (y − x) 2 2n f (x) = 1 Γ V (n + s) d n x D s f (x) = 0 .(27)\nThe first equality in (27) follows from (14), and the second is by the same argument used following (23). To summarize, for Archimedean V and for 2 < s < 4,\nd n xd n y |x − y| n+s (y − x) 2 n (∂φ(x)) 2 − [φ(x) − φ(y)] 2 = 2 Γ V (n + s) d n k φ(−k)|k| s φ(k) .(28)\nΓ V (n + s) > 0 for 2 < s < 4, and so without the (∂φ) 2 on the left hand side of (28) we would have a sign problem. The equality (26) can be checked in a similar manner for s > 4. A key relation is\nm y (φ(x) − φ(y)) 2 x=y = −2φ(x) m φ(x) + m φ(x) 2 .(29)\nTwo take-away lessons are:\n\n• When we write simple |k| s kinetic terms in momentum space, in position space we are combining non-local position space terms and local terms involving derivatives in a precisely tuned ratio.\n\n• There is some freedom in the precise structure of the position space form, as exemplified by the equality of the last integrals in (24) and (25) due to a manipulation which is the non-local version of integration by parts.\n\n\nThe bi-local non-linear sigma model\n\nWe are now in a position to present the action for the bi-local non-linear sigma model. Let M be a smooth D-dimensional manifold with a Riemannian metric g ab , whose Riemann and Ricci tensors are\nR ab c d = ∂ a Γ c bd − ∂ b Γ c ad + Γ c ae Γ e bd − Γ c be Γ e ad R ac = g bd R abcd .(30)\nGiven any two points X and Y on M , let\nQ(X, Y ) = d(X, Y ) 2(31)\nbe the square of the shortest distance between X and Y . Clearly, Q(X, Y ) is a smooth function of X and Y , provided X and Y are not too far apart. For smooth functions φ : V → M whose range is sufficiently localized, we consider the action functional\nS = µ n−s 2γ d n xd n y |x − y| n+s Q(φ(x), φ(y)) ,(32)\nwhere indicates a regulated double integral of the type discussed around (19)-(21). 3 Note that this discussion requires us to avoid positive even integer s when x and y are valued in R n (as opposed to Q p n ). In the Archimedean setting, when s > 2, there are derivative terms like (∂φ) 2 implicitly built into (32), with coefficients tuned so as to ensure convergence of the integral. The parameter µ has dimensions of energy so that we can regard φ and Q(φ(x), φ(y)) as dimensionless. The factor γ is a loop-counting parameter: Classical effects are O(γ −1 ), one-loop amplitudes are O(γ 0 ), two loop amplitudes are O(γ), and so forth. In other words, γ plays the role of . 3 One may wonder whether the primed integral, as defined following (20), spoils coordinate invariance of the integrand. For instance, if s is sufficiently large we may, in light of (21), be required to include a y Q term to the integrand, which if written only in terms of partial derivatives does not appear to be coordinate invariant. In fact, it is easy to convince oneself that, e.g., y Q can be constructed from covariant quantities:\n∂ y a ∂ y b Q = ∂ 2 Q ∂φ i ∂φ j ∂φ i ∂y a ∂φ j ∂y b + ∂Q ∂φ i ∂ 2 φ i ∂y a ∂y b = ∇ φ i ∂Q ∂φ j ∂φ i ∂y a ∂φ j ∂y b + ∂Q ∂φ i ∂φ j ∂y a ∇ φ j ∂φ i ∂y b .\nA close cousin of the action (32) was considered in [7]:\nS = xy d(φ(x), φ(y)) 2 (33)\nwhere now x and y are vertices of a graph and the sum is over undirected edges. The formula actually appears earlier in [6], though it was intended there to be considered on a square lattice, as a regulator for the local non-linear sigma model, rather than on the Bruhat-Tits tree as in [7]. We require the range of the maps φ to be sufficiently localized in order to ensure that we do not encounter any failures of smoothness in Q(X, Y ), and in order to ensure that we can use a single system of Riemann normal coordinates for φ throughout. One can now solve the geodesic equation perturbatively in the curvature and use that to evaluate Q(X, Y ) and expand the action (32),\nS = S 2 + S 4 + . . . ,(34)\nwhere\nS 2 = µ n−s 2γ g ab d n xd n y |x − y| n+s [φ a (x) − φ a (y)] φ b (x) − φ b (y)(35)\nand\nS 4 = − µ n−s 6γ R abcd d n xd n y |x − y| n+s φ a (x)φ b (y)φ c (x)φ d (y) .(36)\nHere g ab and R abcd are evaluated at the origin of the Riemann normal coordinates, which we assume is at X = 0. Often, the definition of Riemann normal coordinates includes the requirement g ab = δ ab , but this is not necessary for our calculations, and we find it more convenient to retain explicit factors of g ab and the inverse metric g ab . Put differently, we are choosing a coordinate system so that all geodesics passing through the origin are linear in the affine parameter:\ng ab (φ) = g ab − 1 3 R acbd φ c φ d + O(φ 3 ) .(37)\nThe ellipsis in (34) indicates higher order interactions, involving five or more powers of φ(x) and/or φ(y), as well as derivatives and powers of the curvature. We will consider up to six point interactions in section 10. quadratic kinetic term in the action (2), where V = R n or Q p n , with\nU (φ) = g 3! φ 3 .(38)\nThe purely cubic theory is unstable, but it serves our purpose because we are only interested in analyzing the behavior of the one-loop correction to the propagator. Using the diagram shown in figure 1a, we obtain the one-loop contribution to the quadratic part of the oneparticle irreducible (1PI) effective action:\nδΓ 2 (k) = − g 2 2 I where I = d n | | s |k − | s .(39)\nWe continue the convention of integrating over all of V except as otherwise indicated. Let's assume n > 2s, so I is UV divergent (and IR convergent). To regulate the divergence, we introduce a hard cutoff: | | ≤ Λ. If V = R n , then Λ can be any positive real number. If V = Q p n , then we will require that Λ is an integer power of p. The ultrametric case is easy to analyze, because when | | > |k| we have | | = |k − | exactly. So, except in the compact region where | | ≤ |k|, the integrand has no k dependence at all. Therefore, any UV divergences are entirely independent of k, and to evaluate them we can set k = 0:\nI(Λ) = | |≤Λ d n | | s |k − | s = | |≤Λ d n | | 2s + (UV finite) = ζ p (n − 2s) ζ p (n) Λ n−2s + (UV finite) .(40)\nThe last equality comes from splitting the integration region into shells with fixed | |; then the integral becomes a geometric sum. Because the divergent part of I(Λ) has no k-dependence, the counterterm required to cancel it is proportional to d n kφ(−k)φ(k) = d n x φ(x) 2 . In other words, it is a mass term. This argument is easy to generalize to the statement that only purely local terms (powers of φ(x)) can be radiatively generated starting from the action (2) over V = Q p n . An essentially equivalent argument was made in a Wilsonian picture in [3]. The Archimedean case is more subtle because of the possibility of subleading divergences. A straightforward approach is to expand\nI(Λ) = | |≤Λ d n | | s |k − | s = | |≤Λ d n | | 2s 1 − 2k ·ˆ | | + k 2 2 −s/2(41)\nin powers of k. Terms with an odd number of powers of k vanish by parity, leaving only terms analytic in k 2 . Of these, only terms proportional to k 2r with r ≤ n 2 − s are UV divergent. In short, the divergent part of I(Λ) is a polynomial in k 2 whose order is n 2 − s . A divergent term proportional to k 2r requires a counterterm proportional to d n x φ(x) r φ(x). These are the radiatively generated derivative terms mentioned in section 1.\n\nWe should note a troublesome feature of the hard momentum cutoff for Archimedean theories: The coefficients one finds for sub-leading divergences depend on how one implements the cutoff. For example, it is easy to check that the coefficient of the k 2 term in I(Λ) changes if instead of requiring | | ≤ Λ we impose the more democratic condition − k 2 ≤ Λ. However, the feature that we care about, namely the fact that the divergent terms have only polynomial dependence on k 2 , doesn't depend on the details of the cutoff. It is perhaps instructive to consider one other alternative, namely dimensional regularization, in which one first computes\nI = Γ R n (2s − n) Γ R n (s) 2 (k 2 ) n−2s 2(42)\nby continuing to a domain of n in which the integral is convergent. (In the current example, s < n < 2s is such a domain.) The only divergences one then tracks are poles of the right hand side of (42) as a function of n. These occur precisely when n−2s 2 is a non-negative integer. It is characteristic of dimensional regularization that there is (at most) one divergent term for a given n, corresponding to a logarithmic divergence in the original integral.\n\nThe loop diagrams we will need to consider in our analysis of the non-local non-linear sigma model are simpler than (39) in one regard: The loop is a single propagator starting and ending at the same vertex. This matters because there is then only one internal momentum , and imposing the hard cutoff | | ≤ Λ is a privileged choice because it corresponds to integrating over an O(n)-invariant region. An example is the diagram shown in figure 1b, which is proportional to\nI 0 = d n | | s ,(43)\nassuming that whatever vertex factor is needed to fully evaluate the diagram doesn't depend on . We also assume n > s so that I 0 is UV divergent but IR convergent. We straightforwardly find\nI 0 = 2 ζ ∞ (n) Λ n−s n − s for V = R n ζ p (n − s) ζ p (n) Λ n−s for V = Q p n .(44)\nThere are obviously no subleading divergences in I 0 .\n\nFor convenience we introduce\n= n − s .(45)\nWe are interested in divergences proportional to log Λ that arise when = n − s = 0. As a technical trick to isolate these divergences, we make small and positive, and we look for divergences of the form Λ / , which in the → 0 + limit give rise to log Λ terms.\n\nTo characterize this limit precisely, given n 0 > 0 and λ ∈ R, we set n = n 0 + λ and s = n 0 + (λ − 1) and then take the → 0 + limit with n 0 and λ held fixed. (Clearly then we are allowing non-integer n, in the spirit of [6].) For the most part, our final results are independent of λ. When is sufficiently small, we may replace (44) with\nI 0 = i 0 Λ where i 0 = 2 ζ ∞ (n 0 ) + O( ) for V = R n 1 ζ p (n 0 ) log p + O( ) for V = Q p n .(46)\nIf we lift the requirement that Λ is an integer power of p when V = Q p n , then (46) is unaltered, because I 0 = ζp( ) ζp(n) p log p Λ , and p log p Λ differs from Λ at most by a factor of p = 1 + O( ). The important point is that in the limit → 0 + , I 0 includes a logarithmic term i 0 log Λ, and isolating this term is our stated objective.\n\nWe will encounter one other loop integral:\nI 2 (k) = d n |k − | s | | s .(47)\nIt comes from graphs similar to the one in (43), but with a vertex prefactor |k − | s . Using the same reasoning that led to (40), we see that when a hard cutoff | | ≤ Λ is imposed, one obtains in the ultrametric case\nI 2 (k) = Λ n + (UV finite) when V = Q p n .(48)\nIf V = R n and n is positive but not an even integer, then\nI 2 (k) = n/2 r=0 c r k 2r Λ n−2r n − 2r + (UV finite)(49)\nfor some coefficients c r . If we choose n 0 positive but not an even integer and fix any finite value of λ, then for sufficiently small , (49) applies, and the least singular power of Λ appearing in it is Λ n−2 n/2 . As → 0 + , this power remains positive and finite, tending to Λ n 0 −2 n 0 /2 . So there is no log Λ behavior, even in the → 0 + limit. If instead we make n 0 a positive even integer, then by choosing the very particular value λ = 1, so that s = n 0 exactly, we find (for sufficiently small > 0) that the least singular term in (49) is c n/2 |k| s Λ , which does contribute a c n/2 |k| s log Λ divergence in the → 0 + limit; moreover, in this case, by calculation, c n/2 = i 0 + O( ). 4 We can summarize the situation, both for the Archimedean and ultrametric cases, by stating that for = n − s positive but sufficiently small, then subject to the restriction that n cannot be a positive even integer when V = R n ,\nI 2 (k) = (higher powers of Λ) + i 2 |k| s Λ + (UV finite) ,(50)\nwhere \ni 2 = i 0 + O( ) when V = R\nThe higher powers of Λ in (50) are accompanied by non-negative integer powers of k 2 , and they correspond to operators which remain relevant in the → 0 + limit. The only log Λ behavior arising from I 2 (k), in the → 0 + limits described above, is the i 2 |k| s log Λ term coming from the i 2 |k| s Λ term shown in (50). We are not concerned about O( ) terms to i 0 and i 2 because they drop out of the log Λ behavior in the → 0 + limit. In the following sections, therefore, we will drop O( ) terms from (46) and (51), and we will evaluate i 0 and i 2 in terms of n rather than n 0 .\n\n\nThe propagator through one loop\n\nTo derive the tree-level propagator, we use an obvious generalization of (26) to multicomponent scalar fields to rewrite the free action in momentum space:\nS 2 = µ 2γ g ab d n kφ a (−k)|k| sφb (k) ,(52)\nwhere we recall that = n − s, and for notational convenience we have introduced 5\nγ = − Γ V (n + s) 2 γ .(53)\nWe immediately extract from (52) the propagator\nG ab (k) =γ g ab µ |k| s G ab (x) = Γ V ( )γ g ab (µ|x|) + (contact terms) .(54)\nWe are primarily interested in small, so that G ab (x) is nearly logarithmic. Informally, we can understand the one-loop correction to the propagator as a contribution to the 1PI effective action coming from all possible Wick contractions of the two of the four factors of φ in S int . The calculation is done most straightforwardly in momentum space, where we can express\nS 4 = − µ 6γ R abcd d n x (φ a φ c )(x) d n y |x − y| n+s (φ b φ d )(y) − (φ b φ d )(x) = − µ 6γΓ V (n + s) R abcd d n x (φ a φ c )(x)(D s φ b φ d )(x) = µ 12γ R abcd d n k ( φ a φ c )(−k)|k| s ( φ b φ d )(k) = µ 12γ R abcd d 4n k δ n 4 i=1 k i φ a (k 1 )φ b (k 2 )φ c (k 3 )φ d (k 4 )|k 2 + k 4 | s ,(55)\nwhere d 4n k = 4 i=1 d n k i . As usual, derivative terms are implied in in the Archimedean 5 A point worthy of remark is that whileγ and γ have the same sign in the ultrametric case, and also in the Archimedean case for 0 < s < 2, for 2 < s < 4 they have the opposite sign. The integral in (52) is well-defined and positive, so to make our theory sensible we should always chooseγ > 0. This means that γ < 0 for 2 < s < 4. As explained in (28) for a single real scalar, the regulated position space integral used to define the action (32) includes a (∂φ) 2 term that enters with the opposite sign of the non-local [φ(x) − φ(y)] 2 term, so positivity conditions are difficult to judge in position space. case when s > 2. Symbolically, the Wick-contracted quartic action is\nS Wick 4 = µ 12γ R abcd d 4n k δ n 4 i=1 k i φ a (k 1 )φ b (k 2 )φ c (k 3 )φ d (k 4 )|k 2 + k 4 | s + (φ aφd contraction) . (56) We understandφ b (k 2 )φ c (k 3 ) to mean a replacement ofφ b (k 2 )φ c (k 3 ) byĜ bc (k 2 )δ n (k 2 +k 3 )\n. We omit theφ aφb andφ cφd contractions from (56) because of the antisymmetry of R abcd in ab and cd. We omit theφ aφc andφ bφd contractions because they include a factor |k 1 +k 3 | s δ n (k 1 +k 3 ), which vanishes when s > 0. After some straightforward algebra, we obtain from (56) the form\nS Wick 4 = − 1 6 R ab d n kφ a (−k)I 2 (k)φ b (k) ,(57)\nwhere I 2 (k) is given in (47).\n\nAs discussed below (47), for suitably small positive = n − s, I 2 (k) includes a term i 2 |k| s Λ iff V = R n and s is a positive even integer. This is the case which leads to local non-linear sigma models. Otherwise-excluding the case V = R n with positive even n-the divergent terms in I 2 (k) are proportional to |k| 2r Λ n−2r for non-negative powers n − 2r which remain finite as → 0 + . Therefore, apart from the case of local non-linear sigma models, the effects of the ultraviolet divergences in (57) are limited to generating relevant, local interactions. We assume that relevant terms of this type can be tuned away.\n\nIt would be tempting at this point to conclude that the non-local action (32) is nonrenormalized, as in the case (2). The reality is more subtle: We will see in section 10 that higher point diagrams generate one loop divergences that require non-local counterterms; however they are not quite of the form (32), involving instead the target space laplacian of Q(X, Y ).\n\n\nCurvature and arc length calculations\n\nHigher point amplitudes in the bi-local non-linear sigma model involve tensors of large rank. We will therefore find it convenient to introduce some abbreviated notation, based on the following equivalences:\nstandard a 1 a 2 ∇ (a 1 ∇ a 2 ) X a 1 X a 2 abbreviated a 12 ∇ a 12 X a 12(58)\nHere (ab) = 1 2 (ab + ba). We employ obvious extensions of (58) to larger index sets, e.g. X a 123 means X a 1 X a 2 X a 3 .\n\nWe will often need to simplify expressions involving the curvature tensor and its covariant derivatives. A primary tool is the Bianchi identities, which we may write using our abbreviated notation as\nR a 1234 + R a 1423 + R a 1342 = 0 ∇ a 5 R a 1234 + ∇ a 1 R a 2534 + ∇ a 2 R a 5134 = 0 .(59)\nA contracted form of the second Bianchi identity,\n∇ b R b a 123 = ∇ a 2 R a 13 − ∇ a 3 R a 12(60)\nshows that any three-index contraction of ∇ a 5 R a 1234 (meaning any contraction leaving three indices free) can be expressed as linear combinations of re-indexed versions of the tensor ∇ a 1 R a 23 ; in this sense ∇ a 1 R a 23 by itself is a basis for all the three-index contractions of ∇ a 1 R a 2345 . This observation will be useful to us when we consider the possible Wick contractions of the five-point interaction vertex in the bi-local non-linear sigma model. Acting on the contracted second Bianchi identity (60) with ∇ b 1 gives\n∇ b 1 b 2 R b 2 a 123 = ∇ b 1 a 2 R a 13 − ∇ b 1 a 3 R a 12 + 1 2 [∇ b 1 , ∇ a 2 ]R a 13 − 1 2 [∇ b 1 , ∇ a 3 ]R a 12 − 1 2 [∇ b 1 , ∇ b 2 ]R b 2 a 123 .(61)\nWe describe the terms in the second line of (61) as commutator terms. Evidently, they can be written as curvature bilinears, meaning contractions of two factors of the Riemann and/or Ricci tensors, with no covariant derivatives. Acting on the uncontracted second Bianchi identity (the second line of (59)) with ∇ a 5 gives\n∇ 2 R a 1234 = ∇ a 13 R a 24 + ∇ a 24 R a 13 − ∇ a 14 R a 23 − ∇ a 23 R a 14 + (commutators) ,(62)\nwhere ∇ 2 = ∇ b ∇ b , and the commutator terms are similar to the ones occurring in (61): In particular, they are curvature bilinears. The results (61) and (62) show that all four-index contractions of ∇ a 56 R a 1234 can be expressed in terms of linear combinations of re-indexed versions of the tensor ∇ a 12 R a 34 , together with curvature bilinears. So far, all formulas in this section have been entirely independent of the choice of coordinate system. We now pass to Riemann normal coordinates in order to study the square of the arc length, Q(X, Y ) = d(X, Y ) 2 between two points X and Y . We have from [10] 6\nQ(X, Y ) = g a 12 (X a 1 − Y a 1 )(X a 2 − Y a 2 ) + r>3 Q r (X, Y ) Q 4 (X, Y ) = − 1 3 R a 1234 X a 13 Y a 24 Q 5 (X, Y ) = − 1 12 ∇ a 5 R a 1234 X a 13 Y a 24 (X a 5 + Y a 5 ) Q 6 (X, Y ) = Q ∇∇R 6 (X, Y ) + Q RR 6 (X, Y ) Q ∇∇R 6 (X, Y ) = − 1 60 ∇ a 56 R a 1234 (X a 1356 Y a 24 + X a 13 Y a 2456 + X a 135 Y a 246 ) Q RR 6 (X, Y ) = 1 45 R b a 123 R ba 456 (4X a 125 Y a 346 − X a 1245 Y a 36 − X a 25 Y a 1346 )(63)\nHere g a 12 , R a 1234 , and its derivatives are all evaluated at the origin of Riemann normal coordinates, which is the origin in terms of the coordinates X a and Y a used in (63).\n\n\nThree-point vertices\n\nThere are no three-point vertices at tree-level provided we employ Riemann normal coordinates. As we will explain in this section, three-point vertices appear to be generated at the one-loop level, by the diagram in figure 2b; however, they can be absorbed through field redefinition.\n\nAs for our discussion in section 6 of the one-loop corrections to the propagator, the one-loop contribution to the three-point function can be obtained efficiently through Wick 6 Note however that the results leading to Q RR contractions in the momentum space of the quintic term in the action:\nS 5 = 1 2γ d n xd n y |x − y| n+s Q 5 (φ(x), φ(y)) = µ 24γ ∇ a 5 R a 1234 5 i=1 d n k iφ a i (k i ) δ n 5 i=1 k i |k 24 | s .(64)\nHigh-dimensional Fourier integrals of the type seen in the second line of (64) are common in our calculations, so we have found it useful to introduce some shorthand notation:\nstandard d n k 1 d n k 2 δ n (k 1 + k 2 ) |k 1 + k 2 |φ a 1 (k 1 )φ a 2 (k 2 ) abbreviated d 2n k 12 δ n (k 12 ) |k 12 |φ a 12 (k 12 )(65)\nwith obvious extensions to larger index sets. If A is any ordered set of indices, like 123, and |A| is the number of indices in the set, then the integrals we see most often are of the form\nJ a A φ,k A [q(k A )] ≡ i∈A d n k iφ a i (k i ) δ n i∈A k i q(k A ) = d |A|n k A δ n (k A )φ a A (k A )q(k A ) ,(66)\nwhere q(k A ) is any function of the k i , and the third expression is just a rewriting of the second using the shorthand notation introduced in (65). Evidently, the J φ integrals are convergent for reasonable integrands q, like powers of norms of sums of momenta, provided the φ a are Schwartz-Bruhat functions. When there is no risk of confusion, we will omit the subscripted k A and just write J a A φ . We can now rewrite (64) as\nS 5 = µ 24γ ∇ a 5 R a 1234 J a 12345 φ [|k 24 | s ] .(67)\nAs in section 6, a Wick contraction amounts to a replacement\nφ a i (k i )φ a j (k j ) →Ĝ a ij (k j )δ n (k ij ) =γ µ g a ij δ n (k ij )|k j | −s .(68)\nIt is helpful to note the following examples of Wick contraction:\nJ a 1 a 2 a A φ,k 12A [|k 12 | s ] = d (|A|+2)n k 12A δ n (k 12A )φ a 1 (k 1 )φ a 2 (k 2 )φ a A (k A )|k 12 | s =γ µ g a 12 d |A|n k A δ n (k A )φ a A (k A ) d 2n k 12 δ n (k 12 )|k 2 | −s |k 12 | s = 0 J a 1 a 2 a A φ,k 12A [|k C | s ] =γ µ g a 12 d |A|n k A δ n (k A )φ a A (k A )|k C | s d 2n k 12 δ n (k 12 )|k 2 | −s =γ µ g a 12 J a A φ,k A [|k C | s ]I 0 = J a 1 a 2 a A φ,k A [|k 12C | s ] J a 1 a 2 a A φ,k 12A [|k 1C | s ] =γ µ g a 12 d |A|n k A δ n (k A )φ a A (k A ) d 2n k 12 δ n (k 12 )|k 2 | −s |k 1C | s =γ µ g a 12 J a A φ,k A [I 2 (k C )] .(69)\nHere A and C are collections of indices, neither including 1 or 2, with C ⊂ A, and we assume s > 0 in order to obtain the vanishing of the first integral. Recall from section 5 that I 0 diverges as Λ / when = n−s is sufficiently small and positive, giving rise to log Λ behavior in the limit → 0 + . There are 5 2 = 10 possible single Wick contractions of S 5 , but the (12) contraction (meaning the contraction of a 1 and a 2 ) vanishes because g a 12 ∇ a 5 R a 1234 = 0; likewise the (34) contraction vanishes. Meanwhile the (14) and (23) \n= 1 24 ∇ a 3 R a 12 J a 123 φ [|k 3 | s ]I 0 S Wick 5,(14) = µ /γ 24 ∇ a 5 R a 1234 J a 1 a 23 a 4 a 5 φ [|k 24 | s ] = − 1 24 ∇ a 5 R a 23 J a 235 φ [I 2 (k 2 )] = − 1 24 ∇ a 1 R a 23 J a 123 φ [I 2 (k 3 )] S Wick 5,(15) = µ /γ 24 ∇ a 5 R a 1234 J a 1 a 234 a 5 φ [|k 24 | s ] = 1 24 (∇ a 3 R a 24 − ∇ a 4 R a 23 )J a 234 φ [|k 24 | s ]I 0 = 1 24 (∇ a 3 R a 12 − ∇ a 1 R a 23 )J a 123 φ [|k 3 | s ]I 0 S Wick 5,(25) = µ /γ 24 ∇ a 5 R a 1234 J a 1 a 2 a 34 a 5 φ [|k 24 | s ] = − 1 24 (∇ a 3 R a 14 − ∇ a 4 R a 13 )J a 134 φ [I 2 (k 4 )] = 1 24 (∇ a 3 R a 12 − ∇ a 1 R a 23 )J a 123 φ [I 2 (k 3 )] .(71)\nA few comments are in order:\n\n• Because S Wick 5, (14) and S Wick 5,(25) are proportional to the I 2 loop integral, they do not contribute logarithmic divergences in the → 0 + limits described in section 5, except when s is a positive even integer. In a position space account, these non-logarithmic terms correspond to contractions of φ(x) with φ(y).\n\n• We are mostly interested in S Wick 5,(13) and S Wick 5,(15) because I 0 does produce a logarithmic divergence in the → 0 + limit. Note that in these terms, the vertex factor |k 13 | s involves only external momenta. This is the crucial feature, noted already in the introduction, which allows non-local counterterms to arise. In position space, the logarithmic terms correspond to contractions of φ(y) with itself.\n\nIn summary,\nS Wick 5 = 1 24 d 3n k 123 δ n (k 123 )φ a 123 (k 123 ) I 2 (k 3 )(2∇ a 3 R a 12 − 4∇ a 1 R a 23 ) + I 0 |k 3 | s (3∇ a 3 R a 12 − 2∇ a 1 R a 23 ) .(72)\nThe minimal counterterm needed to cancel the Λ / divergences in (72) is\nS ct 3 = Λ 2 d 3n k 123 δ n (k 123 )φ a 123 (k 123 )|k 3 | s K 1 ∇ a 3 R a 12 + K 2 ∇ (a 1 R a 2 )a 3(73)\nwhere\nK 1 = − 3i 0 + 2i 2 12 K 2 = i 0 + 2i 2 6 ,(74)\nand i 0 and i 2 are as defined in (46) and (51). We use the notation S ct 3 to denote a counterterm that is cubic in the fields. Our primary interest is in cases where i 2 = 0-namely, cases in which V = Q p n , or V = R n but s is not a positive even integer. However, tracking I 0 , I 2 (k), i 0 , and i 2 throughout our computations is useful as a bookkeeping device in order to simultaneously treat the local and bi-local theories, in the → 0 + limit, with i 2 entering in only to describe the local theories.\n\n\nRenormalization through cubic order in the fields\n\nBefore entering into the more complicated story of quartic terms in the action, let's preview the endgame of our analysis, in which we produce a bare action which incorporates the renormalized action plus the counterterms in a form that we can express entirely in terms of arc length.\n\nFirst, let's rephrase the Wick-contracted quartic action (57) as\nS Wick 4 = − 1 6 d 2n k 12 δ n (k 12 )φ a 12 (k 12 )I 2 (k 2 )R a 12 ,(75)\nfrom which we conclude that we need a counterterm quadratic in the fields of the form\nS ct 2 = Λ 2 d 2n k 12 δ n (k 12 )φ a 12 (k 12 )|k 2 | s K 0 R a 12 ,(76)\nwhere\nK 0 = i 2 3 .(77)\nThe results (76) and (73) together put some constraints on the renormalization procedure, but as we will see they do not completely determine it. The question of renormalizability is whether we can reorganize the renormalized action plus counterterms into a bare action whose form is the same that we started with:\nS[φ] + S ct [φ] = S B [φ B ] .(78)\nwhere\nS ct [φ] = r>1 S ct r(79)\nis the sum of all counterterms, S[φ] is as given in (32), and\nS B [φ B ] = Λ n−s 2γ V ×V d n xd n y |x − y| n+s Q B (φ B (x), φ B (y)) ,(80)\nwhere\nQ B (X B , Y B ) = d B (X B , Y B ) 2 .\nThe bare arc length d B , derived from a bare metric tensor g B ab , may differ from the renormalized arc length d, and the bare coordinates φ a B may likewise differ from the renormalized coordinates φ a . We require however that the points φ B = 0 and φ = 0 coincide. At tree level, where we ignore all counterterms, we have the relations\nΛ µ g B ab (φ B ) = g ab (φ) φ a B = φ a ,(81)\nand our key task is to find perturbative corrections to these relations that render (78) correct.\n\nTo begin, let's examine the quadratic terms in (78), using the counterterm S ct 2 from (76):\nd 2n k 12 δ n (k 12 )φ a 12 (k 12 )|k 2 | s g a 12 +γ Λ µ K 0 R a 12 + O(φ 3 ) + O(γ 2 ) = Λ µ d 2n k 12 δ n (k 12 )φ a 12 B (k 12 )|k 2 | s g B a 12 ,(82)\nwhere for simplicity we multiplied through by 2γ/µ . We surmise from (82) that corrections to (81) can be expressed as a power series in the dimensionless quantitỹ\nγ =γ Λ µ .(83)\nThat is,\nφ a B (x) = φ a (x) +γ U a b φ b (x) + V a b 12 φ b 12 (x) + W a b 123 φ b 123 (x) + O(φ 4 ) + O(γ 2 ) ,(84)\nwhere V a b 12 and W a b 123 are fully symmetric in their lower indices, and U , V , and W are all independent of φ (and φ B ). In other words, (84) is a Taylor expansion of φ a B in the coordinates φ a . Also,\nΛ µ g B ab (φ B ) = g ab (φ) +γT ab (φ) + O(γ 2 )(85)\nfor some tensor T ab (φ). As our notation indicates, T ab (φ) does depend on φ. As with other tensors, if we omit the argument, we mean that T ab is evaluated at φ = 0. Using (84) and\n\n(85), we see that (82) is satisfied provided\ng a 12 +γK 0 R a 12 = (δ b 1 a 1 +γU b 1 a 1 )(δ b 2 a 2 +γU b 2 a 2 )(g b 12 +γT b 12 ) + O(γ 2 ) ,(86)\nor in other words provided\nT a 12 + 2U a (12) = K 0 R a 12 ,(87)\nwhere\nU a 12 = U b a 2 g a 1 b U a (12) = 1 2 (U a 12 + U a 21 ) .(88)\nIt should be kept in mind that T a 12 is the φ = 0 value of a tensor field T a 12 (φ) defined over the whole of M , whereas U a (12) is defined only at φ = 0. Let's assume that\nT a 12 (φ) = t 0 R a 12 (φ) U a (12) = u 0 R a 12 .(89)\n(A term in T a 12 (φ) proportional to R(φ)g a 12 (φ) is also possible, but the divergences we will encounter do not require it.) Then (87) reduces to\nt 0 + 2u 0 = K 0 .(90)\nAs previously noted, based on the treatment of quadratic terms alone, we cannot distinguished between metric renormalization (related to the coefficient t 0 ) and field redefinition (related to the coefficient u 0 ). In order to proceed to higher orders, we require the squared arc length formula for bare quantities:\nQ B (X B , Y B ) = g B a 12 (X a 1 B − Y a 1 B )(X a 2 B − Y a 2 B ) + r>2 Q B r (X B , Y B ) .(91)\nWe do not require φ B to be Riemann normal coordinates for d B , so there are contributions to Q B at cubic order:\nQ B 3 (X B , Y B ) = Γ B a 123 (X a 1 B − Y a 1 B )(X a 2 B − Y a 2 B )(X a 3 B + Y a 3 B ) ,(92)\nwhere Γ B abc = g B ad Γ Bd bc and Γ Ba bc is the Christoffel connection for g B ab . From (85) we have immediately\nΛ µ Γ B a 312 =γ ∇ (a 1 T a 2 )a 3 − 1 2 ∇ a 3 T a 12 + O(γ 2 ) ,(93)\nFrom Q B 3 we obtain a cubic term in the bare action:\nS B 3 = Λ 2γ d n xd n y |x − y| n+s Q B 3 (φ B (x), φ B (y)) = Λ 2γ J a 123 φ B [|k 3 | s ]Γ B a 312 = Λ 2 J a 123 φ [|k 3 | s ] ∇ (a 1 T a 2 )a 3 − 1 2 ∇ a 3 T a 12 + O(γ) .(94)\nAnother term cubic in φ arises in the bare action from plugging the non-linear field redefinition (84) into the quadratic term S B 2 . To work this out, it helps first to note that passing (84) through a Fourier transform yieldŝ\nφ a B (k) =φ a (k) +γδφ a (k) + O(γ 2 ) (95) where δφ a (k) = U a bφ b (k) + V a b 12 (φ b 1 * φ b 2 )(k) + W a b 123 (φ b 1 * φ b 2 * φ b 3 )(k) + O(φ 4 )(96)\nand * denotes convolution. It follows immediately that\nJ a 12 φ B [|k 2 | s ] = J a 12 φ [|k 2 | s ] + 2γ d 2n k 12 δ n (k 12 )δφ (a 1 (k 1 )φ a 2 ) (k 2 )|k 2 | s + O(φ 5 ) + O(γ 2 ) = J a 12 φ [|k 2 | s ] + 2γ U (a 1 b J a 2 )b φ,k 2 [|k 2 | s ] + V (a 1 b 12 J a 2 )b 12 φ,k 2 12 [|k 2 | s ] + W (a 1 b 123 J a 2 )b 123 φ,k 2 123 [|k 2 | s ] + O(φ 5 ) + O(γ 2 ) ,(97)\nand so\nS B 2 = Λ 2γ J a 12 φ B [|k 2 | s ]g B a 12 = S 2 + Λ 2 J a 12 φ [|k 2 | s ] [T a 12 + 2U a 12 ] + Λ J a 123 φ [|k 3 | s ]V a 312 + Λ J a 1234 φ [|k 4 | s ]W a 4123 + O(φ 5 ) + O(γ) ,(98)\nwhere we are lowering indices on V and W with the renormalized metric g ab . The T a 12 +2U a 12 term in (98) is the same combination we saw in (87), with the symmetrization U a 12 → U a (12) implied because we multiply by J a 12 φ [|k 2 | s ], which is symmetric. The next term in (98) is the interesting one for us. The only constraint on V a 312 is symmetry in the 12 indices. This is the same symmetry that J a 123 ically, we expand (78) to cubic order in the renormalized fields, using the expression (73) for S ct 3 , as well as S B 3 from (94) and the O(φ 3 ) term from (98). The result is\nJ a 123 φ [|k 3 | s ] K 1 ∇ a 3 R a 12 + K 2 ∇ (a 1 R a 2 )a 3 = J a 123 φ [|k 3 | s ] ∇ (a 1 T a 2 )a 3 − 1 2 ∇ a 3 T a 12 + 2V a 312 .(99)\nEvidently, we may set\nV a 312 = v 1 ∇ a 3 R a 12 + v 2 ∇ (a 1 R a 2 )a 3 ,(100)\nwhere\n− 1 2 t 0 + 2v 1 = K 1 t 0 + 2v 2 = K 2 .(101)\nThe larger message is that V a 312 is sufficiently unconstrained that we could use it to absorb any counterterm proportional to J a 123 φ [|k 3 | s ]. By the same token, when we get to quartic order, the field redefinition coefficients W a 4123 can be used to absorb any terms proportional to J a 1234 φ [|k 4 | s ]. Therefore, when we do proceed to quartic order, we may simplify our work by systematically dropping all such terms. We will see in section 10 that other terms emerge, proportional to J a 1234 φ [|k 24 | s ]. These are the ones that cannot be absorbed into field redefinitions.\n\n\nQuartic counterterms\n\nFour-point vertices are present at tree-level, and they are also generated by three different types of one-loop diagrams, as shown in figure 3. Our goal in this section is to evaluate one-loop divergences proportional to ∇ a 12 R a 34 (and re-indexings of it). The claim is that only the diagram in figure 3a can contribute. Tracking only target space indices, the vertex factor in this diagram is ∇ a 56 R a 1234 , and the internal propagator can tie any two of the six indices together. So the diagram as a whole must be proportional to four-index contractions of ∇ a 56 R a 1234 . As explained in section 7, re-indexed versions of ∇ a 12 R a 34 , together with cur-vature bilinears, provide a basis for such contractions. The diagrams in figure 3b and 3c are manifestly proportional to curvature bilinears, so they cannot contribute terms proportional to ∇ a 12 R a 34 . (One immediate way to see this is that the Riemann tensor could vanish at φ = 0 without its derivatives vanishing.) In the explicit calculations below, we will encounter and discard many curvature bilinear terms, which we generically write as O(RR), meaning some contraction of R a 1234 R a 5678 .\n\nWe claim that the counterterms needed to cancel the divergences from the diagram in figure 3a proportional to ∇ a 12 R a 34 (and re-indexings of it) are\nS ct 6,∇∇R = Λ 2 J a 1234 φ [|k 24 | s ] (K 3 ∇ a 12 R a 34 + K 4 ∇ a 13 R a 24 ) + J a 1234 φ [|k 4 | s ] (K 5 ∇ a 12 R a 34 + K 6 ∇ a 34 R a 12 ) ,(102)\nwhere\nK 3 = 3i 0 + 2i 2 30 K 4 = − 7i 0 + 3i 2 60 K 5 = i 0 + 3i 2 30 K 6 = − 3i 0 + 4i 2 60 .(103)\nColor-coding in (102) is to help track to which term in (102) each of the many terms in later equations contribute. The remainder of this section is devoted to deriving (103).\n\nTo derive the sixth-order vertex used in figure 3a, we start from the Q ∇∇R 6 term in (63) and extract the following six-order terms in the action:\nS 6,∇∇R = µ 2γ d n xd n y |x − y| n+s Q ∇∇R 6 (φ(x), φ(y)) = S 4+2 6,∇∇R + S 3+3 6,∇∇R(104)\nwhere\nS 4+2 6,∇∇R = µ 120γ ∇ a 56 R a 1234 J a 123456 φ [|k 24 | s ] S 3+3 6,∇∇R = µ 240γ ∇ a 56 R a 1234 J a 123456 φ [|k 246 | s ] .(105)\nOur task is to compute the counterterms for all the single Wick contractions of S 4+2 6,∇∇R and S 3+3 6,∇∇R . For S 4+2 6,∇∇R , relations among Wick contractions that are obvious from symmetries plus the first line of (69) are as follows: \n\nIf we write the counterterm for an expression Q as {Q} ct , then the rules of computation we need are a trivial adaptation of (69):\nJ a 1 a 2 a A φ [|k C | s ] ct = J a 1 a 2 a A φ [|k 12C | s ] ct = −γg a 12 J a A φ [|k C | s ]i 0 J a 1 a 2 a A φ [|k 1C | s ] ct = −γg a 12 J a A φ [|k C | s ]i 2 .(108)\nThe counterterms that we need to cancel divergences coming from the single Wick contractions of S 4+2 6,∇∇R , as shown in (107) \n\nwhere \nI 4+2,ct 6,∇∇R,(13) = 1 γ ∇ a 56 R a 1234 J a 1 a 2 a 3 a 456 φ [|k 24 | s ] ct = −∇ a 56 R a 24 J a 2456 φ [|k 24 | s ]i 0 = −∇ a 13 R act = −∇ ba 6 R b a 234 J a 2346 φ [|k 24 | s ]i 0 = (−∇ a 63 R a 24 + ∇ a 64 R a 23 )J a 2346 φ [|k 24 | s ]i 0 + O(RR) = (∇ a 12 R a 34 − ∇ a 13 R a 24 )J a 1234 φ [|k 24 | s ]i 0 + O(RR) I 4+2,ct 6,∇∇R,(25) = 1 γ ∇ a 56 R a 1234 J a 1 a 2 a 34 a 5 a 6 φ [|k 24 | s ] ct = −∇ ba 6 R a 1 b a 34 J a 1346 φ [|k 4 | s ]i 2 = (∇ a 63 R a 14 − ∇ a 64 R a 13 )J a 1346 φ [|k 4 | s ]i 2 + O(RR) = (∇ a 12 R a 34 − ∇ a 34 R a 12 )J a 1234 φ [|k 4 | s ]i 2 + O(RR) I 4+2,ct 6,∇∇R,(56) = 1 γ ∇ a 56 R a 1234 J a 1234 a 5 a 6 φ [|k 24 | s ] ct = −∇ 2 R a 1234 J a 1234 φ [|k 24 | s ]i 0 = (−2∇ a 13 R a 24 + 2∇ a 12 R a 34 )J a 1234 φ [|k 24 | s ]i 0 + O(RR) ,(110)\nwhere the color coding is to show which terms in (110) contribute to which terms of (102). By direct computation using the rules (108), \nI 3+3,ct 6,∇∇R,(13) = 1 γ ∇ a 56 R a 1234 J a 1 a 2 a 3 a 456 φ [|k 246 | s ] ct = −∇ a 56 R a 24 J a 2456 φ [|k 246 | s ]i 0 = −∇ a 34 R a 12 J a 1234 φ [|k 4 | s ]i 0 I 3+3,ct 6,∇∇R,(14) = 1 γ ∇ a 56 R a 1234 J a 1 a 23 a 4 a 56 φ [|k 246 | s ] ct = ∇ a 56 R a 23 J a 2356 φ [|k 26 | s ]i 2 = ∇ a 12 R a 34 J a 1234 φ [|k 24 | s ]i 2 I 3+3,ct 6,∇∇R,(15) = 1 γ ∇ a 56 R a 1234 J a 1 a 234 a 5 a 6 φ [|k 246 | s ] ct = −∇ ba 6 R b a 234 J a 2346 φ [|k 246 | s ]i 0 = (∇ a 12 R a 34 − ∇ a 34 RaK 3 = 4 60 i 0 + 2 60 i 0 + 2 120 i 2 + 4 120 i 2 + 2 120 i 2 K 4 = − 1 60 i 0 − 4 60 i 0 − 2 60 i 0 − 4 120 i 2 − 2 120 i 2 K 5 = 2 60 i 2 + 4 60 i 2 + 4 120 i 0 K 6 = − 4 60 i 2 − 2 120 i 0 − 4 120 i 0 ,(114)\nwhich agrees with (103).\n\n\nRenormalization at quartic order\n\nTo renormalize at quartic order, we first inquire whether the counterterms (102) can be organized into the bare arc length action (80), using the field redefinition (84) and the relationship (85) between the bare and renormalized metric. For the non-local model, we will find that this is impossible! So we will turn to a generalized form of the bare action that includes a term proportional to the target space laplacian of the square of the arc length.\n\nTo get started, we need the bare arc length formula to quartic order in the bare fields: That is, we need one more term in the series (91) than we computed in section 9. The computation of arc length is less simple than for the renormalized metric because the φ a B are not Riemann normal coordinates for g B ab -due to effects at O(γ), in particular a connection Γ Ba b 12 ∼ O(γ). As a technical device, we therefore introduce a third set of coordinates, φ a , which are Riemann normal coordinates for the bare metric, which in barred coordinates takes the form g ab (φ). We can express φ a in terms of φ a B as\nφ a = φ a B +γ L a b 12 φ b 12 B + M a b 123 φ b 123 B + O(φ 4 B ) + O(γ 2 ) ,(115)\nand we can write g B ab (φ B ) in terms of g ab (φ) as\ng B b 12 (φ B ) = g a 12 (φ) ∂φ a 1 ∂φ b 1 B ∂φ a 2 ∂φ b 2 B .(116)\nNote that, by assumption, g B a 12 = g a 12 at φ B = φ = 0. The condition that φ a are Riemann normal coordinates allows us to concludẽ\nγL a b 12 = 1 2 Γ Ba b 12γ M a b 123 = 1 6 ∂ ∂φ (b 1 B Γ Ba b 23 ) + Γ Ba b(b 1 Γ Bb b 23 ) ;(117)\nsee for example [11] for a derivation. The Γ B Γ B term in the expression (117) for M a b 123 is optional because it is O(γ 2 ), but it arises naturally in the derivation of [11], so we include it.\n\nThe bare arc length coincides between φ and φ B coordinate systems because these are just different coordinate systems for the same metric, as per (116). Explicitly,\nQ B (X B , Y B ) = Q(X, Y ) = g a 12 (X a 1 − Y a 1 )(X a 2 − Y a 2 ) − 1 3 R a 1234 X a 13 Y a 24 + O(φ 5 ) ,(118)\nwhere in the second equality we used the fact that φ a are Riemann normal coordinates. The (118) is short for all terms involving five or more powers of X and Y combined; similar notation is used below. Using the first equation in (115) to eliminate X and Y in favor of X B and Y B , we arrive at\nnotation O(φ 5 ) inQ B (X B ,Y B ) = g B a 12 (X a 1 B − Y a 1 B )(X a 2 B − Y a 2 B ) + 2γ [L a 312 (X a 12 B − Y a 12 B )(X a 3 B − Y a 3 B ) + M a 4123 (X a 123 B − Y a 123 B )(X a 4 B − Y a 4 B )] − 1 3 R B a 1234 X a 13 B Y a 24 B + O(φ 5 B ) + O(γ 2 ) ,(119)\nwhere L a 312 = g B a 3 b L b a 12 and M a 4123 = g B a 4 b M b a 123 . Note that the cubic terms in (119) agree with (92), and recall from the subsequent analysis that the corresponding cubic term S B 3 does not need to match S 3 + S ct 3 , because of the additional term cubic in φ in (98) arising from the O(γ) difference between φ B and φ. Likewise, the term in (119) proportional to M a 4123 gives rise to a term proportional to J φ B [|k 4 | s ] in the action, but we do not need to track it explicitly because the quartic term in (98) shows that it is precisely the sort of term that we can absorb into the field redefinition coefficient W a b 123 . Thus we may write\nS B 4 = Λ 2γ d n xd n y |x − y| n+s Q B 4 (φ B (x), φ B (y)) = Λ 12γ J a 1234 φ B [|k 24 | s ]R B a 1234 + (field redef) ,(120)\nwhere (field redef) indicates field redefinition terms as discussed above. Next we express S B 4 in terms of renormalized quantities in order to compare to (102). Starting from (85), we obtain Λ µ R B a 1234 = R a 1234 +γ 2 (∇ a 14 T a 23 − ∇ a 24 T a 13 − ∇ a 13 T a 24 + ∇ a 23 T a 14 )\n+ O(RR) + O(γ 2 ) .(121)\nThus we find\nS B 4 = S 4 + Λ 12 J a 1234 φ [|k 24 | s ] [t 0 ∇ a 12 R a 34 − t 0 ∇ a 13 R a 24 ] + (field redef) + O(RR) + O(γ) .(122)\nO(RR) terms arise in (122) not just from those in (121), but also from expressing J a 1234 φ B\n\n[|k 24 | s ] in terms of the renormalized field φ. 7 We have color-coded terms in (122) to match the way we did in (102). Comparing the two equations, we can see that S B 4 accommodates the counterterms S ct 6,∇∇R iff K 3 = −K 4 . Based on (103), this happens iff i 0 = i 2 , which means iff s is a positive even integer. As we will develop more fully in section 12, this indeed corresponds to the case of local non-linear sigma models.\n\nLet us pursue further here what happens for the non-local case. Because i 2 = 0, we have K 1 /K 2 = −6/7, there is no hope of rendering the theory renormalizable with just the arc length action we have been using so far. Some generalization of the arc length action is needed. Whatever modification we make should involve two target space derivatives relative to the original action, so as to absorb counterterms that appear with two extra derivativeslike the ∇ a 12 R a 34 structure in S ct 6,∇∇R as compared to R a 1234 in S 4 . Luckily, there is a new term with the right properties which we can add to S B :\nδS B = κ B Λ 2γ d n xd n y |x − y| n+s Q B (φ B (x), φ B (y))(123)\nwhere we define\nQ B (X B , Y B ) ≡ (∇ 2 X B + ∇ 2 Y B )Q B (X B , Y B ) − 4n .(124)\nThe −4n term in (124) is present in order to ensure Q B (X B , Y B ) = 0 when X B and Y B coincide. 8 By explicit calculation (as sketched below (132)),\nδS B = κ B Λ 60γ (−6∇ B a 12 R B a 34 + 7∇ B a 13 R B a 24 ) × d n xd n y |x − y| n+s [φ a 13 B (x) − φ a 13 B (y)] [φ a 24 B (x) − φ a 24 B (y)] + (field redef) + O(φ 5 B ) + O(RR) ,(125)\nPassing to momentum space, we find\nδS B = κ B Λ 60γ (−6∇ B a 12 R B a 34 + 7∇ B a 13 R B a 24 )J a 1234 φ B [|k 24 | s ] + (field redef) + O(φ 5 B ) + O(RR) .(126)\nCombining (122) and (126) and comparing with (102), we see that κ B = −γ i 0 2 . Strikingly, we are forced also to choose t 0 = 0: That is, the metric is not renormalized! Having allowed the two-derivative term (123) in S B , we should allow addition of a similar term to the renormalized action:\nδS = κ µ 2γ d n xd n y |x − y| n+s Q (φ(x), φ(y)) .(127)\nWe restrict κ to be an O(γ) quantity, which makes sense because then the overall scaling with γ of (127) is O(γ 0 ), and this aligns with the invariance of Q (X, Y ) under overall rescaling of the target manifold. The additional term (127) changes all the one-loop amplitudes, but only by O(γ) quantities, relative to the O(γ 0 ) scaling of one-loop amplitudes and counterterms that we obtained in previous sections. In short, the only effect of allowing non-zero κ in our counterterm analysis is to lead to a direct tree-level contribution to κ B , so that in total,\nΛ µ κ B = κ −γ i 0 2 .(128)\nTo rephrase this result in terms of the renormalization group, we can rewrite (128) as\nΛ κ B +γ i 0 2 = µ κ ,(129)\nand then since the right hand side is independent of Λ, we arrive at\nΛ dκ B dΛ = − κ B −γ i 0 2 .(130)\nThe first term on the right hand side of (130) is the tree-level term coming from the engineering dimension factor of (Λ/µ) in (128). The one-loop effects are responsible for the second term in (130). If we now take → 0 in (130), we see that κ B runs logarithmically:\nκ B = −γ i 0 2 log Λ Λ 0 ,(131)\nwhere Λ 0 is a dynamically generated scale. Note thatγ and i 0 are positive, so κ B is positive at scales Λ below Λ 0 and negative above Λ 0 . To see that (125) is correct, let's work on the renormalized side and note that\ng b 12 ∂ 2 ∂X b 1 ∂X b 2 Q ∇∇R 6 = − 1 60 ∇ a 56 R a 1234 Y a 24 g b 12 ∂ 2 X a 1356 ∂X b 1 ∂X b 2 + . . . = − 1 30 (∇ a 13 R a 24 + 4∇ a 1 b R b a 234 + ∇ 2 R a 1234 )X a 13 Y a 24 + . . . = − 1 30 (6∇ a 13 R a 24 − 6∇ a 12 R a 34 + ∇ a 24 R a 13 )X a 13 Y a 24 + . . . .(132)\nThe expression (132) is part of ∇ 2 X Q(X, Y ), and it is easy to see that it is the only part contributing terms of the form (∇∇R)XXY Y . Therefore\nQ (X, Y ) = − 1 30 (−6∇ a 12 R a 34 + 7∇ a 13 R a 24 )(X a 13 Y a 24 + X a 24 Y a 13 ) + O(φ 5 ) + O(RR) + . . . .(133)\nThe ellipses in (132) and (133) indicate terms that are not quadratic in both X and Y , for example terms schematically of the form (∇∇R)XY Y Y or (∇∇R)Y Y Y Y , as well as lower order terms which are either independent of X or Y , or linear in X or Y . Plugging (133) into (127), we find\nδS = −κ µ 60γ (−6∇ a 12 R a 34 + 7∇ a 13 R a 24 ) × d n xd n y |x − y| n+s [φ a 13 (x)φ a 24 (y) + φ a 24 (x)φ a 13 (y)] + (field redef) + O(φ 5 ) + O(RR) = κ µ 60γ (−6∇ a 12 R a 34 + 7∇ a 13 R a 24 ) × d n xd n y |x − y| n+s [φ a 13 (x) − φ a 13 (y)] [φ a 24 (x) − φ a 24 (y)] + (field redef) + O(φ 5 ) + O(RR) .(134)\nThe contributions labeled (field redef) in (134) are linear in φ(x) or φ(y). To see that (134) is correct, we have only to understand why we can freely add or drop from the integrand smooth functions which depend only on x or only on y, such as the direct terms φ a 1234 (x) + φ a 1234 (y) which are present in the last expression in (134) but not the middle expression. As in section 3, this follows from careful use of the regulated integral prescription:\nd n xd n y |x − y| n+s f (y) = d n xd n y |x − y| n+s f (y) − s/2 r=0 b r r f (x)(y − x) 2r = 1 Γ V (n + s) d n x D s f (x) = 0(135)\nfor smooth functions f (x) with suitable falloff conditions at large x. The integral d n xd n y |x−y| n+s f (x) vanishes more trivially by subtraction of the r = 0 term in the sum appearing in square brackets in (135).\n\n\nRecovering the local non-linear sigma model\n\nWhen V = R n and s is a positive even integer, the original position space action (32) is problematic, because our scheme for defining regulated integrals breaks down. However, this is precisely the case where we expect to recover a local theory, expressible in terms of φ(x) and its derivatives. In this section we will verify this expectation for the simplest case, s = 2. We proceed in three steps: First, we start from the momentum space form of the action, obtained for general s from the non-local action (32); then we take s = 2; and finally we Fourier transform back to a local position-space action. It is helpful (though not really necessary) to work with Riemann normal coordinates throughout. We exhibit the process in detail for the quadratic and quartic terms in the action. The first step is to put (52) and (55) together to get\nS 2 + S 4 = µ 2γ g a 12 d n kφ a 1 (−k)|k| sφa 2 (k) + µ 12γ R a 1234 d n k ( φ a 1 φ a 3 )(−k)|k| s ( φ a 2 φ a 4 )(k) .(136)\nHaving set s = 2, we pass to position space, keeping in mind that in our conventions,\nk 2 → − 1 4π 2 .\nThe result is\nS 2 + S 4 = µ 8π 2γ d n x g a 12 ∂ x i φ a 1 (x)∂ x i φ a 2 (x) + 1 6 R a 1234 ∂ x i (φ a 1 (x)φ a 3 (x))∂ x i (φ a 2 (x)φ a 4 (x)) ,(137)\nwhere integration by parts was allowed without generating derivatives of g a 12 or R a 1234 because these tensors are evaluated at φ = 0. The result (137) is to be compared with the standard local sigma model action, expanded to quartic order in fields:\nS local = µ 2σ d n x g a 12 (φ(x))∂ x i φ a 1 (x)∂ x i φ a 2 (x) = µ 2σ d n x g a 12 − 1 3 R a 1324 φ a 3 (x)φ a 4 (x) + O(φ 3 ) ∂ x i φ a 1 (x)∂ x i φ a 2 (x) .(138)\nIt is easy to check that (137) and (138) agree provided σ = 4π 2γ . Note that γ = − 2 Γ R n (n+2)γ has a simple pole at s = 2. This is a reminder that the original action (2) is ill-defined when s = 2. However, the s → 2 limit is well-defined and smooth in Fourier space, withγ held fixed and positive.\n\nIt is straightforward but unilluminating to extend the comparison of (137) and (138) through O(φ 6 ); suffice it to say that one does recover S local order by order in φ through the procedure outlined in the previous paragraph.\n\nLet's now use the results of previous sections to recover the well-known beta function for the local non-linear sigma model on V = R n with s = 2 and n = 2 + , for positive and sufficiently small. Comparing the first line of (102) with the first line of (122), we conclude that the metric renormalization coefficient t 0 must take the value\nt 0 = 6K 3 = i 0 = 2 ζ ∞ (n) = 2π ,(139)\nwhere we used i 2 = i 0 from (51) together with (103) to conclude that K 3 = −K 4 = i 0 /6; then in the third equality of (139) we used (46) to obtain the explicit value of i 0 in dimension n = 2 + . In (139), we are only tracking i 0 and i 2 up to O( ) contributions, consistent with our aim of calculating only the leading behavior in of the one-loop beta function. It is worth noting that in (139), we are extracting t 0 exclusively from the four-point amplitude.\n\nThe two-point amplitude (i.e. the one-loop correction to the propagator) doesn't help at all in determining t 0 ; instead, it results in the condition (90) which determines the field renormalization coefficient u 0 once t 0 is known-specifically, u 0 = −i 0 /3. An important point is that we are able to completely cancel one-loop divergences with metric renormalization (together with field redefinition), consistent with the local non-linear sigma model being renormalizable.\n\nTo go further and extract the beta function for the metric, we start with a rearrangement of (85):\nΛ g B ab (φ B ) −γ t 0 R B ab (φ B ) = µ g ab (φ) + O(γ 2 ) .(140)\nLet's set φ = φ B = 0 and differentiate both sides with respect to Λ, holding renormalized quantities fixed. The result is\nΛ dg B ab dΛ = − g B ab +γt 0 R B ab .(141)\nUsingγ = σ/4π 2 and i 0 = 2π, we arrive at\nΛ dg B ab dΛ = − g B ab + σ 2π R B ab .(142)\nThis accords with the standard one-loop result as quoted for example in [6]. 9 For positive even s > 2, the action of a local, s-derivative non-linear sigma model does not appear to be uniquely determined by symmetries (though it is possible we do not fully understand how diffeomorphism symmetry is implemented in these higher derivative theories). The simplicity of our loop calculations for all positive even s suggests that there 9 When using the results of [6], one must keep in mind that the parameter T (analogous to our σ) appearing in the beta function β ij (T −1 g) = − T −1 g ij + R ij must be replaced by T /2π, so that β ij (g) = − g ij + T 2π R ij , in order to be consistent with the action S = Λ dx 1 2 T −1 g ij (φ(x))∂ µ φ i (x)∂ µ φ j (x); see the comment in this regard on p. 388. may be a privileged local non-linear sigma model in each positive even dimension whose renormalization group flow is characterized by (141).\n\n\nOutlook\n\nLet's start with a recapitulation of the main points of our analysis. The starting point action is\nS = µ 2γ xy Q(φ(x), φ(y)) ,(143)\nwhere Q(X, Y ) = d(X, Y ) 2 is the square of the shortest distance between points X and Y on the target manifold, and we understand that\nxy G(x, y) = V ×V d n xd n y |x − y| n+s G(x, y)(144)\nis defined with a suitable regulation prescription, as in section 3. Recall that = n − s.\n\nFocusing on the limit → 0 + , with n and s converging to some positive n 0 which is not an even integer when V = R n , we find that we are obliged to generalize the action (143) to\nS improved = µ 2γ xy [Q(φ(x), φ(y)) + κQ (φ(x), φ(y))] ,(145)\nwhere Q (X, Y ) = (∇ 2 X + ∇ 2 Y )Q(X, Y ), and κ ∼ O(γ). With this improved action, one-loop amplitudes at O(γ 0 ) have a divergence structure which, as far as we have taken the computations, can be absorbed entirely through field redefinitions and additive renormalization of κ, as given in the form of a renormalization group equation for the bare version of κ in (130). No metric renormalization arises in the non-local model (at one loop). This is in contrast with the local non-linear sigma model, where no improvement terms are needed, and we cancel one-loop divergences instead through field redefinitions and renormalization of the metric.\n\nThe one-loop divergences we encountered are all proportional to the Ricci tensor-more precisely, to covariant derivatives of it. So all of them vanish when R ab = 0. However, we did not analyze diagrams proportional to the square of the Riemann tensor. In the local case, the sum of the divergences of all diagrams (at one loop) vanishes when R ab = 0. It would seem sensible to find the same outcome for the non-local case, but settling this question is left for future work.\n\nIn the non-local theory (143), we expect that additional improvement terms will be needed at each new order in γ, corresponding to higher order target space derivatives of Q(X, Y ). This would be somewhat analogous to the situation in chiral perturbation theory: At each order in a derivative expansion one has finitely many parameters to adjust, but every new order introduces new free parameters. However, an alternative viewpoint may be possible: Instead of renormalizing individual parameters like κ, it may be possible to renormalize Q(X, Y ) as a bi-local function, regarding the improvement term κQ (X, Y ) as an additive renormalization of Q(X, Y ). To find a beta function for Q even at the one-loop level, we would need full control over all field redefinitions, because Λ d dΛ Q B (X B , Y B ) should be computed with the renormalized X and Y held fixed, and that means X B and Y B will have some Λ dependence. If this alternative viewpoint is indeed viable, then the relation Q(X, Y ) = d(X, Y ) 2 would be only an initial condition that we might impose at some scale. Some technical aspects of our analysis might have to change to accommodate a function Q(X, Y ) that is not locked to the square of the distance function; in particular, Q 3 (X, Y ) may not vanish in any convenient coordinate system.\n\nThere is a reason already at the level of the classical action to consider relaxing the relation Q(X, Y ) = d(X, Y ) 2 . Namely, d(X, Y ) 2 is in general not a globally smooth function, though it is certainly smooth when X and Y are sufficiently close to one another. Consider for example making the target space manifold M a circle parametrized by an angle θ ∈ R mod 2π. Then d(0, θ) 2 = θ 2 for θ ∈ [−π, π], but right at π one finds a discontinuity in the first derivative of d(0, θ) 2 , so that e.g. d(0, θ) = (θ − 2π) 2 for θ ∈ [π, 3π]. This example makes it seem like failure of differentiability of Q(X, Y ) is associated with a homology cycle, but that is not necessarily the case. For example, consider first a cylinder, M = S 1 × R. Clearly, d(X, Y ) 2 again has a discontinuity in its first derivative when X and Y are diametrically opposite points. Now consider an asymptotically flat surface which has a sufficiently long cylindrical region so that the same discontinuity in the first derivative of d(X, Y ) 2 shows up for diametrically opposite points in the cylindrical region. The upshot is that if an asymptotically flat, smooth, contractible surface has a sufficiently prominent bulge, d(X, Y ) 2 will not be a smooth function. See figure 4. But-we speculate-renormalization group flow of Q(X, Y ) might smooth away the raggedness of d(X, Y ) 2 without entirely losing the notion of a useful distance function between points on M .\n\nThere are some obvious extensions of our work which do not depend on renormalizing Q(X, Y ) as a function. First, it would be desirable to work out all the O(RR) terms in oneloop renormalization through four-point amplitudes. From a diagrammatic point of view, it seems natural that (∇ 2 X + ∇ 2 Y )Q(X, Y ) should compensate for divergences arising from tying two legs of a vertex with a single propagator. This is because when the derivatives act as ∂ 2 X or ∂ 2 Y , they eliminate two factors of X from a vertex operator, or two factors of Y , just like a Wick contraction. But this intuition leaves out the covariance of ∇ X and ∇ Y , and it doesn't Figure 4: A surface with a bulge which is cylindrical over some sufficient distance and then closes off. The distance function d(X, Y ) 2 on this surface will have discontinuities in its first derivative.\n\nexplain how to compensate for divergences from the type of diagram shown in figure 3c. We hope that some improved understanding is possible that will make it obvious that the detailed calculations in section 10 had to lead to counterterms that can be assembled into Q (X, Y ). Perhaps then it will be easier to track field redefinition terms, O(RR) terms, and even higher loop counterterms. Another obvious extension is to consider more carefully the local limit. As commented in section 12, we have made no attempt to work out local theories with more than two derivatives, though we note that at least the four-derivative O(N ) model in n = 4 dimensions is a long-established theory [12]. We restricted attention to a particular way of taking the → 0 + limit for local theories, namely by setting s exactly equal to a positive even integer and then sliding n = s + down toward s. Also, we have not considered possible competition between local and non local theories. For example, if s < 2 then a local, s = 2, action is classically irrelevant and one would expect that it is not generated along the renormalization group flow. Borrowing from the analysis of [1,13,14] (see also, e.g., [2,15] for a modern perspective) it is possible that non local theories with s < 2 will flow to local theories in the infrared due to quantum corrections.\n\nFinally, we hope that there may be some applicability of non-local theories to a fundamental issue in M-theory, namely the description of M2-branes far from the limit of nearly flat brane configurations. The M2-brane appears to be afflicted by a pathology which is ultraviolet with respect to the brane worldvolume but infrared from the point of view of spacetime [16,17]: It can throw out long thin tendrils with essentially zero action. We speculate that this issue might find its cure through a modification of the M2-brane in the\n\nFigure 1 :\n1(a) One-loop contribution to the 1PI two-point function. (b) A diagram where the loop consists of a single propagator closing on itself.\n\nFigure 2 :\n2(a) The one-loop contribution to the propagator. (b) The one-loop contribution to the 1PI three-point vertex.\n\n\ncontain errors. In particular, 44 should have been 4 in the first line of (11.24).\n\n\ncontractions are equal because both ∇ a 5 R a 1234 and |k 24 | s are symmetrical under the simultaneous exchange of 1 ↔ 3 and 2 ↔ 4. For the same reason, the (15) and (35) contractions are equal, and so are the (25) and (45) contractions. Finally, the (24) contraction vanishes because of the first line of (69).\n\nFigure 3 :\n3φ [|k 3 | s ] possesses. Therefore J a 123 φ [|k 3 | s ]V a 312 is the most general linear combination of terms coming from J a 123 φ [|k 3 | s ] integrals. Likewise, the only constraint on W a 4123 is symmetry in 123, so the last term shown explicitly in (98) is the most general linear combination of terms coming from J a 1234 φ [|k 4 | s ] integrals. We now have all the ingredients needed to calculate the O(φ 3 ) correction to (82). Specif-One-loop contributions to the 1PI four-point vertex: (a) Single Wick contractions of the ∇∇R six-point vertices; (b) Single Wick contractions of RR sixpoint vertices; (c) Diagrams involving only four-point vertices.\n\n\n) = (23) (15) = (35) = (16) = (36) (25) = (45) = (26) = (46) .\n\n\n∇∇R,(13) + 2I 3+3,ct 6,∇∇R,(14) + 4I 3+3,ct 6,∇∇R,(15) + 4I 3+3,ct 6,∇∇R,(25) + I 3+3,ct 6,∇∇R,(56) . (112)\n\n\n56 R a 1234 J a 1 a 2 a 34 a 5 a 6 φ [|k 246 | s ] ct = −∇ ba 6 R a 1 b a 34 J a 1346 φ [|k 46 | s ]i 2 = (∇ a 12 R a 34 − ∇ a 13 R a 24 )J a 1234 φ [|k 24 | s ]i 2 + O(RR) I 3+3,ct 6,∇∇R,(56) = 1 γ ∇ a 56 R a 1234 J a 1234 a 5 a 6 φ [|k 246 | s ] ct = −∇ 2 R a 1234 J a 1234 φ [|k 24 | s ]i 2 = (−2∇ a 13 R a 24 + 2∇ a 12 R a 34 )J a 1234 φ [|k 24 | s ]i 2 + O(RR) .\n\n\n109)-(110) and (112)-(113) together and comparing with (102), we arrive at\nAn exception, as we will see, is when s is an even integer and the base space V = R n . Through a procedure we will outline in section 12, one recovers in this case a local non-linear sigma model, and at least for s = 2 we can use our results to check the standard analysis[6] of the one-loop beta function.\nThe alert reader may notice that the alternative approach using cutoffs is not quite equivalent to adjusting G(x, y) by pure powers of |x − y|: For example, if s is a positive even integer and G(x, y) = |x − y| s , then we get a logarithmic divergence that would obviously be canceled using an appropriate type I function but\nLoop divergences in momentum spaceOur aim in this section is to introduce the main concepts related to divergent loop diagrams that we will need in our analysis of the non-local non-linear sigma model. As a warmup, we first exhibit the simplest manifestation of the non-renormalization theorem of the non-local\nAlthough we have argued that the hard cutoff prescription | | ≤ Λ is the natural one to use, it is interesting to note that if instead we impose |k − | ≤ Λ, then still c n/2 = i 0 + O( ) when s = 2 n/2 and is sufficiently small.\ndoes not contribute terms quartic in φ that we need to track because it starts at O(γ 0 ), so quartic terms coming from expressing J a123 φB [|k 3 | s ] in terms of the renormalized φ enter at O(γ).\nStrictly speaking we do not need the −4n subtraction in (124) when we used a regulated integral in (123), so including it explicitly in (124) is a matter of taste.\nAcknowledgmentsThis work was supported in part by the Department of Energy under Grant No. DE-FG02-91ER40671, and by the Simons Foundation, Grant 511167 (SSG). In addition, AY is supported in part by an Israeli Science Foundation excellence center grant 2289/18 and a Binational Science Foundation grant 2016324. We thank C. Callan, D. Gross, M. Heydeman, and I. Klebanov for useful discussions.ultraviolet into a non-local action, perhaps some supersymmetrization ofwhere d M2 is distance with respect to the intrinsic world-volume metric and d 11d is distance with respect to the spacetime metric. The second line of (146) shows an approximate gaugefixed form of the action for an M2-brane that is stretched out nearly flat in a large elevendimensional geometry. The power |x − y| 6 corresponds to = 0, i.e. to the value required for the theory to be conformal if loop divergences are absent. It is our hope that some supersymmetric version of our calculations can be carried out, starting from an action similar to (146), to recover eleven-dimensional supergravity as a condition of conformal invariance. A theory like (146) is at best a UV description of the M2-brane, and one should anticipate the addition of a two-derivative term, which in gauge-fixed form would readThis term is relevant by the power counting based on (146), allowing us to recover our ordinary understanding of the M2-brane at long distances compared to the eleven-dimensional Planck scale. A theory on the M2-brane world-volume should be capable of being framed in Lorentzian signature. At least naively, one can make the replacementin(146), where ε is a small positive number (unrelated to = n − s). The good news is that the right hand side of (148) is essentially real-as contrasted for example withn+s for general n and s, which has a phase e i(π−ε)(n+s) inside the lightcone. Questions abound which we have not explored. Can causality be maintained? Is there a relationship with generalized free field theory when the target space geometry is flat? How does one handle gauge-fixing and the related ghosts? 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Fiz.78,248(1989)].\n\nExistence of a phase transition in a one-dimensional Ising ferromagnet. F J Dyson, 10.1007/BF01645907Commun. Math. Phys. 12F. J. Dyson, \"Existence of a phase transition in a one-dimensional Ising ferromagnet,\" Commun. Math. Phys. 12 (1969) 91-107.\n\nO(N) and O(N) and O(N). S S Gubser, C Jepsen, S Parikh, B Trundy, 10.1007/JHEP11(2017)107arXiv:1703.04202JHEP. 11107hep-thS. S. Gubser, C. Jepsen, S. Parikh, and B. Trundy, \"O(N) and O(N) and O(N),\" JHEP 11 (2017) 107, arXiv:1703.04202 [hep-th].\n\nNonlinear Models in Two + Epsilon Dimensions. D H Friedan, 10.1016/0003-4916(85)90384-7Annals Phys. 163318D. H. Friedan, \"Nonlinear Models in Two + Epsilon Dimensions,\" Annals Phys. 163 (1985) 318.\n\nA Huang, B Stoica, S.-T Yau, arXiv:1901.02013General relativity from p-adic strings. hep-thA. Huang, B. Stoica, and S.-T. Yau, \"General relativity from p-adic strings,\" arXiv:1901.02013 [hep-th].\n\nS S Gubser, J Knaute, S Parikh, A Samberg, P Witaszczyk, 10.1007/s00220-016-2813-6arXiv:1605.01061p-adic AdS/CFT. hep-thS. S. Gubser, J. Knaute, S. Parikh, A. Samberg, and P. Witaszczyk, \"p-adic AdS/CFT,\" Communications in Mathematical Physics (2017) , arXiv:1605.01061 [hep-th].\n\nTensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS 3 /CFT 2 correspondence. M Heydeman, M Marcolli, I Saberi, B Stoica, 10.4310/ATMP.2018.v22.n1.a4arXiv:1605.07639Adv. Theor. Math. Phys. 22hep-thM. Heydeman, M. Marcolli, I. Saberi, and B. Stoica, \"Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS 3 /CFT 2 correspondence,\" Adv. Theor. Math. Phys. 22 (2018) 93-176, arXiv:1605.07639 [hep-th].\n\nRiemann Normal Coordinate expansions using Cadabra. L Brewin, 10.1088/0264-9381/26/17/175017arXiv:0903.2087Class. Quant. Grav. 26175017gr-qcL. Brewin, \"Riemann Normal Coordinate expansions using Cadabra,\" Class. Quant. Grav. 26 (2009) 175017, arXiv:0903.2087 [gr-qc].\n\nRiemann normal coordinates. L Brewin, L. Brewin, \"Riemann normal coordinates.\" http://users.monash.edu.au/~leo/research/papers/files/lcb96-01.pdf.\n\nA four-dimensional nonlinear σ-model. E Gava, R Jengo, 10.1016/0550-3213(78)90010-XNucl. Phys. 140E. Gava and R. Jengo, \"A four-dimensional nonlinear σ-model,\" Nucl. Phys. B140 (1978) 510-524.\n\nRecursion relations and fixed points for ferromagnets with long-range interactions. J Sak, 10.1103/PhysRevB.8.281Phys. Rev. B. 8J. Sak, \"Recursion relations and fixed points for ferromagnets with long-range interactions,\" Phys. Rev. B 8 (Jul, 1973) 281-285.\n\n. https:/link.aps.org/doi/10.1103/PhysRevB.8.281https://link.aps.org/doi/10.1103/PhysRevB.8.281.\n\nCrossover between field theories with short-range and long-range exchange or correlations. J Honkonen, M Y Nalimov, 10.1088/0305-4470/22/6/024Journal of Physics A: Mathematical and General. 226J. Honkonen and M. Y. Nalimov, \"Crossover between field theories with short-range and long-range exchange or correlations,\" Journal of Physics A: Mathematical and General 22 no. 6, (Mar, 1989) 751-763.\n\nA scaling theory for the long-range to short-range crossover and an infrared duality. C Behan, L Rastelli, S Rychkov, B Zan, 10.1088/1751-8121/aa8099arXiv:1703.05325J. Phys. 5035354002hep-thC. Behan, L. Rastelli, S. Rychkov, and B. Zan, \"A scaling theory for the long-range to short-range crossover and an infrared duality,\" J. Phys. A50 no. 35, (2017) 354002, arXiv:1703.05325 [hep-th].\n\nOn the Quantum Mechanics of Supermembranes. B De Wit, J Hoppe, H Nicolai, 10.1016/0550-3213(88)90116-2Nucl. Phys. 305545B. de Wit, J. Hoppe, and H. Nicolai, \"On the Quantum Mechanics of Supermembranes,\" Nucl. Phys. B305 (1988) 545.\n\nThe Supermembrane Is Unstable. B De Wit, M Luscher, H Nicolai, 10.1016/0550-3213(89)90214-9Nucl. Phys. 320B. de Wit, M. Luscher, and H. Nicolai, \"The Supermembrane Is Unstable,\" Nucl. Phys. B320 (1989) 135-159.\n"},"corpus_id":{"kind":"number","value":195584174,"string":"195,584,174"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1906.10281\",\n \"mag\": 3101387650,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1007/jhep09(2019)005\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/1906.10281v1.pdf\"\n ],\n \"pdf_sha\": \"4e3df0cf3b274c5a4911d1f10ad54a28fa9039d3\",\n \"open_access\": {\n \"license\": \"CCBY\",\n \"url\": \"https://link.springer.com/content/pdf/10.1007/JHEP09(2019)005.pdf\",\n \"status\": \"GOLD\"\n }\n}"},"title":{"kind":"string","value":"Non-local non-linear sigma models"},"authors":{"kind":"list like","value":["Steven S Gubser ssgubser@princeton.edu \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA","Christian B Jepsen cjepsen@princeton.edu \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA","Ziming Ji \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA","Brian Trundy btrundy@princeton.edu \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA","Amos Yarom ayarom@physics.technion.ac.il \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA\n\nDepartment of Physics\n32000Technion, HaifaIsrael"],"string":"[\n \"Steven S Gubser ssgubser@princeton.edu \\nJoseph Henry Laboratories\\nPrinceton University\\n08544PrincetonNJUSA\",\n \"Christian B Jepsen cjepsen@princeton.edu \\nJoseph Henry Laboratories\\nPrinceton University\\n08544PrincetonNJUSA\",\n \"Ziming Ji \\nJoseph Henry Laboratories\\nPrinceton University\\n08544PrincetonNJUSA\",\n \"Brian Trundy btrundy@princeton.edu \\nJoseph Henry Laboratories\\nPrinceton University\\n08544PrincetonNJUSA\",\n \"Amos Yarom ayarom@physics.technion.ac.il \\nJoseph Henry Laboratories\\nPrinceton University\\n08544PrincetonNJUSA\\n\\nDepartment of Physics\\n32000Technion, HaifaIsrael\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We study non-local non-linear sigma models in arbitrary dimension, focusing on the scale invariant limit in which the scalar fields naturally have scaling dimension zero, so that the free propagator is logarithmic. The classical action is a bi-local integral of the square of the arc length between points on the target manifold. One-loop divergences can be canceled by introducing an additional bi-local term in the action, proportional to the target space laplacian of the square of the arc length. The metric renormalization that one encounters in the two-derivative non-linear sigma model is absent in the non-local case. In our analysis, the target space manifold is assumed to be smooth and Archimedean; however, the base space may be either Archimedean or ultrametric. We comment on the relation to higher derivative non-linear sigma models and speculate on a possible application to the dynamics of M2-branes."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Scalar field theories over the reals with bi-local kinetic terms were introduced in [1], and the recent work [2] provides a useful point of entry into the extensive literature. Similar field theories over the p-adic numbers were considered in [3] as a continuum description of Dyson's hierarchical model [4]. A unifying point of view on the bi-local O(N ) vector model was provided in [5], showing that the standard large N development can be framed in terms that are largely independent of whether the theory is formulated over the reals or the p-adics. The present work extends the study of bi-local theories to bi-local non-linear sigma models, starting with the action S = µ n−s 2γ V ×V d n xd n y |x − y| n+s d(φ(x), φ(y)) 2 ,\nwhere |x − y| is the distance function on the n-dimensional base space V and d(φ(x), φ(y)) is the distance function on the target manifold. In the limit s → n, where the theory (1) becomes classically scale invariant, we find logarithmic divergences in one-loop diagrams which can be canceled by counterterms that can be expressed in terms of the target space laplacian of the square of the distance function, together with field redefinitions. 1 Ricci flatness suppresses the one-loop divergences that we encounter, so in a sense (and with significant caveats) we may claim that we are deriving the vacuum Einstein equations from conformal invariance, as in [6]. Our work was partly motivated by the more recent results of [7], which were derived for a nearest neighbor arc length model on the Bruhat-Tits tree-in other words, on the other side of the p-adic AdS/CFT duality [8,9] from our results for field theories over the p-adic numbers. However, the particular structure of counterterm we find suggests that renormalization of our theories have less to do with renormalization of the local metric as normally understood (i.e. Ricci flow) than with an augmentation of the action (1) to include the target space laplacian of d(φ(x), φ(y)) 2 .\nA conservative expectation is that once non-local terms are allowed in a field theory, they proliferate and the theory becomes non-renormalizable. Theories with purely quadratic bi-local kinetic terms, as studied in [1,3] (as well as many subsequent works) avoid such problems through a non-renormalization theorem: If we write\nthen the claim is that the quadratic bi-local term is never renormalized (at least perturbatively), though the purely local term U (φ(x)) certainly is-and depending on details, derivative terms might be radiatively generated. Non-local interaction terms vitiate this non-renormalization theorem, and one's suspicions could be renewed that there is no sensible theory. We will not be able in this work entirely to allay such concerns, because we do not give a demonstration parallel to the one in [6] that Ward identities based on diffeomorphism invariance guarantee that loop divergences can only modify the original form of the action. Indeed, the counterterms we generate at one loop do modify the bi-local action in an unexpected way, but one which appears to be controlled in a derivative expansion, so that higher derivative terms can be radiatively generated at each new order without spoiling results from lower orders. We will revisit the question of renormalizability in section 13. The organization of the rest of this paper is as follows. In section 2 we present the main results in Fourier analysis that we need, both over the reals and the p-adics. In section 3 we explain how double integrals such as the one in (1) can be regulated if divergences arise as |x − y| → 0. In section 4 we introduce the classical action for the bi-local non-linear sigma model. In section 5 we discuss loop divergences in general terms, including an introductory account of the non-renormalization property of the kinetic term in (2). In sections 6-10 we investigate the simplest one-loop divergences of the bi-local non-linear sigma model, and then in section 11 we argue that all these divergences can be canceled by a laplacian counterterm in place of renormalization of the local metric, together with field redefinitions. As a byproduct of our analysis, we recover in section 12 the usual beta function for the two-derivative theory in two dimensions. We conclude in section 13 with a summary of possible future directions."},{"title":"Fourier transforms","content":"In loop calculations we will often need to go back and forth between momentum space expressions similar to the ones presented in (2) and their real space counterparts, using the Fourier transforms\nThe relevant results are fairly similar between real and p-adic cases, so we present them together. When V = R n , the definitions (3) are entirely standard, and k·x can be understood as the ordinary dot product. Likewise, in this case, |x| is understood as the standard L 2 norm on R n . We refer to the real case as Archimedean because the norm | · | has the property that if 0 < |x| < |y|, then there is some n ∈ Z such that |y| < |nx|. The simplest n-dimensional p-adic construction is based on letting V = Q p n be the (unique) unramified n-dimensional extension of Q p . Let N and Tr be the field norm and field trace with respect to the extension Q p n /Q p . Then we define |x| = | N(x)| 1/n p where | · | p is the usual p-adic norm. We will refer to the p-adic case as ultrametric because the norm | · | just defined has the property |x + y| ≤ max{|x|, |y|}. Next we define k · x = 1 n Tr(kx). Note that k · x ∈ Q p , so to give meaning to e 2πik·x we now only need to define e 2πiξ for ξ ∈ Q p . To this end we find the unique p-adic integer ξ such that ξ − ξ ∈ [0, 1) ∩ Q, and we understand that by e 2πiξ we really mean e 2πi(ξ− ξ ) .\nWe are particularly interested in the Fourier transform of powers of |k|: V d n k e 2πik·x |k| s = Γ V (n + s) |x| n+s + (contact terms) .\nHere Γ V (s) is a meromorphic function of s which can be evaluated as\nwhere we set v = ∞ in the Archimedean case and v = p in the ultrametric case, with\nIntuitively, Γ V is a variant of the Euler gamma, specific to the choice of V , and constructed so as to be the coefficient of the 1/|x| n+s term in (4). In the remainder of our discussion, integrals are over V unless otherwise indicated. The contact terms in (4) are somewhat delicate and dependent on detail. When −n < s < 0, the integral in (4) is convergent, and no contact terms are needed. One can easily check that Γ V (n + s) → 0 as s → 0, so when s = 0 the power law term goes away and we recover the obvious result d n k e 2πik·x = δ n (x) .\nFor s > 0, the integral in (4) diverges, and we need a more careful approach. A good first step is to understand (4) in terms of its action on a test function φ : V → R:\nwhere D s is some linear map on functions φ(x). A suitable class of test functions are socalled Schwartz-Bruhat functions. When φ : Q p n → R, we require that φ is locally constant with compact support. For example, the characteristic function of the p-adic integers is a Schwartz-Bruhat function on Q p . When φ : R n → R, the test functions are more appropriately called Schwartz functions, and their defining property is that they go to 0 faster than any power of |x|, as do all their derivatives. An example is a Gaussian. Both in the real and ultrametric cases, the Fourier transformφ(k) of a Schwartz-Bruhat function is again a Schwartz-Bruhat function.\nWith (8) taken as the definition of D s , our task is to find a representation of D s entirely in position space. In the ultrametric case for arbitrarily positive s, one finds\nThis is the Vladimirov derivative. In the Archimedean case, the same expression (9) is valid for 0 < s < 2. There is one more easy case to dispose of: even positive integer s for Archimedean V . Then Γ V (n + s) = 0, which makes sense in (4) because the right hand should be purely distributional, on account of |k| s = (k 2 ) s/2 being analytic in k 2 . Explicitly,\nfor positive even s ,\nwhere = n i=1 ∂ 2 x i . We are left with the task of defining D s for Archimedean V and for s > 2 but not an even integer. Heuristically, the contact terms in (4) are a sum of terms of the form r δ n (x), where 0 ≤ r < s/2, with divergent coefficients. To state this more precisely, we write\nwhere a regulated integral\nis rendered finite (if possible) by allowing the subtraction from G(x, y) of a finite sum of smooth functions of either of the following types:\nI. Pure powers: more precisely, any function whose y dependence comes solely through a factor |x − y| α where α is a real number. This is meant to include, through the case α = 0, functions which have no y dependence.\nII. Higher partial waves: more precisely, any function of the form Y ( y − x)g(|x−y|) where Y (ẑ) is a spherical harmonic on S n−1 other than the s-wave.\nType I functions are never integrable, whereas type II functions may or may not be; so at best there is a unique choice of type I functions that will work, whereas many choices of type II functions are possible. An alternative approach, generalizing the principle value prescription, is to eschew modifications of the integrand and instead carry out y integration in polar coordinates centered around x, as follows. One first performs the angular integrals. Then the radial integral is restricted to run from l to L. One next allows the subtraction of an arbitrary finite sum of negative powers of l and/or positive powers of L, chosen (if possible) so that the limits l → 0 and L → ∞, taken independently, lead to a finite result. Doing the angular integration first obviates the need for type II functions, while the ultraviolet and infrared cutoffs, l and L, obviate the need for type I. 2\nWhile the subtractions described can in principle cure either ultraviolet (UV) or infrared (IR) divergences, we will be interested only in applications where UV divergences matter: that is, divergences arising when |x − y| → 0 (with x held fixed). Type II subtractions are relatively innocuous because they follow automatically from performing angular integrations first; therefore we will use the notation d n y . . . to indicate a y integration with type II subtractions which we usually omit to write explicitly.\nAlthough we have stated our integration prescriptions in the abstract, it is easy to see how to apply them to (11) when φ is a Schwartz function. Consider the case 2 < s < 3, and set x = 0 for simplicity. Then (11) becomes\nThe extra terms in square brackets on the right hand side of (13) evidently render the integral convergent near y = 0 for 2 < s < 3. The term linear in y is clearly a type II function, and the term quadratic in y is a sum of a type II function proportional to y i 1 y i 2 − y 2 n δ i 1 i 2 (a d-wave term) and a type I function proportional to y 2 . If 3 ≤ s < 4, then we would need one additional term in the Taylor series expansion of φ around y = 0, and this additional term is a type II function. In summary, for 2 < s < 4, and omitting type II subtractions,\nEvidently, if 0 < s < 2, a simpler subtraction scheme would work, resulting in (14) with the laplacian term omitted, in agreement with (9). For general s > 0 (other than positive even integers) and Archimedean V ,\nIn principle, one may derive (15) by subtracting an appropriate number of terms in the Taylor series expansion of φ(y) and then finding appropriate type II subtractions to bring the result into the form (15).\ncannot be cured using powers of l and/or L after a cutoff integration. Because we avoid even integer s as well as functions G(x, y) which grow as positive powers of large separation |x − y|, we do not need to specify a resolution to this inequivalence.\nA more efficient way to determine the coefficients b r is to start from (15) and Fourier transform:\nIn the second equality of (17), we have partially carried out the y integral in polar coordinates around the point y = x, introducing a radial variableỹ = |x − y|. In the third equality, we have carried out the angular θ integral and introduced a new radial variable, ρ = 2π|k|ỹ. The ρ integral in the last line of (17) converges, provided s is positive but not an even integer, and provided the coefficients a r are coefficients in the Taylor series expansion of the Bessel function around ρ = 0. These coefficients a r are well known, and from them one can recover the expression (16) for the b r ."},{"title":"Bi-local integrals","content":"We are particularly interested in double integrals of the form\nwhere s > 0 and G(x, y) is piecewise constant if V is ultrametric and smooth if V is Archimedean. Unless otherwise noted, all double integrals over x and y will by taken over all of V × V . In the ultrametric case, for any s > 0, following (9) we define\nIn the Archimedean case, we define d n xd n y |x − y| n+s G(x, y)\nby performing the y integration first and allowing the subtraction of type I and type II functions to G(x, y) in order to achieve a finite result (if possible). As in the previous section, type II subtractions are deemed relatively inconsequential, so even unprimed integration over x and y means to perform the y integration first, allowing the subtraction of type II functions in order to achieve a finite result (if possible). Explicitly, for s not a positive even integer,\nwhere the coefficients b r are as given in (16). We avoid positive even integer s when V is Archimedean because in this case we expect that our constructions will lead instead to purely local theories; also, precisely in this case, the subtleties pointed out in footnote 2 regarding logarithmic divergences come into play.\nOur computational strategy will turn on converting bi-local position space integrals into Fourier space integrals. Let's start with the simplest example of that calculation, valid for ultrametric V and any s > 0, and also for Archimedean V and 0 < s < 2. Let φ : V → R be a Schwartz-Bruhat function. Then\nThe first step is actually the trickiest, because it is not clear from the rules of integration set forth following (20) that we are allowed to add a function like φ(x) 2 − φ(y) 2 to the integrand. To justify this step, we denote f (x) = φ(x) 2 , and we argue that\nThe second integral in (23) is the k = 0 component of the Fourier transform of D s f (x). But this Fourier transform is |k| sf (k), and since s > 0 the k = 0 component indeed vanishes.\nLet's now pursue the same computation for the Archimedean case with 2 < s < 4. On one hand, using (21),\nOn the other hand, using (14),\nIn order to conclude\nwe must therefore argue that the final integrals in (24) and (25) agree. Subtracting (25) from (24) and simplifying slightly with the definition f (x) = φ(x) 2 , we arrive at\nThe first equality in (27) follows from (14), and the second is by the same argument used following (23). To summarize, for Archimedean V and for 2 < s < 4,\nΓ V (n + s) > 0 for 2 < s < 4, and so without the (∂φ) 2 on the left hand side of (28) we would have a sign problem. The equality (26) can be checked in a similar manner for s > 4. A key relation is\nTwo take-away lessons are:\n• When we write simple |k| s kinetic terms in momentum space, in position space we are combining non-local position space terms and local terms involving derivatives in a precisely tuned ratio.\n• There is some freedom in the precise structure of the position space form, as exemplified by the equality of the last integrals in (24) and (25) due to a manipulation which is the non-local version of integration by parts."},{"title":"The bi-local non-linear sigma model","content":"We are now in a position to present the action for the bi-local non-linear sigma model. Let M be a smooth D-dimensional manifold with a Riemannian metric g ab , whose Riemann and Ricci tensors are\nGiven any two points X and Y on M , let\nbe the square of the shortest distance between X and Y . Clearly, Q(X, Y ) is a smooth function of X and Y , provided X and Y are not too far apart. For smooth functions φ : V → M whose range is sufficiently localized, we consider the action functional\nwhere indicates a regulated double integral of the type discussed around (19)-(21). 3 Note that this discussion requires us to avoid positive even integer s when x and y are valued in R n (as opposed to Q p n ). In the Archimedean setting, when s > 2, there are derivative terms like (∂φ) 2 implicitly built into (32), with coefficients tuned so as to ensure convergence of the integral. The parameter µ has dimensions of energy so that we can regard φ and Q(φ(x), φ(y)) as dimensionless. The factor γ is a loop-counting parameter: Classical effects are O(γ −1 ), one-loop amplitudes are O(γ 0 ), two loop amplitudes are O(γ), and so forth. In other words, γ plays the role of . 3 One may wonder whether the primed integral, as defined following (20), spoils coordinate invariance of the integrand. For instance, if s is sufficiently large we may, in light of (21), be required to include a y Q term to the integrand, which if written only in terms of partial derivatives does not appear to be coordinate invariant. In fact, it is easy to convince oneself that, e.g., y Q can be constructed from covariant quantities:\nA close cousin of the action (32) was considered in [7]:\nwhere now x and y are vertices of a graph and the sum is over undirected edges. The formula actually appears earlier in [6], though it was intended there to be considered on a square lattice, as a regulator for the local non-linear sigma model, rather than on the Bruhat-Tits tree as in [7]. We require the range of the maps φ to be sufficiently localized in order to ensure that we do not encounter any failures of smoothness in Q(X, Y ), and in order to ensure that we can use a single system of Riemann normal coordinates for φ throughout. One can now solve the geodesic equation perturbatively in the curvature and use that to evaluate Q(X, Y ) and expand the action (32),\nwhere\nand\nHere g ab and R abcd are evaluated at the origin of the Riemann normal coordinates, which we assume is at X = 0. Often, the definition of Riemann normal coordinates includes the requirement g ab = δ ab , but this is not necessary for our calculations, and we find it more convenient to retain explicit factors of g ab and the inverse metric g ab . Put differently, we are choosing a coordinate system so that all geodesics passing through the origin are linear in the affine parameter:\nThe ellipsis in (34) indicates higher order interactions, involving five or more powers of φ(x) and/or φ(y), as well as derivatives and powers of the curvature. We will consider up to six point interactions in section 10. quadratic kinetic term in the action (2), where V = R n or Q p n , with\nThe purely cubic theory is unstable, but it serves our purpose because we are only interested in analyzing the behavior of the one-loop correction to the propagator. Using the diagram shown in figure 1a, we obtain the one-loop contribution to the quadratic part of the oneparticle irreducible (1PI) effective action:\nWe continue the convention of integrating over all of V except as otherwise indicated. Let's assume n > 2s, so I is UV divergent (and IR convergent). To regulate the divergence, we introduce a hard cutoff: | | ≤ Λ. If V = R n , then Λ can be any positive real number. If V = Q p n , then we will require that Λ is an integer power of p. The ultrametric case is easy to analyze, because when | | > |k| we have | | = |k − | exactly. So, except in the compact region where | | ≤ |k|, the integrand has no k dependence at all. Therefore, any UV divergences are entirely independent of k, and to evaluate them we can set k = 0:\nThe last equality comes from splitting the integration region into shells with fixed | |; then the integral becomes a geometric sum. Because the divergent part of I(Λ) has no k-dependence, the counterterm required to cancel it is proportional to d n kφ(−k)φ(k) = d n x φ(x) 2 . In other words, it is a mass term. This argument is easy to generalize to the statement that only purely local terms (powers of φ(x)) can be radiatively generated starting from the action (2) over V = Q p n . An essentially equivalent argument was made in a Wilsonian picture in [3]. The Archimedean case is more subtle because of the possibility of subleading divergences. A straightforward approach is to expand\nin powers of k. Terms with an odd number of powers of k vanish by parity, leaving only terms analytic in k 2 . Of these, only terms proportional to k 2r with r ≤ n 2 − s are UV divergent. In short, the divergent part of I(Λ) is a polynomial in k 2 whose order is n 2 − s . A divergent term proportional to k 2r requires a counterterm proportional to d n x φ(x) r φ(x). These are the radiatively generated derivative terms mentioned in section 1.\nWe should note a troublesome feature of the hard momentum cutoff for Archimedean theories: The coefficients one finds for sub-leading divergences depend on how one implements the cutoff. For example, it is easy to check that the coefficient of the k 2 term in I(Λ) changes if instead of requiring | | ≤ Λ we impose the more democratic condition − k 2 ≤ Λ. However, the feature that we care about, namely the fact that the divergent terms have only polynomial dependence on k 2 , doesn't depend on the details of the cutoff. It is perhaps instructive to consider one other alternative, namely dimensional regularization, in which one first computes\nby continuing to a domain of n in which the integral is convergent. (In the current example, s < n < 2s is such a domain.) The only divergences one then tracks are poles of the right hand side of (42) as a function of n. These occur precisely when n−2s 2 is a non-negative integer. It is characteristic of dimensional regularization that there is (at most) one divergent term for a given n, corresponding to a logarithmic divergence in the original integral.\nThe loop diagrams we will need to consider in our analysis of the non-local non-linear sigma model are simpler than (39) in one regard: The loop is a single propagator starting and ending at the same vertex. This matters because there is then only one internal momentum , and imposing the hard cutoff | | ≤ Λ is a privileged choice because it corresponds to integrating over an O(n)-invariant region. An example is the diagram shown in figure 1b, which is proportional to\nassuming that whatever vertex factor is needed to fully evaluate the diagram doesn't depend on . We also assume n > s so that I 0 is UV divergent but IR convergent. We straightforwardly find\nThere are obviously no subleading divergences in I 0 .\nFor convenience we introduce\nWe are interested in divergences proportional to log Λ that arise when = n − s = 0. As a technical trick to isolate these divergences, we make small and positive, and we look for divergences of the form Λ / , which in the → 0 + limit give rise to log Λ terms.\nTo characterize this limit precisely, given n 0 > 0 and λ ∈ R, we set n = n 0 + λ and s = n 0 + (λ − 1) and then take the → 0 + limit with n 0 and λ held fixed. (Clearly then we are allowing non-integer n, in the spirit of [6].) For the most part, our final results are independent of λ. When is sufficiently small, we may replace (44) with\nIf we lift the requirement that Λ is an integer power of p when V = Q p n , then (46) is unaltered, because I 0 = ζp( ) ζp(n) p log p Λ , and p log p Λ differs from Λ at most by a factor of p = 1 + O( ). The important point is that in the limit → 0 + , I 0 includes a logarithmic term i 0 log Λ, and isolating this term is our stated objective.\nWe will encounter one other loop integral:\nIt comes from graphs similar to the one in (43), but with a vertex prefactor |k − | s . Using the same reasoning that led to (40), we see that when a hard cutoff | | ≤ Λ is imposed, one obtains in the ultrametric case\nIf V = R n and n is positive but not an even integer, then\nfor some coefficients c r . If we choose n 0 positive but not an even integer and fix any finite value of λ, then for sufficiently small , (49) applies, and the least singular power of Λ appearing in it is Λ n−2 n/2 . As → 0 + , this power remains positive and finite, tending to Λ n 0 −2 n 0 /2 . So there is no log Λ behavior, even in the → 0 + limit. If instead we make n 0 a positive even integer, then by choosing the very particular value λ = 1, so that s = n 0 exactly, we find (for sufficiently small > 0) that the least singular term in (49) is c n/2 |k| s Λ , which does contribute a c n/2 |k| s log Λ divergence in the → 0 + limit; moreover, in this case, by calculation, c n/2 = i 0 + O( ). 4 We can summarize the situation, both for the Archimedean and ultrametric cases, by stating that for = n − s positive but sufficiently small, then subject to the restriction that n cannot be a positive even integer when V = R n ,\nwhere\nThe higher powers of Λ in (50) are accompanied by non-negative integer powers of k 2 , and they correspond to operators which remain relevant in the → 0 + limit. The only log Λ behavior arising from I 2 (k), in the → 0 + limits described above, is the i 2 |k| s log Λ term coming from the i 2 |k| s Λ term shown in (50). We are not concerned about O( ) terms to i 0 and i 2 because they drop out of the log Λ behavior in the → 0 + limit. In the following sections, therefore, we will drop O( ) terms from (46) and (51), and we will evaluate i 0 and i 2 in terms of n rather than n 0 ."},{"title":"The propagator through one loop","content":"To derive the tree-level propagator, we use an obvious generalization of (26) to multicomponent scalar fields to rewrite the free action in momentum space:\nwhere we recall that = n − s, and for notational convenience we have introduced 5\nWe immediately extract from (52) the propagator\nWe are primarily interested in small, so that G ab (x) is nearly logarithmic. Informally, we can understand the one-loop correction to the propagator as a contribution to the 1PI effective action coming from all possible Wick contractions of the two of the four factors of φ in S int . The calculation is done most straightforwardly in momentum space, where we can express\nwhere d 4n k = 4 i=1 d n k i . As usual, derivative terms are implied in in the Archimedean 5 A point worthy of remark is that whileγ and γ have the same sign in the ultrametric case, and also in the Archimedean case for 0 < s < 2, for 2 < s < 4 they have the opposite sign. The integral in (52) is well-defined and positive, so to make our theory sensible we should always chooseγ > 0. This means that γ < 0 for 2 < s < 4. As explained in (28) for a single real scalar, the regulated position space integral used to define the action (32) includes a (∂φ) 2 term that enters with the opposite sign of the non-local [φ(x) − φ(y)] 2 term, so positivity conditions are difficult to judge in position space. case when s > 2. Symbolically, the Wick-contracted quartic action is\n. We omit theφ aφb andφ cφd contractions from (56) because of the antisymmetry of R abcd in ab and cd. We omit theφ aφc andφ bφd contractions because they include a factor |k 1 +k 3 | s δ n (k 1 +k 3 ), which vanishes when s > 0. After some straightforward algebra, we obtain from (56) the form\nwhere I 2 (k) is given in (47).\nAs discussed below (47), for suitably small positive = n − s, I 2 (k) includes a term i 2 |k| s Λ iff V = R n and s is a positive even integer. This is the case which leads to local non-linear sigma models. Otherwise-excluding the case V = R n with positive even n-the divergent terms in I 2 (k) are proportional to |k| 2r Λ n−2r for non-negative powers n − 2r which remain finite as → 0 + . Therefore, apart from the case of local non-linear sigma models, the effects of the ultraviolet divergences in (57) are limited to generating relevant, local interactions. We assume that relevant terms of this type can be tuned away.\nIt would be tempting at this point to conclude that the non-local action (32) is nonrenormalized, as in the case (2). The reality is more subtle: We will see in section 10 that higher point diagrams generate one loop divergences that require non-local counterterms; however they are not quite of the form (32), involving instead the target space laplacian of Q(X, Y )."},{"title":"Curvature and arc length calculations","content":"Higher point amplitudes in the bi-local non-linear sigma model involve tensors of large rank. We will therefore find it convenient to introduce some abbreviated notation, based on the following equivalences:\nHere (ab) = 1 2 (ab + ba). We employ obvious extensions of (58) to larger index sets, e.g. X a 123 means X a 1 X a 2 X a 3 .\nWe will often need to simplify expressions involving the curvature tensor and its covariant derivatives. A primary tool is the Bianchi identities, which we may write using our abbreviated notation as\nA contracted form of the second Bianchi identity,\nshows that any three-index contraction of ∇ a 5 R a 1234 (meaning any contraction leaving three indices free) can be expressed as linear combinations of re-indexed versions of the tensor ∇ a 1 R a 23 ; in this sense ∇ a 1 R a 23 by itself is a basis for all the three-index contractions of ∇ a 1 R a 2345 . This observation will be useful to us when we consider the possible Wick contractions of the five-point interaction vertex in the bi-local non-linear sigma model. Acting on the contracted second Bianchi identity (60) with ∇ b 1 gives\nWe describe the terms in the second line of (61) as commutator terms. Evidently, they can be written as curvature bilinears, meaning contractions of two factors of the Riemann and/or Ricci tensors, with no covariant derivatives. Acting on the uncontracted second Bianchi identity (the second line of (59)) with ∇ a 5 gives\nwhere ∇ 2 = ∇ b ∇ b , and the commutator terms are similar to the ones occurring in (61): In particular, they are curvature bilinears. The results (61) and (62) show that all four-index contractions of ∇ a 56 R a 1234 can be expressed in terms of linear combinations of re-indexed versions of the tensor ∇ a 12 R a 34 , together with curvature bilinears. So far, all formulas in this section have been entirely independent of the choice of coordinate system. We now pass to Riemann normal coordinates in order to study the square of the arc length, Q(X, Y ) = d(X, Y ) 2 between two points X and Y . We have from [10] 6\nHere g a 12 , R a 1234 , and its derivatives are all evaluated at the origin of Riemann normal coordinates, which is the origin in terms of the coordinates X a and Y a used in (63)."},{"title":"Three-point vertices","content":"There are no three-point vertices at tree-level provided we employ Riemann normal coordinates. As we will explain in this section, three-point vertices appear to be generated at the one-loop level, by the diagram in figure 2b; however, they can be absorbed through field redefinition.\nAs for our discussion in section 6 of the one-loop corrections to the propagator, the one-loop contribution to the three-point function can be obtained efficiently through Wick 6 Note however that the results leading to Q RR contractions in the momentum space of the quintic term in the action:\nHigh-dimensional Fourier integrals of the type seen in the second line of (64) are common in our calculations, so we have found it useful to introduce some shorthand notation:\nwith obvious extensions to larger index sets. If A is any ordered set of indices, like 123, and |A| is the number of indices in the set, then the integrals we see most often are of the form\nwhere q(k A ) is any function of the k i , and the third expression is just a rewriting of the second using the shorthand notation introduced in (65). Evidently, the J φ integrals are convergent for reasonable integrands q, like powers of norms of sums of momenta, provided the φ a are Schwartz-Bruhat functions. When there is no risk of confusion, we will omit the subscripted k A and just write J a A φ . We can now rewrite (64) as\nAs in section 6, a Wick contraction amounts to a replacement\nIt is helpful to note the following examples of Wick contraction:\nHere A and C are collections of indices, neither including 1 or 2, with C ⊂ A, and we assume s > 0 in order to obtain the vanishing of the first integral. Recall from section 5 that I 0 diverges as Λ / when = n−s is sufficiently small and positive, giving rise to log Λ behavior in the limit → 0 + . There are 5 2 = 10 possible single Wick contractions of S 5 , but the (12) contraction (meaning the contraction of a 1 and a 2 ) vanishes because g a 12 ∇ a 5 R a 1234 = 0; likewise the (34) contraction vanishes. Meanwhile the (14) and (23)\nA few comments are in order:\n• Because S Wick 5, (14) and S Wick 5,(25) are proportional to the I 2 loop integral, they do not contribute logarithmic divergences in the → 0 + limits described in section 5, except when s is a positive even integer. In a position space account, these non-logarithmic terms correspond to contractions of φ(x) with φ(y).\n• We are mostly interested in S Wick 5,(13) and S Wick 5,(15) because I 0 does produce a logarithmic divergence in the → 0 + limit. Note that in these terms, the vertex factor |k 13 | s involves only external momenta. This is the crucial feature, noted already in the introduction, which allows non-local counterterms to arise. In position space, the logarithmic terms correspond to contractions of φ(y) with itself.\nIn summary,\nThe minimal counterterm needed to cancel the Λ / divergences in (72) is\nwhere\nand i 0 and i 2 are as defined in (46) and (51). We use the notation S ct 3 to denote a counterterm that is cubic in the fields. Our primary interest is in cases where i 2 = 0-namely, cases in which V = Q p n , or V = R n but s is not a positive even integer. However, tracking I 0 , I 2 (k), i 0 , and i 2 throughout our computations is useful as a bookkeeping device in order to simultaneously treat the local and bi-local theories, in the → 0 + limit, with i 2 entering in only to describe the local theories."},{"title":"Renormalization through cubic order in the fields","content":"Before entering into the more complicated story of quartic terms in the action, let's preview the endgame of our analysis, in which we produce a bare action which incorporates the renormalized action plus the counterterms in a form that we can express entirely in terms of arc length.\nFirst, let's rephrase the Wick-contracted quartic action (57) as\nfrom which we conclude that we need a counterterm quadratic in the fields of the form\nwhere\nThe results (76) and (73) together put some constraints on the renormalization procedure, but as we will see they do not completely determine it. The question of renormalizability is whether we can reorganize the renormalized action plus counterterms into a bare action whose form is the same that we started with:\nwhere\nis the sum of all counterterms, S[φ] is as given in (32), and\nwhere\nThe bare arc length d B , derived from a bare metric tensor g B ab , may differ from the renormalized arc length d, and the bare coordinates φ a B may likewise differ from the renormalized coordinates φ a . We require however that the points φ B = 0 and φ = 0 coincide. At tree level, where we ignore all counterterms, we have the relations\nand our key task is to find perturbative corrections to these relations that render (78) correct.\nTo begin, let's examine the quadratic terms in (78), using the counterterm S ct 2 from (76):\nwhere for simplicity we multiplied through by 2γ/µ . We surmise from (82) that corrections to (81) can be expressed as a power series in the dimensionless quantitỹ\nThat is,\nwhere V a b 12 and W a b 123 are fully symmetric in their lower indices, and U , V , and W are all independent of φ (and φ B ). In other words, (84) is a Taylor expansion of φ a B in the coordinates φ a . Also,\nfor some tensor T ab (φ). As our notation indicates, T ab (φ) does depend on φ. As with other tensors, if we omit the argument, we mean that T ab is evaluated at φ = 0. Using (84) and\n(85), we see that (82) is satisfied provided\nor in other words provided\nwhere\nIt should be kept in mind that T a 12 is the φ = 0 value of a tensor field T a 12 (φ) defined over the whole of M , whereas U a (12) is defined only at φ = 0. Let's assume that\n(A term in T a 12 (φ) proportional to R(φ)g a 12 (φ) is also possible, but the divergences we will encounter do not require it.) Then (87) reduces to\nAs previously noted, based on the treatment of quadratic terms alone, we cannot distinguished between metric renormalization (related to the coefficient t 0 ) and field redefinition (related to the coefficient u 0 ). In order to proceed to higher orders, we require the squared arc length formula for bare quantities:\nWe do not require φ B to be Riemann normal coordinates for d B , so there are contributions to Q B at cubic order:\nwhere Γ B abc = g B ad Γ Bd bc and Γ Ba bc is the Christoffel connection for g B ab . From (85) we have immediately\nFrom Q B 3 we obtain a cubic term in the bare action:\nAnother term cubic in φ arises in the bare action from plugging the non-linear field redefinition (84) into the quadratic term S B 2 . To work this out, it helps first to note that passing (84) through a Fourier transform yieldŝ\nand * denotes convolution. It follows immediately that\nand so\nwhere we are lowering indices on V and W with the renormalized metric g ab . The T a 12 +2U a 12 term in (98) is the same combination we saw in (87), with the symmetrization U a 12 → U a (12) implied because we multiply by J a 12 φ [|k 2 | s ], which is symmetric. The next term in (98) is the interesting one for us. The only constraint on V a 312 is symmetry in the 12 indices. This is the same symmetry that J a 123 ically, we expand (78) to cubic order in the renormalized fields, using the expression (73) for S ct 3 , as well as S B 3 from (94) and the O(φ 3 ) term from (98). The result is\nEvidently, we may set\nwhere\nThe larger message is that V a 312 is sufficiently unconstrained that we could use it to absorb any counterterm proportional to J a 123 φ [|k 3 | s ]. By the same token, when we get to quartic order, the field redefinition coefficients W a 4123 can be used to absorb any terms proportional to J a 1234 φ [|k 4 | s ]. Therefore, when we do proceed to quartic order, we may simplify our work by systematically dropping all such terms. We will see in section 10 that other terms emerge, proportional to J a 1234 φ [|k 24 | s ]. These are the ones that cannot be absorbed into field redefinitions."},{"title":"Quartic counterterms","content":"Four-point vertices are present at tree-level, and they are also generated by three different types of one-loop diagrams, as shown in figure 3. Our goal in this section is to evaluate one-loop divergences proportional to ∇ a 12 R a 34 (and re-indexings of it). The claim is that only the diagram in figure 3a can contribute. Tracking only target space indices, the vertex factor in this diagram is ∇ a 56 R a 1234 , and the internal propagator can tie any two of the six indices together. So the diagram as a whole must be proportional to four-index contractions of ∇ a 56 R a 1234 . As explained in section 7, re-indexed versions of ∇ a 12 R a 34 , together with cur-vature bilinears, provide a basis for such contractions. The diagrams in figure 3b and 3c are manifestly proportional to curvature bilinears, so they cannot contribute terms proportional to ∇ a 12 R a 34 . (One immediate way to see this is that the Riemann tensor could vanish at φ = 0 without its derivatives vanishing.) In the explicit calculations below, we will encounter and discard many curvature bilinear terms, which we generically write as O(RR), meaning some contraction of R a 1234 R a 5678 .\nWe claim that the counterterms needed to cancel the divergences from the diagram in figure 3a proportional to ∇ a 12 R a 34 (and re-indexings of it) are\nwhere\nColor-coding in (102) is to help track to which term in (102) each of the many terms in later equations contribute. The remainder of this section is devoted to deriving (103).\nTo derive the sixth-order vertex used in figure 3a, we start from the Q ∇∇R 6 term in (63) and extract the following six-order terms in the action:\nwhere\nOur task is to compute the counterterms for all the single Wick contractions of S 4+2 6,∇∇R and S 3+3 6,∇∇R . For S 4+2 6,∇∇R , relations among Wick contractions that are obvious from symmetries plus the first line of (69) are as follows:\nIf we write the counterterm for an expression Q as {Q} ct , then the rules of computation we need are a trivial adaptation of (69):\nThe counterterms that we need to cancel divergences coming from the single Wick contractions of S 4+2 6,∇∇R , as shown in (107)\nwhere\nwhere the color coding is to show which terms in (110) contribute to which terms of (102). By direct computation using the rules (108),\nwhich agrees with (103)."},{"title":"Renormalization at quartic order","content":"To renormalize at quartic order, we first inquire whether the counterterms (102) can be organized into the bare arc length action (80), using the field redefinition (84) and the relationship (85) between the bare and renormalized metric. For the non-local model, we will find that this is impossible! So we will turn to a generalized form of the bare action that includes a term proportional to the target space laplacian of the square of the arc length.\nTo get started, we need the bare arc length formula to quartic order in the bare fields: That is, we need one more term in the series (91) than we computed in section 9. The computation of arc length is less simple than for the renormalized metric because the φ a B are not Riemann normal coordinates for g B ab -due to effects at O(γ), in particular a connection Γ Ba b 12 ∼ O(γ). As a technical device, we therefore introduce a third set of coordinates, φ a , which are Riemann normal coordinates for the bare metric, which in barred coordinates takes the form g ab (φ). We can express φ a in terms of φ a B as\nand we can write g B ab (φ B ) in terms of g ab (φ) as\nNote that, by assumption, g B a 12 = g a 12 at φ B = φ = 0. The condition that φ a are Riemann normal coordinates allows us to concludẽ\nsee for example [11] for a derivation. The Γ B Γ B term in the expression (117) for M a b 123 is optional because it is O(γ 2 ), but it arises naturally in the derivation of [11], so we include it.\nThe bare arc length coincides between φ and φ B coordinate systems because these are just different coordinate systems for the same metric, as per (116). Explicitly,\nwhere in the second equality we used the fact that φ a are Riemann normal coordinates. The (118) is short for all terms involving five or more powers of X and Y combined; similar notation is used below. Using the first equation in (115) to eliminate X and Y in favor of X B and Y B , we arrive at\nwhere L a 312 = g B a 3 b L b a 12 and M a 4123 = g B a 4 b M b a 123 . Note that the cubic terms in (119) agree with (92), and recall from the subsequent analysis that the corresponding cubic term S B 3 does not need to match S 3 + S ct 3 , because of the additional term cubic in φ in (98) arising from the O(γ) difference between φ B and φ. Likewise, the term in (119) proportional to M a 4123 gives rise to a term proportional to J φ B [|k 4 | s ] in the action, but we do not need to track it explicitly because the quartic term in (98) shows that it is precisely the sort of term that we can absorb into the field redefinition coefficient W a b 123 . Thus we may write\nwhere (field redef) indicates field redefinition terms as discussed above. Next we express S B 4 in terms of renormalized quantities in order to compare to (102). Starting from (85), we obtain Λ µ R B a 1234 = R a 1234 +γ 2 (∇ a 14 T a 23 − ∇ a 24 T a 13 − ∇ a 13 T a 24 + ∇ a 23 T a 14 )\nThus we find\nO(RR) terms arise in (122) not just from those in (121), but also from expressing J a 1234 φ B\n[|k 24 | s ] in terms of the renormalized field φ. 7 We have color-coded terms in (122) to match the way we did in (102). Comparing the two equations, we can see that S B 4 accommodates the counterterms S ct 6,∇∇R iff K 3 = −K 4 . Based on (103), this happens iff i 0 = i 2 , which means iff s is a positive even integer. As we will develop more fully in section 12, this indeed corresponds to the case of local non-linear sigma models.\nLet us pursue further here what happens for the non-local case. Because i 2 = 0, we have K 1 /K 2 = −6/7, there is no hope of rendering the theory renormalizable with just the arc length action we have been using so far. Some generalization of the arc length action is needed. Whatever modification we make should involve two target space derivatives relative to the original action, so as to absorb counterterms that appear with two extra derivativeslike the ∇ a 12 R a 34 structure in S ct 6,∇∇R as compared to R a 1234 in S 4 . Luckily, there is a new term with the right properties which we can add to S B :\nwhere we define\nThe −4n term in (124) is present in order to ensure Q B (X B , Y B ) = 0 when X B and Y B coincide. 8 By explicit calculation (as sketched below (132)),\nPassing to momentum space, we find\nCombining (122) and (126) and comparing with (102), we see that κ B = −γ i 0 2 . Strikingly, we are forced also to choose t 0 = 0: That is, the metric is not renormalized! Having allowed the two-derivative term (123) in S B , we should allow addition of a similar term to the renormalized action:\nWe restrict κ to be an O(γ) quantity, which makes sense because then the overall scaling with γ of (127) is O(γ 0 ), and this aligns with the invariance of Q (X, Y ) under overall rescaling of the target manifold. The additional term (127) changes all the one-loop amplitudes, but only by O(γ) quantities, relative to the O(γ 0 ) scaling of one-loop amplitudes and counterterms that we obtained in previous sections. In short, the only effect of allowing non-zero κ in our counterterm analysis is to lead to a direct tree-level contribution to κ B , so that in total,\nTo rephrase this result in terms of the renormalization group, we can rewrite (128) as\nand then since the right hand side is independent of Λ, we arrive at\nThe first term on the right hand side of (130) is the tree-level term coming from the engineering dimension factor of (Λ/µ) in (128). The one-loop effects are responsible for the second term in (130). If we now take → 0 in (130), we see that κ B runs logarithmically:\nwhere Λ 0 is a dynamically generated scale. Note thatγ and i 0 are positive, so κ B is positive at scales Λ below Λ 0 and negative above Λ 0 . To see that (125) is correct, let's work on the renormalized side and note that\nThe expression (132) is part of ∇ 2 X Q(X, Y ), and it is easy to see that it is the only part contributing terms of the form (∇∇R)XXY Y . Therefore\nThe ellipses in (132) and (133) indicate terms that are not quadratic in both X and Y , for example terms schematically of the form (∇∇R)XY Y Y or (∇∇R)Y Y Y Y , as well as lower order terms which are either independent of X or Y , or linear in X or Y . Plugging (133) into (127), we find\nThe contributions labeled (field redef) in (134) are linear in φ(x) or φ(y). To see that (134) is correct, we have only to understand why we can freely add or drop from the integrand smooth functions which depend only on x or only on y, such as the direct terms φ a 1234 (x) + φ a 1234 (y) which are present in the last expression in (134) but not the middle expression. As in section 3, this follows from careful use of the regulated integral prescription:\nfor smooth functions f (x) with suitable falloff conditions at large x. The integral d n xd n y |x−y| n+s f (x) vanishes more trivially by subtraction of the r = 0 term in the sum appearing in square brackets in (135)."},{"title":"Recovering the local non-linear sigma model","content":"When V = R n and s is a positive even integer, the original position space action (32) is problematic, because our scheme for defining regulated integrals breaks down. However, this is precisely the case where we expect to recover a local theory, expressible in terms of φ(x) and its derivatives. In this section we will verify this expectation for the simplest case, s = 2. We proceed in three steps: First, we start from the momentum space form of the action, obtained for general s from the non-local action (32); then we take s = 2; and finally we Fourier transform back to a local position-space action. It is helpful (though not really necessary) to work with Riemann normal coordinates throughout. We exhibit the process in detail for the quadratic and quartic terms in the action. The first step is to put (52) and (55) together to get\nHaving set s = 2, we pass to position space, keeping in mind that in our conventions,\nThe result is\nwhere integration by parts was allowed without generating derivatives of g a 12 or R a 1234 because these tensors are evaluated at φ = 0. The result (137) is to be compared with the standard local sigma model action, expanded to quartic order in fields:\nIt is easy to check that (137) and (138) agree provided σ = 4π 2γ . Note that γ = − 2 Γ R n (n+2)γ has a simple pole at s = 2. This is a reminder that the original action (2) is ill-defined when s = 2. However, the s → 2 limit is well-defined and smooth in Fourier space, withγ held fixed and positive.\nIt is straightforward but unilluminating to extend the comparison of (137) and (138) through O(φ 6 ); suffice it to say that one does recover S local order by order in φ through the procedure outlined in the previous paragraph.\nLet's now use the results of previous sections to recover the well-known beta function for the local non-linear sigma model on V = R n with s = 2 and n = 2 + , for positive and sufficiently small. Comparing the first line of (102) with the first line of (122), we conclude that the metric renormalization coefficient t 0 must take the value\nwhere we used i 2 = i 0 from (51) together with (103) to conclude that K 3 = −K 4 = i 0 /6; then in the third equality of (139) we used (46) to obtain the explicit value of i 0 in dimension n = 2 + . In (139), we are only tracking i 0 and i 2 up to O( ) contributions, consistent with our aim of calculating only the leading behavior in of the one-loop beta function. It is worth noting that in (139), we are extracting t 0 exclusively from the four-point amplitude.\nThe two-point amplitude (i.e. the one-loop correction to the propagator) doesn't help at all in determining t 0 ; instead, it results in the condition (90) which determines the field renormalization coefficient u 0 once t 0 is known-specifically, u 0 = −i 0 /3. An important point is that we are able to completely cancel one-loop divergences with metric renormalization (together with field redefinition), consistent with the local non-linear sigma model being renormalizable.\nTo go further and extract the beta function for the metric, we start with a rearrangement of (85):\nLet's set φ = φ B = 0 and differentiate both sides with respect to Λ, holding renormalized quantities fixed. The result is\nUsingγ = σ/4π 2 and i 0 = 2π, we arrive at\nThis accords with the standard one-loop result as quoted for example in [6]. 9 For positive even s > 2, the action of a local, s-derivative non-linear sigma model does not appear to be uniquely determined by symmetries (though it is possible we do not fully understand how diffeomorphism symmetry is implemented in these higher derivative theories). The simplicity of our loop calculations for all positive even s suggests that there 9 When using the results of [6], one must keep in mind that the parameter T (analogous to our σ) appearing in the beta function β ij (T −1 g) = − T −1 g ij + R ij must be replaced by T /2π, so that β ij (g) = − g ij + T 2π R ij , in order to be consistent with the action S = Λ dx 1 2 T −1 g ij (φ(x))∂ µ φ i (x)∂ µ φ j (x); see the comment in this regard on p. 388. may be a privileged local non-linear sigma model in each positive even dimension whose renormalization group flow is characterized by (141)."},{"title":"Outlook","content":"Let's start with a recapitulation of the main points of our analysis. The starting point action is\nwhere Q(X, Y ) = d(X, Y ) 2 is the square of the shortest distance between points X and Y on the target manifold, and we understand that\nis defined with a suitable regulation prescription, as in section 3. Recall that = n − s.\nFocusing on the limit → 0 + , with n and s converging to some positive n 0 which is not an even integer when V = R n , we find that we are obliged to generalize the action (143) to\nwhere Q (X, Y ) = (∇ 2 X + ∇ 2 Y )Q(X, Y ), and κ ∼ O(γ). With this improved action, one-loop amplitudes at O(γ 0 ) have a divergence structure which, as far as we have taken the computations, can be absorbed entirely through field redefinitions and additive renormalization of κ, as given in the form of a renormalization group equation for the bare version of κ in (130). No metric renormalization arises in the non-local model (at one loop). This is in contrast with the local non-linear sigma model, where no improvement terms are needed, and we cancel one-loop divergences instead through field redefinitions and renormalization of the metric.\nThe one-loop divergences we encountered are all proportional to the Ricci tensor-more precisely, to covariant derivatives of it. So all of them vanish when R ab = 0. However, we did not analyze diagrams proportional to the square of the Riemann tensor. In the local case, the sum of the divergences of all diagrams (at one loop) vanishes when R ab = 0. It would seem sensible to find the same outcome for the non-local case, but settling this question is left for future work.\nIn the non-local theory (143), we expect that additional improvement terms will be needed at each new order in γ, corresponding to higher order target space derivatives of Q(X, Y ). This would be somewhat analogous to the situation in chiral perturbation theory: At each order in a derivative expansion one has finitely many parameters to adjust, but every new order introduces new free parameters. However, an alternative viewpoint may be possible: Instead of renormalizing individual parameters like κ, it may be possible to renormalize Q(X, Y ) as a bi-local function, regarding the improvement term κQ (X, Y ) as an additive renormalization of Q(X, Y ). To find a beta function for Q even at the one-loop level, we would need full control over all field redefinitions, because Λ d dΛ Q B (X B , Y B ) should be computed with the renormalized X and Y held fixed, and that means X B and Y B will have some Λ dependence. If this alternative viewpoint is indeed viable, then the relation Q(X, Y ) = d(X, Y ) 2 would be only an initial condition that we might impose at some scale. Some technical aspects of our analysis might have to change to accommodate a function Q(X, Y ) that is not locked to the square of the distance function; in particular, Q 3 (X, Y ) may not vanish in any convenient coordinate system.\nThere is a reason already at the level of the classical action to consider relaxing the relation Q(X, Y ) = d(X, Y ) 2 . Namely, d(X, Y ) 2 is in general not a globally smooth function, though it is certainly smooth when X and Y are sufficiently close to one another. Consider for example making the target space manifold M a circle parametrized by an angle θ ∈ R mod 2π. Then d(0, θ) 2 = θ 2 for θ ∈ [−π, π], but right at π one finds a discontinuity in the first derivative of d(0, θ) 2 , so that e.g. d(0, θ) = (θ − 2π) 2 for θ ∈ [π, 3π]. This example makes it seem like failure of differentiability of Q(X, Y ) is associated with a homology cycle, but that is not necessarily the case. For example, consider first a cylinder, M = S 1 × R. Clearly, d(X, Y ) 2 again has a discontinuity in its first derivative when X and Y are diametrically opposite points. Now consider an asymptotically flat surface which has a sufficiently long cylindrical region so that the same discontinuity in the first derivative of d(X, Y ) 2 shows up for diametrically opposite points in the cylindrical region. The upshot is that if an asymptotically flat, smooth, contractible surface has a sufficiently prominent bulge, d(X, Y ) 2 will not be a smooth function. See figure 4. But-we speculate-renormalization group flow of Q(X, Y ) might smooth away the raggedness of d(X, Y ) 2 without entirely losing the notion of a useful distance function between points on M .\nThere are some obvious extensions of our work which do not depend on renormalizing Q(X, Y ) as a function. First, it would be desirable to work out all the O(RR) terms in oneloop renormalization through four-point amplitudes. From a diagrammatic point of view, it seems natural that (∇ 2 X + ∇ 2 Y )Q(X, Y ) should compensate for divergences arising from tying two legs of a vertex with a single propagator. This is because when the derivatives act as ∂ 2 X or ∂ 2 Y , they eliminate two factors of X from a vertex operator, or two factors of Y , just like a Wick contraction. But this intuition leaves out the covariance of ∇ X and ∇ Y , and it doesn't Figure 4: A surface with a bulge which is cylindrical over some sufficient distance and then closes off. The distance function d(X, Y ) 2 on this surface will have discontinuities in its first derivative.\nexplain how to compensate for divergences from the type of diagram shown in figure 3c. We hope that some improved understanding is possible that will make it obvious that the detailed calculations in section 10 had to lead to counterterms that can be assembled into Q (X, Y ). Perhaps then it will be easier to track field redefinition terms, O(RR) terms, and even higher loop counterterms. Another obvious extension is to consider more carefully the local limit. As commented in section 12, we have made no attempt to work out local theories with more than two derivatives, though we note that at least the four-derivative O(N ) model in n = 4 dimensions is a long-established theory [12]. We restricted attention to a particular way of taking the → 0 + limit for local theories, namely by setting s exactly equal to a positive even integer and then sliding n = s + down toward s. Also, we have not considered possible competition between local and non local theories. For example, if s < 2 then a local, s = 2, action is classically irrelevant and one would expect that it is not generated along the renormalization group flow. Borrowing from the analysis of [1,13,14] (see also, e.g., [2,15] for a modern perspective) it is possible that non local theories with s < 2 will flow to local theories in the infrared due to quantum corrections.\nFinally, we hope that there may be some applicability of non-local theories to a fundamental issue in M-theory, namely the description of M2-branes far from the limit of nearly flat brane configurations. The M2-brane appears to be afflicted by a pathology which is ultraviolet with respect to the brane worldvolume but infrared from the point of view of spacetime [16,17]: It can throw out long thin tendrils with essentially zero action. We speculate that this issue might find its cure through a modification of the M2-brane in the"},{"title":"Figure 1 :","content":""},{"title":"Figure 2 :","content":""},{"title":"Figure 3 :","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Scalar field theories over the reals with bi-local kinetic terms were introduced in [1], and the recent work [2] provides a useful point of entry into the extensive literature. Similar field theories over the p-adic numbers were considered in [3] as a continuum description of Dyson's hierarchical model [4]. A unifying point of view on the bi-local O(N ) vector model was provided in [5], showing that the standard large N development can be framed in terms that are largely independent of whether the theory is formulated over the reals or the p-adics. The present work extends the study of bi-local theories to bi-local non-linear sigma models, starting with the action S = µ n−s 2γ V ×V d n xd n y |x − y| n+s d(φ(x), φ(y)) 2 ,\\nwhere |x − y| is the distance function on the n-dimensional base space V and d(φ(x), φ(y)) is the distance function on the target manifold. In the limit s → n, where the theory (1) becomes classically scale invariant, we find logarithmic divergences in one-loop diagrams which can be canceled by counterterms that can be expressed in terms of the target space laplacian of the square of the distance function, together with field redefinitions. 1 Ricci flatness suppresses the one-loop divergences that we encounter, so in a sense (and with significant caveats) we may claim that we are deriving the vacuum Einstein equations from conformal invariance, as in [6]. Our work was partly motivated by the more recent results of [7], which were derived for a nearest neighbor arc length model on the Bruhat-Tits tree-in other words, on the other side of the p-adic AdS/CFT duality [8,9] from our results for field theories over the p-adic numbers. However, the particular structure of counterterm we find suggests that renormalization of our theories have less to do with renormalization of the local metric as normally understood (i.e. Ricci flow) than with an augmentation of the action (1) to include the target space laplacian of d(φ(x), φ(y)) 2 .\\nA conservative expectation is that once non-local terms are allowed in a field theory, they proliferate and the theory becomes non-renormalizable. Theories with purely quadratic bi-local kinetic terms, as studied in [1,3] (as well as many subsequent works) avoid such problems through a non-renormalization theorem: If we write\\nthen the claim is that the quadratic bi-local term is never renormalized (at least perturbatively), though the purely local term U (φ(x)) certainly is-and depending on details, derivative terms might be radiatively generated. Non-local interaction terms vitiate this non-renormalization theorem, and one's suspicions could be renewed that there is no sensible theory. We will not be able in this work entirely to allay such concerns, because we do not give a demonstration parallel to the one in [6] that Ward identities based on diffeomorphism invariance guarantee that loop divergences can only modify the original form of the action. Indeed, the counterterms we generate at one loop do modify the bi-local action in an unexpected way, but one which appears to be controlled in a derivative expansion, so that higher derivative terms can be radiatively generated at each new order without spoiling results from lower orders. We will revisit the question of renormalizability in section 13. The organization of the rest of this paper is as follows. In section 2 we present the main results in Fourier analysis that we need, both over the reals and the p-adics. In section 3 we explain how double integrals such as the one in (1) can be regulated if divergences arise as |x − y| → 0. In section 4 we introduce the classical action for the bi-local non-linear sigma model. In section 5 we discuss loop divergences in general terms, including an introductory account of the non-renormalization property of the kinetic term in (2). In sections 6-10 we investigate the simplest one-loop divergences of the bi-local non-linear sigma model, and then in section 11 we argue that all these divergences can be canceled by a laplacian counterterm in place of renormalization of the local metric, together with field redefinitions. As a byproduct of our analysis, we recover in section 12 the usual beta function for the two-derivative theory in two dimensions. We conclude in section 13 with a summary of possible future directions.\"\n },\n {\n \"title\": \"Fourier transforms\",\n \"content\": \"In loop calculations we will often need to go back and forth between momentum space expressions similar to the ones presented in (2) and their real space counterparts, using the Fourier transforms\\nThe relevant results are fairly similar between real and p-adic cases, so we present them together. When V = R n , the definitions (3) are entirely standard, and k·x can be understood as the ordinary dot product. Likewise, in this case, |x| is understood as the standard L 2 norm on R n . We refer to the real case as Archimedean because the norm | · | has the property that if 0 < |x| < |y|, then there is some n ∈ Z such that |y| < |nx|. The simplest n-dimensional p-adic construction is based on letting V = Q p n be the (unique) unramified n-dimensional extension of Q p . Let N and Tr be the field norm and field trace with respect to the extension Q p n /Q p . Then we define |x| = | N(x)| 1/n p where | · | p is the usual p-adic norm. We will refer to the p-adic case as ultrametric because the norm | · | just defined has the property |x + y| ≤ max{|x|, |y|}. Next we define k · x = 1 n Tr(kx). Note that k · x ∈ Q p , so to give meaning to e 2πik·x we now only need to define e 2πiξ for ξ ∈ Q p . To this end we find the unique p-adic integer ξ such that ξ − ξ ∈ [0, 1) ∩ Q, and we understand that by e 2πiξ we really mean e 2πi(ξ− ξ ) .\\nWe are particularly interested in the Fourier transform of powers of |k|: V d n k e 2πik·x |k| s = Γ V (n + s) |x| n+s + (contact terms) .\\nHere Γ V (s) is a meromorphic function of s which can be evaluated as\\nwhere we set v = ∞ in the Archimedean case and v = p in the ultrametric case, with\\nIntuitively, Γ V is a variant of the Euler gamma, specific to the choice of V , and constructed so as to be the coefficient of the 1/|x| n+s term in (4). In the remainder of our discussion, integrals are over V unless otherwise indicated. The contact terms in (4) are somewhat delicate and dependent on detail. When −n < s < 0, the integral in (4) is convergent, and no contact terms are needed. One can easily check that Γ V (n + s) → 0 as s → 0, so when s = 0 the power law term goes away and we recover the obvious result d n k e 2πik·x = δ n (x) .\\nFor s > 0, the integral in (4) diverges, and we need a more careful approach. A good first step is to understand (4) in terms of its action on a test function φ : V → R:\\nwhere D s is some linear map on functions φ(x). A suitable class of test functions are socalled Schwartz-Bruhat functions. When φ : Q p n → R, we require that φ is locally constant with compact support. For example, the characteristic function of the p-adic integers is a Schwartz-Bruhat function on Q p . When φ : R n → R, the test functions are more appropriately called Schwartz functions, and their defining property is that they go to 0 faster than any power of |x|, as do all their derivatives. An example is a Gaussian. Both in the real and ultrametric cases, the Fourier transformφ(k) of a Schwartz-Bruhat function is again a Schwartz-Bruhat function.\\nWith (8) taken as the definition of D s , our task is to find a representation of D s entirely in position space. In the ultrametric case for arbitrarily positive s, one finds\\nThis is the Vladimirov derivative. In the Archimedean case, the same expression (9) is valid for 0 < s < 2. There is one more easy case to dispose of: even positive integer s for Archimedean V . Then Γ V (n + s) = 0, which makes sense in (4) because the right hand should be purely distributional, on account of |k| s = (k 2 ) s/2 being analytic in k 2 . Explicitly,\\nfor positive even s ,\\nwhere = n i=1 ∂ 2 x i . We are left with the task of defining D s for Archimedean V and for s > 2 but not an even integer. Heuristically, the contact terms in (4) are a sum of terms of the form r δ n (x), where 0 ≤ r < s/2, with divergent coefficients. To state this more precisely, we write\\nwhere a regulated integral\\nis rendered finite (if possible) by allowing the subtraction from G(x, y) of a finite sum of smooth functions of either of the following types:\\nI. Pure powers: more precisely, any function whose y dependence comes solely through a factor |x − y| α where α is a real number. This is meant to include, through the case α = 0, functions which have no y dependence.\\nII. Higher partial waves: more precisely, any function of the form Y ( y − x)g(|x−y|) where Y (ẑ) is a spherical harmonic on S n−1 other than the s-wave.\\nType I functions are never integrable, whereas type II functions may or may not be; so at best there is a unique choice of type I functions that will work, whereas many choices of type II functions are possible. An alternative approach, generalizing the principle value prescription, is to eschew modifications of the integrand and instead carry out y integration in polar coordinates centered around x, as follows. One first performs the angular integrals. Then the radial integral is restricted to run from l to L. One next allows the subtraction of an arbitrary finite sum of negative powers of l and/or positive powers of L, chosen (if possible) so that the limits l → 0 and L → ∞, taken independently, lead to a finite result. Doing the angular integration first obviates the need for type II functions, while the ultraviolet and infrared cutoffs, l and L, obviate the need for type I. 2\\nWhile the subtractions described can in principle cure either ultraviolet (UV) or infrared (IR) divergences, we will be interested only in applications where UV divergences matter: that is, divergences arising when |x − y| → 0 (with x held fixed). Type II subtractions are relatively innocuous because they follow automatically from performing angular integrations first; therefore we will use the notation d n y . . . to indicate a y integration with type II subtractions which we usually omit to write explicitly.\\nAlthough we have stated our integration prescriptions in the abstract, it is easy to see how to apply them to (11) when φ is a Schwartz function. Consider the case 2 < s < 3, and set x = 0 for simplicity. Then (11) becomes\\nThe extra terms in square brackets on the right hand side of (13) evidently render the integral convergent near y = 0 for 2 < s < 3. The term linear in y is clearly a type II function, and the term quadratic in y is a sum of a type II function proportional to y i 1 y i 2 − y 2 n δ i 1 i 2 (a d-wave term) and a type I function proportional to y 2 . If 3 ≤ s < 4, then we would need one additional term in the Taylor series expansion of φ around y = 0, and this additional term is a type II function. In summary, for 2 < s < 4, and omitting type II subtractions,\\nEvidently, if 0 < s < 2, a simpler subtraction scheme would work, resulting in (14) with the laplacian term omitted, in agreement with (9). For general s > 0 (other than positive even integers) and Archimedean V ,\\nIn principle, one may derive (15) by subtracting an appropriate number of terms in the Taylor series expansion of φ(y) and then finding appropriate type II subtractions to bring the result into the form (15).\\ncannot be cured using powers of l and/or L after a cutoff integration. Because we avoid even integer s as well as functions G(x, y) which grow as positive powers of large separation |x − y|, we do not need to specify a resolution to this inequivalence.\\nA more efficient way to determine the coefficients b r is to start from (15) and Fourier transform:\\nIn the second equality of (17), we have partially carried out the y integral in polar coordinates around the point y = x, introducing a radial variableỹ = |x − y|. In the third equality, we have carried out the angular θ integral and introduced a new radial variable, ρ = 2π|k|ỹ. The ρ integral in the last line of (17) converges, provided s is positive but not an even integer, and provided the coefficients a r are coefficients in the Taylor series expansion of the Bessel function around ρ = 0. These coefficients a r are well known, and from them one can recover the expression (16) for the b r .\"\n },\n {\n \"title\": \"Bi-local integrals\",\n \"content\": \"We are particularly interested in double integrals of the form\\nwhere s > 0 and G(x, y) is piecewise constant if V is ultrametric and smooth if V is Archimedean. Unless otherwise noted, all double integrals over x and y will by taken over all of V × V . In the ultrametric case, for any s > 0, following (9) we define\\nIn the Archimedean case, we define d n xd n y |x − y| n+s G(x, y)\\nby performing the y integration first and allowing the subtraction of type I and type II functions to G(x, y) in order to achieve a finite result (if possible). As in the previous section, type II subtractions are deemed relatively inconsequential, so even unprimed integration over x and y means to perform the y integration first, allowing the subtraction of type II functions in order to achieve a finite result (if possible). Explicitly, for s not a positive even integer,\\nwhere the coefficients b r are as given in (16). We avoid positive even integer s when V is Archimedean because in this case we expect that our constructions will lead instead to purely local theories; also, precisely in this case, the subtleties pointed out in footnote 2 regarding logarithmic divergences come into play.\\nOur computational strategy will turn on converting bi-local position space integrals into Fourier space integrals. Let's start with the simplest example of that calculation, valid for ultrametric V and any s > 0, and also for Archimedean V and 0 < s < 2. Let φ : V → R be a Schwartz-Bruhat function. Then\\nThe first step is actually the trickiest, because it is not clear from the rules of integration set forth following (20) that we are allowed to add a function like φ(x) 2 − φ(y) 2 to the integrand. To justify this step, we denote f (x) = φ(x) 2 , and we argue that\\nThe second integral in (23) is the k = 0 component of the Fourier transform of D s f (x). But this Fourier transform is |k| sf (k), and since s > 0 the k = 0 component indeed vanishes.\\nLet's now pursue the same computation for the Archimedean case with 2 < s < 4. On one hand, using (21),\\nOn the other hand, using (14),\\nIn order to conclude\\nwe must therefore argue that the final integrals in (24) and (25) agree. Subtracting (25) from (24) and simplifying slightly with the definition f (x) = φ(x) 2 , we arrive at\\nThe first equality in (27) follows from (14), and the second is by the same argument used following (23). To summarize, for Archimedean V and for 2 < s < 4,\\nΓ V (n + s) > 0 for 2 < s < 4, and so without the (∂φ) 2 on the left hand side of (28) we would have a sign problem. The equality (26) can be checked in a similar manner for s > 4. A key relation is\\nTwo take-away lessons are:\\n• When we write simple |k| s kinetic terms in momentum space, in position space we are combining non-local position space terms and local terms involving derivatives in a precisely tuned ratio.\\n• There is some freedom in the precise structure of the position space form, as exemplified by the equality of the last integrals in (24) and (25) due to a manipulation which is the non-local version of integration by parts.\"\n },\n {\n \"title\": \"The bi-local non-linear sigma model\",\n \"content\": \"We are now in a position to present the action for the bi-local non-linear sigma model. Let M be a smooth D-dimensional manifold with a Riemannian metric g ab , whose Riemann and Ricci tensors are\\nGiven any two points X and Y on M , let\\nbe the square of the shortest distance between X and Y . Clearly, Q(X, Y ) is a smooth function of X and Y , provided X and Y are not too far apart. For smooth functions φ : V → M whose range is sufficiently localized, we consider the action functional\\nwhere indicates a regulated double integral of the type discussed around (19)-(21). 3 Note that this discussion requires us to avoid positive even integer s when x and y are valued in R n (as opposed to Q p n ). In the Archimedean setting, when s > 2, there are derivative terms like (∂φ) 2 implicitly built into (32), with coefficients tuned so as to ensure convergence of the integral. The parameter µ has dimensions of energy so that we can regard φ and Q(φ(x), φ(y)) as dimensionless. The factor γ is a loop-counting parameter: Classical effects are O(γ −1 ), one-loop amplitudes are O(γ 0 ), two loop amplitudes are O(γ), and so forth. In other words, γ plays the role of . 3 One may wonder whether the primed integral, as defined following (20), spoils coordinate invariance of the integrand. For instance, if s is sufficiently large we may, in light of (21), be required to include a y Q term to the integrand, which if written only in terms of partial derivatives does not appear to be coordinate invariant. In fact, it is easy to convince oneself that, e.g., y Q can be constructed from covariant quantities:\\nA close cousin of the action (32) was considered in [7]:\\nwhere now x and y are vertices of a graph and the sum is over undirected edges. The formula actually appears earlier in [6], though it was intended there to be considered on a square lattice, as a regulator for the local non-linear sigma model, rather than on the Bruhat-Tits tree as in [7]. We require the range of the maps φ to be sufficiently localized in order to ensure that we do not encounter any failures of smoothness in Q(X, Y ), and in order to ensure that we can use a single system of Riemann normal coordinates for φ throughout. One can now solve the geodesic equation perturbatively in the curvature and use that to evaluate Q(X, Y ) and expand the action (32),\\nwhere\\nand\\nHere g ab and R abcd are evaluated at the origin of the Riemann normal coordinates, which we assume is at X = 0. Often, the definition of Riemann normal coordinates includes the requirement g ab = δ ab , but this is not necessary for our calculations, and we find it more convenient to retain explicit factors of g ab and the inverse metric g ab . Put differently, we are choosing a coordinate system so that all geodesics passing through the origin are linear in the affine parameter:\\nThe ellipsis in (34) indicates higher order interactions, involving five or more powers of φ(x) and/or φ(y), as well as derivatives and powers of the curvature. We will consider up to six point interactions in section 10. quadratic kinetic term in the action (2), where V = R n or Q p n , with\\nThe purely cubic theory is unstable, but it serves our purpose because we are only interested in analyzing the behavior of the one-loop correction to the propagator. Using the diagram shown in figure 1a, we obtain the one-loop contribution to the quadratic part of the oneparticle irreducible (1PI) effective action:\\nWe continue the convention of integrating over all of V except as otherwise indicated. Let's assume n > 2s, so I is UV divergent (and IR convergent). To regulate the divergence, we introduce a hard cutoff: | | ≤ Λ. If V = R n , then Λ can be any positive real number. If V = Q p n , then we will require that Λ is an integer power of p. The ultrametric case is easy to analyze, because when | | > |k| we have | | = |k − | exactly. So, except in the compact region where | | ≤ |k|, the integrand has no k dependence at all. Therefore, any UV divergences are entirely independent of k, and to evaluate them we can set k = 0:\\nThe last equality comes from splitting the integration region into shells with fixed | |; then the integral becomes a geometric sum. Because the divergent part of I(Λ) has no k-dependence, the counterterm required to cancel it is proportional to d n kφ(−k)φ(k) = d n x φ(x) 2 . In other words, it is a mass term. This argument is easy to generalize to the statement that only purely local terms (powers of φ(x)) can be radiatively generated starting from the action (2) over V = Q p n . An essentially equivalent argument was made in a Wilsonian picture in [3]. The Archimedean case is more subtle because of the possibility of subleading divergences. A straightforward approach is to expand\\nin powers of k. Terms with an odd number of powers of k vanish by parity, leaving only terms analytic in k 2 . Of these, only terms proportional to k 2r with r ≤ n 2 − s are UV divergent. In short, the divergent part of I(Λ) is a polynomial in k 2 whose order is n 2 − s . A divergent term proportional to k 2r requires a counterterm proportional to d n x φ(x) r φ(x). These are the radiatively generated derivative terms mentioned in section 1.\\nWe should note a troublesome feature of the hard momentum cutoff for Archimedean theories: The coefficients one finds for sub-leading divergences depend on how one implements the cutoff. For example, it is easy to check that the coefficient of the k 2 term in I(Λ) changes if instead of requiring | | ≤ Λ we impose the more democratic condition − k 2 ≤ Λ. However, the feature that we care about, namely the fact that the divergent terms have only polynomial dependence on k 2 , doesn't depend on the details of the cutoff. It is perhaps instructive to consider one other alternative, namely dimensional regularization, in which one first computes\\nby continuing to a domain of n in which the integral is convergent. (In the current example, s < n < 2s is such a domain.) The only divergences one then tracks are poles of the right hand side of (42) as a function of n. These occur precisely when n−2s 2 is a non-negative integer. It is characteristic of dimensional regularization that there is (at most) one divergent term for a given n, corresponding to a logarithmic divergence in the original integral.\\nThe loop diagrams we will need to consider in our analysis of the non-local non-linear sigma model are simpler than (39) in one regard: The loop is a single propagator starting and ending at the same vertex. This matters because there is then only one internal momentum , and imposing the hard cutoff | | ≤ Λ is a privileged choice because it corresponds to integrating over an O(n)-invariant region. An example is the diagram shown in figure 1b, which is proportional to\\nassuming that whatever vertex factor is needed to fully evaluate the diagram doesn't depend on . We also assume n > s so that I 0 is UV divergent but IR convergent. We straightforwardly find\\nThere are obviously no subleading divergences in I 0 .\\nFor convenience we introduce\\nWe are interested in divergences proportional to log Λ that arise when = n − s = 0. As a technical trick to isolate these divergences, we make small and positive, and we look for divergences of the form Λ / , which in the → 0 + limit give rise to log Λ terms.\\nTo characterize this limit precisely, given n 0 > 0 and λ ∈ R, we set n = n 0 + λ and s = n 0 + (λ − 1) and then take the → 0 + limit with n 0 and λ held fixed. (Clearly then we are allowing non-integer n, in the spirit of [6].) For the most part, our final results are independent of λ. When is sufficiently small, we may replace (44) with\\nIf we lift the requirement that Λ is an integer power of p when V = Q p n , then (46) is unaltered, because I 0 = ζp( ) ζp(n) p log p Λ , and p log p Λ differs from Λ at most by a factor of p = 1 + O( ). The important point is that in the limit → 0 + , I 0 includes a logarithmic term i 0 log Λ, and isolating this term is our stated objective.\\nWe will encounter one other loop integral:\\nIt comes from graphs similar to the one in (43), but with a vertex prefactor |k − | s . Using the same reasoning that led to (40), we see that when a hard cutoff | | ≤ Λ is imposed, one obtains in the ultrametric case\\nIf V = R n and n is positive but not an even integer, then\\nfor some coefficients c r . If we choose n 0 positive but not an even integer and fix any finite value of λ, then for sufficiently small , (49) applies, and the least singular power of Λ appearing in it is Λ n−2 n/2 . As → 0 + , this power remains positive and finite, tending to Λ n 0 −2 n 0 /2 . So there is no log Λ behavior, even in the → 0 + limit. If instead we make n 0 a positive even integer, then by choosing the very particular value λ = 1, so that s = n 0 exactly, we find (for sufficiently small > 0) that the least singular term in (49) is c n/2 |k| s Λ , which does contribute a c n/2 |k| s log Λ divergence in the → 0 + limit; moreover, in this case, by calculation, c n/2 = i 0 + O( ). 4 We can summarize the situation, both for the Archimedean and ultrametric cases, by stating that for = n − s positive but sufficiently small, then subject to the restriction that n cannot be a positive even integer when V = R n ,\\nwhere\\nThe higher powers of Λ in (50) are accompanied by non-negative integer powers of k 2 , and they correspond to operators which remain relevant in the → 0 + limit. The only log Λ behavior arising from I 2 (k), in the → 0 + limits described above, is the i 2 |k| s log Λ term coming from the i 2 |k| s Λ term shown in (50). We are not concerned about O( ) terms to i 0 and i 2 because they drop out of the log Λ behavior in the → 0 + limit. In the following sections, therefore, we will drop O( ) terms from (46) and (51), and we will evaluate i 0 and i 2 in terms of n rather than n 0 .\"\n },\n {\n \"title\": \"The propagator through one loop\",\n \"content\": \"To derive the tree-level propagator, we use an obvious generalization of (26) to multicomponent scalar fields to rewrite the free action in momentum space:\\nwhere we recall that = n − s, and for notational convenience we have introduced 5\\nWe immediately extract from (52) the propagator\\nWe are primarily interested in small, so that G ab (x) is nearly logarithmic. Informally, we can understand the one-loop correction to the propagator as a contribution to the 1PI effective action coming from all possible Wick contractions of the two of the four factors of φ in S int . The calculation is done most straightforwardly in momentum space, where we can express\\nwhere d 4n k = 4 i=1 d n k i . As usual, derivative terms are implied in in the Archimedean 5 A point worthy of remark is that whileγ and γ have the same sign in the ultrametric case, and also in the Archimedean case for 0 < s < 2, for 2 < s < 4 they have the opposite sign. The integral in (52) is well-defined and positive, so to make our theory sensible we should always chooseγ > 0. This means that γ < 0 for 2 < s < 4. As explained in (28) for a single real scalar, the regulated position space integral used to define the action (32) includes a (∂φ) 2 term that enters with the opposite sign of the non-local [φ(x) − φ(y)] 2 term, so positivity conditions are difficult to judge in position space. case when s > 2. Symbolically, the Wick-contracted quartic action is\\n. We omit theφ aφb andφ cφd contractions from (56) because of the antisymmetry of R abcd in ab and cd. We omit theφ aφc andφ bφd contractions because they include a factor |k 1 +k 3 | s δ n (k 1 +k 3 ), which vanishes when s > 0. After some straightforward algebra, we obtain from (56) the form\\nwhere I 2 (k) is given in (47).\\nAs discussed below (47), for suitably small positive = n − s, I 2 (k) includes a term i 2 |k| s Λ iff V = R n and s is a positive even integer. This is the case which leads to local non-linear sigma models. Otherwise-excluding the case V = R n with positive even n-the divergent terms in I 2 (k) are proportional to |k| 2r Λ n−2r for non-negative powers n − 2r which remain finite as → 0 + . Therefore, apart from the case of local non-linear sigma models, the effects of the ultraviolet divergences in (57) are limited to generating relevant, local interactions. We assume that relevant terms of this type can be tuned away.\\nIt would be tempting at this point to conclude that the non-local action (32) is nonrenormalized, as in the case (2). The reality is more subtle: We will see in section 10 that higher point diagrams generate one loop divergences that require non-local counterterms; however they are not quite of the form (32), involving instead the target space laplacian of Q(X, Y ).\"\n },\n {\n \"title\": \"Curvature and arc length calculations\",\n \"content\": \"Higher point amplitudes in the bi-local non-linear sigma model involve tensors of large rank. We will therefore find it convenient to introduce some abbreviated notation, based on the following equivalences:\\nHere (ab) = 1 2 (ab + ba). We employ obvious extensions of (58) to larger index sets, e.g. X a 123 means X a 1 X a 2 X a 3 .\\nWe will often need to simplify expressions involving the curvature tensor and its covariant derivatives. A primary tool is the Bianchi identities, which we may write using our abbreviated notation as\\nA contracted form of the second Bianchi identity,\\nshows that any three-index contraction of ∇ a 5 R a 1234 (meaning any contraction leaving three indices free) can be expressed as linear combinations of re-indexed versions of the tensor ∇ a 1 R a 23 ; in this sense ∇ a 1 R a 23 by itself is a basis for all the three-index contractions of ∇ a 1 R a 2345 . This observation will be useful to us when we consider the possible Wick contractions of the five-point interaction vertex in the bi-local non-linear sigma model. Acting on the contracted second Bianchi identity (60) with ∇ b 1 gives\\nWe describe the terms in the second line of (61) as commutator terms. Evidently, they can be written as curvature bilinears, meaning contractions of two factors of the Riemann and/or Ricci tensors, with no covariant derivatives. Acting on the uncontracted second Bianchi identity (the second line of (59)) with ∇ a 5 gives\\nwhere ∇ 2 = ∇ b ∇ b , and the commutator terms are similar to the ones occurring in (61): In particular, they are curvature bilinears. The results (61) and (62) show that all four-index contractions of ∇ a 56 R a 1234 can be expressed in terms of linear combinations of re-indexed versions of the tensor ∇ a 12 R a 34 , together with curvature bilinears. So far, all formulas in this section have been entirely independent of the choice of coordinate system. We now pass to Riemann normal coordinates in order to study the square of the arc length, Q(X, Y ) = d(X, Y ) 2 between two points X and Y . We have from [10] 6\\nHere g a 12 , R a 1234 , and its derivatives are all evaluated at the origin of Riemann normal coordinates, which is the origin in terms of the coordinates X a and Y a used in (63).\"\n },\n {\n \"title\": \"Three-point vertices\",\n \"content\": \"There are no three-point vertices at tree-level provided we employ Riemann normal coordinates. As we will explain in this section, three-point vertices appear to be generated at the one-loop level, by the diagram in figure 2b; however, they can be absorbed through field redefinition.\\nAs for our discussion in section 6 of the one-loop corrections to the propagator, the one-loop contribution to the three-point function can be obtained efficiently through Wick 6 Note however that the results leading to Q RR contractions in the momentum space of the quintic term in the action:\\nHigh-dimensional Fourier integrals of the type seen in the second line of (64) are common in our calculations, so we have found it useful to introduce some shorthand notation:\\nwith obvious extensions to larger index sets. If A is any ordered set of indices, like 123, and |A| is the number of indices in the set, then the integrals we see most often are of the form\\nwhere q(k A ) is any function of the k i , and the third expression is just a rewriting of the second using the shorthand notation introduced in (65). Evidently, the J φ integrals are convergent for reasonable integrands q, like powers of norms of sums of momenta, provided the φ a are Schwartz-Bruhat functions. When there is no risk of confusion, we will omit the subscripted k A and just write J a A φ . We can now rewrite (64) as\\nAs in section 6, a Wick contraction amounts to a replacement\\nIt is helpful to note the following examples of Wick contraction:\\nHere A and C are collections of indices, neither including 1 or 2, with C ⊂ A, and we assume s > 0 in order to obtain the vanishing of the first integral. Recall from section 5 that I 0 diverges as Λ / when = n−s is sufficiently small and positive, giving rise to log Λ behavior in the limit → 0 + . There are 5 2 = 10 possible single Wick contractions of S 5 , but the (12) contraction (meaning the contraction of a 1 and a 2 ) vanishes because g a 12 ∇ a 5 R a 1234 = 0; likewise the (34) contraction vanishes. Meanwhile the (14) and (23)\\nA few comments are in order:\\n• Because S Wick 5, (14) and S Wick 5,(25) are proportional to the I 2 loop integral, they do not contribute logarithmic divergences in the → 0 + limits described in section 5, except when s is a positive even integer. In a position space account, these non-logarithmic terms correspond to contractions of φ(x) with φ(y).\\n• We are mostly interested in S Wick 5,(13) and S Wick 5,(15) because I 0 does produce a logarithmic divergence in the → 0 + limit. Note that in these terms, the vertex factor |k 13 | s involves only external momenta. This is the crucial feature, noted already in the introduction, which allows non-local counterterms to arise. In position space, the logarithmic terms correspond to contractions of φ(y) with itself.\\nIn summary,\\nThe minimal counterterm needed to cancel the Λ / divergences in (72) is\\nwhere\\nand i 0 and i 2 are as defined in (46) and (51). We use the notation S ct 3 to denote a counterterm that is cubic in the fields. Our primary interest is in cases where i 2 = 0-namely, cases in which V = Q p n , or V = R n but s is not a positive even integer. However, tracking I 0 , I 2 (k), i 0 , and i 2 throughout our computations is useful as a bookkeeping device in order to simultaneously treat the local and bi-local theories, in the → 0 + limit, with i 2 entering in only to describe the local theories.\"\n },\n {\n \"title\": \"Renormalization through cubic order in the fields\",\n \"content\": \"Before entering into the more complicated story of quartic terms in the action, let's preview the endgame of our analysis, in which we produce a bare action which incorporates the renormalized action plus the counterterms in a form that we can express entirely in terms of arc length.\\nFirst, let's rephrase the Wick-contracted quartic action (57) as\\nfrom which we conclude that we need a counterterm quadratic in the fields of the form\\nwhere\\nThe results (76) and (73) together put some constraints on the renormalization procedure, but as we will see they do not completely determine it. The question of renormalizability is whether we can reorganize the renormalized action plus counterterms into a bare action whose form is the same that we started with:\\nwhere\\nis the sum of all counterterms, S[φ] is as given in (32), and\\nwhere\\nThe bare arc length d B , derived from a bare metric tensor g B ab , may differ from the renormalized arc length d, and the bare coordinates φ a B may likewise differ from the renormalized coordinates φ a . We require however that the points φ B = 0 and φ = 0 coincide. At tree level, where we ignore all counterterms, we have the relations\\nand our key task is to find perturbative corrections to these relations that render (78) correct.\\nTo begin, let's examine the quadratic terms in (78), using the counterterm S ct 2 from (76):\\nwhere for simplicity we multiplied through by 2γ/µ . We surmise from (82) that corrections to (81) can be expressed as a power series in the dimensionless quantitỹ\\nThat is,\\nwhere V a b 12 and W a b 123 are fully symmetric in their lower indices, and U , V , and W are all independent of φ (and φ B ). In other words, (84) is a Taylor expansion of φ a B in the coordinates φ a . Also,\\nfor some tensor T ab (φ). As our notation indicates, T ab (φ) does depend on φ. As with other tensors, if we omit the argument, we mean that T ab is evaluated at φ = 0. Using (84) and\\n(85), we see that (82) is satisfied provided\\nor in other words provided\\nwhere\\nIt should be kept in mind that T a 12 is the φ = 0 value of a tensor field T a 12 (φ) defined over the whole of M , whereas U a (12) is defined only at φ = 0. Let's assume that\\n(A term in T a 12 (φ) proportional to R(φ)g a 12 (φ) is also possible, but the divergences we will encounter do not require it.) Then (87) reduces to\\nAs previously noted, based on the treatment of quadratic terms alone, we cannot distinguished between metric renormalization (related to the coefficient t 0 ) and field redefinition (related to the coefficient u 0 ). In order to proceed to higher orders, we require the squared arc length formula for bare quantities:\\nWe do not require φ B to be Riemann normal coordinates for d B , so there are contributions to Q B at cubic order:\\nwhere Γ B abc = g B ad Γ Bd bc and Γ Ba bc is the Christoffel connection for g B ab . From (85) we have immediately\\nFrom Q B 3 we obtain a cubic term in the bare action:\\nAnother term cubic in φ arises in the bare action from plugging the non-linear field redefinition (84) into the quadratic term S B 2 . To work this out, it helps first to note that passing (84) through a Fourier transform yieldŝ\\nand * denotes convolution. It follows immediately that\\nand so\\nwhere we are lowering indices on V and W with the renormalized metric g ab . The T a 12 +2U a 12 term in (98) is the same combination we saw in (87), with the symmetrization U a 12 → U a (12) implied because we multiply by J a 12 φ [|k 2 | s ], which is symmetric. The next term in (98) is the interesting one for us. The only constraint on V a 312 is symmetry in the 12 indices. This is the same symmetry that J a 123 ically, we expand (78) to cubic order in the renormalized fields, using the expression (73) for S ct 3 , as well as S B 3 from (94) and the O(φ 3 ) term from (98). The result is\\nEvidently, we may set\\nwhere\\nThe larger message is that V a 312 is sufficiently unconstrained that we could use it to absorb any counterterm proportional to J a 123 φ [|k 3 | s ]. By the same token, when we get to quartic order, the field redefinition coefficients W a 4123 can be used to absorb any terms proportional to J a 1234 φ [|k 4 | s ]. Therefore, when we do proceed to quartic order, we may simplify our work by systematically dropping all such terms. We will see in section 10 that other terms emerge, proportional to J a 1234 φ [|k 24 | s ]. These are the ones that cannot be absorbed into field redefinitions.\"\n },\n {\n \"title\": \"Quartic counterterms\",\n \"content\": \"Four-point vertices are present at tree-level, and they are also generated by three different types of one-loop diagrams, as shown in figure 3. Our goal in this section is to evaluate one-loop divergences proportional to ∇ a 12 R a 34 (and re-indexings of it). The claim is that only the diagram in figure 3a can contribute. Tracking only target space indices, the vertex factor in this diagram is ∇ a 56 R a 1234 , and the internal propagator can tie any two of the six indices together. So the diagram as a whole must be proportional to four-index contractions of ∇ a 56 R a 1234 . As explained in section 7, re-indexed versions of ∇ a 12 R a 34 , together with cur-vature bilinears, provide a basis for such contractions. The diagrams in figure 3b and 3c are manifestly proportional to curvature bilinears, so they cannot contribute terms proportional to ∇ a 12 R a 34 . (One immediate way to see this is that the Riemann tensor could vanish at φ = 0 without its derivatives vanishing.) In the explicit calculations below, we will encounter and discard many curvature bilinear terms, which we generically write as O(RR), meaning some contraction of R a 1234 R a 5678 .\\nWe claim that the counterterms needed to cancel the divergences from the diagram in figure 3a proportional to ∇ a 12 R a 34 (and re-indexings of it) are\\nwhere\\nColor-coding in (102) is to help track to which term in (102) each of the many terms in later equations contribute. The remainder of this section is devoted to deriving (103).\\nTo derive the sixth-order vertex used in figure 3a, we start from the Q ∇∇R 6 term in (63) and extract the following six-order terms in the action:\\nwhere\\nOur task is to compute the counterterms for all the single Wick contractions of S 4+2 6,∇∇R and S 3+3 6,∇∇R . For S 4+2 6,∇∇R , relations among Wick contractions that are obvious from symmetries plus the first line of (69) are as follows:\\nIf we write the counterterm for an expression Q as {Q} ct , then the rules of computation we need are a trivial adaptation of (69):\\nThe counterterms that we need to cancel divergences coming from the single Wick contractions of S 4+2 6,∇∇R , as shown in (107)\\nwhere\\nwhere the color coding is to show which terms in (110) contribute to which terms of (102). By direct computation using the rules (108),\\nwhich agrees with (103).\"\n },\n {\n \"title\": \"Renormalization at quartic order\",\n \"content\": \"To renormalize at quartic order, we first inquire whether the counterterms (102) can be organized into the bare arc length action (80), using the field redefinition (84) and the relationship (85) between the bare and renormalized metric. For the non-local model, we will find that this is impossible! So we will turn to a generalized form of the bare action that includes a term proportional to the target space laplacian of the square of the arc length.\\nTo get started, we need the bare arc length formula to quartic order in the bare fields: That is, we need one more term in the series (91) than we computed in section 9. The computation of arc length is less simple than for the renormalized metric because the φ a B are not Riemann normal coordinates for g B ab -due to effects at O(γ), in particular a connection Γ Ba b 12 ∼ O(γ). As a technical device, we therefore introduce a third set of coordinates, φ a , which are Riemann normal coordinates for the bare metric, which in barred coordinates takes the form g ab (φ). We can express φ a in terms of φ a B as\\nand we can write g B ab (φ B ) in terms of g ab (φ) as\\nNote that, by assumption, g B a 12 = g a 12 at φ B = φ = 0. The condition that φ a are Riemann normal coordinates allows us to concludẽ\\nsee for example [11] for a derivation. The Γ B Γ B term in the expression (117) for M a b 123 is optional because it is O(γ 2 ), but it arises naturally in the derivation of [11], so we include it.\\nThe bare arc length coincides between φ and φ B coordinate systems because these are just different coordinate systems for the same metric, as per (116). Explicitly,\\nwhere in the second equality we used the fact that φ a are Riemann normal coordinates. The (118) is short for all terms involving five or more powers of X and Y combined; similar notation is used below. Using the first equation in (115) to eliminate X and Y in favor of X B and Y B , we arrive at\\nwhere L a 312 = g B a 3 b L b a 12 and M a 4123 = g B a 4 b M b a 123 . Note that the cubic terms in (119) agree with (92), and recall from the subsequent analysis that the corresponding cubic term S B 3 does not need to match S 3 + S ct 3 , because of the additional term cubic in φ in (98) arising from the O(γ) difference between φ B and φ. Likewise, the term in (119) proportional to M a 4123 gives rise to a term proportional to J φ B [|k 4 | s ] in the action, but we do not need to track it explicitly because the quartic term in (98) shows that it is precisely the sort of term that we can absorb into the field redefinition coefficient W a b 123 . Thus we may write\\nwhere (field redef) indicates field redefinition terms as discussed above. Next we express S B 4 in terms of renormalized quantities in order to compare to (102). Starting from (85), we obtain Λ µ R B a 1234 = R a 1234 +γ 2 (∇ a 14 T a 23 − ∇ a 24 T a 13 − ∇ a 13 T a 24 + ∇ a 23 T a 14 )\\nThus we find\\nO(RR) terms arise in (122) not just from those in (121), but also from expressing J a 1234 φ B\\n[|k 24 | s ] in terms of the renormalized field φ. 7 We have color-coded terms in (122) to match the way we did in (102). Comparing the two equations, we can see that S B 4 accommodates the counterterms S ct 6,∇∇R iff K 3 = −K 4 . Based on (103), this happens iff i 0 = i 2 , which means iff s is a positive even integer. As we will develop more fully in section 12, this indeed corresponds to the case of local non-linear sigma models.\\nLet us pursue further here what happens for the non-local case. Because i 2 = 0, we have K 1 /K 2 = −6/7, there is no hope of rendering the theory renormalizable with just the arc length action we have been using so far. Some generalization of the arc length action is needed. Whatever modification we make should involve two target space derivatives relative to the original action, so as to absorb counterterms that appear with two extra derivativeslike the ∇ a 12 R a 34 structure in S ct 6,∇∇R as compared to R a 1234 in S 4 . Luckily, there is a new term with the right properties which we can add to S B :\\nwhere we define\\nThe −4n term in (124) is present in order to ensure Q B (X B , Y B ) = 0 when X B and Y B coincide. 8 By explicit calculation (as sketched below (132)),\\nPassing to momentum space, we find\\nCombining (122) and (126) and comparing with (102), we see that κ B = −γ i 0 2 . Strikingly, we are forced also to choose t 0 = 0: That is, the metric is not renormalized! Having allowed the two-derivative term (123) in S B , we should allow addition of a similar term to the renormalized action:\\nWe restrict κ to be an O(γ) quantity, which makes sense because then the overall scaling with γ of (127) is O(γ 0 ), and this aligns with the invariance of Q (X, Y ) under overall rescaling of the target manifold. The additional term (127) changes all the one-loop amplitudes, but only by O(γ) quantities, relative to the O(γ 0 ) scaling of one-loop amplitudes and counterterms that we obtained in previous sections. In short, the only effect of allowing non-zero κ in our counterterm analysis is to lead to a direct tree-level contribution to κ B , so that in total,\\nTo rephrase this result in terms of the renormalization group, we can rewrite (128) as\\nand then since the right hand side is independent of Λ, we arrive at\\nThe first term on the right hand side of (130) is the tree-level term coming from the engineering dimension factor of (Λ/µ) in (128). The one-loop effects are responsible for the second term in (130). If we now take → 0 in (130), we see that κ B runs logarithmically:\\nwhere Λ 0 is a dynamically generated scale. Note thatγ and i 0 are positive, so κ B is positive at scales Λ below Λ 0 and negative above Λ 0 . To see that (125) is correct, let's work on the renormalized side and note that\\nThe expression (132) is part of ∇ 2 X Q(X, Y ), and it is easy to see that it is the only part contributing terms of the form (∇∇R)XXY Y . Therefore\\nThe ellipses in (132) and (133) indicate terms that are not quadratic in both X and Y , for example terms schematically of the form (∇∇R)XY Y Y or (∇∇R)Y Y Y Y , as well as lower order terms which are either independent of X or Y , or linear in X or Y . Plugging (133) into (127), we find\\nThe contributions labeled (field redef) in (134) are linear in φ(x) or φ(y). To see that (134) is correct, we have only to understand why we can freely add or drop from the integrand smooth functions which depend only on x or only on y, such as the direct terms φ a 1234 (x) + φ a 1234 (y) which are present in the last expression in (134) but not the middle expression. As in section 3, this follows from careful use of the regulated integral prescription:\\nfor smooth functions f (x) with suitable falloff conditions at large x. The integral d n xd n y |x−y| n+s f (x) vanishes more trivially by subtraction of the r = 0 term in the sum appearing in square brackets in (135).\"\n },\n {\n \"title\": \"Recovering the local non-linear sigma model\",\n \"content\": \"When V = R n and s is a positive even integer, the original position space action (32) is problematic, because our scheme for defining regulated integrals breaks down. However, this is precisely the case where we expect to recover a local theory, expressible in terms of φ(x) and its derivatives. In this section we will verify this expectation for the simplest case, s = 2. We proceed in three steps: First, we start from the momentum space form of the action, obtained for general s from the non-local action (32); then we take s = 2; and finally we Fourier transform back to a local position-space action. It is helpful (though not really necessary) to work with Riemann normal coordinates throughout. We exhibit the process in detail for the quadratic and quartic terms in the action. The first step is to put (52) and (55) together to get\\nHaving set s = 2, we pass to position space, keeping in mind that in our conventions,\\nThe result is\\nwhere integration by parts was allowed without generating derivatives of g a 12 or R a 1234 because these tensors are evaluated at φ = 0. The result (137) is to be compared with the standard local sigma model action, expanded to quartic order in fields:\\nIt is easy to check that (137) and (138) agree provided σ = 4π 2γ . Note that γ = − 2 Γ R n (n+2)γ has a simple pole at s = 2. This is a reminder that the original action (2) is ill-defined when s = 2. However, the s → 2 limit is well-defined and smooth in Fourier space, withγ held fixed and positive.\\nIt is straightforward but unilluminating to extend the comparison of (137) and (138) through O(φ 6 ); suffice it to say that one does recover S local order by order in φ through the procedure outlined in the previous paragraph.\\nLet's now use the results of previous sections to recover the well-known beta function for the local non-linear sigma model on V = R n with s = 2 and n = 2 + , for positive and sufficiently small. Comparing the first line of (102) with the first line of (122), we conclude that the metric renormalization coefficient t 0 must take the value\\nwhere we used i 2 = i 0 from (51) together with (103) to conclude that K 3 = −K 4 = i 0 /6; then in the third equality of (139) we used (46) to obtain the explicit value of i 0 in dimension n = 2 + . In (139), we are only tracking i 0 and i 2 up to O( ) contributions, consistent with our aim of calculating only the leading behavior in of the one-loop beta function. It is worth noting that in (139), we are extracting t 0 exclusively from the four-point amplitude.\\nThe two-point amplitude (i.e. the one-loop correction to the propagator) doesn't help at all in determining t 0 ; instead, it results in the condition (90) which determines the field renormalization coefficient u 0 once t 0 is known-specifically, u 0 = −i 0 /3. An important point is that we are able to completely cancel one-loop divergences with metric renormalization (together with field redefinition), consistent with the local non-linear sigma model being renormalizable.\\nTo go further and extract the beta function for the metric, we start with a rearrangement of (85):\\nLet's set φ = φ B = 0 and differentiate both sides with respect to Λ, holding renormalized quantities fixed. The result is\\nUsingγ = σ/4π 2 and i 0 = 2π, we arrive at\\nThis accords with the standard one-loop result as quoted for example in [6]. 9 For positive even s > 2, the action of a local, s-derivative non-linear sigma model does not appear to be uniquely determined by symmetries (though it is possible we do not fully understand how diffeomorphism symmetry is implemented in these higher derivative theories). The simplicity of our loop calculations for all positive even s suggests that there 9 When using the results of [6], one must keep in mind that the parameter T (analogous to our σ) appearing in the beta function β ij (T −1 g) = − T −1 g ij + R ij must be replaced by T /2π, so that β ij (g) = − g ij + T 2π R ij , in order to be consistent with the action S = Λ dx 1 2 T −1 g ij (φ(x))∂ µ φ i (x)∂ µ φ j (x); see the comment in this regard on p. 388. may be a privileged local non-linear sigma model in each positive even dimension whose renormalization group flow is characterized by (141).\"\n },\n {\n \"title\": \"Outlook\",\n \"content\": \"Let's start with a recapitulation of the main points of our analysis. The starting point action is\\nwhere Q(X, Y ) = d(X, Y ) 2 is the square of the shortest distance between points X and Y on the target manifold, and we understand that\\nis defined with a suitable regulation prescription, as in section 3. Recall that = n − s.\\nFocusing on the limit → 0 + , with n and s converging to some positive n 0 which is not an even integer when V = R n , we find that we are obliged to generalize the action (143) to\\nwhere Q (X, Y ) = (∇ 2 X + ∇ 2 Y )Q(X, Y ), and κ ∼ O(γ). With this improved action, one-loop amplitudes at O(γ 0 ) have a divergence structure which, as far as we have taken the computations, can be absorbed entirely through field redefinitions and additive renormalization of κ, as given in the form of a renormalization group equation for the bare version of κ in (130). No metric renormalization arises in the non-local model (at one loop). This is in contrast with the local non-linear sigma model, where no improvement terms are needed, and we cancel one-loop divergences instead through field redefinitions and renormalization of the metric.\\nThe one-loop divergences we encountered are all proportional to the Ricci tensor-more precisely, to covariant derivatives of it. So all of them vanish when R ab = 0. However, we did not analyze diagrams proportional to the square of the Riemann tensor. In the local case, the sum of the divergences of all diagrams (at one loop) vanishes when R ab = 0. It would seem sensible to find the same outcome for the non-local case, but settling this question is left for future work.\\nIn the non-local theory (143), we expect that additional improvement terms will be needed at each new order in γ, corresponding to higher order target space derivatives of Q(X, Y ). This would be somewhat analogous to the situation in chiral perturbation theory: At each order in a derivative expansion one has finitely many parameters to adjust, but every new order introduces new free parameters. However, an alternative viewpoint may be possible: Instead of renormalizing individual parameters like κ, it may be possible to renormalize Q(X, Y ) as a bi-local function, regarding the improvement term κQ (X, Y ) as an additive renormalization of Q(X, Y ). To find a beta function for Q even at the one-loop level, we would need full control over all field redefinitions, because Λ d dΛ Q B (X B , Y B ) should be computed with the renormalized X and Y held fixed, and that means X B and Y B will have some Λ dependence. If this alternative viewpoint is indeed viable, then the relation Q(X, Y ) = d(X, Y ) 2 would be only an initial condition that we might impose at some scale. Some technical aspects of our analysis might have to change to accommodate a function Q(X, Y ) that is not locked to the square of the distance function; in particular, Q 3 (X, Y ) may not vanish in any convenient coordinate system.\\nThere is a reason already at the level of the classical action to consider relaxing the relation Q(X, Y ) = d(X, Y ) 2 . Namely, d(X, Y ) 2 is in general not a globally smooth function, though it is certainly smooth when X and Y are sufficiently close to one another. Consider for example making the target space manifold M a circle parametrized by an angle θ ∈ R mod 2π. Then d(0, θ) 2 = θ 2 for θ ∈ [−π, π], but right at π one finds a discontinuity in the first derivative of d(0, θ) 2 , so that e.g. d(0, θ) = (θ − 2π) 2 for θ ∈ [π, 3π]. This example makes it seem like failure of differentiability of Q(X, Y ) is associated with a homology cycle, but that is not necessarily the case. For example, consider first a cylinder, M = S 1 × R. Clearly, d(X, Y ) 2 again has a discontinuity in its first derivative when X and Y are diametrically opposite points. Now consider an asymptotically flat surface which has a sufficiently long cylindrical region so that the same discontinuity in the first derivative of d(X, Y ) 2 shows up for diametrically opposite points in the cylindrical region. The upshot is that if an asymptotically flat, smooth, contractible surface has a sufficiently prominent bulge, d(X, Y ) 2 will not be a smooth function. See figure 4. But-we speculate-renormalization group flow of Q(X, Y ) might smooth away the raggedness of d(X, Y ) 2 without entirely losing the notion of a useful distance function between points on M .\\nThere are some obvious extensions of our work which do not depend on renormalizing Q(X, Y ) as a function. First, it would be desirable to work out all the O(RR) terms in oneloop renormalization through four-point amplitudes. From a diagrammatic point of view, it seems natural that (∇ 2 X + ∇ 2 Y )Q(X, Y ) should compensate for divergences arising from tying two legs of a vertex with a single propagator. This is because when the derivatives act as ∂ 2 X or ∂ 2 Y , they eliminate two factors of X from a vertex operator, or two factors of Y , just like a Wick contraction. But this intuition leaves out the covariance of ∇ X and ∇ Y , and it doesn't Figure 4: A surface with a bulge which is cylindrical over some sufficient distance and then closes off. The distance function d(X, Y ) 2 on this surface will have discontinuities in its first derivative.\\nexplain how to compensate for divergences from the type of diagram shown in figure 3c. We hope that some improved understanding is possible that will make it obvious that the detailed calculations in section 10 had to lead to counterterms that can be assembled into Q (X, Y ). Perhaps then it will be easier to track field redefinition terms, O(RR) terms, and even higher loop counterterms. Another obvious extension is to consider more carefully the local limit. As commented in section 12, we have made no attempt to work out local theories with more than two derivatives, though we note that at least the four-derivative O(N ) model in n = 4 dimensions is a long-established theory [12]. We restricted attention to a particular way of taking the → 0 + limit for local theories, namely by setting s exactly equal to a positive even integer and then sliding n = s + down toward s. Also, we have not considered possible competition between local and non local theories. For example, if s < 2 then a local, s = 2, action is classically irrelevant and one would expect that it is not generated along the renormalization group flow. Borrowing from the analysis of [1,13,14] (see also, e.g., [2,15] for a modern perspective) it is possible that non local theories with s < 2 will flow to local theories in the infrared due to quantum corrections.\\nFinally, we hope that there may be some applicability of non-local theories to a fundamental issue in M-theory, namely the description of M2-branes far from the limit of nearly flat brane configurations. The M2-brane appears to be afflicted by a pathology which is ultraviolet with respect to the brane worldvolume but infrared from the point of view of spacetime [16,17]: It can throw out long thin tendrils with essentially zero action. We speculate that this issue might find its cure through a modification of the M2-brane in the\"\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]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"(a) One-loop contribution to the 1PI two-point function. (b) A diagram where the loop consists of a single propagator closing on itself.","content":"Figure 1 :\n1(a) One-loop contribution to the 1PI two-point function. (b) A diagram where the loop consists of a single propagator closing on itself."},{"id":"","caption":"(a) The one-loop contribution to the propagator. (b) The one-loop contribution to the 1PI three-point vertex.","content":"Figure 2 :\n2(a) The one-loop contribution to the propagator. (b) The one-loop contribution to the 1PI three-point vertex."},{"id":"","caption":"contain errors. In particular, 44 should have been 4 in the first line of (11.24).","content":"contain errors. In particular, 44 should have been 4 in the first line of (11.24)."},{"id":"","caption":"contractions are equal because both ∇ a 5 R a 1234 and |k 24 | s are symmetrical under the simultaneous exchange of 1 ↔ 3 and 2 ↔ 4. For the same reason, the (15) and (35) contractions are equal, and so are the (25) and (45) contractions. Finally, the (24) contraction vanishes because of the first line of (69).","content":"contractions are equal because both ∇ a 5 R a 1234 and |k 24 | s are symmetrical under the simultaneous exchange of 1 ↔ 3 and 2 ↔ 4. For the same reason, the (15) and (35) contractions are equal, and so are the (25) and (45) contractions. Finally, the (24) contraction vanishes because of the first line of (69)."},{"id":"","caption":"φ [|k 3 | s ] possesses. Therefore J a 123 φ [|k 3 | s ]V a 312 is the most general linear combination of terms coming from J a 123 φ [|k 3 | s ] integrals. Likewise, the only constraint on W a 4123 is symmetry in 123, so the last term shown explicitly in (98) is the most general linear combination of terms coming from J a 1234 φ [|k 4 | s ] integrals. We now have all the ingredients needed to calculate the O(φ 3 ) correction to (82). Specif-One-loop contributions to the 1PI four-point vertex: (a) Single Wick contractions of the ∇∇R six-point vertices; (b) Single Wick contractions of RR sixpoint vertices; (c) Diagrams involving only four-point vertices.","content":"Figure 3 :\n3φ [|k 3 | s ] possesses. Therefore J a 123 φ [|k 3 | s ]V a 312 is the most general linear combination of terms coming from J a 123 φ [|k 3 | s ] integrals. Likewise, the only constraint on W a 4123 is symmetry in 123, so the last term shown explicitly in (98) is the most general linear combination of terms coming from J a 1234 φ [|k 4 | s ] integrals. We now have all the ingredients needed to calculate the O(φ 3 ) correction to (82). Specif-One-loop contributions to the 1PI four-point vertex: (a) Single Wick contractions of the ∇∇R six-point vertices; (b) Single Wick contractions of RR sixpoint vertices; (c) Diagrams involving only four-point vertices."},{"id":"","caption":") = (23) (15) = (35) = (16) = (36) (25) = (45) = (26) = (46) .","content":") = (23) (15) = (35) = (16) = (36) (25) = (45) = (26) = (46) ."},{"id":"","caption":"∇∇R,(13) + 2I 3+3,ct 6,∇∇R,(14) + 4I 3+3,ct 6,∇∇R,(15) + 4I 3+3,ct 6,∇∇R,(25) + I 3+3,ct 6,∇∇R,(56) . (112)","content":"∇∇R,(13) + 2I 3+3,ct 6,∇∇R,(14) + 4I 3+3,ct 6,∇∇R,(15) + 4I 3+3,ct 6,∇∇R,(25) + I 3+3,ct 6,∇∇R,(56) . (112)"},{"id":"","caption":"56 R a 1234 J a 1 a 2 a 34 a 5 a 6 φ [|k 246 | s ] ct = −∇ ba 6 R a 1 b a 34 J a 1346 φ [|k 46 | s ]i 2 = (∇ a 12 R a 34 − ∇ a 13 R a 24 )J a 1234 φ [|k 24 | s ]i 2 + O(RR) I 3+3,ct 6,∇∇R,(56) = 1 γ ∇ a 56 R a 1234 J a 1234 a 5 a 6 φ [|k 246 | s ] ct = −∇ 2 R a 1234 J a 1234 φ [|k 24 | s ]i 2 = (−2∇ a 13 R a 24 + 2∇ a 12 R a 34 )J a 1234 φ [|k 24 | s ]i 2 + O(RR) .","content":"56 R a 1234 J a 1 a 2 a 34 a 5 a 6 φ [|k 246 | s ] ct = −∇ ba 6 R a 1 b a 34 J a 1346 φ [|k 46 | s ]i 2 = (∇ a 12 R a 34 − ∇ a 13 R a 24 )J a 1234 φ [|k 24 | s ]i 2 + O(RR) I 3+3,ct 6,∇∇R,(56) = 1 γ ∇ a 56 R a 1234 J a 1234 a 5 a 6 φ [|k 246 | s ] ct = −∇ 2 R a 1234 J a 1234 φ [|k 24 | s ]i 2 = (−2∇ a 13 R a 24 + 2∇ a 12 R a 34 )J a 1234 φ [|k 24 | s ]i 2 + O(RR) ."},{"id":"","caption":"109)-(110) and (112)-(113) together and comparing with (102), we arrive at","content":"109)-(110) and (112)-(113) together and comparing with (102), we arrive at"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"(a) One-loop contribution to the 1PI two-point function. (b) A diagram where the loop consists of a single propagator closing on itself.\",\n \"content\": \"Figure 1 :\\n1(a) One-loop contribution to the 1PI two-point function. (b) A diagram where the loop consists of a single propagator closing on itself.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(a) The one-loop contribution to the propagator. (b) The one-loop contribution to the 1PI three-point vertex.\",\n \"content\": \"Figure 2 :\\n2(a) The one-loop contribution to the propagator. (b) The one-loop contribution to the 1PI three-point vertex.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"contain errors. In particular, 44 should have been 4 in the first line of (11.24).\",\n \"content\": \"contain errors. In particular, 44 should have been 4 in the first line of (11.24).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"contractions are equal because both ∇ a 5 R a 1234 and |k 24 | s are symmetrical under the simultaneous exchange of 1 ↔ 3 and 2 ↔ 4. For the same reason, the (15) and (35) contractions are equal, and so are the (25) and (45) contractions. Finally, the (24) contraction vanishes because of the first line of (69).\",\n \"content\": \"contractions are equal because both ∇ a 5 R a 1234 and |k 24 | s are symmetrical under the simultaneous exchange of 1 ↔ 3 and 2 ↔ 4. For the same reason, the (15) and (35) contractions are equal, and so are the (25) and (45) contractions. Finally, the (24) contraction vanishes because of the first line of (69).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"φ [|k 3 | s ] possesses. Therefore J a 123 φ [|k 3 | s ]V a 312 is the most general linear combination of terms coming from J a 123 φ [|k 3 | s ] integrals. Likewise, the only constraint on W a 4123 is symmetry in 123, so the last term shown explicitly in (98) is the most general linear combination of terms coming from J a 1234 φ [|k 4 | s ] integrals. We now have all the ingredients needed to calculate the O(φ 3 ) correction to (82). Specif-One-loop contributions to the 1PI four-point vertex: (a) Single Wick contractions of the ∇∇R six-point vertices; (b) Single Wick contractions of RR sixpoint vertices; (c) Diagrams involving only four-point vertices.\",\n \"content\": \"Figure 3 :\\n3φ [|k 3 | s ] possesses. Therefore J a 123 φ [|k 3 | s ]V a 312 is the most general linear combination of terms coming from J a 123 φ [|k 3 | s ] integrals. Likewise, the only constraint on W a 4123 is symmetry in 123, so the last term shown explicitly in (98) is the most general linear combination of terms coming from J a 1234 φ [|k 4 | s ] integrals. We now have all the ingredients needed to calculate the O(φ 3 ) correction to (82). Specif-One-loop contributions to the 1PI four-point vertex: (a) Single Wick contractions of the ∇∇R six-point vertices; (b) Single Wick contractions of RR sixpoint vertices; (c) Diagrams involving only four-point vertices.\"\n },\n {\n \"id\": \"\",\n \"caption\": \") = (23) (15) = (35) = (16) = (36) (25) = (45) = (26) = (46) .\",\n \"content\": \") = (23) (15) = (35) = (16) = (36) (25) = (45) = (26) = (46) .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"∇∇R,(13) + 2I 3+3,ct 6,∇∇R,(14) + 4I 3+3,ct 6,∇∇R,(15) + 4I 3+3,ct 6,∇∇R,(25) + I 3+3,ct 6,∇∇R,(56) . (112)\",\n \"content\": \"∇∇R,(13) + 2I 3+3,ct 6,∇∇R,(14) + 4I 3+3,ct 6,∇∇R,(15) + 4I 3+3,ct 6,∇∇R,(25) + I 3+3,ct 6,∇∇R,(56) . (112)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"56 R a 1234 J a 1 a 2 a 34 a 5 a 6 φ [|k 246 | s ] ct = −∇ ba 6 R a 1 b a 34 J a 1346 φ [|k 46 | s ]i 2 = (∇ a 12 R a 34 − ∇ a 13 R a 24 )J a 1234 φ [|k 24 | s ]i 2 + O(RR) I 3+3,ct 6,∇∇R,(56) = 1 γ ∇ a 56 R a 1234 J a 1234 a 5 a 6 φ [|k 246 | s ] ct = −∇ 2 R a 1234 J a 1234 φ [|k 24 | s ]i 2 = (−2∇ a 13 R a 24 + 2∇ a 12 R a 34 )J a 1234 φ [|k 24 | s ]i 2 + O(RR) .\",\n \"content\": \"56 R a 1234 J a 1 a 2 a 34 a 5 a 6 φ [|k 246 | s ] ct = −∇ ba 6 R a 1 b a 34 J a 1346 φ [|k 46 | s ]i 2 = (∇ a 12 R a 34 − ∇ a 13 R a 24 )J a 1234 φ [|k 24 | s ]i 2 + O(RR) I 3+3,ct 6,∇∇R,(56) = 1 γ ∇ a 56 R a 1234 J a 1234 a 5 a 6 φ [|k 246 | s ] ct = −∇ 2 R a 1234 J a 1234 φ [|k 24 | s ]i 2 = (−2∇ a 13 R a 24 + 2∇ a 12 R a 34 )J a 1234 φ [|k 24 | s ]i 2 + O(RR) .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"109)-(110) and (112)-(113) together and comparing with (102), we arrive at\",\n \"content\": \"109)-(110) and (112)-(113) together and comparing with (102), we arrive at\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Critical Exponents for Long-Range Interactions. M E Fisher, S Ma, B G Nickel, 10.1103/PhysRevLett.29.917Phys. Rev. Lett. 29M. E. Fisher, S.-k. Ma, and B. G. Nickel, \"Critical Exponents for Long-Range Interactions,\" Phys. Rev. Lett. 29 (1972) 917-920.","doi":"10.1103/PhysRevLett.29.917"},{"text":"Conformal Invariance in the Long-Range Ising Model. M F Paulos, S Rychkov, B C Van Rees, B Zan, 10.1016/j.nuclphysb.2015.10.018arXiv:1509.00008Nucl. Phys. 902hep-thM. F. Paulos, S. Rychkov, B. C. van Rees, and B. Zan, \"Conformal Invariance in the Long-Range Ising Model,\" Nucl. Phys. B902 (2016) 246-291, arXiv:1509.00008 [hep-th].","doi":"10.1016/j.nuclphysb.2015.10.018"},{"text":"Scalar Models of p-adic Quantum Field Theory and Hierarchical Models. E Yu, M D Lerner, Missarov, 10.1007/BF01018683Theor. Math. Phys. 78Teor. Mat. Fiz.E. Yu. Lerner and M. D. Missarov, \"Scalar Models of p-adic Quantum Field Theory and Hierarchical Models,\" Theor. Math. Phys. 78 (1989) 177-184. [Teor. Mat. Fiz.78,248(1989)].","doi":""},{"text":"Existence of a phase transition in a one-dimensional Ising ferromagnet. F J Dyson, 10.1007/BF01645907Commun. Math. Phys. 12F. J. Dyson, \"Existence of a phase transition in a one-dimensional Ising ferromagnet,\" Commun. Math. Phys. 12 (1969) 91-107.","doi":""},{"text":"O(N) and O(N) and O(N). S S Gubser, C Jepsen, S Parikh, B Trundy, 10.1007/JHEP11(2017)107arXiv:1703.04202JHEP. 11107hep-thS. S. Gubser, C. Jepsen, S. Parikh, and B. Trundy, \"O(N) and O(N) and O(N),\" JHEP 11 (2017) 107, arXiv:1703.04202 [hep-th].","doi":""},{"text":"Nonlinear Models in Two + Epsilon Dimensions. D H Friedan, 10.1016/0003-4916(85)90384-7Annals Phys. 163318D. H. Friedan, \"Nonlinear Models in Two + Epsilon Dimensions,\" Annals Phys. 163 (1985) 318.","doi":""},{"text":"A Huang, B Stoica, S.-T Yau, arXiv:1901.02013General relativity from p-adic strings. hep-thA. Huang, B. Stoica, and S.-T. Yau, \"General relativity from p-adic strings,\" arXiv:1901.02013 [hep-th].","doi":""},{"text":"S S Gubser, J Knaute, S Parikh, A Samberg, P Witaszczyk, 10.1007/s00220-016-2813-6arXiv:1605.01061p-adic AdS/CFT. hep-thS. S. Gubser, J. Knaute, S. Parikh, A. Samberg, and P. Witaszczyk, \"p-adic AdS/CFT,\" Communications in Mathematical Physics (2017) , arXiv:1605.01061 [hep-th].","doi":""},{"text":"Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS 3 /CFT 2 correspondence. M Heydeman, M Marcolli, I Saberi, B Stoica, 10.4310/ATMP.2018.v22.n1.a4arXiv:1605.07639Adv. Theor. Math. Phys. 22hep-thM. Heydeman, M. Marcolli, I. Saberi, and B. Stoica, \"Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS 3 /CFT 2 correspondence,\" Adv. Theor. Math. Phys. 22 (2018) 93-176, arXiv:1605.07639 [hep-th].","doi":""},{"text":"Riemann Normal Coordinate expansions using Cadabra. L Brewin, 10.1088/0264-9381/26/17/175017arXiv:0903.2087Class. Quant. Grav. 26175017gr-qcL. Brewin, \"Riemann Normal Coordinate expansions using Cadabra,\" Class. Quant. Grav. 26 (2009) 175017, arXiv:0903.2087 [gr-qc].","doi":""},{"text":"Riemann normal coordinates. L Brewin, L. Brewin, \"Riemann normal coordinates.\" http://users.monash.edu.au/~leo/research/papers/files/lcb96-01.pdf.","doi":""},{"text":"A four-dimensional nonlinear σ-model. E Gava, R Jengo, 10.1016/0550-3213(78)90010-XNucl. Phys. 140E. Gava and R. Jengo, \"A four-dimensional nonlinear σ-model,\" Nucl. Phys. B140 (1978) 510-524.","doi":""},{"text":"Recursion relations and fixed points for ferromagnets with long-range interactions. J Sak, 10.1103/PhysRevB.8.281Phys. Rev. B. 8J. Sak, \"Recursion relations and fixed points for ferromagnets with long-range interactions,\" Phys. Rev. B 8 (Jul, 1973) 281-285.","doi":""},{"text":". https:/link.aps.org/doi/10.1103/PhysRevB.8.281https://link.aps.org/doi/10.1103/PhysRevB.8.281.","doi":""},{"text":"Crossover between field theories with short-range and long-range exchange or correlations. J Honkonen, M Y Nalimov, 10.1088/0305-4470/22/6/024Journal of Physics A: Mathematical and General. 226J. Honkonen and M. Y. Nalimov, \"Crossover between field theories with short-range and long-range exchange or correlations,\" Journal of Physics A: Mathematical and General 22 no. 6, (Mar, 1989) 751-763.","doi":""},{"text":"A scaling theory for the long-range to short-range crossover and an infrared duality. C Behan, L Rastelli, S Rychkov, B Zan, 10.1088/1751-8121/aa8099arXiv:1703.05325J. Phys. 5035354002hep-thC. Behan, L. Rastelli, S. Rychkov, and B. Zan, \"A scaling theory for the long-range to short-range crossover and an infrared duality,\" J. Phys. A50 no. 35, (2017) 354002, arXiv:1703.05325 [hep-th].","doi":""},{"text":"On the Quantum Mechanics of Supermembranes. B De Wit, J Hoppe, H Nicolai, 10.1016/0550-3213(88)90116-2Nucl. Phys. 305545B. de Wit, J. Hoppe, and H. Nicolai, \"On the Quantum Mechanics of Supermembranes,\" Nucl. Phys. B305 (1988) 545.","doi":""},{"text":"The Supermembrane Is Unstable. B De Wit, M Luscher, H Nicolai, 10.1016/0550-3213(89)90214-9Nucl. Phys. 320B. de Wit, M. Luscher, and H. Nicolai, \"The Supermembrane Is Unstable,\" Nucl. Phys. B320 (1989) 135-159.","doi":""}],"string":"[\n {\n \"text\": \"Critical Exponents for Long-Range Interactions. M E Fisher, S Ma, B G Nickel, 10.1103/PhysRevLett.29.917Phys. Rev. Lett. 29M. E. Fisher, S.-k. Ma, and B. G. Nickel, \\\"Critical Exponents for Long-Range Interactions,\\\" Phys. Rev. Lett. 29 (1972) 917-920.\",\n \"doi\": \"10.1103/PhysRevLett.29.917\"\n },\n {\n \"text\": \"Conformal Invariance in the Long-Range Ising Model. M F Paulos, S Rychkov, B C Van Rees, B Zan, 10.1016/j.nuclphysb.2015.10.018arXiv:1509.00008Nucl. Phys. 902hep-thM. F. Paulos, S. Rychkov, B. C. van Rees, and B. Zan, \\\"Conformal Invariance in the Long-Range Ising Model,\\\" Nucl. Phys. B902 (2016) 246-291, arXiv:1509.00008 [hep-th].\",\n \"doi\": \"10.1016/j.nuclphysb.2015.10.018\"\n },\n {\n \"text\": \"Scalar Models of p-adic Quantum Field Theory and Hierarchical Models. E Yu, M D Lerner, Missarov, 10.1007/BF01018683Theor. Math. Phys. 78Teor. Mat. Fiz.E. Yu. Lerner and M. D. Missarov, \\\"Scalar Models of p-adic Quantum Field Theory and Hierarchical Models,\\\" Theor. Math. Phys. 78 (1989) 177-184. [Teor. Mat. Fiz.78,248(1989)].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Existence of a phase transition in a one-dimensional Ising ferromagnet. F J Dyson, 10.1007/BF01645907Commun. Math. Phys. 12F. J. Dyson, \\\"Existence of a phase transition in a one-dimensional Ising ferromagnet,\\\" Commun. Math. Phys. 12 (1969) 91-107.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"O(N) and O(N) and O(N). S S Gubser, C Jepsen, S Parikh, B Trundy, 10.1007/JHEP11(2017)107arXiv:1703.04202JHEP. 11107hep-thS. S. Gubser, C. Jepsen, S. Parikh, and B. Trundy, \\\"O(N) and O(N) and O(N),\\\" JHEP 11 (2017) 107, arXiv:1703.04202 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Nonlinear Models in Two + Epsilon Dimensions. D H Friedan, 10.1016/0003-4916(85)90384-7Annals Phys. 163318D. H. Friedan, \\\"Nonlinear Models in Two + Epsilon Dimensions,\\\" Annals Phys. 163 (1985) 318.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Huang, B Stoica, S.-T Yau, arXiv:1901.02013General relativity from p-adic strings. hep-thA. Huang, B. Stoica, and S.-T. Yau, \\\"General relativity from p-adic strings,\\\" arXiv:1901.02013 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"S S Gubser, J Knaute, S Parikh, A Samberg, P Witaszczyk, 10.1007/s00220-016-2813-6arXiv:1605.01061p-adic AdS/CFT. hep-thS. S. Gubser, J. Knaute, S. Parikh, A. Samberg, and P. Witaszczyk, \\\"p-adic AdS/CFT,\\\" Communications in Mathematical Physics (2017) , arXiv:1605.01061 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS 3 /CFT 2 correspondence. M Heydeman, M Marcolli, I Saberi, B Stoica, 10.4310/ATMP.2018.v22.n1.a4arXiv:1605.07639Adv. Theor. Math. Phys. 22hep-thM. Heydeman, M. Marcolli, I. Saberi, and B. Stoica, \\\"Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS 3 /CFT 2 correspondence,\\\" Adv. Theor. Math. Phys. 22 (2018) 93-176, arXiv:1605.07639 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Riemann Normal Coordinate expansions using Cadabra. L Brewin, 10.1088/0264-9381/26/17/175017arXiv:0903.2087Class. Quant. Grav. 26175017gr-qcL. Brewin, \\\"Riemann Normal Coordinate expansions using Cadabra,\\\" Class. Quant. Grav. 26 (2009) 175017, arXiv:0903.2087 [gr-qc].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Riemann normal coordinates. L Brewin, L. Brewin, \\\"Riemann normal coordinates.\\\" http://users.monash.edu.au/~leo/research/papers/files/lcb96-01.pdf.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A four-dimensional nonlinear σ-model. E Gava, R Jengo, 10.1016/0550-3213(78)90010-XNucl. Phys. 140E. Gava and R. Jengo, \\\"A four-dimensional nonlinear σ-model,\\\" Nucl. Phys. B140 (1978) 510-524.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Recursion relations and fixed points for ferromagnets with long-range interactions. J Sak, 10.1103/PhysRevB.8.281Phys. Rev. B. 8J. Sak, \\\"Recursion relations and fixed points for ferromagnets with long-range interactions,\\\" Phys. Rev. B 8 (Jul, 1973) 281-285.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". https:/link.aps.org/doi/10.1103/PhysRevB.8.281https://link.aps.org/doi/10.1103/PhysRevB.8.281.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Crossover between field theories with short-range and long-range exchange or correlations. J Honkonen, M Y Nalimov, 10.1088/0305-4470/22/6/024Journal of Physics A: Mathematical and General. 226J. Honkonen and M. Y. Nalimov, \\\"Crossover between field theories with short-range and long-range exchange or correlations,\\\" Journal of Physics A: Mathematical and General 22 no. 6, (Mar, 1989) 751-763.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A scaling theory for the long-range to short-range crossover and an infrared duality. C Behan, L Rastelli, S Rychkov, B Zan, 10.1088/1751-8121/aa8099arXiv:1703.05325J. Phys. 5035354002hep-thC. Behan, L. Rastelli, S. Rychkov, and B. Zan, \\\"A scaling theory for the long-range to short-range crossover and an infrared duality,\\\" J. Phys. A50 no. 35, (2017) 354002, arXiv:1703.05325 [hep-th].\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the Quantum Mechanics of Supermembranes. B De Wit, J Hoppe, H Nicolai, 10.1016/0550-3213(88)90116-2Nucl. Phys. 305545B. de Wit, J. Hoppe, and H. Nicolai, \\\"On the Quantum Mechanics of Supermembranes,\\\" Nucl. Phys. B305 (1988) 545.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Supermembrane Is Unstable. B De Wit, M Luscher, H Nicolai, 10.1016/0550-3213(89)90214-9Nucl. Phys. 320B. de Wit, M. Luscher, and H. Nicolai, \\\"The Supermembrane Is Unstable,\\\" Nucl. Phys. B320 (1989) 135-159.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":583,"cells":{"text":{"kind":"string","value":"\nAn in vivo Comparison Study Between Strontium Nanoparticles and rhBMP2\nJune 2020\n\nSilvia Minardi \nHakan Ceylan \nGloria Belen Ramirez \nBeth Bragdon bragdon@bu.edu \nLivia Visai livia.visai@unipv.it \nGiulia Montagna \nDepartment of Molecular Medicine (DMM)\nCenter for Health Technologies (CHT)\nUdR INSTM\nUniversity of Pavia\nPaviaItaly\n\nDepartment of Electrical, Computer and Biomedical Engineering\nDepartment of Chemistry, Physical-Chemistry Section\nUniversity of Pavia\nPaviaItaly, 3 C.S.G.I\n\nUniversity of Pavia\nPaviaItaly\n\nFrancesco Cristofaro \nDepartment of Molecular Medicine (DMM)\nCenter for Health Technologies (CHT)\nUdR INSTM\nUniversity of Pavia\nPaviaItaly\n\n† \nLorenzo Fassina \nDepartment of Electrical, Computer and Biomedical Engineering\nDepartment of Chemistry, Physical-Chemistry Section\nUniversity of Pavia\nPaviaItaly, 3 C.S.G.I\n\nUniversity of Pavia\nPaviaItaly\n\nGiovanna Bruni \nLucia Cucca \nDepartment of Chemistry\nUniversity of Pavia\nPaviaItaly\n\nAlejandro Kochen \nDepartment of Translational Dental Medicine\nGoldman School of Dental Medicine\nBoston University\nBostonMAUnited States\n\nPaola Divieti Pajevic \nDepartment of Translational Dental Medicine\nGoldman School of Dental Medicine\nBoston University\nBostonMAUnited States\n\nBeth Bragdon \nDepartment of Orthopaedic Surgery\nSchool of Medicine\nBoston University\nBostonMAUnited States\n\nLivia Visai \nDepartment of Molecular Medicine (DMM)\nCenter for Health Technologies (CHT)\nUdR INSTM\nUniversity of Pavia\nPaviaItaly\n\nDepartment of Occupational Medicine, Toxicology and Environmental Risks\nIstituti Clinici Scientifici Maugeri, IRCCS\nPaviaItaly\n\nLouis Gerstenfeld \nDepartment of Orthopaedic Surgery\nSchool of Medicine\nBoston University\nBostonMAUnited States\n\n\nMax Planck Institute for Intelligent Systems\nNorthwestern University\nUnited States, Germany\n\n\nUniversity of Granada\nSpain\n\nAn in vivo Comparison Study Between Strontium Nanoparticles and rhBMP2\n\nFrontiers in Bioengineering and Biotechnology | www.frontiersin.org\n1499June 202010.3389/fbioe.2020.00499Received: 15 February 2020 Accepted: 29 April 2020 Published: 16 June 2020ORIGINAL RESEARCH Edited by: Reviewed by: *Correspondence: † These authors have contributed equally to this work Specialty section: This article was submitted to Nanobiotechnology, a section of the journal Frontiers in Bioengineering and Biotechnology Citation: Montagna G, Cristofaro F, Fassina L, Bruni G, Cucca L, Kochen A, Divieti Pajevic P, Bragdon B, Visai L and Gerstenfeld L (2020) An in vivo Comparison Study Between Strontium Nanoparticles and rhBMP2.strontium hydroxyapatite nanoparticlesBMP2osteoporosisscaffoldendochondral ossificationbone regeneration Montagna et al Strontium Hydroxyapatite Nanoparticles for Bone Regeneration\nThe osteoinductive property of strontium was repeatedly proven in the last decades. Compelling in vitro data demonstrated that strontium hydroxyapatite nanoparticles exert a dual action, by promoting osteoblasts-driven matrix secretion and inhibiting osteoclasts-driven matrix resorption. Recombinant human bone morphogenetic protein 2 (rhBMP2) is a powerful osteoinductive biologic, used for the treatment of vertebral fractures and critically-sized bone defects. Although effective, the use of rhBMP2 has limitations due its recombinant morphogen nature. In this study, we examined the comparison between two osteoinductive agents: rhBMP2 and the innovative strontium-substituted hydroxyapatite nanoparticles. To test their effectiveness, we independently loaded Gelfoam sponges with the two osteoinductive agents and used the sponges as agent-carriers. Gelfoam are FDA-approved biodegradable medical devices used as delivery system for musculoskeletal defects. Their porous structure and spongy morphology make them attractive in orthopedic field. The abiotic characterization of the loaded sponges, involving ion release pattern and structure investigation, was followed by in vivo implantation onto the periosteum of healthy mice and comparison of the effects induced by each implant was performed. Abiotic analysis demonstrated that strontium was continuously released from the sponges over 28 days with a pattern similar to rhBMP2. Histological observations and gene expression analysis showed stronger endochondral ossification elicited by strontium compared to rhBMP2. Osteoclast activity was more inhibited by strontium than by rhBMP2. These results demonstrated the use of sponges loaded with strontium nanoparticles as potential bone grafts might provide better outcomes for complex fractures. Strontium nanoparticles are a novel and effective non-biologic treatment for bone injuries and can be used as novel powerful therapeutics for bone regeneration.\n\nThe osteoinductive property of strontium was repeatedly proven in the last decades. Compelling in vitro data demonstrated that strontium hydroxyapatite nanoparticles exert a dual action, by promoting osteoblasts-driven matrix secretion and inhibiting osteoclasts-driven matrix resorption. Recombinant human bone morphogenetic protein 2 (rhBMP2) is a powerful osteoinductive biologic, used for the treatment of vertebral fractures and critically-sized bone defects. Although effective, the use of rhBMP2 has limitations due its recombinant morphogen nature. In this study, we examined the comparison between two osteoinductive agents: rhBMP2 and the innovative strontium-substituted hydroxyapatite nanoparticles. To test their effectiveness, we independently loaded Gelfoam sponges with the two osteoinductive agents and used the sponges as agent-carriers. Gelfoam are FDA-approved biodegradable medical devices used as delivery system for musculoskeletal defects. Their porous structure and spongy morphology make them attractive in orthopedic field. The abiotic characterization of the loaded sponges, involving ion release pattern and structure investigation, was followed by in vivo implantation onto the periosteum of healthy mice and comparison of the effects induced by each implant was performed. Abiotic analysis demonstrated that strontium was continuously released from the sponges over 28 days with a pattern similar to rhBMP2. Histological observations and gene expression analysis showed stronger endochondral ossification elicited by strontium compared to rhBMP2. Osteoclast activity was more inhibited by strontium than by rhBMP2. These results demonstrated the use of sponges loaded with strontium nanoparticles as potential bone grafts might provide better outcomes for complex fractures. Strontium nanoparticles are a novel and effective non-biologic treatment for bone injuries and can be used as novel powerful therapeutics for bone regeneration.\n\n\nINTRODUCTION\n\nBone tissue constitutes the rigid scaffold that is the skeleton, which provides structural support for vertebrates and confers protection to the most delicate vital organs. The extraordinary regenerative capability of bone tissue to repair and heal without the formation of a fibrotic scar was recognized by Imhotep (2630( -2611 and Hippocrates (460-370 BC) (Lanza and Vegetti, 1974). However, in pathological conditions such as osteoporosis, sever bone trauma or critically-sized bone defects this process can be hindered, leading to delayed healing and/or non-union fractures (Holroyd et al., 2008;Sozen et al., 2017). These unsolicited outcomes constitute an often underestimated burden for our society, in term of quality of life, recovery time and costs for the healthcare system (Gentleman et al., 2010;Marcus, 2007).\n\nBone tissue engineering focuses on the creation of devices capable of providing physical support, activating bone forming cells (concepts, respectively, defined as osteoconductivity and osteoinductivity) with the aim of assisting and accelerating osteogenesis (Dennis et al., 2015;Kowalczewski and Saul, 2018). During the last two decades in vitro and in vivo studies were designed to identify novel scaffolding systems for a topical and controlled delivery of osteoinductive agents (Landi et al., 2007;Visai et al., 2017;Chandran et al., 2018). Many of them focused on the employment of recombinant human bone morphogenic protein 2 (rhBMP2) as a powerful osteoinductive agent (Wang et al., 1990;Buza and Einhorn, 2016). A great amount of effort has been placed in investigating rhBMP2 suitability as treatment for conditions such as spinal fractures and criticallysized bone defects (Noshi et al., 2001;Boix et al., 2005). Several studies highlighted the benefits of using this recombinant morphogen, but also its side effects (Wang et al., 1990;Noshi et al., 2001). Local side effects (such as insufficient or excessive bone formation, heterotopic bone formation, infections and inflammatory responses) as well as systemic ones (cancer in a relatively low percentage of cases) were identified in several independent studies (Boraiah et al., 2009;Latzman et al., 2010;Hoffmann et al., 2013;Woo, 2013;Poon et al., 2016). Compelling experimental results showed the efficacy of strontium cations, as osteoinductive and anti-osteoporotic agents (Rohnke et al., 2016;Carmo et al., 2018;Li et al., 2018). Strontium modulates bone remodeling by enhancing bone formation and suppressing bone resorption (Chattopadhyay et al., 2006;Takaoka et al., 2010;Yang et al., 2011;Saidak and Marie, 2012;Tian et al., 2014) although, to date, the molecular and cellular mechanisms of strontium activity remain partially elusive. Physical-chemical properties of strontium-substituted hydroxyapatite nanoparticles in combination with calcium hydroxyapatite nanoparticles and their in vitro osteoinductivity were assessed in our previous works (Frasnelli et al., 2016;Visai et al., 2017;Cristofaro et al., 2019).\n\nAbbreviations: ECM, extracellular matrix; OBs, osteoblasts; OCs, osteoclasts; CaSR, calcium sensing receptor HAn, calcium hydroxyapatite nanoparticles; SrHAn, mixture of 90% calcium hydroxyapatite nanoparticles and 10% strontium hydroxyapatite nanoparticles.\n\nIn this work we utilized new nanotechnology to design a novel nanomaterial that is absent of recombinant morphogen yet still possess bone healing properties exerted by strontium to be used as an application for orthopedic surgery. This nanomaterial is a combination of strontium-substituted and calcium hydroxyapatite nanoparticles which was delivered in vivo by Gelfoam sponges, an FDA-approved collagen-based sponge commonly used as hemostatic application on bleeding surfaces (Pharmacia and Upjohn Company and Pfizer, 2017). The combination between the porous structure and molecular composition made it attractive in the orthopedic field and suitable either for morphogens or for ceramics delivery (Rohanizadeh et al., 2008;Giorgi et al., 2017).\n\nThe aim of this work is to compare the in vivo osteogenesis of strontium hydroxyapatite nanoparticles (SrHAn) to calcium hydroxyapatite nanoparticles and rhBMP2 (HAn-BMP2) using Gelfoam sponges as carriers with both treatments. An extensive characterization was performed including an abiotic study on the implant structure, as well as analysis of ions and rhBMP2 release patterns. The in vivo study was performed, on a healthy, bone damage-free mouse model to analyze the effectiveness of the described implants in targeting bone tissue. The sponges were implanted adjacent to the periosteum and gene expression of markers of osteocytes and OCs maintenance, MSCs recruitment, osteogenic and chondrogenic differentiation were evaluated, in order to analyze the responses at the periosteal bone. The experimental scheme is shown in Figure 1, with the comparison between SrHAn and HAn-BMP2. Sponges loaded with only calcium hydroxyapatite nanoparticles (HAn) were used as the negative control. In the study we included also the stability assessment of lyophilized samples, as this lyophilization could be suitable for the packaging and distribution of the implants.\n\n\nMATERIALS AND METHODS\n\n\nStandard Sponges Loading Protocol\n\nGelfoam sponges (Gelfoam; Pfizer R , NY, USA) were cut into 1 cm 2 . The sponge absorbance capacity of 200 µL was defined as the volume of water completely absorbed by each sponge. The sponges were loaded with a HCl-acidified MilliQ water solution (pH 5) containing 30% (w/v) of calcium hydroxyapatite nanoparticles (HAn) or with a MilliQ water solution containing 27% (w/v) of HAn and 3% (w/v) of strontium hydroxyapatite nanoparticles (SrHAn). For abiotic characterization unloaded sponges were hydrated with acidified MilliQ and defined as the control (CTRL). For rhBMP2 loaded-sponges, 3 µg of human recombinant BMP2 (R&D Systems) were added to the HAn suspension before the sponge absorption phase (HAn-BMP2). Acidified MilliQ was used to increase nanoparticles solubility, buffer pH suspension (otherwise alkaline) and to prevent rhBMP2 precipitation (Friess et al., 1999;Luca et al., 2010). The CTRL, HAn, HAn-BMP2 and SrHAn samples were incubated at room temperature (RT) in a 24-well plate for 15 min after loading to allow the volume to be adsorbed. Following 30 min of incubation at 37 • C, in a humidified and controlled atmosphere, with 5% CO 2 , the wells were filled with 800 µL of acidified MilliQ. FIGURE 1 | Schematic representation of abiotic and in vivo sponge characterization. The collagen sponges were loaded with calcium hydroxyapatite nanoparticles (HAn), with calcium hydroxyapatite nanoparticles + BMP2 (HAn-BMP2), and a mixture of calcium hydroxyapatite nanoparticles and strontium hydroxyapatite nanoparticles (SrHAn). Sponge characterization was achieved by several physical-chemical techniques. X-ray observations and Fast green/Safranin-O histological staining were performed on tissue samples collected from the in vivo experiments. Expression of genes for osteogenesis, osteocytes and osteoclasts activity, chondrogenesis, and stem cell recruitment was also investigated. Figure created with BioRender.com.\n\nA second set of sponges was prepared exactly as the first set of samples but with an additional step of lyophilization in order to check the system stability. Exception made for HAn-BMP2, which was not prepared in the second set. This second set was labeled with an \"s\" for \"stability\" (sCTRL, sHAn, sSrHAn) and collectively defined as \"treated\" sponges. Following the second incubation period carried out in 800 µL of acidified water, all samples were frozen O/N at −20 • C and lyophilized for 4 h (−50 • C, 0.3 mbar). Following lyophilization samples were rehydrated with 200 µL of acidified MilliQ. For both sets of sponges, water solutions were poured off and fresh non-acidified MilliQ water was added on day 1, 2, 3, 7, 14, 21, and 28.\n\n\nSponge Calcium, Strontium, and BMP-2 Release\n\nWater solution samples were collected at 1, 2, 3, 7, 14, 21, and 28 days from untreated sponges (CTRL, HAn, SrHAn), treated sponges (sCTRL, sHAn, sSrHAn) and from sponges loaded simultaneously with HAn and rhBMP2 (HAn-BMP2). For the analysis, each sample replicate (1mL) was diluted 1:10 and acidified with suprapur nitric acid (Merck -Sigma Aldrich, Merck KGaA, Darmstadt, Germany) at the final concentration of 0.5% (v/v), centrifuged and filtered with 0.45 µm membranes. Acidified water was used as blank (Moonesi Rad et al., 2019). Measurements were performed using an ICP-OES (inductively coupled plasma optical emission spectroscopy) iCAP 7000 Series (Thermo Scientific) following the standard procedures suggested by the manufacturer and an instrumental method already validated. Calcium and strontium were quantified by external calibration with four standard solutions (0, 5, 1, 5, 10, and 20 mg/L) of the alkaline earth standard metal mix TraceCERT R (100 mg/L). The following wavelengths were selected for the investigated elements: 422,673 nm for Ca 2+ and 407,771 nm for Sr 2+ . Time zero corresponds to the acidified water before the introduction of the sponges. Statistical evaluation was performed with one-way ANOVA. 95% confidence intervals showing the differences between averaged values were plotted in separated charts, reported as ( Figure S2). rhBMP2 release rate was evaluated by ELISA. rhBMP2 release in the water solutions was measured after addition of 20 mM monosodium phosphate, in order to displace rhBMP2 from its binding with HAn (Urist et al., 1984;Boix et al., 2005). Samples were then centrifugated to remove excess calcium and 100 µL of each solution was used for immobilization on ELISA wells. Anti-rhBMP2 polyclonal rabbit IgG (ReliaTech GmgH) was used on dilution factor of 1:3,000. Goat anti-rabbit/HRP (Agilent Tech) secondary antibody was used at 1:5,000 dilution. Reaction was developed with TMB (Sigma T0440−100 mL) for 30 min at RT, stopped with sulfuric acid 0.5 M and ELISA microplate was read with Clariostar ELISA reader at 450 nm. (HAn, SrHAn, sHAn and sSrHAn).\n\n\nFT-IR Spectroscopy Investigation of Sponge-Nanoparticles Interactions\n\nFT-IR (Fourier Transform-Infra Red) spectra were obtained using a Nicolet FT-IR iS10 Spectrometer (Nicolet, Madison, WI) equipped with attenuated total reflectance sampling accessory (Smart iTR with diamond plate) by coadding 32 scans in the 4,000-650 cm −1 range with a resolution at 4 cm −1 . FT-IR spectra were recorded on the unloaded sponges, on the unloaded sponges blended together with the nanoparticles (referred to as physical mixture, PM) and on the nanoparticles loaded-sponges.\n\n\nSponge SEM Observations and Mineral Phase Distribution\n\nUntreated (CTRL, HAn, SrHAn) and treated (sCTRL, sHAn, sSrHAn) sponges collected at time 0 days (T0) and time 28 days (T28) were frozen O/N at −20 • C and lyophilized for 3 h (−50 • C, 0.3 mbar). Microscopic structure was investigated using a scanning electron microscope (SEM) Zeiss EVO-MA10 (Carl Zeiss, Oberkochen, Germany). Images were acquired at different magnifications (150x and 350x) and at an accelerating voltage of 20 kV. CTRL, HAn and SrHAn sponges were also investigated with SEM Zeiss EVO-MA10 (Carl Zeiss, Oberkochen, Germany) coupled to an electron dispersive spectroscopy (EDS) (X-max 50 mm 2 , Oxford Instruments, Oxford, UK). Images were acquired at a lower magnification and energy dispersive microanalysis was performed in order to map phosphorus (P), calcium (Ca) and strontium (Sr) and quantify carbon (C), oxygen (O), P, ca and Sr.\n\n\nSponge Degradation Rate Measurement\n\nTo evaluate the sponge degradation rate, untreated unloaded and loaded sponge masses were measured at the following time points: 1, 2, 3, 7, 14, 21, and 28 days. Sponges were prepared as described above. Day 0 corresponded to the untreated and unloaded dried sponge, while day 1 corresponded to loaded sponges. Weights at day 1 were assumed as the initial weights (Wi) and percentage weight loss was calculated for each point. The calculation performed is the following: [(Wi-Wt)/Wi] * 100, were Wt is the weight at each time point. Average and standard deviation of the percentage weight loss were plotted.\n\n\nIn vivo Animal Study\n\nAll animal studies were approved by the Institutional Animal Care and Use Committee at Boston University (BU). Animals enrolled for these studies were crosses of the B6.Cg-Gt(ROSA)26sortm14(CAG-tdTomato)Hze/J (Ai14; stock number 007914) mouse strain with either B6.CG-Pax7tm1(cre/ER2)Gaka/J (Pax7; stock number 017763) or Prx1CreER-GFP (Kawanami et al., 2009). The Ai14 and Pax7 stains were obtained from The Jackson Laboratory (Bar Harbor ME) and housed at the BU animal facilities under standard conditions. All enrolled for this study were healthy, bone damage-free, male mice aged 9-11 weeks. Strains were randomly assigned to treatment and time point.\n\n\nSponge Implant Surgical Procedure and in vivo Analysis\n\nThe previously prepared untreated samples (HAn, SrHAn, and HAn-BMP2) were used for in vivo animal studies. Mice were bilaterally implanted as described before with the exception that the sponges were implanted instead of demineralized bone matrix to induce ectopic bone formation adjacent to the periosteal surface of femurs (Bragdon et al., 2017). Three mice were enrolled per condition per time point (post-operative day 16 and 33) for RNA extraction and histology analysis. Figure 1 shows a scheme of the performed experiment. No complications and/or collateral effects arose in that period. At post-operative day 16 and 33 mice were euthanized by carbon dioxide inhalation followed by cervical dislocation. Immediately following euthanasia, mice were X-rayed using Faxitron MX-20 Specimen Radiography System at 30 kV for 40 s using Kodak BioMax XAR Scientific Film. The implanted sponge and femur from the left limb were collected separately for RNA extraction and gene analysis. Samples harvested for RNA analysis were stored at −80 • C. The right limb was recovered for histology analysis.\n\n\nHistological Analysis of Mouse Limb\n\nAfter fixation in paraformaldehyde (4%), the right limbs were decalcified in 14% w/v EDTA (pH 7.2) for 1 week at 4 • C. Limbs were dehydrated and embedded in paraffin for histology and 5 µm-thick sections were cut across the samples. Fast green and Safranin-O (American Mastertek Inc.) staining was performed on the 5 µm-thick sections for the investigation of ectopic bone formation (Provot and Schipani, 2005).\n\n\nRNA Extraction and Quantitative Reverse Transcriptase PCR\n\nRNA extraction was performed by tissue dissociation and chemical extraction as previously described (Bragdon et al., 2017). Briefly, samples were snap frozen in QIAzol R Lysis Reagent and lysed with the Qiagen R Tissue Lyser II. Chloroform (Sigma-Aldrich) and isopropanol (Sigma-Aldrich) was used to extract and precipitate RNA followed by 70% ethanol washes. The RNA was re-suspended in 30-50 µL of RNase free water and stored at −80 • C. In order to ensure the quality and quantity of the extracted RNA, both spectroscopy and gel electrophoresis were used. For the spectroscopy, a Beckman CoulterTM DU R 530 Life Science UV/Vis Spectrophotometer was used and a 260 nm/280 nm ratio value in the range of 1.7-1.9 indicated an acceptable quality of RNA. The 260 nm absorbance value was used to determine the concentration of RNA in the samples. RNA samples were loaded into a 1% agarose gel, GelStarTM Nucleic Acid Gel Stain from Lonza Group was used to detect the presence of the nucleic acid. The presence of two bands under UV light indicates the RNA is intact and not degraded (data not shown). cDNA was synthesized and qRT-PCR was performed as previously described (Bragdon et al., 2017). Briefly, 1 µg of extracted RNA was reverse transcribed using the TaqMan R Reverse Transcription Reagents kit from Applied Biosystems R . Primers used to probe specific expression of genes are listed in Table 1. The 18S gene was used as a reference gene and in addition to our samples non-operative femurs were used as naïve controls. Target gene expression was normalized using the Ct method of Schmittgen and Livak (2008). Acan Mm00545794_ml\n\n\nOSTEOGENIC-ASSOCIATED PRIMERS\n\n\nBGlap Mm03413826_mH\n\nRunx2 Mm00501578_ml\n\nDmp1 Mm00803833_gl\n\nSp7 Mm04209856_sl\n\nIbsp Mm00492555_ml\n\n\nOSTEOCLASTS-ASSOCIATED PRIMERS\n\n\nAcp5 Mm00475698_ml\n\nRankl Mm00441908_ml\n\nCtsk Mm00484039_ml\n\n\nOSTEOCYTES-ASSOCIATED PRIMERS\n\n\nSost Mm00470479_ml\n\nList of primers used during the in vivo investigation of genes crucial for the differentiation and activity of cells residing in the bone. The genes are divided over their specificity and their catalog number is reported beside.\n\n\nStatistical Analysis\n\nStatistical analyses were performed with Prism 7 (GraphPad Software Inc). Means are plotted and standard deviations (SD) of means, or standard error means (SEM) are represented by error bars. One-way ANOVA corrected by Tukey's honestly significant difference (THSD) test was used to analyze data, if not otherwise specified. A significance level of 0.05 was used for all statistical analyses, unless noted differently.\n\n\nRESULTS\n\n\nPhysical-Chemical Characterization of the Loaded, Untreated, and Treated Sponges\n\nUntreated sponges (CTRL, HAn, and SrHAn) were prepared and physical-chemically characterized for degradation rate, ion release properties by ICP-OES, sponge-nanoparticles interactions by FT-IR and morphology by SEM. Furthermore, a stability assessment was performed on the treated (lyophilized) sponge samples (sCTRL, sHAn, sSrHAn). The absolute amounts of calcium and strontium loaded onto each sponge was then calculated. 60 mg of HAn contains 23.89 mg of calcium whereas 54 mg + 6 mg of SrHAn contains 21.5 mg of calcium and 3.55 mg of strontium. Calcium mass for HAn-BMP2 sponge was calculated as for the HAn sample and corresponds to 23.89 mg. ICP-OES data of the water solutions collected at different time points from untreated samples (Figure 2 and Table 2) was used to evaluate the progressive release of calcium and strontium ions in solution.\n\nFor HAn-BMP2 samples, calcium ions and rhBMP2 release were measured via ICP-OES and indirect ELISA assay, respectively (Figures 2C,D and Table 2). An initial burst of Ca 2+ release was reported for samples HAn and SrHAn (Figures 2A,B), and this was followed by a significant decrement at day 3, after which it stabilized at about 10 mg/L- Table 2. A similar pattern is presented also for strontium ions (Figure 2B), although the concentration values was higher at day 3, ranging between 40 and 50 mg/L- Table 2. In Han-BMP2 samples, calcium ion release trend was flattened and strongly decreased ( Figure 2C) compared to the Han and SrHAn samples. This is probably due to the presence of rhBMP2 in the same solution. Although, the release of rhBMP2 showed a trend very similar to SrHAn samples, with a concentration spike in the first 2 days followed by a decrease at day 3 and constant release until day 28 ( Figure 2D).\n\nThe ICP-OES analyses performed on samples treated for stability assessments showed similar ion release patterns after the lyophilization treatment ( Figure S1 and Table S1). In particular, for the sHAn sponges (Figure S1A), calcium ion release was higher at day 3 reaching 30 and 40 mg/L. Interestingly, sSrHAn sponges showed a less intense burst at day 3 and increasing values from day 7 to day 28 ( Figure 2D). Although increments in the days 7-28 were not significant ( Figure S2). The 95% confidence intervals of the ICP-OES average values are reported in (Figure S2).\n\nTo evaluate sponge-nanoparticle interactions, FT-IR spectroscopy was employed (Figure 3). Unloaded and untreated sponge spectrum (Figures 3A,B) showed characteristic protein peaks (Morris and Finney, 2004;Barth, 2007): NH stretching band at 3271 cm −1 ; C=O stretching band (amide I) at 1,624 cm −1 ; N-H bending band (amide II) at 1,528 cm −1 ; C-N stretching (amide III) at 1,232 cm −1 . FT-IR spectra (Figures 3C-J), showed the comparative analysis between unloaded sponge blended along with nanoparticles (physical mixture = PM; Figures 3C,D,G,H) and untreated sponges loaded with nanoparticles ( Figures 3E,F,I,J). All FT-IR spectra of nanoparticles containing-specimens showed good agreement with the characteristic PO43-(1,023 cm −1 ) and OH-(3,570 cm −1 ) vibrations of hydroxyapatite lattice as reported before (Kim et al., 2002). Due to the low concentration of strontium in the samples, no significant variations were recorded between HAn_PM and SrHAn_PM, in accordance with previous results (Frasnelli et al., 2016). However, protein bands showed a much lower intensity in nanoparticles loaded sponges (Figures 3E,F,I,J) and completely disappeared in PM spectra (Figures 3C,D,G,H). While NH stretching band at 3,271 cm −1 was recorded only in the CTRL, C=O stretching and N-H bending slightly moved from initial values (1,624 and 1,528 cm −1 ) to 1,650-1,631 cm −1 , and 1,531-1,535 cm −1 , respectively. After lyophilization treatment (Figure S3), all protein bands completely disappeared (Figures S3C-F).\n\nSEM images were collected to analyze the morphology of unloaded and loaded sponges (Figure 4). Unloaded sponges at T0 (Figure 4A) were characterized by a spongy structure, with interconnected pores and smooth wall surfaces. Loaded sponge surfaces were rougher and more irregular (Figures 4B-D). A reduction of wall surfaces occurred in the unloaded FIGURE 2 | Calcium, strontium and rhBMP2 release from loaded untreated sponges. ICP-OES data representation of calcium and strontium ions release from HAn, HAn-BMP2, and SrHAn loaded sponges. Indirect ELISA was performed to quantify solubilized rhBMP2. Data were collected during the 28 days in aqueous solution. Ca 2+ released from HAn sponges (A); Ca 2+ and Sr 2+ released from SrHAn sponges (B); Ca 2+ and rhBMP2 release from HAn-BMP2 sponges (C,D) N = 3 samples were measure per each time point. samples after 28 days (Figure 4E), but it was not observed in loaded samples (Figures 4F-H). A morphological change was appreciated comparing the T0 and T28 (Figures 4B,C,E-H).\n\nAt T0 hydroxyapatite aggregates were detected on the sponge walls, while at T28 sponge walls looked smoother and the whole structure became more compact. Energy dispersive microanalysis conducted on HAn and SrHAn samples (Figures 4I-L) showed a homogenization of the phosphorus and calcium distribution on the sponge surfaces after 28 days in aqueous solution (Figure 4, P and Ca distribution maps). The same microanalysis performed on CTRL sample did not show an appreciable level of P, Ca and Sr (data not shown). Moreover, on HAn and SrHAn samples a decrement in the weight percentage of calcium and strontium was recorded at day 28 compared to day 0 (Figures 4M-P).\n\nRegarding the treated samples (Figure S4), unloaded sponges at T0 (Figure S4A) showed a morphology very similar to untreated samples at T0 (Figure S4A). Even though, it is possible to appreciate the formation of minor ripples, probably due to the additional lyophilization phase. At T0, the treated and untreated sponges reported similar structures. At T28, treated and loaded sponges acquired a cement-like morphology (Figures S4F-H).\n\nThe degradation rate was determined by measuring the sponge weight throughout 28 days in aqueous solution (Figure 5). A slow degradation pattern was observed with no significant variations recorded, except for the SrHAn loaded sponges (Figure 5C). SrHAn samples registered a significant difference in percentage weight loss between day 1 and day 28 (p < 0.0001).\n\nA reduction in the quantities of calcium and strontium on the sponge surfaces was recorded at day 28, in accordance with ICP-OES data. Taken together these results indicated the SrHAn loaded sponges are a stable system for a constant and prolonged release of Ca 2+ and Sr 2+ . They are biodegradable and their morphology fits the golden standard parameters for osteoconductive scaffolds. Although lyophilization partially changed the morphology of the sponges at T28, there was no significant change in calcium and strontium release. \n\n\nIn vivo Sponge's Implantation and Characterization of Their Effects\n\nNext, sponges prepared following the standard protocol (HAn, HAn-BMP2, and SrHAn) were utilized for in vivo implantation to determine the osteogenic ability of the nanoparticles. Radiography, macroscopic observations and histological staining of bone tissues and implanted sponges were carried out to analyze the sponge integration and surrounding tissue adaptation at postoperative day 16 and 33. Gene expression analysis from the implants and femurs were also performed at corresponding times to determine the molecular responses.\n\nThe radiographic images support the macroscopic observation that strong integration underwent between the femur, sponge, and surrounding tissue (Figure 6). Within the implanted sponges there were few visible blood vessels, as shown in Figure 7. Implants and ectopic bones showed various degrees of calcification, as shown by the radiographic images. The sponges loaded with HAn-BMP2 and SrHAn appeared to show increased calcified material than in the sponges with HAn alone. The radiographic and macroscopic images also suggested that there was a reduction in implant dimensions at day 33 compared to day 16 (Figures 6A-C,J-L). Since the implants consisted of sponge and hydroxyapatite (black arrowheads in Figure 6), it was difficult to distinguish between the implanted material and the formation of ectopic bone. However, the histological analysis using Fast Green and Safranin-O stain demonstrated the formation of both cartilage and bone tissue at the sponge implant site (Figure 7).\n\nMore extended Safranin-O positive regions (indicated with red arrowheads in Figure 7) were observed in the HAn and SrHAn samples at 16 days. Whereas, at 33 days, no Safranin-O positive tissue could be detected. This indicates that the ectopic bone formed through the endochondral ossification process, with cartilaginous tissue turning to calcified areas. The forming bone tissue at the implant sites also appears to support hematopoiesis, as shown by the recruitment of marrow cells between the femur cortical bone and the sponge (Figure 7E). The newly formed tissues (bone and bone marrow) were present in all implants but in the SrHAn loaded sponges were more represented. At 33 days, (Figures 7G-L) ectopic bone tissue is shown with respect FIGURE 4 | The microscopic structure (150× and 350×) of the untreated CTRL (unloaded sponges), HAn, HAn-BMP2 and SrHAn (loaded sponges) samples is shown in (A-H). Pictures represented samples at T0 (A-D) and after 28 days (E-H) of permanence in aqueous solutions, in a controlled and humidified atmosphere (37 • C, with 5% of CO 2 ). Untreated samples surfaces were also analyzed with SEM-EDS (Energy Dispersive X-Ray Spectroscopy) at 0 (I,J) and 28 days (K,L). Images at low magnifications of the sponges were taken with SEM (I-L). The elements phosphorus (P), calcium (Ca) and strontium (Sr) were mapped on the same pictures using different colors (P = green; Ca = red; Sr = white) and a black background. Quantification of the elements carbon (C), oxygen (O), phosphorus (P), calcium (Ca) and strontium (Sr) present of the samples surface was performed and data were reported as weight percentage in the bar charts (M,N,O,P) histograms. to control, accompanied by the presence of an area enriched in bone marrow cellular components.\n\nTo investigate the changes in gene expression in both the femur and implants after 16 and 33 days, tissues were harvested separately from HAn, HAn-BMP2, and SrHAn implanted mice. Figure 8 represents chondrogenesis (Acan, Col10A1, and Sox9) and stem cell recruitment (Nanog) markers. While Figure 9 shows gene expression of markers for osteogenesis (Runx2, Sp7, Ibsp, BGlap, and Dmp1), osteocytes (Sost) and OCs activity (Acp5, Rankl, and Ctsk). Acan (Aggrecan core protein), Col10A1 (Type 10 collagen) and Sox9 expression was analyzed as chondrogenesis and endochondral ossification markers (Mackie et al., 2008;Dennis et al., 2015). There was a consistent difference between the expression of chondrogenic markers in bone and sponge samples as noted by the sectioned Y axis. Indeed, at 16 days, in the former these three genes were downregulated by both rhBMP2 and strontium, with respect to HAn condition ( Figure 8A). While in the latter, both osteoinductive factors brought to significantly higher expression of Col10A1 and Sox9 (Figure 8C), with an enhanced effect in SrHAn loaded sponges. At 33 days, the difference between the tissue samples is equalized and only significant up-regulation of Sox9 in the bone and Acan in the sponge were shown as strontium-driven effects (Figures 8E,G). Expression of the stem cells recruitment marker, Nanog, showed no significant difference at 16 days and only a mild up-regulation with respect to control at 33 days in the sponge sample. After 16 days, a significant up-regulation of BGlap (Osteocalcin), Ibsp (bone sialoprotein), and Sp7 (Osterix) was observed in the FIGURE 5 | Graphical representation of sponge weights variations. Untreated sample weights were measured with an analytical balance. Weights were recorded at different time points throughout the 28 days in aqueous solution. Day 0 is referred to the unloaded sponge, whereas Day 1 corresponds to the loaded sponges and it was considered the initial weight (Wi). N = 3 different sponges were weighted per each condition. From the recorded weights, we calculated the percentage weight loss. Error bars indicate the standard deviation. Statistically significant differences were assessed with one way ANOVA. Significant differences were reported only against day 1 (*p < 0, 05; **p < 0, 01; ****p < 0, 0001).\n\nfemur of mice that received the SrHAn implant compared to HAn-BMP2 and to HAn samples alone ( Figure 9A). Similar results were observed with BGlap, Ibsp, and Sp7 expression being evaluated in the sponge samples at 16 days ( Figure 9D). In both the femur and the sponge tissues Dmp1 (Dentin matrix protein 1) was upregulated by strontium with respect to the control HAn. A similar trend of expression was also detected at 33 days, where Ibsp in the bone ( Figure 9G) and Ibsp, BGlap, and Sp7 in the sponge ( Figure 9J) were upregulated by strontium. Runx2 didn't show significant differences in expression at given time points. Sost (Sclerostin) expression was analyzed as marker for osteocyte differentiation. After 16 days, Sost was upregulated in the strontium samples only with respect to the control HAn, both in the bone and sponge samples (Figures 9B,E). At 33 days, an up-regulation of Sost was observed only in the bone samples loaded with strontium compared to HAn-BMP2 and HAn (Figures 9H,K). Acp5 (Tartrate-resistant acid phosphatase type 5), Rankl (Tumor necrosis factor ligand superfamily member 11) and Ctsk (Cathepsin K) expression were analyzed as markers for OCs differentiation and activity. At 16 days, Acp5 and Rankl (Figures 9C,F) were downregulated in SrHAn femurs and sponges Ctsk at 16 days was upregulated in femur and sponge by both types of osteoinductive factors (Figures 9C,F). At 33 days, strontium presence led to an upregulation of Acp5 and down-regulation of Rankl only in the bone samples ( Figure 9I). Ctsk was down-regulated in SrHAn implanted mice with respect to control, both in bone and sponge samples (Figures 9I,L).\n\nSrHAn loaded-sponge structure were suitable for the promotion of cell invasion, bone marrow cell recruitment and ossification. Following a lyophilization treatment (which could be useful for sponge long-term storage and packaging), the loaded sponges did not show significant variation from the untreated samples and all physical-chemical properties were conserved. The biodegradability and biocompatibility of the system was proven in vivo, together with the induction of endochondral ossification stages in adult bone. Evidence of endochondral ossification were given in the histological investigation, where proliferating and hypertrophic chondrocytes were more represented in the strontium-containing samples than in the BMP2 ones. Gene expression corroborated histological evidences, showing increased expression of osteogenic and chondrogenesis markers. Moreover, these data were consistent with an enhancement of endochondral ossification in the SrHAn samples if compared to the HAn-BMP2 samples. Antiresorptive properties of strontium were also demonstrated by investigating markers of osteoclasts differentiation and activity. Histological and gene expression data further Statistical significance values were calculated with one way-ANOVA, followed by Tukey's honestly significant difference test. *, significant difference against the HAn condition (p < 0.05). #, significant difference between HAn-BMP2 and Sr conditions (p < 0.05). Statistical significance values were calculated with one way-ANOVA, followed by Tukey's honestly significant difference test. *, significant difference against the HAn condition (p < 0.05). #, significant difference between HAn-BMP2 and SrHAn conditions (p < 0.05).\n\nsupported the osteoinductive and osteoconductive potential of this system.\n\n\nDISCUSSION\n\nBone tissue engineering application in the orthopeadic surgical field is becoming increasingly relevant. In the last decades, rhBMP2 was extensively studied for its osteoinductive potential. Although several successful applications have been achieved, this recombinant morphogen was proven to be not always suitable and safe (Poon et al., 2016;Lykissas and Gkiatas, 2017). Another powerful osteoinductive agent, strontium, was widely studied in the previous years and compelling results were obtained in term of biocompatibility and osteoconductivity (Boanini et al., 2011;Yang et al., 2011). Strontium hydroxyapatite nanoparticles were recently developed, carrying an inorganic osteoinductive agent (Frasnelli et al., 2016). They have been characterized in the last years and in this work, they are presented as a substitute for rhBMP2.\n\nIn this study, the medical grade collagen-based Gelfoam sponges were loaded with a combination of either calcium and strontium hydroxyapatite nanoparticles suspensions (SrHAn) or calcium hydroxyapatite nanoparticles and rhBMP2 (HAn-BMP2) in order to compare their physical-chemical properties. The HAn sponges were loaded only with calcium hydroxyapatite nanoparticles and used as control. A second set of samples were treated with an additional lyophilization step and characterized.\n\nThe hydroxyapatite nanoparticles amount on each sponge was 30% (w/v) of the volume absorbed by the sponge and 3 µg of rhBMP2 per sponge were used for HAn-BMP2. For the SrHAn loaded-sponges, the calcium/strontium hydroxyapatite nanoparticles ratio was 9:1 w/w (Chandran et al., 2016) while the spontaneously occurring Sr:Ca ratio is between 1:1,000 and 1:2,000 in human bone tissue as well as in blood serum (Cabrera et al., 1999).\n\nThe cumulative amount of calcium and strontium ions after 28 days in aqueous solutions were measured as 0.0118 and 0.0358 mg, respectively. Interestingly, several in vitro and in vivo studies reported that strontium elicits rhBMP2-like effects when administered at concentrations up to 1 mM, as reported by Schumacher et al. (2013) and others (Qiu et al., 2006;Li et al., 2010;Schumacher et al., 2013;Wei et al., 2019). Despite the lower water solubility of Sr 2+ compared to Ca 2+ , we detected enhanced strontium solubilization from the SrHAn sponges, compared to the calcium release from the HAn sponges. These data are in agreement with Kaufman and Kleinberg (1979) and Pan et al. (2009) studies. They demonstrated how, due to a larger atomic radius, strontium atoms slightly reduce hydroxyapatite lattice thermodynamic stability, favoring the release of Sr 2+ from the lattice itself. Furthermore, ICP-OES results were comparable to the ones presented by Landi et al. in term of ion release pattern (Landi et al., 2007) and were not significatively influenced by the lyophilization treatment. We investigated the spongenanoparticle system through FT-IR analysis of the untreated and treated systems (HAn, SrHAn, sHAn, and sSrHAn). Significant interactions were not found between the sponge and either HAn or SrHAn. FT-IR spectra of loaded sponges showed no chemical interactions were undergoing between the sponges and the nanoparticles. The minor shifting of the amide I and II bands was attributed to hydrogen bonds breaking in the sponge proteins and consequent modification of the hydrogen bonds network in the sponge structure. We can therefore assume that the ion release patterns studied with ICP-OES were not influenced by chemical interactions between the nanoparticles and the sponge support, either before or after the lyophilization treatment. To illustrate the patterns of ion release in function of time we considered the porous structure of the sponges. We could speculate a bulk of nanoparticles flowed inside the sponge pores, during the loading procedure. Nanoparticles remaining on the sponge surfaces released a high percentage of Ca 2+ and Sr 2+ ions during the first 3 days, while embedded nanoparticles were responsible for the constant release recorded after day 3. The same speculation can also fit the data on rhBMP2 release. This hypothesis was corroborated by SEM images, that showed smoother surfaces and solubilization of hydroxyapatite aggregates at day 28. Surprisingly, even after 28 days in aqueous solution, calcium and traces of strontium were still detected on the sponge surfaces, as demonstrated by SEM-EDS data.\n\nInterestingly, the calcium release pattern of HAn-BMP2 loaded-sponges was different from the other samples. This can be due to the non-covalent interaction between HAn and rhBMP2 (Urist et al., 1984;Boix et al., 2005). We were not able to investigate HAn-BMP2 loaded-sponges with FT-IR, since this technique is not able to discern between different proteins. We can speculate that calcium ions remained bound to solubilized rhBMP2, therefore decreasing the amount of free Ca 2+ released in solution.\n\nPore sizes ranging between 100 and 300 µm are considered the most suitable for osteoinduction improvement as well as for angiogenesis promotion (Karageorgiou and Kaplan, 2005;Scheinpflug et al., 2018). Furthermore, geometry of the investigated sponges reflected the \"gold standard\" in bone tissue engineering, according to Kuboki et al. (2007) and due to the given pores size and geometry the surface area exposed to the surrounding microenvironment is increased, allowing for a wider interaction between the nanoparticles coating and the surrounding microenvironment.\n\nThe degradability and recyclability of the graft is a crucial point, although often disregarded, in tissue engineering applications (Gentleman et al., 2010;Rohnke et al., 2016). In this work, the time-dependent biodegradability of the sponges and the hydroxyapatite nanoparticles both in the abiotic and in vivo conditions, were analyzed. Degradation pattern based on the sponge weights appeared mild but significant for SrHAn sponges, throughout the 28 days in solution. While, in vivo radiographic images and histology suggested an accelerated degradation of SrHAn sponges. Radiographic images also revealed an induction of bone formation and osseointegration by all three types of implants at both time points. Although, a more extensive mineralized tissue in mice implanted with HAn-BMP2 and SrHAn sponges were shown with respect to the controls. Histological analysis revealed new tissue formation between the sponge and the femoral cortex, suggesting a center of active osteogenesis was present in the sponge site. Given the similarities between Ca 2+ and Sr 2+ , it has been shown that this element can substitute calcium in the lattice of the hydroxyapatite crystals present in bones (Ni et al., 2011;Querido et al., 2016). Sr 2+ is processed like Ca 2+ throughout metabolism, being preferentially introduced in active osteogenesis sites (Bauman et al., 2005).\n\nRhBMP2 and Sr 2+ release was quantified using different techniques, therefore it is not possible to compare their quantities/concentrations in solution head-to-head. Despite the similarities in the pattern release between HAn-BMP2 and SrHAn, strontium containing nanoparticles induced significant increase of endochondral ossification markers. It is wellknown how the process of endochondral ossification can be recapitulated in adults by bone fracture and bone tissue damage (Gerstenfeld et al., 2003;Dennis et al., 2015), although in this work evidences of endochondral ossification were induced in healthy implanted mice, with no bone damage. Furthermore, no record of strontium related-endochondral ossification induction is present up to date, but here at 16 days the onset of typical multicellular clusters constitutes by enlarged chondrocyte was demonstrated (Mackie et al., 2008). Hypertrophic chondrocytes can be detected in the sponge surroundings at 16 days, together with smaller proliferating chondrocytes. These were particularly represented in the HAn and SrHAn samples and red stained-cartilage proteoglycans were present all around the cells. Hypertrophic chondrocytes were almost absent in the HAn-BMP2 samples, suggesting a possible delay or acceleration in chondrogenic differentiation. Different phases of endochondral ossification process were shown also by gene expression data: Sox9 upregulation was an indicator of chondrocytes proliferation (Akiyama et al., 2002;Lefebvre and Dvir-Ginzberg, 2017) and after its downregulation, markers of hypertrophic chondrocytes, such as aggrecan core protein (Acan) and type X collagen (Col10A1) were upregulated (Akiyama and Lefebvre, 2011). We can speculate that the different temporal upregulation of Sox9, among bone and sponges, can be attributed to the time dependent Sr 2+ diffusion, which resulted in its earlier expression in the sponge area and delayed expression in the bone one.\n\nThe interaction of Sr 2+ with the calcium-sensing receptor (CaSR) was shown both in OBs and osteoclasts (OCs), resulting in improved anabolic processes and reduced catabolic pathways, respectively (Chattopadhyay et al., 2006). Our gene expression data showed SrHAn double activity on OBs and OCs. Several studies reported that strontium modulates alkaline phosphatase (ALP) activity and Runx-2 expression (Tian et al., 2014) in OBs and OBs progenitors while promoting Wnt/β-catenin pathway in human bone marrow derived mesenchymal stem cells (MSCs), thereby enhancing ECM secretion and osteogenic differentiation (Yang et al., 2011). Interestingly, in our study no significant variation was detected in Runx2 levels. However, our results measured an upregulation of Sp7 (Osterix), indicating OBs differentiation at early and late stage (Santo et al., 2013;Kawane et al., 2018). Sp7 is a transcriptional factor expressed in developing bones regulating the commitment of MSCs toward osteoblastic lineage (Rutkovskiy et al., 2016). Its regulation affects expression of protein-coding genes typically related to osteoblastic differentiation, such as osteocalcin, type I collagen and bone sialoprotein (BGlap, Ibsp, and Dmp1). Increased expression of the ECM proteins BGlap, Ibsp, and Dmp1 was attributed to mature osteoblasts activity (Wrobel et al., 2016) in SrHAn samples. Sp7, BGlap, and Ibsp expression also indicated a stronger osteogenic induction for strontium ions compared to rhBMP2. Interestingly Runx2, the master regulator of osteoblastic differentiation acting above Sp7, was not significantly influenced neither by strontium nor by rhBMP2 (Nakashima et al., 2002). Soluble cytokine promoting OCs differentiation, Rankl, (expressed by OBs) is a key modulator of the resorption cycle, controlling OCs differentiation and activation, and therefore bone resorption. This molecule has already been linked to strontium-mediated modulation by Atkins et al. (2009) andBrennan et al. (2009). Our results showed strontium significantly downregulated Rankl when compared to control and HAn-BMP2 samples. It has been shown that strontium effects are mediated by CaSR and by others, yet unclarified receptors (Chattopadhyay et al., 2006;Takaoka et al., 2010;Saidak and Marie, 2012). Furthermore, being CaSR present also on OCs outer membrane it was demonstrated the action of strontium on OCs activity. Indeed, OCs-secreted endoprotease responsible for ECM degradation (Sage et al., 2012), Ctsk (Cathepsin K) was reduced by strontium presence at 33 days, with respect to the control. Of note, in some of the SrHAn sponges we observed vascularization, suggesting a potential effect of Sr 2+ on neo-vascularization, although further studies are needed to confirm these findings.\n\n\nCONCLUSIONS\n\nMany studies have been conducted to investigate rhBMP2 suitability as treatment for spinal fractures and critically-sized bone defects. Although its indisputable osteoinductive potential, rhBMP2 also showed adverse effects. On the other hand, strontium has already been tested as a treatment for osteoporosis and its activity was demonstrated in a patient population. The biggest advantage of strontium over rhBMP2 however, is that the former is a chemical and not a biologic. Herein, following the abiotic characterization, we presented the results of a comparative in vivo study, where Gelfoam sponges loaded either with strontium hydroxyapatite nanoparticles or with rhBMP2 were implanted in healthy, bone damage-free mouse model. In vivo studies, showed that SrHAn and HAn-BMP2 have comparable effects, driving the onset of endochondral ossification and promoting the bone remodeling process. These results demonstrated that SrHAn loaded-sponges have marked osteogenic potential when applied on the periosteum of long bones, comparable to HAn-BMP2 loaded-sponges, but eliciting a more controlled ossification response. We propose to use Gelfoam sponges enriched in Sr 2+ as an effective therapeutic intervention to treat severe bone defects or open, complicated fractures. The short-term benefit of this nanotechnological system is the availability of novel therapeutic option to treat bone fracture (or bone defect). The use of sponges loaded with strontium hydroxyapatite nanoparticles might also provide better outcomes for complex fractures. Results from these studies can provide novel therapeutic options for active duty personnel and can be beneficial to anyone suffering from trauma, bone defects or severe bone injuries. Future studies shall also evaluate the efficacy of SrHAn loaded sponges for the treatment of spinal fusion.\n\n\nDATA AVAILABILITY STATEMENT\n\nAll datasets generated for this study are included in the article/Supplementary Material.\n\n\nETHICS STATEMENT\n\nAll animal studies were approved by the Institutional Animal Care and Use Committee at Boston University (BU). \n\n\nAUTHOR CONTRIBUTIONS\n\nFIGURE 3 |\n3IR evaluation of sponge-nanoparticles interactions. FT-IR spectra of sponges and nanoparticles physical mixtures (Han_PM and SrHAn_PM) (C,D,G,H) were compared with spectra corresponding to the untreated and loaded sponges HAn and SrHAn at 28 days (E,F,I,J). In the panels a and b, FT-IR spectrum of untreated and unloaded sponge was reported (CTRL). Spectra were divided in two sections: wavenumber range 4,000-3,000 cm −1 was reported in (A,C,E,G,I); while wavenumber range 2,000-650 cm −1 was shown in (B,D,F,H,J).\n\nFIGURE 6 |\n6X-ray images and macroscopic observation of mice implanted with loaded sponges. Images at 16 and 33 days of mice implanted with HAn loaded-sponges (A,D,G,J,M,P, respectively), HAn-BMP2 loaded-sponges (B,E,H,K,N,Q) and SrHAn (C,F,I, L,O,R, respectively). The implanted sponges are shown by black arrowheads. Isolated femurs are shown in panels (D-F) and (M-O). Isolated legs including implants are shown in panels (G-I) and (P-R).\n\nFIGURE 7 |\n7Representative histological images of post-implants tissues of mouse limb. Fast green/Safranin-O staining was used on post-implant tissue sections of limbs implanted with loaded sponges with HAn (A,D,G,J), HAn-BMP2 (B,E,H,K), or SrHAn (C,F,I,L), for 16 and 33 days, respectively. In the images, red spots of cartilaginous tissue are indicated with red arrowheads and gray blurs of ectopic bone are highlighted with *. Black arrows and black arrowheads indicate femur bone and sponges, respectively. The circle highlights a blood vessel.\n\nFIGURE 8 |\n8Gene expression of stem cell recruitment and chondrogenesis markers in bone and sponge post-implants. Gene expression was analyzed in femur bone (A,B,E,F) and sponge post-implant (C,D,G,H) samples at 16 (A-D) and 33 days (E-H), respectively. Expression of chondrogenesis markers Acan, Col10A1 and Sox9 (A,C,E,G) as well as stem cell recruitment marker Nanog (B,D,F,H) has been evaluated. The graphs show the inverse of the Ct at the power of 2. Bars indicate mean values ± SEM of results from 4 experiments.\n\nFIGURE 9 |\n9Gene expression of osteocytes, osteoclasts homeostasis, and osteogenic markers in bone and sponge post-implants. Gene expression was evaluated in femur bone (A,B,C,G,H,I) and sponge post-implant (D,E,F,J,K,L) samples at 16 (A-F) and 33 (G-L) days, respectively. Expression of osteogenic markers Dmp1, Bglap, Ibsp, Sp7 and Runx2 (A,D,G,J), osteocytes marker Sost (B,E,H,K) and osteoclasts relevant markers Acp5, Rankl and Ctsk (C,F,I,L) has been evaluated. The graphs show the inverse of the Ct at the power of 2. Bars indicate mean values ± SEM of results from four experiments.\n\nFUNDING\nThis research was supported by funds from the National Institutes of Health NIAMS K99AR068582 (BB), NIAMS R01AR056637 (LG), and a grant from the Musculoskeletal Transplant Foundation (BB). The content of this study is solely the responsibility of the authors and do not necessarily represent the official views of the NIH. This work also was supported by grants from the Italian Space Agency, Project DC-MIC-2012-024, and contract N. 2013-060-I.O (to LV) and Projects of High Relevance, Bilateral project Italy-Sweden, Ministry of Foreign Affairs and International Cooperation (MAECI), and Ministry of Education, University and Research (MIUR) (2018-2020) titled \"Effect of Microgravity e nanoparticles on bone regeneration in simulated Microgravity (REPAIR)\" (to LV). Furthermore, a grant of the Italian Ministry of Education, University and Research (MIUR) to the Department of Molecular Medicine of the University of Pavia under the initiative \"Dipartimenti di Eccellenza (2018-2022)\" also contributed to this research.\n\nTABLE 1 |\n1qRT-PCR gene primers.Primer \nCatalog number \n\nNORMALIZATION PRIMER \n\n18S \nMm03928990_gl \n\nSTEM CELL RECRUITMENT PRIMER \n\nNanog \nMm02384862_gl \n\nCHONDROGENESIS PRIMERS \n\nSox9 \nMm00448840_ml \n\nCol10A1 \nMm00487041_ml \n\n\n\nTABLE 2 |\n2Tabular representation of the data shown inFigure 2.Average concentration values and standard deviations of the calcium and strontium ions and of rhBMP2 measured in the different sponges.Time (days) \nHAn \nSrHAn \nHAn-BMP2 \n\nCa (mg/L) \nCa (mg/L) \nSr (mg/L) \nCa (mg/L) \nrhBMP2 (µg/mL) \n\n0 \n0 \n0 \n0 \n0 \n0 \n\n1 \n30.6 ± 3.5 \n29.5 ± 3.5 \n71.7 ± 3.8 \n3.8 ± 0.4 \n1.3 ± 0.133 \n\n2 \n16.9 ± 2.5 \n13.6 ± 2.4 \n51.1 ± 5.0 \n4.4 ± 0.5 \n1.29 ± 0.01 \n\n3 \n11.5 ± 1.7 \n7.5 ± 3.2 \n35.2 ± 4.7 \n3.3 ± 0.8 \n0.43 ± 0.06 \n\n7 \n13.0 ± 1.8 \n7.5 ± 2.0 \n46.9 ± 3.5 \n2.2 ± 0.4 \n0.38 ± 0.03 \n\n14 \n7.2 ± 0.6 \n6.3 ± 2.4 \n45.4 ± 2.9 \n2.2 ± 0.6 \n0.38 ± 0.03 \n\n21 \n5.2 ± 0.3 \n4.8 ± 4.0 \n42 ± 2.6 \n2.0 ± 0.6 \n0.44 ± 0.07 \n\n2 \n13.7 ± 2.8 \n10.5 ± 2.8 \n46.2 ± 3.3 \n1.7 ± 0.5 \n0.31 ± 0.06 \n\n\n\n\nGM and FC: conceptualization, validation, formal analysis, investigation, writing-original draft and review and editing, and vaisualization. GB and LC: validation, formal analysis, investigation, resources, writing-review & editing, and visualization. LF and AK: conceptualization, validation, formal analysis, writing-review & editing. PD: conceptualization, formal analysis, resources, writing-original draft & review & editing, visualization, supervision, project administration, and funding acquisition. BB: validation, formal analysis, investigation, writing-original draft & review & editing, visualization, supervision, and project administration. LV and LG: conceptualization, resources, writing-original draft & review & editing, visualization, supervision, project administration, and funding acquisition.\nFrontiers in Bioengineering and Biotechnology | www.frontiersin.org\nJune 2020 | Volume 8 | Article 499\nACKNOWLEDGMENTSWe would like to acknowledge Anne Hinds for technical assistance with sectioning histological samples. We would like to thank professor Paola Petrini (Dipartimento di Chimica, Materiali ed Ingegneria Chimica G. Natta, Politecnico Milano, Italy) for the theoretical contribution on the formulation of the nanoparticles loaded-sponges.SUPPLEMENTARY MATERIALThe Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fbioe. 2020.00499/full#supplementary-materialConflict of Interest:The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.Copyright © 2020 Montagna,Cristofaro, Fassina, Bruni, Cucca, Kochen, Divieti Pajevic, Bragdon, Visai and Gerstenfeld.This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). 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Lett. 21, 1-15. doi: 10.1186/s11658-016-0027-8\n\nStrontium enhances osteogenic differentiation of mesenchymal stem cells and in vivo bone formation by activating Wnt/catenin signaling. F Yang, D Yang, J Tu, Q Zheng, L Cai, Wang , L , 10.1002/stem.646Stem Cell. 29Yang, F., Yang, D., Tu, J., Zheng, Q., Cai, L., and Wang, L. (2011). Strontium enhances osteogenic differentiation of mesenchymal stem cells and in vivo bone formation by activating Wnt/catenin signaling. Stem Cell. 29, 981-991. doi: 10.1002/stem.646\n"},"corpus_id":{"kind":"number","value":219689268,"string":"219,689,268"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2006.12490\",\n \"mag\": 3099288547,\n \"acl\": null,\n \"pubmed\": \"32612980\",\n \"pubmedcentral\": 7308719,\n \"dblp\": null,\n \"doi\": \"10.3389/fbioe.2020.00499\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [],\n \"pdf_sha\": \"987232bb8f65073911bf0ce49dc87ee8ddef54d8\",\n \"open_access\": {\n \"license\": \"CCBY\",\n \"url\": \"https://www.frontiersin.org/articles/10.3389/fbioe.2020.00499/pdf\",\n \"status\": \"GOLD\"\n }\n}"},"title":{"kind":"string","value":"An in vivo Comparison Study Between Strontium Nanoparticles and rhBMP2"},"authors":{"kind":"list like","value":["Silvia Minardi","Hakan Ceylan","Gloria Belen Ramirez","Beth Bragdon bragdon@bu.edu","Livia Visai livia.visai@unipv.it","Giulia Montagna \nDepartment of Molecular Medicine (DMM)\nCenter for Health Technologies (CHT)\nUdR INSTM\nUniversity of Pavia\nPaviaItaly\n\nDepartment of Electrical, Computer and Biomedical Engineering\nDepartment of Chemistry, Physical-Chemistry Section\nUniversity of Pavia\nPaviaItaly, 3 C.S.G.I\n\nUniversity of Pavia\nPaviaItaly","Francesco Cristofaro \nDepartment of Molecular Medicine (DMM)\nCenter for Health Technologies (CHT)\nUdR INSTM\nUniversity of Pavia\nPaviaItaly","†","Lorenzo Fassina \nDepartment of Electrical, Computer and Biomedical Engineering\nDepartment of Chemistry, Physical-Chemistry Section\nUniversity of Pavia\nPaviaItaly, 3 C.S.G.I\n\nUniversity of Pavia\nPaviaItaly","Giovanna Bruni","Lucia Cucca \nDepartment of Chemistry\nUniversity of Pavia\nPaviaItaly","Alejandro Kochen \nDepartment of Translational Dental Medicine\nGoldman School of Dental Medicine\nBoston University\nBostonMAUnited States","Paola Divieti Pajevic \nDepartment of Translational Dental Medicine\nGoldman School of Dental Medicine\nBoston University\nBostonMAUnited States","Beth Bragdon \nDepartment of Orthopaedic Surgery\nSchool of Medicine\nBoston University\nBostonMAUnited States","Livia Visai \nDepartment of Molecular Medicine (DMM)\nCenter for Health Technologies (CHT)\nUdR INSTM\nUniversity of Pavia\nPaviaItaly\n\nDepartment of Occupational Medicine, Toxicology and Environmental Risks\nIstituti Clinici Scientifici Maugeri, IRCCS\nPaviaItaly","Louis Gerstenfeld \nDepartment of Orthopaedic Surgery\nSchool of Medicine\nBoston University\nBostonMAUnited States","Max Planck Institute for Intelligent Systems\nNorthwestern University\nUnited States, Germany","University of Granada\nSpain"],"string":"[\n \"Silvia Minardi\",\n \"Hakan Ceylan\",\n \"Gloria Belen Ramirez\",\n \"Beth Bragdon bragdon@bu.edu\",\n \"Livia Visai livia.visai@unipv.it\",\n \"Giulia Montagna \\nDepartment of Molecular Medicine (DMM)\\nCenter for Health Technologies (CHT)\\nUdR INSTM\\nUniversity of Pavia\\nPaviaItaly\\n\\nDepartment of Electrical, Computer and Biomedical Engineering\\nDepartment of Chemistry, Physical-Chemistry Section\\nUniversity of Pavia\\nPaviaItaly, 3 C.S.G.I\\n\\nUniversity of Pavia\\nPaviaItaly\",\n \"Francesco Cristofaro \\nDepartment of Molecular Medicine (DMM)\\nCenter for Health Technologies (CHT)\\nUdR INSTM\\nUniversity of Pavia\\nPaviaItaly\",\n \"†\",\n \"Lorenzo Fassina \\nDepartment of Electrical, Computer and Biomedical Engineering\\nDepartment of Chemistry, Physical-Chemistry Section\\nUniversity of Pavia\\nPaviaItaly, 3 C.S.G.I\\n\\nUniversity of Pavia\\nPaviaItaly\",\n \"Giovanna Bruni\",\n \"Lucia Cucca \\nDepartment of Chemistry\\nUniversity of Pavia\\nPaviaItaly\",\n \"Alejandro Kochen \\nDepartment of Translational Dental Medicine\\nGoldman School of Dental Medicine\\nBoston University\\nBostonMAUnited States\",\n \"Paola Divieti Pajevic \\nDepartment of Translational Dental Medicine\\nGoldman School of Dental Medicine\\nBoston University\\nBostonMAUnited States\",\n \"Beth Bragdon \\nDepartment of Orthopaedic Surgery\\nSchool of Medicine\\nBoston University\\nBostonMAUnited States\",\n \"Livia Visai \\nDepartment of Molecular Medicine (DMM)\\nCenter for Health Technologies (CHT)\\nUdR INSTM\\nUniversity of Pavia\\nPaviaItaly\\n\\nDepartment of Occupational Medicine, Toxicology and Environmental Risks\\nIstituti Clinici Scientifici Maugeri, IRCCS\\nPaviaItaly\",\n \"Louis Gerstenfeld \\nDepartment of Orthopaedic Surgery\\nSchool of Medicine\\nBoston University\\nBostonMAUnited States\",\n \"Max Planck Institute for Intelligent Systems\\nNorthwestern University\\nUnited States, Germany\",\n \"University of Granada\\nSpain\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The osteoinductive property of strontium was repeatedly proven in the last decades. Compelling in vitro data demonstrated that strontium hydroxyapatite nanoparticles exert a dual action, by promoting osteoblasts-driven matrix secretion and inhibiting osteoclasts-driven matrix resorption. Recombinant human bone morphogenetic protein 2 (rhBMP2) is a powerful osteoinductive biologic, used for the treatment of vertebral fractures and critically-sized bone defects. Although effective, the use of rhBMP2 has limitations due its recombinant morphogen nature. In this study, we examined the comparison between two osteoinductive agents: rhBMP2 and the innovative strontium-substituted hydroxyapatite nanoparticles. To test their effectiveness, we independently loaded Gelfoam sponges with the two osteoinductive agents and used the sponges as agent-carriers. Gelfoam are FDA-approved biodegradable medical devices used as delivery system for musculoskeletal defects. Their porous structure and spongy morphology make them attractive in orthopedic field. The abiotic characterization of the loaded sponges, involving ion release pattern and structure investigation, was followed by in vivo implantation onto the periosteum of healthy mice and comparison of the effects induced by each implant was performed. Abiotic analysis demonstrated that strontium was continuously released from the sponges over 28 days with a pattern similar to rhBMP2. Histological observations and gene expression analysis showed stronger endochondral ossification elicited by strontium compared to rhBMP2. Osteoclast activity was more inhibited by strontium than by rhBMP2. These results demonstrated the use of sponges loaded with strontium nanoparticles as potential bone grafts might provide better outcomes for complex fractures. Strontium nanoparticles are a novel and effective non-biologic treatment for bone injuries and can be used as novel powerful therapeutics for bone regeneration."},"sections":{"kind":"list like","value":[{"title":"INTRODUCTION","content":"Bone tissue constitutes the rigid scaffold that is the skeleton, which provides structural support for vertebrates and confers protection to the most delicate vital organs. The extraordinary regenerative capability of bone tissue to repair and heal without the formation of a fibrotic scar was recognized by Imhotep (2630( -2611 and Hippocrates (460-370 BC) (Lanza and Vegetti, 1974). However, in pathological conditions such as osteoporosis, sever bone trauma or critically-sized bone defects this process can be hindered, leading to delayed healing and/or non-union fractures (Holroyd et al., 2008;Sozen et al., 2017). These unsolicited outcomes constitute an often underestimated burden for our society, in term of quality of life, recovery time and costs for the healthcare system (Gentleman et al., 2010;Marcus, 2007).\nBone tissue engineering focuses on the creation of devices capable of providing physical support, activating bone forming cells (concepts, respectively, defined as osteoconductivity and osteoinductivity) with the aim of assisting and accelerating osteogenesis (Dennis et al., 2015;Kowalczewski and Saul, 2018). During the last two decades in vitro and in vivo studies were designed to identify novel scaffolding systems for a topical and controlled delivery of osteoinductive agents (Landi et al., 2007;Visai et al., 2017;Chandran et al., 2018). Many of them focused on the employment of recombinant human bone morphogenic protein 2 (rhBMP2) as a powerful osteoinductive agent (Wang et al., 1990;Buza and Einhorn, 2016). A great amount of effort has been placed in investigating rhBMP2 suitability as treatment for conditions such as spinal fractures and criticallysized bone defects (Noshi et al., 2001;Boix et al., 2005). Several studies highlighted the benefits of using this recombinant morphogen, but also its side effects (Wang et al., 1990;Noshi et al., 2001). Local side effects (such as insufficient or excessive bone formation, heterotopic bone formation, infections and inflammatory responses) as well as systemic ones (cancer in a relatively low percentage of cases) were identified in several independent studies (Boraiah et al., 2009;Latzman et al., 2010;Hoffmann et al., 2013;Woo, 2013;Poon et al., 2016). Compelling experimental results showed the efficacy of strontium cations, as osteoinductive and anti-osteoporotic agents (Rohnke et al., 2016;Carmo et al., 2018;Li et al., 2018). Strontium modulates bone remodeling by enhancing bone formation and suppressing bone resorption (Chattopadhyay et al., 2006;Takaoka et al., 2010;Yang et al., 2011;Saidak and Marie, 2012;Tian et al., 2014) although, to date, the molecular and cellular mechanisms of strontium activity remain partially elusive. Physical-chemical properties of strontium-substituted hydroxyapatite nanoparticles in combination with calcium hydroxyapatite nanoparticles and their in vitro osteoinductivity were assessed in our previous works (Frasnelli et al., 2016;Visai et al., 2017;Cristofaro et al., 2019).\nAbbreviations: ECM, extracellular matrix; OBs, osteoblasts; OCs, osteoclasts; CaSR, calcium sensing receptor HAn, calcium hydroxyapatite nanoparticles; SrHAn, mixture of 90% calcium hydroxyapatite nanoparticles and 10% strontium hydroxyapatite nanoparticles.\nIn this work we utilized new nanotechnology to design a novel nanomaterial that is absent of recombinant morphogen yet still possess bone healing properties exerted by strontium to be used as an application for orthopedic surgery. This nanomaterial is a combination of strontium-substituted and calcium hydroxyapatite nanoparticles which was delivered in vivo by Gelfoam sponges, an FDA-approved collagen-based sponge commonly used as hemostatic application on bleeding surfaces (Pharmacia and Upjohn Company and Pfizer, 2017). The combination between the porous structure and molecular composition made it attractive in the orthopedic field and suitable either for morphogens or for ceramics delivery (Rohanizadeh et al., 2008;Giorgi et al., 2017).\nThe aim of this work is to compare the in vivo osteogenesis of strontium hydroxyapatite nanoparticles (SrHAn) to calcium hydroxyapatite nanoparticles and rhBMP2 (HAn-BMP2) using Gelfoam sponges as carriers with both treatments. An extensive characterization was performed including an abiotic study on the implant structure, as well as analysis of ions and rhBMP2 release patterns. The in vivo study was performed, on a healthy, bone damage-free mouse model to analyze the effectiveness of the described implants in targeting bone tissue. The sponges were implanted adjacent to the periosteum and gene expression of markers of osteocytes and OCs maintenance, MSCs recruitment, osteogenic and chondrogenic differentiation were evaluated, in order to analyze the responses at the periosteal bone. The experimental scheme is shown in Figure 1, with the comparison between SrHAn and HAn-BMP2. Sponges loaded with only calcium hydroxyapatite nanoparticles (HAn) were used as the negative control. In the study we included also the stability assessment of lyophilized samples, as this lyophilization could be suitable for the packaging and distribution of the implants."},{"title":"MATERIALS AND METHODS","content":""},{"title":"Standard Sponges Loading Protocol","content":"Gelfoam sponges (Gelfoam; Pfizer R , NY, USA) were cut into 1 cm 2 . The sponge absorbance capacity of 200 µL was defined as the volume of water completely absorbed by each sponge. The sponges were loaded with a HCl-acidified MilliQ water solution (pH 5) containing 30% (w/v) of calcium hydroxyapatite nanoparticles (HAn) or with a MilliQ water solution containing 27% (w/v) of HAn and 3% (w/v) of strontium hydroxyapatite nanoparticles (SrHAn). For abiotic characterization unloaded sponges were hydrated with acidified MilliQ and defined as the control (CTRL). For rhBMP2 loaded-sponges, 3 µg of human recombinant BMP2 (R&D Systems) were added to the HAn suspension before the sponge absorption phase (HAn-BMP2). Acidified MilliQ was used to increase nanoparticles solubility, buffer pH suspension (otherwise alkaline) and to prevent rhBMP2 precipitation (Friess et al., 1999;Luca et al., 2010). The CTRL, HAn, HAn-BMP2 and SrHAn samples were incubated at room temperature (RT) in a 24-well plate for 15 min after loading to allow the volume to be adsorbed. Following 30 min of incubation at 37 • C, in a humidified and controlled atmosphere, with 5% CO 2 , the wells were filled with 800 µL of acidified MilliQ. FIGURE 1 | Schematic representation of abiotic and in vivo sponge characterization. The collagen sponges were loaded with calcium hydroxyapatite nanoparticles (HAn), with calcium hydroxyapatite nanoparticles + BMP2 (HAn-BMP2), and a mixture of calcium hydroxyapatite nanoparticles and strontium hydroxyapatite nanoparticles (SrHAn). Sponge characterization was achieved by several physical-chemical techniques. X-ray observations and Fast green/Safranin-O histological staining were performed on tissue samples collected from the in vivo experiments. Expression of genes for osteogenesis, osteocytes and osteoclasts activity, chondrogenesis, and stem cell recruitment was also investigated. Figure created with BioRender.com.\nA second set of sponges was prepared exactly as the first set of samples but with an additional step of lyophilization in order to check the system stability. Exception made for HAn-BMP2, which was not prepared in the second set. This second set was labeled with an \"s\" for \"stability\" (sCTRL, sHAn, sSrHAn) and collectively defined as \"treated\" sponges. Following the second incubation period carried out in 800 µL of acidified water, all samples were frozen O/N at −20 • C and lyophilized for 4 h (−50 • C, 0.3 mbar). Following lyophilization samples were rehydrated with 200 µL of acidified MilliQ. For both sets of sponges, water solutions were poured off and fresh non-acidified MilliQ water was added on day 1, 2, 3, 7, 14, 21, and 28."},{"title":"Sponge Calcium, Strontium, and BMP-2 Release","content":"Water solution samples were collected at 1, 2, 3, 7, 14, 21, and 28 days from untreated sponges (CTRL, HAn, SrHAn), treated sponges (sCTRL, sHAn, sSrHAn) and from sponges loaded simultaneously with HAn and rhBMP2 (HAn-BMP2). For the analysis, each sample replicate (1mL) was diluted 1:10 and acidified with suprapur nitric acid (Merck -Sigma Aldrich, Merck KGaA, Darmstadt, Germany) at the final concentration of 0.5% (v/v), centrifuged and filtered with 0.45 µm membranes. Acidified water was used as blank (Moonesi Rad et al., 2019). Measurements were performed using an ICP-OES (inductively coupled plasma optical emission spectroscopy) iCAP 7000 Series (Thermo Scientific) following the standard procedures suggested by the manufacturer and an instrumental method already validated. Calcium and strontium were quantified by external calibration with four standard solutions (0, 5, 1, 5, 10, and 20 mg/L) of the alkaline earth standard metal mix TraceCERT R (100 mg/L). The following wavelengths were selected for the investigated elements: 422,673 nm for Ca 2+ and 407,771 nm for Sr 2+ . Time zero corresponds to the acidified water before the introduction of the sponges. Statistical evaluation was performed with one-way ANOVA. 95% confidence intervals showing the differences between averaged values were plotted in separated charts, reported as ( Figure S2). rhBMP2 release rate was evaluated by ELISA. rhBMP2 release in the water solutions was measured after addition of 20 mM monosodium phosphate, in order to displace rhBMP2 from its binding with HAn (Urist et al., 1984;Boix et al., 2005). Samples were then centrifugated to remove excess calcium and 100 µL of each solution was used for immobilization on ELISA wells. Anti-rhBMP2 polyclonal rabbit IgG (ReliaTech GmgH) was used on dilution factor of 1:3,000. Goat anti-rabbit/HRP (Agilent Tech) secondary antibody was used at 1:5,000 dilution. Reaction was developed with TMB (Sigma T0440−100 mL) for 30 min at RT, stopped with sulfuric acid 0.5 M and ELISA microplate was read with Clariostar ELISA reader at 450 nm. (HAn, SrHAn, sHAn and sSrHAn)."},{"title":"FT-IR Spectroscopy Investigation of Sponge-Nanoparticles Interactions","content":"FT-IR (Fourier Transform-Infra Red) spectra were obtained using a Nicolet FT-IR iS10 Spectrometer (Nicolet, Madison, WI) equipped with attenuated total reflectance sampling accessory (Smart iTR with diamond plate) by coadding 32 scans in the 4,000-650 cm −1 range with a resolution at 4 cm −1 . FT-IR spectra were recorded on the unloaded sponges, on the unloaded sponges blended together with the nanoparticles (referred to as physical mixture, PM) and on the nanoparticles loaded-sponges."},{"title":"Sponge SEM Observations and Mineral Phase Distribution","content":"Untreated (CTRL, HAn, SrHAn) and treated (sCTRL, sHAn, sSrHAn) sponges collected at time 0 days (T0) and time 28 days (T28) were frozen O/N at −20 • C and lyophilized for 3 h (−50 • C, 0.3 mbar). Microscopic structure was investigated using a scanning electron microscope (SEM) Zeiss EVO-MA10 (Carl Zeiss, Oberkochen, Germany). Images were acquired at different magnifications (150x and 350x) and at an accelerating voltage of 20 kV. CTRL, HAn and SrHAn sponges were also investigated with SEM Zeiss EVO-MA10 (Carl Zeiss, Oberkochen, Germany) coupled to an electron dispersive spectroscopy (EDS) (X-max 50 mm 2 , Oxford Instruments, Oxford, UK). Images were acquired at a lower magnification and energy dispersive microanalysis was performed in order to map phosphorus (P), calcium (Ca) and strontium (Sr) and quantify carbon (C), oxygen (O), P, ca and Sr."},{"title":"Sponge Degradation Rate Measurement","content":"To evaluate the sponge degradation rate, untreated unloaded and loaded sponge masses were measured at the following time points: 1, 2, 3, 7, 14, 21, and 28 days. Sponges were prepared as described above. Day 0 corresponded to the untreated and unloaded dried sponge, while day 1 corresponded to loaded sponges. Weights at day 1 were assumed as the initial weights (Wi) and percentage weight loss was calculated for each point. The calculation performed is the following: [(Wi-Wt)/Wi] * 100, were Wt is the weight at each time point. Average and standard deviation of the percentage weight loss were plotted."},{"title":"In vivo Animal Study","content":"All animal studies were approved by the Institutional Animal Care and Use Committee at Boston University (BU). Animals enrolled for these studies were crosses of the B6.Cg-Gt(ROSA)26sortm14(CAG-tdTomato)Hze/J (Ai14; stock number 007914) mouse strain with either B6.CG-Pax7tm1(cre/ER2)Gaka/J (Pax7; stock number 017763) or Prx1CreER-GFP (Kawanami et al., 2009). The Ai14 and Pax7 stains were obtained from The Jackson Laboratory (Bar Harbor ME) and housed at the BU animal facilities under standard conditions. All enrolled for this study were healthy, bone damage-free, male mice aged 9-11 weeks. Strains were randomly assigned to treatment and time point."},{"title":"Sponge Implant Surgical Procedure and in vivo Analysis","content":"The previously prepared untreated samples (HAn, SrHAn, and HAn-BMP2) were used for in vivo animal studies. Mice were bilaterally implanted as described before with the exception that the sponges were implanted instead of demineralized bone matrix to induce ectopic bone formation adjacent to the periosteal surface of femurs (Bragdon et al., 2017). Three mice were enrolled per condition per time point (post-operative day 16 and 33) for RNA extraction and histology analysis. Figure 1 shows a scheme of the performed experiment. No complications and/or collateral effects arose in that period. At post-operative day 16 and 33 mice were euthanized by carbon dioxide inhalation followed by cervical dislocation. Immediately following euthanasia, mice were X-rayed using Faxitron MX-20 Specimen Radiography System at 30 kV for 40 s using Kodak BioMax XAR Scientific Film. The implanted sponge and femur from the left limb were collected separately for RNA extraction and gene analysis. Samples harvested for RNA analysis were stored at −80 • C. The right limb was recovered for histology analysis."},{"title":"Histological Analysis of Mouse Limb","content":"After fixation in paraformaldehyde (4%), the right limbs were decalcified in 14% w/v EDTA (pH 7.2) for 1 week at 4 • C. Limbs were dehydrated and embedded in paraffin for histology and 5 µm-thick sections were cut across the samples. Fast green and Safranin-O (American Mastertek Inc.) staining was performed on the 5 µm-thick sections for the investigation of ectopic bone formation (Provot and Schipani, 2005)."},{"title":"RNA Extraction and Quantitative Reverse Transcriptase PCR","content":"RNA extraction was performed by tissue dissociation and chemical extraction as previously described (Bragdon et al., 2017). Briefly, samples were snap frozen in QIAzol R Lysis Reagent and lysed with the Qiagen R Tissue Lyser II. Chloroform (Sigma-Aldrich) and isopropanol (Sigma-Aldrich) was used to extract and precipitate RNA followed by 70% ethanol washes. The RNA was re-suspended in 30-50 µL of RNase free water and stored at −80 • C. In order to ensure the quality and quantity of the extracted RNA, both spectroscopy and gel electrophoresis were used. For the spectroscopy, a Beckman CoulterTM DU R 530 Life Science UV/Vis Spectrophotometer was used and a 260 nm/280 nm ratio value in the range of 1.7-1.9 indicated an acceptable quality of RNA. The 260 nm absorbance value was used to determine the concentration of RNA in the samples. RNA samples were loaded into a 1% agarose gel, GelStarTM Nucleic Acid Gel Stain from Lonza Group was used to detect the presence of the nucleic acid. The presence of two bands under UV light indicates the RNA is intact and not degraded (data not shown). cDNA was synthesized and qRT-PCR was performed as previously described (Bragdon et al., 2017). Briefly, 1 µg of extracted RNA was reverse transcribed using the TaqMan R Reverse Transcription Reagents kit from Applied Biosystems R . Primers used to probe specific expression of genes are listed in Table 1. The 18S gene was used as a reference gene and in addition to our samples non-operative femurs were used as naïve controls. Target gene expression was normalized using the Ct method of Schmittgen and Livak (2008). Acan Mm00545794_ml"},{"title":"OSTEOGENIC-ASSOCIATED PRIMERS","content":""},{"title":"BGlap Mm03413826_mH","content":"Runx2 Mm00501578_ml\nDmp1 Mm00803833_gl\nSp7 Mm04209856_sl\nIbsp Mm00492555_ml"},{"title":"OSTEOCLASTS-ASSOCIATED PRIMERS","content":""},{"title":"Acp5 Mm00475698_ml","content":"Rankl Mm00441908_ml\nCtsk Mm00484039_ml"},{"title":"OSTEOCYTES-ASSOCIATED PRIMERS","content":""},{"title":"Sost Mm00470479_ml","content":"List of primers used during the in vivo investigation of genes crucial for the differentiation and activity of cells residing in the bone. The genes are divided over their specificity and their catalog number is reported beside."},{"title":"Statistical Analysis","content":"Statistical analyses were performed with Prism 7 (GraphPad Software Inc). Means are plotted and standard deviations (SD) of means, or standard error means (SEM) are represented by error bars. One-way ANOVA corrected by Tukey's honestly significant difference (THSD) test was used to analyze data, if not otherwise specified. A significance level of 0.05 was used for all statistical analyses, unless noted differently."},{"title":"RESULTS","content":""},{"title":"Physical-Chemical Characterization of the Loaded, Untreated, and Treated Sponges","content":"Untreated sponges (CTRL, HAn, and SrHAn) were prepared and physical-chemically characterized for degradation rate, ion release properties by ICP-OES, sponge-nanoparticles interactions by FT-IR and morphology by SEM. Furthermore, a stability assessment was performed on the treated (lyophilized) sponge samples (sCTRL, sHAn, sSrHAn). The absolute amounts of calcium and strontium loaded onto each sponge was then calculated. 60 mg of HAn contains 23.89 mg of calcium whereas 54 mg + 6 mg of SrHAn contains 21.5 mg of calcium and 3.55 mg of strontium. Calcium mass for HAn-BMP2 sponge was calculated as for the HAn sample and corresponds to 23.89 mg. ICP-OES data of the water solutions collected at different time points from untreated samples (Figure 2 and Table 2) was used to evaluate the progressive release of calcium and strontium ions in solution.\nFor HAn-BMP2 samples, calcium ions and rhBMP2 release were measured via ICP-OES and indirect ELISA assay, respectively (Figures 2C,D and Table 2). An initial burst of Ca 2+ release was reported for samples HAn and SrHAn (Figures 2A,B), and this was followed by a significant decrement at day 3, after which it stabilized at about 10 mg/L- Table 2. A similar pattern is presented also for strontium ions (Figure 2B), although the concentration values was higher at day 3, ranging between 40 and 50 mg/L- Table 2. In Han-BMP2 samples, calcium ion release trend was flattened and strongly decreased ( Figure 2C) compared to the Han and SrHAn samples. This is probably due to the presence of rhBMP2 in the same solution. Although, the release of rhBMP2 showed a trend very similar to SrHAn samples, with a concentration spike in the first 2 days followed by a decrease at day 3 and constant release until day 28 ( Figure 2D).\nThe ICP-OES analyses performed on samples treated for stability assessments showed similar ion release patterns after the lyophilization treatment ( Figure S1 and Table S1). In particular, for the sHAn sponges (Figure S1A), calcium ion release was higher at day 3 reaching 30 and 40 mg/L. Interestingly, sSrHAn sponges showed a less intense burst at day 3 and increasing values from day 7 to day 28 ( Figure 2D). Although increments in the days 7-28 were not significant ( Figure S2). The 95% confidence intervals of the ICP-OES average values are reported in (Figure S2).\nTo evaluate sponge-nanoparticle interactions, FT-IR spectroscopy was employed (Figure 3). Unloaded and untreated sponge spectrum (Figures 3A,B) showed characteristic protein peaks (Morris and Finney, 2004;Barth, 2007): NH stretching band at 3271 cm −1 ; C=O stretching band (amide I) at 1,624 cm −1 ; N-H bending band (amide II) at 1,528 cm −1 ; C-N stretching (amide III) at 1,232 cm −1 . FT-IR spectra (Figures 3C-J), showed the comparative analysis between unloaded sponge blended along with nanoparticles (physical mixture = PM; Figures 3C,D,G,H) and untreated sponges loaded with nanoparticles ( Figures 3E,F,I,J). All FT-IR spectra of nanoparticles containing-specimens showed good agreement with the characteristic PO43-(1,023 cm −1 ) and OH-(3,570 cm −1 ) vibrations of hydroxyapatite lattice as reported before (Kim et al., 2002). Due to the low concentration of strontium in the samples, no significant variations were recorded between HAn_PM and SrHAn_PM, in accordance with previous results (Frasnelli et al., 2016). However, protein bands showed a much lower intensity in nanoparticles loaded sponges (Figures 3E,F,I,J) and completely disappeared in PM spectra (Figures 3C,D,G,H). While NH stretching band at 3,271 cm −1 was recorded only in the CTRL, C=O stretching and N-H bending slightly moved from initial values (1,624 and 1,528 cm −1 ) to 1,650-1,631 cm −1 , and 1,531-1,535 cm −1 , respectively. After lyophilization treatment (Figure S3), all protein bands completely disappeared (Figures S3C-F).\nSEM images were collected to analyze the morphology of unloaded and loaded sponges (Figure 4). Unloaded sponges at T0 (Figure 4A) were characterized by a spongy structure, with interconnected pores and smooth wall surfaces. Loaded sponge surfaces were rougher and more irregular (Figures 4B-D). A reduction of wall surfaces occurred in the unloaded FIGURE 2 | Calcium, strontium and rhBMP2 release from loaded untreated sponges. ICP-OES data representation of calcium and strontium ions release from HAn, HAn-BMP2, and SrHAn loaded sponges. Indirect ELISA was performed to quantify solubilized rhBMP2. Data were collected during the 28 days in aqueous solution. Ca 2+ released from HAn sponges (A); Ca 2+ and Sr 2+ released from SrHAn sponges (B); Ca 2+ and rhBMP2 release from HAn-BMP2 sponges (C,D) N = 3 samples were measure per each time point. samples after 28 days (Figure 4E), but it was not observed in loaded samples (Figures 4F-H). A morphological change was appreciated comparing the T0 and T28 (Figures 4B,C,E-H).\nAt T0 hydroxyapatite aggregates were detected on the sponge walls, while at T28 sponge walls looked smoother and the whole structure became more compact. Energy dispersive microanalysis conducted on HAn and SrHAn samples (Figures 4I-L) showed a homogenization of the phosphorus and calcium distribution on the sponge surfaces after 28 days in aqueous solution (Figure 4, P and Ca distribution maps). The same microanalysis performed on CTRL sample did not show an appreciable level of P, Ca and Sr (data not shown). Moreover, on HAn and SrHAn samples a decrement in the weight percentage of calcium and strontium was recorded at day 28 compared to day 0 (Figures 4M-P).\nRegarding the treated samples (Figure S4), unloaded sponges at T0 (Figure S4A) showed a morphology very similar to untreated samples at T0 (Figure S4A). Even though, it is possible to appreciate the formation of minor ripples, probably due to the additional lyophilization phase. At T0, the treated and untreated sponges reported similar structures. At T28, treated and loaded sponges acquired a cement-like morphology (Figures S4F-H).\nThe degradation rate was determined by measuring the sponge weight throughout 28 days in aqueous solution (Figure 5). A slow degradation pattern was observed with no significant variations recorded, except for the SrHAn loaded sponges (Figure 5C). SrHAn samples registered a significant difference in percentage weight loss between day 1 and day 28 (p < 0.0001).\nA reduction in the quantities of calcium and strontium on the sponge surfaces was recorded at day 28, in accordance with ICP-OES data. Taken together these results indicated the SrHAn loaded sponges are a stable system for a constant and prolonged release of Ca 2+ and Sr 2+ . They are biodegradable and their morphology fits the golden standard parameters for osteoconductive scaffolds. Although lyophilization partially changed the morphology of the sponges at T28, there was no significant change in calcium and strontium release."},{"title":"In vivo Sponge's Implantation and Characterization of Their Effects","content":"Next, sponges prepared following the standard protocol (HAn, HAn-BMP2, and SrHAn) were utilized for in vivo implantation to determine the osteogenic ability of the nanoparticles. Radiography, macroscopic observations and histological staining of bone tissues and implanted sponges were carried out to analyze the sponge integration and surrounding tissue adaptation at postoperative day 16 and 33. Gene expression analysis from the implants and femurs were also performed at corresponding times to determine the molecular responses.\nThe radiographic images support the macroscopic observation that strong integration underwent between the femur, sponge, and surrounding tissue (Figure 6). Within the implanted sponges there were few visible blood vessels, as shown in Figure 7. Implants and ectopic bones showed various degrees of calcification, as shown by the radiographic images. The sponges loaded with HAn-BMP2 and SrHAn appeared to show increased calcified material than in the sponges with HAn alone. The radiographic and macroscopic images also suggested that there was a reduction in implant dimensions at day 33 compared to day 16 (Figures 6A-C,J-L). Since the implants consisted of sponge and hydroxyapatite (black arrowheads in Figure 6), it was difficult to distinguish between the implanted material and the formation of ectopic bone. However, the histological analysis using Fast Green and Safranin-O stain demonstrated the formation of both cartilage and bone tissue at the sponge implant site (Figure 7).\nMore extended Safranin-O positive regions (indicated with red arrowheads in Figure 7) were observed in the HAn and SrHAn samples at 16 days. Whereas, at 33 days, no Safranin-O positive tissue could be detected. This indicates that the ectopic bone formed through the endochondral ossification process, with cartilaginous tissue turning to calcified areas. The forming bone tissue at the implant sites also appears to support hematopoiesis, as shown by the recruitment of marrow cells between the femur cortical bone and the sponge (Figure 7E). The newly formed tissues (bone and bone marrow) were present in all implants but in the SrHAn loaded sponges were more represented. At 33 days, (Figures 7G-L) ectopic bone tissue is shown with respect FIGURE 4 | The microscopic structure (150× and 350×) of the untreated CTRL (unloaded sponges), HAn, HAn-BMP2 and SrHAn (loaded sponges) samples is shown in (A-H). Pictures represented samples at T0 (A-D) and after 28 days (E-H) of permanence in aqueous solutions, in a controlled and humidified atmosphere (37 • C, with 5% of CO 2 ). Untreated samples surfaces were also analyzed with SEM-EDS (Energy Dispersive X-Ray Spectroscopy) at 0 (I,J) and 28 days (K,L). Images at low magnifications of the sponges were taken with SEM (I-L). The elements phosphorus (P), calcium (Ca) and strontium (Sr) were mapped on the same pictures using different colors (P = green; Ca = red; Sr = white) and a black background. Quantification of the elements carbon (C), oxygen (O), phosphorus (P), calcium (Ca) and strontium (Sr) present of the samples surface was performed and data were reported as weight percentage in the bar charts (M,N,O,P) histograms. to control, accompanied by the presence of an area enriched in bone marrow cellular components.\nTo investigate the changes in gene expression in both the femur and implants after 16 and 33 days, tissues were harvested separately from HAn, HAn-BMP2, and SrHAn implanted mice. Figure 8 represents chondrogenesis (Acan, Col10A1, and Sox9) and stem cell recruitment (Nanog) markers. While Figure 9 shows gene expression of markers for osteogenesis (Runx2, Sp7, Ibsp, BGlap, and Dmp1), osteocytes (Sost) and OCs activity (Acp5, Rankl, and Ctsk). Acan (Aggrecan core protein), Col10A1 (Type 10 collagen) and Sox9 expression was analyzed as chondrogenesis and endochondral ossification markers (Mackie et al., 2008;Dennis et al., 2015). There was a consistent difference between the expression of chondrogenic markers in bone and sponge samples as noted by the sectioned Y axis. Indeed, at 16 days, in the former these three genes were downregulated by both rhBMP2 and strontium, with respect to HAn condition ( Figure 8A). While in the latter, both osteoinductive factors brought to significantly higher expression of Col10A1 and Sox9 (Figure 8C), with an enhanced effect in SrHAn loaded sponges. At 33 days, the difference between the tissue samples is equalized and only significant up-regulation of Sox9 in the bone and Acan in the sponge were shown as strontium-driven effects (Figures 8E,G). Expression of the stem cells recruitment marker, Nanog, showed no significant difference at 16 days and only a mild up-regulation with respect to control at 33 days in the sponge sample. After 16 days, a significant up-regulation of BGlap (Osteocalcin), Ibsp (bone sialoprotein), and Sp7 (Osterix) was observed in the FIGURE 5 | Graphical representation of sponge weights variations. Untreated sample weights were measured with an analytical balance. Weights were recorded at different time points throughout the 28 days in aqueous solution. Day 0 is referred to the unloaded sponge, whereas Day 1 corresponds to the loaded sponges and it was considered the initial weight (Wi). N = 3 different sponges were weighted per each condition. From the recorded weights, we calculated the percentage weight loss. Error bars indicate the standard deviation. Statistically significant differences were assessed with one way ANOVA. Significant differences were reported only against day 1 (*p < 0, 05; **p < 0, 01; ****p < 0, 0001).\nfemur of mice that received the SrHAn implant compared to HAn-BMP2 and to HAn samples alone ( Figure 9A). Similar results were observed with BGlap, Ibsp, and Sp7 expression being evaluated in the sponge samples at 16 days ( Figure 9D). In both the femur and the sponge tissues Dmp1 (Dentin matrix protein 1) was upregulated by strontium with respect to the control HAn. A similar trend of expression was also detected at 33 days, where Ibsp in the bone ( Figure 9G) and Ibsp, BGlap, and Sp7 in the sponge ( Figure 9J) were upregulated by strontium. Runx2 didn't show significant differences in expression at given time points. Sost (Sclerostin) expression was analyzed as marker for osteocyte differentiation. After 16 days, Sost was upregulated in the strontium samples only with respect to the control HAn, both in the bone and sponge samples (Figures 9B,E). At 33 days, an up-regulation of Sost was observed only in the bone samples loaded with strontium compared to HAn-BMP2 and HAn (Figures 9H,K). Acp5 (Tartrate-resistant acid phosphatase type 5), Rankl (Tumor necrosis factor ligand superfamily member 11) and Ctsk (Cathepsin K) expression were analyzed as markers for OCs differentiation and activity. At 16 days, Acp5 and Rankl (Figures 9C,F) were downregulated in SrHAn femurs and sponges Ctsk at 16 days was upregulated in femur and sponge by both types of osteoinductive factors (Figures 9C,F). At 33 days, strontium presence led to an upregulation of Acp5 and down-regulation of Rankl only in the bone samples ( Figure 9I). Ctsk was down-regulated in SrHAn implanted mice with respect to control, both in bone and sponge samples (Figures 9I,L).\nSrHAn loaded-sponge structure were suitable for the promotion of cell invasion, bone marrow cell recruitment and ossification. Following a lyophilization treatment (which could be useful for sponge long-term storage and packaging), the loaded sponges did not show significant variation from the untreated samples and all physical-chemical properties were conserved. The biodegradability and biocompatibility of the system was proven in vivo, together with the induction of endochondral ossification stages in adult bone. Evidence of endochondral ossification were given in the histological investigation, where proliferating and hypertrophic chondrocytes were more represented in the strontium-containing samples than in the BMP2 ones. Gene expression corroborated histological evidences, showing increased expression of osteogenic and chondrogenesis markers. Moreover, these data were consistent with an enhancement of endochondral ossification in the SrHAn samples if compared to the HAn-BMP2 samples. Antiresorptive properties of strontium were also demonstrated by investigating markers of osteoclasts differentiation and activity. Histological and gene expression data further Statistical significance values were calculated with one way-ANOVA, followed by Tukey's honestly significant difference test. *, significant difference against the HAn condition (p < 0.05). #, significant difference between HAn-BMP2 and Sr conditions (p < 0.05). Statistical significance values were calculated with one way-ANOVA, followed by Tukey's honestly significant difference test. *, significant difference against the HAn condition (p < 0.05). #, significant difference between HAn-BMP2 and SrHAn conditions (p < 0.05).\nsupported the osteoinductive and osteoconductive potential of this system."},{"title":"DISCUSSION","content":"Bone tissue engineering application in the orthopeadic surgical field is becoming increasingly relevant. In the last decades, rhBMP2 was extensively studied for its osteoinductive potential. Although several successful applications have been achieved, this recombinant morphogen was proven to be not always suitable and safe (Poon et al., 2016;Lykissas and Gkiatas, 2017). Another powerful osteoinductive agent, strontium, was widely studied in the previous years and compelling results were obtained in term of biocompatibility and osteoconductivity (Boanini et al., 2011;Yang et al., 2011). Strontium hydroxyapatite nanoparticles were recently developed, carrying an inorganic osteoinductive agent (Frasnelli et al., 2016). They have been characterized in the last years and in this work, they are presented as a substitute for rhBMP2.\nIn this study, the medical grade collagen-based Gelfoam sponges were loaded with a combination of either calcium and strontium hydroxyapatite nanoparticles suspensions (SrHAn) or calcium hydroxyapatite nanoparticles and rhBMP2 (HAn-BMP2) in order to compare their physical-chemical properties. The HAn sponges were loaded only with calcium hydroxyapatite nanoparticles and used as control. A second set of samples were treated with an additional lyophilization step and characterized.\nThe hydroxyapatite nanoparticles amount on each sponge was 30% (w/v) of the volume absorbed by the sponge and 3 µg of rhBMP2 per sponge were used for HAn-BMP2. For the SrHAn loaded-sponges, the calcium/strontium hydroxyapatite nanoparticles ratio was 9:1 w/w (Chandran et al., 2016) while the spontaneously occurring Sr:Ca ratio is between 1:1,000 and 1:2,000 in human bone tissue as well as in blood serum (Cabrera et al., 1999).\nThe cumulative amount of calcium and strontium ions after 28 days in aqueous solutions were measured as 0.0118 and 0.0358 mg, respectively. Interestingly, several in vitro and in vivo studies reported that strontium elicits rhBMP2-like effects when administered at concentrations up to 1 mM, as reported by Schumacher et al. (2013) and others (Qiu et al., 2006;Li et al., 2010;Schumacher et al., 2013;Wei et al., 2019). Despite the lower water solubility of Sr 2+ compared to Ca 2+ , we detected enhanced strontium solubilization from the SrHAn sponges, compared to the calcium release from the HAn sponges. These data are in agreement with Kaufman and Kleinberg (1979) and Pan et al. (2009) studies. They demonstrated how, due to a larger atomic radius, strontium atoms slightly reduce hydroxyapatite lattice thermodynamic stability, favoring the release of Sr 2+ from the lattice itself. Furthermore, ICP-OES results were comparable to the ones presented by Landi et al. in term of ion release pattern (Landi et al., 2007) and were not significatively influenced by the lyophilization treatment. We investigated the spongenanoparticle system through FT-IR analysis of the untreated and treated systems (HAn, SrHAn, sHAn, and sSrHAn). Significant interactions were not found between the sponge and either HAn or SrHAn. FT-IR spectra of loaded sponges showed no chemical interactions were undergoing between the sponges and the nanoparticles. The minor shifting of the amide I and II bands was attributed to hydrogen bonds breaking in the sponge proteins and consequent modification of the hydrogen bonds network in the sponge structure. We can therefore assume that the ion release patterns studied with ICP-OES were not influenced by chemical interactions between the nanoparticles and the sponge support, either before or after the lyophilization treatment. To illustrate the patterns of ion release in function of time we considered the porous structure of the sponges. We could speculate a bulk of nanoparticles flowed inside the sponge pores, during the loading procedure. Nanoparticles remaining on the sponge surfaces released a high percentage of Ca 2+ and Sr 2+ ions during the first 3 days, while embedded nanoparticles were responsible for the constant release recorded after day 3. The same speculation can also fit the data on rhBMP2 release. This hypothesis was corroborated by SEM images, that showed smoother surfaces and solubilization of hydroxyapatite aggregates at day 28. Surprisingly, even after 28 days in aqueous solution, calcium and traces of strontium were still detected on the sponge surfaces, as demonstrated by SEM-EDS data.\nInterestingly, the calcium release pattern of HAn-BMP2 loaded-sponges was different from the other samples. This can be due to the non-covalent interaction between HAn and rhBMP2 (Urist et al., 1984;Boix et al., 2005). We were not able to investigate HAn-BMP2 loaded-sponges with FT-IR, since this technique is not able to discern between different proteins. We can speculate that calcium ions remained bound to solubilized rhBMP2, therefore decreasing the amount of free Ca 2+ released in solution.\nPore sizes ranging between 100 and 300 µm are considered the most suitable for osteoinduction improvement as well as for angiogenesis promotion (Karageorgiou and Kaplan, 2005;Scheinpflug et al., 2018). Furthermore, geometry of the investigated sponges reflected the \"gold standard\" in bone tissue engineering, according to Kuboki et al. (2007) and due to the given pores size and geometry the surface area exposed to the surrounding microenvironment is increased, allowing for a wider interaction between the nanoparticles coating and the surrounding microenvironment.\nThe degradability and recyclability of the graft is a crucial point, although often disregarded, in tissue engineering applications (Gentleman et al., 2010;Rohnke et al., 2016). In this work, the time-dependent biodegradability of the sponges and the hydroxyapatite nanoparticles both in the abiotic and in vivo conditions, were analyzed. Degradation pattern based on the sponge weights appeared mild but significant for SrHAn sponges, throughout the 28 days in solution. While, in vivo radiographic images and histology suggested an accelerated degradation of SrHAn sponges. Radiographic images also revealed an induction of bone formation and osseointegration by all three types of implants at both time points. Although, a more extensive mineralized tissue in mice implanted with HAn-BMP2 and SrHAn sponges were shown with respect to the controls. Histological analysis revealed new tissue formation between the sponge and the femoral cortex, suggesting a center of active osteogenesis was present in the sponge site. Given the similarities between Ca 2+ and Sr 2+ , it has been shown that this element can substitute calcium in the lattice of the hydroxyapatite crystals present in bones (Ni et al., 2011;Querido et al., 2016). Sr 2+ is processed like Ca 2+ throughout metabolism, being preferentially introduced in active osteogenesis sites (Bauman et al., 2005).\nRhBMP2 and Sr 2+ release was quantified using different techniques, therefore it is not possible to compare their quantities/concentrations in solution head-to-head. Despite the similarities in the pattern release between HAn-BMP2 and SrHAn, strontium containing nanoparticles induced significant increase of endochondral ossification markers. It is wellknown how the process of endochondral ossification can be recapitulated in adults by bone fracture and bone tissue damage (Gerstenfeld et al., 2003;Dennis et al., 2015), although in this work evidences of endochondral ossification were induced in healthy implanted mice, with no bone damage. Furthermore, no record of strontium related-endochondral ossification induction is present up to date, but here at 16 days the onset of typical multicellular clusters constitutes by enlarged chondrocyte was demonstrated (Mackie et al., 2008). Hypertrophic chondrocytes can be detected in the sponge surroundings at 16 days, together with smaller proliferating chondrocytes. These were particularly represented in the HAn and SrHAn samples and red stained-cartilage proteoglycans were present all around the cells. Hypertrophic chondrocytes were almost absent in the HAn-BMP2 samples, suggesting a possible delay or acceleration in chondrogenic differentiation. Different phases of endochondral ossification process were shown also by gene expression data: Sox9 upregulation was an indicator of chondrocytes proliferation (Akiyama et al., 2002;Lefebvre and Dvir-Ginzberg, 2017) and after its downregulation, markers of hypertrophic chondrocytes, such as aggrecan core protein (Acan) and type X collagen (Col10A1) were upregulated (Akiyama and Lefebvre, 2011). We can speculate that the different temporal upregulation of Sox9, among bone and sponges, can be attributed to the time dependent Sr 2+ diffusion, which resulted in its earlier expression in the sponge area and delayed expression in the bone one.\nThe interaction of Sr 2+ with the calcium-sensing receptor (CaSR) was shown both in OBs and osteoclasts (OCs), resulting in improved anabolic processes and reduced catabolic pathways, respectively (Chattopadhyay et al., 2006). Our gene expression data showed SrHAn double activity on OBs and OCs. Several studies reported that strontium modulates alkaline phosphatase (ALP) activity and Runx-2 expression (Tian et al., 2014) in OBs and OBs progenitors while promoting Wnt/β-catenin pathway in human bone marrow derived mesenchymal stem cells (MSCs), thereby enhancing ECM secretion and osteogenic differentiation (Yang et al., 2011). Interestingly, in our study no significant variation was detected in Runx2 levels. However, our results measured an upregulation of Sp7 (Osterix), indicating OBs differentiation at early and late stage (Santo et al., 2013;Kawane et al., 2018). Sp7 is a transcriptional factor expressed in developing bones regulating the commitment of MSCs toward osteoblastic lineage (Rutkovskiy et al., 2016). Its regulation affects expression of protein-coding genes typically related to osteoblastic differentiation, such as osteocalcin, type I collagen and bone sialoprotein (BGlap, Ibsp, and Dmp1). Increased expression of the ECM proteins BGlap, Ibsp, and Dmp1 was attributed to mature osteoblasts activity (Wrobel et al., 2016) in SrHAn samples. Sp7, BGlap, and Ibsp expression also indicated a stronger osteogenic induction for strontium ions compared to rhBMP2. Interestingly Runx2, the master regulator of osteoblastic differentiation acting above Sp7, was not significantly influenced neither by strontium nor by rhBMP2 (Nakashima et al., 2002). Soluble cytokine promoting OCs differentiation, Rankl, (expressed by OBs) is a key modulator of the resorption cycle, controlling OCs differentiation and activation, and therefore bone resorption. This molecule has already been linked to strontium-mediated modulation by Atkins et al. (2009) andBrennan et al. (2009). Our results showed strontium significantly downregulated Rankl when compared to control and HAn-BMP2 samples. It has been shown that strontium effects are mediated by CaSR and by others, yet unclarified receptors (Chattopadhyay et al., 2006;Takaoka et al., 2010;Saidak and Marie, 2012). Furthermore, being CaSR present also on OCs outer membrane it was demonstrated the action of strontium on OCs activity. Indeed, OCs-secreted endoprotease responsible for ECM degradation (Sage et al., 2012), Ctsk (Cathepsin K) was reduced by strontium presence at 33 days, with respect to the control. Of note, in some of the SrHAn sponges we observed vascularization, suggesting a potential effect of Sr 2+ on neo-vascularization, although further studies are needed to confirm these findings."},{"title":"CONCLUSIONS","content":"Many studies have been conducted to investigate rhBMP2 suitability as treatment for spinal fractures and critically-sized bone defects. Although its indisputable osteoinductive potential, rhBMP2 also showed adverse effects. On the other hand, strontium has already been tested as a treatment for osteoporosis and its activity was demonstrated in a patient population. The biggest advantage of strontium over rhBMP2 however, is that the former is a chemical and not a biologic. Herein, following the abiotic characterization, we presented the results of a comparative in vivo study, where Gelfoam sponges loaded either with strontium hydroxyapatite nanoparticles or with rhBMP2 were implanted in healthy, bone damage-free mouse model. In vivo studies, showed that SrHAn and HAn-BMP2 have comparable effects, driving the onset of endochondral ossification and promoting the bone remodeling process. These results demonstrated that SrHAn loaded-sponges have marked osteogenic potential when applied on the periosteum of long bones, comparable to HAn-BMP2 loaded-sponges, but eliciting a more controlled ossification response. We propose to use Gelfoam sponges enriched in Sr 2+ as an effective therapeutic intervention to treat severe bone defects or open, complicated fractures. The short-term benefit of this nanotechnological system is the availability of novel therapeutic option to treat bone fracture (or bone defect). The use of sponges loaded with strontium hydroxyapatite nanoparticles might also provide better outcomes for complex fractures. Results from these studies can provide novel therapeutic options for active duty personnel and can be beneficial to anyone suffering from trauma, bone defects or severe bone injuries. Future studies shall also evaluate the efficacy of SrHAn loaded sponges for the treatment of spinal fusion."},{"title":"DATA AVAILABILITY STATEMENT","content":"All datasets generated for this study are included in the article/Supplementary Material."},{"title":"ETHICS STATEMENT","content":"All animal studies were approved by the Institutional Animal Care and Use Committee at Boston University (BU)."},{"title":"AUTHOR CONTRIBUTIONS","content":""},{"title":"FIGURE 3 |","content":""},{"title":"FIGURE 6 |","content":""},{"title":"FIGURE 7 |","content":""},{"title":"FIGURE 8 |","content":""},{"title":"FIGURE 9 |","content":""},{"title":"FUNDING","content":""},{"title":"TABLE 1 |","content":""},{"title":"TABLE 2 |","content":""}],"string":"[\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"Bone tissue constitutes the rigid scaffold that is the skeleton, which provides structural support for vertebrates and confers protection to the most delicate vital organs. The extraordinary regenerative capability of bone tissue to repair and heal without the formation of a fibrotic scar was recognized by Imhotep (2630( -2611 and Hippocrates (460-370 BC) (Lanza and Vegetti, 1974). However, in pathological conditions such as osteoporosis, sever bone trauma or critically-sized bone defects this process can be hindered, leading to delayed healing and/or non-union fractures (Holroyd et al., 2008;Sozen et al., 2017). These unsolicited outcomes constitute an often underestimated burden for our society, in term of quality of life, recovery time and costs for the healthcare system (Gentleman et al., 2010;Marcus, 2007).\\nBone tissue engineering focuses on the creation of devices capable of providing physical support, activating bone forming cells (concepts, respectively, defined as osteoconductivity and osteoinductivity) with the aim of assisting and accelerating osteogenesis (Dennis et al., 2015;Kowalczewski and Saul, 2018). During the last two decades in vitro and in vivo studies were designed to identify novel scaffolding systems for a topical and controlled delivery of osteoinductive agents (Landi et al., 2007;Visai et al., 2017;Chandran et al., 2018). Many of them focused on the employment of recombinant human bone morphogenic protein 2 (rhBMP2) as a powerful osteoinductive agent (Wang et al., 1990;Buza and Einhorn, 2016). A great amount of effort has been placed in investigating rhBMP2 suitability as treatment for conditions such as spinal fractures and criticallysized bone defects (Noshi et al., 2001;Boix et al., 2005). Several studies highlighted the benefits of using this recombinant morphogen, but also its side effects (Wang et al., 1990;Noshi et al., 2001). Local side effects (such as insufficient or excessive bone formation, heterotopic bone formation, infections and inflammatory responses) as well as systemic ones (cancer in a relatively low percentage of cases) were identified in several independent studies (Boraiah et al., 2009;Latzman et al., 2010;Hoffmann et al., 2013;Woo, 2013;Poon et al., 2016). Compelling experimental results showed the efficacy of strontium cations, as osteoinductive and anti-osteoporotic agents (Rohnke et al., 2016;Carmo et al., 2018;Li et al., 2018). Strontium modulates bone remodeling by enhancing bone formation and suppressing bone resorption (Chattopadhyay et al., 2006;Takaoka et al., 2010;Yang et al., 2011;Saidak and Marie, 2012;Tian et al., 2014) although, to date, the molecular and cellular mechanisms of strontium activity remain partially elusive. Physical-chemical properties of strontium-substituted hydroxyapatite nanoparticles in combination with calcium hydroxyapatite nanoparticles and their in vitro osteoinductivity were assessed in our previous works (Frasnelli et al., 2016;Visai et al., 2017;Cristofaro et al., 2019).\\nAbbreviations: ECM, extracellular matrix; OBs, osteoblasts; OCs, osteoclasts; CaSR, calcium sensing receptor HAn, calcium hydroxyapatite nanoparticles; SrHAn, mixture of 90% calcium hydroxyapatite nanoparticles and 10% strontium hydroxyapatite nanoparticles.\\nIn this work we utilized new nanotechnology to design a novel nanomaterial that is absent of recombinant morphogen yet still possess bone healing properties exerted by strontium to be used as an application for orthopedic surgery. This nanomaterial is a combination of strontium-substituted and calcium hydroxyapatite nanoparticles which was delivered in vivo by Gelfoam sponges, an FDA-approved collagen-based sponge commonly used as hemostatic application on bleeding surfaces (Pharmacia and Upjohn Company and Pfizer, 2017). The combination between the porous structure and molecular composition made it attractive in the orthopedic field and suitable either for morphogens or for ceramics delivery (Rohanizadeh et al., 2008;Giorgi et al., 2017).\\nThe aim of this work is to compare the in vivo osteogenesis of strontium hydroxyapatite nanoparticles (SrHAn) to calcium hydroxyapatite nanoparticles and rhBMP2 (HAn-BMP2) using Gelfoam sponges as carriers with both treatments. An extensive characterization was performed including an abiotic study on the implant structure, as well as analysis of ions and rhBMP2 release patterns. The in vivo study was performed, on a healthy, bone damage-free mouse model to analyze the effectiveness of the described implants in targeting bone tissue. The sponges were implanted adjacent to the periosteum and gene expression of markers of osteocytes and OCs maintenance, MSCs recruitment, osteogenic and chondrogenic differentiation were evaluated, in order to analyze the responses at the periosteal bone. The experimental scheme is shown in Figure 1, with the comparison between SrHAn and HAn-BMP2. Sponges loaded with only calcium hydroxyapatite nanoparticles (HAn) were used as the negative control. In the study we included also the stability assessment of lyophilized samples, as this lyophilization could be suitable for the packaging and distribution of the implants.\"\n },\n {\n \"title\": \"MATERIALS AND METHODS\",\n \"content\": \"\"\n },\n {\n \"title\": \"Standard Sponges Loading Protocol\",\n \"content\": \"Gelfoam sponges (Gelfoam; Pfizer R , NY, USA) were cut into 1 cm 2 . The sponge absorbance capacity of 200 µL was defined as the volume of water completely absorbed by each sponge. The sponges were loaded with a HCl-acidified MilliQ water solution (pH 5) containing 30% (w/v) of calcium hydroxyapatite nanoparticles (HAn) or with a MilliQ water solution containing 27% (w/v) of HAn and 3% (w/v) of strontium hydroxyapatite nanoparticles (SrHAn). For abiotic characterization unloaded sponges were hydrated with acidified MilliQ and defined as the control (CTRL). For rhBMP2 loaded-sponges, 3 µg of human recombinant BMP2 (R&D Systems) were added to the HAn suspension before the sponge absorption phase (HAn-BMP2). Acidified MilliQ was used to increase nanoparticles solubility, buffer pH suspension (otherwise alkaline) and to prevent rhBMP2 precipitation (Friess et al., 1999;Luca et al., 2010). The CTRL, HAn, HAn-BMP2 and SrHAn samples were incubated at room temperature (RT) in a 24-well plate for 15 min after loading to allow the volume to be adsorbed. Following 30 min of incubation at 37 • C, in a humidified and controlled atmosphere, with 5% CO 2 , the wells were filled with 800 µL of acidified MilliQ. FIGURE 1 | Schematic representation of abiotic and in vivo sponge characterization. The collagen sponges were loaded with calcium hydroxyapatite nanoparticles (HAn), with calcium hydroxyapatite nanoparticles + BMP2 (HAn-BMP2), and a mixture of calcium hydroxyapatite nanoparticles and strontium hydroxyapatite nanoparticles (SrHAn). Sponge characterization was achieved by several physical-chemical techniques. X-ray observations and Fast green/Safranin-O histological staining were performed on tissue samples collected from the in vivo experiments. Expression of genes for osteogenesis, osteocytes and osteoclasts activity, chondrogenesis, and stem cell recruitment was also investigated. Figure created with BioRender.com.\\nA second set of sponges was prepared exactly as the first set of samples but with an additional step of lyophilization in order to check the system stability. Exception made for HAn-BMP2, which was not prepared in the second set. This second set was labeled with an \\\"s\\\" for \\\"stability\\\" (sCTRL, sHAn, sSrHAn) and collectively defined as \\\"treated\\\" sponges. Following the second incubation period carried out in 800 µL of acidified water, all samples were frozen O/N at −20 • C and lyophilized for 4 h (−50 • C, 0.3 mbar). Following lyophilization samples were rehydrated with 200 µL of acidified MilliQ. For both sets of sponges, water solutions were poured off and fresh non-acidified MilliQ water was added on day 1, 2, 3, 7, 14, 21, and 28.\"\n },\n {\n \"title\": \"Sponge Calcium, Strontium, and BMP-2 Release\",\n \"content\": \"Water solution samples were collected at 1, 2, 3, 7, 14, 21, and 28 days from untreated sponges (CTRL, HAn, SrHAn), treated sponges (sCTRL, sHAn, sSrHAn) and from sponges loaded simultaneously with HAn and rhBMP2 (HAn-BMP2). For the analysis, each sample replicate (1mL) was diluted 1:10 and acidified with suprapur nitric acid (Merck -Sigma Aldrich, Merck KGaA, Darmstadt, Germany) at the final concentration of 0.5% (v/v), centrifuged and filtered with 0.45 µm membranes. Acidified water was used as blank (Moonesi Rad et al., 2019). Measurements were performed using an ICP-OES (inductively coupled plasma optical emission spectroscopy) iCAP 7000 Series (Thermo Scientific) following the standard procedures suggested by the manufacturer and an instrumental method already validated. Calcium and strontium were quantified by external calibration with four standard solutions (0, 5, 1, 5, 10, and 20 mg/L) of the alkaline earth standard metal mix TraceCERT R (100 mg/L). The following wavelengths were selected for the investigated elements: 422,673 nm for Ca 2+ and 407,771 nm for Sr 2+ . Time zero corresponds to the acidified water before the introduction of the sponges. Statistical evaluation was performed with one-way ANOVA. 95% confidence intervals showing the differences between averaged values were plotted in separated charts, reported as ( Figure S2). rhBMP2 release rate was evaluated by ELISA. rhBMP2 release in the water solutions was measured after addition of 20 mM monosodium phosphate, in order to displace rhBMP2 from its binding with HAn (Urist et al., 1984;Boix et al., 2005). Samples were then centrifugated to remove excess calcium and 100 µL of each solution was used for immobilization on ELISA wells. Anti-rhBMP2 polyclonal rabbit IgG (ReliaTech GmgH) was used on dilution factor of 1:3,000. Goat anti-rabbit/HRP (Agilent Tech) secondary antibody was used at 1:5,000 dilution. Reaction was developed with TMB (Sigma T0440−100 mL) for 30 min at RT, stopped with sulfuric acid 0.5 M and ELISA microplate was read with Clariostar ELISA reader at 450 nm. (HAn, SrHAn, sHAn and sSrHAn).\"\n },\n {\n \"title\": \"FT-IR Spectroscopy Investigation of Sponge-Nanoparticles Interactions\",\n \"content\": \"FT-IR (Fourier Transform-Infra Red) spectra were obtained using a Nicolet FT-IR iS10 Spectrometer (Nicolet, Madison, WI) equipped with attenuated total reflectance sampling accessory (Smart iTR with diamond plate) by coadding 32 scans in the 4,000-650 cm −1 range with a resolution at 4 cm −1 . FT-IR spectra were recorded on the unloaded sponges, on the unloaded sponges blended together with the nanoparticles (referred to as physical mixture, PM) and on the nanoparticles loaded-sponges.\"\n },\n {\n \"title\": \"Sponge SEM Observations and Mineral Phase Distribution\",\n \"content\": \"Untreated (CTRL, HAn, SrHAn) and treated (sCTRL, sHAn, sSrHAn) sponges collected at time 0 days (T0) and time 28 days (T28) were frozen O/N at −20 • C and lyophilized for 3 h (−50 • C, 0.3 mbar). Microscopic structure was investigated using a scanning electron microscope (SEM) Zeiss EVO-MA10 (Carl Zeiss, Oberkochen, Germany). Images were acquired at different magnifications (150x and 350x) and at an accelerating voltage of 20 kV. CTRL, HAn and SrHAn sponges were also investigated with SEM Zeiss EVO-MA10 (Carl Zeiss, Oberkochen, Germany) coupled to an electron dispersive spectroscopy (EDS) (X-max 50 mm 2 , Oxford Instruments, Oxford, UK). Images were acquired at a lower magnification and energy dispersive microanalysis was performed in order to map phosphorus (P), calcium (Ca) and strontium (Sr) and quantify carbon (C), oxygen (O), P, ca and Sr.\"\n },\n {\n \"title\": \"Sponge Degradation Rate Measurement\",\n \"content\": \"To evaluate the sponge degradation rate, untreated unloaded and loaded sponge masses were measured at the following time points: 1, 2, 3, 7, 14, 21, and 28 days. Sponges were prepared as described above. Day 0 corresponded to the untreated and unloaded dried sponge, while day 1 corresponded to loaded sponges. Weights at day 1 were assumed as the initial weights (Wi) and percentage weight loss was calculated for each point. The calculation performed is the following: [(Wi-Wt)/Wi] * 100, were Wt is the weight at each time point. Average and standard deviation of the percentage weight loss were plotted.\"\n },\n {\n \"title\": \"In vivo Animal Study\",\n \"content\": \"All animal studies were approved by the Institutional Animal Care and Use Committee at Boston University (BU). Animals enrolled for these studies were crosses of the B6.Cg-Gt(ROSA)26sortm14(CAG-tdTomato)Hze/J (Ai14; stock number 007914) mouse strain with either B6.CG-Pax7tm1(cre/ER2)Gaka/J (Pax7; stock number 017763) or Prx1CreER-GFP (Kawanami et al., 2009). The Ai14 and Pax7 stains were obtained from The Jackson Laboratory (Bar Harbor ME) and housed at the BU animal facilities under standard conditions. All enrolled for this study were healthy, bone damage-free, male mice aged 9-11 weeks. Strains were randomly assigned to treatment and time point.\"\n },\n {\n \"title\": \"Sponge Implant Surgical Procedure and in vivo Analysis\",\n \"content\": \"The previously prepared untreated samples (HAn, SrHAn, and HAn-BMP2) were used for in vivo animal studies. Mice were bilaterally implanted as described before with the exception that the sponges were implanted instead of demineralized bone matrix to induce ectopic bone formation adjacent to the periosteal surface of femurs (Bragdon et al., 2017). Three mice were enrolled per condition per time point (post-operative day 16 and 33) for RNA extraction and histology analysis. Figure 1 shows a scheme of the performed experiment. No complications and/or collateral effects arose in that period. At post-operative day 16 and 33 mice were euthanized by carbon dioxide inhalation followed by cervical dislocation. Immediately following euthanasia, mice were X-rayed using Faxitron MX-20 Specimen Radiography System at 30 kV for 40 s using Kodak BioMax XAR Scientific Film. The implanted sponge and femur from the left limb were collected separately for RNA extraction and gene analysis. Samples harvested for RNA analysis were stored at −80 • C. The right limb was recovered for histology analysis.\"\n },\n {\n \"title\": \"Histological Analysis of Mouse Limb\",\n \"content\": \"After fixation in paraformaldehyde (4%), the right limbs were decalcified in 14% w/v EDTA (pH 7.2) for 1 week at 4 • C. Limbs were dehydrated and embedded in paraffin for histology and 5 µm-thick sections were cut across the samples. Fast green and Safranin-O (American Mastertek Inc.) staining was performed on the 5 µm-thick sections for the investigation of ectopic bone formation (Provot and Schipani, 2005).\"\n },\n {\n \"title\": \"RNA Extraction and Quantitative Reverse Transcriptase PCR\",\n \"content\": \"RNA extraction was performed by tissue dissociation and chemical extraction as previously described (Bragdon et al., 2017). Briefly, samples were snap frozen in QIAzol R Lysis Reagent and lysed with the Qiagen R Tissue Lyser II. Chloroform (Sigma-Aldrich) and isopropanol (Sigma-Aldrich) was used to extract and precipitate RNA followed by 70% ethanol washes. The RNA was re-suspended in 30-50 µL of RNase free water and stored at −80 • C. In order to ensure the quality and quantity of the extracted RNA, both spectroscopy and gel electrophoresis were used. For the spectroscopy, a Beckman CoulterTM DU R 530 Life Science UV/Vis Spectrophotometer was used and a 260 nm/280 nm ratio value in the range of 1.7-1.9 indicated an acceptable quality of RNA. The 260 nm absorbance value was used to determine the concentration of RNA in the samples. RNA samples were loaded into a 1% agarose gel, GelStarTM Nucleic Acid Gel Stain from Lonza Group was used to detect the presence of the nucleic acid. The presence of two bands under UV light indicates the RNA is intact and not degraded (data not shown). cDNA was synthesized and qRT-PCR was performed as previously described (Bragdon et al., 2017). Briefly, 1 µg of extracted RNA was reverse transcribed using the TaqMan R Reverse Transcription Reagents kit from Applied Biosystems R . Primers used to probe specific expression of genes are listed in Table 1. The 18S gene was used as a reference gene and in addition to our samples non-operative femurs were used as naïve controls. Target gene expression was normalized using the Ct method of Schmittgen and Livak (2008). Acan Mm00545794_ml\"\n },\n {\n \"title\": \"OSTEOGENIC-ASSOCIATED PRIMERS\",\n \"content\": \"\"\n },\n {\n \"title\": \"BGlap Mm03413826_mH\",\n \"content\": \"Runx2 Mm00501578_ml\\nDmp1 Mm00803833_gl\\nSp7 Mm04209856_sl\\nIbsp Mm00492555_ml\"\n },\n {\n \"title\": \"OSTEOCLASTS-ASSOCIATED PRIMERS\",\n \"content\": \"\"\n },\n {\n \"title\": \"Acp5 Mm00475698_ml\",\n \"content\": \"Rankl Mm00441908_ml\\nCtsk Mm00484039_ml\"\n },\n {\n \"title\": \"OSTEOCYTES-ASSOCIATED PRIMERS\",\n \"content\": \"\"\n },\n {\n \"title\": \"Sost Mm00470479_ml\",\n \"content\": \"List of primers used during the in vivo investigation of genes crucial for the differentiation and activity of cells residing in the bone. The genes are divided over their specificity and their catalog number is reported beside.\"\n },\n {\n \"title\": \"Statistical Analysis\",\n \"content\": \"Statistical analyses were performed with Prism 7 (GraphPad Software Inc). Means are plotted and standard deviations (SD) of means, or standard error means (SEM) are represented by error bars. One-way ANOVA corrected by Tukey's honestly significant difference (THSD) test was used to analyze data, if not otherwise specified. A significance level of 0.05 was used for all statistical analyses, unless noted differently.\"\n },\n {\n \"title\": \"RESULTS\",\n \"content\": \"\"\n },\n {\n \"title\": \"Physical-Chemical Characterization of the Loaded, Untreated, and Treated Sponges\",\n \"content\": \"Untreated sponges (CTRL, HAn, and SrHAn) were prepared and physical-chemically characterized for degradation rate, ion release properties by ICP-OES, sponge-nanoparticles interactions by FT-IR and morphology by SEM. Furthermore, a stability assessment was performed on the treated (lyophilized) sponge samples (sCTRL, sHAn, sSrHAn). The absolute amounts of calcium and strontium loaded onto each sponge was then calculated. 60 mg of HAn contains 23.89 mg of calcium whereas 54 mg + 6 mg of SrHAn contains 21.5 mg of calcium and 3.55 mg of strontium. Calcium mass for HAn-BMP2 sponge was calculated as for the HAn sample and corresponds to 23.89 mg. ICP-OES data of the water solutions collected at different time points from untreated samples (Figure 2 and Table 2) was used to evaluate the progressive release of calcium and strontium ions in solution.\\nFor HAn-BMP2 samples, calcium ions and rhBMP2 release were measured via ICP-OES and indirect ELISA assay, respectively (Figures 2C,D and Table 2). An initial burst of Ca 2+ release was reported for samples HAn and SrHAn (Figures 2A,B), and this was followed by a significant decrement at day 3, after which it stabilized at about 10 mg/L- Table 2. A similar pattern is presented also for strontium ions (Figure 2B), although the concentration values was higher at day 3, ranging between 40 and 50 mg/L- Table 2. In Han-BMP2 samples, calcium ion release trend was flattened and strongly decreased ( Figure 2C) compared to the Han and SrHAn samples. This is probably due to the presence of rhBMP2 in the same solution. Although, the release of rhBMP2 showed a trend very similar to SrHAn samples, with a concentration spike in the first 2 days followed by a decrease at day 3 and constant release until day 28 ( Figure 2D).\\nThe ICP-OES analyses performed on samples treated for stability assessments showed similar ion release patterns after the lyophilization treatment ( Figure S1 and Table S1). In particular, for the sHAn sponges (Figure S1A), calcium ion release was higher at day 3 reaching 30 and 40 mg/L. Interestingly, sSrHAn sponges showed a less intense burst at day 3 and increasing values from day 7 to day 28 ( Figure 2D). Although increments in the days 7-28 were not significant ( Figure S2). The 95% confidence intervals of the ICP-OES average values are reported in (Figure S2).\\nTo evaluate sponge-nanoparticle interactions, FT-IR spectroscopy was employed (Figure 3). Unloaded and untreated sponge spectrum (Figures 3A,B) showed characteristic protein peaks (Morris and Finney, 2004;Barth, 2007): NH stretching band at 3271 cm −1 ; C=O stretching band (amide I) at 1,624 cm −1 ; N-H bending band (amide II) at 1,528 cm −1 ; C-N stretching (amide III) at 1,232 cm −1 . FT-IR spectra (Figures 3C-J), showed the comparative analysis between unloaded sponge blended along with nanoparticles (physical mixture = PM; Figures 3C,D,G,H) and untreated sponges loaded with nanoparticles ( Figures 3E,F,I,J). All FT-IR spectra of nanoparticles containing-specimens showed good agreement with the characteristic PO43-(1,023 cm −1 ) and OH-(3,570 cm −1 ) vibrations of hydroxyapatite lattice as reported before (Kim et al., 2002). Due to the low concentration of strontium in the samples, no significant variations were recorded between HAn_PM and SrHAn_PM, in accordance with previous results (Frasnelli et al., 2016). However, protein bands showed a much lower intensity in nanoparticles loaded sponges (Figures 3E,F,I,J) and completely disappeared in PM spectra (Figures 3C,D,G,H). While NH stretching band at 3,271 cm −1 was recorded only in the CTRL, C=O stretching and N-H bending slightly moved from initial values (1,624 and 1,528 cm −1 ) to 1,650-1,631 cm −1 , and 1,531-1,535 cm −1 , respectively. After lyophilization treatment (Figure S3), all protein bands completely disappeared (Figures S3C-F).\\nSEM images were collected to analyze the morphology of unloaded and loaded sponges (Figure 4). Unloaded sponges at T0 (Figure 4A) were characterized by a spongy structure, with interconnected pores and smooth wall surfaces. Loaded sponge surfaces were rougher and more irregular (Figures 4B-D). A reduction of wall surfaces occurred in the unloaded FIGURE 2 | Calcium, strontium and rhBMP2 release from loaded untreated sponges. ICP-OES data representation of calcium and strontium ions release from HAn, HAn-BMP2, and SrHAn loaded sponges. Indirect ELISA was performed to quantify solubilized rhBMP2. Data were collected during the 28 days in aqueous solution. Ca 2+ released from HAn sponges (A); Ca 2+ and Sr 2+ released from SrHAn sponges (B); Ca 2+ and rhBMP2 release from HAn-BMP2 sponges (C,D) N = 3 samples were measure per each time point. samples after 28 days (Figure 4E), but it was not observed in loaded samples (Figures 4F-H). A morphological change was appreciated comparing the T0 and T28 (Figures 4B,C,E-H).\\nAt T0 hydroxyapatite aggregates were detected on the sponge walls, while at T28 sponge walls looked smoother and the whole structure became more compact. Energy dispersive microanalysis conducted on HAn and SrHAn samples (Figures 4I-L) showed a homogenization of the phosphorus and calcium distribution on the sponge surfaces after 28 days in aqueous solution (Figure 4, P and Ca distribution maps). The same microanalysis performed on CTRL sample did not show an appreciable level of P, Ca and Sr (data not shown). Moreover, on HAn and SrHAn samples a decrement in the weight percentage of calcium and strontium was recorded at day 28 compared to day 0 (Figures 4M-P).\\nRegarding the treated samples (Figure S4), unloaded sponges at T0 (Figure S4A) showed a morphology very similar to untreated samples at T0 (Figure S4A). Even though, it is possible to appreciate the formation of minor ripples, probably due to the additional lyophilization phase. At T0, the treated and untreated sponges reported similar structures. At T28, treated and loaded sponges acquired a cement-like morphology (Figures S4F-H).\\nThe degradation rate was determined by measuring the sponge weight throughout 28 days in aqueous solution (Figure 5). A slow degradation pattern was observed with no significant variations recorded, except for the SrHAn loaded sponges (Figure 5C). SrHAn samples registered a significant difference in percentage weight loss between day 1 and day 28 (p < 0.0001).\\nA reduction in the quantities of calcium and strontium on the sponge surfaces was recorded at day 28, in accordance with ICP-OES data. Taken together these results indicated the SrHAn loaded sponges are a stable system for a constant and prolonged release of Ca 2+ and Sr 2+ . They are biodegradable and their morphology fits the golden standard parameters for osteoconductive scaffolds. Although lyophilization partially changed the morphology of the sponges at T28, there was no significant change in calcium and strontium release.\"\n },\n {\n \"title\": \"In vivo Sponge's Implantation and Characterization of Their Effects\",\n \"content\": \"Next, sponges prepared following the standard protocol (HAn, HAn-BMP2, and SrHAn) were utilized for in vivo implantation to determine the osteogenic ability of the nanoparticles. Radiography, macroscopic observations and histological staining of bone tissues and implanted sponges were carried out to analyze the sponge integration and surrounding tissue adaptation at postoperative day 16 and 33. Gene expression analysis from the implants and femurs were also performed at corresponding times to determine the molecular responses.\\nThe radiographic images support the macroscopic observation that strong integration underwent between the femur, sponge, and surrounding tissue (Figure 6). Within the implanted sponges there were few visible blood vessels, as shown in Figure 7. Implants and ectopic bones showed various degrees of calcification, as shown by the radiographic images. The sponges loaded with HAn-BMP2 and SrHAn appeared to show increased calcified material than in the sponges with HAn alone. The radiographic and macroscopic images also suggested that there was a reduction in implant dimensions at day 33 compared to day 16 (Figures 6A-C,J-L). Since the implants consisted of sponge and hydroxyapatite (black arrowheads in Figure 6), it was difficult to distinguish between the implanted material and the formation of ectopic bone. However, the histological analysis using Fast Green and Safranin-O stain demonstrated the formation of both cartilage and bone tissue at the sponge implant site (Figure 7).\\nMore extended Safranin-O positive regions (indicated with red arrowheads in Figure 7) were observed in the HAn and SrHAn samples at 16 days. Whereas, at 33 days, no Safranin-O positive tissue could be detected. This indicates that the ectopic bone formed through the endochondral ossification process, with cartilaginous tissue turning to calcified areas. The forming bone tissue at the implant sites also appears to support hematopoiesis, as shown by the recruitment of marrow cells between the femur cortical bone and the sponge (Figure 7E). The newly formed tissues (bone and bone marrow) were present in all implants but in the SrHAn loaded sponges were more represented. At 33 days, (Figures 7G-L) ectopic bone tissue is shown with respect FIGURE 4 | The microscopic structure (150× and 350×) of the untreated CTRL (unloaded sponges), HAn, HAn-BMP2 and SrHAn (loaded sponges) samples is shown in (A-H). Pictures represented samples at T0 (A-D) and after 28 days (E-H) of permanence in aqueous solutions, in a controlled and humidified atmosphere (37 • C, with 5% of CO 2 ). Untreated samples surfaces were also analyzed with SEM-EDS (Energy Dispersive X-Ray Spectroscopy) at 0 (I,J) and 28 days (K,L). Images at low magnifications of the sponges were taken with SEM (I-L). The elements phosphorus (P), calcium (Ca) and strontium (Sr) were mapped on the same pictures using different colors (P = green; Ca = red; Sr = white) and a black background. Quantification of the elements carbon (C), oxygen (O), phosphorus (P), calcium (Ca) and strontium (Sr) present of the samples surface was performed and data were reported as weight percentage in the bar charts (M,N,O,P) histograms. to control, accompanied by the presence of an area enriched in bone marrow cellular components.\\nTo investigate the changes in gene expression in both the femur and implants after 16 and 33 days, tissues were harvested separately from HAn, HAn-BMP2, and SrHAn implanted mice. Figure 8 represents chondrogenesis (Acan, Col10A1, and Sox9) and stem cell recruitment (Nanog) markers. While Figure 9 shows gene expression of markers for osteogenesis (Runx2, Sp7, Ibsp, BGlap, and Dmp1), osteocytes (Sost) and OCs activity (Acp5, Rankl, and Ctsk). Acan (Aggrecan core protein), Col10A1 (Type 10 collagen) and Sox9 expression was analyzed as chondrogenesis and endochondral ossification markers (Mackie et al., 2008;Dennis et al., 2015). There was a consistent difference between the expression of chondrogenic markers in bone and sponge samples as noted by the sectioned Y axis. Indeed, at 16 days, in the former these three genes were downregulated by both rhBMP2 and strontium, with respect to HAn condition ( Figure 8A). While in the latter, both osteoinductive factors brought to significantly higher expression of Col10A1 and Sox9 (Figure 8C), with an enhanced effect in SrHAn loaded sponges. At 33 days, the difference between the tissue samples is equalized and only significant up-regulation of Sox9 in the bone and Acan in the sponge were shown as strontium-driven effects (Figures 8E,G). Expression of the stem cells recruitment marker, Nanog, showed no significant difference at 16 days and only a mild up-regulation with respect to control at 33 days in the sponge sample. After 16 days, a significant up-regulation of BGlap (Osteocalcin), Ibsp (bone sialoprotein), and Sp7 (Osterix) was observed in the FIGURE 5 | Graphical representation of sponge weights variations. Untreated sample weights were measured with an analytical balance. Weights were recorded at different time points throughout the 28 days in aqueous solution. Day 0 is referred to the unloaded sponge, whereas Day 1 corresponds to the loaded sponges and it was considered the initial weight (Wi). N = 3 different sponges were weighted per each condition. From the recorded weights, we calculated the percentage weight loss. Error bars indicate the standard deviation. Statistically significant differences were assessed with one way ANOVA. Significant differences were reported only against day 1 (*p < 0, 05; **p < 0, 01; ****p < 0, 0001).\\nfemur of mice that received the SrHAn implant compared to HAn-BMP2 and to HAn samples alone ( Figure 9A). Similar results were observed with BGlap, Ibsp, and Sp7 expression being evaluated in the sponge samples at 16 days ( Figure 9D). In both the femur and the sponge tissues Dmp1 (Dentin matrix protein 1) was upregulated by strontium with respect to the control HAn. A similar trend of expression was also detected at 33 days, where Ibsp in the bone ( Figure 9G) and Ibsp, BGlap, and Sp7 in the sponge ( Figure 9J) were upregulated by strontium. Runx2 didn't show significant differences in expression at given time points. Sost (Sclerostin) expression was analyzed as marker for osteocyte differentiation. After 16 days, Sost was upregulated in the strontium samples only with respect to the control HAn, both in the bone and sponge samples (Figures 9B,E). At 33 days, an up-regulation of Sost was observed only in the bone samples loaded with strontium compared to HAn-BMP2 and HAn (Figures 9H,K). Acp5 (Tartrate-resistant acid phosphatase type 5), Rankl (Tumor necrosis factor ligand superfamily member 11) and Ctsk (Cathepsin K) expression were analyzed as markers for OCs differentiation and activity. At 16 days, Acp5 and Rankl (Figures 9C,F) were downregulated in SrHAn femurs and sponges Ctsk at 16 days was upregulated in femur and sponge by both types of osteoinductive factors (Figures 9C,F). At 33 days, strontium presence led to an upregulation of Acp5 and down-regulation of Rankl only in the bone samples ( Figure 9I). Ctsk was down-regulated in SrHAn implanted mice with respect to control, both in bone and sponge samples (Figures 9I,L).\\nSrHAn loaded-sponge structure were suitable for the promotion of cell invasion, bone marrow cell recruitment and ossification. Following a lyophilization treatment (which could be useful for sponge long-term storage and packaging), the loaded sponges did not show significant variation from the untreated samples and all physical-chemical properties were conserved. The biodegradability and biocompatibility of the system was proven in vivo, together with the induction of endochondral ossification stages in adult bone. Evidence of endochondral ossification were given in the histological investigation, where proliferating and hypertrophic chondrocytes were more represented in the strontium-containing samples than in the BMP2 ones. Gene expression corroborated histological evidences, showing increased expression of osteogenic and chondrogenesis markers. Moreover, these data were consistent with an enhancement of endochondral ossification in the SrHAn samples if compared to the HAn-BMP2 samples. Antiresorptive properties of strontium were also demonstrated by investigating markers of osteoclasts differentiation and activity. Histological and gene expression data further Statistical significance values were calculated with one way-ANOVA, followed by Tukey's honestly significant difference test. *, significant difference against the HAn condition (p < 0.05). #, significant difference between HAn-BMP2 and Sr conditions (p < 0.05). Statistical significance values were calculated with one way-ANOVA, followed by Tukey's honestly significant difference test. *, significant difference against the HAn condition (p < 0.05). #, significant difference between HAn-BMP2 and SrHAn conditions (p < 0.05).\\nsupported the osteoinductive and osteoconductive potential of this system.\"\n },\n {\n \"title\": \"DISCUSSION\",\n \"content\": \"Bone tissue engineering application in the orthopeadic surgical field is becoming increasingly relevant. In the last decades, rhBMP2 was extensively studied for its osteoinductive potential. Although several successful applications have been achieved, this recombinant morphogen was proven to be not always suitable and safe (Poon et al., 2016;Lykissas and Gkiatas, 2017). Another powerful osteoinductive agent, strontium, was widely studied in the previous years and compelling results were obtained in term of biocompatibility and osteoconductivity (Boanini et al., 2011;Yang et al., 2011). Strontium hydroxyapatite nanoparticles were recently developed, carrying an inorganic osteoinductive agent (Frasnelli et al., 2016). They have been characterized in the last years and in this work, they are presented as a substitute for rhBMP2.\\nIn this study, the medical grade collagen-based Gelfoam sponges were loaded with a combination of either calcium and strontium hydroxyapatite nanoparticles suspensions (SrHAn) or calcium hydroxyapatite nanoparticles and rhBMP2 (HAn-BMP2) in order to compare their physical-chemical properties. The HAn sponges were loaded only with calcium hydroxyapatite nanoparticles and used as control. A second set of samples were treated with an additional lyophilization step and characterized.\\nThe hydroxyapatite nanoparticles amount on each sponge was 30% (w/v) of the volume absorbed by the sponge and 3 µg of rhBMP2 per sponge were used for HAn-BMP2. For the SrHAn loaded-sponges, the calcium/strontium hydroxyapatite nanoparticles ratio was 9:1 w/w (Chandran et al., 2016) while the spontaneously occurring Sr:Ca ratio is between 1:1,000 and 1:2,000 in human bone tissue as well as in blood serum (Cabrera et al., 1999).\\nThe cumulative amount of calcium and strontium ions after 28 days in aqueous solutions were measured as 0.0118 and 0.0358 mg, respectively. Interestingly, several in vitro and in vivo studies reported that strontium elicits rhBMP2-like effects when administered at concentrations up to 1 mM, as reported by Schumacher et al. (2013) and others (Qiu et al., 2006;Li et al., 2010;Schumacher et al., 2013;Wei et al., 2019). Despite the lower water solubility of Sr 2+ compared to Ca 2+ , we detected enhanced strontium solubilization from the SrHAn sponges, compared to the calcium release from the HAn sponges. These data are in agreement with Kaufman and Kleinberg (1979) and Pan et al. (2009) studies. They demonstrated how, due to a larger atomic radius, strontium atoms slightly reduce hydroxyapatite lattice thermodynamic stability, favoring the release of Sr 2+ from the lattice itself. Furthermore, ICP-OES results were comparable to the ones presented by Landi et al. in term of ion release pattern (Landi et al., 2007) and were not significatively influenced by the lyophilization treatment. We investigated the spongenanoparticle system through FT-IR analysis of the untreated and treated systems (HAn, SrHAn, sHAn, and sSrHAn). Significant interactions were not found between the sponge and either HAn or SrHAn. FT-IR spectra of loaded sponges showed no chemical interactions were undergoing between the sponges and the nanoparticles. The minor shifting of the amide I and II bands was attributed to hydrogen bonds breaking in the sponge proteins and consequent modification of the hydrogen bonds network in the sponge structure. We can therefore assume that the ion release patterns studied with ICP-OES were not influenced by chemical interactions between the nanoparticles and the sponge support, either before or after the lyophilization treatment. To illustrate the patterns of ion release in function of time we considered the porous structure of the sponges. We could speculate a bulk of nanoparticles flowed inside the sponge pores, during the loading procedure. Nanoparticles remaining on the sponge surfaces released a high percentage of Ca 2+ and Sr 2+ ions during the first 3 days, while embedded nanoparticles were responsible for the constant release recorded after day 3. The same speculation can also fit the data on rhBMP2 release. This hypothesis was corroborated by SEM images, that showed smoother surfaces and solubilization of hydroxyapatite aggregates at day 28. Surprisingly, even after 28 days in aqueous solution, calcium and traces of strontium were still detected on the sponge surfaces, as demonstrated by SEM-EDS data.\\nInterestingly, the calcium release pattern of HAn-BMP2 loaded-sponges was different from the other samples. This can be due to the non-covalent interaction between HAn and rhBMP2 (Urist et al., 1984;Boix et al., 2005). We were not able to investigate HAn-BMP2 loaded-sponges with FT-IR, since this technique is not able to discern between different proteins. We can speculate that calcium ions remained bound to solubilized rhBMP2, therefore decreasing the amount of free Ca 2+ released in solution.\\nPore sizes ranging between 100 and 300 µm are considered the most suitable for osteoinduction improvement as well as for angiogenesis promotion (Karageorgiou and Kaplan, 2005;Scheinpflug et al., 2018). Furthermore, geometry of the investigated sponges reflected the \\\"gold standard\\\" in bone tissue engineering, according to Kuboki et al. (2007) and due to the given pores size and geometry the surface area exposed to the surrounding microenvironment is increased, allowing for a wider interaction between the nanoparticles coating and the surrounding microenvironment.\\nThe degradability and recyclability of the graft is a crucial point, although often disregarded, in tissue engineering applications (Gentleman et al., 2010;Rohnke et al., 2016). In this work, the time-dependent biodegradability of the sponges and the hydroxyapatite nanoparticles both in the abiotic and in vivo conditions, were analyzed. Degradation pattern based on the sponge weights appeared mild but significant for SrHAn sponges, throughout the 28 days in solution. While, in vivo radiographic images and histology suggested an accelerated degradation of SrHAn sponges. Radiographic images also revealed an induction of bone formation and osseointegration by all three types of implants at both time points. Although, a more extensive mineralized tissue in mice implanted with HAn-BMP2 and SrHAn sponges were shown with respect to the controls. Histological analysis revealed new tissue formation between the sponge and the femoral cortex, suggesting a center of active osteogenesis was present in the sponge site. Given the similarities between Ca 2+ and Sr 2+ , it has been shown that this element can substitute calcium in the lattice of the hydroxyapatite crystals present in bones (Ni et al., 2011;Querido et al., 2016). Sr 2+ is processed like Ca 2+ throughout metabolism, being preferentially introduced in active osteogenesis sites (Bauman et al., 2005).\\nRhBMP2 and Sr 2+ release was quantified using different techniques, therefore it is not possible to compare their quantities/concentrations in solution head-to-head. Despite the similarities in the pattern release between HAn-BMP2 and SrHAn, strontium containing nanoparticles induced significant increase of endochondral ossification markers. It is wellknown how the process of endochondral ossification can be recapitulated in adults by bone fracture and bone tissue damage (Gerstenfeld et al., 2003;Dennis et al., 2015), although in this work evidences of endochondral ossification were induced in healthy implanted mice, with no bone damage. Furthermore, no record of strontium related-endochondral ossification induction is present up to date, but here at 16 days the onset of typical multicellular clusters constitutes by enlarged chondrocyte was demonstrated (Mackie et al., 2008). Hypertrophic chondrocytes can be detected in the sponge surroundings at 16 days, together with smaller proliferating chondrocytes. These were particularly represented in the HAn and SrHAn samples and red stained-cartilage proteoglycans were present all around the cells. Hypertrophic chondrocytes were almost absent in the HAn-BMP2 samples, suggesting a possible delay or acceleration in chondrogenic differentiation. Different phases of endochondral ossification process were shown also by gene expression data: Sox9 upregulation was an indicator of chondrocytes proliferation (Akiyama et al., 2002;Lefebvre and Dvir-Ginzberg, 2017) and after its downregulation, markers of hypertrophic chondrocytes, such as aggrecan core protein (Acan) and type X collagen (Col10A1) were upregulated (Akiyama and Lefebvre, 2011). We can speculate that the different temporal upregulation of Sox9, among bone and sponges, can be attributed to the time dependent Sr 2+ diffusion, which resulted in its earlier expression in the sponge area and delayed expression in the bone one.\\nThe interaction of Sr 2+ with the calcium-sensing receptor (CaSR) was shown both in OBs and osteoclasts (OCs), resulting in improved anabolic processes and reduced catabolic pathways, respectively (Chattopadhyay et al., 2006). Our gene expression data showed SrHAn double activity on OBs and OCs. Several studies reported that strontium modulates alkaline phosphatase (ALP) activity and Runx-2 expression (Tian et al., 2014) in OBs and OBs progenitors while promoting Wnt/β-catenin pathway in human bone marrow derived mesenchymal stem cells (MSCs), thereby enhancing ECM secretion and osteogenic differentiation (Yang et al., 2011). Interestingly, in our study no significant variation was detected in Runx2 levels. However, our results measured an upregulation of Sp7 (Osterix), indicating OBs differentiation at early and late stage (Santo et al., 2013;Kawane et al., 2018). Sp7 is a transcriptional factor expressed in developing bones regulating the commitment of MSCs toward osteoblastic lineage (Rutkovskiy et al., 2016). Its regulation affects expression of protein-coding genes typically related to osteoblastic differentiation, such as osteocalcin, type I collagen and bone sialoprotein (BGlap, Ibsp, and Dmp1). Increased expression of the ECM proteins BGlap, Ibsp, and Dmp1 was attributed to mature osteoblasts activity (Wrobel et al., 2016) in SrHAn samples. Sp7, BGlap, and Ibsp expression also indicated a stronger osteogenic induction for strontium ions compared to rhBMP2. Interestingly Runx2, the master regulator of osteoblastic differentiation acting above Sp7, was not significantly influenced neither by strontium nor by rhBMP2 (Nakashima et al., 2002). Soluble cytokine promoting OCs differentiation, Rankl, (expressed by OBs) is a key modulator of the resorption cycle, controlling OCs differentiation and activation, and therefore bone resorption. This molecule has already been linked to strontium-mediated modulation by Atkins et al. (2009) andBrennan et al. (2009). Our results showed strontium significantly downregulated Rankl when compared to control and HAn-BMP2 samples. It has been shown that strontium effects are mediated by CaSR and by others, yet unclarified receptors (Chattopadhyay et al., 2006;Takaoka et al., 2010;Saidak and Marie, 2012). Furthermore, being CaSR present also on OCs outer membrane it was demonstrated the action of strontium on OCs activity. Indeed, OCs-secreted endoprotease responsible for ECM degradation (Sage et al., 2012), Ctsk (Cathepsin K) was reduced by strontium presence at 33 days, with respect to the control. Of note, in some of the SrHAn sponges we observed vascularization, suggesting a potential effect of Sr 2+ on neo-vascularization, although further studies are needed to confirm these findings.\"\n },\n {\n \"title\": \"CONCLUSIONS\",\n \"content\": \"Many studies have been conducted to investigate rhBMP2 suitability as treatment for spinal fractures and critically-sized bone defects. Although its indisputable osteoinductive potential, rhBMP2 also showed adverse effects. On the other hand, strontium has already been tested as a treatment for osteoporosis and its activity was demonstrated in a patient population. The biggest advantage of strontium over rhBMP2 however, is that the former is a chemical and not a biologic. Herein, following the abiotic characterization, we presented the results of a comparative in vivo study, where Gelfoam sponges loaded either with strontium hydroxyapatite nanoparticles or with rhBMP2 were implanted in healthy, bone damage-free mouse model. In vivo studies, showed that SrHAn and HAn-BMP2 have comparable effects, driving the onset of endochondral ossification and promoting the bone remodeling process. These results demonstrated that SrHAn loaded-sponges have marked osteogenic potential when applied on the periosteum of long bones, comparable to HAn-BMP2 loaded-sponges, but eliciting a more controlled ossification response. We propose to use Gelfoam sponges enriched in Sr 2+ as an effective therapeutic intervention to treat severe bone defects or open, complicated fractures. The short-term benefit of this nanotechnological system is the availability of novel therapeutic option to treat bone fracture (or bone defect). The use of sponges loaded with strontium hydroxyapatite nanoparticles might also provide better outcomes for complex fractures. Results from these studies can provide novel therapeutic options for active duty personnel and can be beneficial to anyone suffering from trauma, bone defects or severe bone injuries. Future studies shall also evaluate the efficacy of SrHAn loaded sponges for the treatment of spinal fusion.\"\n },\n {\n \"title\": \"DATA AVAILABILITY STATEMENT\",\n \"content\": \"All datasets generated for this study are included in the article/Supplementary Material.\"\n },\n {\n \"title\": \"ETHICS STATEMENT\",\n \"content\": \"All animal studies were approved by the Institutional Animal Care and Use Committee at Boston University (BU).\"\n },\n {\n \"title\": \"AUTHOR CONTRIBUTIONS\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIGURE 3 |\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIGURE 6 |\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIGURE 7 |\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIGURE 8 |\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIGURE 9 |\",\n \"content\": \"\"\n },\n {\n \"title\": \"FUNDING\",\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":"IR evaluation of sponge-nanoparticles interactions. FT-IR spectra of sponges and nanoparticles physical mixtures (Han_PM and SrHAn_PM) (C,D,G,H) were compared with spectra corresponding to the untreated and loaded sponges HAn and SrHAn at 28 days (E,F,I,J). In the panels a and b, FT-IR spectrum of untreated and unloaded sponge was reported (CTRL). Spectra were divided in two sections: wavenumber range 4,000-3,000 cm −1 was reported in (A,C,E,G,I); while wavenumber range 2,000-650 cm −1 was shown in (B,D,F,H,J).","content":"FIGURE 3 |\n3IR evaluation of sponge-nanoparticles interactions. FT-IR spectra of sponges and nanoparticles physical mixtures (Han_PM and SrHAn_PM) (C,D,G,H) were compared with spectra corresponding to the untreated and loaded sponges HAn and SrHAn at 28 days (E,F,I,J). In the panels a and b, FT-IR spectrum of untreated and unloaded sponge was reported (CTRL). Spectra were divided in two sections: wavenumber range 4,000-3,000 cm −1 was reported in (A,C,E,G,I); while wavenumber range 2,000-650 cm −1 was shown in (B,D,F,H,J)."},{"id":"","caption":"X-ray images and macroscopic observation of mice implanted with loaded sponges. Images at 16 and 33 days of mice implanted with HAn loaded-sponges (A,D,G,J,M,P, respectively), HAn-BMP2 loaded-sponges (B,E,H,K,N,Q) and SrHAn (C,F,I, L,O,R, respectively). The implanted sponges are shown by black arrowheads. Isolated femurs are shown in panels (D-F) and (M-O). Isolated legs including implants are shown in panels (G-I) and (P-R).","content":"FIGURE 6 |\n6X-ray images and macroscopic observation of mice implanted with loaded sponges. Images at 16 and 33 days of mice implanted with HAn loaded-sponges (A,D,G,J,M,P, respectively), HAn-BMP2 loaded-sponges (B,E,H,K,N,Q) and SrHAn (C,F,I, L,O,R, respectively). The implanted sponges are shown by black arrowheads. Isolated femurs are shown in panels (D-F) and (M-O). Isolated legs including implants are shown in panels (G-I) and (P-R)."},{"id":"","caption":"Representative histological images of post-implants tissues of mouse limb. Fast green/Safranin-O staining was used on post-implant tissue sections of limbs implanted with loaded sponges with HAn (A,D,G,J), HAn-BMP2 (B,E,H,K), or SrHAn (C,F,I,L), for 16 and 33 days, respectively. In the images, red spots of cartilaginous tissue are indicated with red arrowheads and gray blurs of ectopic bone are highlighted with *. Black arrows and black arrowheads indicate femur bone and sponges, respectively. The circle highlights a blood vessel.","content":"FIGURE 7 |\n7Representative histological images of post-implants tissues of mouse limb. Fast green/Safranin-O staining was used on post-implant tissue sections of limbs implanted with loaded sponges with HAn (A,D,G,J), HAn-BMP2 (B,E,H,K), or SrHAn (C,F,I,L), for 16 and 33 days, respectively. In the images, red spots of cartilaginous tissue are indicated with red arrowheads and gray blurs of ectopic bone are highlighted with *. Black arrows and black arrowheads indicate femur bone and sponges, respectively. The circle highlights a blood vessel."},{"id":"","caption":"Gene expression of stem cell recruitment and chondrogenesis markers in bone and sponge post-implants. Gene expression was analyzed in femur bone (A,B,E,F) and sponge post-implant (C,D,G,H) samples at 16 (A-D) and 33 days (E-H), respectively. Expression of chondrogenesis markers Acan, Col10A1 and Sox9 (A,C,E,G) as well as stem cell recruitment marker Nanog (B,D,F,H) has been evaluated. The graphs show the inverse of the Ct at the power of 2. Bars indicate mean values ± SEM of results from 4 experiments.","content":"FIGURE 8 |\n8Gene expression of stem cell recruitment and chondrogenesis markers in bone and sponge post-implants. Gene expression was analyzed in femur bone (A,B,E,F) and sponge post-implant (C,D,G,H) samples at 16 (A-D) and 33 days (E-H), respectively. Expression of chondrogenesis markers Acan, Col10A1 and Sox9 (A,C,E,G) as well as stem cell recruitment marker Nanog (B,D,F,H) has been evaluated. The graphs show the inverse of the Ct at the power of 2. Bars indicate mean values ± SEM of results from 4 experiments."},{"id":"","caption":"Gene expression of osteocytes, osteoclasts homeostasis, and osteogenic markers in bone and sponge post-implants. Gene expression was evaluated in femur bone (A,B,C,G,H,I) and sponge post-implant (D,E,F,J,K,L) samples at 16 (A-F) and 33 (G-L) days, respectively. Expression of osteogenic markers Dmp1, Bglap, Ibsp, Sp7 and Runx2 (A,D,G,J), osteocytes marker Sost (B,E,H,K) and osteoclasts relevant markers Acp5, Rankl and Ctsk (C,F,I,L) has been evaluated. The graphs show the inverse of the Ct at the power of 2. Bars indicate mean values ± SEM of results from four experiments.","content":"FIGURE 9 |\n9Gene expression of osteocytes, osteoclasts homeostasis, and osteogenic markers in bone and sponge post-implants. Gene expression was evaluated in femur bone (A,B,C,G,H,I) and sponge post-implant (D,E,F,J,K,L) samples at 16 (A-F) and 33 (G-L) days, respectively. Expression of osteogenic markers Dmp1, Bglap, Ibsp, Sp7 and Runx2 (A,D,G,J), osteocytes marker Sost (B,E,H,K) and osteoclasts relevant markers Acp5, Rankl and Ctsk (C,F,I,L) has been evaluated. The graphs show the inverse of the Ct at the power of 2. Bars indicate mean values ± SEM of results from four experiments."},{"id":"","caption":"This research was supported by funds from the National Institutes of Health NIAMS K99AR068582 (BB), NIAMS R01AR056637 (LG), and a grant from the Musculoskeletal Transplant Foundation (BB). The content of this study is solely the responsibility of the authors and do not necessarily represent the official views of the NIH. This work also was supported by grants from the Italian Space Agency, Project DC-MIC-2012-024, and contract N. 2013-060-I.O (to LV) and Projects of High Relevance, Bilateral project Italy-Sweden, Ministry of Foreign Affairs and International Cooperation (MAECI), and Ministry of Education, University and Research (MIUR) (2018-2020) titled \"Effect of Microgravity e nanoparticles on bone regeneration in simulated Microgravity (REPAIR)\" (to LV). Furthermore, a grant of the Italian Ministry of Education, University and Research (MIUR) to the Department of Molecular Medicine of the University of Pavia under the initiative \"Dipartimenti di Eccellenza (2018-2022)\" also contributed to this research.","content":"FUNDING\nThis research was supported by funds from the National Institutes of Health NIAMS K99AR068582 (BB), NIAMS R01AR056637 (LG), and a grant from the Musculoskeletal Transplant Foundation (BB). The content of this study is solely the responsibility of the authors and do not necessarily represent the official views of the NIH. This work also was supported by grants from the Italian Space Agency, Project DC-MIC-2012-024, and contract N. 2013-060-I.O (to LV) and Projects of High Relevance, Bilateral project Italy-Sweden, Ministry of Foreign Affairs and International Cooperation (MAECI), and Ministry of Education, University and Research (MIUR) (2018-2020) titled \"Effect of Microgravity e nanoparticles on bone regeneration in simulated Microgravity (REPAIR)\" (to LV). Furthermore, a grant of the Italian Ministry of Education, University and Research (MIUR) to the Department of Molecular Medicine of the University of Pavia under the initiative \"Dipartimenti di Eccellenza (2018-2022)\" also contributed to this research."},{"id":"","caption":"qRT-PCR gene primers.","content":"TABLE 1 |\n1qRT-PCR gene primers.Primer \nCatalog number \n\nNORMALIZATION PRIMER \n\n18S \nMm03928990_gl \n\nSTEM CELL RECRUITMENT PRIMER \n\nNanog \nMm02384862_gl \n\nCHONDROGENESIS PRIMERS \n\nSox9 \nMm00448840_ml \n\nCol10A1 \nMm00487041_ml"},{"id":"","caption":"Tabular representation of the data shown inFigure 2.Average concentration values and standard deviations of the calcium and strontium ions and of rhBMP2 measured in the different sponges.","content":"TABLE 2 |\n2Tabular representation of the data shown inFigure 2.Average concentration values and standard deviations of the calcium and strontium ions and of rhBMP2 measured in the different sponges.Time (days) \nHAn \nSrHAn \nHAn-BMP2 \n\nCa (mg/L) \nCa (mg/L) \nSr (mg/L) \nCa (mg/L) \nrhBMP2 (µg/mL) \n\n0 \n0 \n0 \n0 \n0 \n0 \n\n1 \n30.6 ± 3.5 \n29.5 ± 3.5 \n71.7 ± 3.8 \n3.8 ± 0.4 \n1.3 ± 0.133 \n\n2 \n16.9 ± 2.5 \n13.6 ± 2.4 \n51.1 ± 5.0 \n4.4 ± 0.5 \n1.29 ± 0.01 \n\n3 \n11.5 ± 1.7 \n7.5 ± 3.2 \n35.2 ± 4.7 \n3.3 ± 0.8 \n0.43 ± 0.06 \n\n7 \n13.0 ± 1.8 \n7.5 ± 2.0 \n46.9 ± 3.5 \n2.2 ± 0.4 \n0.38 ± 0.03 \n\n14 \n7.2 ± 0.6 \n6.3 ± 2.4 \n45.4 ± 2.9 \n2.2 ± 0.6 \n0.38 ± 0.03 \n\n21 \n5.2 ± 0.3 \n4.8 ± 4.0 \n42 ± 2.6 \n2.0 ± 0.6 \n0.44 ± 0.07 \n\n2 \n13.7 ± 2.8 \n10.5 ± 2.8 \n46.2 ± 3.3 \n1.7 ± 0.5 \n0.31 ± 0.06"},{"id":"","caption":"GM and FC: conceptualization, validation, formal analysis, investigation, writing-original draft and review and editing, and vaisualization. GB and LC: validation, formal analysis, investigation, resources, writing-review & editing, and visualization. LF and AK: conceptualization, validation, formal analysis, writing-review & editing. PD: conceptualization, formal analysis, resources, writing-original draft & review & editing, visualization, supervision, project administration, and funding acquisition. BB: validation, formal analysis, investigation, writing-original draft & review & editing, visualization, supervision, and project administration. LV and LG: conceptualization, resources, writing-original draft & review & editing, visualization, supervision, project administration, and funding acquisition.","content":"GM and FC: conceptualization, validation, formal analysis, investigation, writing-original draft and review and editing, and vaisualization. GB and LC: validation, formal analysis, investigation, resources, writing-review & editing, and visualization. LF and AK: conceptualization, validation, formal analysis, writing-review & editing. PD: conceptualization, formal analysis, resources, writing-original draft & review & editing, visualization, supervision, project administration, and funding acquisition. BB: validation, formal analysis, investigation, writing-original draft & review & editing, visualization, supervision, and project administration. LV and LG: conceptualization, resources, writing-original draft & review & editing, visualization, supervision, project administration, and funding acquisition."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"IR evaluation of sponge-nanoparticles interactions. FT-IR spectra of sponges and nanoparticles physical mixtures (Han_PM and SrHAn_PM) (C,D,G,H) were compared with spectra corresponding to the untreated and loaded sponges HAn and SrHAn at 28 days (E,F,I,J). In the panels a and b, FT-IR spectrum of untreated and unloaded sponge was reported (CTRL). Spectra were divided in two sections: wavenumber range 4,000-3,000 cm −1 was reported in (A,C,E,G,I); while wavenumber range 2,000-650 cm −1 was shown in (B,D,F,H,J).\",\n \"content\": \"FIGURE 3 |\\n3IR evaluation of sponge-nanoparticles interactions. FT-IR spectra of sponges and nanoparticles physical mixtures (Han_PM and SrHAn_PM) (C,D,G,H) were compared with spectra corresponding to the untreated and loaded sponges HAn and SrHAn at 28 days (E,F,I,J). In the panels a and b, FT-IR spectrum of untreated and unloaded sponge was reported (CTRL). Spectra were divided in two sections: wavenumber range 4,000-3,000 cm −1 was reported in (A,C,E,G,I); while wavenumber range 2,000-650 cm −1 was shown in (B,D,F,H,J).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"X-ray images and macroscopic observation of mice implanted with loaded sponges. Images at 16 and 33 days of mice implanted with HAn loaded-sponges (A,D,G,J,M,P, respectively), HAn-BMP2 loaded-sponges (B,E,H,K,N,Q) and SrHAn (C,F,I, L,O,R, respectively). The implanted sponges are shown by black arrowheads. Isolated femurs are shown in panels (D-F) and (M-O). Isolated legs including implants are shown in panels (G-I) and (P-R).\",\n \"content\": \"FIGURE 6 |\\n6X-ray images and macroscopic observation of mice implanted with loaded sponges. Images at 16 and 33 days of mice implanted with HAn loaded-sponges (A,D,G,J,M,P, respectively), HAn-BMP2 loaded-sponges (B,E,H,K,N,Q) and SrHAn (C,F,I, L,O,R, respectively). The implanted sponges are shown by black arrowheads. Isolated femurs are shown in panels (D-F) and (M-O). Isolated legs including implants are shown in panels (G-I) and (P-R).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Representative histological images of post-implants tissues of mouse limb. Fast green/Safranin-O staining was used on post-implant tissue sections of limbs implanted with loaded sponges with HAn (A,D,G,J), HAn-BMP2 (B,E,H,K), or SrHAn (C,F,I,L), for 16 and 33 days, respectively. In the images, red spots of cartilaginous tissue are indicated with red arrowheads and gray blurs of ectopic bone are highlighted with *. Black arrows and black arrowheads indicate femur bone and sponges, respectively. The circle highlights a blood vessel.\",\n \"content\": \"FIGURE 7 |\\n7Representative histological images of post-implants tissues of mouse limb. Fast green/Safranin-O staining was used on post-implant tissue sections of limbs implanted with loaded sponges with HAn (A,D,G,J), HAn-BMP2 (B,E,H,K), or SrHAn (C,F,I,L), for 16 and 33 days, respectively. In the images, red spots of cartilaginous tissue are indicated with red arrowheads and gray blurs of ectopic bone are highlighted with *. Black arrows and black arrowheads indicate femur bone and sponges, respectively. The circle highlights a blood vessel.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Gene expression of stem cell recruitment and chondrogenesis markers in bone and sponge post-implants. Gene expression was analyzed in femur bone (A,B,E,F) and sponge post-implant (C,D,G,H) samples at 16 (A-D) and 33 days (E-H), respectively. Expression of chondrogenesis markers Acan, Col10A1 and Sox9 (A,C,E,G) as well as stem cell recruitment marker Nanog (B,D,F,H) has been evaluated. The graphs show the inverse of the Ct at the power of 2. Bars indicate mean values ± SEM of results from 4 experiments.\",\n \"content\": \"FIGURE 8 |\\n8Gene expression of stem cell recruitment and chondrogenesis markers in bone and sponge post-implants. Gene expression was analyzed in femur bone (A,B,E,F) and sponge post-implant (C,D,G,H) samples at 16 (A-D) and 33 days (E-H), respectively. Expression of chondrogenesis markers Acan, Col10A1 and Sox9 (A,C,E,G) as well as stem cell recruitment marker Nanog (B,D,F,H) has been evaluated. The graphs show the inverse of the Ct at the power of 2. Bars indicate mean values ± SEM of results from 4 experiments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Gene expression of osteocytes, osteoclasts homeostasis, and osteogenic markers in bone and sponge post-implants. Gene expression was evaluated in femur bone (A,B,C,G,H,I) and sponge post-implant (D,E,F,J,K,L) samples at 16 (A-F) and 33 (G-L) days, respectively. Expression of osteogenic markers Dmp1, Bglap, Ibsp, Sp7 and Runx2 (A,D,G,J), osteocytes marker Sost (B,E,H,K) and osteoclasts relevant markers Acp5, Rankl and Ctsk (C,F,I,L) has been evaluated. The graphs show the inverse of the Ct at the power of 2. Bars indicate mean values ± SEM of results from four experiments.\",\n \"content\": \"FIGURE 9 |\\n9Gene expression of osteocytes, osteoclasts homeostasis, and osteogenic markers in bone and sponge post-implants. Gene expression was evaluated in femur bone (A,B,C,G,H,I) and sponge post-implant (D,E,F,J,K,L) samples at 16 (A-F) and 33 (G-L) days, respectively. Expression of osteogenic markers Dmp1, Bglap, Ibsp, Sp7 and Runx2 (A,D,G,J), osteocytes marker Sost (B,E,H,K) and osteoclasts relevant markers Acp5, Rankl and Ctsk (C,F,I,L) has been evaluated. The graphs show the inverse of the Ct at the power of 2. Bars indicate mean values ± SEM of results from four experiments.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"This research was supported by funds from the National Institutes of Health NIAMS K99AR068582 (BB), NIAMS R01AR056637 (LG), and a grant from the Musculoskeletal Transplant Foundation (BB). The content of this study is solely the responsibility of the authors and do not necessarily represent the official views of the NIH. This work also was supported by grants from the Italian Space Agency, Project DC-MIC-2012-024, and contract N. 2013-060-I.O (to LV) and Projects of High Relevance, Bilateral project Italy-Sweden, Ministry of Foreign Affairs and International Cooperation (MAECI), and Ministry of Education, University and Research (MIUR) (2018-2020) titled \\\"Effect of Microgravity e nanoparticles on bone regeneration in simulated Microgravity (REPAIR)\\\" (to LV). Furthermore, a grant of the Italian Ministry of Education, University and Research (MIUR) to the Department of Molecular Medicine of the University of Pavia under the initiative \\\"Dipartimenti di Eccellenza (2018-2022)\\\" also contributed to this research.\",\n \"content\": \"FUNDING\\nThis research was supported by funds from the National Institutes of Health NIAMS K99AR068582 (BB), NIAMS R01AR056637 (LG), and a grant from the Musculoskeletal Transplant Foundation (BB). The content of this study is solely the responsibility of the authors and do not necessarily represent the official views of the NIH. This work also was supported by grants from the Italian Space Agency, Project DC-MIC-2012-024, and contract N. 2013-060-I.O (to LV) and Projects of High Relevance, Bilateral project Italy-Sweden, Ministry of Foreign Affairs and International Cooperation (MAECI), and Ministry of Education, University and Research (MIUR) (2018-2020) titled \\\"Effect of Microgravity e nanoparticles on bone regeneration in simulated Microgravity (REPAIR)\\\" (to LV). Furthermore, a grant of the Italian Ministry of Education, University and Research (MIUR) to the Department of Molecular Medicine of the University of Pavia under the initiative \\\"Dipartimenti di Eccellenza (2018-2022)\\\" also contributed to this research.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"qRT-PCR gene primers.\",\n \"content\": \"TABLE 1 |\\n1qRT-PCR gene primers.Primer \\nCatalog number \\n\\nNORMALIZATION PRIMER \\n\\n18S \\nMm03928990_gl \\n\\nSTEM CELL RECRUITMENT PRIMER \\n\\nNanog \\nMm02384862_gl \\n\\nCHONDROGENESIS PRIMERS \\n\\nSox9 \\nMm00448840_ml \\n\\nCol10A1 \\nMm00487041_ml\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Tabular representation of the data shown inFigure 2.Average concentration values and standard deviations of the calcium and strontium ions and of rhBMP2 measured in the different sponges.\",\n \"content\": \"TABLE 2 |\\n2Tabular representation of the data shown inFigure 2.Average concentration values and standard deviations of the calcium and strontium ions and of rhBMP2 measured in the different sponges.Time (days) \\nHAn \\nSrHAn \\nHAn-BMP2 \\n\\nCa (mg/L) \\nCa (mg/L) \\nSr (mg/L) \\nCa (mg/L) \\nrhBMP2 (µg/mL) \\n\\n0 \\n0 \\n0 \\n0 \\n0 \\n0 \\n\\n1 \\n30.6 ± 3.5 \\n29.5 ± 3.5 \\n71.7 ± 3.8 \\n3.8 ± 0.4 \\n1.3 ± 0.133 \\n\\n2 \\n16.9 ± 2.5 \\n13.6 ± 2.4 \\n51.1 ± 5.0 \\n4.4 ± 0.5 \\n1.29 ± 0.01 \\n\\n3 \\n11.5 ± 1.7 \\n7.5 ± 3.2 \\n35.2 ± 4.7 \\n3.3 ± 0.8 \\n0.43 ± 0.06 \\n\\n7 \\n13.0 ± 1.8 \\n7.5 ± 2.0 \\n46.9 ± 3.5 \\n2.2 ± 0.4 \\n0.38 ± 0.03 \\n\\n14 \\n7.2 ± 0.6 \\n6.3 ± 2.4 \\n45.4 ± 2.9 \\n2.2 ± 0.6 \\n0.38 ± 0.03 \\n\\n21 \\n5.2 ± 0.3 \\n4.8 ± 4.0 \\n42 ± 2.6 \\n2.0 ± 0.6 \\n0.44 ± 0.07 \\n\\n2 \\n13.7 ± 2.8 \\n10.5 ± 2.8 \\n46.2 ± 3.3 \\n1.7 ± 0.5 \\n0.31 ± 0.06\"\n },\n {\n \"id\": \"\",\n \"caption\": \"GM and FC: conceptualization, validation, formal analysis, investigation, writing-original draft and review and editing, and vaisualization. GB and LC: validation, formal analysis, investigation, resources, writing-review & editing, and visualization. LF and AK: conceptualization, validation, formal analysis, writing-review & editing. PD: conceptualization, formal analysis, resources, writing-original draft & review & editing, visualization, supervision, project administration, and funding acquisition. BB: validation, formal analysis, investigation, writing-original draft & review & editing, visualization, supervision, and project administration. LV and LG: conceptualization, resources, writing-original draft & review & editing, visualization, supervision, project administration, and funding acquisition.\",\n \"content\": \"GM and FC: conceptualization, validation, formal analysis, investigation, writing-original draft and review and editing, and vaisualization. GB and LC: validation, formal analysis, investigation, resources, writing-review & editing, and visualization. LF and AK: conceptualization, validation, formal analysis, writing-review & editing. PD: conceptualization, formal analysis, resources, writing-original draft & review & editing, visualization, supervision, project administration, and funding acquisition. BB: validation, formal analysis, investigation, writing-original draft & review & editing, visualization, supervision, and project administration. LV and LG: conceptualization, resources, writing-original draft & review & editing, visualization, supervision, project administration, and funding acquisition.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"The transcription factor Sox9 has essential roles in successive steps of the chondrocyte differentiation pathway and is required for expression of Sox5 and Sox6. H Akiyama, M C Chaboissier, J F Martin, A Schedl, B De Crombrugghe, 10.1101/gad.1017802Genes Dev. 16Akiyama, H., Chaboissier, M. C., Martin, J. F., Schedl, A., and De Crombrugghe, B. 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Stem Cell. 29, 981-991. doi: 10.1002/stem.646\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":584,"cells":{"text":{"kind":"string","value":"\nTHE CONTINUOUS SPECTRUM IN DISCRETE SERIES BRANCHING LAWS\n7 Aug 2013\n\nHONGYU HE, AND GESTURÓLAFSSONBenjamin Harris \nTHE CONTINUOUS SPECTRUM IN DISCRETE SERIES BRANCHING LAWS\n7 Aug 2013\nIf G is a reductive Lie group of Harish-Chandra class, H is a symmetric subgroup, and π is a discrete series representation of G, the authors give a condition on the pair (G, H) which guarantees that the direct integral decomposition of π| H contains each irreducible representation of H with finite multiplicity. In addition, if G is a reductive Lie group of Harish-Chandra class, and H ⊂ G is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of π| H is constant along 'continuous parameters'. In obtaining these results, the authors develop a new technique for studying multiplicities in the restriction π| H via convolution with Harish-Chandra characters. This technique has the advantage of being useful for studying the continuous spectrum as well as the discrete spectrum.\n\nIntroduction\n\nIf π is an irreducible representation of a Lie group G and H ⊂ G is a type 1 closed Lie subgroup, it is natural to consider the restriction π| H and to decompose this restriction into irreducible representations of H. This type of problem comes up naturally in physics; see for instance [27], [33]. It also arises naturally in number theory; see for instance section 9 of [29], [2], or [21]. In the first case, the multiplicities in this decomposition should correspond to physical quantities and in the second case they should correspond to dimensions of spaces of automorphic forms. Either way, one observes that many branching laws of particular significance have the property that π| H has finite multiplicities.\n\nIn [18] and [19], Kobayashi gives a sufficient condition for an irreducible representation of a real reductive Lie group G of type A q (λ) to restrict discretely to a closed reductive subgroup of Harish-Chandra class, H. In [20], he gives a necessary condition for infinitesimal discrete decomposability, and he shows that the necessary and sufficient conditions agree when (G, H) is a symmetric pair. In addition, he shows that when (G, H) is a symmetric pair and the restriction is infinitesimally discretely decomposable, the branching law A q (λ)| H contains each irreducible H representation with finite multiplicity. More recently, it has been shown that if (G, H) is a strong Gelfand pair of real, reductive algebraic groups and π is an irreducible, unitary representation of G, then the discrete spectrum of π| H contains each irreducible representation of H with multiplicity at most one (see [35] and the references therein). Finally, in the recent preprint [22], Kobayashi and Oshima give a sufficient condition on pairs of reductive Lie groups (G, H) which guarantees that π| H contains finite multiplicities in the discrete spectrum of H for each irreducible, unitary representation π of G.\n\nThese are all very powerful finite multiplicity theorems for branching laws. However, in each case, the finite multiplicity theorem only applies to the discrete part of the direct integral decomposition π| H . In this paper, we attempt to study the continuous spectrum of branching laws in a very special case. In order to do this, we make use of many recent ideas of Ørsted and Vargas [30], [31], [38].\n\nIn this paper we will work with reductive Lie groups of Harish-Chandra class. For the sake of completeness, we give the definition of a reductive Lie group of Harish-Chandra class (taken from pages 105-106 of [12] where this class of groups was introduced). We say that G is a reductive Lie group of Harish-Chandra class if G satisfies the following four conditions:\n\n• Lie(G) = g is a reductive Lie algebra.\n\n• Ad(G) ⊂ Int g C where g C ∼ = g ⊗ C is the complexification of g and Int g C is the connected, analytic subgroup of Aut g C with Lie algebra Der g C , the set of derivations of g C . • If G 1 is the connected, analytic subgroup of G with Lie algebra g 1 = [g, g], then the center of G 1 is finite. • [G : G e ] < ∞ where G e denotes the identity component of G.\n\nHere are our results. Theorem 1.1. Suppose G is a reductive Lie group of Harish-Chandra class and suppose H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of G. Choose a maximal compact subgroup K ⊂ G such that H ∩ K ⊂ H is maximal compact, and denote by θ the associated Cartan involutions of both G and g. Let s be the −1 eigenspace of θ on g, and let q be the −1 eigenspace of τ on g. Suppose H ∼ = K 0 ×H 1 where K 0 is compact and H 1 is noncompact. Let b ⊂ h 1 ∩ s be a maximal abelian subspace, and let a ⊂ s ∩ q be a maximal abelian subspace. Assume Z H1∩K (a)Z H1∩K (b) = H 1 ∩ K.\n\nIf π is a discrete series representation of G, then the direct integral decomposition of the restriction π| H has finite multiplicities.\n\nObserve that, under slightly weaker hypotheses, it is proved in Theorem 4 of [30] that the discrete spectrum of π| H has finite multiplicities. In the last sentence of page 628 in [30], the authors remark that they have not yet determined whether the irreducible continuous factors occur with finite multiplicity in the restriction of a discrete series representation of Spin(2n, 1) to the subgroup Spin(2k) × Spin(2n − 2k, 1). This generalization of their result answers their question in the affirmative.\n\nOne checks that the symmetric pairs (O(2n, 1), O(k) × O(2n − k, 1)), (U (n, 1), U (k) × U (n − k, 1)), (Sp(n, 1), Sp(k) × Sp(n − k, 1)) are all examples of symmetric pairs satisfying the hypotheses of the Theorem. We note that in [22], a class of pairs (G, H) was identified such that π| H has finite multiplicities in the discrete spectrum for every irreducible unitary representation π of G. The above examples are also examples of the class of pairs (G, H) identified in [22]. However, the class in [22] contains additional pairs including (SO(p, q), SO(p − 1, q)) for p, q > 1 that are not covered by the above Theorem. Most likely these pairs also satisfy a finite multiplicity theorem for the continuous spectrum in discrete series branching laws.\n\nThe proof of the Theorem utilizes many techniques from [30]. One new technique involves convolution of certain Harish-Chandra Schwartz functions on G with Harish-Chandra characters of irreducible, tempered representations of H. This technique is also helpful in proving the following result. Theorem 1.2. Suppose G is a reductive Lie group of Harish-Chandra class, and H ⊂ G is a closed reductive subgroup of Harish-Chandra class. Let P = M AN ⊂ H be a cuspidal parabolic subgroup, let δ be a discrete series representation of M , and let ( A) ′ be the set of H regular characters of A. Let σ(δ, ν) be the family of irreducible tempered representations corresponding to δ ∈ M and ν ∈ ( A) ′ (see Section 3 for notation). Let π be a discrete series representation of G, and let m(σ(δ, ν), π| H ) be the multiplicity of the irreducible tempered representation of σ(δ, ν) in π| H . Then m(σ(δ, ν), π| H ) is constant as a function of ν ∈ ( A) ′ .\n\nIn [38], Vargas considers m(σ(δ, ν), π| H ) as a measurable function of ν for fixed δ. In Proposition 2, he shows that whenever m(σ(δ, ν), π| H ) is not identically zero almost everywhere as a function of ν, it is zero on a set of measure zero. Thus, our Theorem is a slight generalization of the one obtained in [38].\n\nFinally, we attempt to use Kahler geometry to obtain more information about the multiplicities in discrete series branching laws in the special case (U (n, 1), U (1) × U (n − 1, 1)). Let ψ m (e iθ ) = e imθ denote the unitary character of U (1) with parameter m ∈ Z. Proposition 1.3. Suppose π is a discrete series representation of G = U (n, 1), and consider the symmetric subgroup H = U (1) × U (n − 1, 1). Decompose\nπ| U(1) ∼ = ψm∈ U(1) ψ m ⊗ Hom U(1) (ψ m , π) = ψm∈ U(1) ψ m ⊗ τ m\ninto irreducibles under the action of U (1). Here τ m is a unitary representation of U (n − 1, 1). Next, write\nτ m = σ(δ,ν)∈ U(n−1,1) temp. m(σ(δ, ν), τ m )σ(δ, ν)\nas a direct integral of irreducible representations of U (n − 1, 1) (See Section 3 for notation). Then m(σ(δ, ν), τ m ) = 0 for finitely many elliptic parameters δ and m(σ(δ, ν), τ m ) < ∞ for all σ, ν. In particular, at most finitely many discrete series occur in τ m .\n\nOne could ask if a similar Proposition holds for all of the symmetric pairs (O(2n, 1), O(k) × O(2n − k, 1)), (U (n, 1), U (k) × U (n − k, 1)), and (Sp(n, 1), Sp(k) × Sp(n − k, 1)). While this may be the case, attempting to prove this generalization using techniques similar to those in the last section of this paper would be computationally unpleasant at best.\n\nWe end the introduction with a remark. It seems likely that for most symmetric pairs (G, H), there will exist discrete series π such that π| H has only finite multiplicities as well as discrete series π ′ such that π ′ | H has some infinite multiplicities. For example, if G has holomorphic discrete series, then the holomorphic discrete series of G always restrict with finite multiplicities to any symmetric subgroup ( [17], [20], [28]). On the other hand, in many cases one can check using the Mackey restriction principle that large unitary principal series of G will restrict with infinite multiplicities to H, and one expects large discrete series to behave the same way. It would be nice to have a criterion to determine when each of these cases occurs that is analogous to the known criteria for discrete decomposability in [20], [3]. Certainly, the authors of this paper are very far from understanding this problem.\n\n\nA Framework for Studying Multiplicities\n\nSuppose G is a reductive Lie group of Harish-Chandra class and H ⊂ G is a closed reductive subgroup of Harish-Chandra class. Let π be a discrete series representation of G, and let σ be a tempered representation of H. Let χ π be the infinitesimal character of π, and let ψ σ be the infinitesimal character of σ. Let K ⊂ G be a maximal compact subgroup such that H ∩ K ⊂ H is a maximal compact subgroup of H. Let θ be the Cartan involution of g = Lie(G) with fixed points k = Lie(K), let s be the −1 eigenspace of θ, let h = Lie(H) be the Lie algebra of H, and let q be a θ stable and H ∩ K stable complement to h in g. In particular, we have the decomposition\ng = k ∩ h ⊕ k ∩ q ⊕ s ∩ h ⊕ s ∩ q.\nLet (ξ, V ξ ) be the lowest K type of π, let V ξ be the corresponding G equivariant, Hermitian vector bundle on K\\G, and let\nL 2 (K\\G, V ξ ) χπ\nbe the χ π -eigenspace of the action of ZU(g) on space of square integrable sections of V ξ over K\\G. Note that the Casimir operator Ω ∈ ZU(g) acts as an elliptic operator on K\\G; hence, this is a space of analytic functions by the elliptic regularity theorem. The group G acts on L 2 (K\\G, V ξ ) χπ by right translation, and this representation of G is isomorphic to π ( [16], [40]). Let S(K\\G, V ξ ) χπ be the dense subspace of those sections that pullback to Harish-Chandra Schwartz functions on G with values in V ξ . The Harish-Chandra Schwartz space was first introduced on page 19 of [10].\n\nWhenever D ∈ U(g), left translation by D will be denoted by L D , and right translation by D will be denoted by R D . For instance, if X ∈ g, then\n(L X f )(g) = d dt t=0 f (exp(−tX)g), (R X f )(g) = d dt t=0 f (g exp(tX)).\nRecall that if G is a Lie group, then there is a natural bijective linear map exp * : S(g) → U(g).\n\nIf we regard S(g) as the space of translation invariant differential operators on g, and we regard U(g) as the space of left invariant differential operators on G, then this map is given by\nR exp * (u) (f )(g) := (u exp * (l g −1 f ))(0).\nIf we instead regard U(g) as the space of right invariant differential operators on G, then the map can be written L exp * (u) (f )(g) := (u exp * (ι * r g f ))(0).\n\nOf course, l g f (x) = f (g −1 x) (respectively (r g f )(x) = r(xg)) denotes left translation (respectively right translation) by g, (exp * f )(X) = f (exp(X)) denotes the pullback of f by exp, and (ι * f )(x) = f (x −1 ) denotes the pullback of f by the inversion map ι(x) = x −1 . For more details on this map, see pages 281-283 of [15]. If V is a real vector space, let S m (V ) denote the elements of the complex symmetric algebra over V with degree at most m. The following Proposition is a consequence of Theorem 1.3 in [31], and it is a slight modification of Theorem 3 in [30]. \nr m : L 2 (K\\G, V ξ ) χπ −→ L 2 ((H ∩ K)\\H, Hom C (S m (s ∩ q), V ξ | (H∩K)\\H )).\nThe map r m is defined by\nr m (f )(h)(u) = (L exp * (u) f )(h). Moreover, we have ker(r m ) ⊃ ker(r m+1 )\nfor all m and ∞ m=0 ker(r m ) = {0}.\n\nObserve that our notation differs slightly from the notation in [30]. Let r m = U m (r * m r m ) 1/2 denote the polar decomposition of the operator r m . Then {U m } is a family of H equivariant partial isometries with ker r m = ker U m and Image(U m ) = Image(r m ) for every m. In particular, {ker(U m ) ⊥ } is a filtration of π| H via closed, H invariant subspaces. We require the following complementary Proposition, which is a close relative of Lemma 1 of [37]. To prove the proposition, we will need a Lemma. Since H ∩ K ⊂ H is maximally compact, we have compatible Cartan decompositions g = k ⊕ s and h = (h ∩ k) ⊕ (h ∩ s). Let a H ⊂ h ∩ s be a maximal abelian subspace, and let a ⊂ s be a maximal abelian subspace satisfying a ∩ (h ∩ s) = a H . Let A = exp(a) and A H = exp(a H ). Let ∆(g, a) denote the roots of g with respect to a, and choose positive roots ∆ + (g, a) ⊂ ∆(g, a). Let a + be the positive Weyl chamber for this choice, let A + = exp(a + ), and let\nρ a + = 1 2 α∈∆ + (g,a)\nα.\n\nLet q : a * → a * H be the pullback of the inclusion, and observe q(∆(g, a)) ⊃ ∆(h, a H ).\n\nNote that we can choose positive roots ∆ + (h, a) ⊂ ∆(h, a) so that each positive root α of h with respect to a H is of the form q(β) for a positive root β ∈ ∆ + (g, a).\n\nFix such a choice and let a + H be the positive Weyl chamber for these positive roots. Let A + H = exp(a + H ), and let\nρ a + H = 1 2 α∈∆ + (h,aH ) α.\nLet W G be the Weyl group of the root system ∆(g, a), and let W H be the Weyl group of the root system ∆(h, a H ). a)). Writing things out, we obtain\nLemma 2.3. If a ∈ A + H and w ∈ W G such that wa ∈ A + , then ρ a + (w log(a)) − ρ a + H (log(a)) ≥ 0. Proof. Note ∆ + (h, a H ) ⊂ q(w −1 ∆ + (g,ρ a + (w log(a)) − ρ a + H (log(a)) = α∈∆ + (g,h) (w −1 α)(log(a)) − α∈∆ + (h,aH ) α(log(a)) = α∈w −1 ∆ + (g,a)−∆ + (h,aH ) α(log(a)).\nThe final sum is nonnegative since wa ∈ A + implies α(log(a)) ≥ 0 for all α ∈ w −1 ∆ + (g, a).\n\nNext, we prove the proposition.\n\nProof. To prove this Proposition, it is enough to check that the restriction to H of a Harish-Chandra Schwartz function on G is a Harish-Chandra Schwartz function on H. Let Ξ G be the Harish-Chandra spherical function on G, and let Ξ H be the Harish-Chandra spherical function on H (see for instance page 186 of [23] for a definition; in his notation Ξ G = ϕ G 0 ). Recall that f ∈ C ∞ (G) lies in the Harish-Chandra Schwartz space, S(G), of G iff for every D 1 , D 2 ∈ U(g) and every m ∈ N, there exists a constant C D1,D2,m > 0 such that\n|(1 + |g|) m (R D1 L D2 f )(g)| ≤ C D1,D2,m |Ξ G (g)| for all g ∈ G.\nTo define | · |, we choose a W G invariant norm | · | a on a, and we define\n|g| = |kak| = | log(a)| a .\nOf course, the definition of the Harish-Chandra Schwartz space is independent of the choice of norm on a. Using well-known upper and lower bounds on Ξ G (see Theorem 3 on page 279 of [7] and (8.83) on page 259 of [23]), we observe that f is in the Harish-Chandra Schwartz space of G iff for every D 1 , D 2 ∈ U(g) and m ∈ N, there exists a constant C D1,D2,m such that\n|(R D1 L D2 f )(k 1 ak 2 )| ≤ C D1,D2,m (1 + | log(a)| a ) −m e −ρ a + (log(a))\nfor all k 1 , k 2 ∈ K and a ∈ A + . Now, fix a ∈ A + H , and choose w ∈ W G such that wa ∈ A + . Choose k ′ ∈ K that acts on A by w, and define | · | aH to be the restriction of | · | a to a H . If D 1 , D 2 ∈ U(h) ⊂ U(g), then we obtain\n|(R D1 L D2 f )(k 1 ak 2 )| = (R D1 L D2 f )(k 1 k ′ (k ′−1 ak ′ )k ′−1 k 2 ) ≤ C D1,D2,m (1 + | log(wa)| a ) −m e −ρ a + (log(wa)) = C D1,D2,m (1 + | log(a)| aH ) −m e −ρ a + (log(wa)) for k 1 , k 2 ∈ K ∩ H. Applying the Lemma, we obtain |(R D1 L D2 f )(k 1 ak 2 )| ≤ C D1,D2,m (1 + | log(a)| aH ) −m e −ρ a + H (log(a)) whenever D 1 , D 2 ∈ ZU(h), m ∈ N, a ∈ A + H , k 1 , k 2 ∈ K ∩ H, and f ∈ S(G)\n. This implies that the restriction of f to H is in S(H).\n\nAs a corollary of the proof, we observe that the restriction of the map r m to Harish-Chandra Schwartz spaces is continuous with respect to the locally convex topologies of the Harish-Chandra Schwartz spaces.\n\nNext, we wish to understand the part of the image of the map r m that transforms by an irreducible tempered representation σ of H. This will help us understand the multiplicity of the representation σ in π| H . We need to introduce an additional map.\nSuppose (η, V ) is a finite dimensional, unitary representation of H ∩ K, and let V → (H ∩ K)\\H be the corresponding H equivariant, Hermitian vector bundle. Let L 2 ((H ∩ K)\\H, V) denote the space of L 2 sections of V over (H ∩ K)\\H with respect to the H invariant measure on (H ∩ K)\\H, and let S((H ∩ K)\\H, V) ⊂ L 2 ((H ∩ K)\\H, V) be the dense subspace of functions that pullback to V valued Harish-Chandra Schwartz functions on H. Let Θ σ ∈ C −∞ (H) be the character of σ on H.\nRecall that in [6], [8] it is proved that Θ σ is given by integration against an analytic function on the dense subset of regular, semisimple elements H ′ ⊂ H that is locally L 1 on all of H. We will write Θ σ for the distribution as well as the analytic function on H ′ .\n\nDefine C ω ((H ∩ K)\\H, V) ψσ to be the space of analytic sections of V → (H ∩ K)\\H satisfying\nR D f = ψ σ (D)f for all D ∈ ZU(h). Further, if µ ∈ H ∩ K is an H ∩ K type, let C ω ((H ∩ K)\\H, V) ψσ (µ) ⊂ C ω ((H ∩ K)\\H, V) ψσ\nbe the subspace that transforms by µ under right translation by H ∩ K.\n\nWe define a map * Θ σ :\nS((H ∩ K)\\H, V) → C ω ((H ∩ K)\\H, V) ψσ by f → f * Θ σ where (f * Θ σ )(h) = H f (hh −1 1 )Θ σ (h 1 )dh 1 .\nThe integral converges because integration against Θ σ defines a tempered distribution (see the remarks on page 45 of [10] together with Lemma 27 of [9] or page 456 of [23]). Note that we may also write\n(f * Θ σ )(h) = H f (h 1 )Θ σ (hh −1 1 )dh 1 = H f (h −1 1 h)Θ σ (h 1 )dh 1 .\nHere we have used substitution and the fact that Θ σ is conjugation invariant. From these formulas, it is clear that * Θ σ is an H × H equivariant map. In particular,\n(f * Θ σ )(kh) = η(k) −1 (f * Θ σ )(h) for every k ∈ H ∩K. Moreover, if D ∈ U(h) H , write D = exp * (D) with D ∈ S(h) H , and observe R exp * D (f * Θ σ )(h) = D X H f (h exp(X)h −1 1 )Θ σ (h 1 )dh 1 = D X H f (hh −1 1 exp(Ad h1 X))Θ σ (h 1 )dh 1 = (Ad h1 D) X H f (hh −1 1 exp(X))Θ σ (h 1 )dh 1 = D X H f (hh −1 1 )Θ σ (h 1 exp(X))dh 1 = R exp * D Θ σ , ι * l h −1 f = ψ σ ( D)(f * Θ σ )(h).\nHere we have used that integration against Θ σ is an invariant eigendistribution with infinitesimal character ψ σ . Finally, the function f * Θ σ is analytic because it is an eigenfunction for the Casimir operator of H, which is elliptic on the Riemannian symmetric space (H ∩ K)\\H.\n\nLet σ ∈ H temp. , and let µ ∈ H ∩ K be a minimal K type for σ. Let φ π,σ,m be the composition of the map r m and the map * Θ σ from\nS((H ∩ K)\\H, Hom C (S m (s ∩ q), V ξ | (H∩K)\\H )) to C ω ((H ∩ K)\\H, S m (s ∩ q) * ⊗ V ξ ) ψσ . Let m(σ, π| H ) denote the multiplicity of σ in the direct integral decomposition of π| H . Lemma 2.4. We have the equality m(σ, π| H ) = lim m→∞ dim φ π,σ,m (S(K\\G, V ξ ) χπ (µ)) = lim m→∞ dim φ π,σ,m (S(K\\G, V ξ ) χπ )(µ).\nBefore beginning the proof of the Lemma, we recall the Plancherel formula for H ( [34], [25], [13] for the original papers, [11], [5], [41] for expository pieces). First, we have an abstract direct integral decomposition\nL 2 (H) ∼ = Htemp. V * σ ⊗ V σ .\nRecall that we may view V * σ ⊗ V σ ∼ = End(V σ ) HS as the space of Hilbert Schmidt operators on V σ . Let the isomorphism between L 2 (H) and the direct integral be\ndenoted f → f . Then for f ∈ (L 1 ∩ L 2 )(H), we have f (σ) = σ(f ) ∈ End(V σ ) HS .\nIn fact, when f is in the Harish-Chandra Schwartz space, we can make things even more explicit. We have a unique H × H equivariant, continuous map\nM σ : (V * σ ⊗ V σ ) ∞ → C ∞ (H) satisfying M σ (w ⊗ v)(h) = σ(h)v, w for v ∈ V ∞ σ , w ∈ (V * σ ) ∞ . Let C ∞ (H) σ * ⊗σ denote the image of this map. If Θ σ is the Harish-Chandra char- acter of σ, then we have an H × H equivariant map S(H) → C ∞ (H) σ * ⊗σ given by f → f * Θ σ .\nMoreover, for f ∈ S(H), we have\nM σ (σ(f )) = f * Θ σ . Now, let V be a finite dimensional complex vector space, and let W ⊂ L 2 (H)⊗V be a closed, right H invariant subspace such that S(W ) := W ∩ (S(H) ⊗ V ) ⊂ W is dense. Then for each σ ∈ H temp. , we must have W σ ⊂ V * σ ⊗ V such that W ∼ = Htemp. W σ ⊗ V σ .\nThe point is that we can use our explicit Fourier transform to describe W σ . Extend M σ to a map on S(H) ⊗ V (trivially on V ), and note that\n{f * Θ σ | f ∈ S(W )} ⊂ M σ (V σ ⊗ W σ ) ∞\nis a dense subspace. Since the maps * Θ σ and M σ are H × H equivariant, we may look at the part of both sides that transforms on the right by the minimal K type µ of σ. Then we obtain\n{f * Θ σ | f ∈ S(W )(µ)} ⊂ M σ (V σ (µ) ⊗ W σ ) ∞ is dense. Since µ is a minimal H ∩K type of σ, it follows from [39] that dim V σ (µ) = 1. In particular, dim W σ = dim{f * Θ σ | f ∈ S(W )(µ)}.\nFinally, note that dim W σ = m(σ, W ), the multiplicity of σ in W .\n\nProof. Now, to compute the multiplicity of σ in π| H , we first realize π as\nL 2 (K\\G, V ξ ) χπ . Recall {ker(U m ) ⊥ } is a filtration of π| H by H invariant, closed subspaces. Since each {ker(U m ) ⊥ } can be identified with a closed subpace of L 2 ((H ∩ K)\\H, Hom C (S m (s ∩ q), V ξ | (H∩K)\\H )) ⊂ L 2 (H) ⊗ Hom C (S m (s ∩ q), V ξ ),\nwe may take the measure on H in the direct integral decomposition of ker(U m ) ⊥ to be the Plancherel measure on H for every m. In particular, this implies\nm(σ, π| H ) = lim m→∞ m(σ, ker(U m ) ⊥ ).\nWe may identify ker(U m ) ⊥ as an H representation with\nr m (L 2 (K\\G, V ξ ) χπ ) ⊂ L 2 ((H ∩ K)\\H, S m (s ∩ q) * ⊗ V ξ ).\nSince Harish-Chandra Schwartz functions on G are dense in L 2 (K\\G, V ξ ) χπ and the restriction of Harish-Chandra Schwartz functions to H are Harish-Chandra Schwartz functions on H by Proposition 2.2, we may use the above discussion of the Plancherel formula to compute the multiplicity of σ in ker(U m ) ⊥ . Explicitly, we have\nm(σ, ker(U m ) ⊥ ) = dim φ π,σ,m (S(K\\G, V ξ ) χπ (µ)).\nThe lemma follows.\n\n\nA Constant Multiplicity Theorem\n\nGiven a cuspidal parabolic subgroup P = M AN ⊂ H, a discrete series representation δ ∈ M and a unitary character ν ∈ A, we may form the (possibly infinite dimensional) vector bundle on G/P corresponding to the tensor product of δ ⊗ ν ⊗ 1 with the square root of the density bundle on G/P . The space of L 2 sections of this vector bundle is a tempered representation of H, which we will call σ(δ, ν); if ν is regular it is irreducible [13]. Moreover, every irreducible tempered representation σ ∈ H with regular infinitesimal character is of this form with ν ∈ A regular [36].\n\nLet ( A) ′ denote the set of regular characters of A, and consider {σ(δ, ν)| ν ∈ ( A) ′ } for fixed δ ∈ M . This set of irreducible tempered representations is an open subset of H temp. and the union of all such open subsets has full measure in H temp. [4], [11], [13], [23]. In particular, we may study the multiplicity function m(σ(δ, ν), π) as a function of ν ∈ ( A) ′ for fixed π ∈ G disc. and δ ∈ M disc. . Theorem 3.1. Let π ∈ G disc. be a discrete series representation of G, a reductive Lie group of Harish-Chandra class, let H ⊂ G be a closed reductive subgroup of G of Harish-Chandra class, and let P = M AN ⊂ H be a cuspidal parabolic subgroup of H. Let δ ∈ M disc. be a discrete series representation of M , and let ν ∈ ( A) ′ be a regular, unitary character of A. Let σ(δ, ν) be the irreducible tempered represen-\ntation of H induced from the representation δ ⊗ ν ⊗ 1 of P = M AN . Let m(σ(δ, ν), π| H )\nbe the multiplicity function that records the multiplicity of σ(δ, ν) in the direct integral decomposition of π| H . Then, as a function of ν ∈ ( A) ′ , the multiplicity function m(σ(δ, ν), π| H ) is constant.\n\nIn [38], Vargas considers m(σ(δ, ν), π| H ) as a measurable function of ν for fixed δ. In Proposition 2, he shows that whenever m(σ(δ, ν), π| H ) is not identically zero almost everywhere as a function of ν, it is zero on a set of measure zero. In particular, this Theorem is a generalization of Proposition 2 of [38].\n\nProof. By Lemma 2.4, we have the formula m(σ(δ, ν), π| H ) = lim m→∞ dim φ π,σ,m (S(K\\G, V ξ ) χπ (µ)).\n\nFix δ and π, and first assume that m(σ(δ, ν), π| H ) is bounded almost everywhere\nfor ν ∈ ( A) ′ with maximum N . Choose ν 0 ∈ ( A) ′ such that N = m(σ(δ, ν 0 ), π| H ), and choose f 1 , . . . , f N ∈ S(K\\G, V ξ ) χπ (µ) such that φ π,σ(δ,ν0),m (f j ) = r m (f j ) * Θ σ(δ,ν0)\nis linearly independent for sufficiently large m. Let m 0 be a natural number that is sufficiently large. Recall\nr m (f j ) * Θ σ(δ,ν0) ∈ C ω ((H ∩ K)\\H, S m (s ∩ q) * ⊗ V ξ ) ψσ (µ).\nIn particular, one may think of each r m (f j ) * Θ σ(δ,ν0) as a complex valued function on H × S m (s ∩ q) ⊗ V * ξ . We require the following Lemma. Lemma 3.2. Let X be a set, and let {F 1 , . . . , F N } be a linearly independent set of complex valued functions on X. Then there exists a subset {x 1 , . . . , x N } ⊂ X such that det(F i (x j )) = 0.\n\nThe proof of this Lemma is an easy linear algebra exercise that is left to the reader. In our case, we may find (h j ,\nw j ) ∈ H × S m0 (s ∩ q) ⊗ V * ξ for j = 1, . . . , N such that det((r m0 (f i ) * Θ σ(δ,ν0) )(h j , w j )) = 0. Now, observe that ν → det((r m0 (f i ) * Θ σ(δ,ν) )(h j , w j ))\nis an analytic function on A since Θ σ(δ,ν) is analytic as a function of ν (Observe that Θ σ(δ,ν) is well-defined and analytic function for all ν ∈ A even if σ(δ, ν) isn't irreducible for singular ν). In particular, since this function is non-zero somewhere, it is non-zero almost everywhere. Therefore,\n{r m0 (f j ) * Θ σ(δ,ν) }\nis a linearly independent set for almost all ν ∈ ( A) ′ . Since µ is a lowest K type of σ(δ, ν 0 ), it is also a lowest K type for σ(δ, ν) for every ν ∈ ( A) ′ (an easy calculation shows that all of these representations have the same K types). In particular, Lemma 2.4 implies that m(σ(δ, ν), π| H ) ≥ N for almost every ν ∈ ( A) ′ . Since N was chosen to be the maximum, m(σ(δ, ν)) is the constant function N almost everywhere on the subset {σ(δ, ν)| ν ∈ ( A) ′ }. Because m is a measurable function that is only well-defined up to subsets of measure zero anyway, we might as well say m is the constant function N on {σ(δ, ν)| ν ∈ ( A) ′ }. Next, suppose {m(σ(δ, ν), π| H )| ν ∈ ( A) ′ } is unbounded. Thus, for any N , we may find ν 0 such that m(σ(δ, ν 0 ), π| H ) ≥ N , and the above argument then implies m(σ(δ, ν), π| H ) ≥ N for almost every ν ∈ ( A) ′ . Since this is true for every N , we observe m(σ(δ, ν), π| H ) = ∞ for almost every ν ∈ ( A) ′ . Again, since m is a measurable function, we might as well call it the constant function ∞ on all of {σ(δ, ν)| ν ∈ ( A) ′ }.\n\n\nA Space of Spherical Functions\n\nNow, we specialize to the case where H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of G. Let χ G be a character of the center of the universal enveloping algebra ZU(g), and let ψ H be a character of ZU(h). Let (ξ, V ξ ) be an irreducible representation of K, and let µ ∈ K ∩ H. Define\nC ω (K\\G, V ξ ) χG,ψH (µ)\nto be the vector space of analytic sections of the bundle V ξ → K\\G satisfying (1)\nR D f = χ G (D)f for all D ∈ ZU(g). (2) R D f = ψ H (D)f for all D ∈ ZU(h).\n(3) If we let K ∩ H act on the space of functions on G,\nSpan C {f (gk ′ )| k ′ ∈ K ∩ H}\nby right translation, then the corresponding representation is isomorphic to a Hilbert space sum of copies of µ.\n\nRecall that we have the decomposition\ng = k ∩ h ⊕ k ∩ q ⊕ s ∩ h ⊕ s ∩ q\nHere s is the −1 eigenspace of the Cartan involution of g with respect to k, and q is the −1 eigenspace of the action of τ on g (we write τ for both the involution on G and its differential on g). The following Theorem is a slight generalization of Lemma 1 on page 623 of [30]. Then\nC ω (K\\G, V ξ ) χG,ψH (µ)\nis finite dimensional for every µ ∈ K ∩ H.\n\nOne observes that the symmetric pairs (O(2n, 1), O(k) × O(2n − k, 1)), (U (n, 1), U (k)×U (n−k, 1)), and (Sp(n, 1), Sp(k)×Sp(n−k, 1)) are all examples of symmetric pairs that satisfy the hypothesis of the above Theorem.\n\nTechnically, this Theorem is a very slight generalization of Lemma 1 of [30]. Thus, we point out how to prove this version. The key is a Lemma of Harish-Chandra.\n\n\nLemma 4.2 (Harish-Chandra).\n\nSuppose H 1 is a reductive Lie group of Harish-Chandra class with maximal compact subgroup K 1 ⊂ H 1 , and let h 1 = k 1 + s 1 be the corresponding Cartan decomposition of h 1 . Let b ⊂ s 1 be a maximal abelian subspace, and let M = Z K1 (b). Suppose K 2 ⊂ K 1 is a subgroup satisfying K 2 M = K 1 . Let (µ 1 , V 1 ) ∈ K 1 , let (µ 2 , V 2 ) be a finite dimensional representation of K 2 , and let ψ be a character of ZU(h 1 ). Let\nC ω (K 2 \\H 1 , V 2 ) ψ (µ 1 )\nbe the space of analytic sections of the vector bundle V 2 → K 2 \\H that transform by µ 1 on the right and by ψ under ZU(h). Then\nC ω (K 2 \\H 1 , V 2 ) ψ (µ 1 ) is finite dimensional.\nThis Lemma is a very slight modification and generalization of Corollary 2 on page 65 of [14] or Lemma 8 on page 67 of [14]. For the sake of completeness, we give a proof.\n\nProof. Choose b ∈ exp(b) regular. Then Corollary 1 of [14] says that there exists a finite collection\n{v i } r i=1 ⊂ U(b) such that U(h 1 ) = Ad(b −1 )U(k 2 ) r i=1 ZU(h 1 )v i U(k 1 ).\nNow, view C ω (K 2 \\H 1 , V 2 ) ψ (µ 1 ) as a space of analytic functions from H 1 to V 2 . Denote the annihilator of V 1 in U(k 1 ) by Ann(V 1 ), and let {w j } m j=1 be a basis for the finite dimensional vector space U(k 1 )/ Ann(V 1 ). We have a map\nC ω (K 2 \\H 1 , V 2 ) ψ (µ 1 ) → V 2 ⊗ C r+m by f → (R vi R wj f )(b)\n. If we show that this map is injective, then the Lemma follows. Therefore, we sup-\npose f ∈ C ω (K 2 \\H 1 , V 2 ) ψ (µ 1 ) with (R vi R wj f )(b) = 0 for all i, j. Since f is ana- lytic, to conclude f = 0, it is enough to show (R D f )(b) = 0 for all D ∈ U(h 1 ). Using the above decomposition of U(h 1 ), we write D = (Ad(b −1 )D 2 )( z i v i )D 1 with D 2 ∈ U(k 2 ), z i ∈ ZU(h 1 ) and D 2 ∈ U(k 1 ). Further, write D 1 = m j=1 a j w j mod(Ann(V 1 )). Then (R D f )(b) = dµ 2 (D 1 ) i,j ψ(z i )a j R vi R wj f (b) = 0.\nThe Lemma follows. Now, we explain how to modify the proof on pages 623-625 of [30] to obtain Theorem 3.1 under our slightly weaker assumptions. Let a ⊂ s ∩ q be a maximal abelian subspace, and let a ∈ exp(a) be regular. Given f ∈ C ω (K\\G, V ξ ) χG,ψH (µ), Ørsted and Vargas define the middle differentiation of f by X ∈ g at (a, h) to be\nf (a; X; h) = d dt t=0 f (a exp(tX)h).\nThey extend this definition to any X ∈ U(g) by iteration. Then, utilizing arguments on page 624, Ørsted and Vargas show that there is a finite set {v i } ⊂ ZU(Z g (a)) such that f ∈ C ω (K\\G, V ξ ) χG ,ψH (µ) is determined by the set of functions H 1 → V 1 of the form h 1 → g f,k0 (h 1 ) = f (a; v j ; k 0 h 1 ) for all k 0 ∈ K 0 and j. This part of their argument is unchanged by our weaker set of hypotheses. Let ψ H1 be the restriction of ψ H to ZU(h 1 ), and let ζ i for i = 1, . . . , r be the finite collection of irreducible H 1 ∩ K = K 1 representations in µ| H1∩K . Now, it is clear that\ng f,k0 ∈ Span r i=1 C ω (K 2 \\H 1 , V 2 ) ψH 1 (ζ i ) ⊂ C ω (K 2 \\H 1 , V 2 ) ψH 1 .\nIn the last Lemma, we showed that this space is finite dimensional. Following [30], we then consider g f,k0 (h 1 ) as a function on K 0 with values in the finite dimensional space Span r i=1 C ω (K 2 \\H 1 , V 2 ) ψH 1 (ζ i ). Note that ZU(k 0 ) acts by a fixed character on k 0 → g f,k0 . Moreover, the space of vector valued functions on a compact group K 0 in a fixed eigenspace for ZU(k 0 ) is always a finite dimensional space. The Theorem follows.\n\n\nA Finite Multiplicity Theorem\n\nWe return to the assumptions of the first section. In particular, G is a reductive Lie group of Harish-Chandra class, H ⊂ G is a closed reductive subgroup of Harish-Chandra class, and K ⊂ G is a maximal compact subgroup, chosen in such a way that H ∩ K ⊂ H is maximal compact. We wish to use the space C ω (K\\G, V ξ ) χG,ψH (µ) introduced in the last section to study multiplicities. To do this, we need to add two additional maps to the framework developed in section one.\n\nDefine\n* H Θ σ : S(K\\G, V ξ ) χπ → C ω (K\\G, V ξ ) χπ ,ψσ by f → f * H Θ σ where (f * H Θ σ )(g) = H f (gh −1 )Θ σ (h)dh.\nThis definition depends on a choice of Haar measure on H; thus, our map is welldefined only up to a positive constant. However, this constant is inconsequential for our applications; thus, we will ignore this ambiguity. The integral converges because l g −1 ι * f is in the Harish-Chandra Schwartz space of G for every g ∈ G (see Lemma 13 of [10]); thus, by the proof of Proposition 2.2, its restriction to H is in the Harish-Chandra Schwartz space of H. Convergence now follows from the fact that integration against Θ σ defines a tempered distribution (see the remarks on page 45 of [10] together with Lemma 27 of [9] or page 456 of [23]).\n\nWe must check that the image of * H Θ σ is indeed contained in C ω (K\\G, V ξ ) χπ ,ψσ . Note that f → f * H Θ σ is left G invariant as a map from functions on G to functions on G; in particular, f * H Θ σ transforms by ξ on the left and can be thought of as a section of the appropriate vector bundle. To check the χ π condition, observe that the map exp * defined earlier is equivariant for the adjoint action. In particular, every D ∈ ZU(g) = U(g) G is of the form exp * (D) for D ∈ S(g) G . Choosing such a D, we observe\nR exp * (D) (f * Θ σ )(g) = D X H f (g exp(X)h −1 )Θ σ (h)dh = D X H f (gh −1 exp(Ad h X))Θ σ (h)dh = (Ad −1 h D) X H f (gh −1 exp(X))Θ σ (h)dh = H D X f (gh −1 exp(X))Θ σ (h)dh = χ π ( D)(f * Θ σ )(g).\nTo check the ψ σ condition, let D ∈ S(h) H be homogeneous and observe\nR exp * (D) (f * Θ σ )(g) = D X H f (g exp(X)h −1 )Θ σ (h)dh = D X H f (g(h exp(−X)) −1 )Θ σ (h)dh = Θ σ , (−1) deg D R exp * (D) l g −1 ι * f R exp * (D) Θ σ , l g −1 ι * f = ψ σ (exp * D) Θ σ , l g −1 ι * f = ψ σ (exp * D)(f * Θ σ )(g).\nHere we have used that Θ σ is an invariant eigendistribution with infitesimal character ψ σ . Note that f * H Θ σ is analytic by the elliptic regularity theorem, since the Casimir operator of G acts on K\\G by an elliptic operator.\n\nTo complete our commutative diagram, we need one final map. Define\nr m : C ω (K\\G, V ξ ) χπ ,ψσ −→ C ω ((H ∩ K)\\H, S m (s ∩ q) * ⊗ V ξ ) ψσ by r m (f )(h)(D) = (L exp * (D) f )(h) for D ∈ S m (s ∩ q)\n. It is not difficult to verify that this map is right H equivariant and left H ∩ K equivariant. Hence, its image does indeed lie in\nC ω ((H ∩ K)\\H, S m (s ∩ q) * ⊗ V ξ ) ψσ .\nHere is the commutative diagram formed from the four maps we have introduced. D). Now, since all four of the maps, r m , r m , * H Θ σ , and * Θ σ are right H equivariant, they all preserve H ∩ K types. Hence, we may take the piece of each of our four spaces that transforms on the right by a minimal H ∩ K type µ of σ, and we obtain a commutative diagram.\nS(K\\G, V ξ ) χπ rm − −−− → S((H ∩ K)\\H, Hom C (S m (s ∩ q), V ξ )) * H Θσ * Θσ C ω (K\\G, V ξ ) χπ ,ψσ rm − −−− → C ω ((H ∩ K)\\H, S m (s ∩ q) * ⊗ V ξ ) ψσ Next, we check that this diagram commutes. Observe r m (f * Θ σ )(D, h) = L exp * (D) (f * H Θ σ )(h) = L exp * (D) H f (hh −1 1 )Θ σ (h 1 )dh 1 = D X H f (exp(−X)hh −1 1 )Θ σ (h 1 )dh 1 = H (D X f )(exp(−X)hh −1 1 )Θ σ (h 1 )dh 1 = H (L exp * (D) )(hh −1 1 )Θ σ (h 1 )dh 1 = (r m f * Θ σ )(h,S(K\\G, V ξ ) χπ (µ) rm − −−− → S((H ∩ K)\\H, Hom C (S m (s ∩ q), V ξ ))(µ) * H Θσ * Θσ C ω (K\\G, V ξ ) χπ ,ψσ (µ) rm − −−− → C ω ((H ∩ K)\\H, S m (s ∩ q) * ⊗ V ξ ) ψσ (µ)\nUsing this commutative diagram together with Lemma 2.4, we obtain the following.\n\nProposition 5.1. Suppose π is a discrete series representation of a reductive Lie group of Harish-Chandra class G with lowest K type (ξ, V ξ ) and infinitesimal character χ π , and suppose σ is a tempered representation of a closed reductive subgroup H ⊂ G of Harish-Chandra class with minimal H ∩ K type µ and infinitesimal character ψ σ . If m(σ, π| H ) denotes the multiplicity of σ in the direct integral decomposition of π| H , then\nm(σ, π| H ) ≤ dim C ω (K\\G, V ξ ) χπ ,ψσ (µ).\nIn particular, if C ω (K\\G, V ξ ) χπ ,ψσ (µ) is finite dimensional, then the multiplicity of σ in π| H is finite.\n\nProof. To prove the proposition, we utilize Lemma 2.4 and the above commutative diagram. We may write φ π,σ,m = r m • * H Θ σ , and the dimension of the image of r m cannot exceed the dimension of its domain. Taking the limit and applying Lemma 2.4, the result follows.\n\nTheorem 5.2. Suppose G is a reductive Lie group of Harish-Chandra class and suppose H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of G. Choose a maximal compact subgroup K ⊂ G such that H ∩ K ⊂ H is maximal compact, and denote by θ the associated Cartan involutions of both G and g. Let s be the −1 eigenspace of θ on g, and let q be the −1 eigenspace of τ on g. Suppose H ∼ = K 0 ×H 1 where K 0 is compact and H 1 is noncompact. Let b ⊂ h 1 ∩ s be a maximal abelian subspace, and let a ⊂ s ∩ q be a maximal abelian subspace. Assume\nZ H1∩K (a)Z H1∩K (b) = H 1 ∩ K.\nIf π is a discrete series representation of G, then the direct integral decomposition of the restriction π| H has finite multiplicities.\n\nObserve that H may not be of Harish-Chandra class even though we have assumed G to be of Harish-Chandra class. However, we observe that the identity component H e is always of Harish-Chandra class. Indeed, referring to the definition of Harish-Chandra class in the introduction, the first condition follows from the well-known fact that the fixed point set of a complex reductive Lie algebra under an involutive automorphism is a complex reductive Lie algebra. For the second condition, we observe\nAd(G τ ) ⊂ Ad(G) τ ⊂ (Int g C ) τ .\n\nTherefore,\n\nAd(H e ) = Ad((G τ ) e ) ⊂ (Int(g C ) τ ) e = Int(g τ C ). The third condition follows from part (b) of Proposition 7.20 of [24]. The fourth condition is automatically satisfied since H e is connected.\n\nThe statement of the Theorem for H e then follows immediately from Theorem 4.1 and Proposition 5.1, and the statement for H follows immediately from the statement for H e . Observe that the symmetric pairs (O(2n, 1), O(k)×O(2n−k, 1)), (U (n, 1), U (k) × U (n − k, 1)), and (Sp(n, 1), Sp(k) × Sp(n − k, 1)) all satisfy the hypothesis of the Theorem.\n\nIt was proven in [30] that the discrete series occurring in π| H have finite multiplicities. Our Theorem says that the continuous spectrum has finite multiplicities as well. 6. The Pair (U (n, 1), U (1) × U (n − 1, 1)) In this section, we prove the following slightly stronger result for the symmetric pair (U (n, 1), U (1) × U (n − 1, 1)). \nτ m = σ(δ,ν)∈ U(n−1,1) temp. m(σ(δ, ν), τ m )σ(δ, ν)\nas a direct integral of irreducible representations of U (n − 1, 1) (See Section 3 for notation). Then m(σ(δ, ν), τ m ) = 0 for finitely many elliptic parameters δ and m(σ(δ, ν), τ m ) < ∞ for all σ, ν. In particular, at most finitely many discrete series occur in τ m .\n\nThe proof of this result will require some notation. First, recall the bounded domain model for U (n, 1)/U (n) × U (1). Let G = U (n, 1), let T ⊂ U (n, 1) be the diagonal torus, and let K = U (n) × U (1) ⊂ G be the maximal compact subgroup of G. Let G C = GL(n + 1, C) denote the complexification of G, and let K C = GL(n, C) × GL(1, C) denote the complexification of K. Let\nP + C = I B 0 1 B ∈ C n ⊂ GL(n + 1, C)\nand let\nP − C = I 0 C 1 C ∈ C n ⊂ GL(n + 1, C).\nThen the product map\nP + C × K C × P − C ֒→ GL(n + 1, C) is a diffeomorphism onto an open subset of G C , which contains G = U(n, 1). Suppose g = A B C d ∈ U (n, 1)\nwith A ∈ M (n, C) an n by n matrix, B an n by 1 matrix, C a 1 by n matrix, and d a single entry. Then we may decompose\ng = A B C d = I Bd −1 0 1 A − Bd −1 C 0 0 d I 0 d −1 C 1\nwith respect to the above decomposition. We will call the first term p + C (g) and the second term κ C (g). One checks that the set of possible Bd −1 that can arise in this way is precisely the set of points in the unit disc, D n , in C n . We define an action of U (n, 1) on D n by thinking of D n ⊂ P + C and taking the P + C part of g · z for g ∈ U (n, 1) and z ∈ D n . Written out this action looks like\nA B C d · Z = AZ + B CZ + d .\nOne checks that the isotropy group at zero is K = U (n) × U (1); thus, we have an identification U (n, 1)/(U (n) × U (1)) ∼ = D n (For more, see for instance page 152 of [23]).\n\nLet λ ∈ T = Hom(T, S 1 ) be a regular, unitary character of T . Given µ ∈ t * , define H µ ∈ t C by µ(H) = B(H, H µ ) for all H ∈ t. Then define an inner product on t * by (µ 1 , µ 2 ) = B(H µ1 , H µ2 ) for µ 1 , µ 2 ∈ t * . Let ∆ be the set of roots of g C with respect to t C . Define a set of positive roots,\n∆ + = {α ∈ ∆| (α, λ) > 0}.\nLet ∆ + c denote the compact, imaginary roots in ∆ + . The G invariant complex structure on G/K gives rise to a K-equivariant splitting\nT e (G/K) ⊗ C ∼ = T (1,0) e (G/K) ⊕ T (0,1) e (G/K) that is preserved under the bracket. Note that T (1,0) e (G/K) ⊂ g C /k C is of the form b C /k C where b C ⊂ g C is a\nLie subalgebra containing k C . The noncompact, imaginary root spaces contained in b C determine a choice of positive noncompact imaginary roots, Φ + n . Let Φ + be a choice of positive roots with Φ + ∩ ∆ n = Φ + n and Φ + c = ∆ + c . Define\nρ = 1 2 α∈Φ + n α.\nLet (ξ λ− ρ , V λ− ρ ) be the irreducible representations of K = U (n) × U (1) with highest weight λ − ρ. Let V λ− ρ be the corresponding G-equivariant, holomorphic vector bundle on G/K. The discrete series with Harish-Chandra parameter λ is isomorphic to\nL 2 (G/K, V λ− ρ ⊗ T (0,q λ ) (G/K)) where = ∂∂ * + ∂ * ∂ and q λ = # (Φ + n ∩ ∆ + ).\nHere the subscript means that we are taking the kernel of the elliptic differential operator . This result is wellknown. It is proved for sufficiently regular parameter in [26], and Schmid remarks in [32] that the general statement follows from his proof of the Blattner formula for groups of Hermitian type. Although Schmid does not carry out the details of the proof in this paper, they are very similar to his argument for the Paratharasy Dirac operator model on page 138 of [32].\n\nNote that we have isomorphisms of vector bundles\nV λ− ρ = G × K V λ− ρ ∼ = GK C P − C × K C P − C V λ− ρ ∼ = D n × V λ− ρ .\nIn particular, the bundle V λ− ρ is holomorphically (though not equivariantly) trivial. The group G = U(n, 1) acts on D n × V λ− ρ by g 1 · (z, v) = (g 1 · z, ξ C (κ C (g 1 g))ξ C (κ C (g)) −1 v).\n\nHere we have chosen g ∈ U(n, 1) with p + C (g) = z and g 1 · z is the above action of U(n, 1) on D n . One checks that this definition is independent of the choice of g. Above, ξ C denotes the holomorphic representation of K C that is the complexification of the representation ξ λ− ρ of K.\n\nLet V λ− ρ denote the trivial bundle on D n with fiber V λ− ρ . We obtain a holomorphic isomorphism of vector bundles\nV λ− ρ ⊗ T (0,q λ ) (G/K) ∼ = V λ− ρ ⊗ T (0,q λ ) D n\nby tensoring the above isomorphism with the map on tangent bundles induced by\nG/K ∼ = D n . Suppose F ∈ L 2 (G/K, V λ− ρ ⊗ T (0,q λ ) (G/K)) ∼ = L 2 (D n , V λ− ρ ⊗ T (0,q λ ) D n ) .\nLet w 1 , . . . , w n−1 , w n = z be holomorphic coordinates on D n ⊂ C n induced from the standard coordinates on C n . Write\nF = I F 0,I dw I + J F 1,J dw J ∧ dz.\nHere I ⊂ {1, . . . , n − 1} is a multi-index of cardinality q λ , J ⊂ {1, . . . , n − 1} is a multi-index of cardinality q λ − 1, and F 0,I , F 1,J : D n → V λ− ρ are analytic functions for each I and J.\n\nThen\n∂F = 0 implies ∂F 0,I ∂z = i∈I ǫ i,I ∂F 1,I−i ∂w i\nwhere ǫ i,I = ±1 for each i. Next, we must calculate ∂ * . We will calculate ∂ * by using the formula on page 86 of [1],\n∂ * = −# * ∂ * #.\nTo use this formula, we must define the operators #, * , and then calculate them in local coordinates. Let * :\nC ω (D n , V λ− ρ ⊗ T (0,q λ ) D n ) −→ C ω (D n , V λ− ρ ⊗ T (0,q λ ) D n )\nbe the usual Hodge * operator. Let m = (m ij ) be the invariant Hermitian metric on T * D n associated to the invariant Riemannian metric on D n (well-defined up to multiplication by a positive scalar) written in the above coordinates, ie\nm = n i,j=1 m ij dw i dw j .\nThen\nm ii = 1 − k =i |w k | 2 (1 − |w| 2 ) 2 , m ij = w i w j (1 − |w| 2 ) 2 if i = j.\nLet m −1 = (m ij ) be the inverse matrix. Applying Cramer's rule, we observe that the diagonal entries of m −1 have the constant term 1 in their Taylor expansions at zero while the off diagonal ij term is of the form\nm ij = w i w j n ij .\nHere n ij is an analytic function on D n for i = j.\n\nNote that m −1 ∈ End(T * D n ) naturally gives rise to an endomorphism M q λ ∈ End(T (0,q λ ) D n ). We express M q λ = (M I,J q λ ) in the coordinates dw I with |I| = q λ a multi-index. The diagonal entries of M q λ are analytic functions with constant term one in their Taylor expansions at w = 0. Suppose I, J are multi-indices with I ∩ J = K. Then the I, J off diagonal entry of M q λ is of the form\nM I,J q λ = w I−K w J−K N I,J q λ . The function N I,J\nq λ is analytic on D n . In coordinates, we obtain * (F 0,I dw I )\n= det(m ij ) I∩K=L, n∈K F 0,I ǫ I,K w I−L zw K−L−n N I,K q λ dw K c + det(m ij ) I∩K=L, n / ∈K F 0,I ǫ I,K M I,K q λ dw c K .\nHere K c = {1, . . . , n} − K is the complement of K and ǫ I,K = ±1. Similarly, we obtain\n* (F 1,J dw J ∧ dz) = det(m ij ) K∩(J∪n)=L, n / ∈K F 1,J ǫ 1,J,K M J∪n,K q λ dw K c + det(m ij ) K∩(J∪n)=L, n∈K F 1,J ǫ 1,J,K M J∪n,K q λ dw K c .\nHere ǫ I,J,K = ±1.\n\nIn order to define ∂ * , we need another operator as well. Let\nA ∈ C ω (D n , Hom C−anti (V λ− ρ , V λ− ρ * ))\nbe the Hermitian metric on the bundle V λ− ρ . Here\nHom C−anti (V λ− ρ , V * λ− ρ ) denotes the space of C antilinear homomorphisms from V λ− ρ to V * λ− ρ . Define # : C ω (D n , V λ− ρ ⊗ T (0,q λ ) D n ) −→ C ω (D n , V λ− ρ ⊗ T (0,q λ ) D n ) by\nF → AF . (One might write (A ⊗ I)F to be more precise).\n\nFollowing [1] page 86, we define\n∂ * = −# * ∂ * #. Lemma 6.2. Suppose F ∈ C ω (D n , V λ− ρ ⊗ T (0,q λ ) D n ) , assume F | Dn−1 = 0, and write F = I F 0,I dw I + J F 1,J dw J ∧ dz = F 0 + F 1 .\nIn the Taylor expansion of F at zero in the variables z = w n , z = w n , the leading term is holomorphic in z in the F 0 part and antiholomorphic in z in the F 1 part.\n\nA more precise statement of the Lemma is as follows. Taylor expand each F 0,I and F 1,J in the variables z = w n and z = w n at zero, and suppose that the first nonvanishing term in any of the expansions is of order k > 0. Then F 0,I ∼ z k v 1 +· · · with v 1 ∈ V λ− ρ and F 1,J ∼ z k v 2 + · · · with v 2 ∈ V λ− ρ .\n\nThe first part follows from the equation ∂F = 0. Recall that this implied ∂F 0,I ∂z = i∈I ǫ i,I ∂F 1,I−i ∂w i where ǫ i,I = ±1 for each i. Choose k 1 + k 2 = k − 1, and differentiate the above equation to obtain\n∂ k1+k2+1 F 0,I ∂z k1 ∂z k2+1 = i∈I ǫ i,I ∂ k1+k2+1 F 1,I−i ∂w i ∂ k1 z∂ k2 z .\nIf we plug in z = z = 0 in the right hand side, we get zero, since we know that every F 0,J and F 1,J vanishes to degree k − 1 on D n−1 by assumption. Thus, the left hand side vanishes on D n−1 as well. The only possibility left for a degree k term is z k . Thus, F 0,I ∼ z k v 1 + terms of higher order. For the second part, we must use the ∂ * operator, which is a bit more complicated. First, observe that * and # are injective maps; hence, ∂ * F = 0 iff This follows from the calculation of * in coordinates on the last page together with the definition of #. Since ∂ * F = 0, we know that the above expression is zero for every K. Now, choose k 1 + k 2 = k − 1, and apply the differential operator ∂ k1 ∂z k1 ∂ k2 ∂z k2 to the above expression and restrict to D n−1 . The second and third expressions are zero since F 0,I and F 1,J vanish to order k − 1 in z and z on D n−1 . The first expression vanishes because of the extra factor of z. If we use one of our z derivatives to differentiate this z, then the F 1,J vanishes. If we do not differentiate the z, then plugging in z = 0 makes the expression vanish. Thus, we are left with\n∂ * #F = 0. Write F = F 0,I dw I + F 1,J dw J ∧ dz. Suppose K ⊂ {1, . . . , n − 1} is a multi-index. The coefficient of dw K c in ∂ * #F = 0 is I∩K=L ǫ I,K ǫ 1,J,K ∂ k ∂z k1 ∂z k2+1 det(m ij )AF 1,J M J∪n,K∪n q λ = 0 on D n−1 .\nApplying the product rule and using the facts that F 1,J vanishes to degree k − 1 on D n−1 , we obtain ∂ k ∂z k1 ∂z k2+1 F 1,J = 0 in a neighborhood of zero inside D n−1 for every K and every k 1 + k 2 = k − 1. But, A is invertible everywhere; therefore, ∂ k ∂z k1 ∂z k2+1 F 1,J = 0 in a neighborhood of zero in D n−1 . By analyticity of F 1,J , this expression is zero on all of D n−1 , and taking the complex conjugate, we obtain ∂ k ∂z k1 ∂z k2+1 F 1,J = 0 on D n−1 . We conclude F 1,J ∼ z k v 2 + higher order terms for every J. The Lemma has been proven. Now, we restrict π λ to H = U (1) × U (n − 1, 1) ⊂ G = U (n, 1). Note that H has a two dimensional center. We consider the one dimensional subgroup U (1) = k θ = e iθ 0 0 I n where I n is the n by n identity matrix and the zeroes are n by 1 and 1 by n matrices. Note that this subgroup is central in U (1) × U (n − 1, 1), but it is not central in U (n, 1). Note that the irreducible, unitary characters of U (1) are the characters ψ m : e iθ → e imθ for m ∈ Z. We may decompose\nπ λ = m∈Z (π λ ) m ∼ = m∈Z ψ m ⊗ Hom(ψ m , π λ ) = m∈Z ψ m ⊗ τ m\nunder the action of U (1). Observe τ m is a representation of U (n− 1, 1). We identify π λ ∼ = L 2 (D n , V λ− ρ ⊗ T (0,q λ ) D n ) , and (π λ ) m is therefore a subspace of these L 2sections. Proof. We begin the proof of the Lemma by calculating the action of U (1) on V λ− ρ ⊗ T (0,q λ ) D n . We compute k θ · (z, w 1 , . . . , w n−1 , v ⊗ dw I ) = (e iθ z, w 1 , . . . , w n−1 , ξ C (κ C (k θ g))ξ C (κ C (g)) −1 v ⊗ dw I ) = (e iθ z, w 1 , . . . , w n−1 , ξ λ− ρ (k θ )v ⊗ dw I ). Here g ∈ U (n, 1) is chosen so that g · 0 = (z, w 1 , . . . , w n−1 ) and we have used ξ C (κ C (k θ g)) = ξ λ− ρ (k θ )ξ C (κ C (g)). Similarly, we have k θ · (z, w 1 , . . . , w n−1 , v ⊗ dw J ∧ dz) = (e iθ z, w 1 , . . . , w n−1 , ξ C (κ C (k θ g))ξ C (κ C (g)) −1 v ⊗ e −iθ dw I ∧ dz) = (e iθ z, w 1 , . . . , w n−1 , ξ λ− ρ (k θ )v ⊗ e −iθ dw I ∧ dz). Now, decompose V λ− ρ = ⊕V s λ− ρ as in the statement of the Lemma, and similarly decompose F 0,I dw I = F s 0,I dw I . Fix s and Taylor expand F s 0,I dw I = j,l F s,j,l 0,I z j z l at zero in z and z. Here F s,j,l 0,I is an analytic function on D n−1 . Then we have k θ · F s,j,l 0,I z j z l dw I = ξ λ− ρ (k θ ) F s,j,l 0,I e i(l−j)θ z j z l dw I = F s,j,l 0,I e i(s+l−j)θ z j z l dw I . Now, we assume F ∈ (π λ ) m . Thus, k θ · F s,j,l 0,I z j z l dw I = e imθ F s,j,l 0,I z j z l dw I .\nLemma 6.3. Suppose F ∈ (π λ ) m ⊂ L 2 (D n , V λ− ρ ⊗ T (0,q λ ) D n ) , and write F = I F 0,I dw I + J F 1,J dw J ∧ dz = F 0 + F 1 in coordinates. Decompose V λ− ρ = V s λ− ρ under U (1),\nIn particular, F s,j,l 0,I = 0 implies s + l − j = m or l − j = m − s. Thus, the Taylor expansion of F 0,I in the variables z and z at zero is of the form Now, consider the case F s 1,J dw J ∧ dz = F s,j,l 1,J z j z l dw J ∧ dz. Then we have k θ · F s,j,l 1,J z j z l dw I ∧ dz = ξ λ− ρ (k θ ) F s,j,l 0,I e i(l−j)θ z j z l dw I = F s,j,l 0,I e i(s+l−j−1)θ z j z l dw I . And since F ∈ (π λ ) m , we must have s + l − j − 1 = m or l − j = 1 + m − s. The lemma follows. Now, we have all of the Lemmas we need to prove the Proposition at the beginning of this section.\n\nProof. We write H = U (1) × U (n − 1, 1), and we decompose π λ = m (π λ ) m as a representation of U (1). Let τ be the involution of G with G τ = H, and let θ be the Cartan involution of G with G θ = K. Differentiate τ and θ to obtain involutions of the Lie algebra g; we will abuse notation and call these involutions τ and θ as well. Let g = k ⊕ p be the decomposition of g into +1 and −1 eigenspaces of θ, and let g = h ⊕ q be the decomposition of g into +1 and −1 eigenspaces of g. Then we have the simultaneous decomposition\ng = k ∩ h ⊕ k ∩ q ⊕ p ∩ h ⊕ p ∩ q.\nLet s be a highest weight of V λ− ρ | U (1) . Consider the H-equivariant map introduced in Proposition 2.1 and [30], [31] r l : L 2 (D n , V λ− ρ ⊗ T (0,q λ ) D n ) −→ L 2 (D n−1 , Hom C (S l (p ∩ q), (V λ− ρ ⊗ T (0,q λ ) D n )| Dn−1 )) where l = |m − s| + 1. We claim that the restriction of r l to (π λ ) m is injective. To see this, let F ∈ (π λ ) m , and write F = F 0,I dw I + F 1,J dw J ∧ dz. By Lemma 6.2, we know that the leading term of F is holomorphic in F 0 and antiholomorphic in F 1 . And by Lemma 6.3, we know that the terms in the Taylor expansions of F 0,I and F 1,J at zero in z and z must be of the form z i z i+m−s or z i z i+m−s+1 .\n\nIn particular, a holomorphic or antiholomorphic term cannot be of degree greater than |m−s|+1. Thus, F cannot vanish to degree greater than |m−s|+1 at zero, and our map is indeed injective. As in section one, we utilize the Polar decomposition to obtain a unitary map U l : (π λ ) m −→ L 2 (D n−1 , Hom C (S l (p ∩ q), (V λ− ρ ⊗ T (0,q λ ) D n )| Dn−1 )). Now, suppose G is a reductive Lie group of Harish-Chandra class, suppose K ⊂ G is a maximal compact subgroup, suppose (η, V ) is a finite dimensional representation of K, and let V → G/K be the corresponding G-equivariant vector bundle. Then σ(δ, ν) occurs in L 2 (G/K, V) iff one of the irreducible constituents of (η, V ) occurs as a K type of σ(δ, ν) by Frobenius reciprocity. In addition, all such representations occur with finite multiplicity. One observes that there are a finite number of elliptic parameters δ such that σ(δ, ν) contains a fixed K type. Thus, L 2 (G/K, V) contains σ(δ, ν) for a finite number of elliptic parameters δ and each σ(δ, ν) occurs with finite multiplicity. Finally, since we have an injective, unitary map of (π λ ) m into a finite number of such spaces, the Theorem follows.\n\n\nAcknowledgments\n\nThe first author would like to thank all of the participants and organizers of the conference \"Branching Problems for Unitary Representations\" at the Max Planck Institute in 2011. Some conversations at that conference, especially ones with Jorge Vargas and Michel Duflo, contributed to the motivation behind beginning this project. The first author would also like to thank Toshiyuki Kobayashi for giving a course at Harvard in 2008 that first introduced him to branching problems.\n\nProposition 2 . 1 (\n21Ørsted-Vargas). There exists a family of continuous, H equivariant maps\n\nProposition 2 . 2 .\n22The map r m restricts to an H equivariant map between the dense subspaces of Harish-Chandra Schwartz functions r m : S(K\\G, V ξ ) χπ −→ S((H ∩ K)\\H, Hom C (S m (s ∩ q), V ξ | (H∩K)\\H )).\n\nTheorem 4. 1 (\n1Ørsted-Vargas). Suppose H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of a reductive Lie group of Harish-Chandra class G, let χ G be a character of ZU(g), and let ψ H be a character of ZU(h). Further, suppose H ∼ = K 0 × H 1 where K 0 is compact and H 1 is noncompact. Let b ⊂ h 1 ∩ s be a maximal abelian subspace, and let a ⊂ s ∩ q be a maximal abelian subspace. Suppose Z H1∩K (b)Z H1∩K (a) = H 1 ∩ K.\n\nProposition 6 . 1 .\n61Suppose π is a discrete series representation of G = U (n, 1), and consider the symmetric subgroup H = U (1) × U (n − 1, 1). Decomposeπ| U(1) ∼ = ψm∈ U(1) ψ m ⊗ τ minto irreducibles under the action of U (1). Here τ m is a unitary representation of U (n − 1, 1). Next, write\n\n\nK∪n ∂ ∂z det(m ij )AF 0,I w I−L zw K−L N I,\n\n\n∂z k1 ∂z k2+1 F 1,J det(m ij )M n−1 . Now, let M be the matrix that takes a vector (v J ) to the vector (det(m ij ) K ǫ K,J,1 M J∪n,K∪n q λ v J ). At zero, M is a diagonal matrix with entries ±1 along the diagonal. Since the entries of M are analytic, we conclude that M must be invertible in a neighborhood of zero. In particular, (det(m ij )\n\n\nand let F 0,I = F s 0,I and F 1,J = F s 1,J be the corresponding decompositions into components. Now, Taylor expand F s 0,I and F s 1,J in the variables z and z at zero. 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Speh, Branching Laws for Some Unitary Representations of SL(4, R), SIGMA 4 (2008), Paper 017.\n\nB Ørsted, J Vargas, Restriction of Square Integrable Representations: Discrete Spectrum. 123B. Ørsted, J. Vargas, Restriction of Square Integrable Representations: Discrete Spectrum, Duke Mathematical Journal 123 (2004), no. 3, 609-633.\n\nA Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a Generalized Berezin Transform. B Ørsted, J Vargas, Annales de L'Institut Fourier (Grenoble). 573B. Ørsted, J. Vargas, A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a Generalized Berezin Transform, Annales de L'Institut Fourier (Grenoble) 57 (2007), No. 3, 693-702.\n\nOn the Characters of the Discrete Series. W Schmid, Inventiones Mathematicae. 30W. Schmid, On the Characters of the Discrete Series, Inventiones Mathematicae 30 (1975), 47-144.\n\nUnitary Representations, Branching Rules, and Matrix Elements for the Non-Compact Symplectic Groups. D J Rowe, B G Whybourne, P H Butler, Journal of Physics A: Mathematical and General. 186D.J. Rowe, B.G. Whybourne, P.H. Butler, Unitary Representations, Branching Rules, and Matrix Elements for the Non-Compact Symplectic Groups, Journal of Physics A: Mathemat- ical and General, Volume 18, Number 6, 1985, 939-953.\n\nAn Extension of Plancherel's Formula to Separable Unimodular Groups. I E Segal, Annals of Mathematics. 2I.E. Segal, An Extension of Plancherel's Formula to Separable Unimodular Groups, Annals of Mathematics (2) 52 (1950), 272-292.\n\nMultiplicity One Theorems: The Archimedean Case. B Sun, C.-B Zhu, Annals of Mathematics. 2B. Sun, C.-B. Zhu, Multiplicity One Theorems: The Archimedean Case, Annals of Mathe- matics (2) 175 (2012), no. 1, 23-44.\n\nThe Tempered Spectrum of a Real Semisimple Lie Group. P C Trombi, American Journal of Mathematics. 991P.C. Trombi, The Tempered Spectrum of a Real Semisimple Lie Group, American Journal of Mathematics 99 (1977), no. 1, 57-75.\n\nJ Vargas, Restriction of Some Discrete Series Representations, Algebras, Groups, and Geometries. 18J. Vargas, Restriction of Some Discrete Series Representations, Algebras, Groups, and Ge- ometries 18 (2001), 85-100.\n\nJ Vargas, Restriction of Discrete Series of a Semisimple Lie Group to Reductive Subgroups, New Developments in Lie Theory and Its Applications. Providence, RIAmerican Mathematical Society544J. Vargas, Restriction of Discrete Series of a Semisimple Lie Group to Reductive Subgroups, New Developments in Lie Theory and Its Applications, Contemporary Mathematics 544, American Mathematical Society, Providence, RI, 2011.\n\nThe Algebraic Structure of the Representations of Semisimple Lie Groups I. D Vogan, Annals of Mathematics. 2D. Vogan, The Algebraic Structure of the Representations of Semisimple Lie Groups I, Annals of Mathematics (2) 109 (1979), no. 1, 1-60.\n\nN Wallach, On the Enright-Varadarajan Modules, Annales Scientifiques de L'Ecole Normale Superieure. 4N. Wallach, On the Enright-Varadarajan Modules, Annales Scientifiques de L'Ecole Normale Superieure 4 9 (1976), no. 1, 81-101.\n\nN Wallach, Real Reductive Groups II. San DiegoAcademic PressN. Wallach, Real Reductive Groups II, Academic Press, San Diego, 1992.\n\nR O Wells, Differential Analysis on Complex Manifolds. New YorkSpringer VerlagR.O. Wells, Differential Analysis on Complex Manifolds, Springer Verlag, New York, 1980.\n"},"corpus_id":{"kind":"number","value":119609556,"string":"119,609,556"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1209.4125\",\n \"mag\": 2952238100,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1142/s0129167x13500493\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1209.4125v3.pdf\"\n ],\n \"pdf_sha\": \"7df02ebc0e9382b0d3d0e2ef8aebe2a26d3c34b5\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1209.4125.pdf\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"THE CONTINUOUS SPECTRUM IN DISCRETE SERIES BRANCHING LAWS"},"authors":{"kind":"list like","value":["HONGYU HE, AND GESTURÓLAFSSONBenjamin Harris"],"string":"[\n \"HONGYU HE, AND GESTURÓLAFSSONBenjamin Harris\"\n]","isConversational":false},"abstract":{"kind":"string","value":"If G is a reductive Lie group of Harish-Chandra class, H is a symmetric subgroup, and π is a discrete series representation of G, the authors give a condition on the pair (G, H) which guarantees that the direct integral decomposition of π| H contains each irreducible representation of H with finite multiplicity. In addition, if G is a reductive Lie group of Harish-Chandra class, and H ⊂ G is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of π| H is constant along 'continuous parameters'. In obtaining these results, the authors develop a new technique for studying multiplicities in the restriction π| H via convolution with Harish-Chandra characters. This technique has the advantage of being useful for studying the continuous spectrum as well as the discrete spectrum."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"If π is an irreducible representation of a Lie group G and H ⊂ G is a type 1 closed Lie subgroup, it is natural to consider the restriction π| H and to decompose this restriction into irreducible representations of H. This type of problem comes up naturally in physics; see for instance [27], [33]. It also arises naturally in number theory; see for instance section 9 of [29], [2], or [21]. In the first case, the multiplicities in this decomposition should correspond to physical quantities and in the second case they should correspond to dimensions of spaces of automorphic forms. Either way, one observes that many branching laws of particular significance have the property that π| H has finite multiplicities.\nIn [18] and [19], Kobayashi gives a sufficient condition for an irreducible representation of a real reductive Lie group G of type A q (λ) to restrict discretely to a closed reductive subgroup of Harish-Chandra class, H. In [20], he gives a necessary condition for infinitesimal discrete decomposability, and he shows that the necessary and sufficient conditions agree when (G, H) is a symmetric pair. In addition, he shows that when (G, H) is a symmetric pair and the restriction is infinitesimally discretely decomposable, the branching law A q (λ)| H contains each irreducible H representation with finite multiplicity. More recently, it has been shown that if (G, H) is a strong Gelfand pair of real, reductive algebraic groups and π is an irreducible, unitary representation of G, then the discrete spectrum of π| H contains each irreducible representation of H with multiplicity at most one (see [35] and the references therein). Finally, in the recent preprint [22], Kobayashi and Oshima give a sufficient condition on pairs of reductive Lie groups (G, H) which guarantees that π| H contains finite multiplicities in the discrete spectrum of H for each irreducible, unitary representation π of G.\nThese are all very powerful finite multiplicity theorems for branching laws. However, in each case, the finite multiplicity theorem only applies to the discrete part of the direct integral decomposition π| H . In this paper, we attempt to study the continuous spectrum of branching laws in a very special case. In order to do this, we make use of many recent ideas of Ørsted and Vargas [30], [31], [38].\nIn this paper we will work with reductive Lie groups of Harish-Chandra class. For the sake of completeness, we give the definition of a reductive Lie group of Harish-Chandra class (taken from pages 105-106 of [12] where this class of groups was introduced). We say that G is a reductive Lie group of Harish-Chandra class if G satisfies the following four conditions:\n• Lie(G) = g is a reductive Lie algebra.\n• Ad(G) ⊂ Int g C where g C ∼ = g ⊗ C is the complexification of g and Int g C is the connected, analytic subgroup of Aut g C with Lie algebra Der g C , the set of derivations of g C . • If G 1 is the connected, analytic subgroup of G with Lie algebra g 1 = [g, g], then the center of G 1 is finite. • [G : G e ] < ∞ where G e denotes the identity component of G.\nHere are our results. Theorem 1.1. Suppose G is a reductive Lie group of Harish-Chandra class and suppose H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of G. Choose a maximal compact subgroup K ⊂ G such that H ∩ K ⊂ H is maximal compact, and denote by θ the associated Cartan involutions of both G and g. Let s be the −1 eigenspace of θ on g, and let q be the −1 eigenspace of τ on g. Suppose H ∼ = K 0 ×H 1 where K 0 is compact and H 1 is noncompact. Let b ⊂ h 1 ∩ s be a maximal abelian subspace, and let a ⊂ s ∩ q be a maximal abelian subspace. Assume Z H1∩K (a)Z H1∩K (b) = H 1 ∩ K.\nIf π is a discrete series representation of G, then the direct integral decomposition of the restriction π| H has finite multiplicities.\nObserve that, under slightly weaker hypotheses, it is proved in Theorem 4 of [30] that the discrete spectrum of π| H has finite multiplicities. In the last sentence of page 628 in [30], the authors remark that they have not yet determined whether the irreducible continuous factors occur with finite multiplicity in the restriction of a discrete series representation of Spin(2n, 1) to the subgroup Spin(2k) × Spin(2n − 2k, 1). This generalization of their result answers their question in the affirmative.\nOne checks that the symmetric pairs (O(2n, 1), O(k) × O(2n − k, 1)), (U (n, 1), U (k) × U (n − k, 1)), (Sp(n, 1), Sp(k) × Sp(n − k, 1)) are all examples of symmetric pairs satisfying the hypotheses of the Theorem. We note that in [22], a class of pairs (G, H) was identified such that π| H has finite multiplicities in the discrete spectrum for every irreducible unitary representation π of G. The above examples are also examples of the class of pairs (G, H) identified in [22]. However, the class in [22] contains additional pairs including (SO(p, q), SO(p − 1, q)) for p, q > 1 that are not covered by the above Theorem. Most likely these pairs also satisfy a finite multiplicity theorem for the continuous spectrum in discrete series branching laws.\nThe proof of the Theorem utilizes many techniques from [30]. One new technique involves convolution of certain Harish-Chandra Schwartz functions on G with Harish-Chandra characters of irreducible, tempered representations of H. This technique is also helpful in proving the following result. Theorem 1.2. Suppose G is a reductive Lie group of Harish-Chandra class, and H ⊂ G is a closed reductive subgroup of Harish-Chandra class. Let P = M AN ⊂ H be a cuspidal parabolic subgroup, let δ be a discrete series representation of M , and let ( A) ′ be the set of H regular characters of A. Let σ(δ, ν) be the family of irreducible tempered representations corresponding to δ ∈ M and ν ∈ ( A) ′ (see Section 3 for notation). Let π be a discrete series representation of G, and let m(σ(δ, ν), π| H ) be the multiplicity of the irreducible tempered representation of σ(δ, ν) in π| H . Then m(σ(δ, ν), π| H ) is constant as a function of ν ∈ ( A) ′ .\nIn [38], Vargas considers m(σ(δ, ν), π| H ) as a measurable function of ν for fixed δ. In Proposition 2, he shows that whenever m(σ(δ, ν), π| H ) is not identically zero almost everywhere as a function of ν, it is zero on a set of measure zero. Thus, our Theorem is a slight generalization of the one obtained in [38].\nFinally, we attempt to use Kahler geometry to obtain more information about the multiplicities in discrete series branching laws in the special case (U (n, 1), U (1) × U (n − 1, 1)). Let ψ m (e iθ ) = e imθ denote the unitary character of U (1) with parameter m ∈ Z. Proposition 1.3. Suppose π is a discrete series representation of G = U (n, 1), and consider the symmetric subgroup H = U (1) × U (n − 1, 1). Decompose\ninto irreducibles under the action of U (1). Here τ m is a unitary representation of U (n − 1, 1). Next, write\nas a direct integral of irreducible representations of U (n − 1, 1) (See Section 3 for notation). Then m(σ(δ, ν), τ m ) = 0 for finitely many elliptic parameters δ and m(σ(δ, ν), τ m ) < ∞ for all σ, ν. In particular, at most finitely many discrete series occur in τ m .\nOne could ask if a similar Proposition holds for all of the symmetric pairs (O(2n, 1), O(k) × O(2n − k, 1)), (U (n, 1), U (k) × U (n − k, 1)), and (Sp(n, 1), Sp(k) × Sp(n − k, 1)). While this may be the case, attempting to prove this generalization using techniques similar to those in the last section of this paper would be computationally unpleasant at best.\nWe end the introduction with a remark. It seems likely that for most symmetric pairs (G, H), there will exist discrete series π such that π| H has only finite multiplicities as well as discrete series π ′ such that π ′ | H has some infinite multiplicities. For example, if G has holomorphic discrete series, then the holomorphic discrete series of G always restrict with finite multiplicities to any symmetric subgroup ( [17], [20], [28]). On the other hand, in many cases one can check using the Mackey restriction principle that large unitary principal series of G will restrict with infinite multiplicities to H, and one expects large discrete series to behave the same way. It would be nice to have a criterion to determine when each of these cases occurs that is analogous to the known criteria for discrete decomposability in [20], [3]. Certainly, the authors of this paper are very far from understanding this problem."},{"title":"A Framework for Studying Multiplicities","content":"Suppose G is a reductive Lie group of Harish-Chandra class and H ⊂ G is a closed reductive subgroup of Harish-Chandra class. Let π be a discrete series representation of G, and let σ be a tempered representation of H. Let χ π be the infinitesimal character of π, and let ψ σ be the infinitesimal character of σ. Let K ⊂ G be a maximal compact subgroup such that H ∩ K ⊂ H is a maximal compact subgroup of H. Let θ be the Cartan involution of g = Lie(G) with fixed points k = Lie(K), let s be the −1 eigenspace of θ, let h = Lie(H) be the Lie algebra of H, and let q be a θ stable and H ∩ K stable complement to h in g. In particular, we have the decomposition\nLet (ξ, V ξ ) be the lowest K type of π, let V ξ be the corresponding G equivariant, Hermitian vector bundle on K\\G, and let\nbe the χ π -eigenspace of the action of ZU(g) on space of square integrable sections of V ξ over K\\G. Note that the Casimir operator Ω ∈ ZU(g) acts as an elliptic operator on K\\G; hence, this is a space of analytic functions by the elliptic regularity theorem. The group G acts on L 2 (K\\G, V ξ ) χπ by right translation, and this representation of G is isomorphic to π ( [16], [40]). Let S(K\\G, V ξ ) χπ be the dense subspace of those sections that pullback to Harish-Chandra Schwartz functions on G with values in V ξ . The Harish-Chandra Schwartz space was first introduced on page 19 of [10].\nWhenever D ∈ U(g), left translation by D will be denoted by L D , and right translation by D will be denoted by R D . For instance, if X ∈ g, then\nRecall that if G is a Lie group, then there is a natural bijective linear map exp * : S(g) → U(g).\nIf we regard S(g) as the space of translation invariant differential operators on g, and we regard U(g) as the space of left invariant differential operators on G, then this map is given by\nIf we instead regard U(g) as the space of right invariant differential operators on G, then the map can be written L exp * (u) (f )(g) := (u exp * (ι * r g f ))(0).\nOf course, l g f (x) = f (g −1 x) (respectively (r g f )(x) = r(xg)) denotes left translation (respectively right translation) by g, (exp * f )(X) = f (exp(X)) denotes the pullback of f by exp, and (ι * f )(x) = f (x −1 ) denotes the pullback of f by the inversion map ι(x) = x −1 . For more details on this map, see pages 281-283 of [15]. If V is a real vector space, let S m (V ) denote the elements of the complex symmetric algebra over V with degree at most m. The following Proposition is a consequence of Theorem 1.3 in [31], and it is a slight modification of Theorem 3 in [30].\nThe map r m is defined by\nfor all m and ∞ m=0 ker(r m ) = {0}.\nObserve that our notation differs slightly from the notation in [30]. Let r m = U m (r * m r m ) 1/2 denote the polar decomposition of the operator r m . Then {U m } is a family of H equivariant partial isometries with ker r m = ker U m and Image(U m ) = Image(r m ) for every m. In particular, {ker(U m ) ⊥ } is a filtration of π| H via closed, H invariant subspaces. We require the following complementary Proposition, which is a close relative of Lemma 1 of [37]. To prove the proposition, we will need a Lemma. Since H ∩ K ⊂ H is maximally compact, we have compatible Cartan decompositions g = k ⊕ s and h = (h ∩ k) ⊕ (h ∩ s). Let a H ⊂ h ∩ s be a maximal abelian subspace, and let a ⊂ s be a maximal abelian subspace satisfying a ∩ (h ∩ s) = a H . Let A = exp(a) and A H = exp(a H ). Let ∆(g, a) denote the roots of g with respect to a, and choose positive roots ∆ + (g, a) ⊂ ∆(g, a). Let a + be the positive Weyl chamber for this choice, let A + = exp(a + ), and let\nα.\nLet q : a * → a * H be the pullback of the inclusion, and observe q(∆(g, a)) ⊃ ∆(h, a H ).\nNote that we can choose positive roots ∆ + (h, a) ⊂ ∆(h, a) so that each positive root α of h with respect to a H is of the form q(β) for a positive root β ∈ ∆ + (g, a).\nFix such a choice and let a + H be the positive Weyl chamber for these positive roots. Let A + H = exp(a + H ), and let\nLet W G be the Weyl group of the root system ∆(g, a), and let W H be the Weyl group of the root system ∆(h, a H ). a)). Writing things out, we obtain\nThe final sum is nonnegative since wa ∈ A + implies α(log(a)) ≥ 0 for all α ∈ w −1 ∆ + (g, a).\nNext, we prove the proposition.\nProof. To prove this Proposition, it is enough to check that the restriction to H of a Harish-Chandra Schwartz function on G is a Harish-Chandra Schwartz function on H. Let Ξ G be the Harish-Chandra spherical function on G, and let Ξ H be the Harish-Chandra spherical function on H (see for instance page 186 of [23] for a definition; in his notation Ξ G = ϕ G 0 ). Recall that f ∈ C ∞ (G) lies in the Harish-Chandra Schwartz space, S(G), of G iff for every D 1 , D 2 ∈ U(g) and every m ∈ N, there exists a constant C D1,D2,m > 0 such that\nTo define | · |, we choose a W G invariant norm | · | a on a, and we define\nOf course, the definition of the Harish-Chandra Schwartz space is independent of the choice of norm on a. Using well-known upper and lower bounds on Ξ G (see Theorem 3 on page 279 of [7] and (8.83) on page 259 of [23]), we observe that f is in the Harish-Chandra Schwartz space of G iff for every D 1 , D 2 ∈ U(g) and m ∈ N, there exists a constant C D1,D2,m such that\nfor all k 1 , k 2 ∈ K and a ∈ A + . Now, fix a ∈ A + H , and choose w ∈ W G such that wa ∈ A + . Choose k ′ ∈ K that acts on A by w, and define | · | aH to be the restriction of | · | a to a H . If D 1 , D 2 ∈ U(h) ⊂ U(g), then we obtain\n. This implies that the restriction of f to H is in S(H).\nAs a corollary of the proof, we observe that the restriction of the map r m to Harish-Chandra Schwartz spaces is continuous with respect to the locally convex topologies of the Harish-Chandra Schwartz spaces.\nNext, we wish to understand the part of the image of the map r m that transforms by an irreducible tempered representation σ of H. This will help us understand the multiplicity of the representation σ in π| H . We need to introduce an additional map.\nRecall that in [6], [8] it is proved that Θ σ is given by integration against an analytic function on the dense subset of regular, semisimple elements H ′ ⊂ H that is locally L 1 on all of H. We will write Θ σ for the distribution as well as the analytic function on H ′ .\nDefine C ω ((H ∩ K)\\H, V) ψσ to be the space of analytic sections of V → (H ∩ K)\\H satisfying\nbe the subspace that transforms by µ under right translation by H ∩ K.\nWe define a map * Θ σ :\nThe integral converges because integration against Θ σ defines a tempered distribution (see the remarks on page 45 of [10] together with Lemma 27 of [9] or page 456 of [23]). Note that we may also write\nHere we have used substitution and the fact that Θ σ is conjugation invariant. From these formulas, it is clear that * Θ σ is an H × H equivariant map. In particular,\nHere we have used that integration against Θ σ is an invariant eigendistribution with infinitesimal character ψ σ . Finally, the function f * Θ σ is analytic because it is an eigenfunction for the Casimir operator of H, which is elliptic on the Riemannian symmetric space (H ∩ K)\\H.\nLet σ ∈ H temp. , and let µ ∈ H ∩ K be a minimal K type for σ. Let φ π,σ,m be the composition of the map r m and the map * Θ σ from\nBefore beginning the proof of the Lemma, we recall the Plancherel formula for H ( [34], [25], [13] for the original papers, [11], [5], [41] for expository pieces). First, we have an abstract direct integral decomposition\nRecall that we may view V * σ ⊗ V σ ∼ = End(V σ ) HS as the space of Hilbert Schmidt operators on V σ . Let the isomorphism between L 2 (H) and the direct integral be\nIn fact, when f is in the Harish-Chandra Schwartz space, we can make things even more explicit. We have a unique H × H equivariant, continuous map\nMoreover, for f ∈ S(H), we have\nThe point is that we can use our explicit Fourier transform to describe W σ . Extend M σ to a map on S(H) ⊗ V (trivially on V ), and note that\nis a dense subspace. Since the maps * Θ σ and M σ are H × H equivariant, we may look at the part of both sides that transforms on the right by the minimal K type µ of σ. Then we obtain\nFinally, note that dim W σ = m(σ, W ), the multiplicity of σ in W .\nProof. Now, to compute the multiplicity of σ in π| H , we first realize π as\nwe may take the measure on H in the direct integral decomposition of ker(U m ) ⊥ to be the Plancherel measure on H for every m. In particular, this implies\nWe may identify ker(U m ) ⊥ as an H representation with\nSince Harish-Chandra Schwartz functions on G are dense in L 2 (K\\G, V ξ ) χπ and the restriction of Harish-Chandra Schwartz functions to H are Harish-Chandra Schwartz functions on H by Proposition 2.2, we may use the above discussion of the Plancherel formula to compute the multiplicity of σ in ker(U m ) ⊥ . Explicitly, we have\nThe lemma follows."},{"title":"A Constant Multiplicity Theorem","content":"Given a cuspidal parabolic subgroup P = M AN ⊂ H, a discrete series representation δ ∈ M and a unitary character ν ∈ A, we may form the (possibly infinite dimensional) vector bundle on G/P corresponding to the tensor product of δ ⊗ ν ⊗ 1 with the square root of the density bundle on G/P . The space of L 2 sections of this vector bundle is a tempered representation of H, which we will call σ(δ, ν); if ν is regular it is irreducible [13]. Moreover, every irreducible tempered representation σ ∈ H with regular infinitesimal character is of this form with ν ∈ A regular [36].\nLet ( A) ′ denote the set of regular characters of A, and consider {σ(δ, ν)| ν ∈ ( A) ′ } for fixed δ ∈ M . This set of irreducible tempered representations is an open subset of H temp. and the union of all such open subsets has full measure in H temp. [4], [11], [13], [23]. In particular, we may study the multiplicity function m(σ(δ, ν), π) as a function of ν ∈ ( A) ′ for fixed π ∈ G disc. and δ ∈ M disc. . Theorem 3.1. Let π ∈ G disc. be a discrete series representation of G, a reductive Lie group of Harish-Chandra class, let H ⊂ G be a closed reductive subgroup of G of Harish-Chandra class, and let P = M AN ⊂ H be a cuspidal parabolic subgroup of H. Let δ ∈ M disc. be a discrete series representation of M , and let ν ∈ ( A) ′ be a regular, unitary character of A. Let σ(δ, ν) be the irreducible tempered represen-\nbe the multiplicity function that records the multiplicity of σ(δ, ν) in the direct integral decomposition of π| H . Then, as a function of ν ∈ ( A) ′ , the multiplicity function m(σ(δ, ν), π| H ) is constant.\nIn [38], Vargas considers m(σ(δ, ν), π| H ) as a measurable function of ν for fixed δ. In Proposition 2, he shows that whenever m(σ(δ, ν), π| H ) is not identically zero almost everywhere as a function of ν, it is zero on a set of measure zero. In particular, this Theorem is a generalization of Proposition 2 of [38].\nProof. By Lemma 2.4, we have the formula m(σ(δ, ν), π| H ) = lim m→∞ dim φ π,σ,m (S(K\\G, V ξ ) χπ (µ)).\nFix δ and π, and first assume that m(σ(δ, ν), π| H ) is bounded almost everywhere\nis linearly independent for sufficiently large m. Let m 0 be a natural number that is sufficiently large. Recall\nIn particular, one may think of each r m (f j ) * Θ σ(δ,ν0) as a complex valued function on H × S m (s ∩ q) ⊗ V * ξ . We require the following Lemma. Lemma 3.2. Let X be a set, and let {F 1 , . . . , F N } be a linearly independent set of complex valued functions on X. Then there exists a subset {x 1 , . . . , x N } ⊂ X such that det(F i (x j )) = 0.\nThe proof of this Lemma is an easy linear algebra exercise that is left to the reader. In our case, we may find (h j ,\nis an analytic function on A since Θ σ(δ,ν) is analytic as a function of ν (Observe that Θ σ(δ,ν) is well-defined and analytic function for all ν ∈ A even if σ(δ, ν) isn't irreducible for singular ν). In particular, since this function is non-zero somewhere, it is non-zero almost everywhere. Therefore,\nis a linearly independent set for almost all ν ∈ ( A) ′ . Since µ is a lowest K type of σ(δ, ν 0 ), it is also a lowest K type for σ(δ, ν) for every ν ∈ ( A) ′ (an easy calculation shows that all of these representations have the same K types). In particular, Lemma 2.4 implies that m(σ(δ, ν), π| H ) ≥ N for almost every ν ∈ ( A) ′ . Since N was chosen to be the maximum, m(σ(δ, ν)) is the constant function N almost everywhere on the subset {σ(δ, ν)| ν ∈ ( A) ′ }. Because m is a measurable function that is only well-defined up to subsets of measure zero anyway, we might as well say m is the constant function N on {σ(δ, ν)| ν ∈ ( A) ′ }. Next, suppose {m(σ(δ, ν), π| H )| ν ∈ ( A) ′ } is unbounded. Thus, for any N , we may find ν 0 such that m(σ(δ, ν 0 ), π| H ) ≥ N , and the above argument then implies m(σ(δ, ν), π| H ) ≥ N for almost every ν ∈ ( A) ′ . Since this is true for every N , we observe m(σ(δ, ν), π| H ) = ∞ for almost every ν ∈ ( A) ′ . Again, since m is a measurable function, we might as well call it the constant function ∞ on all of {σ(δ, ν)| ν ∈ ( A) ′ }."},{"title":"A Space of Spherical Functions","content":"Now, we specialize to the case where H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of G. Let χ G be a character of the center of the universal enveloping algebra ZU(g), and let ψ H be a character of ZU(h). Let (ξ, V ξ ) be an irreducible representation of K, and let µ ∈ K ∩ H. Define\nto be the vector space of analytic sections of the bundle V ξ → K\\G satisfying (1)\n(3) If we let K ∩ H act on the space of functions on G,\nby right translation, then the corresponding representation is isomorphic to a Hilbert space sum of copies of µ.\nRecall that we have the decomposition\nHere s is the −1 eigenspace of the Cartan involution of g with respect to k, and q is the −1 eigenspace of the action of τ on g (we write τ for both the involution on G and its differential on g). The following Theorem is a slight generalization of Lemma 1 on page 623 of [30]. Then\nis finite dimensional for every µ ∈ K ∩ H.\nOne observes that the symmetric pairs (O(2n, 1), O(k) × O(2n − k, 1)), (U (n, 1), U (k)×U (n−k, 1)), and (Sp(n, 1), Sp(k)×Sp(n−k, 1)) are all examples of symmetric pairs that satisfy the hypothesis of the above Theorem.\nTechnically, this Theorem is a very slight generalization of Lemma 1 of [30]. Thus, we point out how to prove this version. The key is a Lemma of Harish-Chandra."},{"title":"Lemma 4.2 (Harish-Chandra).","content":"Suppose H 1 is a reductive Lie group of Harish-Chandra class with maximal compact subgroup K 1 ⊂ H 1 , and let h 1 = k 1 + s 1 be the corresponding Cartan decomposition of h 1 . Let b ⊂ s 1 be a maximal abelian subspace, and let M = Z K1 (b). Suppose K 2 ⊂ K 1 is a subgroup satisfying K 2 M = K 1 . Let (µ 1 , V 1 ) ∈ K 1 , let (µ 2 , V 2 ) be a finite dimensional representation of K 2 , and let ψ be a character of ZU(h 1 ). Let\nbe the space of analytic sections of the vector bundle V 2 → K 2 \\H that transform by µ 1 on the right and by ψ under ZU(h). Then\nThis Lemma is a very slight modification and generalization of Corollary 2 on page 65 of [14] or Lemma 8 on page 67 of [14]. For the sake of completeness, we give a proof.\nProof. Choose b ∈ exp(b) regular. Then Corollary 1 of [14] says that there exists a finite collection\nNow, view C ω (K 2 \\H 1 , V 2 ) ψ (µ 1 ) as a space of analytic functions from H 1 to V 2 . Denote the annihilator of V 1 in U(k 1 ) by Ann(V 1 ), and let {w j } m j=1 be a basis for the finite dimensional vector space U(k 1 )/ Ann(V 1 ). We have a map\n. If we show that this map is injective, then the Lemma follows. Therefore, we sup-\nThe Lemma follows. Now, we explain how to modify the proof on pages 623-625 of [30] to obtain Theorem 3.1 under our slightly weaker assumptions. Let a ⊂ s ∩ q be a maximal abelian subspace, and let a ∈ exp(a) be regular. Given f ∈ C ω (K\\G, V ξ ) χG,ψH (µ), Ørsted and Vargas define the middle differentiation of f by X ∈ g at (a, h) to be\nThey extend this definition to any X ∈ U(g) by iteration. Then, utilizing arguments on page 624, Ørsted and Vargas show that there is a finite set {v i } ⊂ ZU(Z g (a)) such that f ∈ C ω (K\\G, V ξ ) χG ,ψH (µ) is determined by the set of functions H 1 → V 1 of the form h 1 → g f,k0 (h 1 ) = f (a; v j ; k 0 h 1 ) for all k 0 ∈ K 0 and j. This part of their argument is unchanged by our weaker set of hypotheses. Let ψ H1 be the restriction of ψ H to ZU(h 1 ), and let ζ i for i = 1, . . . , r be the finite collection of irreducible H 1 ∩ K = K 1 representations in µ| H1∩K . Now, it is clear that\nIn the last Lemma, we showed that this space is finite dimensional. Following [30], we then consider g f,k0 (h 1 ) as a function on K 0 with values in the finite dimensional space Span r i=1 C ω (K 2 \\H 1 , V 2 ) ψH 1 (ζ i ). Note that ZU(k 0 ) acts by a fixed character on k 0 → g f,k0 . Moreover, the space of vector valued functions on a compact group K 0 in a fixed eigenspace for ZU(k 0 ) is always a finite dimensional space. The Theorem follows."},{"title":"A Finite Multiplicity Theorem","content":"We return to the assumptions of the first section. In particular, G is a reductive Lie group of Harish-Chandra class, H ⊂ G is a closed reductive subgroup of Harish-Chandra class, and K ⊂ G is a maximal compact subgroup, chosen in such a way that H ∩ K ⊂ H is maximal compact. We wish to use the space C ω (K\\G, V ξ ) χG,ψH (µ) introduced in the last section to study multiplicities. To do this, we need to add two additional maps to the framework developed in section one.\nDefine\nThis definition depends on a choice of Haar measure on H; thus, our map is welldefined only up to a positive constant. However, this constant is inconsequential for our applications; thus, we will ignore this ambiguity. The integral converges because l g −1 ι * f is in the Harish-Chandra Schwartz space of G for every g ∈ G (see Lemma 13 of [10]); thus, by the proof of Proposition 2.2, its restriction to H is in the Harish-Chandra Schwartz space of H. Convergence now follows from the fact that integration against Θ σ defines a tempered distribution (see the remarks on page 45 of [10] together with Lemma 27 of [9] or page 456 of [23]).\nWe must check that the image of * H Θ σ is indeed contained in C ω (K\\G, V ξ ) χπ ,ψσ . Note that f → f * H Θ σ is left G invariant as a map from functions on G to functions on G; in particular, f * H Θ σ transforms by ξ on the left and can be thought of as a section of the appropriate vector bundle. To check the χ π condition, observe that the map exp * defined earlier is equivariant for the adjoint action. In particular, every D ∈ ZU(g) = U(g) G is of the form exp * (D) for D ∈ S(g) G . Choosing such a D, we observe\nTo check the ψ σ condition, let D ∈ S(h) H be homogeneous and observe\nHere we have used that Θ σ is an invariant eigendistribution with infitesimal character ψ σ . Note that f * H Θ σ is analytic by the elliptic regularity theorem, since the Casimir operator of G acts on K\\G by an elliptic operator.\nTo complete our commutative diagram, we need one final map. Define\n. It is not difficult to verify that this map is right H equivariant and left H ∩ K equivariant. Hence, its image does indeed lie in\nHere is the commutative diagram formed from the four maps we have introduced. D). Now, since all four of the maps, r m , r m , * H Θ σ , and * Θ σ are right H equivariant, they all preserve H ∩ K types. Hence, we may take the piece of each of our four spaces that transforms on the right by a minimal H ∩ K type µ of σ, and we obtain a commutative diagram.\nUsing this commutative diagram together with Lemma 2.4, we obtain the following.\nProposition 5.1. Suppose π is a discrete series representation of a reductive Lie group of Harish-Chandra class G with lowest K type (ξ, V ξ ) and infinitesimal character χ π , and suppose σ is a tempered representation of a closed reductive subgroup H ⊂ G of Harish-Chandra class with minimal H ∩ K type µ and infinitesimal character ψ σ . If m(σ, π| H ) denotes the multiplicity of σ in the direct integral decomposition of π| H , then\nIn particular, if C ω (K\\G, V ξ ) χπ ,ψσ (µ) is finite dimensional, then the multiplicity of σ in π| H is finite.\nProof. To prove the proposition, we utilize Lemma 2.4 and the above commutative diagram. We may write φ π,σ,m = r m • * H Θ σ , and the dimension of the image of r m cannot exceed the dimension of its domain. Taking the limit and applying Lemma 2.4, the result follows.\nTheorem 5.2. Suppose G is a reductive Lie group of Harish-Chandra class and suppose H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of G. Choose a maximal compact subgroup K ⊂ G such that H ∩ K ⊂ H is maximal compact, and denote by θ the associated Cartan involutions of both G and g. Let s be the −1 eigenspace of θ on g, and let q be the −1 eigenspace of τ on g. Suppose H ∼ = K 0 ×H 1 where K 0 is compact and H 1 is noncompact. Let b ⊂ h 1 ∩ s be a maximal abelian subspace, and let a ⊂ s ∩ q be a maximal abelian subspace. Assume\nIf π is a discrete series representation of G, then the direct integral decomposition of the restriction π| H has finite multiplicities.\nObserve that H may not be of Harish-Chandra class even though we have assumed G to be of Harish-Chandra class. However, we observe that the identity component H e is always of Harish-Chandra class. Indeed, referring to the definition of Harish-Chandra class in the introduction, the first condition follows from the well-known fact that the fixed point set of a complex reductive Lie algebra under an involutive automorphism is a complex reductive Lie algebra. For the second condition, we observe"},{"title":"Therefore,","content":"Ad(H e ) = Ad((G τ ) e ) ⊂ (Int(g C ) τ ) e = Int(g τ C ). The third condition follows from part (b) of Proposition 7.20 of [24]. The fourth condition is automatically satisfied since H e is connected.\nThe statement of the Theorem for H e then follows immediately from Theorem 4.1 and Proposition 5.1, and the statement for H follows immediately from the statement for H e . Observe that the symmetric pairs (O(2n, 1), O(k)×O(2n−k, 1)), (U (n, 1), U (k) × U (n − k, 1)), and (Sp(n, 1), Sp(k) × Sp(n − k, 1)) all satisfy the hypothesis of the Theorem.\nIt was proven in [30] that the discrete series occurring in π| H have finite multiplicities. Our Theorem says that the continuous spectrum has finite multiplicities as well. 6. The Pair (U (n, 1), U (1) × U (n − 1, 1)) In this section, we prove the following slightly stronger result for the symmetric pair (U (n, 1), U (1) × U (n − 1, 1)).\nas a direct integral of irreducible representations of U (n − 1, 1) (See Section 3 for notation). Then m(σ(δ, ν), τ m ) = 0 for finitely many elliptic parameters δ and m(σ(δ, ν), τ m ) < ∞ for all σ, ν. In particular, at most finitely many discrete series occur in τ m .\nThe proof of this result will require some notation. First, recall the bounded domain model for U (n, 1)/U (n) × U (1). Let G = U (n, 1), let T ⊂ U (n, 1) be the diagonal torus, and let K = U (n) × U (1) ⊂ G be the maximal compact subgroup of G. Let G C = GL(n + 1, C) denote the complexification of G, and let K C = GL(n, C) × GL(1, C) denote the complexification of K. Let\nand let\nThen the product map\nwith A ∈ M (n, C) an n by n matrix, B an n by 1 matrix, C a 1 by n matrix, and d a single entry. Then we may decompose\nwith respect to the above decomposition. We will call the first term p + C (g) and the second term κ C (g). One checks that the set of possible Bd −1 that can arise in this way is precisely the set of points in the unit disc, D n , in C n . We define an action of U (n, 1) on D n by thinking of D n ⊂ P + C and taking the P + C part of g · z for g ∈ U (n, 1) and z ∈ D n . Written out this action looks like\nOne checks that the isotropy group at zero is K = U (n) × U (1); thus, we have an identification U (n, 1)/(U (n) × U (1)) ∼ = D n (For more, see for instance page 152 of [23]).\nLet λ ∈ T = Hom(T, S 1 ) be a regular, unitary character of T . Given µ ∈ t * , define H µ ∈ t C by µ(H) = B(H, H µ ) for all H ∈ t. Then define an inner product on t * by (µ 1 , µ 2 ) = B(H µ1 , H µ2 ) for µ 1 , µ 2 ∈ t * . Let ∆ be the set of roots of g C with respect to t C . Define a set of positive roots,\nLet ∆ + c denote the compact, imaginary roots in ∆ + . The G invariant complex structure on G/K gives rise to a K-equivariant splitting\nLie subalgebra containing k C . The noncompact, imaginary root spaces contained in b C determine a choice of positive noncompact imaginary roots, Φ + n . Let Φ + be a choice of positive roots with Φ + ∩ ∆ n = Φ + n and Φ + c = ∆ + c . Define\nLet (ξ λ− ρ , V λ− ρ ) be the irreducible representations of K = U (n) × U (1) with highest weight λ − ρ. Let V λ− ρ be the corresponding G-equivariant, holomorphic vector bundle on G/K. The discrete series with Harish-Chandra parameter λ is isomorphic to\nHere the subscript means that we are taking the kernel of the elliptic differential operator . This result is wellknown. It is proved for sufficiently regular parameter in [26], and Schmid remarks in [32] that the general statement follows from his proof of the Blattner formula for groups of Hermitian type. Although Schmid does not carry out the details of the proof in this paper, they are very similar to his argument for the Paratharasy Dirac operator model on page 138 of [32].\nNote that we have isomorphisms of vector bundles\nIn particular, the bundle V λ− ρ is holomorphically (though not equivariantly) trivial. The group G = U(n, 1) acts on D n × V λ− ρ by g 1 · (z, v) = (g 1 · z, ξ C (κ C (g 1 g))ξ C (κ C (g)) −1 v).\nHere we have chosen g ∈ U(n, 1) with p + C (g) = z and g 1 · z is the above action of U(n, 1) on D n . One checks that this definition is independent of the choice of g. Above, ξ C denotes the holomorphic representation of K C that is the complexification of the representation ξ λ− ρ of K.\nLet V λ− ρ denote the trivial bundle on D n with fiber V λ− ρ . We obtain a holomorphic isomorphism of vector bundles\nby tensoring the above isomorphism with the map on tangent bundles induced by\nLet w 1 , . . . , w n−1 , w n = z be holomorphic coordinates on D n ⊂ C n induced from the standard coordinates on C n . Write\nHere I ⊂ {1, . . . , n − 1} is a multi-index of cardinality q λ , J ⊂ {1, . . . , n − 1} is a multi-index of cardinality q λ − 1, and F 0,I , F 1,J : D n → V λ− ρ are analytic functions for each I and J.\nThen\nwhere ǫ i,I = ±1 for each i. Next, we must calculate ∂ * . We will calculate ∂ * by using the formula on page 86 of [1],\nTo use this formula, we must define the operators #, * , and then calculate them in local coordinates. Let * :\nbe the usual Hodge * operator. Let m = (m ij ) be the invariant Hermitian metric on T * D n associated to the invariant Riemannian metric on D n (well-defined up to multiplication by a positive scalar) written in the above coordinates, ie\nThen\nLet m −1 = (m ij ) be the inverse matrix. Applying Cramer's rule, we observe that the diagonal entries of m −1 have the constant term 1 in their Taylor expansions at zero while the off diagonal ij term is of the form\nHere n ij is an analytic function on D n for i = j.\nNote that m −1 ∈ End(T * D n ) naturally gives rise to an endomorphism M q λ ∈ End(T (0,q λ ) D n ). We express M q λ = (M I,J q λ ) in the coordinates dw I with |I| = q λ a multi-index. The diagonal entries of M q λ are analytic functions with constant term one in their Taylor expansions at w = 0. Suppose I, J are multi-indices with I ∩ J = K. Then the I, J off diagonal entry of M q λ is of the form\nq λ is analytic on D n . In coordinates, we obtain * (F 0,I dw I )\nHere K c = {1, . . . , n} − K is the complement of K and ǫ I,K = ±1. Similarly, we obtain\nHere ǫ I,J,K = ±1.\nIn order to define ∂ * , we need another operator as well. Let\nbe the Hermitian metric on the bundle V λ− ρ . Here\nF → AF . (One might write (A ⊗ I)F to be more precise).\nFollowing [1] page 86, we define\nIn the Taylor expansion of F at zero in the variables z = w n , z = w n , the leading term is holomorphic in z in the F 0 part and antiholomorphic in z in the F 1 part.\nA more precise statement of the Lemma is as follows. Taylor expand each F 0,I and F 1,J in the variables z = w n and z = w n at zero, and suppose that the first nonvanishing term in any of the expansions is of order k > 0. Then F 0,I ∼ z k v 1 +· · · with v 1 ∈ V λ− ρ and F 1,J ∼ z k v 2 + · · · with v 2 ∈ V λ− ρ .\nThe first part follows from the equation ∂F = 0. Recall that this implied ∂F 0,I ∂z = i∈I ǫ i,I ∂F 1,I−i ∂w i where ǫ i,I = ±1 for each i. Choose k 1 + k 2 = k − 1, and differentiate the above equation to obtain\nIf we plug in z = z = 0 in the right hand side, we get zero, since we know that every F 0,J and F 1,J vanishes to degree k − 1 on D n−1 by assumption. Thus, the left hand side vanishes on D n−1 as well. The only possibility left for a degree k term is z k . Thus, F 0,I ∼ z k v 1 + terms of higher order. For the second part, we must use the ∂ * operator, which is a bit more complicated. First, observe that * and # are injective maps; hence, ∂ * F = 0 iff This follows from the calculation of * in coordinates on the last page together with the definition of #. Since ∂ * F = 0, we know that the above expression is zero for every K. Now, choose k 1 + k 2 = k − 1, and apply the differential operator ∂ k1 ∂z k1 ∂ k2 ∂z k2 to the above expression and restrict to D n−1 . The second and third expressions are zero since F 0,I and F 1,J vanish to order k − 1 in z and z on D n−1 . The first expression vanishes because of the extra factor of z. If we use one of our z derivatives to differentiate this z, then the F 1,J vanishes. If we do not differentiate the z, then plugging in z = 0 makes the expression vanish. Thus, we are left with\nApplying the product rule and using the facts that F 1,J vanishes to degree k − 1 on D n−1 , we obtain ∂ k ∂z k1 ∂z k2+1 F 1,J = 0 in a neighborhood of zero inside D n−1 for every K and every k 1 + k 2 = k − 1. But, A is invertible everywhere; therefore, ∂ k ∂z k1 ∂z k2+1 F 1,J = 0 in a neighborhood of zero in D n−1 . By analyticity of F 1,J , this expression is zero on all of D n−1 , and taking the complex conjugate, we obtain ∂ k ∂z k1 ∂z k2+1 F 1,J = 0 on D n−1 . We conclude F 1,J ∼ z k v 2 + higher order terms for every J. The Lemma has been proven. Now, we restrict π λ to H = U (1) × U (n − 1, 1) ⊂ G = U (n, 1). Note that H has a two dimensional center. We consider the one dimensional subgroup U (1) = k θ = e iθ 0 0 I n where I n is the n by n identity matrix and the zeroes are n by 1 and 1 by n matrices. Note that this subgroup is central in U (1) × U (n − 1, 1), but it is not central in U (n, 1). Note that the irreducible, unitary characters of U (1) are the characters ψ m : e iθ → e imθ for m ∈ Z. We may decompose\nunder the action of U (1). Observe τ m is a representation of U (n− 1, 1). We identify π λ ∼ = L 2 (D n , V λ− ρ ⊗ T (0,q λ ) D n ) , and (π λ ) m is therefore a subspace of these L 2sections. Proof. We begin the proof of the Lemma by calculating the action of U (1) on V λ− ρ ⊗ T (0,q λ ) D n . We compute k θ · (z, w 1 , . . . , w n−1 , v ⊗ dw I ) = (e iθ z, w 1 , . . . , w n−1 , ξ C (κ C (k θ g))ξ C (κ C (g)) −1 v ⊗ dw I ) = (e iθ z, w 1 , . . . , w n−1 , ξ λ− ρ (k θ )v ⊗ dw I ). Here g ∈ U (n, 1) is chosen so that g · 0 = (z, w 1 , . . . , w n−1 ) and we have used ξ C (κ C (k θ g)) = ξ λ− ρ (k θ )ξ C (κ C (g)). Similarly, we have k θ · (z, w 1 , . . . , w n−1 , v ⊗ dw J ∧ dz) = (e iθ z, w 1 , . . . , w n−1 , ξ C (κ C (k θ g))ξ C (κ C (g)) −1 v ⊗ e −iθ dw I ∧ dz) = (e iθ z, w 1 , . . . , w n−1 , ξ λ− ρ (k θ )v ⊗ e −iθ dw I ∧ dz). Now, decompose V λ− ρ = ⊕V s λ− ρ as in the statement of the Lemma, and similarly decompose F 0,I dw I = F s 0,I dw I . Fix s and Taylor expand F s 0,I dw I = j,l F s,j,l 0,I z j z l at zero in z and z. Here F s,j,l 0,I is an analytic function on D n−1 . Then we have k θ · F s,j,l 0,I z j z l dw I = ξ λ− ρ (k θ ) F s,j,l 0,I e i(l−j)θ z j z l dw I = F s,j,l 0,I e i(s+l−j)θ z j z l dw I . Now, we assume F ∈ (π λ ) m . Thus, k θ · F s,j,l 0,I z j z l dw I = e imθ F s,j,l 0,I z j z l dw I .\nIn particular, F s,j,l 0,I = 0 implies s + l − j = m or l − j = m − s. Thus, the Taylor expansion of F 0,I in the variables z and z at zero is of the form Now, consider the case F s 1,J dw J ∧ dz = F s,j,l 1,J z j z l dw J ∧ dz. Then we have k θ · F s,j,l 1,J z j z l dw I ∧ dz = ξ λ− ρ (k θ ) F s,j,l 0,I e i(l−j)θ z j z l dw I = F s,j,l 0,I e i(s+l−j−1)θ z j z l dw I . And since F ∈ (π λ ) m , we must have s + l − j − 1 = m or l − j = 1 + m − s. The lemma follows. Now, we have all of the Lemmas we need to prove the Proposition at the beginning of this section.\nProof. We write H = U (1) × U (n − 1, 1), and we decompose π λ = m (π λ ) m as a representation of U (1). Let τ be the involution of G with G τ = H, and let θ be the Cartan involution of G with G θ = K. Differentiate τ and θ to obtain involutions of the Lie algebra g; we will abuse notation and call these involutions τ and θ as well. Let g = k ⊕ p be the decomposition of g into +1 and −1 eigenspaces of θ, and let g = h ⊕ q be the decomposition of g into +1 and −1 eigenspaces of g. Then we have the simultaneous decomposition\nLet s be a highest weight of V λ− ρ | U (1) . Consider the H-equivariant map introduced in Proposition 2.1 and [30], [31] r l : L 2 (D n , V λ− ρ ⊗ T (0,q λ ) D n ) −→ L 2 (D n−1 , Hom C (S l (p ∩ q), (V λ− ρ ⊗ T (0,q λ ) D n )| Dn−1 )) where l = |m − s| + 1. We claim that the restriction of r l to (π λ ) m is injective. To see this, let F ∈ (π λ ) m , and write F = F 0,I dw I + F 1,J dw J ∧ dz. By Lemma 6.2, we know that the leading term of F is holomorphic in F 0 and antiholomorphic in F 1 . And by Lemma 6.3, we know that the terms in the Taylor expansions of F 0,I and F 1,J at zero in z and z must be of the form z i z i+m−s or z i z i+m−s+1 .\nIn particular, a holomorphic or antiholomorphic term cannot be of degree greater than |m−s|+1. Thus, F cannot vanish to degree greater than |m−s|+1 at zero, and our map is indeed injective. As in section one, we utilize the Polar decomposition to obtain a unitary map U l : (π λ ) m −→ L 2 (D n−1 , Hom C (S l (p ∩ q), (V λ− ρ ⊗ T (0,q λ ) D n )| Dn−1 )). Now, suppose G is a reductive Lie group of Harish-Chandra class, suppose K ⊂ G is a maximal compact subgroup, suppose (η, V ) is a finite dimensional representation of K, and let V → G/K be the corresponding G-equivariant vector bundle. Then σ(δ, ν) occurs in L 2 (G/K, V) iff one of the irreducible constituents of (η, V ) occurs as a K type of σ(δ, ν) by Frobenius reciprocity. In addition, all such representations occur with finite multiplicity. One observes that there are a finite number of elliptic parameters δ such that σ(δ, ν) contains a fixed K type. Thus, L 2 (G/K, V) contains σ(δ, ν) for a finite number of elliptic parameters δ and each σ(δ, ν) occurs with finite multiplicity. Finally, since we have an injective, unitary map of (π λ ) m into a finite number of such spaces, the Theorem follows."},{"title":"Acknowledgments","content":"The first author would like to thank all of the participants and organizers of the conference \"Branching Problems for Unitary Representations\" at the Max Planck Institute in 2011. Some conversations at that conference, especially ones with Jorge Vargas and Michel Duflo, contributed to the motivation behind beginning this project. The first author would also like to thank Toshiyuki Kobayashi for giving a course at Harvard in 2008 that first introduced him to branching problems."},{"title":"Proposition 2 . 1 (","content":""},{"title":"Proposition 2 . 2 .","content":""},{"title":"Theorem 4. 1 (","content":""},{"title":"Proposition 6 . 1 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"If π is an irreducible representation of a Lie group G and H ⊂ G is a type 1 closed Lie subgroup, it is natural to consider the restriction π| H and to decompose this restriction into irreducible representations of H. This type of problem comes up naturally in physics; see for instance [27], [33]. It also arises naturally in number theory; see for instance section 9 of [29], [2], or [21]. In the first case, the multiplicities in this decomposition should correspond to physical quantities and in the second case they should correspond to dimensions of spaces of automorphic forms. Either way, one observes that many branching laws of particular significance have the property that π| H has finite multiplicities.\\nIn [18] and [19], Kobayashi gives a sufficient condition for an irreducible representation of a real reductive Lie group G of type A q (λ) to restrict discretely to a closed reductive subgroup of Harish-Chandra class, H. In [20], he gives a necessary condition for infinitesimal discrete decomposability, and he shows that the necessary and sufficient conditions agree when (G, H) is a symmetric pair. In addition, he shows that when (G, H) is a symmetric pair and the restriction is infinitesimally discretely decomposable, the branching law A q (λ)| H contains each irreducible H representation with finite multiplicity. More recently, it has been shown that if (G, H) is a strong Gelfand pair of real, reductive algebraic groups and π is an irreducible, unitary representation of G, then the discrete spectrum of π| H contains each irreducible representation of H with multiplicity at most one (see [35] and the references therein). Finally, in the recent preprint [22], Kobayashi and Oshima give a sufficient condition on pairs of reductive Lie groups (G, H) which guarantees that π| H contains finite multiplicities in the discrete spectrum of H for each irreducible, unitary representation π of G.\\nThese are all very powerful finite multiplicity theorems for branching laws. However, in each case, the finite multiplicity theorem only applies to the discrete part of the direct integral decomposition π| H . In this paper, we attempt to study the continuous spectrum of branching laws in a very special case. In order to do this, we make use of many recent ideas of Ørsted and Vargas [30], [31], [38].\\nIn this paper we will work with reductive Lie groups of Harish-Chandra class. For the sake of completeness, we give the definition of a reductive Lie group of Harish-Chandra class (taken from pages 105-106 of [12] where this class of groups was introduced). We say that G is a reductive Lie group of Harish-Chandra class if G satisfies the following four conditions:\\n• Lie(G) = g is a reductive Lie algebra.\\n• Ad(G) ⊂ Int g C where g C ∼ = g ⊗ C is the complexification of g and Int g C is the connected, analytic subgroup of Aut g C with Lie algebra Der g C , the set of derivations of g C . • If G 1 is the connected, analytic subgroup of G with Lie algebra g 1 = [g, g], then the center of G 1 is finite. • [G : G e ] < ∞ where G e denotes the identity component of G.\\nHere are our results. Theorem 1.1. Suppose G is a reductive Lie group of Harish-Chandra class and suppose H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of G. Choose a maximal compact subgroup K ⊂ G such that H ∩ K ⊂ H is maximal compact, and denote by θ the associated Cartan involutions of both G and g. Let s be the −1 eigenspace of θ on g, and let q be the −1 eigenspace of τ on g. Suppose H ∼ = K 0 ×H 1 where K 0 is compact and H 1 is noncompact. Let b ⊂ h 1 ∩ s be a maximal abelian subspace, and let a ⊂ s ∩ q be a maximal abelian subspace. Assume Z H1∩K (a)Z H1∩K (b) = H 1 ∩ K.\\nIf π is a discrete series representation of G, then the direct integral decomposition of the restriction π| H has finite multiplicities.\\nObserve that, under slightly weaker hypotheses, it is proved in Theorem 4 of [30] that the discrete spectrum of π| H has finite multiplicities. In the last sentence of page 628 in [30], the authors remark that they have not yet determined whether the irreducible continuous factors occur with finite multiplicity in the restriction of a discrete series representation of Spin(2n, 1) to the subgroup Spin(2k) × Spin(2n − 2k, 1). This generalization of their result answers their question in the affirmative.\\nOne checks that the symmetric pairs (O(2n, 1), O(k) × O(2n − k, 1)), (U (n, 1), U (k) × U (n − k, 1)), (Sp(n, 1), Sp(k) × Sp(n − k, 1)) are all examples of symmetric pairs satisfying the hypotheses of the Theorem. We note that in [22], a class of pairs (G, H) was identified such that π| H has finite multiplicities in the discrete spectrum for every irreducible unitary representation π of G. The above examples are also examples of the class of pairs (G, H) identified in [22]. However, the class in [22] contains additional pairs including (SO(p, q), SO(p − 1, q)) for p, q > 1 that are not covered by the above Theorem. Most likely these pairs also satisfy a finite multiplicity theorem for the continuous spectrum in discrete series branching laws.\\nThe proof of the Theorem utilizes many techniques from [30]. One new technique involves convolution of certain Harish-Chandra Schwartz functions on G with Harish-Chandra characters of irreducible, tempered representations of H. This technique is also helpful in proving the following result. Theorem 1.2. Suppose G is a reductive Lie group of Harish-Chandra class, and H ⊂ G is a closed reductive subgroup of Harish-Chandra class. Let P = M AN ⊂ H be a cuspidal parabolic subgroup, let δ be a discrete series representation of M , and let ( A) ′ be the set of H regular characters of A. Let σ(δ, ν) be the family of irreducible tempered representations corresponding to δ ∈ M and ν ∈ ( A) ′ (see Section 3 for notation). Let π be a discrete series representation of G, and let m(σ(δ, ν), π| H ) be the multiplicity of the irreducible tempered representation of σ(δ, ν) in π| H . Then m(σ(δ, ν), π| H ) is constant as a function of ν ∈ ( A) ′ .\\nIn [38], Vargas considers m(σ(δ, ν), π| H ) as a measurable function of ν for fixed δ. In Proposition 2, he shows that whenever m(σ(δ, ν), π| H ) is not identically zero almost everywhere as a function of ν, it is zero on a set of measure zero. Thus, our Theorem is a slight generalization of the one obtained in [38].\\nFinally, we attempt to use Kahler geometry to obtain more information about the multiplicities in discrete series branching laws in the special case (U (n, 1), U (1) × U (n − 1, 1)). Let ψ m (e iθ ) = e imθ denote the unitary character of U (1) with parameter m ∈ Z. Proposition 1.3. Suppose π is a discrete series representation of G = U (n, 1), and consider the symmetric subgroup H = U (1) × U (n − 1, 1). Decompose\\ninto irreducibles under the action of U (1). Here τ m is a unitary representation of U (n − 1, 1). Next, write\\nas a direct integral of irreducible representations of U (n − 1, 1) (See Section 3 for notation). Then m(σ(δ, ν), τ m ) = 0 for finitely many elliptic parameters δ and m(σ(δ, ν), τ m ) < ∞ for all σ, ν. In particular, at most finitely many discrete series occur in τ m .\\nOne could ask if a similar Proposition holds for all of the symmetric pairs (O(2n, 1), O(k) × O(2n − k, 1)), (U (n, 1), U (k) × U (n − k, 1)), and (Sp(n, 1), Sp(k) × Sp(n − k, 1)). While this may be the case, attempting to prove this generalization using techniques similar to those in the last section of this paper would be computationally unpleasant at best.\\nWe end the introduction with a remark. It seems likely that for most symmetric pairs (G, H), there will exist discrete series π such that π| H has only finite multiplicities as well as discrete series π ′ such that π ′ | H has some infinite multiplicities. For example, if G has holomorphic discrete series, then the holomorphic discrete series of G always restrict with finite multiplicities to any symmetric subgroup ( [17], [20], [28]). On the other hand, in many cases one can check using the Mackey restriction principle that large unitary principal series of G will restrict with infinite multiplicities to H, and one expects large discrete series to behave the same way. It would be nice to have a criterion to determine when each of these cases occurs that is analogous to the known criteria for discrete decomposability in [20], [3]. Certainly, the authors of this paper are very far from understanding this problem.\"\n },\n {\n \"title\": \"A Framework for Studying Multiplicities\",\n \"content\": \"Suppose G is a reductive Lie group of Harish-Chandra class and H ⊂ G is a closed reductive subgroup of Harish-Chandra class. Let π be a discrete series representation of G, and let σ be a tempered representation of H. Let χ π be the infinitesimal character of π, and let ψ σ be the infinitesimal character of σ. Let K ⊂ G be a maximal compact subgroup such that H ∩ K ⊂ H is a maximal compact subgroup of H. Let θ be the Cartan involution of g = Lie(G) with fixed points k = Lie(K), let s be the −1 eigenspace of θ, let h = Lie(H) be the Lie algebra of H, and let q be a θ stable and H ∩ K stable complement to h in g. In particular, we have the decomposition\\nLet (ξ, V ξ ) be the lowest K type of π, let V ξ be the corresponding G equivariant, Hermitian vector bundle on K\\\\G, and let\\nbe the χ π -eigenspace of the action of ZU(g) on space of square integrable sections of V ξ over K\\\\G. Note that the Casimir operator Ω ∈ ZU(g) acts as an elliptic operator on K\\\\G; hence, this is a space of analytic functions by the elliptic regularity theorem. The group G acts on L 2 (K\\\\G, V ξ ) χπ by right translation, and this representation of G is isomorphic to π ( [16], [40]). Let S(K\\\\G, V ξ ) χπ be the dense subspace of those sections that pullback to Harish-Chandra Schwartz functions on G with values in V ξ . The Harish-Chandra Schwartz space was first introduced on page 19 of [10].\\nWhenever D ∈ U(g), left translation by D will be denoted by L D , and right translation by D will be denoted by R D . For instance, if X ∈ g, then\\nRecall that if G is a Lie group, then there is a natural bijective linear map exp * : S(g) → U(g).\\nIf we regard S(g) as the space of translation invariant differential operators on g, and we regard U(g) as the space of left invariant differential operators on G, then this map is given by\\nIf we instead regard U(g) as the space of right invariant differential operators on G, then the map can be written L exp * (u) (f )(g) := (u exp * (ι * r g f ))(0).\\nOf course, l g f (x) = f (g −1 x) (respectively (r g f )(x) = r(xg)) denotes left translation (respectively right translation) by g, (exp * f )(X) = f (exp(X)) denotes the pullback of f by exp, and (ι * f )(x) = f (x −1 ) denotes the pullback of f by the inversion map ι(x) = x −1 . For more details on this map, see pages 281-283 of [15]. If V is a real vector space, let S m (V ) denote the elements of the complex symmetric algebra over V with degree at most m. The following Proposition is a consequence of Theorem 1.3 in [31], and it is a slight modification of Theorem 3 in [30].\\nThe map r m is defined by\\nfor all m and ∞ m=0 ker(r m ) = {0}.\\nObserve that our notation differs slightly from the notation in [30]. Let r m = U m (r * m r m ) 1/2 denote the polar decomposition of the operator r m . Then {U m } is a family of H equivariant partial isometries with ker r m = ker U m and Image(U m ) = Image(r m ) for every m. In particular, {ker(U m ) ⊥ } is a filtration of π| H via closed, H invariant subspaces. We require the following complementary Proposition, which is a close relative of Lemma 1 of [37]. To prove the proposition, we will need a Lemma. Since H ∩ K ⊂ H is maximally compact, we have compatible Cartan decompositions g = k ⊕ s and h = (h ∩ k) ⊕ (h ∩ s). Let a H ⊂ h ∩ s be a maximal abelian subspace, and let a ⊂ s be a maximal abelian subspace satisfying a ∩ (h ∩ s) = a H . Let A = exp(a) and A H = exp(a H ). Let ∆(g, a) denote the roots of g with respect to a, and choose positive roots ∆ + (g, a) ⊂ ∆(g, a). Let a + be the positive Weyl chamber for this choice, let A + = exp(a + ), and let\\nα.\\nLet q : a * → a * H be the pullback of the inclusion, and observe q(∆(g, a)) ⊃ ∆(h, a H ).\\nNote that we can choose positive roots ∆ + (h, a) ⊂ ∆(h, a) so that each positive root α of h with respect to a H is of the form q(β) for a positive root β ∈ ∆ + (g, a).\\nFix such a choice and let a + H be the positive Weyl chamber for these positive roots. Let A + H = exp(a + H ), and let\\nLet W G be the Weyl group of the root system ∆(g, a), and let W H be the Weyl group of the root system ∆(h, a H ). a)). Writing things out, we obtain\\nThe final sum is nonnegative since wa ∈ A + implies α(log(a)) ≥ 0 for all α ∈ w −1 ∆ + (g, a).\\nNext, we prove the proposition.\\nProof. To prove this Proposition, it is enough to check that the restriction to H of a Harish-Chandra Schwartz function on G is a Harish-Chandra Schwartz function on H. Let Ξ G be the Harish-Chandra spherical function on G, and let Ξ H be the Harish-Chandra spherical function on H (see for instance page 186 of [23] for a definition; in his notation Ξ G = ϕ G 0 ). Recall that f ∈ C ∞ (G) lies in the Harish-Chandra Schwartz space, S(G), of G iff for every D 1 , D 2 ∈ U(g) and every m ∈ N, there exists a constant C D1,D2,m > 0 such that\\nTo define | · |, we choose a W G invariant norm | · | a on a, and we define\\nOf course, the definition of the Harish-Chandra Schwartz space is independent of the choice of norm on a. Using well-known upper and lower bounds on Ξ G (see Theorem 3 on page 279 of [7] and (8.83) on page 259 of [23]), we observe that f is in the Harish-Chandra Schwartz space of G iff for every D 1 , D 2 ∈ U(g) and m ∈ N, there exists a constant C D1,D2,m such that\\nfor all k 1 , k 2 ∈ K and a ∈ A + . Now, fix a ∈ A + H , and choose w ∈ W G such that wa ∈ A + . Choose k ′ ∈ K that acts on A by w, and define | · | aH to be the restriction of | · | a to a H . If D 1 , D 2 ∈ U(h) ⊂ U(g), then we obtain\\n. This implies that the restriction of f to H is in S(H).\\nAs a corollary of the proof, we observe that the restriction of the map r m to Harish-Chandra Schwartz spaces is continuous with respect to the locally convex topologies of the Harish-Chandra Schwartz spaces.\\nNext, we wish to understand the part of the image of the map r m that transforms by an irreducible tempered representation σ of H. This will help us understand the multiplicity of the representation σ in π| H . We need to introduce an additional map.\\nRecall that in [6], [8] it is proved that Θ σ is given by integration against an analytic function on the dense subset of regular, semisimple elements H ′ ⊂ H that is locally L 1 on all of H. We will write Θ σ for the distribution as well as the analytic function on H ′ .\\nDefine C ω ((H ∩ K)\\\\H, V) ψσ to be the space of analytic sections of V → (H ∩ K)\\\\H satisfying\\nbe the subspace that transforms by µ under right translation by H ∩ K.\\nWe define a map * Θ σ :\\nThe integral converges because integration against Θ σ defines a tempered distribution (see the remarks on page 45 of [10] together with Lemma 27 of [9] or page 456 of [23]). Note that we may also write\\nHere we have used substitution and the fact that Θ σ is conjugation invariant. From these formulas, it is clear that * Θ σ is an H × H equivariant map. In particular,\\nHere we have used that integration against Θ σ is an invariant eigendistribution with infinitesimal character ψ σ . Finally, the function f * Θ σ is analytic because it is an eigenfunction for the Casimir operator of H, which is elliptic on the Riemannian symmetric space (H ∩ K)\\\\H.\\nLet σ ∈ H temp. , and let µ ∈ H ∩ K be a minimal K type for σ. Let φ π,σ,m be the composition of the map r m and the map * Θ σ from\\nBefore beginning the proof of the Lemma, we recall the Plancherel formula for H ( [34], [25], [13] for the original papers, [11], [5], [41] for expository pieces). First, we have an abstract direct integral decomposition\\nRecall that we may view V * σ ⊗ V σ ∼ = End(V σ ) HS as the space of Hilbert Schmidt operators on V σ . Let the isomorphism between L 2 (H) and the direct integral be\\nIn fact, when f is in the Harish-Chandra Schwartz space, we can make things even more explicit. We have a unique H × H equivariant, continuous map\\nMoreover, for f ∈ S(H), we have\\nThe point is that we can use our explicit Fourier transform to describe W σ . Extend M σ to a map on S(H) ⊗ V (trivially on V ), and note that\\nis a dense subspace. Since the maps * Θ σ and M σ are H × H equivariant, we may look at the part of both sides that transforms on the right by the minimal K type µ of σ. Then we obtain\\nFinally, note that dim W σ = m(σ, W ), the multiplicity of σ in W .\\nProof. Now, to compute the multiplicity of σ in π| H , we first realize π as\\nwe may take the measure on H in the direct integral decomposition of ker(U m ) ⊥ to be the Plancherel measure on H for every m. In particular, this implies\\nWe may identify ker(U m ) ⊥ as an H representation with\\nSince Harish-Chandra Schwartz functions on G are dense in L 2 (K\\\\G, V ξ ) χπ and the restriction of Harish-Chandra Schwartz functions to H are Harish-Chandra Schwartz functions on H by Proposition 2.2, we may use the above discussion of the Plancherel formula to compute the multiplicity of σ in ker(U m ) ⊥ . Explicitly, we have\\nThe lemma follows.\"\n },\n {\n \"title\": \"A Constant Multiplicity Theorem\",\n \"content\": \"Given a cuspidal parabolic subgroup P = M AN ⊂ H, a discrete series representation δ ∈ M and a unitary character ν ∈ A, we may form the (possibly infinite dimensional) vector bundle on G/P corresponding to the tensor product of δ ⊗ ν ⊗ 1 with the square root of the density bundle on G/P . The space of L 2 sections of this vector bundle is a tempered representation of H, which we will call σ(δ, ν); if ν is regular it is irreducible [13]. Moreover, every irreducible tempered representation σ ∈ H with regular infinitesimal character is of this form with ν ∈ A regular [36].\\nLet ( A) ′ denote the set of regular characters of A, and consider {σ(δ, ν)| ν ∈ ( A) ′ } for fixed δ ∈ M . This set of irreducible tempered representations is an open subset of H temp. and the union of all such open subsets has full measure in H temp. [4], [11], [13], [23]. In particular, we may study the multiplicity function m(σ(δ, ν), π) as a function of ν ∈ ( A) ′ for fixed π ∈ G disc. and δ ∈ M disc. . Theorem 3.1. Let π ∈ G disc. be a discrete series representation of G, a reductive Lie group of Harish-Chandra class, let H ⊂ G be a closed reductive subgroup of G of Harish-Chandra class, and let P = M AN ⊂ H be a cuspidal parabolic subgroup of H. Let δ ∈ M disc. be a discrete series representation of M , and let ν ∈ ( A) ′ be a regular, unitary character of A. Let σ(δ, ν) be the irreducible tempered represen-\\nbe the multiplicity function that records the multiplicity of σ(δ, ν) in the direct integral decomposition of π| H . Then, as a function of ν ∈ ( A) ′ , the multiplicity function m(σ(δ, ν), π| H ) is constant.\\nIn [38], Vargas considers m(σ(δ, ν), π| H ) as a measurable function of ν for fixed δ. In Proposition 2, he shows that whenever m(σ(δ, ν), π| H ) is not identically zero almost everywhere as a function of ν, it is zero on a set of measure zero. In particular, this Theorem is a generalization of Proposition 2 of [38].\\nProof. By Lemma 2.4, we have the formula m(σ(δ, ν), π| H ) = lim m→∞ dim φ π,σ,m (S(K\\\\G, V ξ ) χπ (µ)).\\nFix δ and π, and first assume that m(σ(δ, ν), π| H ) is bounded almost everywhere\\nis linearly independent for sufficiently large m. Let m 0 be a natural number that is sufficiently large. Recall\\nIn particular, one may think of each r m (f j ) * Θ σ(δ,ν0) as a complex valued function on H × S m (s ∩ q) ⊗ V * ξ . We require the following Lemma. Lemma 3.2. Let X be a set, and let {F 1 , . . . , F N } be a linearly independent set of complex valued functions on X. Then there exists a subset {x 1 , . . . , x N } ⊂ X such that det(F i (x j )) = 0.\\nThe proof of this Lemma is an easy linear algebra exercise that is left to the reader. In our case, we may find (h j ,\\nis an analytic function on A since Θ σ(δ,ν) is analytic as a function of ν (Observe that Θ σ(δ,ν) is well-defined and analytic function for all ν ∈ A even if σ(δ, ν) isn't irreducible for singular ν). In particular, since this function is non-zero somewhere, it is non-zero almost everywhere. Therefore,\\nis a linearly independent set for almost all ν ∈ ( A) ′ . Since µ is a lowest K type of σ(δ, ν 0 ), it is also a lowest K type for σ(δ, ν) for every ν ∈ ( A) ′ (an easy calculation shows that all of these representations have the same K types). In particular, Lemma 2.4 implies that m(σ(δ, ν), π| H ) ≥ N for almost every ν ∈ ( A) ′ . Since N was chosen to be the maximum, m(σ(δ, ν)) is the constant function N almost everywhere on the subset {σ(δ, ν)| ν ∈ ( A) ′ }. Because m is a measurable function that is only well-defined up to subsets of measure zero anyway, we might as well say m is the constant function N on {σ(δ, ν)| ν ∈ ( A) ′ }. Next, suppose {m(σ(δ, ν), π| H )| ν ∈ ( A) ′ } is unbounded. Thus, for any N , we may find ν 0 such that m(σ(δ, ν 0 ), π| H ) ≥ N , and the above argument then implies m(σ(δ, ν), π| H ) ≥ N for almost every ν ∈ ( A) ′ . Since this is true for every N , we observe m(σ(δ, ν), π| H ) = ∞ for almost every ν ∈ ( A) ′ . Again, since m is a measurable function, we might as well call it the constant function ∞ on all of {σ(δ, ν)| ν ∈ ( A) ′ }.\"\n },\n {\n \"title\": \"A Space of Spherical Functions\",\n \"content\": \"Now, we specialize to the case where H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of G. Let χ G be a character of the center of the universal enveloping algebra ZU(g), and let ψ H be a character of ZU(h). Let (ξ, V ξ ) be an irreducible representation of K, and let µ ∈ K ∩ H. Define\\nto be the vector space of analytic sections of the bundle V ξ → K\\\\G satisfying (1)\\n(3) If we let K ∩ H act on the space of functions on G,\\nby right translation, then the corresponding representation is isomorphic to a Hilbert space sum of copies of µ.\\nRecall that we have the decomposition\\nHere s is the −1 eigenspace of the Cartan involution of g with respect to k, and q is the −1 eigenspace of the action of τ on g (we write τ for both the involution on G and its differential on g). The following Theorem is a slight generalization of Lemma 1 on page 623 of [30]. Then\\nis finite dimensional for every µ ∈ K ∩ H.\\nOne observes that the symmetric pairs (O(2n, 1), O(k) × O(2n − k, 1)), (U (n, 1), U (k)×U (n−k, 1)), and (Sp(n, 1), Sp(k)×Sp(n−k, 1)) are all examples of symmetric pairs that satisfy the hypothesis of the above Theorem.\\nTechnically, this Theorem is a very slight generalization of Lemma 1 of [30]. Thus, we point out how to prove this version. The key is a Lemma of Harish-Chandra.\"\n },\n {\n \"title\": \"Lemma 4.2 (Harish-Chandra).\",\n \"content\": \"Suppose H 1 is a reductive Lie group of Harish-Chandra class with maximal compact subgroup K 1 ⊂ H 1 , and let h 1 = k 1 + s 1 be the corresponding Cartan decomposition of h 1 . Let b ⊂ s 1 be a maximal abelian subspace, and let M = Z K1 (b). Suppose K 2 ⊂ K 1 is a subgroup satisfying K 2 M = K 1 . Let (µ 1 , V 1 ) ∈ K 1 , let (µ 2 , V 2 ) be a finite dimensional representation of K 2 , and let ψ be a character of ZU(h 1 ). Let\\nbe the space of analytic sections of the vector bundle V 2 → K 2 \\\\H that transform by µ 1 on the right and by ψ under ZU(h). Then\\nThis Lemma is a very slight modification and generalization of Corollary 2 on page 65 of [14] or Lemma 8 on page 67 of [14]. For the sake of completeness, we give a proof.\\nProof. Choose b ∈ exp(b) regular. Then Corollary 1 of [14] says that there exists a finite collection\\nNow, view C ω (K 2 \\\\H 1 , V 2 ) ψ (µ 1 ) as a space of analytic functions from H 1 to V 2 . Denote the annihilator of V 1 in U(k 1 ) by Ann(V 1 ), and let {w j } m j=1 be a basis for the finite dimensional vector space U(k 1 )/ Ann(V 1 ). We have a map\\n. If we show that this map is injective, then the Lemma follows. Therefore, we sup-\\nThe Lemma follows. Now, we explain how to modify the proof on pages 623-625 of [30] to obtain Theorem 3.1 under our slightly weaker assumptions. Let a ⊂ s ∩ q be a maximal abelian subspace, and let a ∈ exp(a) be regular. Given f ∈ C ω (K\\\\G, V ξ ) χG,ψH (µ), Ørsted and Vargas define the middle differentiation of f by X ∈ g at (a, h) to be\\nThey extend this definition to any X ∈ U(g) by iteration. Then, utilizing arguments on page 624, Ørsted and Vargas show that there is a finite set {v i } ⊂ ZU(Z g (a)) such that f ∈ C ω (K\\\\G, V ξ ) χG ,ψH (µ) is determined by the set of functions H 1 → V 1 of the form h 1 → g f,k0 (h 1 ) = f (a; v j ; k 0 h 1 ) for all k 0 ∈ K 0 and j. This part of their argument is unchanged by our weaker set of hypotheses. Let ψ H1 be the restriction of ψ H to ZU(h 1 ), and let ζ i for i = 1, . . . , r be the finite collection of irreducible H 1 ∩ K = K 1 representations in µ| H1∩K . Now, it is clear that\\nIn the last Lemma, we showed that this space is finite dimensional. Following [30], we then consider g f,k0 (h 1 ) as a function on K 0 with values in the finite dimensional space Span r i=1 C ω (K 2 \\\\H 1 , V 2 ) ψH 1 (ζ i ). Note that ZU(k 0 ) acts by a fixed character on k 0 → g f,k0 . Moreover, the space of vector valued functions on a compact group K 0 in a fixed eigenspace for ZU(k 0 ) is always a finite dimensional space. The Theorem follows.\"\n },\n {\n \"title\": \"A Finite Multiplicity Theorem\",\n \"content\": \"We return to the assumptions of the first section. In particular, G is a reductive Lie group of Harish-Chandra class, H ⊂ G is a closed reductive subgroup of Harish-Chandra class, and K ⊂ G is a maximal compact subgroup, chosen in such a way that H ∩ K ⊂ H is maximal compact. We wish to use the space C ω (K\\\\G, V ξ ) χG,ψH (µ) introduced in the last section to study multiplicities. To do this, we need to add two additional maps to the framework developed in section one.\\nDefine\\nThis definition depends on a choice of Haar measure on H; thus, our map is welldefined only up to a positive constant. However, this constant is inconsequential for our applications; thus, we will ignore this ambiguity. The integral converges because l g −1 ι * f is in the Harish-Chandra Schwartz space of G for every g ∈ G (see Lemma 13 of [10]); thus, by the proof of Proposition 2.2, its restriction to H is in the Harish-Chandra Schwartz space of H. Convergence now follows from the fact that integration against Θ σ defines a tempered distribution (see the remarks on page 45 of [10] together with Lemma 27 of [9] or page 456 of [23]).\\nWe must check that the image of * H Θ σ is indeed contained in C ω (K\\\\G, V ξ ) χπ ,ψσ . Note that f → f * H Θ σ is left G invariant as a map from functions on G to functions on G; in particular, f * H Θ σ transforms by ξ on the left and can be thought of as a section of the appropriate vector bundle. To check the χ π condition, observe that the map exp * defined earlier is equivariant for the adjoint action. In particular, every D ∈ ZU(g) = U(g) G is of the form exp * (D) for D ∈ S(g) G . Choosing such a D, we observe\\nTo check the ψ σ condition, let D ∈ S(h) H be homogeneous and observe\\nHere we have used that Θ σ is an invariant eigendistribution with infitesimal character ψ σ . Note that f * H Θ σ is analytic by the elliptic regularity theorem, since the Casimir operator of G acts on K\\\\G by an elliptic operator.\\nTo complete our commutative diagram, we need one final map. Define\\n. It is not difficult to verify that this map is right H equivariant and left H ∩ K equivariant. Hence, its image does indeed lie in\\nHere is the commutative diagram formed from the four maps we have introduced. D). Now, since all four of the maps, r m , r m , * H Θ σ , and * Θ σ are right H equivariant, they all preserve H ∩ K types. Hence, we may take the piece of each of our four spaces that transforms on the right by a minimal H ∩ K type µ of σ, and we obtain a commutative diagram.\\nUsing this commutative diagram together with Lemma 2.4, we obtain the following.\\nProposition 5.1. Suppose π is a discrete series representation of a reductive Lie group of Harish-Chandra class G with lowest K type (ξ, V ξ ) and infinitesimal character χ π , and suppose σ is a tempered representation of a closed reductive subgroup H ⊂ G of Harish-Chandra class with minimal H ∩ K type µ and infinitesimal character ψ σ . If m(σ, π| H ) denotes the multiplicity of σ in the direct integral decomposition of π| H , then\\nIn particular, if C ω (K\\\\G, V ξ ) χπ ,ψσ (µ) is finite dimensional, then the multiplicity of σ in π| H is finite.\\nProof. To prove the proposition, we utilize Lemma 2.4 and the above commutative diagram. We may write φ π,σ,m = r m • * H Θ σ , and the dimension of the image of r m cannot exceed the dimension of its domain. Taking the limit and applying Lemma 2.4, the result follows.\\nTheorem 5.2. Suppose G is a reductive Lie group of Harish-Chandra class and suppose H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of G. Choose a maximal compact subgroup K ⊂ G such that H ∩ K ⊂ H is maximal compact, and denote by θ the associated Cartan involutions of both G and g. Let s be the −1 eigenspace of θ on g, and let q be the −1 eigenspace of τ on g. Suppose H ∼ = K 0 ×H 1 where K 0 is compact and H 1 is noncompact. Let b ⊂ h 1 ∩ s be a maximal abelian subspace, and let a ⊂ s ∩ q be a maximal abelian subspace. Assume\\nIf π is a discrete series representation of G, then the direct integral decomposition of the restriction π| H has finite multiplicities.\\nObserve that H may not be of Harish-Chandra class even though we have assumed G to be of Harish-Chandra class. However, we observe that the identity component H e is always of Harish-Chandra class. Indeed, referring to the definition of Harish-Chandra class in the introduction, the first condition follows from the well-known fact that the fixed point set of a complex reductive Lie algebra under an involutive automorphism is a complex reductive Lie algebra. For the second condition, we observe\"\n },\n {\n \"title\": \"Therefore,\",\n \"content\": \"Ad(H e ) = Ad((G τ ) e ) ⊂ (Int(g C ) τ ) e = Int(g τ C ). The third condition follows from part (b) of Proposition 7.20 of [24]. The fourth condition is automatically satisfied since H e is connected.\\nThe statement of the Theorem for H e then follows immediately from Theorem 4.1 and Proposition 5.1, and the statement for H follows immediately from the statement for H e . Observe that the symmetric pairs (O(2n, 1), O(k)×O(2n−k, 1)), (U (n, 1), U (k) × U (n − k, 1)), and (Sp(n, 1), Sp(k) × Sp(n − k, 1)) all satisfy the hypothesis of the Theorem.\\nIt was proven in [30] that the discrete series occurring in π| H have finite multiplicities. Our Theorem says that the continuous spectrum has finite multiplicities as well. 6. The Pair (U (n, 1), U (1) × U (n − 1, 1)) In this section, we prove the following slightly stronger result for the symmetric pair (U (n, 1), U (1) × U (n − 1, 1)).\\nas a direct integral of irreducible representations of U (n − 1, 1) (See Section 3 for notation). Then m(σ(δ, ν), τ m ) = 0 for finitely many elliptic parameters δ and m(σ(δ, ν), τ m ) < ∞ for all σ, ν. In particular, at most finitely many discrete series occur in τ m .\\nThe proof of this result will require some notation. First, recall the bounded domain model for U (n, 1)/U (n) × U (1). Let G = U (n, 1), let T ⊂ U (n, 1) be the diagonal torus, and let K = U (n) × U (1) ⊂ G be the maximal compact subgroup of G. Let G C = GL(n + 1, C) denote the complexification of G, and let K C = GL(n, C) × GL(1, C) denote the complexification of K. Let\\nand let\\nThen the product map\\nwith A ∈ M (n, C) an n by n matrix, B an n by 1 matrix, C a 1 by n matrix, and d a single entry. Then we may decompose\\nwith respect to the above decomposition. We will call the first term p + C (g) and the second term κ C (g). One checks that the set of possible Bd −1 that can arise in this way is precisely the set of points in the unit disc, D n , in C n . We define an action of U (n, 1) on D n by thinking of D n ⊂ P + C and taking the P + C part of g · z for g ∈ U (n, 1) and z ∈ D n . Written out this action looks like\\nOne checks that the isotropy group at zero is K = U (n) × U (1); thus, we have an identification U (n, 1)/(U (n) × U (1)) ∼ = D n (For more, see for instance page 152 of [23]).\\nLet λ ∈ T = Hom(T, S 1 ) be a regular, unitary character of T . Given µ ∈ t * , define H µ ∈ t C by µ(H) = B(H, H µ ) for all H ∈ t. Then define an inner product on t * by (µ 1 , µ 2 ) = B(H µ1 , H µ2 ) for µ 1 , µ 2 ∈ t * . Let ∆ be the set of roots of g C with respect to t C . Define a set of positive roots,\\nLet ∆ + c denote the compact, imaginary roots in ∆ + . The G invariant complex structure on G/K gives rise to a K-equivariant splitting\\nLie subalgebra containing k C . The noncompact, imaginary root spaces contained in b C determine a choice of positive noncompact imaginary roots, Φ + n . Let Φ + be a choice of positive roots with Φ + ∩ ∆ n = Φ + n and Φ + c = ∆ + c . Define\\nLet (ξ λ− ρ , V λ− ρ ) be the irreducible representations of K = U (n) × U (1) with highest weight λ − ρ. Let V λ− ρ be the corresponding G-equivariant, holomorphic vector bundle on G/K. The discrete series with Harish-Chandra parameter λ is isomorphic to\\nHere the subscript means that we are taking the kernel of the elliptic differential operator . This result is wellknown. It is proved for sufficiently regular parameter in [26], and Schmid remarks in [32] that the general statement follows from his proof of the Blattner formula for groups of Hermitian type. Although Schmid does not carry out the details of the proof in this paper, they are very similar to his argument for the Paratharasy Dirac operator model on page 138 of [32].\\nNote that we have isomorphisms of vector bundles\\nIn particular, the bundle V λ− ρ is holomorphically (though not equivariantly) trivial. The group G = U(n, 1) acts on D n × V λ− ρ by g 1 · (z, v) = (g 1 · z, ξ C (κ C (g 1 g))ξ C (κ C (g)) −1 v).\\nHere we have chosen g ∈ U(n, 1) with p + C (g) = z and g 1 · z is the above action of U(n, 1) on D n . One checks that this definition is independent of the choice of g. Above, ξ C denotes the holomorphic representation of K C that is the complexification of the representation ξ λ− ρ of K.\\nLet V λ− ρ denote the trivial bundle on D n with fiber V λ− ρ . We obtain a holomorphic isomorphism of vector bundles\\nby tensoring the above isomorphism with the map on tangent bundles induced by\\nLet w 1 , . . . , w n−1 , w n = z be holomorphic coordinates on D n ⊂ C n induced from the standard coordinates on C n . Write\\nHere I ⊂ {1, . . . , n − 1} is a multi-index of cardinality q λ , J ⊂ {1, . . . , n − 1} is a multi-index of cardinality q λ − 1, and F 0,I , F 1,J : D n → V λ− ρ are analytic functions for each I and J.\\nThen\\nwhere ǫ i,I = ±1 for each i. Next, we must calculate ∂ * . We will calculate ∂ * by using the formula on page 86 of [1],\\nTo use this formula, we must define the operators #, * , and then calculate them in local coordinates. Let * :\\nbe the usual Hodge * operator. Let m = (m ij ) be the invariant Hermitian metric on T * D n associated to the invariant Riemannian metric on D n (well-defined up to multiplication by a positive scalar) written in the above coordinates, ie\\nThen\\nLet m −1 = (m ij ) be the inverse matrix. Applying Cramer's rule, we observe that the diagonal entries of m −1 have the constant term 1 in their Taylor expansions at zero while the off diagonal ij term is of the form\\nHere n ij is an analytic function on D n for i = j.\\nNote that m −1 ∈ End(T * D n ) naturally gives rise to an endomorphism M q λ ∈ End(T (0,q λ ) D n ). We express M q λ = (M I,J q λ ) in the coordinates dw I with |I| = q λ a multi-index. The diagonal entries of M q λ are analytic functions with constant term one in their Taylor expansions at w = 0. Suppose I, J are multi-indices with I ∩ J = K. Then the I, J off diagonal entry of M q λ is of the form\\nq λ is analytic on D n . In coordinates, we obtain * (F 0,I dw I )\\nHere K c = {1, . . . , n} − K is the complement of K and ǫ I,K = ±1. Similarly, we obtain\\nHere ǫ I,J,K = ±1.\\nIn order to define ∂ * , we need another operator as well. Let\\nbe the Hermitian metric on the bundle V λ− ρ . Here\\nF → AF . (One might write (A ⊗ I)F to be more precise).\\nFollowing [1] page 86, we define\\nIn the Taylor expansion of F at zero in the variables z = w n , z = w n , the leading term is holomorphic in z in the F 0 part and antiholomorphic in z in the F 1 part.\\nA more precise statement of the Lemma is as follows. Taylor expand each F 0,I and F 1,J in the variables z = w n and z = w n at zero, and suppose that the first nonvanishing term in any of the expansions is of order k > 0. Then F 0,I ∼ z k v 1 +· · · with v 1 ∈ V λ− ρ and F 1,J ∼ z k v 2 + · · · with v 2 ∈ V λ− ρ .\\nThe first part follows from the equation ∂F = 0. Recall that this implied ∂F 0,I ∂z = i∈I ǫ i,I ∂F 1,I−i ∂w i where ǫ i,I = ±1 for each i. Choose k 1 + k 2 = k − 1, and differentiate the above equation to obtain\\nIf we plug in z = z = 0 in the right hand side, we get zero, since we know that every F 0,J and F 1,J vanishes to degree k − 1 on D n−1 by assumption. Thus, the left hand side vanishes on D n−1 as well. The only possibility left for a degree k term is z k . Thus, F 0,I ∼ z k v 1 + terms of higher order. For the second part, we must use the ∂ * operator, which is a bit more complicated. First, observe that * and # are injective maps; hence, ∂ * F = 0 iff This follows from the calculation of * in coordinates on the last page together with the definition of #. Since ∂ * F = 0, we know that the above expression is zero for every K. Now, choose k 1 + k 2 = k − 1, and apply the differential operator ∂ k1 ∂z k1 ∂ k2 ∂z k2 to the above expression and restrict to D n−1 . The second and third expressions are zero since F 0,I and F 1,J vanish to order k − 1 in z and z on D n−1 . The first expression vanishes because of the extra factor of z. If we use one of our z derivatives to differentiate this z, then the F 1,J vanishes. If we do not differentiate the z, then plugging in z = 0 makes the expression vanish. Thus, we are left with\\nApplying the product rule and using the facts that F 1,J vanishes to degree k − 1 on D n−1 , we obtain ∂ k ∂z k1 ∂z k2+1 F 1,J = 0 in a neighborhood of zero inside D n−1 for every K and every k 1 + k 2 = k − 1. But, A is invertible everywhere; therefore, ∂ k ∂z k1 ∂z k2+1 F 1,J = 0 in a neighborhood of zero in D n−1 . By analyticity of F 1,J , this expression is zero on all of D n−1 , and taking the complex conjugate, we obtain ∂ k ∂z k1 ∂z k2+1 F 1,J = 0 on D n−1 . We conclude F 1,J ∼ z k v 2 + higher order terms for every J. The Lemma has been proven. Now, we restrict π λ to H = U (1) × U (n − 1, 1) ⊂ G = U (n, 1). Note that H has a two dimensional center. We consider the one dimensional subgroup U (1) = k θ = e iθ 0 0 I n where I n is the n by n identity matrix and the zeroes are n by 1 and 1 by n matrices. Note that this subgroup is central in U (1) × U (n − 1, 1), but it is not central in U (n, 1). Note that the irreducible, unitary characters of U (1) are the characters ψ m : e iθ → e imθ for m ∈ Z. We may decompose\\nunder the action of U (1). Observe τ m is a representation of U (n− 1, 1). We identify π λ ∼ = L 2 (D n , V λ− ρ ⊗ T (0,q λ ) D n ) , and (π λ ) m is therefore a subspace of these L 2sections. Proof. We begin the proof of the Lemma by calculating the action of U (1) on V λ− ρ ⊗ T (0,q λ ) D n . We compute k θ · (z, w 1 , . . . , w n−1 , v ⊗ dw I ) = (e iθ z, w 1 , . . . , w n−1 , ξ C (κ C (k θ g))ξ C (κ C (g)) −1 v ⊗ dw I ) = (e iθ z, w 1 , . . . , w n−1 , ξ λ− ρ (k θ )v ⊗ dw I ). Here g ∈ U (n, 1) is chosen so that g · 0 = (z, w 1 , . . . , w n−1 ) and we have used ξ C (κ C (k θ g)) = ξ λ− ρ (k θ )ξ C (κ C (g)). Similarly, we have k θ · (z, w 1 , . . . , w n−1 , v ⊗ dw J ∧ dz) = (e iθ z, w 1 , . . . , w n−1 , ξ C (κ C (k θ g))ξ C (κ C (g)) −1 v ⊗ e −iθ dw I ∧ dz) = (e iθ z, w 1 , . . . , w n−1 , ξ λ− ρ (k θ )v ⊗ e −iθ dw I ∧ dz). Now, decompose V λ− ρ = ⊕V s λ− ρ as in the statement of the Lemma, and similarly decompose F 0,I dw I = F s 0,I dw I . Fix s and Taylor expand F s 0,I dw I = j,l F s,j,l 0,I z j z l at zero in z and z. Here F s,j,l 0,I is an analytic function on D n−1 . Then we have k θ · F s,j,l 0,I z j z l dw I = ξ λ− ρ (k θ ) F s,j,l 0,I e i(l−j)θ z j z l dw I = F s,j,l 0,I e i(s+l−j)θ z j z l dw I . Now, we assume F ∈ (π λ ) m . Thus, k θ · F s,j,l 0,I z j z l dw I = e imθ F s,j,l 0,I z j z l dw I .\\nIn particular, F s,j,l 0,I = 0 implies s + l − j = m or l − j = m − s. Thus, the Taylor expansion of F 0,I in the variables z and z at zero is of the form Now, consider the case F s 1,J dw J ∧ dz = F s,j,l 1,J z j z l dw J ∧ dz. Then we have k θ · F s,j,l 1,J z j z l dw I ∧ dz = ξ λ− ρ (k θ ) F s,j,l 0,I e i(l−j)θ z j z l dw I = F s,j,l 0,I e i(s+l−j−1)θ z j z l dw I . And since F ∈ (π λ ) m , we must have s + l − j − 1 = m or l − j = 1 + m − s. The lemma follows. Now, we have all of the Lemmas we need to prove the Proposition at the beginning of this section.\\nProof. We write H = U (1) × U (n − 1, 1), and we decompose π λ = m (π λ ) m as a representation of U (1). Let τ be the involution of G with G τ = H, and let θ be the Cartan involution of G with G θ = K. Differentiate τ and θ to obtain involutions of the Lie algebra g; we will abuse notation and call these involutions τ and θ as well. Let g = k ⊕ p be the decomposition of g into +1 and −1 eigenspaces of θ, and let g = h ⊕ q be the decomposition of g into +1 and −1 eigenspaces of g. Then we have the simultaneous decomposition\\nLet s be a highest weight of V λ− ρ | U (1) . Consider the H-equivariant map introduced in Proposition 2.1 and [30], [31] r l : L 2 (D n , V λ− ρ ⊗ T (0,q λ ) D n ) −→ L 2 (D n−1 , Hom C (S l (p ∩ q), (V λ− ρ ⊗ T (0,q λ ) D n )| Dn−1 )) where l = |m − s| + 1. We claim that the restriction of r l to (π λ ) m is injective. To see this, let F ∈ (π λ ) m , and write F = F 0,I dw I + F 1,J dw J ∧ dz. By Lemma 6.2, we know that the leading term of F is holomorphic in F 0 and antiholomorphic in F 1 . And by Lemma 6.3, we know that the terms in the Taylor expansions of F 0,I and F 1,J at zero in z and z must be of the form z i z i+m−s or z i z i+m−s+1 .\\nIn particular, a holomorphic or antiholomorphic term cannot be of degree greater than |m−s|+1. Thus, F cannot vanish to degree greater than |m−s|+1 at zero, and our map is indeed injective. As in section one, we utilize the Polar decomposition to obtain a unitary map U l : (π λ ) m −→ L 2 (D n−1 , Hom C (S l (p ∩ q), (V λ− ρ ⊗ T (0,q λ ) D n )| Dn−1 )). Now, suppose G is a reductive Lie group of Harish-Chandra class, suppose K ⊂ G is a maximal compact subgroup, suppose (η, V ) is a finite dimensional representation of K, and let V → G/K be the corresponding G-equivariant vector bundle. Then σ(δ, ν) occurs in L 2 (G/K, V) iff one of the irreducible constituents of (η, V ) occurs as a K type of σ(δ, ν) by Frobenius reciprocity. In addition, all such representations occur with finite multiplicity. One observes that there are a finite number of elliptic parameters δ such that σ(δ, ν) contains a fixed K type. Thus, L 2 (G/K, V) contains σ(δ, ν) for a finite number of elliptic parameters δ and each σ(δ, ν) occurs with finite multiplicity. Finally, since we have an injective, unitary map of (π λ ) m into a finite number of such spaces, the Theorem follows.\"\n },\n {\n \"title\": \"Acknowledgments\",\n \"content\": \"The first author would like to thank all of the participants and organizers of the conference \\\"Branching Problems for Unitary Representations\\\" at the Max Planck Institute in 2011. Some conversations at that conference, especially ones with Jorge Vargas and Michel Duflo, contributed to the motivation behind beginning this project. The first author would also like to thank Toshiyuki Kobayashi for giving a course at Harvard in 2008 that first introduced him to branching problems.\"\n },\n {\n \"title\": \"Proposition 2 . 1 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 2 . 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 4. 1 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"Proposition 6 . 1 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Ørsted-Vargas). There exists a family of continuous, H equivariant maps","content":"Proposition 2 . 1 (\n21Ørsted-Vargas). There exists a family of continuous, H equivariant maps"},{"id":"","caption":"The map r m restricts to an H equivariant map between the dense subspaces of Harish-Chandra Schwartz functions r m : S(K\\G, V ξ ) χπ −→ S((H ∩ K)\\H, Hom C (S m (s ∩ q), V ξ | (H∩K)\\H )).","content":"Proposition 2 . 2 .\n22The map r m restricts to an H equivariant map between the dense subspaces of Harish-Chandra Schwartz functions r m : S(K\\G, V ξ ) χπ −→ S((H ∩ K)\\H, Hom C (S m (s ∩ q), V ξ | (H∩K)\\H ))."},{"id":"","caption":"Ørsted-Vargas). Suppose H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of a reductive Lie group of Harish-Chandra class G, let χ G be a character of ZU(g), and let ψ H be a character of ZU(h). Further, suppose H ∼ = K 0 × H 1 where K 0 is compact and H 1 is noncompact. Let b ⊂ h 1 ∩ s be a maximal abelian subspace, and let a ⊂ s ∩ q be a maximal abelian subspace. Suppose Z H1∩K (b)Z H1∩K (a) = H 1 ∩ K.","content":"Theorem 4. 1 (\n1Ørsted-Vargas). Suppose H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of a reductive Lie group of Harish-Chandra class G, let χ G be a character of ZU(g), and let ψ H be a character of ZU(h). Further, suppose H ∼ = K 0 × H 1 where K 0 is compact and H 1 is noncompact. Let b ⊂ h 1 ∩ s be a maximal abelian subspace, and let a ⊂ s ∩ q be a maximal abelian subspace. Suppose Z H1∩K (b)Z H1∩K (a) = H 1 ∩ K."},{"id":"","caption":"Suppose π is a discrete series representation of G = U (n, 1), and consider the symmetric subgroup H = U (1) × U (n − 1, 1). Decomposeπ| U(1) ∼ = ψm∈ U(1) ψ m ⊗ τ minto irreducibles under the action of U (1). Here τ m is a unitary representation of U (n − 1, 1). Next, write","content":"Proposition 6 . 1 .\n61Suppose π is a discrete series representation of G = U (n, 1), and consider the symmetric subgroup H = U (1) × U (n − 1, 1). Decomposeπ| U(1) ∼ = ψm∈ U(1) ψ m ⊗ τ minto irreducibles under the action of U (1). Here τ m is a unitary representation of U (n − 1, 1). Next, write"},{"id":"","caption":"K∪n ∂ ∂z det(m ij )AF 0,I w I−L zw K−L N I,","content":"K∪n ∂ ∂z det(m ij )AF 0,I w I−L zw K−L N I,"},{"id":"","caption":"∂z k1 ∂z k2+1 F 1,J det(m ij )M n−1 . Now, let M be the matrix that takes a vector (v J ) to the vector (det(m ij ) K ǫ K,J,1 M J∪n,K∪n q λ v J ). At zero, M is a diagonal matrix with entries ±1 along the diagonal. Since the entries of M are analytic, we conclude that M must be invertible in a neighborhood of zero. In particular, (det(m ij )","content":"∂z k1 ∂z k2+1 F 1,J det(m ij )M n−1 . Now, let M be the matrix that takes a vector (v J ) to the vector (det(m ij ) K ǫ K,J,1 M J∪n,K∪n q λ v J ). At zero, M is a diagonal matrix with entries ±1 along the diagonal. Since the entries of M are analytic, we conclude that M must be invertible in a neighborhood of zero. In particular, (det(m ij )"},{"id":"","caption":"and let F 0,I = F s 0,I and F 1,J = F s 1,J be the corresponding decompositions into components. Now, Taylor expand F s 0,I and F s 1,J in the variables z and z at zero. Then","content":"and let F 0,I = F s 0,I and F 1,J = F s 1,J be the corresponding decompositions into components. Now, Taylor expand F s 0,I and F s 1,J in the variables z and z at zero. Then"},{"id":"","caption":"J z i z m−s+1+i .","content":"J z i z m−s+1+i ."},{"id":"","caption":"z j+m−s dw I .","content":"z j+m−s dw I ."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Ørsted-Vargas). There exists a family of continuous, H equivariant maps\",\n \"content\": \"Proposition 2 . 1 (\\n21Ørsted-Vargas). There exists a family of continuous, H equivariant maps\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The map r m restricts to an H equivariant map between the dense subspaces of Harish-Chandra Schwartz functions r m : S(K\\\\G, V ξ ) χπ −→ S((H ∩ K)\\\\H, Hom C (S m (s ∩ q), V ξ | (H∩K)\\\\H )).\",\n \"content\": \"Proposition 2 . 2 .\\n22The map r m restricts to an H equivariant map between the dense subspaces of Harish-Chandra Schwartz functions r m : S(K\\\\G, V ξ ) χπ −→ S((H ∩ K)\\\\H, Hom C (S m (s ∩ q), V ξ | (H∩K)\\\\H )).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Ørsted-Vargas). Suppose H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of a reductive Lie group of Harish-Chandra class G, let χ G be a character of ZU(g), and let ψ H be a character of ZU(h). Further, suppose H ∼ = K 0 × H 1 where K 0 is compact and H 1 is noncompact. Let b ⊂ h 1 ∩ s be a maximal abelian subspace, and let a ⊂ s ∩ q be a maximal abelian subspace. Suppose Z H1∩K (b)Z H1∩K (a) = H 1 ∩ K.\",\n \"content\": \"Theorem 4. 1 (\\n1Ørsted-Vargas). Suppose H ⊂ G τ ⊂ G is an open subgroup of a symmetric subgroup of a reductive Lie group of Harish-Chandra class G, let χ G be a character of ZU(g), and let ψ H be a character of ZU(h). Further, suppose H ∼ = K 0 × H 1 where K 0 is compact and H 1 is noncompact. Let b ⊂ h 1 ∩ s be a maximal abelian subspace, and let a ⊂ s ∩ q be a maximal abelian subspace. Suppose Z H1∩K (b)Z H1∩K (a) = H 1 ∩ K.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Suppose π is a discrete series representation of G = U (n, 1), and consider the symmetric subgroup H = U (1) × U (n − 1, 1). Decomposeπ| U(1) ∼ = ψm∈ U(1) ψ m ⊗ τ minto irreducibles under the action of U (1). Here τ m is a unitary representation of U (n − 1, 1). Next, write\",\n \"content\": \"Proposition 6 . 1 .\\n61Suppose π is a discrete series representation of G = U (n, 1), and consider the symmetric subgroup H = U (1) × U (n − 1, 1). Decomposeπ| U(1) ∼ = ψm∈ U(1) ψ m ⊗ τ minto irreducibles under the action of U (1). Here τ m is a unitary representation of U (n − 1, 1). Next, write\"\n },\n {\n \"id\": \"\",\n \"caption\": \"K∪n ∂ ∂z det(m ij )AF 0,I w I−L zw K−L N I,\",\n \"content\": \"K∪n ∂ ∂z det(m ij )AF 0,I w I−L zw K−L N I,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"∂z k1 ∂z k2+1 F 1,J det(m ij )M n−1 . Now, let M be the matrix that takes a vector (v J ) to the vector (det(m ij ) K ǫ K,J,1 M J∪n,K∪n q λ v J ). At zero, M is a diagonal matrix with entries ±1 along the diagonal. Since the entries of M are analytic, we conclude that M must be invertible in a neighborhood of zero. In particular, (det(m ij )\",\n \"content\": \"∂z k1 ∂z k2+1 F 1,J det(m ij )M n−1 . Now, let M be the matrix that takes a vector (v J ) to the vector (det(m ij ) K ǫ K,J,1 M J∪n,K∪n q λ v J ). At zero, M is a diagonal matrix with entries ±1 along the diagonal. Since the entries of M are analytic, we conclude that M must be invertible in a neighborhood of zero. In particular, (det(m ij )\"\n },\n {\n \"id\": \"\",\n \"caption\": \"and let F 0,I = F s 0,I and F 1,J = F s 1,J be the corresponding decompositions into components. Now, Taylor expand F s 0,I and F s 1,J in the variables z and z at zero. Then\",\n \"content\": \"and let F 0,I = F s 0,I and F 1,J = F s 1,J be the corresponding decompositions into components. Now, Taylor expand F s 0,I and F s 1,J in the variables z and z at zero. Then\"\n },\n {\n \"id\": \"\",\n \"caption\": \"J z i z m−s+1+i .\",\n \"content\": \"J z i z m−s+1+i .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"z j+m−s dw I .\",\n \"content\": \"z j+m−s dw I .\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"A Andreotti, E Vesentini, Carleman Estimates for the Laplace-Beltrami Equations on Complex Manifolds, Institute Hautes Etudes Scientifique Publications Mathematiques. 25A. Andreotti, E. Vesentini, Carleman Estimates for the Laplace-Beltrami Equations on Com- plex Manifolds, Institute Hautes Etudes Scientifique Publications Mathematiques 25 (1965), 313-362.","doi":""},{"text":"N Bergeron, Proprits de Lefschetz Automorphes pour les Groupes Unitaires et Orthogonaux, Mmoires de la Socit Mathmatique de France. 106125N. Bergeron, Proprits de Lefschetz Automorphes pour les Groupes Unitaires et Orthogonaux, Mmoires de la Socit Mathmatique de France 106 (2006), vi-125.","doi":""},{"text":"Branching Laws for Square Integrable Representations. M Duflo, J Vargas, Series A. 86M. Duflo, J. Vargas, Branching Laws for Square Integrable Representations, Proceedings of the Japan Academy 86 (2010), Series A, 49-54.","doi":""},{"text":"The Dual Spaces of C * Algebras. J M G Fell, Transactions of the American Mathematical Society. 94J.M.G. Fell, The Dual Spaces of C * Algebras, Transactions of the American Mathematical Society 94 (1960), 365-403.","doi":""},{"text":"G Folland, A Course in Abstract Harmonic Analysis. Boca RatonCRC PressG. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, 1995.","doi":""},{"text":"The Characters of Semisimple Lie Groups. Harish-Chandra, Transactions of the American Mathematical Society. 83Harish-Chandra, The Characters of Semisimple Lie Groups, Transactions of the American Mathematical Society 83 (1956), 98-163.","doi":""},{"text":"Spherical Functions on a Semisimple Lie Group I. Harish-Chandra, American Journal of Mathematics. 80Harish-Chandra, Spherical Functions on a Semisimple Lie Group I, American Journal of Mathematics 80 (1958), 553-613.","doi":""},{"text":". Harish-Chandra, Transactions of the American Mathematical Society. 119Invariant Eigendistributions on a Semisimple Lie GroupHarish-Chandra, Invariant Eigendistributions on a Semisimple Lie Group, Transactions of the American Mathematical Society 119 (1965), 457-508.","doi":""},{"text":"Harish-Chandra, Two Theorems on Semi-simple Lie Groups. Harish-Chandra, Two Theorems on Semi-simple Lie Groups, Annals of Mathematics (2) 83 (1966), 74-128.","doi":""},{"text":"Harish-Chandra, The Discrete Series for Semisimple Lie Groups II: Explicit Determination of the Characters. 116Harish-Chandra, The Discrete Series for Semisimple Lie Groups II: Explicit Determination of the Characters, Acta Mathematica 116 (1966), 1-111.","doi":""},{"text":"Harish-Chandra, Harmonic Analysis on Semisimple Lie Groups. 76Harish-Chandra, Harmonic Analysis on Semisimple Lie Groups, Bulletin of the American Mathematical Society 76 (1970), 529-551.","doi":""},{"text":"Harmonic Analysis on Real Reductive Groups I. Harish-Chandra, Journal of Functional Analysis. 192Harish-Chandra, Harmonic Analysis on Real Reductive Groups I, Journal of Functional Anal- ysis, 19 (1975), no. 2, 104-204.","doi":""},{"text":"Harish-Chandra, Harmonic Analysis on Real Reductive Groups III. Harish-Chandra, Harmonic Analysis on Real Reductive Groups III, Annals of Mathematics (2) 104 (1976), no. 1, 117-201.","doi":""},{"text":"Harish-Chandra, IIIDifferential Equations and Semisimple Lie Groups, Collected Papers. V.S. VaradarajanNew YorkSpringerHarish-Chandra, Differential Equations and Semisimple Lie Groups, Collected Papers, Vol- ume III, ed. V.S. Varadarajan, Springer, New York, 1984.","doi":""},{"text":"Groups and Geometric Analysis. S Helgason, Mathematical Surveys and Monographs. 83American Mathematical SocietyS. Helgason, Groups and Geometric Analysis, Mathematical Surveys and Monographs, Vol- ume 83, American Mathematical Society, Providence, RI, 1984.","doi":""},{"text":"On Realization of the Discrete Series for Semisimple Lie Groups. R Hotta, Journal of the Mathematical Society of Japan. 232R. Hotta, On Realization of the Discrete Series for Semisimple Lie Groups, Journal of the Mathematical Society of Japan 23 (1971), no. 2, 384-407.","doi":""},{"text":"Restrictions and Expansions of Holomorphic Representations. H P Jakobsen, M Vergne, Journal of Functional Analysis. 34H.P. Jakobsen, M. Vergne, Restrictions and Expansions of Holomorphic Representations, Journal of Functional Analysis 34 (1979), 29-53.","doi":""},{"text":"Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups and its applications. T Kobayashi, Inventiones Mathematicae. 1172T. Kobayashi, Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups and its applications, Inventiones Mathematicae 117 (1994), no. 2, 181-205.","doi":""},{"text":"Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups. II. Micro-local analysis and asymptotic K-support. T Kobayashi, Annals of Math. 2T. Kobayashi, Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups. II. Micro-local analysis and asymptotic K-support, Annals of Math (2) 147 (1998), no. 3, 709-729.","doi":""},{"text":"Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties. T Kobayashi, Inventiones Mathematicae. 1312T. Kobayashi, Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Inventiones Mathematicae 131 (1998), no. 2, 229-256.","doi":""},{"text":"A vanishing theorem for modular symbols on locally symmetric spaces. T Kobayashi, T Oda, Commentarii Mathematici Helvetici. 731T. Kobayashi, T. Oda, A vanishing theorem for modular symbols on locally symmetric spaces, Commentarii Mathematici Helvetici 73 (1998), no. 1, 45-70.","doi":""},{"text":"T Kobayashi, T Oshima, arXiv:1108.3477Finite Multiplicity Theorems. PreprintT. Kobayashi, T. Oshima, Finite Multiplicity Theorems, Preprint: arXiv:1108.3477.","doi":"arXiv:1108.3477"},{"text":"A Knapp, Representation Theory of Semisimple Lie Groups. Princeton, NJPrinceton University PressA. Knapp, Representation Theory of Semisimple Lie Groups, Princeton University Press, Princeton, NJ, 1986.","doi":""},{"text":"Lie Groups Beyond an Introduction. A Knapp, Progress in Mathematics. 140BirkhauserA. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics, Volume 140, Birkhauser, Boston, MA, 2004.","doi":""},{"text":"F I Mautner, Unitary Representations of Locally Compact Groups II. F.I. Mautner, Unitary Representations of Locally Compact Groups II, Annals of Mathematics (2) 52 (1950), 528-556.","doi":""},{"text":"An Analogue of the Borel-Weil-Bott Theorem for Hermitian Symmetric Pairs of Non-Compact Type. M S Narasimhan, K Okamoto, Annals of Mathematics. 2M.S. Narasimhan, K. Okamoto, An Analogue of the Borel-Weil-Bott Theorem for Hermitian Symmetric Pairs of Non-Compact Type, Annals of Mathematics (2) 91 (1970), no. 3, 486-511.","doi":""},{"text":"Why Would Multiplicities be Log-Concave?. A Okounkov, The Orbit Method in Geometry and Physics. 213BirkhauserProgress in MathematicsA. Okounkov, Why Would Multiplicities be Log-Concave?, In \"The Orbit Method in Geometry and Physics\" (Merseille 2000), Progress in Mathematics Volume 213, Birkhauser, Boston, 2003.","doi":""},{"text":"Generalizations of the Bargmann Transform. G Ólafsson, B Ørsted, Lie Theory and Its Applications to Physics. River Edge, NJClausthal 1995G.Ólafsson, B. Ørsted, Generalizations of the Bargmann Transform, In \"Lie Theory and Its Applications to Physics\" (Clausthal 1995), 3-14, World Sci. Publ., River Edge, NJ, 1996.","doi":""},{"text":"Branching Laws for Some Unitary Representations of SL(4, R). B Ørsted, B Speh, SIGMA. 4Paper 017B. Ørsted, B. Speh, Branching Laws for Some Unitary Representations of SL(4, R), SIGMA 4 (2008), Paper 017.","doi":""},{"text":"B Ørsted, J Vargas, Restriction of Square Integrable Representations: Discrete Spectrum. 123B. Ørsted, J. Vargas, Restriction of Square Integrable Representations: Discrete Spectrum, Duke Mathematical Journal 123 (2004), no. 3, 609-633.","doi":""},{"text":"A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a Generalized Berezin Transform. B Ørsted, J Vargas, Annales de L'Institut Fourier (Grenoble). 573B. Ørsted, J. Vargas, A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a Generalized Berezin Transform, Annales de L'Institut Fourier (Grenoble) 57 (2007), No. 3, 693-702.","doi":""},{"text":"On the Characters of the Discrete Series. W Schmid, Inventiones Mathematicae. 30W. Schmid, On the Characters of the Discrete Series, Inventiones Mathematicae 30 (1975), 47-144.","doi":""},{"text":"Unitary Representations, Branching Rules, and Matrix Elements for the Non-Compact Symplectic Groups. D J Rowe, B G Whybourne, P H Butler, Journal of Physics A: Mathematical and General. 186D.J. Rowe, B.G. Whybourne, P.H. Butler, Unitary Representations, Branching Rules, and Matrix Elements for the Non-Compact Symplectic Groups, Journal of Physics A: Mathemat- ical and General, Volume 18, Number 6, 1985, 939-953.","doi":""},{"text":"An Extension of Plancherel's Formula to Separable Unimodular Groups. I E Segal, Annals of Mathematics. 2I.E. Segal, An Extension of Plancherel's Formula to Separable Unimodular Groups, Annals of Mathematics (2) 52 (1950), 272-292.","doi":""},{"text":"Multiplicity One Theorems: The Archimedean Case. B Sun, C.-B Zhu, Annals of Mathematics. 2B. Sun, C.-B. Zhu, Multiplicity One Theorems: The Archimedean Case, Annals of Mathe- matics (2) 175 (2012), no. 1, 23-44.","doi":""},{"text":"The Tempered Spectrum of a Real Semisimple Lie Group. P C Trombi, American Journal of Mathematics. 991P.C. Trombi, The Tempered Spectrum of a Real Semisimple Lie Group, American Journal of Mathematics 99 (1977), no. 1, 57-75.","doi":""},{"text":"J Vargas, Restriction of Some Discrete Series Representations, Algebras, Groups, and Geometries. 18J. Vargas, Restriction of Some Discrete Series Representations, Algebras, Groups, and Ge- ometries 18 (2001), 85-100.","doi":""},{"text":"J Vargas, Restriction of Discrete Series of a Semisimple Lie Group to Reductive Subgroups, New Developments in Lie Theory and Its Applications. Providence, RIAmerican Mathematical Society544J. Vargas, Restriction of Discrete Series of a Semisimple Lie Group to Reductive Subgroups, New Developments in Lie Theory and Its Applications, Contemporary Mathematics 544, American Mathematical Society, Providence, RI, 2011.","doi":""},{"text":"The Algebraic Structure of the Representations of Semisimple Lie Groups I. D Vogan, Annals of Mathematics. 2D. Vogan, The Algebraic Structure of the Representations of Semisimple Lie Groups I, Annals of Mathematics (2) 109 (1979), no. 1, 1-60.","doi":""},{"text":"N Wallach, On the Enright-Varadarajan Modules, Annales Scientifiques de L'Ecole Normale Superieure. 4N. Wallach, On the Enright-Varadarajan Modules, Annales Scientifiques de L'Ecole Normale Superieure 4 9 (1976), no. 1, 81-101.","doi":""},{"text":"N Wallach, Real Reductive Groups II. San DiegoAcademic PressN. Wallach, Real Reductive Groups II, Academic Press, San Diego, 1992.","doi":""},{"text":"R O Wells, Differential Analysis on Complex Manifolds. New YorkSpringer VerlagR.O. Wells, Differential Analysis on Complex Manifolds, Springer Verlag, New York, 1980.","doi":""}],"string":"[\n {\n \"text\": \"A Andreotti, E Vesentini, Carleman Estimates for the Laplace-Beltrami Equations on Complex Manifolds, Institute Hautes Etudes Scientifique Publications Mathematiques. 25A. Andreotti, E. Vesentini, Carleman Estimates for the Laplace-Beltrami Equations on Com- plex Manifolds, Institute Hautes Etudes Scientifique Publications Mathematiques 25 (1965), 313-362.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N Bergeron, Proprits de Lefschetz Automorphes pour les Groupes Unitaires et Orthogonaux, Mmoires de la Socit Mathmatique de France. 106125N. Bergeron, Proprits de Lefschetz Automorphes pour les Groupes Unitaires et Orthogonaux, Mmoires de la Socit Mathmatique de France 106 (2006), vi-125.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Branching Laws for Square Integrable Representations. M Duflo, J Vargas, Series A. 86M. Duflo, J. Vargas, Branching Laws for Square Integrable Representations, Proceedings of the Japan Academy 86 (2010), Series A, 49-54.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Dual Spaces of C * Algebras. J M G Fell, Transactions of the American Mathematical Society. 94J.M.G. Fell, The Dual Spaces of C * Algebras, Transactions of the American Mathematical Society 94 (1960), 365-403.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"G Folland, A Course in Abstract Harmonic Analysis. Boca RatonCRC PressG. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Characters of Semisimple Lie Groups. Harish-Chandra, Transactions of the American Mathematical Society. 83Harish-Chandra, The Characters of Semisimple Lie Groups, Transactions of the American Mathematical Society 83 (1956), 98-163.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spherical Functions on a Semisimple Lie Group I. Harish-Chandra, American Journal of Mathematics. 80Harish-Chandra, Spherical Functions on a Semisimple Lie Group I, American Journal of Mathematics 80 (1958), 553-613.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Harish-Chandra, Transactions of the American Mathematical Society. 119Invariant Eigendistributions on a Semisimple Lie GroupHarish-Chandra, Invariant Eigendistributions on a Semisimple Lie Group, Transactions of the American Mathematical Society 119 (1965), 457-508.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Harish-Chandra, Two Theorems on Semi-simple Lie Groups. Harish-Chandra, Two Theorems on Semi-simple Lie Groups, Annals of Mathematics (2) 83 (1966), 74-128.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Harish-Chandra, The Discrete Series for Semisimple Lie Groups II: Explicit Determination of the Characters. 116Harish-Chandra, The Discrete Series for Semisimple Lie Groups II: Explicit Determination of the Characters, Acta Mathematica 116 (1966), 1-111.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Harish-Chandra, Harmonic Analysis on Semisimple Lie Groups. 76Harish-Chandra, Harmonic Analysis on Semisimple Lie Groups, Bulletin of the American Mathematical Society 76 (1970), 529-551.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Harmonic Analysis on Real Reductive Groups I. Harish-Chandra, Journal of Functional Analysis. 192Harish-Chandra, Harmonic Analysis on Real Reductive Groups I, Journal of Functional Anal- ysis, 19 (1975), no. 2, 104-204.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Harish-Chandra, Harmonic Analysis on Real Reductive Groups III. Harish-Chandra, Harmonic Analysis on Real Reductive Groups III, Annals of Mathematics (2) 104 (1976), no. 1, 117-201.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Harish-Chandra, IIIDifferential Equations and Semisimple Lie Groups, Collected Papers. V.S. VaradarajanNew YorkSpringerHarish-Chandra, Differential Equations and Semisimple Lie Groups, Collected Papers, Vol- ume III, ed. V.S. Varadarajan, Springer, New York, 1984.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Groups and Geometric Analysis. S Helgason, Mathematical Surveys and Monographs. 83American Mathematical SocietyS. Helgason, Groups and Geometric Analysis, Mathematical Surveys and Monographs, Vol- ume 83, American Mathematical Society, Providence, RI, 1984.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On Realization of the Discrete Series for Semisimple Lie Groups. R Hotta, Journal of the Mathematical Society of Japan. 232R. Hotta, On Realization of the Discrete Series for Semisimple Lie Groups, Journal of the Mathematical Society of Japan 23 (1971), no. 2, 384-407.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Restrictions and Expansions of Holomorphic Representations. H P Jakobsen, M Vergne, Journal of Functional Analysis. 34H.P. Jakobsen, M. Vergne, Restrictions and Expansions of Holomorphic Representations, Journal of Functional Analysis 34 (1979), 29-53.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups and its applications. T Kobayashi, Inventiones Mathematicae. 1172T. Kobayashi, Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups and its applications, Inventiones Mathematicae 117 (1994), no. 2, 181-205.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups. II. Micro-local analysis and asymptotic K-support. T Kobayashi, Annals of Math. 2T. Kobayashi, Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups. II. Micro-local analysis and asymptotic K-support, Annals of Math (2) 147 (1998), no. 3, 709-729.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties. T Kobayashi, Inventiones Mathematicae. 1312T. Kobayashi, Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Inventiones Mathematicae 131 (1998), no. 2, 229-256.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A vanishing theorem for modular symbols on locally symmetric spaces. T Kobayashi, T Oda, Commentarii Mathematici Helvetici. 731T. Kobayashi, T. Oda, A vanishing theorem for modular symbols on locally symmetric spaces, Commentarii Mathematici Helvetici 73 (1998), no. 1, 45-70.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"T Kobayashi, T Oshima, arXiv:1108.3477Finite Multiplicity Theorems. PreprintT. Kobayashi, T. Oshima, Finite Multiplicity Theorems, Preprint: arXiv:1108.3477.\",\n \"doi\": \"arXiv:1108.3477\"\n },\n {\n \"text\": \"A Knapp, Representation Theory of Semisimple Lie Groups. Princeton, NJPrinceton University PressA. Knapp, Representation Theory of Semisimple Lie Groups, Princeton University Press, Princeton, NJ, 1986.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lie Groups Beyond an Introduction. A Knapp, Progress in Mathematics. 140BirkhauserA. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics, Volume 140, Birkhauser, Boston, MA, 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"F I Mautner, Unitary Representations of Locally Compact Groups II. F.I. Mautner, Unitary Representations of Locally Compact Groups II, Annals of Mathematics (2) 52 (1950), 528-556.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An Analogue of the Borel-Weil-Bott Theorem for Hermitian Symmetric Pairs of Non-Compact Type. M S Narasimhan, K Okamoto, Annals of Mathematics. 2M.S. Narasimhan, K. Okamoto, An Analogue of the Borel-Weil-Bott Theorem for Hermitian Symmetric Pairs of Non-Compact Type, Annals of Mathematics (2) 91 (1970), no. 3, 486-511.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Why Would Multiplicities be Log-Concave?. A Okounkov, The Orbit Method in Geometry and Physics. 213BirkhauserProgress in MathematicsA. Okounkov, Why Would Multiplicities be Log-Concave?, In \\\"The Orbit Method in Geometry and Physics\\\" (Merseille 2000), Progress in Mathematics Volume 213, Birkhauser, Boston, 2003.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Generalizations of the Bargmann Transform. G Ólafsson, B Ørsted, Lie Theory and Its Applications to Physics. River Edge, NJClausthal 1995G.Ólafsson, B. Ørsted, Generalizations of the Bargmann Transform, In \\\"Lie Theory and Its Applications to Physics\\\" (Clausthal 1995), 3-14, World Sci. Publ., River Edge, NJ, 1996.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Branching Laws for Some Unitary Representations of SL(4, R). B Ørsted, B Speh, SIGMA. 4Paper 017B. Ørsted, B. Speh, Branching Laws for Some Unitary Representations of SL(4, R), SIGMA 4 (2008), Paper 017.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"B Ørsted, J Vargas, Restriction of Square Integrable Representations: Discrete Spectrum. 123B. Ørsted, J. Vargas, Restriction of Square Integrable Representations: Discrete Spectrum, Duke Mathematical Journal 123 (2004), no. 3, 609-633.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a Generalized Berezin Transform. B Ørsted, J Vargas, Annales de L'Institut Fourier (Grenoble). 573B. Ørsted, J. Vargas, A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a Generalized Berezin Transform, Annales de L'Institut Fourier (Grenoble) 57 (2007), No. 3, 693-702.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the Characters of the Discrete Series. W Schmid, Inventiones Mathematicae. 30W. Schmid, On the Characters of the Discrete Series, Inventiones Mathematicae 30 (1975), 47-144.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Unitary Representations, Branching Rules, and Matrix Elements for the Non-Compact Symplectic Groups. D J Rowe, B G Whybourne, P H Butler, Journal of Physics A: Mathematical and General. 186D.J. Rowe, B.G. Whybourne, P.H. Butler, Unitary Representations, Branching Rules, and Matrix Elements for the Non-Compact Symplectic Groups, Journal of Physics A: Mathemat- ical and General, Volume 18, Number 6, 1985, 939-953.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An Extension of Plancherel's Formula to Separable Unimodular Groups. I E Segal, Annals of Mathematics. 2I.E. Segal, An Extension of Plancherel's Formula to Separable Unimodular Groups, Annals of Mathematics (2) 52 (1950), 272-292.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Multiplicity One Theorems: The Archimedean Case. B Sun, C.-B Zhu, Annals of Mathematics. 2B. Sun, C.-B. Zhu, Multiplicity One Theorems: The Archimedean Case, Annals of Mathe- matics (2) 175 (2012), no. 1, 23-44.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Tempered Spectrum of a Real Semisimple Lie Group. P C Trombi, American Journal of Mathematics. 991P.C. Trombi, The Tempered Spectrum of a Real Semisimple Lie Group, American Journal of Mathematics 99 (1977), no. 1, 57-75.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Vargas, Restriction of Some Discrete Series Representations, Algebras, Groups, and Geometries. 18J. Vargas, Restriction of Some Discrete Series Representations, Algebras, Groups, and Ge- ometries 18 (2001), 85-100.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"J Vargas, Restriction of Discrete Series of a Semisimple Lie Group to Reductive Subgroups, New Developments in Lie Theory and Its Applications. Providence, RIAmerican Mathematical Society544J. Vargas, Restriction of Discrete Series of a Semisimple Lie Group to Reductive Subgroups, New Developments in Lie Theory and Its Applications, Contemporary Mathematics 544, American Mathematical Society, Providence, RI, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Algebraic Structure of the Representations of Semisimple Lie Groups I. D Vogan, Annals of Mathematics. 2D. Vogan, The Algebraic Structure of the Representations of Semisimple Lie Groups I, Annals of Mathematics (2) 109 (1979), no. 1, 1-60.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N Wallach, On the Enright-Varadarajan Modules, Annales Scientifiques de L'Ecole Normale Superieure. 4N. Wallach, On the Enright-Varadarajan Modules, Annales Scientifiques de L'Ecole Normale Superieure 4 9 (1976), no. 1, 81-101.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"N Wallach, Real Reductive Groups II. San DiegoAcademic PressN. Wallach, Real Reductive Groups II, Academic Press, San Diego, 1992.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R O Wells, Differential Analysis on Complex Manifolds. New YorkSpringer VerlagR.O. Wells, Differential Analysis on Complex Manifolds, Springer Verlag, New York, 1980.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":585,"cells":{"text":{"kind":"string","value":"\nTHE CHANDRA COSMOS LEGACY SURVEY: CLUSTERING OF X-RAY SELECTED AGN AT 2.9≤z≤5.5 USING PHOTOMETRIC REDSHIFT PROBABILITY DISTRIBUTION FUNCTIONS\n16 Sep 2016\n\nV Allevato \nF Civano \nA Finoguenov \nS Marchesi \nF Shankar \nG Zamorani \nG Hasinger \nM Salvato \nT Miyaji \nR Gilli \nN Cappelluti \nM Brusa \nH Suh \nG Lanzuisi \nB Trakhtenbrot \nR Griffiths \nC Vignali \nK Schawinski \nA Karim \nTHE CHANDRA COSMOS LEGACY SURVEY: CLUSTERING OF X-RAY SELECTED AGN AT 2.9≤z≤5.5 USING PHOTOMETRIC REDSHIFT PROBABILITY DISTRIBUTION FUNCTIONS\n16 Sep 2016Accepted for publication in The Astrophysical JournalAccepted for publication in The Astrophysical Journal Preprint typeset using L A T E X style emulateapj v. 5/2/11Subject headings: Surveys -Galaxies: active -X-rays: general -Cosmology: Large-scale structure of Universe -Dark Matter\nWe present the measurement of the projected and redshift space 2-point correlation function (2pcf) of the new catalog of Chandra COSMOS-Legacy AGN at 2.9≤z≤5.5 ( L bol ∼10 46 erg/s) using the generalized clustering estimator based on phot-z probability distribution functions (Pdfs) in addition to any available spec-z. We model the projected 2pcf estimated using π max = 200 h −1 Mpc with the 2-halo term and we derive a bias at z∼3.4 equal to b = 6.6 +0.60 −0.55 , which corresponds to a typical mass of the hosting halos of log M h = 12.83 +0.12 −0.11 h −1 M ⊙ . A similar bias is derived using the redshiftspace 2pcf, modelled including the typical phot-z error σ z = 0.052 of our sample at z≥2.9. Once we integrate the projected 2pcf up to π max = 200 h −1 Mpc, the bias of XMM and Chandra COSMOS at z=2.8 used inAllevato et al. (2014)is consistent with our results at higher redshift. The results suggest only a slight increase of the bias factor of COSMOS AGN at z 3 with the typical hosting halo mass of moderate luminosity AGN almost constant with redshift and equal to logM h = 12.92 +0.13 −0.18 at z=2.8 and log M h = 12.83 +0.12 −0.11 at z∼3.4, respectively. The observed redshift evolution of the bias of COSMOS AGN implies that moderate luminosity AGN still inhabit group-sized halos at z 3, but slightly less massive than observed in different independent studies using X-ray AGN at z≤ 2.\n\nINTRODUCTION\n\nThe presence of a nuclear supermassive black hole (BH) in almost all galaxies in the present day Universe is an accepted paradigm in astronomy (e.g. Kormendy & Richstone 1995;Kormendy & Bender 2011). Despite major observational and theoretical efforts over the last two decades, a clear explanation for the origin and evolution of BHs and their actual role in galaxy evolution remains elusive. Diverse scenarios have been proposed. One pos-sible picture includes major galaxy merger as the main triggering mechanism (e.g. Hopkins et al. 2006;Volonteri et al. 2003, Menci et al. 2003,2004. On the other hand, there is mounting observational evidence suggesting that moderate levels of AGN activity might not be always causally connected to galaxy interactions (Lutz et al. 2010, Mullaney et al. 2012, Rosario et al. 2013, Villforth et al. 2014. Several works on the morphology of the AGN host galaxies suggest that, even at moderate luminosities, a large fraction of AGN is not associated with morphologically disturbed galaxies. This trend has been observed both at low (z ∼ 1, e.g., Georgakakis et al. 2009;Cisternas et al. 2011) and high (z ∼ 2, e.g., Schawinski et al. 2011Schawinski et al. , 2012Kocevski et al. 2012, Treister et al. 2012 redshift. Theoretically, in-situ processes, such as disk instabilities or stochastic accretion of gas clouds, have also been invoked as triggers of AGN activity (e.g. Genzel et al. 2008, Dekel et al. 2009, Bournaud et al. 2011.\n\nAGN clustering analysis provides a unique way to unravel the knots of this complex situation, providing important, independent constraints on the BH/galaxy formation and co-evolution. In the cold dark matterdominated Universe galaxies and their BHs are believed to populate the collapsed dark matter halos, thus reflecting the spatial distribution of dark matter in the Universe. The most common statistical estimator for large-scale clustering is the two-point correlation function (2pcf, Davis & Peebles 1983). This quantity measures the excess probability above random to find pairs of galaxies/AGN separated by a given scale r. By matching the observed 2pcf to detailed outputs of dark matter numerical simulations, one can infer the typical mass of Mean normalized phot-z Pdf of all CCL AGN with best-fit phot-z > 2.9. Right Panel: Normalized phot-z Pdf for the source lid766. This source has a best-fit photo-z value < 2.9, but a phot-z Pdf(z i >2.9)>0.001 (red thick line). The redshifts above this threshold, weighted by their Pdf, have been taken in account in the catalog used to estimate the 2pcf. the hosting dark matter halos. This is derived through the so called AGN bias b, enabling then to pin down the typical environment where AGN live. This in turn can provide new insights into the physical mechanisms responsible for triggering AGN activity.\n\nThe 2pcf of AGN has been measured in optical large area surveys, such as the 2dF (2QZ, Croom et al. 2005;Porciani & Norberg 2006) and the Sloan Digital Sky Survey (SDSS, Li et al. 2006;Shen et al. 2009;Ross et al. 2009). These optical surveys are thousands of square degree fields, mainly sampling rare and high luminosity quasars. The amplitude of the 2PCF of quasars suggests that these luminous AGN are hosted by halos of roughly constant mass, a few times 10 12 M ⊙ , out to z =3-4 (Shanks et al. 2011). Models of major mergers between gas-rich galaxies appear to naturally reproduce the clustering properties of optically selected quasars as a function of luminosity and redshift (Hopkins et al. 2007a(Hopkins et al. , 2008Shen 2009;Bonoli et al. 2009). This supports the scenario in which major mergers dominate the luminous quasar population (Scannapieco et al. 2004;Neistein & Netzer 2014;Treister et al. 2012).\n\nChandra surveys have contributed significantly to the study of the AGN clustering (e.g. CDFS-N, Gilli et al. 2009; Chandra/Bootes, Starikova et al. 2011, Allevato et al. 2014. Deep X-ray data can be used to draw conclusions on the faint portions of the AGN luminosity function, where a significant fraction of obscured sources is present. In particular, the Chandra survey in the two square degree COSMOS field (C-COSMOS, Elvis et al. 2009, Civano et al. 2012; Chandra COSMOS Legacy Survey, Civano et al. 2016) has allowed the investigation of the redshift evolution of the clustering properties of Xray AGN, for the first time up to z∼3. Interestingly, over a broad redshift range (z ∼ 0 -2) moderate luminosity AGN occupy DM halo masses of log M h ∼ 12.5-13.5 M ⊙ h −1 . The clustering strength of X-ray selected AGN has been measured by independent studies to be higher than that of optical quasars. Merger models usually fail in reproducing the data from X-ray surveys, opening the possibility of additional AGN triggering mechanisms (e.g., Allevato et al. 2011;Mountrichas & Georgakakis 2012) and/or multiple modes of BH accretion (e.g., Fanidakis et al. 2013). Recently, Mendez et al. (2015) and Gatti et al. (2016) have suggested that selection cuts in terms of AGN luminosity, host galaxy properties and redshift interval, might have a more relevant role in driving the differences often observed in the bias factor inferred from different surveys. The measurement of the AGN bias is crucial at high redshifts, especially at z>2-3, i.e. at the peak in the accretion history of the Universe. At z>3, Shen et al. (2007Shen et al. ( ,2009) measured for the first time the 2pcf of luminous SDSS-DR5 quasars (log L bol ∼ 10 47 erg/s) at z = 3.2 and 3.8. Even if with very large uncertainty, they found that these objects live in massive halos of the order of 10 13 M ⊙ h −1 . This result is consistent with models invoking galaxy major mergers as the main triggering mechanism for very luminous AGN. Recently, Eftekharzadeh et al. (2015) studying a sample of spectroscopically confirmed SDSS-III/BOSS quasars at 2.2≤z≤3.4, performed a more precise estimation of the quasar bias at high redshift. They found no evolution of the bias in three redshift bins, with halo masses equal to 3× and ∼0.6× 10 12 M ⊙ h −1 at z∼2.3 and ∼ 3, respectively.\n\nThere are only a few attempts of measuring the clustering properties of X-ray AGN at z ∼ 3. Francke et al. (2008) estimated the bias of a small sample of X-ray AGNs (L bol ∼ 10 44.8 erg s −1 ) in the Extended Chandra Deep Field South (ECDFS), with very large uncertainty. They found indications that X-ray ECDFS AGNs reside in dark matter halos with minimum mass of log M min = 12.6 +0.5 −0.8 h −1 M ⊙ . On the other hand, Allevato et al. (2014) used a sample of Chandra and XMM-Newton AGN in COSMOS with moderate luminosity (log L bol ∼ 10 45.3 erg/s) at z =2.86. For the first time they estimated the bias of X-ray selected AGN at high redshift, suggesting that they inhabit halos of logM h = 12.37±0.10 M ⊙ h −1 . They also extended to z∼3 the result that Type 1 AGN reside in more massive halos than Type 2 AGN. Recently, Ikeda et al. (2015) estimated the clustering properties of low-luminosity quasars in COS-MOS at 3.1≤z≤4.5, using the cross-correlation between Lyman-Break Galaxies (LBGs) and 25 quasars with spectroscopic and photometric redshifts. They derived a 86% upper limit of 5.63 for the bias at z∼ 4.\n\nIn this paper we want to extend the study of the clus- tering properties of X-ray selected AGN to z>3 using the new Chandra COSMOS-Legacy data. To this goal, we perform clustering measurements using techniques based on photometric redshift in the form of probability distribution functions (Pdfs), in addition to any available spectroscopy. This is motivated by the development in the last years of clustering measurement techniques based on photometric redshift Pdfs by Myers, White & Ball (2009), Hickox et al. (2011 and Mountrichas et al. (2013) and Georgakakis et al. (2014). One of the advantages of this new clustering estimator is that one can use in the analysis all sources not just the optically brighter ones for which spectroscopy is available. For this reason it is well suited to clustering investigations using future large X-ray AGN surveys, where the fraction of spectroscopic redshifts might be small.\n\nThroughout the paper, all distances are measured in comoving coordinates and are given in units of Mpc h −1 , where h = H 0 /100km/s. We use a ΛCDM cosmology with Ω M = 0.3, Ω Λ = 0.7, Ω b = 0.045, σ 8 = 0.8. The symbol log signifies a base-10 logarithm. (Civano et al. 2016) for a total coverage of 2.2 deg 2 of the COSMOS field (Scoville et al. 2007). The X-ray source catalog consists of 4016 sources. 2076 (∼52%) have a secure spectroscopic redshift (spec-z) and for ∼96% the photometric redshift (photo-z) is available. As shown in Marchesi et al. (2016a), the spectroscopic redshifts have been obtained with different observing programs, as the zCOSMOS survey (Very Large Telescope/VIMOS; Lilly et al. 2007) and the Magellan/IMACS survey (Trump et al. 2007(Trump et al. , 2009). Other programs, many of which have been specifically targeting the CCLS have been The photo-zs are estimated following the procedure described in Salvato et al. (2011). Following Marchesi et al. (2016b), the accuracy of the photometric redshifts with respect to the whole spectroscopic redshift sample is σ ∆z/(1+zspec) =0.02, with a fraction of outliers ≃11%. At z>2.9 there are 9 outliers (∆z/(1 + z spec ) > 0.15), but for the remaining sources the agreement between specz and photo-z has the same quality of the whole sample. In detail, the normalized median absolute deviation σ N MAD =1.48×median( zspec− zphot /(1 + zspec)) = 0.012.\n\nThe CCL AGN sample at 2.9<z<5.5 consists of 212 AGN detected in the 0.5-10 keV band, 107/212 with spec-zs and 105/212 with only phot-zs. To each of the 105 AGN with best-fit phot-z in the range 2.9≤z<5.5, is associated a probability distribution function (Pdf), which gives the probability of the source to be in the redshift range z i ± ∆z/2 with a binsize of ∆z = 0.01. The integrated area of the Pdf on all redshift bins z i is normalized to 1, i.e. i P df (z i ) = 1 for each AGN. We take into account for this analysis the redshift bins z i with Pdf(z i ) >0.001. Figure 1 shows the mean normalized phot-z Pdf for 105 sources with best-fit phot-z 2.9≤z<5.5. The effective contribution to the number of AGN at z≥2.9 of these 105 AGN weighted by the Pdf is 78.32 sources, i.e. 105 j=1 P df j (z ≥ 2.9)=78.32. In the CCL sample there are also 246 sources with phot-z < 2.9 but which contribute to the Pdf at 2.9≤ z ≤ 5.5 (i.e., with Pdf>0.001 at 2.9 ≤ z i ≤5.5 (see for example the Pdf of source lid766, whose nominal best-fit phot-z value is 2.51, in Figure 1). All these 246 sources have been taken in account in our analysis, using for each of them the Pdf of each bin of redshift 2.9 ≤ z i ≤5.5. The weighted contribution of these sources, i.e. the sum of all weights, is equal to 36.3 AGN (\nSample N ΣPdf j (z≥2.9) z log L bol b logM h b logM h erg s −1 Eq. 7 h −1 M ⊙ Eq. 14 h −1 M ⊙246 j=1 P df j (z ≥ 2.9)=36.3).\nTo all the 107 sources with known spec-z we assign a Pdf j = 1 to the spec-z value ( 107 j=1 P df j =107). To summarize, the total effective number of CCL AGN at 2.9≤z<5.5 weighted by the Pdf and used for the clustering measurements is 78.3 + 36.3 + 107 = 221.6 objects. Figure 2 shows the normalized redshift and 2-10 keV rest-frame X-ray luminosity distribution for our sample of CCL AGN at 2.9<z<5.5, when the phot-z Pdfs are used (black dotted line, z =3.36). The mean bolometric luminosity of this sample derived using the bolometric correction defined in Equation (21) of Marconi et al. (2004) is log L bol = 45.99 erg s −1 . For comparison, we also show the normalized distributions of our AGN sample when only the best-fit phot-zs are taken into account in addition to any available spec-z (gray solid line, z =3.34) and for the sample with known spec-z (reddashed line, z =3.35).\n\n3. 2PCF USING PHOT-z PDFS 3.1. Projected 2pcf The most commonly used quantitative measure of large scale structure is the 2pcf, ξ(r), which traces the amplitude of AGN clustering as a function of scale. ξ(r) is defined as a measure of the excess probability dP , above what is expected for an unclustered random Poisson distribution, of finding an AGN in a volume element dV at a separation r from another AGN:\ndP = n[1 + ξ(r)]dV(1)\nwhere n is the mean number density of the AGN sample (Peebles 1980). Measurements of ξ(r) are generally performed in comoving space, with r having units of h −1 Mpc. With a redshift survey, we cannot directly measure ξ(r) in physical space, because peculiar motions of galaxies distort the line-of-sight distances inferred from redshift. To separate the effects of redshift distortions, the spatial correlation function is measured in two dimensions r p and π, where r p and π are the projected comoving separations between the considered objects in the directions perpendicular and parallel, respectively, to the mean line-of-sight between the two sources. Following Davis & Peebles (1983), r 1 and r 2 are the redshift positions of a pair of objects, s is the redshift-space separation (r 1 − r 2 ), and l = 1 2 (r 1 + r 2 ) is the mean distance to the pair. The separations between the two considered objects across r p and π are defined as:\nπ = s · l |l| (2) r p = (s · s − π 2 )(3)\nRedshift space distortions only affect the correlation function along the line of sight, so we estimate the socalled projected correlation function w p (r p ) (Davis & Peebles 1983):\nw(σ) = 2 πmax 0 ξ(σ, π)dπ(4)\nwhere ξ(r p , π) is the two-point correlation function in terms of r p and π, measured using the Landy & Szalay (1993, LS) estimator:\nξ = 1 RR ′ [DD ′ − 2DR ′ + RR ′ ](5)\nwhere DD', DR' and RR' are the normalized data-data, data-random and random-random pairs. In this classic approach of estimating the redshiftspace correlation function, in presence of accurate speczs, when a data-data pair with separation (r p , π) is found, the pair number is incremented by one, i.e. DD(r p ,π) = DD(r p , π) + 1. Following Georgakakis et al. (2014), in the generalized clustering estimator the number of datadata pairs with projected and line of sight separation (r p , π) is, instead, incremented by the product Pdf 1 (z i ) × Pdf 2 (z j ):\nDD(r p , π) = DD(r p , π) + P df 1 (z i ) × P df 2 (z j )(6)\nwhere Pdf 1 (z i ) and Pdf 2 (z j ) are the Pdf values (per redshift bin) of the source 1 at z = z i and of the source 2 at z = z j respectively.\n\nThe measurements of the 2pcf requires the construction of a random catalog with the same selection criteria and observational effects as the data. To this end, we constructed a random catalog where each simulated source is placed at a random position in the sky, with its flux randomly extracted from the catalog of real source fluxes. The simulated source is kept in the random sample if its flux is above the sensitivity map value at that random position (Miyaji et al. 2007;Cappelluti et al. 2009). The corresponding redshift for each random object is then assigned based on the smoothed redshift distribution of the AGN sample, where each redshift is weighted by the Pdf associated to that redshift for the particular source. Since the phot-z Pdfs are already Bias as a function of πmax for 221.6 CCL AGN at 2.9≤z<5.5, when the projected 2pcf is measured using the generalized clustering estimator based on phot-z Pdfs in addition to any available spec-z (black dots). For comparison, the gray triangles and red squares show the bias of 212 AGN with known spec or best-fit phot-z and 107 AGN with known spec-zs at 2.9<z<5.5 when the classic LS estimator (i.e. no phot-z Pdfs) is used. Right Panel: Projected 2pcf for πmax=200 h −1 Mpc. The 1σ errors on wp(rp) are the square root of the diagonal components of the covariance matrix. The continuous line represents the DM projected 2pcf estimated at z =3.36.\n\ntaken into account in the generation of the random redshifts, we decided to assign Pdf=1 to each random source.\n\nIn the halo model approach, the large scale amplitude signal is due to the correlation between objects in distinct halos and the bias parameter defines the relation between the large scale clustering amplitude of the AGN correlation function and the DM 2-halo term:\nb 2−h (r p ) = (w AGN (r p )/w 2−h DM (r p )) 1/2(7)\nWe first estimated the DM 2-halo term at the median redshift of the sample, using:\nw 2−h DM (r p ) = 2 ∞ rp ξ 2−h DM (r)rdr r 2 − r 2 p(8)\nintegrating up to 200 h −1 Mpc along π, where:\nξ 2−h DM (r) = 1 2π 2 P 2−h (k)k 2 sin(kr) kr dk (9) P 2−h (k)\nis the linear power spectrum, assuming a power spectrum shape parameter Γ = Ω m h = 0.2 (Efstathiou et al. 1992) which corresponds to h = 0.7.\n\n3.2. z-space correlation function Similarly, we can estimate the z-space correlation function ξ(s) using Equation (5) and (6), written now as a function of the redshift-space separation s = (r 1 − r 2 ) between the sources. ξ(s) is affected by perturbations in the cosmological redshifts due to peculiar velocities and redshift errors. The z-space power spectrum can be modelled in polar coordinates as follow (e.g. Kaiser 1987, Peacock et al. 2001:\nP (k, µ) = P DM (k)(b + f µ 2 ) 2 exp(−k 2 µ 2 σ 2 )(10)\nwhere k = k 2 ⊥ + k 2 , k ⊥ and k are the wavevector components perpendicular and parallel to the line of sight, respectively. µ = k ⊥ /k , P DM (k) is the dark matter power spectrum, b is the linear bias factor, f is the growth rate of density fluctuations. σ is the displacement along the line of sight due to random perturbations of cosmological redshifts.Assuming standard gravity, we approximated the growth rate f ≃ Ω M (z) γ , with γ = 0.545 (e.g. Sereno et al. 2015).\n\nThe f µ 2 term parametrizes the coherent motions due to large-scale structures, enhancing the clustering signal on all scales. The exponential cut-off term describes the random perturbations of the redshifts caused by both non-linear stochastic motions and redshift errors. The integration of Eq.(7) over the angle µ, and then the Fourier anti-transformation gives:\nξ(s) = b 2 ξ ′ (s) + bξ ′′ (s) + ξ ′′′ (s)(11)\nThe main term, ξ ′ (s), is the Fourier anti-transform of the monopole P ′ (k):\nP ′ (k) = P DM (k) √ π 2kσ erf (kσ),(12)\nthat corresponds to the model given by Equation (10) when neglecting the dynamic distortion term. In our case, photo-z errors perturb the most the distance measurements along the line of sight. Therefore the small-scale random motions are negligible with respect to photo-z errors. The cut-off scale in Eq. (12) can thus be written as:\nσ = cσ z H(z)(13)\nwhere H(z) is the Hubble function computed at the median redshift of the sample, and σ z is the typical photo-z error.\n\nIn this case, knowing the cut-off scale, the AGN bias can be derived from the Fourier anti-transform of the monopole P ′ (k), i.e.:\nb 2 = ξ ′ (s) ξ(s)(14)\nwhere ξ(s) is the observed z-space 2pcf of our AGN sample.\n\n4. RESULTS 4.1. w p (r p ) and Bias We have measured the 2pcf of 221.6 CCL AGN at 2.9≤z<5.5, using the generalized clustering estimator defined in Equation (6), based on phot-z Pdfs in addition to any available spec-z. The projected 2pcf w p (r p ) is then estimated using Equation (4).\n\nThe typical value of π max used in clustering measurements of both optically-selected luminous quasars and X-ray selected AGN is ∼20-100 h −1 Mpc (e.g. Zehavi et al. 2005, Coil et al. 2009, Krumpe et al. 2010, Allevato et al. 2011. The optimum π max value can be determined by measuring the 2pcf for different π max and then adopting the value at which the amplitude of the signal appears to level off. Figure 3 (Left Panel) shows the bias factor estimated for different values of π max in Equation (4), when the phot-z Pdfs are used in addition to any available specz. For comparison, we also estimated the bias for case i) 107 AGN with known spec-zs; case ii) 107+105 AGN with known spec-z or best-fit phot-zs. In these cases the 2pcf is measured using the classic LS estimator and the Pdf is set to unity for each source.\n\nAs expected, when including phot-zs in the analysis, the bias levels-off only at large scales, because of the large uncertainties in the redshifts measured via photometric methods (Georgakakis et al. 2014). Surprisingly, even if the error bars are large, an increase of the bias factor at π max >100 h −1 Mpc is suggested also when only spec-zs are used with the classic 2pcf estimator. This suggests that a fraction of spec-zs might be affected by large errors (see Section 4.2).\n\nThe amplitude of the projected 2pcf of CCL AGN measured using the generalized clustering estimator converges at π max ≥ 200 h −1 Mpc. We decide to use π max = 200 h −1 Mpc in order to balance the advantage of integrating out redshift-space distortions against the disadvantage of introducing noise from uncorrelated line-ofsight structure. Figure 3 (Right Panel) shows the projected 2pcf estimated using the generalized clustering estimator, whith π max =200 h −1 Mpc. The 1σ errors on w p (r p ) are the square root of the diagonal components of the covariance matrix (Miyaji et al. 2007, Krumpe et al. 2010) estimated using the bootstrap method. The latter quantifies the level of correlation between different bins. For comparison, we also estimate the projected 2pcf for case i) 107 AGN with known spec-zs; case ii) 107+105 AGN with known spec-z or best-fit phot-zs. Note that in these cases the classic LS estimator is used (i.e. Pdf = 1 for each source) and π max is fixed to 200 h −1 Mpc also in these cases. Following Eq. 7, we derive the best-fit bias by using a χ 2 minimization technique with 1 free parameter in the range r p = 1 -30 h −1 Mpc, where χ 2 = ∆ T M −1 cov ∆. In detail, ∆ is a vector composed of w p (r p ) − w mod (r p ) (see Equations 4 and 7), ∆ T is its transpose and M −1 cov is the inverse of covariance matrix. The latter full covariance matrix is used in the fit to take into account the correlation between errors.\n\nAs shown in Table 1, we derived a bias for our sample of CCL AGN equal to b = 6.6 +0.60 −0.55 at z =3.36. Following the bias-mass relation b(M h , z) described in van den Bosch (2002) and Sheth et al. (2001), the AGN bias corresponds to a typical mass of the hosting halos of log M h = 12.83 +0.12 −0.11 h −1 M ⊙ . It is worth noticing that this is a typical/characteristic mass of the halos hosting CCL AGN. Only the HOD modelling of the clustering signal at all scales can provide the entire hosting halo mass distribution for this sample.\n\nThe bias has a larger uncertainty when derived from the 2pcf estimated without the phot-z Pdfs. In detail, we find b = 7.5 +1.6 −1.7 at z =3.35 for 107 AGN with known spec-zs (case i) and b = 6.48 +1.27 −1.36 at z =3.34 for 107+105 AGN with known spec or best-fit phot-zs (case ii). Note that in these cases the 2pcf is measured using the classic LS estimator and the Pdf is set to unity for each source.\n\n\nξ(s) and phot-z errors\n\nTo investigate the effect of phot-z errors on the 2pcf, we also measured the z-space correlation function ξ(s). dition to any available spec-z. For comparison, the gray triangles show ξ(s) for case i) and case ii).\n\nAs described in section 3.2, ξ(s) is affected by the Kaiser effect that enhances the clustering signal at all scales and by phot-z errors, that are modelled by using an exponential cut-off in the z-space power spectrum. For our sample of CCL AGN the typical error on phot-zs is σ z =0.012×(1 + z spec ) = 0.052 at z =3.4. This implies a cut-off scale σ = 43.45 h −1 Mpc (see Equation 13). Including the phot-z damping in the modeling of ξ(s), we derived the best-fit bias by using Equation 14 and the χ 2 minimization technique with 1 free parameter in the range s=1-50 h −1 Mpc. In particular, for the z-space 2pcf measured using the generalized clustering estimator we derived b = 6.53 +0.52 −0.58 . For the sample of 212 AGN with known spec or best-fit phot-zs for which the classic clustering estimator is used, we derived b = 6.96 +0.72 −0.73 . These values are in perfect agreement with the bias obtained from the projected 2pcf with π max = 200 h −1 Mpc. This confirms that the convergence of the projected 2pcf observed only at large scales (π max ≥200 h −1 Mpc) is due to large phot-z errors. In fact, for large redshift errors and small survey area it is necessary to integrate the correlation function up to large scales to fully correct for them.\n\nWe also estimated the bias using the z-space 2pcf for 107 AGN with known spec-zs. In general, we do not expect spec-zs to be affected by large errors. If we do not include the phot-z damping in the model of ξ(s), we obtain for this sample a bias b = 5.86 +1.13 −1.05 , which is lower than b = 7.5 +1.6 −1.7 obtained using the projected 2pcf and π max = 200 h −1 Mpc. A better fit of ξ(s) and a larger bias can be obtained only if we include in the model specz errors of the order of σ z = 0.02-0.025. In particular, for σ z = 0.020 (cut-off scale σ = 16.7 h −1 Mpc) and σ z = 0.025 (σ = 20.9 h −1 Mpc), we derived b = 7.4 +1.35 −1.40 and b = 8.01 +1.4 −1.5 , respectively. However, given the low statistics, smaller values of σ z and then of the bias can not be ruled-out. This error redshift is larger than what is expected for a spectroscopic sample of AGN. However, the presence in the spectroscopic sample of ∼20% of the objects (21/107) with a low quality flag (i.e. flag = 1.5, corresponding to low quality spectra, and therefore not fully reliable redshift, but with known phot-z such that σ ∆z/(1+zspec) <0.1) could explain both the improvement of the fit including such an error in the analysis and the increase of the bias with increasing π max up to ∼200 h −1 Mpc when using only AGN with spectroscopic redshift.\n\n\nDISCUSSION\n\nIn this section we compare our results with previous measurements using COSMOS AGN at z∼3 and with previous studies at similar redshift. We also interpret our results in terms of AGN triggering mechanisms. Figure 5 (right panel) shows the redshift evolution of the AGN bias estimated using moderate luminosity AGN detected in different X-ray surveys. Interestingly, moderate luminosity AGN occupy DM halo masses of log M h ∼ 12.5-13.5 M ⊙ h −1 up to z∼2, tracing a constant group-sized halo mass. Allevato et al. (2011) have shown that XMM COSMOS AGN (L bol ∼ 10 45.2 erg s −1 ) reside in DM halos with constant mass equal to logM h = 13.12±0.07 M ⊙ h −1 up to z = 2. They also argue that this high bias can not be reproduced assuming that major merger between gas-rich galaxies (Shen 2009) is the main triggering scenario for moderate luminosity AGN. By contrast, at z∼3, Allevato et al. (2014) found a drop in the mass of the hosting halos, with Chandra and XMM-Newton COSMOS AGN (L bol ∼ 10 45.3 erg s −1 ), inhabiting halos of logM h = 12.37±0.10 M ⊙ h −1 .\n\n\nRedshift evolution of the AGN bias\n\nIn the present paper we measure a bias for 221.6 CCL AGN at 2.9≤z≤5.5 equal to b = 6.6 +0.6 −0.55 , that corresponds to a typical mass of the hosting dark matter halos of logM h = 12.83 +0.12 −0.11 h −1 M ⊙ . This result suggests a higher bias for CCL AGN compared to previous studies in COSMOS at z∼ 3. In fact, Allevato et al. (2014) found a bias of 3.85 +0.22 −0.21 at z =2.8, using a sample of XMM and Chandra AGN. Although the two samples only partially overlap, we argue that the most likely explanation of these differences lies in the small value of π max (= 40 h −1 Mpc) used in Allevato et al. (2014). As shown in Figure 3, the bias strongly increases with π max due to the large phot-z errors and the use of 40 h −1 Mpc might produce an underestimated clustering signal.\n\nTo verify this effect, we took the same sample used in Allevato et al. (2014), i.e. 346 XMM and Chandra COS-MOS AGN with known spec or phot-z > 2.2. As already mentioned, Allevato et al. (2014) used the classic LS estimator where the phot-zs Pdfs are not taken into account. Using their same classic approach, we estimated the projected 2pcf for different values of π max and found that the clustering signal converges only at π max ≥ 200 h −1 Mpc. In particular, we derived a bias b = 5.8 +0.61 −0.65 for π max = 200 h −1 Mpc, which corresponds to a typical hosting halo mass logM h = 12.92 +0.13 −0.18 . As shown in Figure 5 (left panel) the bias estimated using π max = 200 h −1 Mpc is in agreement with the bias of CCL AGN at z∼3.4 as derived in the present work. It is worth noticing that also the mean luminosity of the samples is increasing with z, with mean L bol = 10 45.5 erg/s for XMM and Chandra COSMOS AGN at z∼3 and ∼46 erg/s for CCL AGN at higher z.\n\nThese results imply that at z>3: i) the typical hosting halo mass of moderate luminosity AGN remains almost constant with redshift, going from ∼8.3 × 10 12 at z=2.8 to ∼ 6.7× 10 12 M ⊙ h −1 at z∼3.4, since a lower mass is required to yield the same bias at a higher redshift; ii) moderate luminosity AGN still inhabit group-sized halos at high redshift, but slightly less massive than observed in different independent studies using X-ray selected AGN at z≤2.\n\n\nPrevious studies at high redshift\n\nThe evolution of the bias with redshit has been studied in Eftekharzadeh et al. (2015) for SDSS-III/BOSS quasars at 2.2 ≤z ≤3.4. They investigated the redshift dependence of quasar clustering in three redshift bins and found no evolution of the correlation length and bias. In terms of halo mass, this corresponds to a characteristic halo mass that decreases with redshift, with halo masses of 3× (6×) and ∼0.6× (1.3×) 10 12 M ⊙ h −1 at z∼2.3 and ∼ 3, respectively, where the dark matter halo masses are estimated using Tinker et al. 2010(Sheth et al. 2001).\n\nThese results are surprisingly different in terms of bias Shen (2009) at z = 3 (in black) and z = 3.5 (in gold), Hopkins et al. (2007b) at z=3 (blue dotted line) and Conroy & White (2013) at z=3.4 (green dashed line) compared to previously estimated bias factors for optically selected quasars at z=3.2 , gray hexagon), X-ray selected AGN at z=3 (Francke et al. 2008, open triangle), Chandra and XMM-COSMOS AGN at z=2.9 as estimated in Allevato et al. (2014, filled black square) and as re-estimated in the present work (black empty square) and our results (red circle). The errors on the L bol axis correspond to the dispersion of the bolometric luminosity distributions for the different subsets. and halo mass when compared to Shen et al. (2009). At z∼3, Eftekharzadeh et al. (2015) derived halo masses that are close to an order-of-magnitude smaller than those presented in Shen et al. (2009). In this latter study, they measured the bias of SDSS-DR5 quasars with mean L bol ∼ 10 47 erg/s, at z = 3.2. Even if with large uncertainties, their results suggest that luminous quasars reside in massive halos with mass few times 10 13 h −1 M ⊙ (based on Sheth et al. 2001). Although the two samples do not completely overlap, Eftekharzadeh et al. (2015) argue that the most likely explanation of these differences lies in the improvements in SDSS photometric calibration and target selection algorithms as well as in the much larger number of quasars that afford greater measurements precision compared to Shen et al. (2009). Our results are in disagreement with the bias factor equal to b = 3.57±0.09 at z∼3 derived in Eftekharzadeh et al. (2015). This disagreement might be due to the slightly different average redshift and the significantly different luminosity (almost one order of magnitude) of the samples used in the two different studies. An additional important difference between the samples is that our catalog of CCL AGN includes both Type 1 and Type 2 AGN, while Eftekharzadeh et al. (2015) use Type 1 BOSS luminous quasars. It is also worth noticing that in Eftekharzadeh et al. (2015) the bias is derived modelling the z-space 2pcf in the Kaiser formalism, i.e. not including the effect of random peculiar velocities and redshift errors. The r 0 value (the correlation length of the projected 2pcf) reported in their Table 5 would suggest, instead, a higher bias (∼5) when derived assuming that w p (r p ) is modelled by a power law with index γ= -2 in the range r p =4-30 h −1 Mpc. At a slightly lower redshift, Francke et al. (2008) estimated the correlation function of a small sample of X-ray AGN with L bol ∼ 10 44.8 erg s −1 , in the Extended Chandra Deep Field South (ECDFS). Given the small number of sources they only infer a minimum mass of halos hosting X-ray ECDFS AGN of logM min = 12.6 +0.5 −0.8 h −1 M ⊙ (based on Sheth et al. 2001 formalism). Our result is in agreement with this study at z∼3, but measured with higher accuracy.\n\nRecently, Ikeda et al. (2015) investigated the clustering properties of low-luminosity quasars at z∼4 using the cross-correlation function of quasars and LBGs in the COSMOS field. They estimated the bias factor for a spectroscopic sample of 16 quasars and a total sample of 25 quasars including sources with photometric redshifts. They obtained a 86% upper limit for the bias of 5.63 and 10.50 for the total and spectroscopic sample, respectively. Figure 6 shows the predicted evolution of the AGN bias as a function of the bolometric luminosity, computed according to the framework of the growth and evolution of BHs presented in Shen (2009, see also Shankar 2010) at z = 3 and 3.5 Their model assumes that quasar activity is triggered by major mergers of host halos (e.g. Kauffmann & Haehnelt 2000).\n\n\nComparison to theoretical models\n\nThe major merger model is quite successful in predicting the bias of COSMOS AGN at z=2.8 as presented in Allevato et al. (2014), but underpredicts the bias re-estimated in the present work using the same AGN sample and π max = 200 h −1 Mpc. Given the large error bars, the model is in broad agreement with the bias of luminous quasars at similar redshift as measured in Shen et al. (2009) and X-ray AGN as estimated in Francke et al. (2008).\n\nThe prediction from the model slightly underpredicts our results for CCL AGN at z∼3.4. We verified that the mismatch between merger models and our data does not change if a few parameters, such as the light curve or the host halo mass distribution are changed in the major merger model. In fact, our result is still not well reproduced by the predictions from a modified Shen (2009) model in which the post-peak descending phase is cut out, with all other parameters held fixed. On the other hand, a model characterized by a steepening in the L peak -M h relation mainly implying that preferentially lowerluminosity quasars are now mapped to more massive, less numerous host DM halos, still underpredicts our results.\n\nA similar mismatch has also been found for a sample of CCL AGN at z=3-6.5 in terms of observed number counts (Marchesi et al. 2016b). In fact, they verified that the reference model overproduces the observed number counts by a factor of 3 to 10, depending on the redshift.\n\nWe also compare the observations with the theoretical model presented in Hopkins et al. (2007b), that adopts the feedback-regulated quasar light-curve/lifetime models from Hopkins at al. (2006) derived from numerical simulations of galaxy mergers that incorporate BH growth. Even if we assume an evolution with redshift, this model underpredicts the bias factor of CCL AGN.\n\nA similar tension is also observed when comparing with the semi-empirical model presented in Conroy & White (2013). In the latter, the BH mass is linearly related to galaxy mass and connected to dark matter halos via empirical constrained relations. This model makes no assumption about what triggers the AGN activity and includes a scatter in the AGN luminosityhalo mass relation, contrary to Hopkins et al. (2007b) and Shen (2009) . Conroy & White (2013) show that this semi-empirical model naturally reproduces the clustering properties of quasars at z<3, but shows some tension at higher redshift. They argue that this disagreement can be explained if AGN have a duty cycle close to unity at z>3, indicating that we approach the era of rapid BH growth in the early universe.\n\nRecently, Gatti et al. (2016) have used advanced semi analytic models (SAMs) of galaxy formation, coupled to halo occupation modelling, to investigate AGN triggering mechanisms such as galaxy interactions and disk instabilities. They compared the predictions with high redshift clustering measurements from Allevato et al. (2014), Shen et al. (2009) andEftekharzadeh et al. (2015). Their SAMs underpredict the bias of luminous quasars shown in Shen et al. (2009). The mismatch is reduced when the models are compared to Eftekharzadeh et al. (2015). They pointed out that, irrespective of the exact implementations in their SAMs, at low-z moderateluminosity AGN (L bol ∼10 44−46 erg/s) mainly inhabit halos with mass ∼10 12−13 M ⊙ for both galaxy interaction and disk instabilities models (even if disk instabilities do not trigger the most luminous AGN with L bol ≥10 47 erg/s). At higher redshift (z∼2.5), structures with mass greater than M h > 10 13 M ⊙ become significantly rarer, relegating active galaxies to live mainly in less massive environment. Moreover, in all models only galaxies with stellar masses above 10 11 M ⊙ would be able to host AGN with luminosity of L bol ∼10 46 erg/s and highly biased such as COSMOS AGN at z>2-3. This would imply that the characteristic M star /M h ratio in AGN hosts should increase with lookback time, as expected from basic considerations on number densities evolution between the halo mass function and AGN luminosity function (e.g., ).\n\n6. CONCLUSIONS We use the new CCL catalog to probe the projected and redshift-space 2pcf of X-ray selected AGN for the first time at 2.9≤z≤5.5, using the generalized clustering estimator based on phot-z Pdfs in addition to any available spec-z. We model the clustering signal with the 2-halo model and we derive the bias factor and the typical mass of the hosting halos. Our key results are:\n\n1. At z∼3.4, CCL AGN have a bias b = 6.6 +0.60 −0.55 , which corresponds to a typical mass of the hosting halos of log M h = 12.83 +0.12 −0.11 h −1 M ⊙ . A similar bias is derived using the z-space 2pcf, modelled including the typical phot-z error σ z = 0.052 of our sample. This confirms that the convergence of the projected 2pcf observed only at large scales (π max ≥ 200 h −1 Mpc) is due to large phot-z errors.\n\n2. A slightly larger bias b = 7.5 +1.6 −1.7 (but consistent within the error bars) is found using a sample of 107 CCL AGN with known spec-z. The modelling of ξ(s) suggests that this larger bias can be explained assuming that spec-zs are affected by errors of the order of σ z = 0.02 − 0.025. This would explain the convergence of the projected 2pcf surprisingly observed only at π max ≥ 200 h −1 Mpc, even when phot-zs are not included in the analysis. However, given the low statistics smaller spec-z errors and then bias can not be ruled-out.\n\n3. We estimate the bias factor for the sample of 346 XMM and Chandra AGN used in Allevato et al. (2014) using π max = 200 h −1 Mpc in estimating the projected 2pcf and then accounting for the large phot-z errors. In particular we found b = 5.8 +0.61 −0.55 , which is significantly larger than the AGN bias measured in Allevato et al. (2014) and corresponds to logM h = 12.92 +0.13 −0.18 at z=2.8.\n\n\n4.\n\nOur results suggest only a slight increase of the bias factor of COSMOS AGN at z 3, with the typical hosting halo mass of moderate luminosity AGN almost constant with redshift and equal to logM h = 12.92 +0.13 −0.18 at z=2.8 and log M h = 12.83 +0.12 −0.11 at z∼3.4, respectively.\n\n5. The observed redshift evolution of the bias of COS-MOS AGN implies that moderate luminosity AGN still inhabit group-sized halos, but slightly less massive than observed in different independent studies using X-ray AGN at z≤ 2.\n\n6. Theoretical models presented in Shen (2009) and Hopkins et al. (2007b) that assume an AGN activity mainly triggered by major mergers of host halos underpredict our results at z∼3.4 for CCL AGN with mean L bol ∼ 10 46 erg s −1 . A similar tension is also observed when comparing to the semi-empirical models presented in Conroy & White (2013). In the latter model, this disagreement can be explained if AGN have a duty cycle approaching unity at z>3. On the other hand, following the semi-analytic models presented in Gatti et al. (2016), in both galaxy interaction and disk instability models only galaxies with stellar masses above 10 11 M ⊙ would be able to host AGN with luminosity of L bol ∼10 46 erg/s and highly biased such as COSMOS AGN at z>2-3.\n\nOnly future facilities, like the X-ray Surveyor (Vikhlinin 2015) and Athena (PI K. P. Nandra), will be able to collect sizable samples (∼1000s) of low luminosity (L X <10 43 erg/s) AGN at z>3 (Civano 2015), allowing to explore the clustering for significantly less luminous source and to test AGN triggering scenarios at different AGN luminosities.\n\nFig. 1 .\n1-Left Panel:\n\nFig. 2 .\n2-Redshift (Left Panel) and 2-10 keV X-ray luminosity (Right Panel) distribution for 107 CCL AGN with known spec-z (red dashed line), 212 AGN with known spec or best phot-z (solid grey line) and 221.6 AGN with known spec-z or phot-z weighted by the Pdf (black dotted line), at 2.9≤z≤5.5\n\n2 .\n2AGN SAMPLE AT 2.9≤z≤5.5 The Chandra-COSMOS-Legacy survey (CCLS) is the combination of the 1.8 Ms C-COSMOS survey (Elvis et al 2009) with 2.8 Ms of new Chandra ACIS-I observations\n\n\ncarried with Keck-MOSFIRE (P.I. F. Civano, N. Scoville), Keck-DEIMOS (P.I.s Capak, Kartaltepe, Salvato, Sanders, Scoville, Hasinger), Subaru-FMOS (P.I. J. Silverman), VLT-FORS2 (P.I. J. Coparat) and Magellan-PRIMUS (public data).\n\nFig. 3 .\n3-Left Panel:\n\nFig. 4 .\n4-Redshift-space correlation function of 221.6 CCL AGN at 2.9<z<5.5 derived using the generalized clustering estimator based on phot-z Pdfs in addition to any available spec-z (black circles in the bottom panel). For comparison, the gray triangles (Middle Panel) and the red squares (Upper Panel) show the 2pcf of 212 AGN with known spec or best-fit phot-z and 107 AGN with known spec-zs at 2.9<z<5.5 estimated using the classic LS estimator (i.e. Pdf=1 for each source). The dotted lines show the best-fit models obtained including the dominant term b 2 ξ ′ in Equation 11, while the dashed lines is the model without the photo-z damping term. The error bars show the square roots of the diagonal values of the covariance matrix.\n\nFig. 5 .\n5Figure 4shows ξ(s) for 221.6 CCL AGN using the generalized clustering estimator based on phot-z Pdfs in ad--Left Panel: Bias of CCL AGN at z∼3.4 estimated using the projected 2pcf (empty points) and the z-space 2pcf (filled points). The empty symbols are offset in the horizontal direction by +0.04 for clarity. For comparison the bias of XMM and Chandra COSMOS AGN at z = 2.8 is shown as presented inAllevato et al. (2014) using πmax = 40 h −1 Mpc (filled hexagon) and when the bias is re-estimated using πmax = 200 h −1 Mpc. The dashed lines show the expected b(z) of typical DM halo with mass of 12, 12.5, 13, h −1 M ⊙ in log scale (from bottom to top), based onSheth et al. (2001) formalism. Right Panel: Bias parameter as a function of redshift for X-ray selected AGN (black) from previous studies as described in the legend. The red circle at z∼3.4 show the bias factors as estimated in this work for CCL AGN. The red square shows the bias factor as re-estimated in the present paper using the same catalog of XMM and Chandra-COSMOS AGN used inAllevato et al. (2014), but with πmax=200 h −1 Mpc. The dashed lines show the expected b(z) of typical DM halo with mass of 11.5, 12, 12.5, 13, 13.5 h −1 M ⊙ in log scale (from bottom to top), based onSheth et al. (2001). Bias factors from different studies are converted to a common cosmology (Ω Λ = 0.7, Ωm = 0.3, σ 8 = 0.8).\n\nFig. 6 .\n6-Predicted bias as a function of bolometric luminosity, computed according to\n\nTABLE 1\n1Properties of the AGN Samples\nWe thank the anonymous referee for helpful comments.We gratefully thank Federico Marulli for helpful discussions and inputs.We acknowledge the contributions of the entire COSMOS collaboration consisting of more than 100 scientists. 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I Zehavi, ApJ. 62122Zehavi, I., et al. 2005, ApJ, 621, 22\n"},"corpus_id":{"kind":"number","value":209867532,"string":"209,867,532"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1609.02553\",\n \"mag\": 2953019277,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.3847/0004-637x/832/1/70\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/1609.02553v2.pdf\"\n ],\n \"pdf_sha\": \"f73bef1233875dd4457983337863d0387d9eea9d\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://helda.helsinki.fi/bitstream/10138/187230/1/Allevato_2016_ApJ_832_70.pdf\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"THE CHANDRA COSMOS LEGACY SURVEY: CLUSTERING OF X-RAY SELECTED AGN AT 2.9≤z≤5.5 USING PHOTOMETRIC REDSHIFT PROBABILITY DISTRIBUTION FUNCTIONS"},"authors":{"kind":"list like","value":["V Allevato","F Civano","A Finoguenov","S Marchesi","F Shankar","G Zamorani","G Hasinger","M Salvato","T Miyaji","R Gilli","N Cappelluti","M Brusa","H Suh","G Lanzuisi","B Trakhtenbrot","R Griffiths","C Vignali","K Schawinski","A Karim"],"string":"[\n \"V Allevato\",\n \"F Civano\",\n \"A Finoguenov\",\n \"S Marchesi\",\n \"F Shankar\",\n \"G Zamorani\",\n \"G Hasinger\",\n \"M Salvato\",\n \"T Miyaji\",\n \"R Gilli\",\n \"N Cappelluti\",\n \"M Brusa\",\n \"H Suh\",\n \"G Lanzuisi\",\n \"B Trakhtenbrot\",\n \"R Griffiths\",\n \"C Vignali\",\n \"K Schawinski\",\n \"A Karim\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We present the measurement of the projected and redshift space 2-point correlation function (2pcf) of the new catalog of Chandra COSMOS-Legacy AGN at 2.9≤z≤5.5 ( L bol ∼10 46 erg/s) using the generalized clustering estimator based on phot-z probability distribution functions (Pdfs) in addition to any available spec-z. We model the projected 2pcf estimated using π max = 200 h −1 Mpc with the 2-halo term and we derive a bias at z∼3.4 equal to b = 6.6 +0.60 −0.55 , which corresponds to a typical mass of the hosting halos of log M h = 12.83 +0.12 −0.11 h −1 M ⊙ . A similar bias is derived using the redshiftspace 2pcf, modelled including the typical phot-z error σ z = 0.052 of our sample at z≥2.9. Once we integrate the projected 2pcf up to π max = 200 h −1 Mpc, the bias of XMM and Chandra COSMOS at z=2.8 used inAllevato et al. (2014)is consistent with our results at higher redshift. The results suggest only a slight increase of the bias factor of COSMOS AGN at z 3 with the typical hosting halo mass of moderate luminosity AGN almost constant with redshift and equal to logM h = 12.92 +0.13 −0.18 at z=2.8 and log M h = 12.83 +0.12 −0.11 at z∼3.4, respectively. The observed redshift evolution of the bias of COSMOS AGN implies that moderate luminosity AGN still inhabit group-sized halos at z 3, but slightly less massive than observed in different independent studies using X-ray AGN at z≤ 2."},"sections":{"kind":"list like","value":[{"title":"INTRODUCTION","content":"The presence of a nuclear supermassive black hole (BH) in almost all galaxies in the present day Universe is an accepted paradigm in astronomy (e.g. Kormendy & Richstone 1995;Kormendy & Bender 2011). Despite major observational and theoretical efforts over the last two decades, a clear explanation for the origin and evolution of BHs and their actual role in galaxy evolution remains elusive. Diverse scenarios have been proposed. One pos-sible picture includes major galaxy merger as the main triggering mechanism (e.g. Hopkins et al. 2006;Volonteri et al. 2003, Menci et al. 2003,2004. On the other hand, there is mounting observational evidence suggesting that moderate levels of AGN activity might not be always causally connected to galaxy interactions (Lutz et al. 2010, Mullaney et al. 2012, Rosario et al. 2013, Villforth et al. 2014. Several works on the morphology of the AGN host galaxies suggest that, even at moderate luminosities, a large fraction of AGN is not associated with morphologically disturbed galaxies. This trend has been observed both at low (z ∼ 1, e.g., Georgakakis et al. 2009;Cisternas et al. 2011) and high (z ∼ 2, e.g., Schawinski et al. 2011Schawinski et al. , 2012Kocevski et al. 2012, Treister et al. 2012 redshift. Theoretically, in-situ processes, such as disk instabilities or stochastic accretion of gas clouds, have also been invoked as triggers of AGN activity (e.g. Genzel et al. 2008, Dekel et al. 2009, Bournaud et al. 2011.\nAGN clustering analysis provides a unique way to unravel the knots of this complex situation, providing important, independent constraints on the BH/galaxy formation and co-evolution. In the cold dark matterdominated Universe galaxies and their BHs are believed to populate the collapsed dark matter halos, thus reflecting the spatial distribution of dark matter in the Universe. The most common statistical estimator for large-scale clustering is the two-point correlation function (2pcf, Davis & Peebles 1983). This quantity measures the excess probability above random to find pairs of galaxies/AGN separated by a given scale r. By matching the observed 2pcf to detailed outputs of dark matter numerical simulations, one can infer the typical mass of Mean normalized phot-z Pdf of all CCL AGN with best-fit phot-z > 2.9. Right Panel: Normalized phot-z Pdf for the source lid766. This source has a best-fit photo-z value < 2.9, but a phot-z Pdf(z i >2.9)>0.001 (red thick line). The redshifts above this threshold, weighted by their Pdf, have been taken in account in the catalog used to estimate the 2pcf. the hosting dark matter halos. This is derived through the so called AGN bias b, enabling then to pin down the typical environment where AGN live. This in turn can provide new insights into the physical mechanisms responsible for triggering AGN activity.\nThe 2pcf of AGN has been measured in optical large area surveys, such as the 2dF (2QZ, Croom et al. 2005;Porciani & Norberg 2006) and the Sloan Digital Sky Survey (SDSS, Li et al. 2006;Shen et al. 2009;Ross et al. 2009). These optical surveys are thousands of square degree fields, mainly sampling rare and high luminosity quasars. The amplitude of the 2PCF of quasars suggests that these luminous AGN are hosted by halos of roughly constant mass, a few times 10 12 M ⊙ , out to z =3-4 (Shanks et al. 2011). Models of major mergers between gas-rich galaxies appear to naturally reproduce the clustering properties of optically selected quasars as a function of luminosity and redshift (Hopkins et al. 2007a(Hopkins et al. , 2008Shen 2009;Bonoli et al. 2009). This supports the scenario in which major mergers dominate the luminous quasar population (Scannapieco et al. 2004;Neistein & Netzer 2014;Treister et al. 2012).\nChandra surveys have contributed significantly to the study of the AGN clustering (e.g. CDFS-N, Gilli et al. 2009; Chandra/Bootes, Starikova et al. 2011, Allevato et al. 2014. Deep X-ray data can be used to draw conclusions on the faint portions of the AGN luminosity function, where a significant fraction of obscured sources is present. In particular, the Chandra survey in the two square degree COSMOS field (C-COSMOS, Elvis et al. 2009, Civano et al. 2012; Chandra COSMOS Legacy Survey, Civano et al. 2016) has allowed the investigation of the redshift evolution of the clustering properties of Xray AGN, for the first time up to z∼3. Interestingly, over a broad redshift range (z ∼ 0 -2) moderate luminosity AGN occupy DM halo masses of log M h ∼ 12.5-13.5 M ⊙ h −1 . The clustering strength of X-ray selected AGN has been measured by independent studies to be higher than that of optical quasars. Merger models usually fail in reproducing the data from X-ray surveys, opening the possibility of additional AGN triggering mechanisms (e.g., Allevato et al. 2011;Mountrichas & Georgakakis 2012) and/or multiple modes of BH accretion (e.g., Fanidakis et al. 2013). Recently, Mendez et al. (2015) and Gatti et al. (2016) have suggested that selection cuts in terms of AGN luminosity, host galaxy properties and redshift interval, might have a more relevant role in driving the differences often observed in the bias factor inferred from different surveys. The measurement of the AGN bias is crucial at high redshifts, especially at z>2-3, i.e. at the peak in the accretion history of the Universe. At z>3, Shen et al. (2007Shen et al. ( ,2009) measured for the first time the 2pcf of luminous SDSS-DR5 quasars (log L bol ∼ 10 47 erg/s) at z = 3.2 and 3.8. Even if with very large uncertainty, they found that these objects live in massive halos of the order of 10 13 M ⊙ h −1 . This result is consistent with models invoking galaxy major mergers as the main triggering mechanism for very luminous AGN. Recently, Eftekharzadeh et al. (2015) studying a sample of spectroscopically confirmed SDSS-III/BOSS quasars at 2.2≤z≤3.4, performed a more precise estimation of the quasar bias at high redshift. They found no evolution of the bias in three redshift bins, with halo masses equal to 3× and ∼0.6× 10 12 M ⊙ h −1 at z∼2.3 and ∼ 3, respectively.\nThere are only a few attempts of measuring the clustering properties of X-ray AGN at z ∼ 3. Francke et al. (2008) estimated the bias of a small sample of X-ray AGNs (L bol ∼ 10 44.8 erg s −1 ) in the Extended Chandra Deep Field South (ECDFS), with very large uncertainty. They found indications that X-ray ECDFS AGNs reside in dark matter halos with minimum mass of log M min = 12.6 +0.5 −0.8 h −1 M ⊙ . On the other hand, Allevato et al. (2014) used a sample of Chandra and XMM-Newton AGN in COSMOS with moderate luminosity (log L bol ∼ 10 45.3 erg/s) at z =2.86. For the first time they estimated the bias of X-ray selected AGN at high redshift, suggesting that they inhabit halos of logM h = 12.37±0.10 M ⊙ h −1 . They also extended to z∼3 the result that Type 1 AGN reside in more massive halos than Type 2 AGN. Recently, Ikeda et al. (2015) estimated the clustering properties of low-luminosity quasars in COS-MOS at 3.1≤z≤4.5, using the cross-correlation between Lyman-Break Galaxies (LBGs) and 25 quasars with spectroscopic and photometric redshifts. They derived a 86% upper limit of 5.63 for the bias at z∼ 4.\nIn this paper we want to extend the study of the clus- tering properties of X-ray selected AGN to z>3 using the new Chandra COSMOS-Legacy data. To this goal, we perform clustering measurements using techniques based on photometric redshift in the form of probability distribution functions (Pdfs), in addition to any available spectroscopy. This is motivated by the development in the last years of clustering measurement techniques based on photometric redshift Pdfs by Myers, White & Ball (2009), Hickox et al. (2011 and Mountrichas et al. (2013) and Georgakakis et al. (2014). One of the advantages of this new clustering estimator is that one can use in the analysis all sources not just the optically brighter ones for which spectroscopy is available. For this reason it is well suited to clustering investigations using future large X-ray AGN surveys, where the fraction of spectroscopic redshifts might be small.\nThroughout the paper, all distances are measured in comoving coordinates and are given in units of Mpc h −1 , where h = H 0 /100km/s. We use a ΛCDM cosmology with Ω M = 0.3, Ω Λ = 0.7, Ω b = 0.045, σ 8 = 0.8. The symbol log signifies a base-10 logarithm. (Civano et al. 2016) for a total coverage of 2.2 deg 2 of the COSMOS field (Scoville et al. 2007). The X-ray source catalog consists of 4016 sources. 2076 (∼52%) have a secure spectroscopic redshift (spec-z) and for ∼96% the photometric redshift (photo-z) is available. As shown in Marchesi et al. (2016a), the spectroscopic redshifts have been obtained with different observing programs, as the zCOSMOS survey (Very Large Telescope/VIMOS; Lilly et al. 2007) and the Magellan/IMACS survey (Trump et al. 2007(Trump et al. , 2009). Other programs, many of which have been specifically targeting the CCLS have been The photo-zs are estimated following the procedure described in Salvato et al. (2011). Following Marchesi et al. (2016b), the accuracy of the photometric redshifts with respect to the whole spectroscopic redshift sample is σ ∆z/(1+zspec) =0.02, with a fraction of outliers ≃11%. At z>2.9 there are 9 outliers (∆z/(1 + z spec ) > 0.15), but for the remaining sources the agreement between specz and photo-z has the same quality of the whole sample. In detail, the normalized median absolute deviation σ N MAD =1.48×median( zspec− zphot /(1 + zspec)) = 0.012.\nThe CCL AGN sample at 2.9<z<5.5 consists of 212 AGN detected in the 0.5-10 keV band, 107/212 with spec-zs and 105/212 with only phot-zs. To each of the 105 AGN with best-fit phot-z in the range 2.9≤z<5.5, is associated a probability distribution function (Pdf), which gives the probability of the source to be in the redshift range z i ± ∆z/2 with a binsize of ∆z = 0.01. The integrated area of the Pdf on all redshift bins z i is normalized to 1, i.e. i P df (z i ) = 1 for each AGN. We take into account for this analysis the redshift bins z i with Pdf(z i ) >0.001. Figure 1 shows the mean normalized phot-z Pdf for 105 sources with best-fit phot-z 2.9≤z<5.5. The effective contribution to the number of AGN at z≥2.9 of these 105 AGN weighted by the Pdf is 78.32 sources, i.e. 105 j=1 P df j (z ≥ 2.9)=78.32. In the CCL sample there are also 246 sources with phot-z < 2.9 but which contribute to the Pdf at 2.9≤ z ≤ 5.5 (i.e., with Pdf>0.001 at 2.9 ≤ z i ≤5.5 (see for example the Pdf of source lid766, whose nominal best-fit phot-z value is 2.51, in Figure 1). All these 246 sources have been taken in account in our analysis, using for each of them the Pdf of each bin of redshift 2.9 ≤ z i ≤5.5. The weighted contribution of these sources, i.e. the sum of all weights, is equal to 36.3 AGN (\nTo all the 107 sources with known spec-z we assign a Pdf j = 1 to the spec-z value ( 107 j=1 P df j =107). To summarize, the total effective number of CCL AGN at 2.9≤z<5.5 weighted by the Pdf and used for the clustering measurements is 78.3 + 36.3 + 107 = 221.6 objects. Figure 2 shows the normalized redshift and 2-10 keV rest-frame X-ray luminosity distribution for our sample of CCL AGN at 2.9<z<5.5, when the phot-z Pdfs are used (black dotted line, z =3.36). The mean bolometric luminosity of this sample derived using the bolometric correction defined in Equation (21) of Marconi et al. (2004) is log L bol = 45.99 erg s −1 . For comparison, we also show the normalized distributions of our AGN sample when only the best-fit phot-zs are taken into account in addition to any available spec-z (gray solid line, z =3.34) and for the sample with known spec-z (reddashed line, z =3.35).\n3. 2PCF USING PHOT-z PDFS 3.1. Projected 2pcf The most commonly used quantitative measure of large scale structure is the 2pcf, ξ(r), which traces the amplitude of AGN clustering as a function of scale. ξ(r) is defined as a measure of the excess probability dP , above what is expected for an unclustered random Poisson distribution, of finding an AGN in a volume element dV at a separation r from another AGN:\nwhere n is the mean number density of the AGN sample (Peebles 1980). Measurements of ξ(r) are generally performed in comoving space, with r having units of h −1 Mpc. With a redshift survey, we cannot directly measure ξ(r) in physical space, because peculiar motions of galaxies distort the line-of-sight distances inferred from redshift. To separate the effects of redshift distortions, the spatial correlation function is measured in two dimensions r p and π, where r p and π are the projected comoving separations between the considered objects in the directions perpendicular and parallel, respectively, to the mean line-of-sight between the two sources. Following Davis & Peebles (1983), r 1 and r 2 are the redshift positions of a pair of objects, s is the redshift-space separation (r 1 − r 2 ), and l = 1 2 (r 1 + r 2 ) is the mean distance to the pair. The separations between the two considered objects across r p and π are defined as:\nRedshift space distortions only affect the correlation function along the line of sight, so we estimate the socalled projected correlation function w p (r p ) (Davis & Peebles 1983):\nwhere ξ(r p , π) is the two-point correlation function in terms of r p and π, measured using the Landy & Szalay (1993, LS) estimator:\nwhere DD', DR' and RR' are the normalized data-data, data-random and random-random pairs. In this classic approach of estimating the redshiftspace correlation function, in presence of accurate speczs, when a data-data pair with separation (r p , π) is found, the pair number is incremented by one, i.e. DD(r p ,π) = DD(r p , π) + 1. Following Georgakakis et al. (2014), in the generalized clustering estimator the number of datadata pairs with projected and line of sight separation (r p , π) is, instead, incremented by the product Pdf 1 (z i ) × Pdf 2 (z j ):\nwhere Pdf 1 (z i ) and Pdf 2 (z j ) are the Pdf values (per redshift bin) of the source 1 at z = z i and of the source 2 at z = z j respectively.\nThe measurements of the 2pcf requires the construction of a random catalog with the same selection criteria and observational effects as the data. To this end, we constructed a random catalog where each simulated source is placed at a random position in the sky, with its flux randomly extracted from the catalog of real source fluxes. The simulated source is kept in the random sample if its flux is above the sensitivity map value at that random position (Miyaji et al. 2007;Cappelluti et al. 2009). The corresponding redshift for each random object is then assigned based on the smoothed redshift distribution of the AGN sample, where each redshift is weighted by the Pdf associated to that redshift for the particular source. Since the phot-z Pdfs are already Bias as a function of πmax for 221.6 CCL AGN at 2.9≤z<5.5, when the projected 2pcf is measured using the generalized clustering estimator based on phot-z Pdfs in addition to any available spec-z (black dots). For comparison, the gray triangles and red squares show the bias of 212 AGN with known spec or best-fit phot-z and 107 AGN with known spec-zs at 2.9<z<5.5 when the classic LS estimator (i.e. no phot-z Pdfs) is used. Right Panel: Projected 2pcf for πmax=200 h −1 Mpc. The 1σ errors on wp(rp) are the square root of the diagonal components of the covariance matrix. The continuous line represents the DM projected 2pcf estimated at z =3.36.\ntaken into account in the generation of the random redshifts, we decided to assign Pdf=1 to each random source.\nIn the halo model approach, the large scale amplitude signal is due to the correlation between objects in distinct halos and the bias parameter defines the relation between the large scale clustering amplitude of the AGN correlation function and the DM 2-halo term:\nWe first estimated the DM 2-halo term at the median redshift of the sample, using:\nintegrating up to 200 h −1 Mpc along π, where:\nis the linear power spectrum, assuming a power spectrum shape parameter Γ = Ω m h = 0.2 (Efstathiou et al. 1992) which corresponds to h = 0.7.\n3.2. z-space correlation function Similarly, we can estimate the z-space correlation function ξ(s) using Equation (5) and (6), written now as a function of the redshift-space separation s = (r 1 − r 2 ) between the sources. ξ(s) is affected by perturbations in the cosmological redshifts due to peculiar velocities and redshift errors. The z-space power spectrum can be modelled in polar coordinates as follow (e.g. Kaiser 1987, Peacock et al. 2001:\nwhere k = k 2 ⊥ + k 2 , k ⊥ and k are the wavevector components perpendicular and parallel to the line of sight, respectively. µ = k ⊥ /k , P DM (k) is the dark matter power spectrum, b is the linear bias factor, f is the growth rate of density fluctuations. σ is the displacement along the line of sight due to random perturbations of cosmological redshifts.Assuming standard gravity, we approximated the growth rate f ≃ Ω M (z) γ , with γ = 0.545 (e.g. Sereno et al. 2015).\nThe f µ 2 term parametrizes the coherent motions due to large-scale structures, enhancing the clustering signal on all scales. The exponential cut-off term describes the random perturbations of the redshifts caused by both non-linear stochastic motions and redshift errors. The integration of Eq.(7) over the angle µ, and then the Fourier anti-transformation gives:\nThe main term, ξ ′ (s), is the Fourier anti-transform of the monopole P ′ (k):\nthat corresponds to the model given by Equation (10) when neglecting the dynamic distortion term. In our case, photo-z errors perturb the most the distance measurements along the line of sight. Therefore the small-scale random motions are negligible with respect to photo-z errors. The cut-off scale in Eq. (12) can thus be written as:\nwhere H(z) is the Hubble function computed at the median redshift of the sample, and σ z is the typical photo-z error.\nIn this case, knowing the cut-off scale, the AGN bias can be derived from the Fourier anti-transform of the monopole P ′ (k), i.e.:\nwhere ξ(s) is the observed z-space 2pcf of our AGN sample.\n4. RESULTS 4.1. w p (r p ) and Bias We have measured the 2pcf of 221.6 CCL AGN at 2.9≤z<5.5, using the generalized clustering estimator defined in Equation (6), based on phot-z Pdfs in addition to any available spec-z. The projected 2pcf w p (r p ) is then estimated using Equation (4).\nThe typical value of π max used in clustering measurements of both optically-selected luminous quasars and X-ray selected AGN is ∼20-100 h −1 Mpc (e.g. Zehavi et al. 2005, Coil et al. 2009, Krumpe et al. 2010, Allevato et al. 2011. The optimum π max value can be determined by measuring the 2pcf for different π max and then adopting the value at which the amplitude of the signal appears to level off. Figure 3 (Left Panel) shows the bias factor estimated for different values of π max in Equation (4), when the phot-z Pdfs are used in addition to any available specz. For comparison, we also estimated the bias for case i) 107 AGN with known spec-zs; case ii) 107+105 AGN with known spec-z or best-fit phot-zs. In these cases the 2pcf is measured using the classic LS estimator and the Pdf is set to unity for each source.\nAs expected, when including phot-zs in the analysis, the bias levels-off only at large scales, because of the large uncertainties in the redshifts measured via photometric methods (Georgakakis et al. 2014). Surprisingly, even if the error bars are large, an increase of the bias factor at π max >100 h −1 Mpc is suggested also when only spec-zs are used with the classic 2pcf estimator. This suggests that a fraction of spec-zs might be affected by large errors (see Section 4.2).\nThe amplitude of the projected 2pcf of CCL AGN measured using the generalized clustering estimator converges at π max ≥ 200 h −1 Mpc. We decide to use π max = 200 h −1 Mpc in order to balance the advantage of integrating out redshift-space distortions against the disadvantage of introducing noise from uncorrelated line-ofsight structure. Figure 3 (Right Panel) shows the projected 2pcf estimated using the generalized clustering estimator, whith π max =200 h −1 Mpc. The 1σ errors on w p (r p ) are the square root of the diagonal components of the covariance matrix (Miyaji et al. 2007, Krumpe et al. 2010) estimated using the bootstrap method. The latter quantifies the level of correlation between different bins. For comparison, we also estimate the projected 2pcf for case i) 107 AGN with known spec-zs; case ii) 107+105 AGN with known spec-z or best-fit phot-zs. Note that in these cases the classic LS estimator is used (i.e. Pdf = 1 for each source) and π max is fixed to 200 h −1 Mpc also in these cases. Following Eq. 7, we derive the best-fit bias by using a χ 2 minimization technique with 1 free parameter in the range r p = 1 -30 h −1 Mpc, where χ 2 = ∆ T M −1 cov ∆. In detail, ∆ is a vector composed of w p (r p ) − w mod (r p ) (see Equations 4 and 7), ∆ T is its transpose and M −1 cov is the inverse of covariance matrix. The latter full covariance matrix is used in the fit to take into account the correlation between errors.\nAs shown in Table 1, we derived a bias for our sample of CCL AGN equal to b = 6.6 +0.60 −0.55 at z =3.36. Following the bias-mass relation b(M h , z) described in van den Bosch (2002) and Sheth et al. (2001), the AGN bias corresponds to a typical mass of the hosting halos of log M h = 12.83 +0.12 −0.11 h −1 M ⊙ . It is worth noticing that this is a typical/characteristic mass of the halos hosting CCL AGN. Only the HOD modelling of the clustering signal at all scales can provide the entire hosting halo mass distribution for this sample.\nThe bias has a larger uncertainty when derived from the 2pcf estimated without the phot-z Pdfs. In detail, we find b = 7.5 +1.6 −1.7 at z =3.35 for 107 AGN with known spec-zs (case i) and b = 6.48 +1.27 −1.36 at z =3.34 for 107+105 AGN with known spec or best-fit phot-zs (case ii). Note that in these cases the 2pcf is measured using the classic LS estimator and the Pdf is set to unity for each source."},{"title":"ξ(s) and phot-z errors","content":"To investigate the effect of phot-z errors on the 2pcf, we also measured the z-space correlation function ξ(s). dition to any available spec-z. For comparison, the gray triangles show ξ(s) for case i) and case ii).\nAs described in section 3.2, ξ(s) is affected by the Kaiser effect that enhances the clustering signal at all scales and by phot-z errors, that are modelled by using an exponential cut-off in the z-space power spectrum. For our sample of CCL AGN the typical error on phot-zs is σ z =0.012×(1 + z spec ) = 0.052 at z =3.4. This implies a cut-off scale σ = 43.45 h −1 Mpc (see Equation 13). Including the phot-z damping in the modeling of ξ(s), we derived the best-fit bias by using Equation 14 and the χ 2 minimization technique with 1 free parameter in the range s=1-50 h −1 Mpc. In particular, for the z-space 2pcf measured using the generalized clustering estimator we derived b = 6.53 +0.52 −0.58 . For the sample of 212 AGN with known spec or best-fit phot-zs for which the classic clustering estimator is used, we derived b = 6.96 +0.72 −0.73 . These values are in perfect agreement with the bias obtained from the projected 2pcf with π max = 200 h −1 Mpc. This confirms that the convergence of the projected 2pcf observed only at large scales (π max ≥200 h −1 Mpc) is due to large phot-z errors. In fact, for large redshift errors and small survey area it is necessary to integrate the correlation function up to large scales to fully correct for them.\nWe also estimated the bias using the z-space 2pcf for 107 AGN with known spec-zs. In general, we do not expect spec-zs to be affected by large errors. If we do not include the phot-z damping in the model of ξ(s), we obtain for this sample a bias b = 5.86 +1.13 −1.05 , which is lower than b = 7.5 +1.6 −1.7 obtained using the projected 2pcf and π max = 200 h −1 Mpc. A better fit of ξ(s) and a larger bias can be obtained only if we include in the model specz errors of the order of σ z = 0.02-0.025. In particular, for σ z = 0.020 (cut-off scale σ = 16.7 h −1 Mpc) and σ z = 0.025 (σ = 20.9 h −1 Mpc), we derived b = 7.4 +1.35 −1.40 and b = 8.01 +1.4 −1.5 , respectively. However, given the low statistics, smaller values of σ z and then of the bias can not be ruled-out. This error redshift is larger than what is expected for a spectroscopic sample of AGN. However, the presence in the spectroscopic sample of ∼20% of the objects (21/107) with a low quality flag (i.e. flag = 1.5, corresponding to low quality spectra, and therefore not fully reliable redshift, but with known phot-z such that σ ∆z/(1+zspec) <0.1) could explain both the improvement of the fit including such an error in the analysis and the increase of the bias with increasing π max up to ∼200 h −1 Mpc when using only AGN with spectroscopic redshift."},{"title":"DISCUSSION","content":"In this section we compare our results with previous measurements using COSMOS AGN at z∼3 and with previous studies at similar redshift. We also interpret our results in terms of AGN triggering mechanisms. Figure 5 (right panel) shows the redshift evolution of the AGN bias estimated using moderate luminosity AGN detected in different X-ray surveys. Interestingly, moderate luminosity AGN occupy DM halo masses of log M h ∼ 12.5-13.5 M ⊙ h −1 up to z∼2, tracing a constant group-sized halo mass. Allevato et al. (2011) have shown that XMM COSMOS AGN (L bol ∼ 10 45.2 erg s −1 ) reside in DM halos with constant mass equal to logM h = 13.12±0.07 M ⊙ h −1 up to z = 2. They also argue that this high bias can not be reproduced assuming that major merger between gas-rich galaxies (Shen 2009) is the main triggering scenario for moderate luminosity AGN. By contrast, at z∼3, Allevato et al. (2014) found a drop in the mass of the hosting halos, with Chandra and XMM-Newton COSMOS AGN (L bol ∼ 10 45.3 erg s −1 ), inhabiting halos of logM h = 12.37±0.10 M ⊙ h −1 ."},{"title":"Redshift evolution of the AGN bias","content":"In the present paper we measure a bias for 221.6 CCL AGN at 2.9≤z≤5.5 equal to b = 6.6 +0.6 −0.55 , that corresponds to a typical mass of the hosting dark matter halos of logM h = 12.83 +0.12 −0.11 h −1 M ⊙ . This result suggests a higher bias for CCL AGN compared to previous studies in COSMOS at z∼ 3. In fact, Allevato et al. (2014) found a bias of 3.85 +0.22 −0.21 at z =2.8, using a sample of XMM and Chandra AGN. Although the two samples only partially overlap, we argue that the most likely explanation of these differences lies in the small value of π max (= 40 h −1 Mpc) used in Allevato et al. (2014). As shown in Figure 3, the bias strongly increases with π max due to the large phot-z errors and the use of 40 h −1 Mpc might produce an underestimated clustering signal.\nTo verify this effect, we took the same sample used in Allevato et al. (2014), i.e. 346 XMM and Chandra COS-MOS AGN with known spec or phot-z > 2.2. As already mentioned, Allevato et al. (2014) used the classic LS estimator where the phot-zs Pdfs are not taken into account. Using their same classic approach, we estimated the projected 2pcf for different values of π max and found that the clustering signal converges only at π max ≥ 200 h −1 Mpc. In particular, we derived a bias b = 5.8 +0.61 −0.65 for π max = 200 h −1 Mpc, which corresponds to a typical hosting halo mass logM h = 12.92 +0.13 −0.18 . As shown in Figure 5 (left panel) the bias estimated using π max = 200 h −1 Mpc is in agreement with the bias of CCL AGN at z∼3.4 as derived in the present work. It is worth noticing that also the mean luminosity of the samples is increasing with z, with mean L bol = 10 45.5 erg/s for XMM and Chandra COSMOS AGN at z∼3 and ∼46 erg/s for CCL AGN at higher z.\nThese results imply that at z>3: i) the typical hosting halo mass of moderate luminosity AGN remains almost constant with redshift, going from ∼8.3 × 10 12 at z=2.8 to ∼ 6.7× 10 12 M ⊙ h −1 at z∼3.4, since a lower mass is required to yield the same bias at a higher redshift; ii) moderate luminosity AGN still inhabit group-sized halos at high redshift, but slightly less massive than observed in different independent studies using X-ray selected AGN at z≤2."},{"title":"Previous studies at high redshift","content":"The evolution of the bias with redshit has been studied in Eftekharzadeh et al. (2015) for SDSS-III/BOSS quasars at 2.2 ≤z ≤3.4. They investigated the redshift dependence of quasar clustering in three redshift bins and found no evolution of the correlation length and bias. In terms of halo mass, this corresponds to a characteristic halo mass that decreases with redshift, with halo masses of 3× (6×) and ∼0.6× (1.3×) 10 12 M ⊙ h −1 at z∼2.3 and ∼ 3, respectively, where the dark matter halo masses are estimated using Tinker et al. 2010(Sheth et al. 2001).\nThese results are surprisingly different in terms of bias Shen (2009) at z = 3 (in black) and z = 3.5 (in gold), Hopkins et al. (2007b) at z=3 (blue dotted line) and Conroy & White (2013) at z=3.4 (green dashed line) compared to previously estimated bias factors for optically selected quasars at z=3.2 , gray hexagon), X-ray selected AGN at z=3 (Francke et al. 2008, open triangle), Chandra and XMM-COSMOS AGN at z=2.9 as estimated in Allevato et al. (2014, filled black square) and as re-estimated in the present work (black empty square) and our results (red circle). The errors on the L bol axis correspond to the dispersion of the bolometric luminosity distributions for the different subsets. and halo mass when compared to Shen et al. (2009). At z∼3, Eftekharzadeh et al. (2015) derived halo masses that are close to an order-of-magnitude smaller than those presented in Shen et al. (2009). In this latter study, they measured the bias of SDSS-DR5 quasars with mean L bol ∼ 10 47 erg/s, at z = 3.2. Even if with large uncertainties, their results suggest that luminous quasars reside in massive halos with mass few times 10 13 h −1 M ⊙ (based on Sheth et al. 2001). Although the two samples do not completely overlap, Eftekharzadeh et al. (2015) argue that the most likely explanation of these differences lies in the improvements in SDSS photometric calibration and target selection algorithms as well as in the much larger number of quasars that afford greater measurements precision compared to Shen et al. (2009). Our results are in disagreement with the bias factor equal to b = 3.57±0.09 at z∼3 derived in Eftekharzadeh et al. (2015). This disagreement might be due to the slightly different average redshift and the significantly different luminosity (almost one order of magnitude) of the samples used in the two different studies. An additional important difference between the samples is that our catalog of CCL AGN includes both Type 1 and Type 2 AGN, while Eftekharzadeh et al. (2015) use Type 1 BOSS luminous quasars. It is also worth noticing that in Eftekharzadeh et al. (2015) the bias is derived modelling the z-space 2pcf in the Kaiser formalism, i.e. not including the effect of random peculiar velocities and redshift errors. The r 0 value (the correlation length of the projected 2pcf) reported in their Table 5 would suggest, instead, a higher bias (∼5) when derived assuming that w p (r p ) is modelled by a power law with index γ= -2 in the range r p =4-30 h −1 Mpc. At a slightly lower redshift, Francke et al. (2008) estimated the correlation function of a small sample of X-ray AGN with L bol ∼ 10 44.8 erg s −1 , in the Extended Chandra Deep Field South (ECDFS). Given the small number of sources they only infer a minimum mass of halos hosting X-ray ECDFS AGN of logM min = 12.6 +0.5 −0.8 h −1 M ⊙ (based on Sheth et al. 2001 formalism). Our result is in agreement with this study at z∼3, but measured with higher accuracy.\nRecently, Ikeda et al. (2015) investigated the clustering properties of low-luminosity quasars at z∼4 using the cross-correlation function of quasars and LBGs in the COSMOS field. They estimated the bias factor for a spectroscopic sample of 16 quasars and a total sample of 25 quasars including sources with photometric redshifts. They obtained a 86% upper limit for the bias of 5.63 and 10.50 for the total and spectroscopic sample, respectively. Figure 6 shows the predicted evolution of the AGN bias as a function of the bolometric luminosity, computed according to the framework of the growth and evolution of BHs presented in Shen (2009, see also Shankar 2010) at z = 3 and 3.5 Their model assumes that quasar activity is triggered by major mergers of host halos (e.g. Kauffmann & Haehnelt 2000)."},{"title":"Comparison to theoretical models","content":"The major merger model is quite successful in predicting the bias of COSMOS AGN at z=2.8 as presented in Allevato et al. (2014), but underpredicts the bias re-estimated in the present work using the same AGN sample and π max = 200 h −1 Mpc. Given the large error bars, the model is in broad agreement with the bias of luminous quasars at similar redshift as measured in Shen et al. (2009) and X-ray AGN as estimated in Francke et al. (2008).\nThe prediction from the model slightly underpredicts our results for CCL AGN at z∼3.4. We verified that the mismatch between merger models and our data does not change if a few parameters, such as the light curve or the host halo mass distribution are changed in the major merger model. In fact, our result is still not well reproduced by the predictions from a modified Shen (2009) model in which the post-peak descending phase is cut out, with all other parameters held fixed. On the other hand, a model characterized by a steepening in the L peak -M h relation mainly implying that preferentially lowerluminosity quasars are now mapped to more massive, less numerous host DM halos, still underpredicts our results.\nA similar mismatch has also been found for a sample of CCL AGN at z=3-6.5 in terms of observed number counts (Marchesi et al. 2016b). In fact, they verified that the reference model overproduces the observed number counts by a factor of 3 to 10, depending on the redshift.\nWe also compare the observations with the theoretical model presented in Hopkins et al. (2007b), that adopts the feedback-regulated quasar light-curve/lifetime models from Hopkins at al. (2006) derived from numerical simulations of galaxy mergers that incorporate BH growth. Even if we assume an evolution with redshift, this model underpredicts the bias factor of CCL AGN.\nA similar tension is also observed when comparing with the semi-empirical model presented in Conroy & White (2013). In the latter, the BH mass is linearly related to galaxy mass and connected to dark matter halos via empirical constrained relations. This model makes no assumption about what triggers the AGN activity and includes a scatter in the AGN luminosityhalo mass relation, contrary to Hopkins et al. (2007b) and Shen (2009) . Conroy & White (2013) show that this semi-empirical model naturally reproduces the clustering properties of quasars at z<3, but shows some tension at higher redshift. They argue that this disagreement can be explained if AGN have a duty cycle close to unity at z>3, indicating that we approach the era of rapid BH growth in the early universe.\nRecently, Gatti et al. (2016) have used advanced semi analytic models (SAMs) of galaxy formation, coupled to halo occupation modelling, to investigate AGN triggering mechanisms such as galaxy interactions and disk instabilities. They compared the predictions with high redshift clustering measurements from Allevato et al. (2014), Shen et al. (2009) andEftekharzadeh et al. (2015). Their SAMs underpredict the bias of luminous quasars shown in Shen et al. (2009). The mismatch is reduced when the models are compared to Eftekharzadeh et al. (2015). They pointed out that, irrespective of the exact implementations in their SAMs, at low-z moderateluminosity AGN (L bol ∼10 44−46 erg/s) mainly inhabit halos with mass ∼10 12−13 M ⊙ for both galaxy interaction and disk instabilities models (even if disk instabilities do not trigger the most luminous AGN with L bol ≥10 47 erg/s). At higher redshift (z∼2.5), structures with mass greater than M h > 10 13 M ⊙ become significantly rarer, relegating active galaxies to live mainly in less massive environment. Moreover, in all models only galaxies with stellar masses above 10 11 M ⊙ would be able to host AGN with luminosity of L bol ∼10 46 erg/s and highly biased such as COSMOS AGN at z>2-3. This would imply that the characteristic M star /M h ratio in AGN hosts should increase with lookback time, as expected from basic considerations on number densities evolution between the halo mass function and AGN luminosity function (e.g., ).\n6. CONCLUSIONS We use the new CCL catalog to probe the projected and redshift-space 2pcf of X-ray selected AGN for the first time at 2.9≤z≤5.5, using the generalized clustering estimator based on phot-z Pdfs in addition to any available spec-z. We model the clustering signal with the 2-halo model and we derive the bias factor and the typical mass of the hosting halos. Our key results are:\n1. At z∼3.4, CCL AGN have a bias b = 6.6 +0.60 −0.55 , which corresponds to a typical mass of the hosting halos of log M h = 12.83 +0.12 −0.11 h −1 M ⊙ . A similar bias is derived using the z-space 2pcf, modelled including the typical phot-z error σ z = 0.052 of our sample. This confirms that the convergence of the projected 2pcf observed only at large scales (π max ≥ 200 h −1 Mpc) is due to large phot-z errors.\n2. A slightly larger bias b = 7.5 +1.6 −1.7 (but consistent within the error bars) is found using a sample of 107 CCL AGN with known spec-z. The modelling of ξ(s) suggests that this larger bias can be explained assuming that spec-zs are affected by errors of the order of σ z = 0.02 − 0.025. This would explain the convergence of the projected 2pcf surprisingly observed only at π max ≥ 200 h −1 Mpc, even when phot-zs are not included in the analysis. However, given the low statistics smaller spec-z errors and then bias can not be ruled-out.\n3. We estimate the bias factor for the sample of 346 XMM and Chandra AGN used in Allevato et al. (2014) using π max = 200 h −1 Mpc in estimating the projected 2pcf and then accounting for the large phot-z errors. In particular we found b = 5.8 +0.61 −0.55 , which is significantly larger than the AGN bias measured in Allevato et al. (2014) and corresponds to logM h = 12.92 +0.13 −0.18 at z=2.8."},{"title":"4.","content":"Our results suggest only a slight increase of the bias factor of COSMOS AGN at z 3, with the typical hosting halo mass of moderate luminosity AGN almost constant with redshift and equal to logM h = 12.92 +0.13 −0.18 at z=2.8 and log M h = 12.83 +0.12 −0.11 at z∼3.4, respectively.\n5. The observed redshift evolution of the bias of COS-MOS AGN implies that moderate luminosity AGN still inhabit group-sized halos, but slightly less massive than observed in different independent studies using X-ray AGN at z≤ 2.\n6. Theoretical models presented in Shen (2009) and Hopkins et al. (2007b) that assume an AGN activity mainly triggered by major mergers of host halos underpredict our results at z∼3.4 for CCL AGN with mean L bol ∼ 10 46 erg s −1 . A similar tension is also observed when comparing to the semi-empirical models presented in Conroy & White (2013). In the latter model, this disagreement can be explained if AGN have a duty cycle approaching unity at z>3. On the other hand, following the semi-analytic models presented in Gatti et al. (2016), in both galaxy interaction and disk instability models only galaxies with stellar masses above 10 11 M ⊙ would be able to host AGN with luminosity of L bol ∼10 46 erg/s and highly biased such as COSMOS AGN at z>2-3.\nOnly future facilities, like the X-ray Surveyor (Vikhlinin 2015) and Athena (PI K. P. Nandra), will be able to collect sizable samples (∼1000s) of low luminosity (L X <10 43 erg/s) AGN at z>3 (Civano 2015), allowing to explore the clustering for significantly less luminous source and to test AGN triggering scenarios at different AGN luminosities."},{"title":"Fig. 1 .","content":""},{"title":"Fig. 2 .","content":""},{"title":"2 .","content":""},{"title":"Fig. 3 .","content":""},{"title":"Fig. 4 .","content":""},{"title":"Fig. 5 .","content":""},{"title":"Fig. 6 .","content":""},{"title":"TABLE 1","content":""}],"string":"[\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"The presence of a nuclear supermassive black hole (BH) in almost all galaxies in the present day Universe is an accepted paradigm in astronomy (e.g. Kormendy & Richstone 1995;Kormendy & Bender 2011). Despite major observational and theoretical efforts over the last two decades, a clear explanation for the origin and evolution of BHs and their actual role in galaxy evolution remains elusive. Diverse scenarios have been proposed. One pos-sible picture includes major galaxy merger as the main triggering mechanism (e.g. Hopkins et al. 2006;Volonteri et al. 2003, Menci et al. 2003,2004. On the other hand, there is mounting observational evidence suggesting that moderate levels of AGN activity might not be always causally connected to galaxy interactions (Lutz et al. 2010, Mullaney et al. 2012, Rosario et al. 2013, Villforth et al. 2014. Several works on the morphology of the AGN host galaxies suggest that, even at moderate luminosities, a large fraction of AGN is not associated with morphologically disturbed galaxies. This trend has been observed both at low (z ∼ 1, e.g., Georgakakis et al. 2009;Cisternas et al. 2011) and high (z ∼ 2, e.g., Schawinski et al. 2011Schawinski et al. , 2012Kocevski et al. 2012, Treister et al. 2012 redshift. Theoretically, in-situ processes, such as disk instabilities or stochastic accretion of gas clouds, have also been invoked as triggers of AGN activity (e.g. Genzel et al. 2008, Dekel et al. 2009, Bournaud et al. 2011.\\nAGN clustering analysis provides a unique way to unravel the knots of this complex situation, providing important, independent constraints on the BH/galaxy formation and co-evolution. In the cold dark matterdominated Universe galaxies and their BHs are believed to populate the collapsed dark matter halos, thus reflecting the spatial distribution of dark matter in the Universe. The most common statistical estimator for large-scale clustering is the two-point correlation function (2pcf, Davis & Peebles 1983). This quantity measures the excess probability above random to find pairs of galaxies/AGN separated by a given scale r. By matching the observed 2pcf to detailed outputs of dark matter numerical simulations, one can infer the typical mass of Mean normalized phot-z Pdf of all CCL AGN with best-fit phot-z > 2.9. Right Panel: Normalized phot-z Pdf for the source lid766. This source has a best-fit photo-z value < 2.9, but a phot-z Pdf(z i >2.9)>0.001 (red thick line). The redshifts above this threshold, weighted by their Pdf, have been taken in account in the catalog used to estimate the 2pcf. the hosting dark matter halos. This is derived through the so called AGN bias b, enabling then to pin down the typical environment where AGN live. This in turn can provide new insights into the physical mechanisms responsible for triggering AGN activity.\\nThe 2pcf of AGN has been measured in optical large area surveys, such as the 2dF (2QZ, Croom et al. 2005;Porciani & Norberg 2006) and the Sloan Digital Sky Survey (SDSS, Li et al. 2006;Shen et al. 2009;Ross et al. 2009). These optical surveys are thousands of square degree fields, mainly sampling rare and high luminosity quasars. The amplitude of the 2PCF of quasars suggests that these luminous AGN are hosted by halos of roughly constant mass, a few times 10 12 M ⊙ , out to z =3-4 (Shanks et al. 2011). Models of major mergers between gas-rich galaxies appear to naturally reproduce the clustering properties of optically selected quasars as a function of luminosity and redshift (Hopkins et al. 2007a(Hopkins et al. , 2008Shen 2009;Bonoli et al. 2009). This supports the scenario in which major mergers dominate the luminous quasar population (Scannapieco et al. 2004;Neistein & Netzer 2014;Treister et al. 2012).\\nChandra surveys have contributed significantly to the study of the AGN clustering (e.g. CDFS-N, Gilli et al. 2009; Chandra/Bootes, Starikova et al. 2011, Allevato et al. 2014. Deep X-ray data can be used to draw conclusions on the faint portions of the AGN luminosity function, where a significant fraction of obscured sources is present. In particular, the Chandra survey in the two square degree COSMOS field (C-COSMOS, Elvis et al. 2009, Civano et al. 2012; Chandra COSMOS Legacy Survey, Civano et al. 2016) has allowed the investigation of the redshift evolution of the clustering properties of Xray AGN, for the first time up to z∼3. Interestingly, over a broad redshift range (z ∼ 0 -2) moderate luminosity AGN occupy DM halo masses of log M h ∼ 12.5-13.5 M ⊙ h −1 . The clustering strength of X-ray selected AGN has been measured by independent studies to be higher than that of optical quasars. Merger models usually fail in reproducing the data from X-ray surveys, opening the possibility of additional AGN triggering mechanisms (e.g., Allevato et al. 2011;Mountrichas & Georgakakis 2012) and/or multiple modes of BH accretion (e.g., Fanidakis et al. 2013). Recently, Mendez et al. (2015) and Gatti et al. (2016) have suggested that selection cuts in terms of AGN luminosity, host galaxy properties and redshift interval, might have a more relevant role in driving the differences often observed in the bias factor inferred from different surveys. The measurement of the AGN bias is crucial at high redshifts, especially at z>2-3, i.e. at the peak in the accretion history of the Universe. At z>3, Shen et al. (2007Shen et al. ( ,2009) measured for the first time the 2pcf of luminous SDSS-DR5 quasars (log L bol ∼ 10 47 erg/s) at z = 3.2 and 3.8. Even if with very large uncertainty, they found that these objects live in massive halos of the order of 10 13 M ⊙ h −1 . This result is consistent with models invoking galaxy major mergers as the main triggering mechanism for very luminous AGN. Recently, Eftekharzadeh et al. (2015) studying a sample of spectroscopically confirmed SDSS-III/BOSS quasars at 2.2≤z≤3.4, performed a more precise estimation of the quasar bias at high redshift. They found no evolution of the bias in three redshift bins, with halo masses equal to 3× and ∼0.6× 10 12 M ⊙ h −1 at z∼2.3 and ∼ 3, respectively.\\nThere are only a few attempts of measuring the clustering properties of X-ray AGN at z ∼ 3. Francke et al. (2008) estimated the bias of a small sample of X-ray AGNs (L bol ∼ 10 44.8 erg s −1 ) in the Extended Chandra Deep Field South (ECDFS), with very large uncertainty. They found indications that X-ray ECDFS AGNs reside in dark matter halos with minimum mass of log M min = 12.6 +0.5 −0.8 h −1 M ⊙ . On the other hand, Allevato et al. (2014) used a sample of Chandra and XMM-Newton AGN in COSMOS with moderate luminosity (log L bol ∼ 10 45.3 erg/s) at z =2.86. For the first time they estimated the bias of X-ray selected AGN at high redshift, suggesting that they inhabit halos of logM h = 12.37±0.10 M ⊙ h −1 . They also extended to z∼3 the result that Type 1 AGN reside in more massive halos than Type 2 AGN. Recently, Ikeda et al. (2015) estimated the clustering properties of low-luminosity quasars in COS-MOS at 3.1≤z≤4.5, using the cross-correlation between Lyman-Break Galaxies (LBGs) and 25 quasars with spectroscopic and photometric redshifts. They derived a 86% upper limit of 5.63 for the bias at z∼ 4.\\nIn this paper we want to extend the study of the clus- tering properties of X-ray selected AGN to z>3 using the new Chandra COSMOS-Legacy data. To this goal, we perform clustering measurements using techniques based on photometric redshift in the form of probability distribution functions (Pdfs), in addition to any available spectroscopy. This is motivated by the development in the last years of clustering measurement techniques based on photometric redshift Pdfs by Myers, White & Ball (2009), Hickox et al. (2011 and Mountrichas et al. (2013) and Georgakakis et al. (2014). One of the advantages of this new clustering estimator is that one can use in the analysis all sources not just the optically brighter ones for which spectroscopy is available. For this reason it is well suited to clustering investigations using future large X-ray AGN surveys, where the fraction of spectroscopic redshifts might be small.\\nThroughout the paper, all distances are measured in comoving coordinates and are given in units of Mpc h −1 , where h = H 0 /100km/s. We use a ΛCDM cosmology with Ω M = 0.3, Ω Λ = 0.7, Ω b = 0.045, σ 8 = 0.8. The symbol log signifies a base-10 logarithm. (Civano et al. 2016) for a total coverage of 2.2 deg 2 of the COSMOS field (Scoville et al. 2007). The X-ray source catalog consists of 4016 sources. 2076 (∼52%) have a secure spectroscopic redshift (spec-z) and for ∼96% the photometric redshift (photo-z) is available. As shown in Marchesi et al. (2016a), the spectroscopic redshifts have been obtained with different observing programs, as the zCOSMOS survey (Very Large Telescope/VIMOS; Lilly et al. 2007) and the Magellan/IMACS survey (Trump et al. 2007(Trump et al. , 2009). Other programs, many of which have been specifically targeting the CCLS have been The photo-zs are estimated following the procedure described in Salvato et al. (2011). Following Marchesi et al. (2016b), the accuracy of the photometric redshifts with respect to the whole spectroscopic redshift sample is σ ∆z/(1+zspec) =0.02, with a fraction of outliers ≃11%. At z>2.9 there are 9 outliers (∆z/(1 + z spec ) > 0.15), but for the remaining sources the agreement between specz and photo-z has the same quality of the whole sample. In detail, the normalized median absolute deviation σ N MAD =1.48×median( zspec− zphot /(1 + zspec)) = 0.012.\\nThe CCL AGN sample at 2.9<z<5.5 consists of 212 AGN detected in the 0.5-10 keV band, 107/212 with spec-zs and 105/212 with only phot-zs. To each of the 105 AGN with best-fit phot-z in the range 2.9≤z<5.5, is associated a probability distribution function (Pdf), which gives the probability of the source to be in the redshift range z i ± ∆z/2 with a binsize of ∆z = 0.01. The integrated area of the Pdf on all redshift bins z i is normalized to 1, i.e. i P df (z i ) = 1 for each AGN. We take into account for this analysis the redshift bins z i with Pdf(z i ) >0.001. Figure 1 shows the mean normalized phot-z Pdf for 105 sources with best-fit phot-z 2.9≤z<5.5. The effective contribution to the number of AGN at z≥2.9 of these 105 AGN weighted by the Pdf is 78.32 sources, i.e. 105 j=1 P df j (z ≥ 2.9)=78.32. In the CCL sample there are also 246 sources with phot-z < 2.9 but which contribute to the Pdf at 2.9≤ z ≤ 5.5 (i.e., with Pdf>0.001 at 2.9 ≤ z i ≤5.5 (see for example the Pdf of source lid766, whose nominal best-fit phot-z value is 2.51, in Figure 1). All these 246 sources have been taken in account in our analysis, using for each of them the Pdf of each bin of redshift 2.9 ≤ z i ≤5.5. The weighted contribution of these sources, i.e. the sum of all weights, is equal to 36.3 AGN (\\nTo all the 107 sources with known spec-z we assign a Pdf j = 1 to the spec-z value ( 107 j=1 P df j =107). To summarize, the total effective number of CCL AGN at 2.9≤z<5.5 weighted by the Pdf and used for the clustering measurements is 78.3 + 36.3 + 107 = 221.6 objects. Figure 2 shows the normalized redshift and 2-10 keV rest-frame X-ray luminosity distribution for our sample of CCL AGN at 2.9<z<5.5, when the phot-z Pdfs are used (black dotted line, z =3.36). The mean bolometric luminosity of this sample derived using the bolometric correction defined in Equation (21) of Marconi et al. (2004) is log L bol = 45.99 erg s −1 . For comparison, we also show the normalized distributions of our AGN sample when only the best-fit phot-zs are taken into account in addition to any available spec-z (gray solid line, z =3.34) and for the sample with known spec-z (reddashed line, z =3.35).\\n3. 2PCF USING PHOT-z PDFS 3.1. Projected 2pcf The most commonly used quantitative measure of large scale structure is the 2pcf, ξ(r), which traces the amplitude of AGN clustering as a function of scale. ξ(r) is defined as a measure of the excess probability dP , above what is expected for an unclustered random Poisson distribution, of finding an AGN in a volume element dV at a separation r from another AGN:\\nwhere n is the mean number density of the AGN sample (Peebles 1980). Measurements of ξ(r) are generally performed in comoving space, with r having units of h −1 Mpc. With a redshift survey, we cannot directly measure ξ(r) in physical space, because peculiar motions of galaxies distort the line-of-sight distances inferred from redshift. To separate the effects of redshift distortions, the spatial correlation function is measured in two dimensions r p and π, where r p and π are the projected comoving separations between the considered objects in the directions perpendicular and parallel, respectively, to the mean line-of-sight between the two sources. Following Davis & Peebles (1983), r 1 and r 2 are the redshift positions of a pair of objects, s is the redshift-space separation (r 1 − r 2 ), and l = 1 2 (r 1 + r 2 ) is the mean distance to the pair. The separations between the two considered objects across r p and π are defined as:\\nRedshift space distortions only affect the correlation function along the line of sight, so we estimate the socalled projected correlation function w p (r p ) (Davis & Peebles 1983):\\nwhere ξ(r p , π) is the two-point correlation function in terms of r p and π, measured using the Landy & Szalay (1993, LS) estimator:\\nwhere DD', DR' and RR' are the normalized data-data, data-random and random-random pairs. In this classic approach of estimating the redshiftspace correlation function, in presence of accurate speczs, when a data-data pair with separation (r p , π) is found, the pair number is incremented by one, i.e. DD(r p ,π) = DD(r p , π) + 1. Following Georgakakis et al. (2014), in the generalized clustering estimator the number of datadata pairs with projected and line of sight separation (r p , π) is, instead, incremented by the product Pdf 1 (z i ) × Pdf 2 (z j ):\\nwhere Pdf 1 (z i ) and Pdf 2 (z j ) are the Pdf values (per redshift bin) of the source 1 at z = z i and of the source 2 at z = z j respectively.\\nThe measurements of the 2pcf requires the construction of a random catalog with the same selection criteria and observational effects as the data. To this end, we constructed a random catalog where each simulated source is placed at a random position in the sky, with its flux randomly extracted from the catalog of real source fluxes. The simulated source is kept in the random sample if its flux is above the sensitivity map value at that random position (Miyaji et al. 2007;Cappelluti et al. 2009). The corresponding redshift for each random object is then assigned based on the smoothed redshift distribution of the AGN sample, where each redshift is weighted by the Pdf associated to that redshift for the particular source. Since the phot-z Pdfs are already Bias as a function of πmax for 221.6 CCL AGN at 2.9≤z<5.5, when the projected 2pcf is measured using the generalized clustering estimator based on phot-z Pdfs in addition to any available spec-z (black dots). For comparison, the gray triangles and red squares show the bias of 212 AGN with known spec or best-fit phot-z and 107 AGN with known spec-zs at 2.9<z<5.5 when the classic LS estimator (i.e. no phot-z Pdfs) is used. Right Panel: Projected 2pcf for πmax=200 h −1 Mpc. The 1σ errors on wp(rp) are the square root of the diagonal components of the covariance matrix. The continuous line represents the DM projected 2pcf estimated at z =3.36.\\ntaken into account in the generation of the random redshifts, we decided to assign Pdf=1 to each random source.\\nIn the halo model approach, the large scale amplitude signal is due to the correlation between objects in distinct halos and the bias parameter defines the relation between the large scale clustering amplitude of the AGN correlation function and the DM 2-halo term:\\nWe first estimated the DM 2-halo term at the median redshift of the sample, using:\\nintegrating up to 200 h −1 Mpc along π, where:\\nis the linear power spectrum, assuming a power spectrum shape parameter Γ = Ω m h = 0.2 (Efstathiou et al. 1992) which corresponds to h = 0.7.\\n3.2. z-space correlation function Similarly, we can estimate the z-space correlation function ξ(s) using Equation (5) and (6), written now as a function of the redshift-space separation s = (r 1 − r 2 ) between the sources. ξ(s) is affected by perturbations in the cosmological redshifts due to peculiar velocities and redshift errors. The z-space power spectrum can be modelled in polar coordinates as follow (e.g. Kaiser 1987, Peacock et al. 2001:\\nwhere k = k 2 ⊥ + k 2 , k ⊥ and k are the wavevector components perpendicular and parallel to the line of sight, respectively. µ = k ⊥ /k , P DM (k) is the dark matter power spectrum, b is the linear bias factor, f is the growth rate of density fluctuations. σ is the displacement along the line of sight due to random perturbations of cosmological redshifts.Assuming standard gravity, we approximated the growth rate f ≃ Ω M (z) γ , with γ = 0.545 (e.g. Sereno et al. 2015).\\nThe f µ 2 term parametrizes the coherent motions due to large-scale structures, enhancing the clustering signal on all scales. The exponential cut-off term describes the random perturbations of the redshifts caused by both non-linear stochastic motions and redshift errors. The integration of Eq.(7) over the angle µ, and then the Fourier anti-transformation gives:\\nThe main term, ξ ′ (s), is the Fourier anti-transform of the monopole P ′ (k):\\nthat corresponds to the model given by Equation (10) when neglecting the dynamic distortion term. In our case, photo-z errors perturb the most the distance measurements along the line of sight. Therefore the small-scale random motions are negligible with respect to photo-z errors. The cut-off scale in Eq. (12) can thus be written as:\\nwhere H(z) is the Hubble function computed at the median redshift of the sample, and σ z is the typical photo-z error.\\nIn this case, knowing the cut-off scale, the AGN bias can be derived from the Fourier anti-transform of the monopole P ′ (k), i.e.:\\nwhere ξ(s) is the observed z-space 2pcf of our AGN sample.\\n4. RESULTS 4.1. w p (r p ) and Bias We have measured the 2pcf of 221.6 CCL AGN at 2.9≤z<5.5, using the generalized clustering estimator defined in Equation (6), based on phot-z Pdfs in addition to any available spec-z. The projected 2pcf w p (r p ) is then estimated using Equation (4).\\nThe typical value of π max used in clustering measurements of both optically-selected luminous quasars and X-ray selected AGN is ∼20-100 h −1 Mpc (e.g. Zehavi et al. 2005, Coil et al. 2009, Krumpe et al. 2010, Allevato et al. 2011. The optimum π max value can be determined by measuring the 2pcf for different π max and then adopting the value at which the amplitude of the signal appears to level off. Figure 3 (Left Panel) shows the bias factor estimated for different values of π max in Equation (4), when the phot-z Pdfs are used in addition to any available specz. For comparison, we also estimated the bias for case i) 107 AGN with known spec-zs; case ii) 107+105 AGN with known spec-z or best-fit phot-zs. In these cases the 2pcf is measured using the classic LS estimator and the Pdf is set to unity for each source.\\nAs expected, when including phot-zs in the analysis, the bias levels-off only at large scales, because of the large uncertainties in the redshifts measured via photometric methods (Georgakakis et al. 2014). Surprisingly, even if the error bars are large, an increase of the bias factor at π max >100 h −1 Mpc is suggested also when only spec-zs are used with the classic 2pcf estimator. This suggests that a fraction of spec-zs might be affected by large errors (see Section 4.2).\\nThe amplitude of the projected 2pcf of CCL AGN measured using the generalized clustering estimator converges at π max ≥ 200 h −1 Mpc. We decide to use π max = 200 h −1 Mpc in order to balance the advantage of integrating out redshift-space distortions against the disadvantage of introducing noise from uncorrelated line-ofsight structure. Figure 3 (Right Panel) shows the projected 2pcf estimated using the generalized clustering estimator, whith π max =200 h −1 Mpc. The 1σ errors on w p (r p ) are the square root of the diagonal components of the covariance matrix (Miyaji et al. 2007, Krumpe et al. 2010) estimated using the bootstrap method. The latter quantifies the level of correlation between different bins. For comparison, we also estimate the projected 2pcf for case i) 107 AGN with known spec-zs; case ii) 107+105 AGN with known spec-z or best-fit phot-zs. Note that in these cases the classic LS estimator is used (i.e. Pdf = 1 for each source) and π max is fixed to 200 h −1 Mpc also in these cases. Following Eq. 7, we derive the best-fit bias by using a χ 2 minimization technique with 1 free parameter in the range r p = 1 -30 h −1 Mpc, where χ 2 = ∆ T M −1 cov ∆. In detail, ∆ is a vector composed of w p (r p ) − w mod (r p ) (see Equations 4 and 7), ∆ T is its transpose and M −1 cov is the inverse of covariance matrix. The latter full covariance matrix is used in the fit to take into account the correlation between errors.\\nAs shown in Table 1, we derived a bias for our sample of CCL AGN equal to b = 6.6 +0.60 −0.55 at z =3.36. Following the bias-mass relation b(M h , z) described in van den Bosch (2002) and Sheth et al. (2001), the AGN bias corresponds to a typical mass of the hosting halos of log M h = 12.83 +0.12 −0.11 h −1 M ⊙ . It is worth noticing that this is a typical/characteristic mass of the halos hosting CCL AGN. Only the HOD modelling of the clustering signal at all scales can provide the entire hosting halo mass distribution for this sample.\\nThe bias has a larger uncertainty when derived from the 2pcf estimated without the phot-z Pdfs. In detail, we find b = 7.5 +1.6 −1.7 at z =3.35 for 107 AGN with known spec-zs (case i) and b = 6.48 +1.27 −1.36 at z =3.34 for 107+105 AGN with known spec or best-fit phot-zs (case ii). Note that in these cases the 2pcf is measured using the classic LS estimator and the Pdf is set to unity for each source.\"\n },\n {\n \"title\": \"ξ(s) and phot-z errors\",\n \"content\": \"To investigate the effect of phot-z errors on the 2pcf, we also measured the z-space correlation function ξ(s). dition to any available spec-z. For comparison, the gray triangles show ξ(s) for case i) and case ii).\\nAs described in section 3.2, ξ(s) is affected by the Kaiser effect that enhances the clustering signal at all scales and by phot-z errors, that are modelled by using an exponential cut-off in the z-space power spectrum. For our sample of CCL AGN the typical error on phot-zs is σ z =0.012×(1 + z spec ) = 0.052 at z =3.4. This implies a cut-off scale σ = 43.45 h −1 Mpc (see Equation 13). Including the phot-z damping in the modeling of ξ(s), we derived the best-fit bias by using Equation 14 and the χ 2 minimization technique with 1 free parameter in the range s=1-50 h −1 Mpc. In particular, for the z-space 2pcf measured using the generalized clustering estimator we derived b = 6.53 +0.52 −0.58 . For the sample of 212 AGN with known spec or best-fit phot-zs for which the classic clustering estimator is used, we derived b = 6.96 +0.72 −0.73 . These values are in perfect agreement with the bias obtained from the projected 2pcf with π max = 200 h −1 Mpc. This confirms that the convergence of the projected 2pcf observed only at large scales (π max ≥200 h −1 Mpc) is due to large phot-z errors. In fact, for large redshift errors and small survey area it is necessary to integrate the correlation function up to large scales to fully correct for them.\\nWe also estimated the bias using the z-space 2pcf for 107 AGN with known spec-zs. In general, we do not expect spec-zs to be affected by large errors. If we do not include the phot-z damping in the model of ξ(s), we obtain for this sample a bias b = 5.86 +1.13 −1.05 , which is lower than b = 7.5 +1.6 −1.7 obtained using the projected 2pcf and π max = 200 h −1 Mpc. A better fit of ξ(s) and a larger bias can be obtained only if we include in the model specz errors of the order of σ z = 0.02-0.025. In particular, for σ z = 0.020 (cut-off scale σ = 16.7 h −1 Mpc) and σ z = 0.025 (σ = 20.9 h −1 Mpc), we derived b = 7.4 +1.35 −1.40 and b = 8.01 +1.4 −1.5 , respectively. However, given the low statistics, smaller values of σ z and then of the bias can not be ruled-out. This error redshift is larger than what is expected for a spectroscopic sample of AGN. However, the presence in the spectroscopic sample of ∼20% of the objects (21/107) with a low quality flag (i.e. flag = 1.5, corresponding to low quality spectra, and therefore not fully reliable redshift, but with known phot-z such that σ ∆z/(1+zspec) <0.1) could explain both the improvement of the fit including such an error in the analysis and the increase of the bias with increasing π max up to ∼200 h −1 Mpc when using only AGN with spectroscopic redshift.\"\n },\n {\n \"title\": \"DISCUSSION\",\n \"content\": \"In this section we compare our results with previous measurements using COSMOS AGN at z∼3 and with previous studies at similar redshift. We also interpret our results in terms of AGN triggering mechanisms. Figure 5 (right panel) shows the redshift evolution of the AGN bias estimated using moderate luminosity AGN detected in different X-ray surveys. Interestingly, moderate luminosity AGN occupy DM halo masses of log M h ∼ 12.5-13.5 M ⊙ h −1 up to z∼2, tracing a constant group-sized halo mass. Allevato et al. (2011) have shown that XMM COSMOS AGN (L bol ∼ 10 45.2 erg s −1 ) reside in DM halos with constant mass equal to logM h = 13.12±0.07 M ⊙ h −1 up to z = 2. They also argue that this high bias can not be reproduced assuming that major merger between gas-rich galaxies (Shen 2009) is the main triggering scenario for moderate luminosity AGN. By contrast, at z∼3, Allevato et al. (2014) found a drop in the mass of the hosting halos, with Chandra and XMM-Newton COSMOS AGN (L bol ∼ 10 45.3 erg s −1 ), inhabiting halos of logM h = 12.37±0.10 M ⊙ h −1 .\"\n },\n {\n \"title\": \"Redshift evolution of the AGN bias\",\n \"content\": \"In the present paper we measure a bias for 221.6 CCL AGN at 2.9≤z≤5.5 equal to b = 6.6 +0.6 −0.55 , that corresponds to a typical mass of the hosting dark matter halos of logM h = 12.83 +0.12 −0.11 h −1 M ⊙ . This result suggests a higher bias for CCL AGN compared to previous studies in COSMOS at z∼ 3. In fact, Allevato et al. (2014) found a bias of 3.85 +0.22 −0.21 at z =2.8, using a sample of XMM and Chandra AGN. Although the two samples only partially overlap, we argue that the most likely explanation of these differences lies in the small value of π max (= 40 h −1 Mpc) used in Allevato et al. (2014). As shown in Figure 3, the bias strongly increases with π max due to the large phot-z errors and the use of 40 h −1 Mpc might produce an underestimated clustering signal.\\nTo verify this effect, we took the same sample used in Allevato et al. (2014), i.e. 346 XMM and Chandra COS-MOS AGN with known spec or phot-z > 2.2. As already mentioned, Allevato et al. (2014) used the classic LS estimator where the phot-zs Pdfs are not taken into account. Using their same classic approach, we estimated the projected 2pcf for different values of π max and found that the clustering signal converges only at π max ≥ 200 h −1 Mpc. In particular, we derived a bias b = 5.8 +0.61 −0.65 for π max = 200 h −1 Mpc, which corresponds to a typical hosting halo mass logM h = 12.92 +0.13 −0.18 . As shown in Figure 5 (left panel) the bias estimated using π max = 200 h −1 Mpc is in agreement with the bias of CCL AGN at z∼3.4 as derived in the present work. It is worth noticing that also the mean luminosity of the samples is increasing with z, with mean L bol = 10 45.5 erg/s for XMM and Chandra COSMOS AGN at z∼3 and ∼46 erg/s for CCL AGN at higher z.\\nThese results imply that at z>3: i) the typical hosting halo mass of moderate luminosity AGN remains almost constant with redshift, going from ∼8.3 × 10 12 at z=2.8 to ∼ 6.7× 10 12 M ⊙ h −1 at z∼3.4, since a lower mass is required to yield the same bias at a higher redshift; ii) moderate luminosity AGN still inhabit group-sized halos at high redshift, but slightly less massive than observed in different independent studies using X-ray selected AGN at z≤2.\"\n },\n {\n \"title\": \"Previous studies at high redshift\",\n \"content\": \"The evolution of the bias with redshit has been studied in Eftekharzadeh et al. (2015) for SDSS-III/BOSS quasars at 2.2 ≤z ≤3.4. They investigated the redshift dependence of quasar clustering in three redshift bins and found no evolution of the correlation length and bias. In terms of halo mass, this corresponds to a characteristic halo mass that decreases with redshift, with halo masses of 3× (6×) and ∼0.6× (1.3×) 10 12 M ⊙ h −1 at z∼2.3 and ∼ 3, respectively, where the dark matter halo masses are estimated using Tinker et al. 2010(Sheth et al. 2001).\\nThese results are surprisingly different in terms of bias Shen (2009) at z = 3 (in black) and z = 3.5 (in gold), Hopkins et al. (2007b) at z=3 (blue dotted line) and Conroy & White (2013) at z=3.4 (green dashed line) compared to previously estimated bias factors for optically selected quasars at z=3.2 , gray hexagon), X-ray selected AGN at z=3 (Francke et al. 2008, open triangle), Chandra and XMM-COSMOS AGN at z=2.9 as estimated in Allevato et al. (2014, filled black square) and as re-estimated in the present work (black empty square) and our results (red circle). The errors on the L bol axis correspond to the dispersion of the bolometric luminosity distributions for the different subsets. and halo mass when compared to Shen et al. (2009). At z∼3, Eftekharzadeh et al. (2015) derived halo masses that are close to an order-of-magnitude smaller than those presented in Shen et al. (2009). In this latter study, they measured the bias of SDSS-DR5 quasars with mean L bol ∼ 10 47 erg/s, at z = 3.2. Even if with large uncertainties, their results suggest that luminous quasars reside in massive halos with mass few times 10 13 h −1 M ⊙ (based on Sheth et al. 2001). Although the two samples do not completely overlap, Eftekharzadeh et al. (2015) argue that the most likely explanation of these differences lies in the improvements in SDSS photometric calibration and target selection algorithms as well as in the much larger number of quasars that afford greater measurements precision compared to Shen et al. (2009). Our results are in disagreement with the bias factor equal to b = 3.57±0.09 at z∼3 derived in Eftekharzadeh et al. (2015). This disagreement might be due to the slightly different average redshift and the significantly different luminosity (almost one order of magnitude) of the samples used in the two different studies. An additional important difference between the samples is that our catalog of CCL AGN includes both Type 1 and Type 2 AGN, while Eftekharzadeh et al. (2015) use Type 1 BOSS luminous quasars. It is also worth noticing that in Eftekharzadeh et al. (2015) the bias is derived modelling the z-space 2pcf in the Kaiser formalism, i.e. not including the effect of random peculiar velocities and redshift errors. The r 0 value (the correlation length of the projected 2pcf) reported in their Table 5 would suggest, instead, a higher bias (∼5) when derived assuming that w p (r p ) is modelled by a power law with index γ= -2 in the range r p =4-30 h −1 Mpc. At a slightly lower redshift, Francke et al. (2008) estimated the correlation function of a small sample of X-ray AGN with L bol ∼ 10 44.8 erg s −1 , in the Extended Chandra Deep Field South (ECDFS). Given the small number of sources they only infer a minimum mass of halos hosting X-ray ECDFS AGN of logM min = 12.6 +0.5 −0.8 h −1 M ⊙ (based on Sheth et al. 2001 formalism). Our result is in agreement with this study at z∼3, but measured with higher accuracy.\\nRecently, Ikeda et al. (2015) investigated the clustering properties of low-luminosity quasars at z∼4 using the cross-correlation function of quasars and LBGs in the COSMOS field. They estimated the bias factor for a spectroscopic sample of 16 quasars and a total sample of 25 quasars including sources with photometric redshifts. They obtained a 86% upper limit for the bias of 5.63 and 10.50 for the total and spectroscopic sample, respectively. Figure 6 shows the predicted evolution of the AGN bias as a function of the bolometric luminosity, computed according to the framework of the growth and evolution of BHs presented in Shen (2009, see also Shankar 2010) at z = 3 and 3.5 Their model assumes that quasar activity is triggered by major mergers of host halos (e.g. Kauffmann & Haehnelt 2000).\"\n },\n {\n \"title\": \"Comparison to theoretical models\",\n \"content\": \"The major merger model is quite successful in predicting the bias of COSMOS AGN at z=2.8 as presented in Allevato et al. (2014), but underpredicts the bias re-estimated in the present work using the same AGN sample and π max = 200 h −1 Mpc. Given the large error bars, the model is in broad agreement with the bias of luminous quasars at similar redshift as measured in Shen et al. (2009) and X-ray AGN as estimated in Francke et al. (2008).\\nThe prediction from the model slightly underpredicts our results for CCL AGN at z∼3.4. We verified that the mismatch between merger models and our data does not change if a few parameters, such as the light curve or the host halo mass distribution are changed in the major merger model. In fact, our result is still not well reproduced by the predictions from a modified Shen (2009) model in which the post-peak descending phase is cut out, with all other parameters held fixed. On the other hand, a model characterized by a steepening in the L peak -M h relation mainly implying that preferentially lowerluminosity quasars are now mapped to more massive, less numerous host DM halos, still underpredicts our results.\\nA similar mismatch has also been found for a sample of CCL AGN at z=3-6.5 in terms of observed number counts (Marchesi et al. 2016b). In fact, they verified that the reference model overproduces the observed number counts by a factor of 3 to 10, depending on the redshift.\\nWe also compare the observations with the theoretical model presented in Hopkins et al. (2007b), that adopts the feedback-regulated quasar light-curve/lifetime models from Hopkins at al. (2006) derived from numerical simulations of galaxy mergers that incorporate BH growth. Even if we assume an evolution with redshift, this model underpredicts the bias factor of CCL AGN.\\nA similar tension is also observed when comparing with the semi-empirical model presented in Conroy & White (2013). In the latter, the BH mass is linearly related to galaxy mass and connected to dark matter halos via empirical constrained relations. This model makes no assumption about what triggers the AGN activity and includes a scatter in the AGN luminosityhalo mass relation, contrary to Hopkins et al. (2007b) and Shen (2009) . Conroy & White (2013) show that this semi-empirical model naturally reproduces the clustering properties of quasars at z<3, but shows some tension at higher redshift. They argue that this disagreement can be explained if AGN have a duty cycle close to unity at z>3, indicating that we approach the era of rapid BH growth in the early universe.\\nRecently, Gatti et al. (2016) have used advanced semi analytic models (SAMs) of galaxy formation, coupled to halo occupation modelling, to investigate AGN triggering mechanisms such as galaxy interactions and disk instabilities. They compared the predictions with high redshift clustering measurements from Allevato et al. (2014), Shen et al. (2009) andEftekharzadeh et al. (2015). Their SAMs underpredict the bias of luminous quasars shown in Shen et al. (2009). The mismatch is reduced when the models are compared to Eftekharzadeh et al. (2015). They pointed out that, irrespective of the exact implementations in their SAMs, at low-z moderateluminosity AGN (L bol ∼10 44−46 erg/s) mainly inhabit halos with mass ∼10 12−13 M ⊙ for both galaxy interaction and disk instabilities models (even if disk instabilities do not trigger the most luminous AGN with L bol ≥10 47 erg/s). At higher redshift (z∼2.5), structures with mass greater than M h > 10 13 M ⊙ become significantly rarer, relegating active galaxies to live mainly in less massive environment. Moreover, in all models only galaxies with stellar masses above 10 11 M ⊙ would be able to host AGN with luminosity of L bol ∼10 46 erg/s and highly biased such as COSMOS AGN at z>2-3. This would imply that the characteristic M star /M h ratio in AGN hosts should increase with lookback time, as expected from basic considerations on number densities evolution between the halo mass function and AGN luminosity function (e.g., ).\\n6. CONCLUSIONS We use the new CCL catalog to probe the projected and redshift-space 2pcf of X-ray selected AGN for the first time at 2.9≤z≤5.5, using the generalized clustering estimator based on phot-z Pdfs in addition to any available spec-z. We model the clustering signal with the 2-halo model and we derive the bias factor and the typical mass of the hosting halos. Our key results are:\\n1. At z∼3.4, CCL AGN have a bias b = 6.6 +0.60 −0.55 , which corresponds to a typical mass of the hosting halos of log M h = 12.83 +0.12 −0.11 h −1 M ⊙ . A similar bias is derived using the z-space 2pcf, modelled including the typical phot-z error σ z = 0.052 of our sample. This confirms that the convergence of the projected 2pcf observed only at large scales (π max ≥ 200 h −1 Mpc) is due to large phot-z errors.\\n2. A slightly larger bias b = 7.5 +1.6 −1.7 (but consistent within the error bars) is found using a sample of 107 CCL AGN with known spec-z. The modelling of ξ(s) suggests that this larger bias can be explained assuming that spec-zs are affected by errors of the order of σ z = 0.02 − 0.025. This would explain the convergence of the projected 2pcf surprisingly observed only at π max ≥ 200 h −1 Mpc, even when phot-zs are not included in the analysis. However, given the low statistics smaller spec-z errors and then bias can not be ruled-out.\\n3. We estimate the bias factor for the sample of 346 XMM and Chandra AGN used in Allevato et al. (2014) using π max = 200 h −1 Mpc in estimating the projected 2pcf and then accounting for the large phot-z errors. In particular we found b = 5.8 +0.61 −0.55 , which is significantly larger than the AGN bias measured in Allevato et al. (2014) and corresponds to logM h = 12.92 +0.13 −0.18 at z=2.8.\"\n },\n {\n \"title\": \"4.\",\n \"content\": \"Our results suggest only a slight increase of the bias factor of COSMOS AGN at z 3, with the typical hosting halo mass of moderate luminosity AGN almost constant with redshift and equal to logM h = 12.92 +0.13 −0.18 at z=2.8 and log M h = 12.83 +0.12 −0.11 at z∼3.4, respectively.\\n5. The observed redshift evolution of the bias of COS-MOS AGN implies that moderate luminosity AGN still inhabit group-sized halos, but slightly less massive than observed in different independent studies using X-ray AGN at z≤ 2.\\n6. Theoretical models presented in Shen (2009) and Hopkins et al. (2007b) that assume an AGN activity mainly triggered by major mergers of host halos underpredict our results at z∼3.4 for CCL AGN with mean L bol ∼ 10 46 erg s −1 . A similar tension is also observed when comparing to the semi-empirical models presented in Conroy & White (2013). In the latter model, this disagreement can be explained if AGN have a duty cycle approaching unity at z>3. On the other hand, following the semi-analytic models presented in Gatti et al. (2016), in both galaxy interaction and disk instability models only galaxies with stellar masses above 10 11 M ⊙ would be able to host AGN with luminosity of L bol ∼10 46 erg/s and highly biased such as COSMOS AGN at z>2-3.\\nOnly future facilities, like the X-ray Surveyor (Vikhlinin 2015) and Athena (PI K. P. Nandra), will be able to collect sizable samples (∼1000s) of low luminosity (L X <10 43 erg/s) AGN at z>3 (Civano 2015), allowing to explore the clustering for significantly less luminous source and to test AGN triggering scenarios at different AGN luminosities.\"\n },\n {\n \"title\": \"Fig. 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Fig. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"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 {\n \"title\": \"TABLE 1\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"-Left Panel:","content":"Fig. 1 .\n1-Left Panel:"},{"id":"","caption":"-Redshift (Left Panel) and 2-10 keV X-ray luminosity (Right Panel) distribution for 107 CCL AGN with known spec-z (red dashed line), 212 AGN with known spec or best phot-z (solid grey line) and 221.6 AGN with known spec-z or phot-z weighted by the Pdf (black dotted line), at 2.9≤z≤5.5","content":"Fig. 2 .\n2-Redshift (Left Panel) and 2-10 keV X-ray luminosity (Right Panel) distribution for 107 CCL AGN with known spec-z (red dashed line), 212 AGN with known spec or best phot-z (solid grey line) and 221.6 AGN with known spec-z or phot-z weighted by the Pdf (black dotted line), at 2.9≤z≤5.5"},{"id":"","caption":"AGN SAMPLE AT 2.9≤z≤5.5 The Chandra-COSMOS-Legacy survey (CCLS) is the combination of the 1.8 Ms C-COSMOS survey (Elvis et al 2009) with 2.8 Ms of new Chandra ACIS-I observations","content":"2 .\n2AGN SAMPLE AT 2.9≤z≤5.5 The Chandra-COSMOS-Legacy survey (CCLS) is the combination of the 1.8 Ms C-COSMOS survey (Elvis et al 2009) with 2.8 Ms of new Chandra ACIS-I observations"},{"id":"","caption":"carried with Keck-MOSFIRE (P.I. F. Civano, N. Scoville), Keck-DEIMOS (P.I.s Capak, Kartaltepe, Salvato, Sanders, Scoville, Hasinger), Subaru-FMOS (P.I. J. Silverman), VLT-FORS2 (P.I. J. Coparat) and Magellan-PRIMUS (public data).","content":"carried with Keck-MOSFIRE (P.I. F. Civano, N. Scoville), Keck-DEIMOS (P.I.s Capak, Kartaltepe, Salvato, Sanders, Scoville, Hasinger), Subaru-FMOS (P.I. J. Silverman), VLT-FORS2 (P.I. J. Coparat) and Magellan-PRIMUS (public data)."},{"id":"","caption":"-Left Panel:","content":"Fig. 3 .\n3-Left Panel:"},{"id":"","caption":"-Redshift-space correlation function of 221.6 CCL AGN at 2.9<z<5.5 derived using the generalized clustering estimator based on phot-z Pdfs in addition to any available spec-z (black circles in the bottom panel). For comparison, the gray triangles (Middle Panel) and the red squares (Upper Panel) show the 2pcf of 212 AGN with known spec or best-fit phot-z and 107 AGN with known spec-zs at 2.9<z<5.5 estimated using the classic LS estimator (i.e. Pdf=1 for each source). The dotted lines show the best-fit models obtained including the dominant term b 2 ξ ′ in Equation 11, while the dashed lines is the model without the photo-z damping term. The error bars show the square roots of the diagonal values of the covariance matrix.","content":"Fig. 4 .\n4-Redshift-space correlation function of 221.6 CCL AGN at 2.9<z<5.5 derived using the generalized clustering estimator based on phot-z Pdfs in addition to any available spec-z (black circles in the bottom panel). For comparison, the gray triangles (Middle Panel) and the red squares (Upper Panel) show the 2pcf of 212 AGN with known spec or best-fit phot-z and 107 AGN with known spec-zs at 2.9<z<5.5 estimated using the classic LS estimator (i.e. Pdf=1 for each source). The dotted lines show the best-fit models obtained including the dominant term b 2 ξ ′ in Equation 11, while the dashed lines is the model without the photo-z damping term. The error bars show the square roots of the diagonal values of the covariance matrix."},{"id":"","caption":"Figure 4shows ξ(s) for 221.6 CCL AGN using the generalized clustering estimator based on phot-z Pdfs in ad--Left Panel: Bias of CCL AGN at z∼3.4 estimated using the projected 2pcf (empty points) and the z-space 2pcf (filled points). The empty symbols are offset in the horizontal direction by +0.04 for clarity. For comparison the bias of XMM and Chandra COSMOS AGN at z = 2.8 is shown as presented inAllevato et al. (2014) using πmax = 40 h −1 Mpc (filled hexagon) and when the bias is re-estimated using πmax = 200 h −1 Mpc. The dashed lines show the expected b(z) of typical DM halo with mass of 12, 12.5, 13, h −1 M ⊙ in log scale (from bottom to top), based onSheth et al. (2001) formalism. Right Panel: Bias parameter as a function of redshift for X-ray selected AGN (black) from previous studies as described in the legend. The red circle at z∼3.4 show the bias factors as estimated in this work for CCL AGN. The red square shows the bias factor as re-estimated in the present paper using the same catalog of XMM and Chandra-COSMOS AGN used inAllevato et al. (2014), but with πmax=200 h −1 Mpc. The dashed lines show the expected b(z) of typical DM halo with mass of 11.5, 12, 12.5, 13, 13.5 h −1 M ⊙ in log scale (from bottom to top), based onSheth et al. (2001). Bias factors from different studies are converted to a common cosmology (Ω Λ = 0.7, Ωm = 0.3, σ 8 = 0.8).","content":"Fig. 5 .\n5Figure 4shows ξ(s) for 221.6 CCL AGN using the generalized clustering estimator based on phot-z Pdfs in ad--Left Panel: Bias of CCL AGN at z∼3.4 estimated using the projected 2pcf (empty points) and the z-space 2pcf (filled points). The empty symbols are offset in the horizontal direction by +0.04 for clarity. For comparison the bias of XMM and Chandra COSMOS AGN at z = 2.8 is shown as presented inAllevato et al. (2014) using πmax = 40 h −1 Mpc (filled hexagon) and when the bias is re-estimated using πmax = 200 h −1 Mpc. The dashed lines show the expected b(z) of typical DM halo with mass of 12, 12.5, 13, h −1 M ⊙ in log scale (from bottom to top), based onSheth et al. (2001) formalism. Right Panel: Bias parameter as a function of redshift for X-ray selected AGN (black) from previous studies as described in the legend. The red circle at z∼3.4 show the bias factors as estimated in this work for CCL AGN. The red square shows the bias factor as re-estimated in the present paper using the same catalog of XMM and Chandra-COSMOS AGN used inAllevato et al. (2014), but with πmax=200 h −1 Mpc. The dashed lines show the expected b(z) of typical DM halo with mass of 11.5, 12, 12.5, 13, 13.5 h −1 M ⊙ in log scale (from bottom to top), based onSheth et al. (2001). Bias factors from different studies are converted to a common cosmology (Ω Λ = 0.7, Ωm = 0.3, σ 8 = 0.8)."},{"id":"","caption":"-Predicted bias as a function of bolometric luminosity, computed according to","content":"Fig. 6 .\n6-Predicted bias as a function of bolometric luminosity, computed according to"},{"id":"","caption":"Properties of the AGN Samples","content":"TABLE 1\n1Properties of the AGN Samples"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"-Left Panel:\",\n \"content\": \"Fig. 1 .\\n1-Left Panel:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Redshift (Left Panel) and 2-10 keV X-ray luminosity (Right Panel) distribution for 107 CCL AGN with known spec-z (red dashed line), 212 AGN with known spec or best phot-z (solid grey line) and 221.6 AGN with known spec-z or phot-z weighted by the Pdf (black dotted line), at 2.9≤z≤5.5\",\n \"content\": \"Fig. 2 .\\n2-Redshift (Left Panel) and 2-10 keV X-ray luminosity (Right Panel) distribution for 107 CCL AGN with known spec-z (red dashed line), 212 AGN with known spec or best phot-z (solid grey line) and 221.6 AGN with known spec-z or phot-z weighted by the Pdf (black dotted line), at 2.9≤z≤5.5\"\n },\n {\n \"id\": \"\",\n \"caption\": \"AGN SAMPLE AT 2.9≤z≤5.5 The Chandra-COSMOS-Legacy survey (CCLS) is the combination of the 1.8 Ms C-COSMOS survey (Elvis et al 2009) with 2.8 Ms of new Chandra ACIS-I observations\",\n \"content\": \"2 .\\n2AGN SAMPLE AT 2.9≤z≤5.5 The Chandra-COSMOS-Legacy survey (CCLS) is the combination of the 1.8 Ms C-COSMOS survey (Elvis et al 2009) with 2.8 Ms of new Chandra ACIS-I observations\"\n },\n {\n \"id\": \"\",\n \"caption\": \"carried with Keck-MOSFIRE (P.I. F. Civano, N. Scoville), Keck-DEIMOS (P.I.s Capak, Kartaltepe, Salvato, Sanders, Scoville, Hasinger), Subaru-FMOS (P.I. J. Silverman), VLT-FORS2 (P.I. J. Coparat) and Magellan-PRIMUS (public data).\",\n \"content\": \"carried with Keck-MOSFIRE (P.I. F. Civano, N. Scoville), Keck-DEIMOS (P.I.s Capak, Kartaltepe, Salvato, Sanders, Scoville, Hasinger), Subaru-FMOS (P.I. J. Silverman), VLT-FORS2 (P.I. J. Coparat) and Magellan-PRIMUS (public data).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Left Panel:\",\n \"content\": \"Fig. 3 .\\n3-Left Panel:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Redshift-space correlation function of 221.6 CCL AGN at 2.9<z<5.5 derived using the generalized clustering estimator based on phot-z Pdfs in addition to any available spec-z (black circles in the bottom panel). For comparison, the gray triangles (Middle Panel) and the red squares (Upper Panel) show the 2pcf of 212 AGN with known spec or best-fit phot-z and 107 AGN with known spec-zs at 2.9<z<5.5 estimated using the classic LS estimator (i.e. Pdf=1 for each source). The dotted lines show the best-fit models obtained including the dominant term b 2 ξ ′ in Equation 11, while the dashed lines is the model without the photo-z damping term. The error bars show the square roots of the diagonal values of the covariance matrix.\",\n \"content\": \"Fig. 4 .\\n4-Redshift-space correlation function of 221.6 CCL AGN at 2.9<z<5.5 derived using the generalized clustering estimator based on phot-z Pdfs in addition to any available spec-z (black circles in the bottom panel). For comparison, the gray triangles (Middle Panel) and the red squares (Upper Panel) show the 2pcf of 212 AGN with known spec or best-fit phot-z and 107 AGN with known spec-zs at 2.9<z<5.5 estimated using the classic LS estimator (i.e. Pdf=1 for each source). The dotted lines show the best-fit models obtained including the dominant term b 2 ξ ′ in Equation 11, while the dashed lines is the model without the photo-z damping term. The error bars show the square roots of the diagonal values of the covariance matrix.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 4shows ξ(s) for 221.6 CCL AGN using the generalized clustering estimator based on phot-z Pdfs in ad--Left Panel: Bias of CCL AGN at z∼3.4 estimated using the projected 2pcf (empty points) and the z-space 2pcf (filled points). The empty symbols are offset in the horizontal direction by +0.04 for clarity. For comparison the bias of XMM and Chandra COSMOS AGN at z = 2.8 is shown as presented inAllevato et al. (2014) using πmax = 40 h −1 Mpc (filled hexagon) and when the bias is re-estimated using πmax = 200 h −1 Mpc. The dashed lines show the expected b(z) of typical DM halo with mass of 12, 12.5, 13, h −1 M ⊙ in log scale (from bottom to top), based onSheth et al. (2001) formalism. Right Panel: Bias parameter as a function of redshift for X-ray selected AGN (black) from previous studies as described in the legend. The red circle at z∼3.4 show the bias factors as estimated in this work for CCL AGN. The red square shows the bias factor as re-estimated in the present paper using the same catalog of XMM and Chandra-COSMOS AGN used inAllevato et al. (2014), but with πmax=200 h −1 Mpc. The dashed lines show the expected b(z) of typical DM halo with mass of 11.5, 12, 12.5, 13, 13.5 h −1 M ⊙ in log scale (from bottom to top), based onSheth et al. (2001). Bias factors from different studies are converted to a common cosmology (Ω Λ = 0.7, Ωm = 0.3, σ 8 = 0.8).\",\n \"content\": \"Fig. 5 .\\n5Figure 4shows ξ(s) for 221.6 CCL AGN using the generalized clustering estimator based on phot-z Pdfs in ad--Left Panel: Bias of CCL AGN at z∼3.4 estimated using the projected 2pcf (empty points) and the z-space 2pcf (filled points). The empty symbols are offset in the horizontal direction by +0.04 for clarity. For comparison the bias of XMM and Chandra COSMOS AGN at z = 2.8 is shown as presented inAllevato et al. (2014) using πmax = 40 h −1 Mpc (filled hexagon) and when the bias is re-estimated using πmax = 200 h −1 Mpc. The dashed lines show the expected b(z) of typical DM halo with mass of 12, 12.5, 13, h −1 M ⊙ in log scale (from bottom to top), based onSheth et al. (2001) formalism. Right Panel: Bias parameter as a function of redshift for X-ray selected AGN (black) from previous studies as described in the legend. The red circle at z∼3.4 show the bias factors as estimated in this work for CCL AGN. The red square shows the bias factor as re-estimated in the present paper using the same catalog of XMM and Chandra-COSMOS AGN used inAllevato et al. (2014), but with πmax=200 h −1 Mpc. The dashed lines show the expected b(z) of typical DM halo with mass of 11.5, 12, 12.5, 13, 13.5 h −1 M ⊙ in log scale (from bottom to top), based onSheth et al. (2001). Bias factors from different studies are converted to a common cosmology (Ω Λ = 0.7, Ωm = 0.3, σ 8 = 0.8).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Predicted bias as a function of bolometric luminosity, computed according to\",\n \"content\": \"Fig. 6 .\\n6-Predicted bias as a function of bolometric luminosity, computed according to\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Properties of the AGN Samples\",\n \"content\": \"TABLE 1\\n1Properties of the AGN Samples\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". V Allevato, ApJ. 73699Allevato V., et al. 2011, ApJ, 736, 99","doi":""},{"text":". V Allevato, ApJ. 7964Allevato V., et al. 2014, ApJ, 796, 4","doi":""},{"text":". F Bournaud, A Dekel, R Teyssier, ApJ. 74133Bournaud, F., Dekel, A., Teyssier, R., et al. 2011, ApJ, 741, 33","doi":""},{"text":". S Bonoli, F Marulli, V Springel, MNRAS. 396423Bonoli, S., Marulli, F., Springel, V., et al. 2009, MNRAS, 396, 423","doi":""},{"text":". 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I Zehavi, ApJ. 62122Zehavi, I., et al. 2005, ApJ, 621, 22\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":586,"cells":{"text":{"kind":"string","value":"\nSMOOTH SQUAREFREE AND SQUARE-FULL INTEGERS IN ARITHMETIC PROGRESSIONS\n5 Oct 2018\n\nMarc Munsch \nANDIgor E Shparlinski \nKam Hung Yau \nSMOOTH SQUAREFREE AND SQUARE-FULL INTEGERS IN ARITHMETIC PROGRESSIONS\n5 Oct 2018\nWe obtain new lower bounds on the number of smooth squarefree integers up to x in residue classes modulo a prime p, relatively large compared to x, which in some ranges of p and x improve that of A.Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prime p.\n\n1. Introduction 1.1. Background and motivation. Let p be a prime. For any integer n ≥ 2 we denote P + (n) to be the largest prime factor of n. For any positive real number y , we say that an integer is y -smooth if P + (n) ≤ y .\n\nStudying the distribution of y -smooth numbers n ≤ x in progressions modulo an integer q ≥ 2 has always been a very active subject of research, see [3,8,11,16,18] and references therein. For instance, as pointed out in [18], a very good level of distribution would imply the truth of Vinogradov's conjecture about the smallest quadratic nonresidue.\n\nAs usual, we denote by ψ(x, y; p, a) the number of positive integers n ≤ x which are y -smooth and satisfy n ≡ a mod p. Furthermore, we use ψ ♯ (x, y; p, a) for the number of those integers which are also squarefree.\n\nDue to its link with Euclidean prime generators, the positivity of ψ ♯ (x, y; p, a) in the special case of y = p is of special interest, see [5]. Thus, following Booker and Pomerance [5], we use M(p) to denote the least x such that ψ ♯ (x, p; p, a) > 0 for every integer a. The quantity M(p) has been considered in [16], where in particular the conjecture of Booker and Pomerance [5] that M(p) = p O(1) is established in a stronger form M(p) ≤ p 3/2+o (1) , for all primes p, and M(p) ≤ p 4/3+o (1) , for all, but a set of primes p of relative zero density.\n\nHere we use similar ideas to obtain lower bounds on ψ ♯ (x, y; p, a) of essentially the right order of magnitude in a broader range of y . These bounds, even without taking into account the squarefreeness condition, that is, using ψ(x, y; p, a) ≥ ψ ♯ (x, y; p, a), improve the range in which the result of Balog and Pomerance [3] applies.\n\nSubsequently, we also address a question about squarefull numbers in arithmetic progressions (that is numbers, which are divisible by squares of all its prime divisors). This question is significantly less studied, see however [7,15,17]. In particular, Chan [7] obtained an asymptotic formula for the number of squarefull numbers in an arithmetic progression, however, due to a rather complicated structure of the main term, it is not immediately clear when the main term starts to exceed the error term. Here we consider a Linnik-type version of this question. Namely, using very different arguments compared to the case of squarefree numbers (and also to [7]), we investigate the quantity F (a, p) which is defined as the smallest positive squarefull number n ≡ a mod p.\n\n\n1.2.\n\nResults for squarefree numbers. We start with a lower bound on ψ ♯ (x, y; p, a) which holds for any prime p. Theorem 1.1. For any fixed real numbers α and β with β ∈ (23/24, 1] and α ∈ (9/2 − 3β, 3β], for x = p α+o(1) and y = p β+o(1) we have\nψ ♯ (x, y; p, a) ≥ x 1+o(1) /p as p → ∞.\nTaking y = p β with 23/24 < β ≤ 1 and q = p in the main result of Balog & Pomerance [3] gives the existence of a p β -smooth integer (not necessary squarefree) n ≤ p max{3β/2,3/4+β}+o(1) = p 3/4+β+o(1) since β ≤ 1. We notice that 9/2 − 3β < 3/4 + β, under the condition 23/24 < β . Therefore, Theorem 1.1 always improves on the bound given by the main result of Balog & Pomerance [3]. We remark that removing the squarefreeness condition does not help us to improve on Theorem 1.1 due to the method used.\n\nWe also obtain a result for almost all primes. Firstly, we define the interval for any η > η 0 and a sufficiently large p. Noticing that an −3 p is a quadratic residue modulo p, we now obtain F (a, p) ≤ n 3 p (p − 1) 2 . Hence, we have the trivial bound F (a, p) ≤ p 2+3η 0 +o(1) for any a, which we unfortunately do not know how to improve. However, we remark that assuming the Vinogradov's conjecture that n p ≤ p o(1) (which is implied by the Generalised Riemann Hypothesis in the stronger form n p ≪ log 2 p proved by Ankeny [2], see also [12,Section 5.9] for a discussion), we have the bound F (a, p) ≤ p 2+o (1) . Even though we cannot reach such a bound, we obtain an unconditional better bound for almost all a ∈ {0, . . . , p − 1}.\nI(β) = (α 0 (β), β + 1) where α 0 (β) = 5(2 − β)/3 if β ∈ (\nWe also note that from a result on counting squarefull integers [19], for any set A of A distinct residues modulo p we have\nmax a∈A F (a, p) ≫ A 2 ,\nwhere, as usual, we use A ≪ B and B ≫ A as an equivalent to the inequality |A| ≤ cB with some constant c > 0, which occasionally, where obvious, may depend on the real parameter ε > 0. We slightly refine this result:\nTheorem 1.3. For all but o(p) quadratic non-residues a ∈ [0, p − 1], we have p 2 n p f (p) ≪ F (a, p) ≤ p 2+η 0 +o(1) for any function f (p) such that f (p) → 0 as p → ∞ and max a mod p F (a, p) ≫ p 2 n p ,\nwhere n p denotes the least quadratic non-residue modulo p.\n\nUsing the lower bound in Theorem 1.3, together with an unconditional result of Graham and Ringrose [10] on primes with large values of n p and a conditional result on the Generalised Riemann Hypothesis (GRH) of Montgomery [14], we immediately derive Corollary 1.4. For infinitely many primes p we have max a mod p F (a, p) ≫ p 2 (log p)(log log log p), unconditionally, p 2 (log p)(log log p), under the GRH.\n\nIn Section 2, we collect some results which will be used to prove the main results in Section 3.\n\n\nPreparation lemmas\n\n2.1. Exponential sums with reciprocals of primes. As usual, we define e p (z) = exp(2iπz/p). For an integer k with gcd(k, p) = 1 we use k to denote the multiplicative inverse of k modulo p, that is, the unique integer with kk ≡ 1 mod p and 1 ≤ k < p. It is convenient to introduce the quantity\n(2.1) B(p, L) = L 3/2 p 1/8 , if L < p 1/3 , L 15/8 if p 1/3 ≤ L < p.\nThe following bound of the double exponential sum over primes is a combination of [16, Lemma 3.5] (for L ≤ p 1/3 ) and of [9, Lemma 2.4] (for p 1/3 ≤ L < p).\nLemma 2.1. For any real L ≤ p, we have max gcd(a,p)=1 ℓ 1 ,ℓ 2 ∈L e p aℓ 1 ℓ 2 ≤ B(p, L)p o(1) , as p → ∞, where L is the set of primes ℓ ∈ [L, 2L].\n2.2. Some congruences with products of primes. We denote N a,p (L, h) to be the number of solutions to the congruence\n(2.2) ℓ 1 ℓ 2 u ≡ a mod p, ℓ 1 , ℓ 2 ∈ L, 1 ≤ u ≤ h,\nwhere h and L are two positive real numbers and L is the set of primes ℓ ∈ [L, 2L]. We now use Lemma 2.1 to derive an analogue of [16, Lemma 3.10] (which also applies to L ≥ p 1/3 ).\n\nLemma 2.2. For any integer a and prime p with gcd(a, p) = 1 and reals h and L with 1 ≤ h, L < p, we have\nN a,p (L, h) = K 2 h p + O B(p, L)p o(1) ,\nwhere K = #L is the cardinality of L and B(p, L) is defined by (2.1).\n\nWe also recall that by [ \n1 ≤ L, h < p and 1 ≤ D ≤ h 1/2 , we have N ♯ a,p (L, h) = K 2 h ζ(2)p + O L 2 h Dp + DB(p, L) + h 1/2 p o(1) ,\nwhere K = #L is the cardinality of L and B(p, L) is defined by (2.1).\n\nWe also need the bound of [16, Lemma 3.14] on the number of solutions Q a,p (L, h) to the congruence\nℓ 1 ℓ 2 2 v ≡ a mod p, ℓ 1 , ℓ 2 ∈ L, 1 ≤ v ≤ h.\nLemma 2.5. For any integer a and prime p with gcd(a, p) = 1 and\nreals 1 ≤ L, h ≤ p with 2Lh ≤ p we have Q a,p (L, h) ≤ (Lh/p + 1) Lp o(1) .\nIt is shown in [16,Lemma 3.11], that for almost all primes p, the asymptotic formula of Lemma 2.2 can be improved as follows.\nLemma 2.6. As Q → ∞ for any fixed integer k ≥ 1, for 1 ≤ L ≤ Q for all but o(Q/ log Q) primes p ∈ [Q, 2Q], for any integer a with gcd(a, p) = 1 and real h with 1 ≤ h ≤ p, we have N a,p (L, h) = K 2 h p + O((L (3k−1)/2k p 1/2k + L (4k−1)/(2k) )p o(1) )\nwhere K = #L is the cardinality of L.\n\nFinally, we also recall that by [ \nR a,p (F, L, h) = F ≤d≤2F N ad −2 ,p (L, h) we have R a,p (F, L, h) ≤ max{F (L 2 h) 1/4 p −1/4 , F 1/2 (L 2 h) 1/4 }p o(1) .\n\nMoments of character sums.\n\nLet Ω p denote the set of all Dirichlet characters modulo p and let Ω * p = Ω p \\ {χ 0 } denote the set of all nonprincipal Dirichlet characters modulo p.\n\nWe need the following result of Ayyad, Cochrane and Zheng [1, Theorem 2], see also [13] for a slightly sharper bound (which however does not change our final result). \n\n\n2.4.\n\nQuadratic non-residues in short intervals. Let T p (K) denote the number of quadratic non-residues modulo p in the interval [1, K].\n\nWe need an extension of (1.2). The following bound is given in [4, Theorem 2.1]. Lemma 2.9. For any real η > η 0 , where η 0 is given by (1.1), there is a constant c > 0, such that for a sufficiently large p and K ≥ p η we have T p (K) ≥ cK.\n\n\nProofs of main results\n\n3.1. Proof of Theorem 1.1. For a sufficiently small ε > 0, we set\nL = p (α−β)/2−ε/2 and h = p β .\nSince L ≤ h ≤ y , N ♯ a,p (L, h) counts a subset of y -smooth integers in an arithmetic progression. Noticing that L 2 h ≤ p α−ε = x 1−ε+o(1) , we see that for a sufficiently large p we have\nψ ♯ (x, y; p, a) ≥ N ♯ a,p (L, h) + O (h/p + 1) Lp o(1) = N ♯ a,p (L, h) + O Lp o(1) ,(3.1)\nwhere we estimated the contribution coming from non-squarefree products ℓ 1 ℓ 2 u (precisely products with ℓ 1 = ℓ 2 or with ℓ 1 | u or with ℓ 2 | u) using Lemma 2.5 with h/L replacing h as in the end of the proof of [16, Theorem 1.4]. We use a crude estimate for the main term: (1) . Choosing D = p ε/2 and using Lemma 2.4, we derive\n(3.2) K 2 h p ∼ L 2 h p (log L) 2 = p α−1−ε+oψ ♯ (x, y; p, a) ≫ K 2 h p + O p ε/2 B(p, L) + L + h 1/2 p o(1) = K 2 h p + O p ε/2 B(p, L) + h 1/2 p o(1) (3.3)\nsince B(p, L) dominates L and the main term (3.2) dominates the first error term L 2 h(Dp) −1 in Lemma 2.4. To begin, we remark that the term h 1/2 in (3.3) is dominated by the main term due to the inequality α − 1 > 9/2 − 3β − 1 > β/2 for β ≤ 1. We split the discussion on the contribution of B(p, L) into two cases depending on α.\n\nFirstly, suppose that α ∈ (9/2−3β, 2/3+β]. Since α ≤ 2/3+β , this implies L < p 1/3 and hence B(p, L) = L 3/2 p 1/8 by (2.1). Therefore, recalling (3.3) and (3.2), we obtain (1) ).\n(3.4) ψ ♯ (x, y; p, a) ≫ p α−1−ε+o(1) + O(p 3(α−β)/4−ε/4+1/8+o\nFor ε sufficiently small, we have α > 9/2 − 3β + 3ε which implies that the main term dominates trivially the remainder term in (3.4). Secondly, assume that α ∈ (2/3 + β, 3β]. In particular, since β ≤ 1 we have (1) ).\n2/3 + β + ε ≤ α < 3β < 2 + β + ε,\nNotice that we have\nα > 2/3 + β ≥ 16 − 15β + 9ε + o(1)\nwhen β ∈ (23/24, 1] and ε > 0 is sufficiently small. It follows that the main term dominates the remainder term in (3.5). Therefore, in all cases we conclude ψ ♯ (x, y; p, a) ≥ p α−ε−1+o (1) .\n\nSince this is valid for all sufficiently small ε > 0, the result follows.\n\n\n3.2.\n\nProof of Theorem 1.2. We follow the proof of [16,Theorem 1.6].\n\nFor ε > 0, we set\n(3.6) L = Q (α−β)/2−ε/2 , h = Q β , D = Q ε/2 , E = Q (α−1)/2 .\nWe note that (α − 1)/2 > 0 for α ∈ I(β) and so D < E if ε > 0 is sufficiently small. We also have E < h 1/2 since α < β + 1.\n\nSince α < β + 1 ≤ 3β in the range β ∈ (7/8, 1], we get L ≤ h. In particular, we have as before the inequality (3.1).\n\nBy inclusion and exclusion, we have\n(3.7) N ♯ a,p (L, h) = d≤h 1/2 µ(d)N ad −2 ,p (L, h/d 2 ) = Σ 1 + Σ 2 + Σ 3 , where Σ 1 = d≤D µ(d)N ad −2 ,p (L, h/d 2 ), Σ 2 = D<d≤E µ(d)N ad −2 ,p (L, h/d 2 ),Σ 3 = E<d≤h 1/2 µ(d)N ad −2 ,p (L, h/d 2 ).\nTo abstain from clutter, all the bounds below are valid for all but o(Q/ log Q) primes p ∈ [Q, 2Q].\n\nSince α < β + 1 < 2 + β + ε + o(1) and β ≤ 1, we obtain respectively L ≤ Q and h ≤ p. By Lemma 2.6 \nΣ 1 = K 2 h ζ(2)p + O K 2 h Dp + D(L (3k−1)/(2k) p 1/(2k) + L (4k−1)/(2k) )p o(1)(L 2 h/d 2 ≤ L 2 h/F 2 ≤ L 2 h/E 2 < p, hence by Lemma 2.7 R a,p (F, L, h/F 2 ) ≤ max{F (L 2 h/F 2 ) 1/4 p −1/4 , F 1/2 (L 2 h/F 2 ) 1/4 }p o(1) = (L 2 h) 1/4 p o(1) since F ≤ h 1/2 ≤ p 1/2 and so (3.10) Σ 3 ≤ (L 2 h) 1/4 p o(1) .\nSubstituting (3.8), (3.9) and (3.10) in (3.7), we obtain\nN ♯ a,p (L, h) = K 2 h ζ(2)p + O Rp o(1)\nwhere we set R = D L (3k−1)/(2k) p 1/(2k) + L (4k−1)/(2k) + L 2 h Dp + E + (L 2 h) 1/4 and the main term verifies an analogue of (3.2), precisely, (1) .\n(3.11) K 2 h p ∼ L 2 h p (log L) 2 = Q α−1−ε+o\nNotice that the choice of E in (3.6) implies that E is smaller than the main term (3.11). We now see from (3.11) that if\n(3.12) α − 1 > max 3k − 1 2k α − β 2 + 1 2k , 4k − 1 2k α − β 2 , α 4 , α − β 2\nfor some positive integer k , then for a sufficiently small ε the main term dominates the remainder term in (3.1) and the result follows. Rearranging (3.12) gives\n(3.13) α > max (1 − 3k)β + 2 + 4k k + 1 , (1 − 4k)β + 4k, 4/3, 2 − β .\nFirst, we remark that 2 − β ≤ (1 − 4k)β + 4k since β ≤ 1 and we can discard 2 − β from the maximum in (3.13).\n\nFurthermore, for k ≤ 5, we see that 4/3 is dominated by the first term of the right hand side of (3.13). In this case, a quick computation shows that (1 − 3k)β + 2 + 4k\nk + 1 ≥ (1 − 4k)β + 4k\nif and only if β ≥ 1−1/2k 2 . Thus, in the interval (1−1/2k 2 , 1−1/2(k+ 1) 2 ], the maximum is given either by ((1 − 3k)β + 2 + 4k) /(k + 1) or by (1 − 4m)β + 4m with m ≥ k + 1. Since the function f (z) = (1 − 4z)β + 4z is a monotonically increasing function of z , we check only the case m = k + 1 and verify that\n(1 − 3k)β + 2 + 4k k + 1 ≥ f (k + 1) = (1 − 4(k + 1))β + 4(k + 1)\nif and only if\nβ ≥ β 0 (k) where β 0 (k) = 1 − 1 2(k 2 + k + 1)\n.\n\n\nSplitting the interval\nI k = 1 − 1 2k 2 , 1 − 1 2(k + 1) 2\ninto two intervals as follows\nI k = 1 − 1 2k 2 , 1 − 1 2(k 2 + k + 1) 1 − 1 2(k 2 + k + 1) , 1 − 1 2(k + 1) 2\nand recalling that k ≤ 5, we deduce after short computations the result for β ≤ β 0 (5) = 61/62.\n\nFor k ≥ 6, noticing that (1 − 4β) + 4k ≥ 4/3 in the range\nβ ≤ 1 − 1 3(4k − 1) ,\nwe also deduce the case β ∈ (61/62, 68/69]. For the remaining case β ∈ (68/69, 1] and k ≥ 6, we see that\n(1 − 3k)β + 2 + 4k k + 1 ≤ 4/3.\nBased on the above argument, we now give explicit choices of k and corresponding intervals which optimise our bound.\n\n• If β ∈ (7/8, 13/14], we take k = 2 and (3.13) simplifies to Therefore in all cases, where we also recall the condition α < β + 1, we have ψ ♯ (x, y; p, a) ≥ p α−1−ε+o (1) .\n\nSince this is true for all ε > 0, the result follows immediately.\n\n3.3. Proof of Theorem 1.3. Let M be a parameter which will be fixed later. We introduce the subset of residues modulo p S = {a : a quadratic non-residue such that F (a, p) ≤ M} .\n\nFirstly, we remark that every squarefull integer n can be written as n = r 2 s with s | r . Furthermore, if a is a quadratic non-residue, we notice that s has to be a quadratic non-residue in this representation; in particular s ≥ n p . Let us count the number of products r 2 s ≤ M with s | r and s ≥ n p the smallest quadratic non-residue modulo p. So we now turn our attention to the upper bound.\n\nClearly, if a ≡ u 2 mod p, 0 ≤ u < p, is quadratic residue (or a = 0), then F (a, p) ≤ u 2 ≤ p 2 .\n\nWe now fix some ε > 0 and denote by A the set of quadratic nonresidues for which F (a, p) ≥ p 2+η 0 +ε .\n\nIt is enough to show that the cardinality of A satisfies (3.14) #A = o(p).\n\nWe set K = p η 0 +ε/2 and U = p 1−η 0 and let N be the set of quadratic non-residues in the interval [1, K].\n\nIn particular #N = T (K).\n\nClearly for a ∈ A the congruence (3.15) a ≡ n 3 u 2 mod p, n ∈ N , 1 ≤ u ≤ U, has no solution. Thus expressing the number of solutions to (3.15) via characters we see that n∈N 1≤u≤U\n1 p − 1 χ∈Ωp χ a −1 n 3 u 2 = 0.\nSumming over all a ∈ A and using the multiplicativity of characters, we arrive to (3.16) χ∈Ωp a∈A\nχ (a) n∈N χ (n) 3 U u=1 χ (u) 2 = 0,\nwhere χ denotes the complex conjugate character of χ. Now, the contribution to (3.16) from the principal character is obviously #AT (K)U .\n\nFurthermore, since all elements of A are quadratic non-residues, the contribution to (3.16) from the quadratic character, that is, from the Legendre symbol is with Ω ♯ p being the subset of Ω * p where we removed the quadratic character. Now, for χ ∈ Ω ♯ p we have by definition χ 2 = χ 0 . Furthermore, each character from Ω * p occurs at most twice as χ 2 and each character from Ω p (including also χ 0 in this case) occurs at most three times as χ 3 for χ ∈ Ω ♯ p . Using the Hölder inequality, we now derive from (3.17) that For Σ 2 , using again the orthogonality of characters, we write Σ 2 = 3(p − 1)# {(n 1 , n 2 , n 3 , n 4 ) ∈ N : n 1 n 2 ≡ n 3 n 4 mod p} ≤ 3(p − 1)# {(n 1 , n 2 , n 3 , n 4 ) ∈ [1, K] : n 1 n 2 ≡ n 3 n 4 mod p} Applying Lemma 2.8 and using that K 2 ≤ p provided that ε is small enough, we derive (3.20) Σ 2 ≤ K 2 p 1+o (1) .\n\nFinally, we also estimate Σ 3 , directly by Lemma 2.8 getting which together with Lemma 2.9 yields #A ≤ K −1 U −1 p 2+o(1) = p 1−ε/2+o (1) .\n\nWe now see that (3.14) holds which concludes the proof.\n\nLemma 2. 4 .\n4For any integer a and prime p with gcd(a, p) = 1 and reals h, D and L with\n\nLemma 2. 8 .\n8For any integer K ≥ 1, we have χ∈Ω * p 1≤n≤Kχ(n) 4 ≤ K 2 p 1+o(1) .\n\n\nfor ε > 0 chosen sufficiently small. Hence p 1/3 ≤ L < p and we have B(p, L) = L 15/8 by (2.1). Therefore, recalling (3.3) and (3.2), we obtain (3.5) ψ ♯ (x, y; p, a) ≥ p α−1−ε+o(1) + O(p 15(α−β)/16−7ε/16+o\n\n\nα > max {5(2 − β)/3, 8 − 7β, 4/3, 2 − β} = 5(2 − β)/3.• If β ∈ (13/14, 17/18], we take k = 3 and (3.13) simplifies to α > max {(7 − 4β)/2, 12 − 11β, 4/3, 2 − β} = 12 − 11β.• If β ∈ (17/18, 25/26], we take k = 3 and (3.13) simplifies to α > max {(7 − 4β)/2, 12 − 11β, 4/3, 2 − β} = (7 − 4β)/2.• If β ∈ (25/26, 31/32], we take k = 4 and (3.13) simplifies to α > max {(18 − 11β)/5, 16 − 15β, 4/3, 2 − β} = 16 − 15β.• If β ∈ (31/32, 41/42], we take k = 4 and (3.13) simplifies to α > max {(18 − 11β)/5, 16 − 15β, 4/3, 2 − β} = (18 − 11β)/5.• If β ∈ (41/42, 49/50], we take k = 5 and (3.13) simplifies to α > max {(11 − 7β)/3, 20 − 19β, 4/3, 2 − β} = 20 − 19β.• If β ∈ (49/50, 61/62], we take k = 5 and (3.13) simplifies to α > max {(11 − 7β)/3, 20 − 19β, 4/3, 2 − β} = (11 − 7β)/3. • If β ∈ (61/62, 68/69], we take k = 6 and (3.13) simplifies to α > max {(26 − 17β)/7, 24 − 23β, 4/3, 2 − β} = 24 − 23β.• If β ∈ (68/69, 1], we take k = 6 and (3.13) simplifies to α > max {(26 − 17β)/7, 24 − 23β, 4/3, 2 − β} = 4/3.\n\n.\nNoticing that r ≤ (M/s) 1/2 , we have at most M 1/2 s −3/2 possible values of r . Thus the number of different products r 2 s is bounded by Setting M = p 2 n p f (p), we get #S = o(p) which concludes the proof. The assertion max a mod p F (a, p) ≫ p 2 n p follows by the same argument by setting S = {a : a quadratic non-residue} and M = max a∈S F (a, p).\n\n\nupper bounds come from the discussion above. We now see by the orthogonality of characters that we have(3.19) Σ 1 ≤ (p − 1)#A.\n\n( 3 .\n321) Σ 3 ≤ U 2 p 1+o(1) .Substituting(3.19), (3.20) and (3.21) in (3.18), we now derive#AT (K)U ≤ (#A) 1/2 K 1/2 U 1/2 p 1+o(1)\n\n\nand recall that by the classical bound of Burgess[6] on the smallest quadratic non-residue n p we have7/8, 13/14], \n12 − 11β \nif β ∈ (13/14, 17/18], \n(7 − 4β)/2 \nif β ∈ (17/18, 25/26], \n16 − 15β \nif β ∈ (25/26, 31/32], \n(18 − 11β)/5 if β ∈ (31/32, 41/42], \n20 − 19β \nif β ∈ (41/42, 49/50], \n(11 − 7β)/3 \nif β ∈ (49/50, 61/62], \n24 − 23β \nif β ∈ (61/62, 68/69], \n4/3 \nif β ∈ (68/69, 1]. \n\nTheorem 1.2. Fix real numbers α and β such that β ∈ (7/8, 1] and \nα ∈ I(β). Letting x = Q α+o(1) and y = Q β+o(1) , as Q → ∞, we have \n\nψ ♯ (x, y; p, a) ≥ x 1+o(1) /p \n\nfor all but o(Q/ log Q) primes p ∈ [Q, 2Q]. \n\n1.3. Results for squarefull numbers. First we observe that if a is a \nquadratic residue modulo p (or a = 0), then a ≡ b 2 mod p for some \ninteger b ∈ [0, p − 1] and so we have trivially F (a, p) ≤ (p − 1) 2 in this \ncase. \nTo estimate F (a, p) for a quadratic non-residue a we denote \n\n(1.1) \nη 0 = \n1 \n4 \n√ e \n\n(1.2) \nn p ≤ p η \n\n\n\n\n16, Lemma 3.12] we have: Lemma 2.3. For any integer a and prime p with gcd(a, p) = 1 and reals 1 ≤ h, L < p we have N a,p (L, h) ≤ (L 2 h/p + 1)p o(1) .Furthermore, let N ♯ a,p count the number of squarefree solutions to the congruence (2.2). Following the proof of [16, Theorem 1.4], but using a more general bound of Lemma 2.2 instead of [16, Lemma 3.10] as well as Lemma 2.3 (exactly as in [16]), we derive\n\nThe congruence x 1 x 2 ≡ x 3 x 4 mod p, the equation x 1 x 2 = x 3 x 4 and mean values of character sums. A Ayyad, T Cochrane, Z Zheng, J. Number Theory. 596A. Ayyad, T. Cochrane and Z. Zheng, 'The congruence x 1 x 2 ≡ x 3 x 4 mod p, the equation x 1 x 2 = x 3 x 4 and mean values of character sums', J. Number Theory 59 (1996), 398-413. 6\n\nThe least quadratic non residue. N C Ankeny, Ann. of Math. 552N. C. Ankeny, 'The least quadratic non residue', Ann. of Math. (2), 55 (1952), 65-72. 3\n\nThe distribution of smooth numbers in arithmetic progressions. A Balog, C Pomerance, Proc. Amer. Math. Soc. 1152A. Balog and C. Pomerance, 'The distribution of smooth numbers in arithmetic progressions', Proc. Amer. Math. Soc., 115 (1992), 33-43. 1, 2\n\nDensity of non-residues in Burgess-type intervals and applications. W D Banks, M Z Garaev, R Heath-Brown, I E Shparlinski, Bull. London Math. Soc. 407W. D. Banks, M. Z. Garaev, R. Heath-Brown and I. E. Shparlinski, 'Density of non-residues in Burgess-type intervals and applications', Bull. London Math. Soc. , 40 (2008), 88-96. 7\n\nSquarefree smooth numbers and Euclidean prime generators. A Booker, C Pomerance, Proc. Amer. Math. Soc. 1451A. Booker and C. Pomerance, 'Squarefree smooth numbers and Euclidean prime generators', Proc. Amer. Math. Soc., 145 (2017), 5035-5042. 1\n\nThe distribution of quadratic residues and non-residues. D A Burgess, 4MathematikaD. A. Burgess, 'The distribution of quadratic residues and non-residues', Math- ematika, 4 (1957), 106-112. 3\n\nSquarefull numbers in arithmetic progression, II. T H Chan, J. Number Theory. 1522T. H. Chan, 'Squarefull numbers in arithmetic progression, II', J. Number Theory 152 (2015), 90-104. 2\n\nThéorémes de type Fouvry-Iwaniec pour les entiers friables. S Drappeau, Compos. Math. 1511S. Drappeau, 'Théorémes de type Fouvry-Iwaniec pour les entiers friables', Compos. Math., 151 (2015), 828-862. 1\n\nAn estimate of Kloosterman sums with prime numbers and an application. M Z Garaev, Matem. Zametki. 884in Russian. translation in Math. NotesM. Z. Garaev, 'An estimate of Kloosterman sums with prime numbers and an application', Matem. Zametki, 88 (2010), 365-373, (in Russian); translation in Math. Notes, 88 (2010), 330-337. 4\n\nLower bounds for least quadratic nonresidues' Analytic number theory. S W Graham, C J Ringrose, Progr. Math. 854S. W. Graham and C.J. Ringrose, 'Lower bounds for least quadratic non- residues' Analytic number theory, Progr. Math., 85(1990), 269-309. 4\n\nOn a paper of K. Soundararajan on smooth numbers in arithmetic progressions. A J Harper, J. Number Theory. 1321A. J. Harper, 'On a paper of K. Soundararajan on smooth numbers in arith- metic progressions', J. Number Theory, 132 (2012), 182-199. 1\n\nAnalytic number theory. H Iwaniec, E Kowalski, American Mathematical SocietyProvidence, RIH. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society, Providence, RI, 2004. 3\n\nOn the congruence x 1 x 2 ≡ x 3 x 4 mod q. B Kerr, J. Number Theory. 6B. Kerr, 'On the congruence x 1 x 2 ≡ x 3 x 4 mod q ', J. Number Theory, 180 (2017), 154-168. 6\n\nTopics in multiplicative number theory. H L Montgomery, Lecture Notes in Mathematics. 2274Springer-VerlagH. L. Montgomery, 'Topics in multiplicative number theory', Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, 1971. 4\n\nCharacter sums over squarefree and squarefull numbers. M Munsch, Arch. Math. 1022M. Munsch, 'Character sums over squarefree and squarefull numbers', Arch. Math., 102 (2014), 555-563. 2\n\nOn smooth square-free numbers in arithmetic progressions. M Munsch, I E Shparlinski, 5M. Munsch and I. E. Shparlinski, 'On smooth square-free numbers in arith- metic progressions', Preprint , 2017, http://arxiv.org/abs/1710.04705. 1, 4, 5, 6, 7, 8\n\nSquare-full primitive roots. M Munsch, T Trudgian, Intern. J. Number Theory. 142M. Munsch and T. Trudgian, 'Square-full primitive roots', Intern. J. Number Theory, 14 (2018), 1013-1021. 2\n\nThe distribution of smooth numbers in arithmetic progressions', Anatomy of integers. K Soundararajan, CRM Proc. Lecture Notes. Providence, RIAmer. Math. Soc46K. Soundararajan, 'The distribution of smooth numbers in arithmetic progres- sions', Anatomy of integers, CRM Proc. Lecture Notes, v.46, Amer. Math. Soc., Providence, RI, 2008, 115-128. 1\n\nThe distribution of squarefull integers. D Suryanarayana, R , Sitaramachandra Rao, Arkiv för Matematik. 114D. Suryanarayana and R. Sitaramachandra Rao, 'The distribution of square- full integers', Arkiv för Matematik , 11 (1973), 195-201. 4\n\nInstitut für Analysis und Zahlentheorie 8010 Graz. Steyr-Institut für Analysis und Zahlentheorie 8010 Graz, Steyr-\n"},"corpus_id":{"kind":"number","value":119712364,"string":"119,712,364"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1810.02573\",\n \"mag\": 2990938197,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1112/mtk.12012\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1810.02573v2.pdf\"\n ],\n \"pdf_sha\": \"e3d054f22d8c14ecb2c8705df24f5e5e603f56d3\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1810.02573\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"SMOOTH SQUAREFREE AND SQUARE-FULL INTEGERS IN ARITHMETIC PROGRESSIONS"},"authors":{"kind":"list like","value":["Marc Munsch","ANDIgor E Shparlinski","Kam Hung Yau"],"string":"[\n \"Marc Munsch\",\n \"ANDIgor E Shparlinski\",\n \"Kam Hung Yau\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We obtain new lower bounds on the number of smooth squarefree integers up to x in residue classes modulo a prime p, relatively large compared to x, which in some ranges of p and x improve that of A.Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prime p."},"sections":{"kind":"list like","value":[{"title":"1.2.","content":"Results for squarefree numbers. We start with a lower bound on ψ ♯ (x, y; p, a) which holds for any prime p. Theorem 1.1. For any fixed real numbers α and β with β ∈ (23/24, 1] and α ∈ (9/2 − 3β, 3β], for x = p α+o(1) and y = p β+o(1) we have\nTaking y = p β with 23/24 < β ≤ 1 and q = p in the main result of Balog & Pomerance [3] gives the existence of a p β -smooth integer (not necessary squarefree) n ≤ p max{3β/2,3/4+β}+o(1) = p 3/4+β+o(1) since β ≤ 1. We notice that 9/2 − 3β < 3/4 + β, under the condition 23/24 < β . Therefore, Theorem 1.1 always improves on the bound given by the main result of Balog & Pomerance [3]. We remark that removing the squarefreeness condition does not help us to improve on Theorem 1.1 due to the method used.\nWe also obtain a result for almost all primes. Firstly, we define the interval for any η > η 0 and a sufficiently large p. Noticing that an −3 p is a quadratic residue modulo p, we now obtain F (a, p) ≤ n 3 p (p − 1) 2 . Hence, we have the trivial bound F (a, p) ≤ p 2+3η 0 +o(1) for any a, which we unfortunately do not know how to improve. However, we remark that assuming the Vinogradov's conjecture that n p ≤ p o(1) (which is implied by the Generalised Riemann Hypothesis in the stronger form n p ≪ log 2 p proved by Ankeny [2], see also [12,Section 5.9] for a discussion), we have the bound F (a, p) ≤ p 2+o (1) . Even though we cannot reach such a bound, we obtain an unconditional better bound for almost all a ∈ {0, . . . , p − 1}.\nWe also note that from a result on counting squarefull integers [19], for any set A of A distinct residues modulo p we have\nwhere, as usual, we use A ≪ B and B ≫ A as an equivalent to the inequality |A| ≤ cB with some constant c > 0, which occasionally, where obvious, may depend on the real parameter ε > 0. We slightly refine this result:\nwhere n p denotes the least quadratic non-residue modulo p.\nUsing the lower bound in Theorem 1.3, together with an unconditional result of Graham and Ringrose [10] on primes with large values of n p and a conditional result on the Generalised Riemann Hypothesis (GRH) of Montgomery [14], we immediately derive Corollary 1.4. For infinitely many primes p we have max a mod p F (a, p) ≫ p 2 (log p)(log log log p), unconditionally, p 2 (log p)(log log p), under the GRH.\nIn Section 2, we collect some results which will be used to prove the main results in Section 3."},{"title":"Preparation lemmas","content":"2.1. Exponential sums with reciprocals of primes. As usual, we define e p (z) = exp(2iπz/p). For an integer k with gcd(k, p) = 1 we use k to denote the multiplicative inverse of k modulo p, that is, the unique integer with kk ≡ 1 mod p and 1 ≤ k < p. It is convenient to introduce the quantity\nThe following bound of the double exponential sum over primes is a combination of [16, Lemma 3.5] (for L ≤ p 1/3 ) and of [9, Lemma 2.4] (for p 1/3 ≤ L < p).\n2.2. Some congruences with products of primes. We denote N a,p (L, h) to be the number of solutions to the congruence\nwhere h and L are two positive real numbers and L is the set of primes ℓ ∈ [L, 2L]. We now use Lemma 2.1 to derive an analogue of [16, Lemma 3.10] (which also applies to L ≥ p 1/3 ).\nLemma 2.2. For any integer a and prime p with gcd(a, p) = 1 and reals h and L with 1 ≤ h, L < p, we have\nwhere K = #L is the cardinality of L and B(p, L) is defined by (2.1).\nWe also recall that by [\nwhere K = #L is the cardinality of L and B(p, L) is defined by (2.1).\nWe also need the bound of [16, Lemma 3.14] on the number of solutions Q a,p (L, h) to the congruence\nLemma 2.5. For any integer a and prime p with gcd(a, p) = 1 and\nIt is shown in [16,Lemma 3.11], that for almost all primes p, the asymptotic formula of Lemma 2.2 can be improved as follows.\nwhere K = #L is the cardinality of L.\nFinally, we also recall that by ["},{"title":"Moments of character sums.","content":"Let Ω p denote the set of all Dirichlet characters modulo p and let Ω * p = Ω p \\ {χ 0 } denote the set of all nonprincipal Dirichlet characters modulo p.\nWe need the following result of Ayyad, Cochrane and Zheng [1, Theorem 2], see also [13] for a slightly sharper bound (which however does not change our final result)."},{"title":"2.4.","content":"Quadratic non-residues in short intervals. Let T p (K) denote the number of quadratic non-residues modulo p in the interval [1, K].\nWe need an extension of (1.2). The following bound is given in [4, Theorem 2.1]. Lemma 2.9. For any real η > η 0 , where η 0 is given by (1.1), there is a constant c > 0, such that for a sufficiently large p and K ≥ p η we have T p (K) ≥ cK."},{"title":"Proofs of main results","content":"3.1. Proof of Theorem 1.1. For a sufficiently small ε > 0, we set\nSince L ≤ h ≤ y , N ♯ a,p (L, h) counts a subset of y -smooth integers in an arithmetic progression. Noticing that L 2 h ≤ p α−ε = x 1−ε+o(1) , we see that for a sufficiently large p we have\nwhere we estimated the contribution coming from non-squarefree products ℓ 1 ℓ 2 u (precisely products with ℓ 1 = ℓ 2 or with ℓ 1 | u or with ℓ 2 | u) using Lemma 2.5 with h/L replacing h as in the end of the proof of [16, Theorem 1.4]. We use a crude estimate for the main term: (1) . Choosing D = p ε/2 and using Lemma 2.4, we derive\nsince B(p, L) dominates L and the main term (3.2) dominates the first error term L 2 h(Dp) −1 in Lemma 2.4. To begin, we remark that the term h 1/2 in (3.3) is dominated by the main term due to the inequality α − 1 > 9/2 − 3β − 1 > β/2 for β ≤ 1. We split the discussion on the contribution of B(p, L) into two cases depending on α.\nFirstly, suppose that α ∈ (9/2−3β, 2/3+β]. Since α ≤ 2/3+β , this implies L < p 1/3 and hence B(p, L) = L 3/2 p 1/8 by (2.1). Therefore, recalling (3.3) and (3.2), we obtain (1) ).\nFor ε sufficiently small, we have α > 9/2 − 3β + 3ε which implies that the main term dominates trivially the remainder term in (3.4). Secondly, assume that α ∈ (2/3 + β, 3β]. In particular, since β ≤ 1 we have (1) ).\nNotice that we have\nwhen β ∈ (23/24, 1] and ε > 0 is sufficiently small. It follows that the main term dominates the remainder term in (3.5). Therefore, in all cases we conclude ψ ♯ (x, y; p, a) ≥ p α−ε−1+o (1) .\nSince this is valid for all sufficiently small ε > 0, the result follows."},{"title":"3.2.","content":"Proof of Theorem 1.2. We follow the proof of [16,Theorem 1.6].\nFor ε > 0, we set\nWe note that (α − 1)/2 > 0 for α ∈ I(β) and so D < E if ε > 0 is sufficiently small. We also have E < h 1/2 since α < β + 1.\nSince α < β + 1 ≤ 3β in the range β ∈ (7/8, 1], we get L ≤ h. In particular, we have as before the inequality (3.1).\nBy inclusion and exclusion, we have\nTo abstain from clutter, all the bounds below are valid for all but o(Q/ log Q) primes p ∈ [Q, 2Q].\nSince α < β + 1 < 2 + β + ε + o(1) and β ≤ 1, we obtain respectively L ≤ Q and h ≤ p. By Lemma 2.6\nSubstituting (3.8), (3.9) and (3.10) in (3.7), we obtain\nwhere we set R = D L (3k−1)/(2k) p 1/(2k) + L (4k−1)/(2k) + L 2 h Dp + E + (L 2 h) 1/4 and the main term verifies an analogue of (3.2), precisely, (1) .\nNotice that the choice of E in (3.6) implies that E is smaller than the main term (3.11). We now see from (3.11) that if\nfor some positive integer k , then for a sufficiently small ε the main term dominates the remainder term in (3.1) and the result follows. Rearranging (3.12) gives\nFirst, we remark that 2 − β ≤ (1 − 4k)β + 4k since β ≤ 1 and we can discard 2 − β from the maximum in (3.13).\nFurthermore, for k ≤ 5, we see that 4/3 is dominated by the first term of the right hand side of (3.13). In this case, a quick computation shows that (1 − 3k)β + 2 + 4k\nif and only if β ≥ 1−1/2k 2 . Thus, in the interval (1−1/2k 2 , 1−1/2(k+ 1) 2 ], the maximum is given either by ((1 − 3k)β + 2 + 4k) /(k + 1) or by (1 − 4m)β + 4m with m ≥ k + 1. Since the function f (z) = (1 − 4z)β + 4z is a monotonically increasing function of z , we check only the case m = k + 1 and verify that\nif and only if\n."},{"title":"Splitting the interval","content":"into two intervals as follows\nand recalling that k ≤ 5, we deduce after short computations the result for β ≤ β 0 (5) = 61/62.\nFor k ≥ 6, noticing that (1 − 4β) + 4k ≥ 4/3 in the range\nwe also deduce the case β ∈ (61/62, 68/69]. For the remaining case β ∈ (68/69, 1] and k ≥ 6, we see that\nBased on the above argument, we now give explicit choices of k and corresponding intervals which optimise our bound.\n• If β ∈ (7/8, 13/14], we take k = 2 and (3.13) simplifies to Therefore in all cases, where we also recall the condition α < β + 1, we have ψ ♯ (x, y; p, a) ≥ p α−1−ε+o (1) .\nSince this is true for all ε > 0, the result follows immediately.\n3.3. Proof of Theorem 1.3. Let M be a parameter which will be fixed later. We introduce the subset of residues modulo p S = {a : a quadratic non-residue such that F (a, p) ≤ M} .\nFirstly, we remark that every squarefull integer n can be written as n = r 2 s with s | r . Furthermore, if a is a quadratic non-residue, we notice that s has to be a quadratic non-residue in this representation; in particular s ≥ n p . Let us count the number of products r 2 s ≤ M with s | r and s ≥ n p the smallest quadratic non-residue modulo p. So we now turn our attention to the upper bound.\nClearly, if a ≡ u 2 mod p, 0 ≤ u < p, is quadratic residue (or a = 0), then F (a, p) ≤ u 2 ≤ p 2 .\nWe now fix some ε > 0 and denote by A the set of quadratic nonresidues for which F (a, p) ≥ p 2+η 0 +ε .\nIt is enough to show that the cardinality of A satisfies (3.14) #A = o(p).\nWe set K = p η 0 +ε/2 and U = p 1−η 0 and let N be the set of quadratic non-residues in the interval [1, K].\nIn particular #N = T (K).\nClearly for a ∈ A the congruence (3.15) a ≡ n 3 u 2 mod p, n ∈ N , 1 ≤ u ≤ U, has no solution. Thus expressing the number of solutions to (3.15) via characters we see that n∈N 1≤u≤U\nSumming over all a ∈ A and using the multiplicativity of characters, we arrive to (3.16) χ∈Ωp a∈A\nwhere χ denotes the complex conjugate character of χ. Now, the contribution to (3.16) from the principal character is obviously #AT (K)U .\nFurthermore, since all elements of A are quadratic non-residues, the contribution to (3.16) from the quadratic character, that is, from the Legendre symbol is with Ω ♯ p being the subset of Ω * p where we removed the quadratic character. Now, for χ ∈ Ω ♯ p we have by definition χ 2 = χ 0 . Furthermore, each character from Ω * p occurs at most twice as χ 2 and each character from Ω p (including also χ 0 in this case) occurs at most three times as χ 3 for χ ∈ Ω ♯ p . Using the Hölder inequality, we now derive from (3.17) that For Σ 2 , using again the orthogonality of characters, we write Σ 2 = 3(p − 1)# {(n 1 , n 2 , n 3 , n 4 ) ∈ N : n 1 n 2 ≡ n 3 n 4 mod p} ≤ 3(p − 1)# {(n 1 , n 2 , n 3 , n 4 ) ∈ [1, K] : n 1 n 2 ≡ n 3 n 4 mod p} Applying Lemma 2.8 and using that K 2 ≤ p provided that ε is small enough, we derive (3.20) Σ 2 ≤ K 2 p 1+o (1) .\nFinally, we also estimate Σ 3 , directly by Lemma 2.8 getting which together with Lemma 2.9 yields #A ≤ K −1 U −1 p 2+o(1) = p 1−ε/2+o (1) .\nWe now see that (3.14) holds which concludes the proof."},{"title":"Lemma 2. 4 .","content":""},{"title":"Lemma 2. 8 .","content":""},{"title":".","content":""},{"title":"( 3 .","content":""}],"string":"[\n {\n \"title\": \"1.2.\",\n \"content\": \"Results for squarefree numbers. We start with a lower bound on ψ ♯ (x, y; p, a) which holds for any prime p. Theorem 1.1. For any fixed real numbers α and β with β ∈ (23/24, 1] and α ∈ (9/2 − 3β, 3β], for x = p α+o(1) and y = p β+o(1) we have\\nTaking y = p β with 23/24 < β ≤ 1 and q = p in the main result of Balog & Pomerance [3] gives the existence of a p β -smooth integer (not necessary squarefree) n ≤ p max{3β/2,3/4+β}+o(1) = p 3/4+β+o(1) since β ≤ 1. We notice that 9/2 − 3β < 3/4 + β, under the condition 23/24 < β . Therefore, Theorem 1.1 always improves on the bound given by the main result of Balog & Pomerance [3]. We remark that removing the squarefreeness condition does not help us to improve on Theorem 1.1 due to the method used.\\nWe also obtain a result for almost all primes. Firstly, we define the interval for any η > η 0 and a sufficiently large p. Noticing that an −3 p is a quadratic residue modulo p, we now obtain F (a, p) ≤ n 3 p (p − 1) 2 . Hence, we have the trivial bound F (a, p) ≤ p 2+3η 0 +o(1) for any a, which we unfortunately do not know how to improve. However, we remark that assuming the Vinogradov's conjecture that n p ≤ p o(1) (which is implied by the Generalised Riemann Hypothesis in the stronger form n p ≪ log 2 p proved by Ankeny [2], see also [12,Section 5.9] for a discussion), we have the bound F (a, p) ≤ p 2+o (1) . Even though we cannot reach such a bound, we obtain an unconditional better bound for almost all a ∈ {0, . . . , p − 1}.\\nWe also note that from a result on counting squarefull integers [19], for any set A of A distinct residues modulo p we have\\nwhere, as usual, we use A ≪ B and B ≫ A as an equivalent to the inequality |A| ≤ cB with some constant c > 0, which occasionally, where obvious, may depend on the real parameter ε > 0. We slightly refine this result:\\nwhere n p denotes the least quadratic non-residue modulo p.\\nUsing the lower bound in Theorem 1.3, together with an unconditional result of Graham and Ringrose [10] on primes with large values of n p and a conditional result on the Generalised Riemann Hypothesis (GRH) of Montgomery [14], we immediately derive Corollary 1.4. For infinitely many primes p we have max a mod p F (a, p) ≫ p 2 (log p)(log log log p), unconditionally, p 2 (log p)(log log p), under the GRH.\\nIn Section 2, we collect some results which will be used to prove the main results in Section 3.\"\n },\n {\n \"title\": \"Preparation lemmas\",\n \"content\": \"2.1. Exponential sums with reciprocals of primes. As usual, we define e p (z) = exp(2iπz/p). For an integer k with gcd(k, p) = 1 we use k to denote the multiplicative inverse of k modulo p, that is, the unique integer with kk ≡ 1 mod p and 1 ≤ k < p. It is convenient to introduce the quantity\\nThe following bound of the double exponential sum over primes is a combination of [16, Lemma 3.5] (for L ≤ p 1/3 ) and of [9, Lemma 2.4] (for p 1/3 ≤ L < p).\\n2.2. Some congruences with products of primes. We denote N a,p (L, h) to be the number of solutions to the congruence\\nwhere h and L are two positive real numbers and L is the set of primes ℓ ∈ [L, 2L]. We now use Lemma 2.1 to derive an analogue of [16, Lemma 3.10] (which also applies to L ≥ p 1/3 ).\\nLemma 2.2. For any integer a and prime p with gcd(a, p) = 1 and reals h and L with 1 ≤ h, L < p, we have\\nwhere K = #L is the cardinality of L and B(p, L) is defined by (2.1).\\nWe also recall that by [\\nwhere K = #L is the cardinality of L and B(p, L) is defined by (2.1).\\nWe also need the bound of [16, Lemma 3.14] on the number of solutions Q a,p (L, h) to the congruence\\nLemma 2.5. For any integer a and prime p with gcd(a, p) = 1 and\\nIt is shown in [16,Lemma 3.11], that for almost all primes p, the asymptotic formula of Lemma 2.2 can be improved as follows.\\nwhere K = #L is the cardinality of L.\\nFinally, we also recall that by [\"\n },\n {\n \"title\": \"Moments of character sums.\",\n \"content\": \"Let Ω p denote the set of all Dirichlet characters modulo p and let Ω * p = Ω p \\\\ {χ 0 } denote the set of all nonprincipal Dirichlet characters modulo p.\\nWe need the following result of Ayyad, Cochrane and Zheng [1, Theorem 2], see also [13] for a slightly sharper bound (which however does not change our final result).\"\n },\n {\n \"title\": \"2.4.\",\n \"content\": \"Quadratic non-residues in short intervals. Let T p (K) denote the number of quadratic non-residues modulo p in the interval [1, K].\\nWe need an extension of (1.2). The following bound is given in [4, Theorem 2.1]. Lemma 2.9. For any real η > η 0 , where η 0 is given by (1.1), there is a constant c > 0, such that for a sufficiently large p and K ≥ p η we have T p (K) ≥ cK.\"\n },\n {\n \"title\": \"Proofs of main results\",\n \"content\": \"3.1. Proof of Theorem 1.1. For a sufficiently small ε > 0, we set\\nSince L ≤ h ≤ y , N ♯ a,p (L, h) counts a subset of y -smooth integers in an arithmetic progression. Noticing that L 2 h ≤ p α−ε = x 1−ε+o(1) , we see that for a sufficiently large p we have\\nwhere we estimated the contribution coming from non-squarefree products ℓ 1 ℓ 2 u (precisely products with ℓ 1 = ℓ 2 or with ℓ 1 | u or with ℓ 2 | u) using Lemma 2.5 with h/L replacing h as in the end of the proof of [16, Theorem 1.4]. We use a crude estimate for the main term: (1) . Choosing D = p ε/2 and using Lemma 2.4, we derive\\nsince B(p, L) dominates L and the main term (3.2) dominates the first error term L 2 h(Dp) −1 in Lemma 2.4. To begin, we remark that the term h 1/2 in (3.3) is dominated by the main term due to the inequality α − 1 > 9/2 − 3β − 1 > β/2 for β ≤ 1. We split the discussion on the contribution of B(p, L) into two cases depending on α.\\nFirstly, suppose that α ∈ (9/2−3β, 2/3+β]. Since α ≤ 2/3+β , this implies L < p 1/3 and hence B(p, L) = L 3/2 p 1/8 by (2.1). Therefore, recalling (3.3) and (3.2), we obtain (1) ).\\nFor ε sufficiently small, we have α > 9/2 − 3β + 3ε which implies that the main term dominates trivially the remainder term in (3.4). Secondly, assume that α ∈ (2/3 + β, 3β]. In particular, since β ≤ 1 we have (1) ).\\nNotice that we have\\nwhen β ∈ (23/24, 1] and ε > 0 is sufficiently small. It follows that the main term dominates the remainder term in (3.5). Therefore, in all cases we conclude ψ ♯ (x, y; p, a) ≥ p α−ε−1+o (1) .\\nSince this is valid for all sufficiently small ε > 0, the result follows.\"\n },\n {\n \"title\": \"3.2.\",\n \"content\": \"Proof of Theorem 1.2. We follow the proof of [16,Theorem 1.6].\\nFor ε > 0, we set\\nWe note that (α − 1)/2 > 0 for α ∈ I(β) and so D < E if ε > 0 is sufficiently small. We also have E < h 1/2 since α < β + 1.\\nSince α < β + 1 ≤ 3β in the range β ∈ (7/8, 1], we get L ≤ h. In particular, we have as before the inequality (3.1).\\nBy inclusion and exclusion, we have\\nTo abstain from clutter, all the bounds below are valid for all but o(Q/ log Q) primes p ∈ [Q, 2Q].\\nSince α < β + 1 < 2 + β + ε + o(1) and β ≤ 1, we obtain respectively L ≤ Q and h ≤ p. By Lemma 2.6\\nSubstituting (3.8), (3.9) and (3.10) in (3.7), we obtain\\nwhere we set R = D L (3k−1)/(2k) p 1/(2k) + L (4k−1)/(2k) + L 2 h Dp + E + (L 2 h) 1/4 and the main term verifies an analogue of (3.2), precisely, (1) .\\nNotice that the choice of E in (3.6) implies that E is smaller than the main term (3.11). We now see from (3.11) that if\\nfor some positive integer k , then for a sufficiently small ε the main term dominates the remainder term in (3.1) and the result follows. Rearranging (3.12) gives\\nFirst, we remark that 2 − β ≤ (1 − 4k)β + 4k since β ≤ 1 and we can discard 2 − β from the maximum in (3.13).\\nFurthermore, for k ≤ 5, we see that 4/3 is dominated by the first term of the right hand side of (3.13). In this case, a quick computation shows that (1 − 3k)β + 2 + 4k\\nif and only if β ≥ 1−1/2k 2 . Thus, in the interval (1−1/2k 2 , 1−1/2(k+ 1) 2 ], the maximum is given either by ((1 − 3k)β + 2 + 4k) /(k + 1) or by (1 − 4m)β + 4m with m ≥ k + 1. Since the function f (z) = (1 − 4z)β + 4z is a monotonically increasing function of z , we check only the case m = k + 1 and verify that\\nif and only if\\n.\"\n },\n {\n \"title\": \"Splitting the interval\",\n \"content\": \"into two intervals as follows\\nand recalling that k ≤ 5, we deduce after short computations the result for β ≤ β 0 (5) = 61/62.\\nFor k ≥ 6, noticing that (1 − 4β) + 4k ≥ 4/3 in the range\\nwe also deduce the case β ∈ (61/62, 68/69]. For the remaining case β ∈ (68/69, 1] and k ≥ 6, we see that\\nBased on the above argument, we now give explicit choices of k and corresponding intervals which optimise our bound.\\n• If β ∈ (7/8, 13/14], we take k = 2 and (3.13) simplifies to Therefore in all cases, where we also recall the condition α < β + 1, we have ψ ♯ (x, y; p, a) ≥ p α−1−ε+o (1) .\\nSince this is true for all ε > 0, the result follows immediately.\\n3.3. Proof of Theorem 1.3. Let M be a parameter which will be fixed later. We introduce the subset of residues modulo p S = {a : a quadratic non-residue such that F (a, p) ≤ M} .\\nFirstly, we remark that every squarefull integer n can be written as n = r 2 s with s | r . Furthermore, if a is a quadratic non-residue, we notice that s has to be a quadratic non-residue in this representation; in particular s ≥ n p . Let us count the number of products r 2 s ≤ M with s | r and s ≥ n p the smallest quadratic non-residue modulo p. So we now turn our attention to the upper bound.\\nClearly, if a ≡ u 2 mod p, 0 ≤ u < p, is quadratic residue (or a = 0), then F (a, p) ≤ u 2 ≤ p 2 .\\nWe now fix some ε > 0 and denote by A the set of quadratic nonresidues for which F (a, p) ≥ p 2+η 0 +ε .\\nIt is enough to show that the cardinality of A satisfies (3.14) #A = o(p).\\nWe set K = p η 0 +ε/2 and U = p 1−η 0 and let N be the set of quadratic non-residues in the interval [1, K].\\nIn particular #N = T (K).\\nClearly for a ∈ A the congruence (3.15) a ≡ n 3 u 2 mod p, n ∈ N , 1 ≤ u ≤ U, has no solution. Thus expressing the number of solutions to (3.15) via characters we see that n∈N 1≤u≤U\\nSumming over all a ∈ A and using the multiplicativity of characters, we arrive to (3.16) χ∈Ωp a∈A\\nwhere χ denotes the complex conjugate character of χ. Now, the contribution to (3.16) from the principal character is obviously #AT (K)U .\\nFurthermore, since all elements of A are quadratic non-residues, the contribution to (3.16) from the quadratic character, that is, from the Legendre symbol is with Ω ♯ p being the subset of Ω * p where we removed the quadratic character. Now, for χ ∈ Ω ♯ p we have by definition χ 2 = χ 0 . Furthermore, each character from Ω * p occurs at most twice as χ 2 and each character from Ω p (including also χ 0 in this case) occurs at most three times as χ 3 for χ ∈ Ω ♯ p . Using the Hölder inequality, we now derive from (3.17) that For Σ 2 , using again the orthogonality of characters, we write Σ 2 = 3(p − 1)# {(n 1 , n 2 , n 3 , n 4 ) ∈ N : n 1 n 2 ≡ n 3 n 4 mod p} ≤ 3(p − 1)# {(n 1 , n 2 , n 3 , n 4 ) ∈ [1, K] : n 1 n 2 ≡ n 3 n 4 mod p} Applying Lemma 2.8 and using that K 2 ≤ p provided that ε is small enough, we derive (3.20) Σ 2 ≤ K 2 p 1+o (1) .\\nFinally, we also estimate Σ 3 , directly by Lemma 2.8 getting which together with Lemma 2.9 yields #A ≤ K −1 U −1 p 2+o(1) = p 1−ε/2+o (1) .\\nWe now see that (3.14) holds which concludes the proof.\"\n },\n {\n \"title\": \"Lemma 2. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 2. 8 .\",\n \"content\": \"\"\n },\n {\n \"title\": \".\",\n \"content\": \"\"\n },\n {\n \"title\": \"( 3 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"For any integer a and prime p with gcd(a, p) = 1 and reals h, D and L with","content":"Lemma 2. 4 .\n4For any integer a and prime p with gcd(a, p) = 1 and reals h, D and L with"},{"id":"","caption":"For any integer K ≥ 1, we have χ∈Ω * p 1≤n≤Kχ(n) 4 ≤ K 2 p 1+o(1) .","content":"Lemma 2. 8 .\n8For any integer K ≥ 1, we have χ∈Ω * p 1≤n≤Kχ(n) 4 ≤ K 2 p 1+o(1) ."},{"id":"","caption":"for ε > 0 chosen sufficiently small. Hence p 1/3 ≤ L < p and we have B(p, L) = L 15/8 by (2.1). Therefore, recalling (3.3) and (3.2), we obtain (3.5) ψ ♯ (x, y; p, a) ≥ p α−1−ε+o(1) + O(p 15(α−β)/16−7ε/16+o","content":"for ε > 0 chosen sufficiently small. Hence p 1/3 ≤ L < p and we have B(p, L) = L 15/8 by (2.1). Therefore, recalling (3.3) and (3.2), we obtain (3.5) ψ ♯ (x, y; p, a) ≥ p α−1−ε+o(1) + O(p 15(α−β)/16−7ε/16+o"},{"id":"","caption":"α > max {5(2 − β)/3, 8 − 7β, 4/3, 2 − β} = 5(2 − β)/3.• If β ∈ (13/14, 17/18], we take k = 3 and (3.13) simplifies to α > max {(7 − 4β)/2, 12 − 11β, 4/3, 2 − β} = 12 − 11β.• If β ∈ (17/18, 25/26], we take k = 3 and (3.13) simplifies to α > max {(7 − 4β)/2, 12 − 11β, 4/3, 2 − β} = (7 − 4β)/2.• If β ∈ (25/26, 31/32], we take k = 4 and (3.13) simplifies to α > max {(18 − 11β)/5, 16 − 15β, 4/3, 2 − β} = 16 − 15β.• If β ∈ (31/32, 41/42], we take k = 4 and (3.13) simplifies to α > max {(18 − 11β)/5, 16 − 15β, 4/3, 2 − β} = (18 − 11β)/5.• If β ∈ (41/42, 49/50], we take k = 5 and (3.13) simplifies to α > max {(11 − 7β)/3, 20 − 19β, 4/3, 2 − β} = 20 − 19β.• If β ∈ (49/50, 61/62], we take k = 5 and (3.13) simplifies to α > max {(11 − 7β)/3, 20 − 19β, 4/3, 2 − β} = (11 − 7β)/3. • If β ∈ (61/62, 68/69], we take k = 6 and (3.13) simplifies to α > max {(26 − 17β)/7, 24 − 23β, 4/3, 2 − β} = 24 − 23β.• If β ∈ (68/69, 1], we take k = 6 and (3.13) simplifies to α > max {(26 − 17β)/7, 24 − 23β, 4/3, 2 − β} = 4/3.","content":"α > max {5(2 − β)/3, 8 − 7β, 4/3, 2 − β} = 5(2 − β)/3.• If β ∈ (13/14, 17/18], we take k = 3 and (3.13) simplifies to α > max {(7 − 4β)/2, 12 − 11β, 4/3, 2 − β} = 12 − 11β.• If β ∈ (17/18, 25/26], we take k = 3 and (3.13) simplifies to α > max {(7 − 4β)/2, 12 − 11β, 4/3, 2 − β} = (7 − 4β)/2.• If β ∈ (25/26, 31/32], we take k = 4 and (3.13) simplifies to α > max {(18 − 11β)/5, 16 − 15β, 4/3, 2 − β} = 16 − 15β.• If β ∈ (31/32, 41/42], we take k = 4 and (3.13) simplifies to α > max {(18 − 11β)/5, 16 − 15β, 4/3, 2 − β} = (18 − 11β)/5.• If β ∈ (41/42, 49/50], we take k = 5 and (3.13) simplifies to α > max {(11 − 7β)/3, 20 − 19β, 4/3, 2 − β} = 20 − 19β.• If β ∈ (49/50, 61/62], we take k = 5 and (3.13) simplifies to α > max {(11 − 7β)/3, 20 − 19β, 4/3, 2 − β} = (11 − 7β)/3. • If β ∈ (61/62, 68/69], we take k = 6 and (3.13) simplifies to α > max {(26 − 17β)/7, 24 − 23β, 4/3, 2 − β} = 24 − 23β.• If β ∈ (68/69, 1], we take k = 6 and (3.13) simplifies to α > max {(26 − 17β)/7, 24 − 23β, 4/3, 2 − β} = 4/3."},{"id":"","caption":"Noticing that r ≤ (M/s) 1/2 , we have at most M 1/2 s −3/2 possible values of r . Thus the number of different products r 2 s is bounded by Setting M = p 2 n p f (p), we get #S = o(p) which concludes the proof. The assertion max a mod p F (a, p) ≫ p 2 n p follows by the same argument by setting S = {a : a quadratic non-residue} and M = max a∈S F (a, p).","content":".\nNoticing that r ≤ (M/s) 1/2 , we have at most M 1/2 s −3/2 possible values of r . Thus the number of different products r 2 s is bounded by Setting M = p 2 n p f (p), we get #S = o(p) which concludes the proof. The assertion max a mod p F (a, p) ≫ p 2 n p follows by the same argument by setting S = {a : a quadratic non-residue} and M = max a∈S F (a, p)."},{"id":"","caption":"upper bounds come from the discussion above. We now see by the orthogonality of characters that we have(3.19) Σ 1 ≤ (p − 1)#A.","content":"upper bounds come from the discussion above. We now see by the orthogonality of characters that we have(3.19) Σ 1 ≤ (p − 1)#A."},{"id":"","caption":"21) Σ 3 ≤ U 2 p 1+o(1) .Substituting(3.19), (3.20) and (3.21) in (3.18), we now derive#AT (K)U ≤ (#A) 1/2 K 1/2 U 1/2 p 1+o(1)","content":"( 3 .\n321) Σ 3 ≤ U 2 p 1+o(1) .Substituting(3.19), (3.20) and (3.21) in (3.18), we now derive#AT (K)U ≤ (#A) 1/2 K 1/2 U 1/2 p 1+o(1)"},{"id":"","caption":"and recall that by the classical bound of Burgess[6] on the smallest quadratic non-residue n p we have","content":"and recall that by the classical bound of Burgess[6] on the smallest quadratic non-residue n p we have7/8, 13/14], \n12 − 11β \nif β ∈ (13/14, 17/18], \n(7 − 4β)/2 \nif β ∈ (17/18, 25/26], \n16 − 15β \nif β ∈ (25/26, 31/32], \n(18 − 11β)/5 if β ∈ (31/32, 41/42], \n20 − 19β \nif β ∈ (41/42, 49/50], \n(11 − 7β)/3 \nif β ∈ (49/50, 61/62], \n24 − 23β \nif β ∈ (61/62, 68/69], \n4/3 \nif β ∈ (68/69, 1]. \n\nTheorem 1.2. Fix real numbers α and β such that β ∈ (7/8, 1] and \nα ∈ I(β). Letting x = Q α+o(1) and y = Q β+o(1) , as Q → ∞, we have \n\nψ ♯ (x, y; p, a) ≥ x 1+o(1) /p \n\nfor all but o(Q/ log Q) primes p ∈ [Q, 2Q]. \n\n1.3. Results for squarefull numbers. First we observe that if a is a \nquadratic residue modulo p (or a = 0), then a ≡ b 2 mod p for some \ninteger b ∈ [0, p − 1] and so we have trivially F (a, p) ≤ (p − 1) 2 in this \ncase. \nTo estimate F (a, p) for a quadratic non-residue a we denote \n\n(1.1) \nη 0 = \n1 \n4 \n√ e \n\n(1.2) \nn p ≤ p η"},{"id":"","caption":"16, Lemma 3.12] we have: Lemma 2.3. For any integer a and prime p with gcd(a, p) = 1 and reals 1 ≤ h, L < p we have N a,p (L, h) ≤ (L 2 h/p + 1)p o(1) .Furthermore, let N ♯ a,p count the number of squarefree solutions to the congruence (2.2). Following the proof of [16, Theorem 1.4], but using a more general bound of Lemma 2.2 instead of [16, Lemma 3.10] as well as Lemma 2.3 (exactly as in [16]), we derive","content":"16, Lemma 3.12] we have: Lemma 2.3. For any integer a and prime p with gcd(a, p) = 1 and reals 1 ≤ h, L < p we have N a,p (L, h) ≤ (L 2 h/p + 1)p o(1) .Furthermore, let N ♯ a,p count the number of squarefree solutions to the congruence (2.2). Following the proof of [16, Theorem 1.4], but using a more general bound of Lemma 2.2 instead of [16, Lemma 3.10] as well as Lemma 2.3 (exactly as in [16]), we derive"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"For any integer a and prime p with gcd(a, p) = 1 and reals h, D and L with\",\n \"content\": \"Lemma 2. 4 .\\n4For any integer a and prime p with gcd(a, p) = 1 and reals h, D and L with\"\n },\n {\n \"id\": \"\",\n \"caption\": \"For any integer K ≥ 1, we have χ∈Ω * p 1≤n≤Kχ(n) 4 ≤ K 2 p 1+o(1) .\",\n \"content\": \"Lemma 2. 8 .\\n8For any integer K ≥ 1, we have χ∈Ω * p 1≤n≤Kχ(n) 4 ≤ K 2 p 1+o(1) .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"for ε > 0 chosen sufficiently small. Hence p 1/3 ≤ L < p and we have B(p, L) = L 15/8 by (2.1). Therefore, recalling (3.3) and (3.2), we obtain (3.5) ψ ♯ (x, y; p, a) ≥ p α−1−ε+o(1) + O(p 15(α−β)/16−7ε/16+o\",\n \"content\": \"for ε > 0 chosen sufficiently small. Hence p 1/3 ≤ L < p and we have B(p, L) = L 15/8 by (2.1). Therefore, recalling (3.3) and (3.2), we obtain (3.5) ψ ♯ (x, y; p, a) ≥ p α−1−ε+o(1) + O(p 15(α−β)/16−7ε/16+o\"\n },\n {\n \"id\": \"\",\n \"caption\": \"α > max {5(2 − β)/3, 8 − 7β, 4/3, 2 − β} = 5(2 − β)/3.• If β ∈ (13/14, 17/18], we take k = 3 and (3.13) simplifies to α > max {(7 − 4β)/2, 12 − 11β, 4/3, 2 − β} = 12 − 11β.• If β ∈ (17/18, 25/26], we take k = 3 and (3.13) simplifies to α > max {(7 − 4β)/2, 12 − 11β, 4/3, 2 − β} = (7 − 4β)/2.• If β ∈ (25/26, 31/32], we take k = 4 and (3.13) simplifies to α > max {(18 − 11β)/5, 16 − 15β, 4/3, 2 − β} = 16 − 15β.• If β ∈ (31/32, 41/42], we take k = 4 and (3.13) simplifies to α > max {(18 − 11β)/5, 16 − 15β, 4/3, 2 − β} = (18 − 11β)/5.• If β ∈ (41/42, 49/50], we take k = 5 and (3.13) simplifies to α > max {(11 − 7β)/3, 20 − 19β, 4/3, 2 − β} = 20 − 19β.• If β ∈ (49/50, 61/62], we take k = 5 and (3.13) simplifies to α > max {(11 − 7β)/3, 20 − 19β, 4/3, 2 − β} = (11 − 7β)/3. • If β ∈ (61/62, 68/69], we take k = 6 and (3.13) simplifies to α > max {(26 − 17β)/7, 24 − 23β, 4/3, 2 − β} = 24 − 23β.• If β ∈ (68/69, 1], we take k = 6 and (3.13) simplifies to α > max {(26 − 17β)/7, 24 − 23β, 4/3, 2 − β} = 4/3.\",\n \"content\": \"α > max {5(2 − β)/3, 8 − 7β, 4/3, 2 − β} = 5(2 − β)/3.• If β ∈ (13/14, 17/18], we take k = 3 and (3.13) simplifies to α > max {(7 − 4β)/2, 12 − 11β, 4/3, 2 − β} = 12 − 11β.• If β ∈ (17/18, 25/26], we take k = 3 and (3.13) simplifies to α > max {(7 − 4β)/2, 12 − 11β, 4/3, 2 − β} = (7 − 4β)/2.• If β ∈ (25/26, 31/32], we take k = 4 and (3.13) simplifies to α > max {(18 − 11β)/5, 16 − 15β, 4/3, 2 − β} = 16 − 15β.• If β ∈ (31/32, 41/42], we take k = 4 and (3.13) simplifies to α > max {(18 − 11β)/5, 16 − 15β, 4/3, 2 − β} = (18 − 11β)/5.• If β ∈ (41/42, 49/50], we take k = 5 and (3.13) simplifies to α > max {(11 − 7β)/3, 20 − 19β, 4/3, 2 − β} = 20 − 19β.• If β ∈ (49/50, 61/62], we take k = 5 and (3.13) simplifies to α > max {(11 − 7β)/3, 20 − 19β, 4/3, 2 − β} = (11 − 7β)/3. • If β ∈ (61/62, 68/69], we take k = 6 and (3.13) simplifies to α > max {(26 − 17β)/7, 24 − 23β, 4/3, 2 − β} = 24 − 23β.• If β ∈ (68/69, 1], we take k = 6 and (3.13) simplifies to α > max {(26 − 17β)/7, 24 − 23β, 4/3, 2 − β} = 4/3.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Noticing that r ≤ (M/s) 1/2 , we have at most M 1/2 s −3/2 possible values of r . Thus the number of different products r 2 s is bounded by Setting M = p 2 n p f (p), we get #S = o(p) which concludes the proof. The assertion max a mod p F (a, p) ≫ p 2 n p follows by the same argument by setting S = {a : a quadratic non-residue} and M = max a∈S F (a, p).\",\n \"content\": \".\\nNoticing that r ≤ (M/s) 1/2 , we have at most M 1/2 s −3/2 possible values of r . Thus the number of different products r 2 s is bounded by Setting M = p 2 n p f (p), we get #S = o(p) which concludes the proof. The assertion max a mod p F (a, p) ≫ p 2 n p follows by the same argument by setting S = {a : a quadratic non-residue} and M = max a∈S F (a, p).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"upper bounds come from the discussion above. We now see by the orthogonality of characters that we have(3.19) Σ 1 ≤ (p − 1)#A.\",\n \"content\": \"upper bounds come from the discussion above. We now see by the orthogonality of characters that we have(3.19) Σ 1 ≤ (p − 1)#A.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"21) Σ 3 ≤ U 2 p 1+o(1) .Substituting(3.19), (3.20) and (3.21) in (3.18), we now derive#AT (K)U ≤ (#A) 1/2 K 1/2 U 1/2 p 1+o(1)\",\n \"content\": \"( 3 .\\n321) Σ 3 ≤ U 2 p 1+o(1) .Substituting(3.19), (3.20) and (3.21) in (3.18), we now derive#AT (K)U ≤ (#A) 1/2 K 1/2 U 1/2 p 1+o(1)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"and recall that by the classical bound of Burgess[6] on the smallest quadratic non-residue n p we have\",\n \"content\": \"and recall that by the classical bound of Burgess[6] on the smallest quadratic non-residue n p we have7/8, 13/14], \\n12 − 11β \\nif β ∈ (13/14, 17/18], \\n(7 − 4β)/2 \\nif β ∈ (17/18, 25/26], \\n16 − 15β \\nif β ∈ (25/26, 31/32], \\n(18 − 11β)/5 if β ∈ (31/32, 41/42], \\n20 − 19β \\nif β ∈ (41/42, 49/50], \\n(11 − 7β)/3 \\nif β ∈ (49/50, 61/62], \\n24 − 23β \\nif β ∈ (61/62, 68/69], \\n4/3 \\nif β ∈ (68/69, 1]. \\n\\nTheorem 1.2. Fix real numbers α and β such that β ∈ (7/8, 1] and \\nα ∈ I(β). Letting x = Q α+o(1) and y = Q β+o(1) , as Q → ∞, we have \\n\\nψ ♯ (x, y; p, a) ≥ x 1+o(1) /p \\n\\nfor all but o(Q/ log Q) primes p ∈ [Q, 2Q]. \\n\\n1.3. Results for squarefull numbers. First we observe that if a is a \\nquadratic residue modulo p (or a = 0), then a ≡ b 2 mod p for some \\ninteger b ∈ [0, p − 1] and so we have trivially F (a, p) ≤ (p − 1) 2 in this \\ncase. \\nTo estimate F (a, p) for a quadratic non-residue a we denote \\n\\n(1.1) \\nη 0 = \\n1 \\n4 \\n√ e \\n\\n(1.2) \\nn p ≤ p η\"\n },\n {\n \"id\": \"\",\n \"caption\": \"16, Lemma 3.12] we have: Lemma 2.3. For any integer a and prime p with gcd(a, p) = 1 and reals 1 ≤ h, L < p we have N a,p (L, h) ≤ (L 2 h/p + 1)p o(1) .Furthermore, let N ♯ a,p count the number of squarefree solutions to the congruence (2.2). Following the proof of [16, Theorem 1.4], but using a more general bound of Lemma 2.2 instead of [16, Lemma 3.10] as well as Lemma 2.3 (exactly as in [16]), we derive\",\n \"content\": \"16, Lemma 3.12] we have: Lemma 2.3. For any integer a and prime p with gcd(a, p) = 1 and reals 1 ≤ h, L < p we have N a,p (L, h) ≤ (L 2 h/p + 1)p o(1) .Furthermore, let N ♯ a,p count the number of squarefree solutions to the congruence (2.2). Following the proof of [16, Theorem 1.4], but using a more general bound of Lemma 2.2 instead of [16, Lemma 3.10] as well as Lemma 2.3 (exactly as in [16]), we derive\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"The congruence x 1 x 2 ≡ x 3 x 4 mod p, the equation x 1 x 2 = x 3 x 4 and mean values of character sums. A Ayyad, T Cochrane, Z Zheng, J. Number Theory. 596A. Ayyad, T. Cochrane and Z. Zheng, 'The congruence x 1 x 2 ≡ x 3 x 4 mod p, the equation x 1 x 2 = x 3 x 4 and mean values of character sums', J. Number Theory 59 (1996), 398-413. 6","doi":""},{"text":"The least quadratic non residue. N C Ankeny, Ann. of Math. 552N. C. Ankeny, 'The least quadratic non residue', Ann. of Math. (2), 55 (1952), 65-72. 3","doi":""},{"text":"The distribution of smooth numbers in arithmetic progressions. A Balog, C Pomerance, Proc. Amer. Math. Soc. 1152A. Balog and C. Pomerance, 'The distribution of smooth numbers in arithmetic progressions', Proc. Amer. Math. Soc., 115 (1992), 33-43. 1, 2","doi":""},{"text":"Density of non-residues in Burgess-type intervals and applications. W D Banks, M Z Garaev, R Heath-Brown, I E Shparlinski, Bull. London Math. Soc. 407W. D. Banks, M. Z. Garaev, R. Heath-Brown and I. E. Shparlinski, 'Density of non-residues in Burgess-type intervals and applications', Bull. London Math. Soc. , 40 (2008), 88-96. 7","doi":""},{"text":"Squarefree smooth numbers and Euclidean prime generators. A Booker, C Pomerance, Proc. Amer. Math. Soc. 1451A. Booker and C. Pomerance, 'Squarefree smooth numbers and Euclidean prime generators', Proc. Amer. Math. Soc., 145 (2017), 5035-5042. 1","doi":""},{"text":"The distribution of quadratic residues and non-residues. D A Burgess, 4MathematikaD. A. Burgess, 'The distribution of quadratic residues and non-residues', Math- ematika, 4 (1957), 106-112. 3","doi":""},{"text":"Squarefull numbers in arithmetic progression, II. T H Chan, J. Number Theory. 1522T. H. Chan, 'Squarefull numbers in arithmetic progression, II', J. Number Theory 152 (2015), 90-104. 2","doi":""},{"text":"Théorémes de type Fouvry-Iwaniec pour les entiers friables. S Drappeau, Compos. Math. 1511S. Drappeau, 'Théorémes de type Fouvry-Iwaniec pour les entiers friables', Compos. Math., 151 (2015), 828-862. 1","doi":""},{"text":"An estimate of Kloosterman sums with prime numbers and an application. M Z Garaev, Matem. Zametki. 884in Russian. translation in Math. NotesM. Z. Garaev, 'An estimate of Kloosterman sums with prime numbers and an application', Matem. Zametki, 88 (2010), 365-373, (in Russian); translation in Math. Notes, 88 (2010), 330-337. 4","doi":""},{"text":"Lower bounds for least quadratic nonresidues' Analytic number theory. S W Graham, C J Ringrose, Progr. Math. 854S. W. Graham and C.J. Ringrose, 'Lower bounds for least quadratic non- residues' Analytic number theory, Progr. Math., 85(1990), 269-309. 4","doi":""},{"text":"On a paper of K. Soundararajan on smooth numbers in arithmetic progressions. A J Harper, J. Number Theory. 1321A. J. Harper, 'On a paper of K. Soundararajan on smooth numbers in arith- metic progressions', J. Number Theory, 132 (2012), 182-199. 1","doi":""},{"text":"Analytic number theory. H Iwaniec, E Kowalski, American Mathematical SocietyProvidence, RIH. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society, Providence, RI, 2004. 3","doi":""},{"text":"On the congruence x 1 x 2 ≡ x 3 x 4 mod q. B Kerr, J. Number Theory. 6B. Kerr, 'On the congruence x 1 x 2 ≡ x 3 x 4 mod q ', J. Number Theory, 180 (2017), 154-168. 6","doi":""},{"text":"Topics in multiplicative number theory. H L Montgomery, Lecture Notes in Mathematics. 2274Springer-VerlagH. L. Montgomery, 'Topics in multiplicative number theory', Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, 1971. 4","doi":""},{"text":"Character sums over squarefree and squarefull numbers. M Munsch, Arch. Math. 1022M. Munsch, 'Character sums over squarefree and squarefull numbers', Arch. Math., 102 (2014), 555-563. 2","doi":""},{"text":"On smooth square-free numbers in arithmetic progressions. M Munsch, I E Shparlinski, 5M. Munsch and I. E. Shparlinski, 'On smooth square-free numbers in arith- metic progressions', Preprint , 2017, http://arxiv.org/abs/1710.04705. 1, 4, 5, 6, 7, 8","doi":""},{"text":"Square-full primitive roots. M Munsch, T Trudgian, Intern. J. Number Theory. 142M. Munsch and T. Trudgian, 'Square-full primitive roots', Intern. J. Number Theory, 14 (2018), 1013-1021. 2","doi":""},{"text":"The distribution of smooth numbers in arithmetic progressions', Anatomy of integers. K Soundararajan, CRM Proc. Lecture Notes. Providence, RIAmer. Math. Soc46K. Soundararajan, 'The distribution of smooth numbers in arithmetic progres- sions', Anatomy of integers, CRM Proc. Lecture Notes, v.46, Amer. Math. Soc., Providence, RI, 2008, 115-128. 1","doi":""},{"text":"The distribution of squarefull integers. D Suryanarayana, R , Sitaramachandra Rao, Arkiv för Matematik. 114D. Suryanarayana and R. Sitaramachandra Rao, 'The distribution of square- full integers', Arkiv för Matematik , 11 (1973), 195-201. 4","doi":""},{"text":"Institut für Analysis und Zahlentheorie 8010 Graz. Steyr-Institut für Analysis und Zahlentheorie 8010 Graz, Steyr-","doi":""}],"string":"[\n {\n \"text\": \"The congruence x 1 x 2 ≡ x 3 x 4 mod p, the equation x 1 x 2 = x 3 x 4 and mean values of character sums. A Ayyad, T Cochrane, Z Zheng, J. Number Theory. 596A. Ayyad, T. Cochrane and Z. Zheng, 'The congruence x 1 x 2 ≡ x 3 x 4 mod p, the equation x 1 x 2 = x 3 x 4 and mean values of character sums', J. Number Theory 59 (1996), 398-413. 6\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The least quadratic non residue. N C Ankeny, Ann. of Math. 552N. C. Ankeny, 'The least quadratic non residue', Ann. of Math. (2), 55 (1952), 65-72. 3\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The distribution of smooth numbers in arithmetic progressions. A Balog, C Pomerance, Proc. Amer. Math. Soc. 1152A. Balog and C. Pomerance, 'The distribution of smooth numbers in arithmetic progressions', Proc. Amer. Math. Soc., 115 (1992), 33-43. 1, 2\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Density of non-residues in Burgess-type intervals and applications. W D Banks, M Z Garaev, R Heath-Brown, I E Shparlinski, Bull. London Math. Soc. 407W. D. Banks, M. Z. Garaev, R. Heath-Brown and I. E. Shparlinski, 'Density of non-residues in Burgess-type intervals and applications', Bull. London Math. Soc. , 40 (2008), 88-96. 7\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Squarefree smooth numbers and Euclidean prime generators. A Booker, C Pomerance, Proc. Amer. Math. Soc. 1451A. Booker and C. Pomerance, 'Squarefree smooth numbers and Euclidean prime generators', Proc. Amer. Math. Soc., 145 (2017), 5035-5042. 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The distribution of quadratic residues and non-residues. D A Burgess, 4MathematikaD. A. Burgess, 'The distribution of quadratic residues and non-residues', Math- ematika, 4 (1957), 106-112. 3\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Squarefull numbers in arithmetic progression, II. T H Chan, J. Number Theory. 1522T. H. Chan, 'Squarefull numbers in arithmetic progression, II', J. Number Theory 152 (2015), 90-104. 2\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Théorémes de type Fouvry-Iwaniec pour les entiers friables. S Drappeau, Compos. Math. 1511S. Drappeau, 'Théorémes de type Fouvry-Iwaniec pour les entiers friables', Compos. Math., 151 (2015), 828-862. 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"An estimate of Kloosterman sums with prime numbers and an application. M Z Garaev, Matem. Zametki. 884in Russian. translation in Math. NotesM. Z. Garaev, 'An estimate of Kloosterman sums with prime numbers and an application', Matem. Zametki, 88 (2010), 365-373, (in Russian); translation in Math. Notes, 88 (2010), 330-337. 4\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lower bounds for least quadratic nonresidues' Analytic number theory. S W Graham, C J Ringrose, Progr. Math. 854S. W. Graham and C.J. Ringrose, 'Lower bounds for least quadratic non- residues' Analytic number theory, Progr. Math., 85(1990), 269-309. 4\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On a paper of K. Soundararajan on smooth numbers in arithmetic progressions. A J Harper, J. Number Theory. 1321A. J. Harper, 'On a paper of K. Soundararajan on smooth numbers in arith- metic progressions', J. Number Theory, 132 (2012), 182-199. 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Analytic number theory. H Iwaniec, E Kowalski, American Mathematical SocietyProvidence, RIH. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society, Providence, RI, 2004. 3\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the congruence x 1 x 2 ≡ x 3 x 4 mod q. B Kerr, J. Number Theory. 6B. Kerr, 'On the congruence x 1 x 2 ≡ x 3 x 4 mod q ', J. Number Theory, 180 (2017), 154-168. 6\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Topics in multiplicative number theory. H L Montgomery, Lecture Notes in Mathematics. 2274Springer-VerlagH. L. Montgomery, 'Topics in multiplicative number theory', Lecture Notes in Mathematics, Vol. 227, Springer-Verlag, 1971. 4\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Character sums over squarefree and squarefull numbers. M Munsch, Arch. Math. 1022M. Munsch, 'Character sums over squarefree and squarefull numbers', Arch. Math., 102 (2014), 555-563. 2\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On smooth square-free numbers in arithmetic progressions. M Munsch, I E Shparlinski, 5M. Munsch and I. E. Shparlinski, 'On smooth square-free numbers in arith- metic progressions', Preprint , 2017, http://arxiv.org/abs/1710.04705. 1, 4, 5, 6, 7, 8\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Square-full primitive roots. M Munsch, T Trudgian, Intern. J. Number Theory. 142M. Munsch and T. Trudgian, 'Square-full primitive roots', Intern. J. Number Theory, 14 (2018), 1013-1021. 2\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The distribution of smooth numbers in arithmetic progressions', Anatomy of integers. K Soundararajan, CRM Proc. Lecture Notes. Providence, RIAmer. Math. Soc46K. Soundararajan, 'The distribution of smooth numbers in arithmetic progres- sions', Anatomy of integers, CRM Proc. Lecture Notes, v.46, Amer. Math. Soc., Providence, RI, 2008, 115-128. 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The distribution of squarefull integers. D Suryanarayana, R , Sitaramachandra Rao, Arkiv för Matematik. 114D. Suryanarayana and R. Sitaramachandra Rao, 'The distribution of square- full integers', Arkiv för Matematik , 11 (1973), 195-201. 4\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Institut für Analysis und Zahlentheorie 8010 Graz. Steyr-Institut für Analysis und Zahlentheorie 8010 Graz, Steyr-\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":587,"cells":{"text":{"kind":"string","value":"\nProbing high order dependencies with information theory\n\n\nC Granero-Belinchón \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nUniv Claude Bernard\nCNRS\nF-69342LyonFrance\n\nS G Roux \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nUniv Claude Bernard\nCNRS\nF-69342LyonFrance\n\nIEEE Fellow) *P Abry \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nUniv Claude Bernard\nCNRS\nF-69342LyonFrance\n\nN B Garnier \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nUniv Claude Bernard\nCNRS\nF-69342LyonFrance\n\nProbing high order dependencies with information theory\n1Index Terms-Information theoryEntropy ratesMutual In- formationhigher-order temporal dependenciesNon Gaussian processes\nInformation theoretic measures (entropies, entropy rates, mutual information) are nowadays commonly used in statistical signal processing for real-world data analysis. The present work proposes the use of Auto Mutual Information (Mutual Information between subsets of the same signal) and entropy rate as powerful tools to assess refined dependencies of any order in signal temporal dynamics. Notably, it is shown how two-point Auto Mutual Information and entropy rate unveil information conveyed by higher order statistic and thus capture details of temporal dynamics that are overlooked by the (twopoint) correlation function. Statistical performance of relevant estimators for Auto Mutual Information and entropy rate are studied numerically, by means of Monte Carlo simulations, as functions of sample size, dependence structures and hyper parameters that enter their definition. Further, it is shown how Auto Mutual Information permits to discriminate between several different non Gaussian processes, having exactly the same marginal distribution and covariance function. Assessing higher order statistics via multipoint Auto Mutual Information is also shown to unveil the global dependence structure fo these processes, indicating that one of the non Gaussian actually has temporal dynamics that ressembles that of a Gaussian process with same covariance while the other does not.\n\nI. INTRODUCTION\n\nContext. To characterize a (stationary) process X(t), evolving along a given dimension t, e.g., time, the precise assessment of both its static properties (marginal distribution or one-point statistics) and temporal dynamics (joint distributions or multipoint statistics) is crucial.\n\nCorrelation and spectral estimates, providing practitioners with valuable insight into temporal dynamics are key tools in statistics and data analysis and have been widely used in many different fields. In neurosciences, the analysis of correlation between neuron firings [1], [2] or between stimulus and brain activity [3], [4] is essential. In fluid mechanics, Kolmogorov's empirical theory of turbulence [5] characterizes the multiscale distribution of energy with the power spectrum of the velocity field [6]. The analysis of correlations also supports useful descriptions in terms of complex networks for, e.g., ecology [7], climatology [8] or traffic [9].\n\nIt is however well known that, while the correlation function, or equivalently the power spectral density, exhaust the description of joint statistics and hence of temporal This work was supported by the LABEX iMUST (ANR-10-LABX-0064) of Université de Lyon, within the program \"Investissements d'Avenir\" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). dynamics for jointly Gaussian processes, they provide characterizations of temporal dynamics for general processes, that are limited in at least two respects: not only they are restricted to two-point joint statistics but also to a specific measure of these two point-statistics. In general, the full analysis of temporal dynamics require the use of multi-point, or higher order, joint statistics. The goal of the present work is to address such an issue using information theoretic measures.\n\nRelated work. To assess dependencies beyond two-point correlations, multipoint correlation functions were defined. For instance, three-point correlation functions have been used in cosmology, where they were related to cosmological parameters [10], [11]. Alternatively, higher order-spectra, that combine Fourier transforms at several different frequencies and therefore takes into account higher order statistics, have also been used to generalize the power spectrum beyond second-order statistics [12], [13]. For instance, bi-spectrum and tri-spectrum were analyzed in depth in e.g., [14], [15] and used in applications as different as cosmology [16], [17], or medicine [18], [19] and other fields [20], [21]. While of popular use in applications, higher-order correlation functions and spectra suffers from major caveats, mostly related to estimation issues. Indeed, the assessment of order m-statistics with higher-order correlation functions and spectra requires the estimation of m-time point statistics, which, for a given fixed sample size, can only yield decreasing estimation performance when order m is increased.\n\nAlternatively, Shannon's information theory, and the corresponding information theoretic measures, entropy, entropy rate and mutual information, can be considered as an efficient way to characterize high order statistics [22], [23], [24]. Applications of entropy rate were yet reported in a wide range of domains, such as biology [25], dynamical systems [26], fluid turbulence [27] or analysis of languages [28]. Mutual information is also very popular, especially in computer sciences, e.g., in machine learning [29], and other very different domains [30], [31], [32], [33], [34], [35]. Auto-Mutual Information (AMI) -sometimes called Information Storage [36] -, defined as the mutual information between the analyzed process and a delayed version of itself, has been used in different domains [37], [38], [39], [40]. Information theory additionally provides tools to explore causality relationships between processes [41], [42]. However, within this paper we are mainly interested in the characterization of their dependence structures.\n\nEntropy rate and AMI have been envisaged as tools to describe temporal dynamics in time series beyond second-order statistics, or even at second-order statistics, in a richer way than correlation or Fourier spectrum, see [43], [37], [39], [27]. Goals, contributions and outline. The main goals of the article are i) to show the ability of AMI and entropy rate to describe high order dependences, even when considering only two-point interactions. Even though this property of information theory tools is well known, we are able to show it by analyzing processes with identical correlation and marginal distribution but different high order dependences. AMI and entropy rate appear as straightforward generalizations of correlation function and power spectrum when considering non-Gaussian processes analysis. ii) to show the effect of analysing interactions between more than two points, combining Takens embedding [44] with AMI. We characterize the complexity of the dependence structure of a process by measuring AMI for embedding dimension higher than 2. More precisely, we analyze the differences between power law decay and exponential decay dependence structures.\n\nTo demonstrate the ability of information theory to describe high order statistics, we illustrate the use of entropy rate and auto-mutual information to characterize non Gaussian processes across scales. As an example, we consider two synthetic log-normal processes with different dependence structures but with identical correlation function and identical marginal statistics. In section II, we introduce and define the theoretical information theory measures and our estimators. In section III, we interpret the information theory tools used along the article, we present the processes with identical correlation function and one-point statistics but different dependence structure that we study and finally we illustrate how AMI and entropy rate allow to distinguish between them. In section IV-A, we report the convergence and the performance of our estimators of AMI and entropy rate and their ability to measure high order dependencies. We present their evolution across scales and emphasized that, contrary to classical correlation analysis, they allow a fine characterization of non Gaussian processes.\n\n\nII. ENTROPY, ENTROPY RATE AND AUTO MUTUAL INFORMATION\n\n\nA. Definitions\n\nTo analyze the temporal dynamics of univariate stationary processes X = {x t } t∈R , it has been proposed in [44] to rely on the time delay-embedding procedure. It amounts to constructing the m-dimensional process X (m,τ ) , whose elements are m-dimensional vectors:\nx (m,τ ) t = x t , x t−τ , · · · , x t−(m−1)τ ,(1)\nwhere the delay τ implicitly defines an analysis scale and where the embbeding dimension m controls the order of the statistics of X which are analyzed. The information theoretic measures defined below aim to characterize temporal dynamics via the analysis of the m-time point joint statistical distributions p(x (m,τ ) t ). Within this article, we study ergodic stationary processes and then p(x (m,τ ) t ) depend on m and τ but not on t. Conse-\nquently p(x (m,τ ) t ) = p(X (m,τ ) ).\nShannon entropy, is defined as a functional of p(x (m,τ ) t ) [22]:\nH(x (m,τ ) t ) = − R m p(x (m,τ ) t ) ln p(x (m,τ ) t )dx (m,τ ) t = H (m,τ ) (X)\nBy nature, it depends on the full joint distribution and therefore on statistics of any order. However, for stationary processes it does not depend on the mean value of the marginal distribution. For m = 1, Shannon entropy does not relate to temporal dynamics but to one-point statistics and thus obviously does not depend on τ and is hence denoted H(X).\n\nEntropy rate of order m, h (m,τ ) , quantify variations of Shannon entropy between two successive time-embedded versions of a stationary process [22], [24]:\nh (m,τ ) (x t ) = H(x (m+1,τ ) t ) − H(x (m,τ ) t−τ ) = H (m+1,τ ) (X) − H (m,τ ) (X) = h (m,τ ) (X).(2)\nEntropy rate h (m,τ ) , like H (m,τ ) , probes joint statistics of any order, but in addition quantifies how much information is gained in the analysis of temporal dynamics by increasing by one the order of the embedding.\nMutual information I(x (m,τ ) t , y (p,τ ) t\n) quantifies the amount of information shared by two processes X and Y , via their embbeded versions x (m,τ ) t and y (p,τ ) t [22], [24]:\nI(x (m,τ ) t ,y (p,τ ) t ) = R m+p p x (m,τ ) t , y (p,τ ) t ln p x (m,τ ) t , y (p,τ ) t p(x (m,τ ) t )p(y (p,τ ) t ) dy (p,τ ) t dx (m,τ ) t(3)\nWhen X and Y are jointly stationary processes,\nI(x (m,τ ) t , y (p,τ ) t ) = I(X (m,τ ) t , Y (p,τ ) t ) obviously depends on t − t only.\nMutual information can be read as the Kullback-Leibler divergence between the joint probability p(x \n\n\n).\n\nTo analyze the higher statistical order temporal dynamics (also referred to as nonlinear temporal dynamics) of a single stationary process X, one can use auto-mutual information [24], [36], defined as the mutual information between x \nI (m,p,τ ) (X) = I(x (m,τ ) t , x (p,τ ) t+pτ ) .(4)\nBecause the concatenation of x t+pτ :\n\nx\n(m,τ ) t , x (p,τ ) t+pτ = x (m+p,τ ) t+pτ ,\nI (m,p,τ ) (X) quantifies the shared information between the past m-point dynamics and the future p-point dynamics. Auto mutual information quantifies the deviation from independence between the two arguments, beyond the simpler decorrelation. These information theoretic quantities all depend on the chosen time lag τ . When τ → 0, dependences between two consecutive coordinates of the embedding vector increase, therefore inducing an increase in I (m,1,τ ) (X) and a decrease in h (m,τ ) (X). Conversely, when τ → +∞, dependencies decrease to 0, therefore inducing an increase of h (m,τ ) (X) up to the Shannon entropy H(X) of the process, its largest value, and a decrease of I (m,1,τ ) (X) → 0.\n\nFurthermore, entropy rate can then be expressed as [27]:\nh (m,τ ) (X) = H(X) − I (m,1,τ ) (X) ,(5)\nwhich shows that there are two different contributions to the entropy rate: First, H(X), which only depends on the onepoint distribution and hence is a static property ; second, I (m,1,τ ) (X) which gathers all information conveyed by linear and non linear temporal dynamics, irrespective of the variance of the distribution [27].\n\n\nB. Gaussian process\n\nWhen X is a stationary jointly Gaussian process, hence fully defined by its variance σ 2\n\nx and normalized correlation function c X (τ ), analytical expressions for entropy, entropy rate and auto mutual information are available [45]:\nH(X) = 1 2 ln(2πeσ 2 x ) (6) I (m,p,τ ) (X) = 1 2 ln |Σ (m) ||Σ (p) | |Σ (m+p) | (7) h (m,τ ) (X) = 1 2 ln(2πeσ 2 x ) − 1 2 ln |Σ (m) | |Σ (m+1) |(8)\nwith Σ (m) the m-dimensional correlation matrix, and || denoting the determinant. By definition, Σ (1) = 1.\n\n\nC. Estimation procedures\n\nTo estimate these information theoretic quantities, two main categories of non-parametric procedures are available, based on either kernel density estimations (KDE) or nearest neighbor search (NNS). While the former essentially counts the number k( ) of the observed samples in interval of a priori set size , the latter rather measure the size (k) of box that contain an a priori set number of k samples.\n\nBecause this second strategy has been documented as providing low bias estimates [46], [47], we chose k-nearest neighbors procedures to estimate the entropy H(X) [48], [49], [50]:\nH(X) = −ψ(k) + ψ(N ) + d N N i log( i (k))(9)\nwith ψ the Digamma function, d the dimension of X, N the number of samples of X, and using the Chebyshev distance (L ∞ metric).\n\nTo compute the mutual information, we use the k-nearest neighbors procedure presented by Kraskov, et al. [46]:\nI(X, Y ) = ψ(k) + ψ(N ) − ψ(n x ) + ψ(n y )(10)\nThe procedure chooses k for the joint distribution Z(X, Y ) and hence obtains the distance to the kth nearest neighbour.\n\nThen using , it calculates the number of neighbours n x + 1 lying on a segment x = x± 2 of the X dimension, i.e. lying on −∞ ≤ Y ≤ ∞, x − /2 ≤ X ≤ x + /2. Consequently, /2 is the distance to the n x + 1st neighbour of x. The procedure uses the same development for Y and obtains that /2 is the distance to the n y + 1st neighbour of y.\n\nBecause benefits against a direct use of eq.(2) were documented in [51], h (m,τ ) (X) is estimated plugging the above estimates into eq.(5).\n\nFor an stationary process X, p(x (m,τ ) t ) does not depend on t, and consequently, we can exploit this property to estimate the information theoretic quantities of X by averaging on all the temporal samples of the process.\n\nTo perform an analysis across scales by varying the parameter τ we use an adapted Theiler prescription [52] to eliminate the contribution of dependencies from scales smaller than τ .\n\n\nIII. TEMPORAL DYNAMICS AND DEPENDENCE STRUCTURE\n\nThe aim of this section is to try to characterize what part of the statistics of a given stationary process Y contribute to the values taken by information theoretic measures H, h (m,τ ) and I (m,p,τ ) . To that end, we conduct a formal study of the specific yet very instructive case where processes Y with correlation function c Y (τ ) and marginal distributions p Y are obtained as point bijective transformation F , Y = F (X) of an underlying Gaussian process X with correlation function c X , designed to reach the targeted covariance c Y . Processes Y constructed in this way depend on F and c X (τ ) [53], [54].\n\n\nA. Analytical calculations and interpretations\n\nGiven any bijective transformation F that maps a Gaussian signal X into a signal Y = F (X), the information theoretic quantities of Y can be computed as:\nH(Y ) = H(X) + ln F X = 1 2 ln(2πeσ 2 X ) + ln F X(11)I (m,p,τ ) (Y ) = I (m,p,τ ) (X) = 1 2 ln |Σ (m) X ||Σ (p) X | |Σ (m+p) X | (12) h (m,τ ) (Y ) = 1 2 ln(2πeσ 2 X ) + ln F X − 1 2 ln |Σ (m) X | |Σ (m+1) X | (13) with ln F X = p(x (1) t ) ln F (x)dx, hence a constant, σ 2 X and Σ (m) X\nare the variance and the m-dimensional covariance matrix of the underlying Gaussian process X.\n\nThese analytical calculations permit to yield understanding of general validity regarding the nature of the information conveyed by each of these three information theoretic quantities: i) Entropy H is only related to the marginal distribution of the process p Y , hence is quantifying static properties only. ii) Entropy rate h (m,τ ) mixes up marginal distribution features (variance and pointwise transform F ), hence static properties with dynamical properties and dependencies of any order, as indeed Σ (m) X depends jointly on c Y and on F , hence on the 4 joint distribution of y (m,τ ) t . The fact that h (m,τ ) gather both static and dynamics aspects of time series likely explains why it has been observed in many applications to be a often discriminant feature, compared to other features [40]. However, the entanglement of static and dynamic properties may also be considered a drawback for the analysis, characterization and understanding of data properties. iii) Interestingly, auto mutual information I (m,p,τ ) performs a characterization mainly related to the dependence structure of Y , as indeed Σ (m) X depends on the joint distribution of y (m,τ ) t . It therefore quantifies temporal dynamics, irrespective of static properties. Thus, AMI is the sole measure that focuses exclusively on higher order dynamics and on the joint dependence structure, beyond correlation.\n\nOften in the scientific literature, h (m,τ ) and I (m,p,τ ) are referred to as nonlinear features, to emphasize that they characterize temporal dynamics beyond the mere use of the correlation functions or power spectral densities. This is indeed correct. Even when two-point statistics are analyzed, i.e., m = p = 1, I (1,1,τ ) reflects dependencies, including and encompassing correlations as the whole joint two-point distribution function is involved. Nevertheless, I (m,p,τ ) remains mostly driven by the correlation function c Y , and hence by linear effects. This is also the case for h (m,τ ) , driven mostly both by the static variance σ 2 Y and dynamical correlation function c Y .\n\nB. Process definition 1) Analytical derivations: To further illustrate that indeed I (m,p,τ ) and h (m,τ ) may capture differences in any-order dependencies beyond correlations, let us study the cases where two stationary processes Y F1 and Y F2 have exact same marginal distribution p Y and exact same covariance c Y , but are yet obtained from two different point transformations F 1 and F 2 -applied to Gaussian processes X 1 and X 2 -and hence have different joint distributions and hence different dependence structures.\n\nLet denote F Y the cumulative distribution function associated to the marginal distribution p Y . A natural candidate to obtain a process with marginal p Y is to apply the point transform\nF 1 (x) = F −1 Y • Φ(x),(14)\nto a Gaussian process X 1 , where F Y is the cumulative distribution function of the marginal distribution p Y and Φ(x) the cumulative distribution function of a centered standard Gaussian variable, and where X 1 is a jointly Gaussian process with covariance function c X1 (τ ) to reach the targeted c Y (τ ) [53], [54]. However, many other transforms can yield processes Y with the exact same marginal distribution p Y and the exact same covariance c Y . In [54], it was for example proposed to use\nF 2 (x) = F −1 Y • (2(Φ(|x|) − 1/2)) ,(15)\non a jointly Gaussian process X 2 . This thus yields a radically different joint dependence structure for Y F2 , despite having the same prescribed correlation function as Y F1 . As a consequence, Y F1 and Y F2 have different AMI functions I (m,p,τ ) (Y F1 ) = I (m,p,τ ) (Y F2 ), and entropy rates, and hence different joint dependence structure and thus temporal dynamics. Conversely, their entropies are identical H(Y F1 ) = H(Y F2 ). This is not surprising , as the entropy is completely defined by the one-point distribution.\n\n2) The log-normal example: To quantify the qualitative analysis proposed in Section III-A , let us consider stationary processes Y F1 and Y F2 having the same log-normal marginal distribution p Y , chosen as an example of distributions that may complicate estimation because of its being a frontier between distributions with all moments finite and heavy-tail distributions.\n\nFor process Y F1 obtained with the natural transform (14), F 1 simplifies to F 1 (x) = exp(x), which is a bijective transformation. One can therefore use eqs. (11) to (13) above with ln F 1 X1 = µ X1 , σ X1 and Σ (m) X1 being the mean, variance and m-dimensional covariance matrix of the Gaussian variable X 1 [45]. The entropy H(Y F1 ) can be computed using the mean µ Y and variance σ 2 Y of Y F1 :\nµ X1 = ln µ 2 Y σ 2 Y + µ 2 Y , σ X1 = 2 ln 1 + σ 2 Y µ 2 Y .(16)\nWriting I (m,p,τ ) (Y F1 ) thus only requires to express the correlation function of the underlying Gaussian process X 1 as a function of the targeted c Y (τ ):\nc X1 (τ ) = 1 σ 2 X1 ln((e σ 2 X 1 − 1)c Y (τ ) + 1).(17)\nThis permits to compute h (m,τ ) and I (m,p,τ ) analytically, using eqs. (12) and (13). For stationary log-normal process Y F2 , F 2 is not bijective. The entropy H can be computed analytically, but this is not the case for I (m,p,τ ) and h (m,τ ) .\n\n3) Correlation structures: short-range versus long range dependencies: To study the impact of the correlation structure, processes with short-range dependencies are compared against processes with long-range dependencies, the former being well-represented by an exponentially decaying covariance,\nc Y (τ ) = σ 2 Y e −fc|τ | , f c > 0,(18)\nand the latter by an algebraically decaying covariance,\nc Y (τ ) = σ 2 Y 2 (τ − 1) 2H − 2τ 2H + (τ + 1) 2H ,(19)\nwith 0.5 < H < 1.\n\n\nC. Two-point dependencies: Auto mutual information versus correlation\n\nFor both short-range and long range correlations, we compare the two LN processes obtained with F 1 and F 2 , denoted Y F1 , Y F2 , against a Gaussian process with same correlation function, denoted Y I , where I stands for the identity transform. First order statistics (probability density function) and second order statistics measured by the correlation function or the power spectral density as well as two point statistics (i.e., m = p = 1) entropy, entropy rate and AMI estimates are compared on Fig. 1 for Y I , Y F1 and Y F2 with short-range dependences and on Fig.2 for long-range dependences.\n\nAs expected, for each pair of short range and long range processes, the two log-normal processes, Y F1 and Y F2 cannot be distinguished one from the other using either PDF, correlation or power spectral estimation. Entropy being fully prescribed by the sole PDF, it is further not surprising that the two log-normal processes are also undistinguishable when using entropy estimates H (Fig (1,d)).\n\nHowever, and interestingly, the estimates I (1,1,τ ) and h (1,τ ) of two-point AMI and entropy rate show differences as functions of τ between processes Y F1 and Y F2 , both for the shortrange and long-range correlation cases. I (1,1,τ ) decreases and h (1,τ ) increases when τ → ∞, a predictable result as all dependencies vanish for large τ for stationary processes. Both functions (AMI and entropy rate) for processes Y F1 and Y F2 differ for small τ , thus quantifying the difference between the joint distributions of these two processes, despite identical correlation functions and marginal distributions. Fig. 1 and Fig. 2 show that the two-point AMI estimateŝ I (1,1,τ ) for log-normal process Y F1 ressembles far more to that of the Gaussian process Y I than to that of the companion log-normal process Y F2 , even they all three have the same correlation function. Very interestingly, it provides a deep insight into the temporal dynamics of the time series: processes Y F1 and Y F2 have same marginal distributions and same correlations functions, but the temporal dynamics of Y F1 is close to that of the Gaussian process Y I while that of Y F2 is not. This also illustrates the benefits of AMI I (1,1,τ ) over entropy rate h (1,τ ) that mixes the impact of both marginal distribution and joint dynamical structure, thus making Y F2 comparable to Y I at small τ when dependencies matter and Y F2 comparable to Y F1 at large τ when only the marginal distribution contribute to h (1,τ ) . Figs 1,d,e,f and 2,d,e,f also report theoretical H, h and AMI for processes Y I and Y F1 , while theoretical h and AMI are not available for process Y F2 . All estimates well match the predicted theoretical behaviors (solid lines in Figs 1, 2). The theoretical behaviors of AMI and entropy rate of process Y F1 are obtained using eqs. (12) and (13) with the analytical expression (17).\n\nThese observations essentially hold both for the short-range (Fig. 1) and long-range (Fig. 2) correlations and yield the first key conclusion of this work: as opposed to correlation or spectral estimation, estimates of the two-points AMI (I (1,1,τ ) ) and entropy rate (h (1,τ ) ) are able to quantify differences in the temporal dynamics of processes even when they share the exact same marginal distributions and correlation functions, as long as they have different two-point joint statistics.\n\n\nIV. ESTIMATION PERFORMANCE AND ILLUSTRATIONS\n\nThe aim of this section is to assess, by means of Monte Carlo simulations, the performance of the estimatorsĤ, I (m,p,τ ) andĥ (m,τ ) for processes with different statistical structures and to study the impacts of key parameters either entering the definition of the estimators (number of nearest neighbors k) or characterizing the dependence structures of the processes (strength of the correlation c Y (τ )) as well as with sample sizes N . A. Estimation performance 1) Monte Carlo simulations Set-up: The processes are synthetized numerically using circulant matrix embedding strategies, with the toolbox designed in [54], [53] and made available at www.hermir.org. As we have access to analytical curves only for Y I and Y F1 , these are the processes used to characterize the performance of the estimators.\n\nFor short and long memory, parameters are set f c = 0.1 and H = 0.7 respectively. All variances are set to σ Y ≡ 1. For each case, 100 realizations of processes are synthesized and performances are assessed as averages (and standard deviations) across realizations.\n\nTo study the impact of the marginal distribution on estimation performance, process Y F1 is compared to process Y I .\n\n2) Dependence with the sample size N : Fig. 3 reports performances as functions of the sample size N , for 2 8 ≤ N ≤ 2 16 , for a fixed k = 5. It clearly shows that the three estimators are asymptotically consistent (with vanishing biases and variances) irrespective of the marginal distribution and of the correlation being short-range or long-range. Unsurprisingly, variances are larger for the log-normal processes than they are for the Gaussian ones. Also, long-range dependencies imply much larger variances compared to short-range dependencies for a same c Y (τ ) = 0.32 (which corresponds to τ = 1 for the long range correlation process and τ = 11 for the short range one), thus clearly showing that long memory does not induce biases but does imply larger variances. The proposed estimators are overall robust to marginal distributions and dependence structures [46], [47], and have satisfactory performance as soon as N ≥ 2 10 , hence N = 2 16 is used hereafter.\n\n3) Dependence with the number of nearest neighbors k: Fig. 4 reports estimation performance as functions of k, for N = 2 16 and same correlation c Y = 0.32 as above. Performances barely depend on k, but for a mild increase of the shows that all biases slightly increase when c Y (τ ) increases. The variance appears to increase a little faster than linearly in c Y . For both short-range and long-range correlations, the lognormal processes show larger standard deviations compared to the Gaussian ones, yet with similar biases. Further, for a given correlation level c Y (τ ) the long-range correlation structure shows larger standard deviations compared to the short-range correlation.\n\nFrom this characterization, we can conclude that the entropy, AMI and entropy rate estimators show a low bias and standard deviation for the combination k = 5 and N = 2 16 of parameters used along the article. In addition, we can conclude that the standard deviation of all the three estimators is larger for strong correlations and non-Gaussian distributions, but their biases depend slightly on the marginal distribution and the correlation strength.\n\n\nB. Process discrimination with auto mutual information\n\nIn Figs 1,e,f and 2,e,f, we show that both AMI and entropy rate across scales are able to discriminate between processes Y F1 and Y F2 . The discrimination is far more effective for the smaller scales for both correlation structures (Long and short range).\n\nTo quantify the differences between Y F1 and Y F2 that we found in Figs 1 and 2, we first present the histograms of the estimations of the entropy rate (Fig. 6a) and of the AMI (Fig. 6b) for fixed N = 2 16 and τ = 1. We perform a statistical ranksum test using Mann-Whitney U-test (Wilcoxon rank-sum test) and compute the p-value, for varying τ and N . As seen in Fig. 6d, the p-value quickly decreases with increasing N , as expected. Fig. 6c quantifies the distinguishability between both log-normal processes across scales, showing that the discrimination is far better for smaller scales. This confirms the observations of Figs 1,e,f and 2,e,f.\n\n\nC. Dependence structure beyond correlations\n\nThis sections further explores the ability of AMI I (m,p,τ ) , with m ≥ 1 and/or p ≥ 1 to probe multiple-point dependences and hence fine details in temporal dynamics. Fig. 7 reports higher-order AMI estimatesÎ (m,p,τ ) for several combination of m and p, as functions of the embedding dimension p for both short-range (left) and long-range (right) correlation functions, for processes Y I , Y F1 and Y F2 . From top to bottom the m value is equal to 1, 2, 3 and p. Consequently, the first three lines study the behaviour of AMI on p for fixed value of m, while the fourth line shows the behavior of AMI for m = p.\n\nWhile two-point AMI yielded essentially identical conclusions for short-range and long-range correlation processes, higher order AMI reveals clear differences: For short-range correlation processes (left column), Y I , Y F1 and Y F2 ,Î (m,p,τ ) and I (m,p,τ ) when it can be computed analytically, show a very mild dependence of AMI with m and p for a fixed τ . To the converse, for long-range correlation processes (right column), Y I , Y F1 and Y F2 ,Î (m,p,τ ) and I (m,p,τ ) when it can be computed analytically, show constant increase of AMI when p or m increase, for a fixed τ . This fundamental observation clearly indicates that for short-range correlation, whatever τ , there is little or no gain in the analysis of temporal dynamics brought by the study of higher order AMI. Conversely, for long-range correlation processes, I (m,p,τ ) captures more and more information in the temporal dynamics as the embedding dimensions are increased thus revealing a significant difference between short-range and long-range correlation structures. Analytical formulae for I (m,p,τ ) when m, p > 1, are only available for the identity I and the natural transform F 1 , namely for Y I and Y F1 . In order to quantify the evolution of AMI when the embedding increases, we compute the p-value of Wilcoxon test between the statistic ofÎ (1,1,1) andÎ (m,p,1) for the same processes as in Fig. 7. Fig. 8 quantifies the increase of AMI when the embedding increases. This evolution of AMI is far more important in the long range correlations case, as expected from Fig. 7. Further, table I shows that, for fixed m + p, AMI evolves differently depending on the precise values of m and p separately. For a fixed m + p AMI is larger when m and p are close, which can be understood as a closer similarity between the two embedded vectors appearing as the arguments of the AMI in eq.(4). Fig. 8. Statistical test. p-value of Wilcoxon rank-sum test between the statistics ofÎ (1,1) (1) andÎ (m,p) (1) for processes with exponentially decaying correlation (top line) or with a power law decay (bottom line). Left column is for Y I , middle column for Y F 1 and right column for Y F 2 .\nY I Y F 1 Y F 2Short\n\nV. CONCLUSIONS AND PERSPECTIVES\n\nWe explored the evolution of AMI (I (m,p,τ ) ) and entropy rate (h (m,τ ) ) on the scale parameter τ . Even when considering only two point interactions (m = p = 1) AMI and entropy rate probe statistics of any order, and hence non-linear dependences, and as such appear as unambiguous generalizations of the correlation function and the power spectrum.\n\nAs an illustration, we analyzed two log-normal processes with identical 1-point distribution and identical correlation function. AMI (I (1,1,τ ) ) and entropy rate (h (1,τ ) ) are able to discriminate these two processes, which appear as identical for classical analysis, by enlightening differences in high order dependences. The behavior of these quantities along scales reveal the existence of stronger high-order dependences at smaller scale, which allows an easier separation of processes.\n\nIn addition, AMI generalizes easily to consider explicitly interactions between more than two points. We thus probe the complexity of the dependence structure above linear order, i.e., the additional information measured by AMI when considering an extra point should indicate the existence and relevance of next-order interactions. As an example, comparing the effect of increasing the order of interactions for two dependence structures showed that while exponential decay -short rangedependences do not involve next order interactions, power law -long range -dependences do.\n\nThe same qualitative results were obtained when considering others one-point distribution, which supports the validity of the methodology for any distribution and dependence structure. This generality should make the methodology very interesting to perform non-Gaussian processes characterization in several different applications.\n\n\nthe (m + p)-dimensional time-embedded vector x (m+p)\n\nFig. 1 .\n1Short-range dependencies. Classical analysis: histogram (a), power spectrum (b) and correlation function (c). Information theory analysis for m = 1, p = 1: entropy (d), entropy rate (e) and auto mutual information (f). The color indicates the process: blue for standard log-normal (Y F 1 ), red for Y F 2 log-normal and black for Gaussian (Y I ). The continuous lines in (e,d,f) correspond to analytical expressions. We used N = 216 and k = 5.\n\nFig. 2 .\n2Long-range dependencies. Classical analysis: histogram (a), power spectrum (b) and correlation function (c). Information theory analysis for m = 1, p = 1: entropy (d), entropy rate (e) and auto mutual information (f). The color indicates the process: blue for standard log-normal (Y F 1 ), red for Y F 2 log-normal and black for Gaussian (Y I ). The continuous lines in (e,d,f) correspond to analytical expressions. We used N = 216 and k = 5.\n\nFig. 3 .\n3Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of N , for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, c(τ ) = 0.32 and k = 5.\n\nFig. 4 .\n4Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of k, for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, c(τ ) = 0.32 and N = 2 16 . biases, and this irrespective of the marginal distributions and of short-range or long-range nature of the correlations. Because increasing k further implies increasing the computational load, k = 5 is used hereafter. 4) Influence of the correlation strength c Y (τ ): Fig. 5 reports estimation performance as a function of c Y (τ ) and\n\nFig. 5 .\n5Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of the correlation strength c Y , for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, k = 5 and N = 2 12 .\n\nFig. 6 .\n6Statistical test. Histogram of the estimation ofĥ(1,1) (1) (a) and I (1,1) (1) (b) for the scale invariant processes (blue for Y F 1 and red for Y F 2 of size N = 2 16 . (c) p-value of the Wilcoxon rank-sum test between the two log-nomal processes Y F 1 and Y F 2 as fonction of the scale for the statistics ofĥ (1,1) (τ ) (dotted line) andÎ (1,1) (τ ) (full line) with N = 2 16 . (d) same quantities as (c) for scale τ = 1 as functions of the process size N . The statistics has been done on 100 realizations of both process. In (c) and (d) the gray curves correspond to the results obtained on the Exponential Decay processes. Mann-Whitney U-test ranksum.\n\nFig. 7 .\n7Evolution of AMI with embedding.Î (m,p) (τ = 1) for processes with exponentially decaying correlation (left column) and with power law decay (right column). Black is for Y I , blue is for Y F 1 and red for Y F 2 . First three lines are for fixed m and increasing p:m = 1 (a,b), m = 2 (c,d) and m = 3 (e,f). Last line is for m = p. 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Leonenko, \"Sample estimate of entropy of a random vector,\" Problems of Information Transmission, vol. 23, pp. 95- 100, 1987.\n\nNearest neighbor estimates of entropy. H Singh, N Misra, V Hnizdo, A Fedorowicz, E Demchuk, American Journal of Mathematical and Management Sciences. 233and 4H. Singh, N. Misra, V. Hnizdo, A. Fedorowicz, and E. Demchuk, \"Near- est neighbor estimates of entropy,\" American Journal of Mathematical and Management Sciences, vol. 23, no. 3 and 4, pp. 301-321, 2003.\n\nOn the estimation of the entropy using k-th nearest neighbors. P O Amblard, S Zozor, O J J Michel, A M Cuculescu, IMA conf Maths and signals Processing. P. O. Amblard, S. Zozor, O. J. J. Michel, and A. M. Cuculescu, \"On the estimation of the entropy using k-th nearest neighbors,\" in IMA conf Maths and signals Processing, 2008.\n\nUn estimateur du taux d'entropie basé sur l'information mutuelle. C Granero-Belinchón, S Roux, N Garnier, Colloque GretsiC. Granero-Belinchón, S. Roux, and N. Garnier, \"Un estimateur du taux d'entropie basé sur l'information mutuelle,\" in Colloque Gretsi, 2017.\n\nSpurious dimension from correlation algorithms applied to limited time-series data. J Theiler, Physical Review A. 34J. Theiler, \"Spurious dimension from correlation algorithms applied to limited time-series data,\" Physical Review A, vol. 34, pp. 2427-2432, 1986.\n\nFast and exact synthesis of stationary multivariate gaussian time series using circulant embedding. H Helgason, V Pipiras, P Abry, Signal Process. 915H. Helgason, V. Pipiras, and P. Abry, \"Fast and exact synthesis of stationary multivariate gaussian time series using circulant embedding,\" Signal Process., vol. 91, no. 5, pp. 1123-1133, May 2011.\n\nSynthesis of multivariate stationary series with prescribed marginal distributions and covariance using circulant matrix embedding. Signal Processing. 91--, \"Synthesis of multivariate stationary series with prescribed marginal distributions and covariance using circulant matrix embedding,\" Signal Processing, vol. 91, pp. 1741-1758, August 2011.\n"},"corpus_id":{"kind":"number","value":119402330,"string":"119,402,330"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1812.05325\",\n \"mag\": 2951885956,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"journals/tsp/Granero-Belinchon19\",\n \"doi\": \"10.1109/tsp.2019.2920472\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1812.05325v1.pdf\"\n ],\n \"pdf_sha\": \"a71c20c5fd7970ea86a7313ce33f8b049e177745\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1812.05325\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Probing high order dependencies with information theory"},"authors":{"kind":"list like","value":["C Granero-Belinchón \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nUniv Claude Bernard\nCNRS\nF-69342LyonFrance","S G Roux \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nUniv Claude Bernard\nCNRS\nF-69342LyonFrance","IEEE Fellow) *P Abry \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nUniv Claude Bernard\nCNRS\nF-69342LyonFrance","N B Garnier \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nUniv Claude Bernard\nCNRS\nF-69342LyonFrance"],"string":"[\n \"C Granero-Belinchón \\nUniv Lyon\\nEns de Lyon\\n\\nLaboratoire de Physique\\nUniv Claude Bernard\\nCNRS\\nF-69342LyonFrance\",\n \"S G Roux \\nUniv Lyon\\nEns de Lyon\\n\\nLaboratoire de Physique\\nUniv Claude Bernard\\nCNRS\\nF-69342LyonFrance\",\n \"IEEE Fellow) *P Abry \\nUniv Lyon\\nEns de Lyon\\n\\nLaboratoire de Physique\\nUniv Claude Bernard\\nCNRS\\nF-69342LyonFrance\",\n \"N B Garnier \\nUniv Lyon\\nEns de Lyon\\n\\nLaboratoire de Physique\\nUniv Claude Bernard\\nCNRS\\nF-69342LyonFrance\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Information theoretic measures (entropies, entropy rates, mutual information) are nowadays commonly used in statistical signal processing for real-world data analysis. The present work proposes the use of Auto Mutual Information (Mutual Information between subsets of the same signal) and entropy rate as powerful tools to assess refined dependencies of any order in signal temporal dynamics. Notably, it is shown how two-point Auto Mutual Information and entropy rate unveil information conveyed by higher order statistic and thus capture details of temporal dynamics that are overlooked by the (twopoint) correlation function. Statistical performance of relevant estimators for Auto Mutual Information and entropy rate are studied numerically, by means of Monte Carlo simulations, as functions of sample size, dependence structures and hyper parameters that enter their definition. Further, it is shown how Auto Mutual Information permits to discriminate between several different non Gaussian processes, having exactly the same marginal distribution and covariance function. Assessing higher order statistics via multipoint Auto Mutual Information is also shown to unveil the global dependence structure fo these processes, indicating that one of the non Gaussian actually has temporal dynamics that ressembles that of a Gaussian process with same covariance while the other does not."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"Context. To characterize a (stationary) process X(t), evolving along a given dimension t, e.g., time, the precise assessment of both its static properties (marginal distribution or one-point statistics) and temporal dynamics (joint distributions or multipoint statistics) is crucial.\nCorrelation and spectral estimates, providing practitioners with valuable insight into temporal dynamics are key tools in statistics and data analysis and have been widely used in many different fields. In neurosciences, the analysis of correlation between neuron firings [1], [2] or between stimulus and brain activity [3], [4] is essential. In fluid mechanics, Kolmogorov's empirical theory of turbulence [5] characterizes the multiscale distribution of energy with the power spectrum of the velocity field [6]. The analysis of correlations also supports useful descriptions in terms of complex networks for, e.g., ecology [7], climatology [8] or traffic [9].\nIt is however well known that, while the correlation function, or equivalently the power spectral density, exhaust the description of joint statistics and hence of temporal This work was supported by the LABEX iMUST (ANR-10-LABX-0064) of Université de Lyon, within the program \"Investissements d'Avenir\" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). dynamics for jointly Gaussian processes, they provide characterizations of temporal dynamics for general processes, that are limited in at least two respects: not only they are restricted to two-point joint statistics but also to a specific measure of these two point-statistics. In general, the full analysis of temporal dynamics require the use of multi-point, or higher order, joint statistics. The goal of the present work is to address such an issue using information theoretic measures.\nRelated work. To assess dependencies beyond two-point correlations, multipoint correlation functions were defined. For instance, three-point correlation functions have been used in cosmology, where they were related to cosmological parameters [10], [11]. Alternatively, higher order-spectra, that combine Fourier transforms at several different frequencies and therefore takes into account higher order statistics, have also been used to generalize the power spectrum beyond second-order statistics [12], [13]. For instance, bi-spectrum and tri-spectrum were analyzed in depth in e.g., [14], [15] and used in applications as different as cosmology [16], [17], or medicine [18], [19] and other fields [20], [21]. While of popular use in applications, higher-order correlation functions and spectra suffers from major caveats, mostly related to estimation issues. Indeed, the assessment of order m-statistics with higher-order correlation functions and spectra requires the estimation of m-time point statistics, which, for a given fixed sample size, can only yield decreasing estimation performance when order m is increased.\nAlternatively, Shannon's information theory, and the corresponding information theoretic measures, entropy, entropy rate and mutual information, can be considered as an efficient way to characterize high order statistics [22], [23], [24]. Applications of entropy rate were yet reported in a wide range of domains, such as biology [25], dynamical systems [26], fluid turbulence [27] or analysis of languages [28]. Mutual information is also very popular, especially in computer sciences, e.g., in machine learning [29], and other very different domains [30], [31], [32], [33], [34], [35]. Auto-Mutual Information (AMI) -sometimes called Information Storage [36] -, defined as the mutual information between the analyzed process and a delayed version of itself, has been used in different domains [37], [38], [39], [40]. Information theory additionally provides tools to explore causality relationships between processes [41], [42]. However, within this paper we are mainly interested in the characterization of their dependence structures.\nEntropy rate and AMI have been envisaged as tools to describe temporal dynamics in time series beyond second-order statistics, or even at second-order statistics, in a richer way than correlation or Fourier spectrum, see [43], [37], [39], [27]. Goals, contributions and outline. The main goals of the article are i) to show the ability of AMI and entropy rate to describe high order dependences, even when considering only two-point interactions. Even though this property of information theory tools is well known, we are able to show it by analyzing processes with identical correlation and marginal distribution but different high order dependences. AMI and entropy rate appear as straightforward generalizations of correlation function and power spectrum when considering non-Gaussian processes analysis. ii) to show the effect of analysing interactions between more than two points, combining Takens embedding [44] with AMI. We characterize the complexity of the dependence structure of a process by measuring AMI for embedding dimension higher than 2. More precisely, we analyze the differences between power law decay and exponential decay dependence structures.\nTo demonstrate the ability of information theory to describe high order statistics, we illustrate the use of entropy rate and auto-mutual information to characterize non Gaussian processes across scales. As an example, we consider two synthetic log-normal processes with different dependence structures but with identical correlation function and identical marginal statistics. In section II, we introduce and define the theoretical information theory measures and our estimators. In section III, we interpret the information theory tools used along the article, we present the processes with identical correlation function and one-point statistics but different dependence structure that we study and finally we illustrate how AMI and entropy rate allow to distinguish between them. In section IV-A, we report the convergence and the performance of our estimators of AMI and entropy rate and their ability to measure high order dependencies. We present their evolution across scales and emphasized that, contrary to classical correlation analysis, they allow a fine characterization of non Gaussian processes."},{"title":"II. ENTROPY, ENTROPY RATE AND AUTO MUTUAL INFORMATION","content":""},{"title":"A. Definitions","content":"To analyze the temporal dynamics of univariate stationary processes X = {x t } t∈R , it has been proposed in [44] to rely on the time delay-embedding procedure. It amounts to constructing the m-dimensional process X (m,τ ) , whose elements are m-dimensional vectors:\nwhere the delay τ implicitly defines an analysis scale and where the embbeding dimension m controls the order of the statistics of X which are analyzed. The information theoretic measures defined below aim to characterize temporal dynamics via the analysis of the m-time point joint statistical distributions p(x (m,τ ) t ). Within this article, we study ergodic stationary processes and then p(x (m,τ ) t ) depend on m and τ but not on t. Conse-\nShannon entropy, is defined as a functional of p(x (m,τ ) t ) [22]:\nBy nature, it depends on the full joint distribution and therefore on statistics of any order. However, for stationary processes it does not depend on the mean value of the marginal distribution. For m = 1, Shannon entropy does not relate to temporal dynamics but to one-point statistics and thus obviously does not depend on τ and is hence denoted H(X).\nEntropy rate of order m, h (m,τ ) , quantify variations of Shannon entropy between two successive time-embedded versions of a stationary process [22], [24]:\nEntropy rate h (m,τ ) , like H (m,τ ) , probes joint statistics of any order, but in addition quantifies how much information is gained in the analysis of temporal dynamics by increasing by one the order of the embedding.\n) quantifies the amount of information shared by two processes X and Y , via their embbeded versions x (m,τ ) t and y (p,τ ) t [22], [24]:\nWhen X and Y are jointly stationary processes,\nMutual information can be read as the Kullback-Leibler divergence between the joint probability p(x"},{"title":").","content":"To analyze the higher statistical order temporal dynamics (also referred to as nonlinear temporal dynamics) of a single stationary process X, one can use auto-mutual information [24], [36], defined as the mutual information between x\nBecause the concatenation of x t+pτ :\nx\nI (m,p,τ ) (X) quantifies the shared information between the past m-point dynamics and the future p-point dynamics. Auto mutual information quantifies the deviation from independence between the two arguments, beyond the simpler decorrelation. These information theoretic quantities all depend on the chosen time lag τ . When τ → 0, dependences between two consecutive coordinates of the embedding vector increase, therefore inducing an increase in I (m,1,τ ) (X) and a decrease in h (m,τ ) (X). Conversely, when τ → +∞, dependencies decrease to 0, therefore inducing an increase of h (m,τ ) (X) up to the Shannon entropy H(X) of the process, its largest value, and a decrease of I (m,1,τ ) (X) → 0.\nFurthermore, entropy rate can then be expressed as [27]:\nwhich shows that there are two different contributions to the entropy rate: First, H(X), which only depends on the onepoint distribution and hence is a static property ; second, I (m,1,τ ) (X) which gathers all information conveyed by linear and non linear temporal dynamics, irrespective of the variance of the distribution [27]."},{"title":"B. Gaussian process","content":"When X is a stationary jointly Gaussian process, hence fully defined by its variance σ 2\nx and normalized correlation function c X (τ ), analytical expressions for entropy, entropy rate and auto mutual information are available [45]:\nwith Σ (m) the m-dimensional correlation matrix, and || denoting the determinant. By definition, Σ (1) = 1."},{"title":"C. Estimation procedures","content":"To estimate these information theoretic quantities, two main categories of non-parametric procedures are available, based on either kernel density estimations (KDE) or nearest neighbor search (NNS). While the former essentially counts the number k( ) of the observed samples in interval of a priori set size , the latter rather measure the size (k) of box that contain an a priori set number of k samples.\nBecause this second strategy has been documented as providing low bias estimates [46], [47], we chose k-nearest neighbors procedures to estimate the entropy H(X) [48], [49], [50]:\nwith ψ the Digamma function, d the dimension of X, N the number of samples of X, and using the Chebyshev distance (L ∞ metric).\nTo compute the mutual information, we use the k-nearest neighbors procedure presented by Kraskov, et al. [46]:\nThe procedure chooses k for the joint distribution Z(X, Y ) and hence obtains the distance to the kth nearest neighbour.\nThen using , it calculates the number of neighbours n x + 1 lying on a segment x = x± 2 of the X dimension, i.e. lying on −∞ ≤ Y ≤ ∞, x − /2 ≤ X ≤ x + /2. Consequently, /2 is the distance to the n x + 1st neighbour of x. The procedure uses the same development for Y and obtains that /2 is the distance to the n y + 1st neighbour of y.\nBecause benefits against a direct use of eq.(2) were documented in [51], h (m,τ ) (X) is estimated plugging the above estimates into eq.(5).\nFor an stationary process X, p(x (m,τ ) t ) does not depend on t, and consequently, we can exploit this property to estimate the information theoretic quantities of X by averaging on all the temporal samples of the process.\nTo perform an analysis across scales by varying the parameter τ we use an adapted Theiler prescription [52] to eliminate the contribution of dependencies from scales smaller than τ ."},{"title":"III. TEMPORAL DYNAMICS AND DEPENDENCE STRUCTURE","content":"The aim of this section is to try to characterize what part of the statistics of a given stationary process Y contribute to the values taken by information theoretic measures H, h (m,τ ) and I (m,p,τ ) . To that end, we conduct a formal study of the specific yet very instructive case where processes Y with correlation function c Y (τ ) and marginal distributions p Y are obtained as point bijective transformation F , Y = F (X) of an underlying Gaussian process X with correlation function c X , designed to reach the targeted covariance c Y . Processes Y constructed in this way depend on F and c X (τ ) [53], [54]."},{"title":"A. Analytical calculations and interpretations","content":"Given any bijective transformation F that maps a Gaussian signal X into a signal Y = F (X), the information theoretic quantities of Y can be computed as:\nare the variance and the m-dimensional covariance matrix of the underlying Gaussian process X.\nThese analytical calculations permit to yield understanding of general validity regarding the nature of the information conveyed by each of these three information theoretic quantities: i) Entropy H is only related to the marginal distribution of the process p Y , hence is quantifying static properties only. ii) Entropy rate h (m,τ ) mixes up marginal distribution features (variance and pointwise transform F ), hence static properties with dynamical properties and dependencies of any order, as indeed Σ (m) X depends jointly on c Y and on F , hence on the 4 joint distribution of y (m,τ ) t . The fact that h (m,τ ) gather both static and dynamics aspects of time series likely explains why it has been observed in many applications to be a often discriminant feature, compared to other features [40]. However, the entanglement of static and dynamic properties may also be considered a drawback for the analysis, characterization and understanding of data properties. iii) Interestingly, auto mutual information I (m,p,τ ) performs a characterization mainly related to the dependence structure of Y , as indeed Σ (m) X depends on the joint distribution of y (m,τ ) t . It therefore quantifies temporal dynamics, irrespective of static properties. Thus, AMI is the sole measure that focuses exclusively on higher order dynamics and on the joint dependence structure, beyond correlation.\nOften in the scientific literature, h (m,τ ) and I (m,p,τ ) are referred to as nonlinear features, to emphasize that they characterize temporal dynamics beyond the mere use of the correlation functions or power spectral densities. This is indeed correct. Even when two-point statistics are analyzed, i.e., m = p = 1, I (1,1,τ ) reflects dependencies, including and encompassing correlations as the whole joint two-point distribution function is involved. Nevertheless, I (m,p,τ ) remains mostly driven by the correlation function c Y , and hence by linear effects. This is also the case for h (m,τ ) , driven mostly both by the static variance σ 2 Y and dynamical correlation function c Y .\nB. Process definition 1) Analytical derivations: To further illustrate that indeed I (m,p,τ ) and h (m,τ ) may capture differences in any-order dependencies beyond correlations, let us study the cases where two stationary processes Y F1 and Y F2 have exact same marginal distribution p Y and exact same covariance c Y , but are yet obtained from two different point transformations F 1 and F 2 -applied to Gaussian processes X 1 and X 2 -and hence have different joint distributions and hence different dependence structures.\nLet denote F Y the cumulative distribution function associated to the marginal distribution p Y . A natural candidate to obtain a process with marginal p Y is to apply the point transform\nto a Gaussian process X 1 , where F Y is the cumulative distribution function of the marginal distribution p Y and Φ(x) the cumulative distribution function of a centered standard Gaussian variable, and where X 1 is a jointly Gaussian process with covariance function c X1 (τ ) to reach the targeted c Y (τ ) [53], [54]. However, many other transforms can yield processes Y with the exact same marginal distribution p Y and the exact same covariance c Y . In [54], it was for example proposed to use\non a jointly Gaussian process X 2 . This thus yields a radically different joint dependence structure for Y F2 , despite having the same prescribed correlation function as Y F1 . As a consequence, Y F1 and Y F2 have different AMI functions I (m,p,τ ) (Y F1 ) = I (m,p,τ ) (Y F2 ), and entropy rates, and hence different joint dependence structure and thus temporal dynamics. Conversely, their entropies are identical H(Y F1 ) = H(Y F2 ). This is not surprising , as the entropy is completely defined by the one-point distribution.\n2) The log-normal example: To quantify the qualitative analysis proposed in Section III-A , let us consider stationary processes Y F1 and Y F2 having the same log-normal marginal distribution p Y , chosen as an example of distributions that may complicate estimation because of its being a frontier between distributions with all moments finite and heavy-tail distributions.\nFor process Y F1 obtained with the natural transform (14), F 1 simplifies to F 1 (x) = exp(x), which is a bijective transformation. One can therefore use eqs. (11) to (13) above with ln F 1 X1 = µ X1 , σ X1 and Σ (m) X1 being the mean, variance and m-dimensional covariance matrix of the Gaussian variable X 1 [45]. The entropy H(Y F1 ) can be computed using the mean µ Y and variance σ 2 Y of Y F1 :\nWriting I (m,p,τ ) (Y F1 ) thus only requires to express the correlation function of the underlying Gaussian process X 1 as a function of the targeted c Y (τ ):\nThis permits to compute h (m,τ ) and I (m,p,τ ) analytically, using eqs. (12) and (13). For stationary log-normal process Y F2 , F 2 is not bijective. The entropy H can be computed analytically, but this is not the case for I (m,p,τ ) and h (m,τ ) .\n3) Correlation structures: short-range versus long range dependencies: To study the impact of the correlation structure, processes with short-range dependencies are compared against processes with long-range dependencies, the former being well-represented by an exponentially decaying covariance,\nand the latter by an algebraically decaying covariance,\nwith 0.5 < H < 1."},{"title":"C. Two-point dependencies: Auto mutual information versus correlation","content":"For both short-range and long range correlations, we compare the two LN processes obtained with F 1 and F 2 , denoted Y F1 , Y F2 , against a Gaussian process with same correlation function, denoted Y I , where I stands for the identity transform. First order statistics (probability density function) and second order statistics measured by the correlation function or the power spectral density as well as two point statistics (i.e., m = p = 1) entropy, entropy rate and AMI estimates are compared on Fig. 1 for Y I , Y F1 and Y F2 with short-range dependences and on Fig.2 for long-range dependences.\nAs expected, for each pair of short range and long range processes, the two log-normal processes, Y F1 and Y F2 cannot be distinguished one from the other using either PDF, correlation or power spectral estimation. Entropy being fully prescribed by the sole PDF, it is further not surprising that the two log-normal processes are also undistinguishable when using entropy estimates H (Fig (1,d)).\nHowever, and interestingly, the estimates I (1,1,τ ) and h (1,τ ) of two-point AMI and entropy rate show differences as functions of τ between processes Y F1 and Y F2 , both for the shortrange and long-range correlation cases. I (1,1,τ ) decreases and h (1,τ ) increases when τ → ∞, a predictable result as all dependencies vanish for large τ for stationary processes. Both functions (AMI and entropy rate) for processes Y F1 and Y F2 differ for small τ , thus quantifying the difference between the joint distributions of these two processes, despite identical correlation functions and marginal distributions. Fig. 1 and Fig. 2 show that the two-point AMI estimateŝ I (1,1,τ ) for log-normal process Y F1 ressembles far more to that of the Gaussian process Y I than to that of the companion log-normal process Y F2 , even they all three have the same correlation function. Very interestingly, it provides a deep insight into the temporal dynamics of the time series: processes Y F1 and Y F2 have same marginal distributions and same correlations functions, but the temporal dynamics of Y F1 is close to that of the Gaussian process Y I while that of Y F2 is not. This also illustrates the benefits of AMI I (1,1,τ ) over entropy rate h (1,τ ) that mixes the impact of both marginal distribution and joint dynamical structure, thus making Y F2 comparable to Y I at small τ when dependencies matter and Y F2 comparable to Y F1 at large τ when only the marginal distribution contribute to h (1,τ ) . Figs 1,d,e,f and 2,d,e,f also report theoretical H, h and AMI for processes Y I and Y F1 , while theoretical h and AMI are not available for process Y F2 . All estimates well match the predicted theoretical behaviors (solid lines in Figs 1, 2). The theoretical behaviors of AMI and entropy rate of process Y F1 are obtained using eqs. (12) and (13) with the analytical expression (17).\nThese observations essentially hold both for the short-range (Fig. 1) and long-range (Fig. 2) correlations and yield the first key conclusion of this work: as opposed to correlation or spectral estimation, estimates of the two-points AMI (I (1,1,τ ) ) and entropy rate (h (1,τ ) ) are able to quantify differences in the temporal dynamics of processes even when they share the exact same marginal distributions and correlation functions, as long as they have different two-point joint statistics."},{"title":"IV. ESTIMATION PERFORMANCE AND ILLUSTRATIONS","content":"The aim of this section is to assess, by means of Monte Carlo simulations, the performance of the estimatorsĤ, I (m,p,τ ) andĥ (m,τ ) for processes with different statistical structures and to study the impacts of key parameters either entering the definition of the estimators (number of nearest neighbors k) or characterizing the dependence structures of the processes (strength of the correlation c Y (τ )) as well as with sample sizes N . A. Estimation performance 1) Monte Carlo simulations Set-up: The processes are synthetized numerically using circulant matrix embedding strategies, with the toolbox designed in [54], [53] and made available at www.hermir.org. As we have access to analytical curves only for Y I and Y F1 , these are the processes used to characterize the performance of the estimators.\nFor short and long memory, parameters are set f c = 0.1 and H = 0.7 respectively. All variances are set to σ Y ≡ 1. For each case, 100 realizations of processes are synthesized and performances are assessed as averages (and standard deviations) across realizations.\nTo study the impact of the marginal distribution on estimation performance, process Y F1 is compared to process Y I .\n2) Dependence with the sample size N : Fig. 3 reports performances as functions of the sample size N , for 2 8 ≤ N ≤ 2 16 , for a fixed k = 5. It clearly shows that the three estimators are asymptotically consistent (with vanishing biases and variances) irrespective of the marginal distribution and of the correlation being short-range or long-range. Unsurprisingly, variances are larger for the log-normal processes than they are for the Gaussian ones. Also, long-range dependencies imply much larger variances compared to short-range dependencies for a same c Y (τ ) = 0.32 (which corresponds to τ = 1 for the long range correlation process and τ = 11 for the short range one), thus clearly showing that long memory does not induce biases but does imply larger variances. The proposed estimators are overall robust to marginal distributions and dependence structures [46], [47], and have satisfactory performance as soon as N ≥ 2 10 , hence N = 2 16 is used hereafter.\n3) Dependence with the number of nearest neighbors k: Fig. 4 reports estimation performance as functions of k, for N = 2 16 and same correlation c Y = 0.32 as above. Performances barely depend on k, but for a mild increase of the shows that all biases slightly increase when c Y (τ ) increases. The variance appears to increase a little faster than linearly in c Y . For both short-range and long-range correlations, the lognormal processes show larger standard deviations compared to the Gaussian ones, yet with similar biases. Further, for a given correlation level c Y (τ ) the long-range correlation structure shows larger standard deviations compared to the short-range correlation.\nFrom this characterization, we can conclude that the entropy, AMI and entropy rate estimators show a low bias and standard deviation for the combination k = 5 and N = 2 16 of parameters used along the article. In addition, we can conclude that the standard deviation of all the three estimators is larger for strong correlations and non-Gaussian distributions, but their biases depend slightly on the marginal distribution and the correlation strength."},{"title":"B. Process discrimination with auto mutual information","content":"In Figs 1,e,f and 2,e,f, we show that both AMI and entropy rate across scales are able to discriminate between processes Y F1 and Y F2 . The discrimination is far more effective for the smaller scales for both correlation structures (Long and short range).\nTo quantify the differences between Y F1 and Y F2 that we found in Figs 1 and 2, we first present the histograms of the estimations of the entropy rate (Fig. 6a) and of the AMI (Fig. 6b) for fixed N = 2 16 and τ = 1. We perform a statistical ranksum test using Mann-Whitney U-test (Wilcoxon rank-sum test) and compute the p-value, for varying τ and N . As seen in Fig. 6d, the p-value quickly decreases with increasing N , as expected. Fig. 6c quantifies the distinguishability between both log-normal processes across scales, showing that the discrimination is far better for smaller scales. This confirms the observations of Figs 1,e,f and 2,e,f."},{"title":"C. Dependence structure beyond correlations","content":"This sections further explores the ability of AMI I (m,p,τ ) , with m ≥ 1 and/or p ≥ 1 to probe multiple-point dependences and hence fine details in temporal dynamics. Fig. 7 reports higher-order AMI estimatesÎ (m,p,τ ) for several combination of m and p, as functions of the embedding dimension p for both short-range (left) and long-range (right) correlation functions, for processes Y I , Y F1 and Y F2 . From top to bottom the m value is equal to 1, 2, 3 and p. Consequently, the first three lines study the behaviour of AMI on p for fixed value of m, while the fourth line shows the behavior of AMI for m = p.\nWhile two-point AMI yielded essentially identical conclusions for short-range and long-range correlation processes, higher order AMI reveals clear differences: For short-range correlation processes (left column), Y I , Y F1 and Y F2 ,Î (m,p,τ ) and I (m,p,τ ) when it can be computed analytically, show a very mild dependence of AMI with m and p for a fixed τ . To the converse, for long-range correlation processes (right column), Y I , Y F1 and Y F2 ,Î (m,p,τ ) and I (m,p,τ ) when it can be computed analytically, show constant increase of AMI when p or m increase, for a fixed τ . This fundamental observation clearly indicates that for short-range correlation, whatever τ , there is little or no gain in the analysis of temporal dynamics brought by the study of higher order AMI. Conversely, for long-range correlation processes, I (m,p,τ ) captures more and more information in the temporal dynamics as the embedding dimensions are increased thus revealing a significant difference between short-range and long-range correlation structures. Analytical formulae for I (m,p,τ ) when m, p > 1, are only available for the identity I and the natural transform F 1 , namely for Y I and Y F1 . In order to quantify the evolution of AMI when the embedding increases, we compute the p-value of Wilcoxon test between the statistic ofÎ (1,1,1) andÎ (m,p,1) for the same processes as in Fig. 7. Fig. 8 quantifies the increase of AMI when the embedding increases. This evolution of AMI is far more important in the long range correlations case, as expected from Fig. 7. Further, table I shows that, for fixed m + p, AMI evolves differently depending on the precise values of m and p separately. For a fixed m + p AMI is larger when m and p are close, which can be understood as a closer similarity between the two embedded vectors appearing as the arguments of the AMI in eq.(4). Fig. 8. Statistical test. p-value of Wilcoxon rank-sum test between the statistics ofÎ (1,1) (1) andÎ (m,p) (1) for processes with exponentially decaying correlation (top line) or with a power law decay (bottom line). Left column is for Y I , middle column for Y F 1 and right column for Y F 2 ."},{"title":"V. CONCLUSIONS AND PERSPECTIVES","content":"We explored the evolution of AMI (I (m,p,τ ) ) and entropy rate (h (m,τ ) ) on the scale parameter τ . Even when considering only two point interactions (m = p = 1) AMI and entropy rate probe statistics of any order, and hence non-linear dependences, and as such appear as unambiguous generalizations of the correlation function and the power spectrum.\nAs an illustration, we analyzed two log-normal processes with identical 1-point distribution and identical correlation function. AMI (I (1,1,τ ) ) and entropy rate (h (1,τ ) ) are able to discriminate these two processes, which appear as identical for classical analysis, by enlightening differences in high order dependences. The behavior of these quantities along scales reveal the existence of stronger high-order dependences at smaller scale, which allows an easier separation of processes.\nIn addition, AMI generalizes easily to consider explicitly interactions between more than two points. We thus probe the complexity of the dependence structure above linear order, i.e., the additional information measured by AMI when considering an extra point should indicate the existence and relevance of next-order interactions. As an example, comparing the effect of increasing the order of interactions for two dependence structures showed that while exponential decay -short rangedependences do not involve next order interactions, power law -long range -dependences do.\nThe same qualitative results were obtained when considering others one-point distribution, which supports the validity of the methodology for any distribution and dependence structure. This generality should make the methodology very interesting to perform non-Gaussian processes characterization in several different applications."},{"title":"Fig. 1 .","content":""},{"title":"Fig. 2 .","content":""},{"title":"Fig. 3 .","content":""},{"title":"Fig. 4 .","content":""},{"title":"Fig. 5 .","content":""},{"title":"Fig. 6 .","content":""},{"title":"Fig. 7 .","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"Context. To characterize a (stationary) process X(t), evolving along a given dimension t, e.g., time, the precise assessment of both its static properties (marginal distribution or one-point statistics) and temporal dynamics (joint distributions or multipoint statistics) is crucial.\\nCorrelation and spectral estimates, providing practitioners with valuable insight into temporal dynamics are key tools in statistics and data analysis and have been widely used in many different fields. In neurosciences, the analysis of correlation between neuron firings [1], [2] or between stimulus and brain activity [3], [4] is essential. In fluid mechanics, Kolmogorov's empirical theory of turbulence [5] characterizes the multiscale distribution of energy with the power spectrum of the velocity field [6]. The analysis of correlations also supports useful descriptions in terms of complex networks for, e.g., ecology [7], climatology [8] or traffic [9].\\nIt is however well known that, while the correlation function, or equivalently the power spectral density, exhaust the description of joint statistics and hence of temporal This work was supported by the LABEX iMUST (ANR-10-LABX-0064) of Université de Lyon, within the program \\\"Investissements d'Avenir\\\" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). dynamics for jointly Gaussian processes, they provide characterizations of temporal dynamics for general processes, that are limited in at least two respects: not only they are restricted to two-point joint statistics but also to a specific measure of these two point-statistics. In general, the full analysis of temporal dynamics require the use of multi-point, or higher order, joint statistics. The goal of the present work is to address such an issue using information theoretic measures.\\nRelated work. To assess dependencies beyond two-point correlations, multipoint correlation functions were defined. For instance, three-point correlation functions have been used in cosmology, where they were related to cosmological parameters [10], [11]. Alternatively, higher order-spectra, that combine Fourier transforms at several different frequencies and therefore takes into account higher order statistics, have also been used to generalize the power spectrum beyond second-order statistics [12], [13]. For instance, bi-spectrum and tri-spectrum were analyzed in depth in e.g., [14], [15] and used in applications as different as cosmology [16], [17], or medicine [18], [19] and other fields [20], [21]. While of popular use in applications, higher-order correlation functions and spectra suffers from major caveats, mostly related to estimation issues. Indeed, the assessment of order m-statistics with higher-order correlation functions and spectra requires the estimation of m-time point statistics, which, for a given fixed sample size, can only yield decreasing estimation performance when order m is increased.\\nAlternatively, Shannon's information theory, and the corresponding information theoretic measures, entropy, entropy rate and mutual information, can be considered as an efficient way to characterize high order statistics [22], [23], [24]. Applications of entropy rate were yet reported in a wide range of domains, such as biology [25], dynamical systems [26], fluid turbulence [27] or analysis of languages [28]. Mutual information is also very popular, especially in computer sciences, e.g., in machine learning [29], and other very different domains [30], [31], [32], [33], [34], [35]. Auto-Mutual Information (AMI) -sometimes called Information Storage [36] -, defined as the mutual information between the analyzed process and a delayed version of itself, has been used in different domains [37], [38], [39], [40]. Information theory additionally provides tools to explore causality relationships between processes [41], [42]. However, within this paper we are mainly interested in the characterization of their dependence structures.\\nEntropy rate and AMI have been envisaged as tools to describe temporal dynamics in time series beyond second-order statistics, or even at second-order statistics, in a richer way than correlation or Fourier spectrum, see [43], [37], [39], [27]. Goals, contributions and outline. The main goals of the article are i) to show the ability of AMI and entropy rate to describe high order dependences, even when considering only two-point interactions. Even though this property of information theory tools is well known, we are able to show it by analyzing processes with identical correlation and marginal distribution but different high order dependences. AMI and entropy rate appear as straightforward generalizations of correlation function and power spectrum when considering non-Gaussian processes analysis. ii) to show the effect of analysing interactions between more than two points, combining Takens embedding [44] with AMI. We characterize the complexity of the dependence structure of a process by measuring AMI for embedding dimension higher than 2. More precisely, we analyze the differences between power law decay and exponential decay dependence structures.\\nTo demonstrate the ability of information theory to describe high order statistics, we illustrate the use of entropy rate and auto-mutual information to characterize non Gaussian processes across scales. As an example, we consider two synthetic log-normal processes with different dependence structures but with identical correlation function and identical marginal statistics. In section II, we introduce and define the theoretical information theory measures and our estimators. In section III, we interpret the information theory tools used along the article, we present the processes with identical correlation function and one-point statistics but different dependence structure that we study and finally we illustrate how AMI and entropy rate allow to distinguish between them. In section IV-A, we report the convergence and the performance of our estimators of AMI and entropy rate and their ability to measure high order dependencies. We present their evolution across scales and emphasized that, contrary to classical correlation analysis, they allow a fine characterization of non Gaussian processes.\"\n },\n {\n \"title\": \"II. ENTROPY, ENTROPY RATE AND AUTO MUTUAL INFORMATION\",\n \"content\": \"\"\n },\n {\n \"title\": \"A. Definitions\",\n \"content\": \"To analyze the temporal dynamics of univariate stationary processes X = {x t } t∈R , it has been proposed in [44] to rely on the time delay-embedding procedure. It amounts to constructing the m-dimensional process X (m,τ ) , whose elements are m-dimensional vectors:\\nwhere the delay τ implicitly defines an analysis scale and where the embbeding dimension m controls the order of the statistics of X which are analyzed. The information theoretic measures defined below aim to characterize temporal dynamics via the analysis of the m-time point joint statistical distributions p(x (m,τ ) t ). Within this article, we study ergodic stationary processes and then p(x (m,τ ) t ) depend on m and τ but not on t. Conse-\\nShannon entropy, is defined as a functional of p(x (m,τ ) t ) [22]:\\nBy nature, it depends on the full joint distribution and therefore on statistics of any order. However, for stationary processes it does not depend on the mean value of the marginal distribution. For m = 1, Shannon entropy does not relate to temporal dynamics but to one-point statistics and thus obviously does not depend on τ and is hence denoted H(X).\\nEntropy rate of order m, h (m,τ ) , quantify variations of Shannon entropy between two successive time-embedded versions of a stationary process [22], [24]:\\nEntropy rate h (m,τ ) , like H (m,τ ) , probes joint statistics of any order, but in addition quantifies how much information is gained in the analysis of temporal dynamics by increasing by one the order of the embedding.\\n) quantifies the amount of information shared by two processes X and Y , via their embbeded versions x (m,τ ) t and y (p,τ ) t [22], [24]:\\nWhen X and Y are jointly stationary processes,\\nMutual information can be read as the Kullback-Leibler divergence between the joint probability p(x\"\n },\n {\n \"title\": \").\",\n \"content\": \"To analyze the higher statistical order temporal dynamics (also referred to as nonlinear temporal dynamics) of a single stationary process X, one can use auto-mutual information [24], [36], defined as the mutual information between x\\nBecause the concatenation of x t+pτ :\\nx\\nI (m,p,τ ) (X) quantifies the shared information between the past m-point dynamics and the future p-point dynamics. Auto mutual information quantifies the deviation from independence between the two arguments, beyond the simpler decorrelation. These information theoretic quantities all depend on the chosen time lag τ . When τ → 0, dependences between two consecutive coordinates of the embedding vector increase, therefore inducing an increase in I (m,1,τ ) (X) and a decrease in h (m,τ ) (X). Conversely, when τ → +∞, dependencies decrease to 0, therefore inducing an increase of h (m,τ ) (X) up to the Shannon entropy H(X) of the process, its largest value, and a decrease of I (m,1,τ ) (X) → 0.\\nFurthermore, entropy rate can then be expressed as [27]:\\nwhich shows that there are two different contributions to the entropy rate: First, H(X), which only depends on the onepoint distribution and hence is a static property ; second, I (m,1,τ ) (X) which gathers all information conveyed by linear and non linear temporal dynamics, irrespective of the variance of the distribution [27].\"\n },\n {\n \"title\": \"B. Gaussian process\",\n \"content\": \"When X is a stationary jointly Gaussian process, hence fully defined by its variance σ 2\\nx and normalized correlation function c X (τ ), analytical expressions for entropy, entropy rate and auto mutual information are available [45]:\\nwith Σ (m) the m-dimensional correlation matrix, and || denoting the determinant. By definition, Σ (1) = 1.\"\n },\n {\n \"title\": \"C. Estimation procedures\",\n \"content\": \"To estimate these information theoretic quantities, two main categories of non-parametric procedures are available, based on either kernel density estimations (KDE) or nearest neighbor search (NNS). While the former essentially counts the number k( ) of the observed samples in interval of a priori set size , the latter rather measure the size (k) of box that contain an a priori set number of k samples.\\nBecause this second strategy has been documented as providing low bias estimates [46], [47], we chose k-nearest neighbors procedures to estimate the entropy H(X) [48], [49], [50]:\\nwith ψ the Digamma function, d the dimension of X, N the number of samples of X, and using the Chebyshev distance (L ∞ metric).\\nTo compute the mutual information, we use the k-nearest neighbors procedure presented by Kraskov, et al. [46]:\\nThe procedure chooses k for the joint distribution Z(X, Y ) and hence obtains the distance to the kth nearest neighbour.\\nThen using , it calculates the number of neighbours n x + 1 lying on a segment x = x± 2 of the X dimension, i.e. lying on −∞ ≤ Y ≤ ∞, x − /2 ≤ X ≤ x + /2. Consequently, /2 is the distance to the n x + 1st neighbour of x. The procedure uses the same development for Y and obtains that /2 is the distance to the n y + 1st neighbour of y.\\nBecause benefits against a direct use of eq.(2) were documented in [51], h (m,τ ) (X) is estimated plugging the above estimates into eq.(5).\\nFor an stationary process X, p(x (m,τ ) t ) does not depend on t, and consequently, we can exploit this property to estimate the information theoretic quantities of X by averaging on all the temporal samples of the process.\\nTo perform an analysis across scales by varying the parameter τ we use an adapted Theiler prescription [52] to eliminate the contribution of dependencies from scales smaller than τ .\"\n },\n {\n \"title\": \"III. TEMPORAL DYNAMICS AND DEPENDENCE STRUCTURE\",\n \"content\": \"The aim of this section is to try to characterize what part of the statistics of a given stationary process Y contribute to the values taken by information theoretic measures H, h (m,τ ) and I (m,p,τ ) . To that end, we conduct a formal study of the specific yet very instructive case where processes Y with correlation function c Y (τ ) and marginal distributions p Y are obtained as point bijective transformation F , Y = F (X) of an underlying Gaussian process X with correlation function c X , designed to reach the targeted covariance c Y . Processes Y constructed in this way depend on F and c X (τ ) [53], [54].\"\n },\n {\n \"title\": \"A. Analytical calculations and interpretations\",\n \"content\": \"Given any bijective transformation F that maps a Gaussian signal X into a signal Y = F (X), the information theoretic quantities of Y can be computed as:\\nare the variance and the m-dimensional covariance matrix of the underlying Gaussian process X.\\nThese analytical calculations permit to yield understanding of general validity regarding the nature of the information conveyed by each of these three information theoretic quantities: i) Entropy H is only related to the marginal distribution of the process p Y , hence is quantifying static properties only. ii) Entropy rate h (m,τ ) mixes up marginal distribution features (variance and pointwise transform F ), hence static properties with dynamical properties and dependencies of any order, as indeed Σ (m) X depends jointly on c Y and on F , hence on the 4 joint distribution of y (m,τ ) t . The fact that h (m,τ ) gather both static and dynamics aspects of time series likely explains why it has been observed in many applications to be a often discriminant feature, compared to other features [40]. However, the entanglement of static and dynamic properties may also be considered a drawback for the analysis, characterization and understanding of data properties. iii) Interestingly, auto mutual information I (m,p,τ ) performs a characterization mainly related to the dependence structure of Y , as indeed Σ (m) X depends on the joint distribution of y (m,τ ) t . It therefore quantifies temporal dynamics, irrespective of static properties. Thus, AMI is the sole measure that focuses exclusively on higher order dynamics and on the joint dependence structure, beyond correlation.\\nOften in the scientific literature, h (m,τ ) and I (m,p,τ ) are referred to as nonlinear features, to emphasize that they characterize temporal dynamics beyond the mere use of the correlation functions or power spectral densities. This is indeed correct. Even when two-point statistics are analyzed, i.e., m = p = 1, I (1,1,τ ) reflects dependencies, including and encompassing correlations as the whole joint two-point distribution function is involved. Nevertheless, I (m,p,τ ) remains mostly driven by the correlation function c Y , and hence by linear effects. This is also the case for h (m,τ ) , driven mostly both by the static variance σ 2 Y and dynamical correlation function c Y .\\nB. Process definition 1) Analytical derivations: To further illustrate that indeed I (m,p,τ ) and h (m,τ ) may capture differences in any-order dependencies beyond correlations, let us study the cases where two stationary processes Y F1 and Y F2 have exact same marginal distribution p Y and exact same covariance c Y , but are yet obtained from two different point transformations F 1 and F 2 -applied to Gaussian processes X 1 and X 2 -and hence have different joint distributions and hence different dependence structures.\\nLet denote F Y the cumulative distribution function associated to the marginal distribution p Y . A natural candidate to obtain a process with marginal p Y is to apply the point transform\\nto a Gaussian process X 1 , where F Y is the cumulative distribution function of the marginal distribution p Y and Φ(x) the cumulative distribution function of a centered standard Gaussian variable, and where X 1 is a jointly Gaussian process with covariance function c X1 (τ ) to reach the targeted c Y (τ ) [53], [54]. However, many other transforms can yield processes Y with the exact same marginal distribution p Y and the exact same covariance c Y . In [54], it was for example proposed to use\\non a jointly Gaussian process X 2 . This thus yields a radically different joint dependence structure for Y F2 , despite having the same prescribed correlation function as Y F1 . As a consequence, Y F1 and Y F2 have different AMI functions I (m,p,τ ) (Y F1 ) = I (m,p,τ ) (Y F2 ), and entropy rates, and hence different joint dependence structure and thus temporal dynamics. Conversely, their entropies are identical H(Y F1 ) = H(Y F2 ). This is not surprising , as the entropy is completely defined by the one-point distribution.\\n2) The log-normal example: To quantify the qualitative analysis proposed in Section III-A , let us consider stationary processes Y F1 and Y F2 having the same log-normal marginal distribution p Y , chosen as an example of distributions that may complicate estimation because of its being a frontier between distributions with all moments finite and heavy-tail distributions.\\nFor process Y F1 obtained with the natural transform (14), F 1 simplifies to F 1 (x) = exp(x), which is a bijective transformation. One can therefore use eqs. (11) to (13) above with ln F 1 X1 = µ X1 , σ X1 and Σ (m) X1 being the mean, variance and m-dimensional covariance matrix of the Gaussian variable X 1 [45]. The entropy H(Y F1 ) can be computed using the mean µ Y and variance σ 2 Y of Y F1 :\\nWriting I (m,p,τ ) (Y F1 ) thus only requires to express the correlation function of the underlying Gaussian process X 1 as a function of the targeted c Y (τ ):\\nThis permits to compute h (m,τ ) and I (m,p,τ ) analytically, using eqs. (12) and (13). For stationary log-normal process Y F2 , F 2 is not bijective. The entropy H can be computed analytically, but this is not the case for I (m,p,τ ) and h (m,τ ) .\\n3) Correlation structures: short-range versus long range dependencies: To study the impact of the correlation structure, processes with short-range dependencies are compared against processes with long-range dependencies, the former being well-represented by an exponentially decaying covariance,\\nand the latter by an algebraically decaying covariance,\\nwith 0.5 < H < 1.\"\n },\n {\n \"title\": \"C. Two-point dependencies: Auto mutual information versus correlation\",\n \"content\": \"For both short-range and long range correlations, we compare the two LN processes obtained with F 1 and F 2 , denoted Y F1 , Y F2 , against a Gaussian process with same correlation function, denoted Y I , where I stands for the identity transform. First order statistics (probability density function) and second order statistics measured by the correlation function or the power spectral density as well as two point statistics (i.e., m = p = 1) entropy, entropy rate and AMI estimates are compared on Fig. 1 for Y I , Y F1 and Y F2 with short-range dependences and on Fig.2 for long-range dependences.\\nAs expected, for each pair of short range and long range processes, the two log-normal processes, Y F1 and Y F2 cannot be distinguished one from the other using either PDF, correlation or power spectral estimation. Entropy being fully prescribed by the sole PDF, it is further not surprising that the two log-normal processes are also undistinguishable when using entropy estimates H (Fig (1,d)).\\nHowever, and interestingly, the estimates I (1,1,τ ) and h (1,τ ) of two-point AMI and entropy rate show differences as functions of τ between processes Y F1 and Y F2 , both for the shortrange and long-range correlation cases. I (1,1,τ ) decreases and h (1,τ ) increases when τ → ∞, a predictable result as all dependencies vanish for large τ for stationary processes. Both functions (AMI and entropy rate) for processes Y F1 and Y F2 differ for small τ , thus quantifying the difference between the joint distributions of these two processes, despite identical correlation functions and marginal distributions. Fig. 1 and Fig. 2 show that the two-point AMI estimateŝ I (1,1,τ ) for log-normal process Y F1 ressembles far more to that of the Gaussian process Y I than to that of the companion log-normal process Y F2 , even they all three have the same correlation function. Very interestingly, it provides a deep insight into the temporal dynamics of the time series: processes Y F1 and Y F2 have same marginal distributions and same correlations functions, but the temporal dynamics of Y F1 is close to that of the Gaussian process Y I while that of Y F2 is not. This also illustrates the benefits of AMI I (1,1,τ ) over entropy rate h (1,τ ) that mixes the impact of both marginal distribution and joint dynamical structure, thus making Y F2 comparable to Y I at small τ when dependencies matter and Y F2 comparable to Y F1 at large τ when only the marginal distribution contribute to h (1,τ ) . Figs 1,d,e,f and 2,d,e,f also report theoretical H, h and AMI for processes Y I and Y F1 , while theoretical h and AMI are not available for process Y F2 . All estimates well match the predicted theoretical behaviors (solid lines in Figs 1, 2). The theoretical behaviors of AMI and entropy rate of process Y F1 are obtained using eqs. (12) and (13) with the analytical expression (17).\\nThese observations essentially hold both for the short-range (Fig. 1) and long-range (Fig. 2) correlations and yield the first key conclusion of this work: as opposed to correlation or spectral estimation, estimates of the two-points AMI (I (1,1,τ ) ) and entropy rate (h (1,τ ) ) are able to quantify differences in the temporal dynamics of processes even when they share the exact same marginal distributions and correlation functions, as long as they have different two-point joint statistics.\"\n },\n {\n \"title\": \"IV. ESTIMATION PERFORMANCE AND ILLUSTRATIONS\",\n \"content\": \"The aim of this section is to assess, by means of Monte Carlo simulations, the performance of the estimatorsĤ, I (m,p,τ ) andĥ (m,τ ) for processes with different statistical structures and to study the impacts of key parameters either entering the definition of the estimators (number of nearest neighbors k) or characterizing the dependence structures of the processes (strength of the correlation c Y (τ )) as well as with sample sizes N . A. Estimation performance 1) Monte Carlo simulations Set-up: The processes are synthetized numerically using circulant matrix embedding strategies, with the toolbox designed in [54], [53] and made available at www.hermir.org. As we have access to analytical curves only for Y I and Y F1 , these are the processes used to characterize the performance of the estimators.\\nFor short and long memory, parameters are set f c = 0.1 and H = 0.7 respectively. All variances are set to σ Y ≡ 1. For each case, 100 realizations of processes are synthesized and performances are assessed as averages (and standard deviations) across realizations.\\nTo study the impact of the marginal distribution on estimation performance, process Y F1 is compared to process Y I .\\n2) Dependence with the sample size N : Fig. 3 reports performances as functions of the sample size N , for 2 8 ≤ N ≤ 2 16 , for a fixed k = 5. It clearly shows that the three estimators are asymptotically consistent (with vanishing biases and variances) irrespective of the marginal distribution and of the correlation being short-range or long-range. Unsurprisingly, variances are larger for the log-normal processes than they are for the Gaussian ones. Also, long-range dependencies imply much larger variances compared to short-range dependencies for a same c Y (τ ) = 0.32 (which corresponds to τ = 1 for the long range correlation process and τ = 11 for the short range one), thus clearly showing that long memory does not induce biases but does imply larger variances. The proposed estimators are overall robust to marginal distributions and dependence structures [46], [47], and have satisfactory performance as soon as N ≥ 2 10 , hence N = 2 16 is used hereafter.\\n3) Dependence with the number of nearest neighbors k: Fig. 4 reports estimation performance as functions of k, for N = 2 16 and same correlation c Y = 0.32 as above. Performances barely depend on k, but for a mild increase of the shows that all biases slightly increase when c Y (τ ) increases. The variance appears to increase a little faster than linearly in c Y . For both short-range and long-range correlations, the lognormal processes show larger standard deviations compared to the Gaussian ones, yet with similar biases. Further, for a given correlation level c Y (τ ) the long-range correlation structure shows larger standard deviations compared to the short-range correlation.\\nFrom this characterization, we can conclude that the entropy, AMI and entropy rate estimators show a low bias and standard deviation for the combination k = 5 and N = 2 16 of parameters used along the article. In addition, we can conclude that the standard deviation of all the three estimators is larger for strong correlations and non-Gaussian distributions, but their biases depend slightly on the marginal distribution and the correlation strength.\"\n },\n {\n \"title\": \"B. Process discrimination with auto mutual information\",\n \"content\": \"In Figs 1,e,f and 2,e,f, we show that both AMI and entropy rate across scales are able to discriminate between processes Y F1 and Y F2 . The discrimination is far more effective for the smaller scales for both correlation structures (Long and short range).\\nTo quantify the differences between Y F1 and Y F2 that we found in Figs 1 and 2, we first present the histograms of the estimations of the entropy rate (Fig. 6a) and of the AMI (Fig. 6b) for fixed N = 2 16 and τ = 1. We perform a statistical ranksum test using Mann-Whitney U-test (Wilcoxon rank-sum test) and compute the p-value, for varying τ and N . As seen in Fig. 6d, the p-value quickly decreases with increasing N , as expected. Fig. 6c quantifies the distinguishability between both log-normal processes across scales, showing that the discrimination is far better for smaller scales. This confirms the observations of Figs 1,e,f and 2,e,f.\"\n },\n {\n \"title\": \"C. Dependence structure beyond correlations\",\n \"content\": \"This sections further explores the ability of AMI I (m,p,τ ) , with m ≥ 1 and/or p ≥ 1 to probe multiple-point dependences and hence fine details in temporal dynamics. Fig. 7 reports higher-order AMI estimatesÎ (m,p,τ ) for several combination of m and p, as functions of the embedding dimension p for both short-range (left) and long-range (right) correlation functions, for processes Y I , Y F1 and Y F2 . From top to bottom the m value is equal to 1, 2, 3 and p. Consequently, the first three lines study the behaviour of AMI on p for fixed value of m, while the fourth line shows the behavior of AMI for m = p.\\nWhile two-point AMI yielded essentially identical conclusions for short-range and long-range correlation processes, higher order AMI reveals clear differences: For short-range correlation processes (left column), Y I , Y F1 and Y F2 ,Î (m,p,τ ) and I (m,p,τ ) when it can be computed analytically, show a very mild dependence of AMI with m and p for a fixed τ . To the converse, for long-range correlation processes (right column), Y I , Y F1 and Y F2 ,Î (m,p,τ ) and I (m,p,τ ) when it can be computed analytically, show constant increase of AMI when p or m increase, for a fixed τ . This fundamental observation clearly indicates that for short-range correlation, whatever τ , there is little or no gain in the analysis of temporal dynamics brought by the study of higher order AMI. Conversely, for long-range correlation processes, I (m,p,τ ) captures more and more information in the temporal dynamics as the embedding dimensions are increased thus revealing a significant difference between short-range and long-range correlation structures. Analytical formulae for I (m,p,τ ) when m, p > 1, are only available for the identity I and the natural transform F 1 , namely for Y I and Y F1 . In order to quantify the evolution of AMI when the embedding increases, we compute the p-value of Wilcoxon test between the statistic ofÎ (1,1,1) andÎ (m,p,1) for the same processes as in Fig. 7. Fig. 8 quantifies the increase of AMI when the embedding increases. This evolution of AMI is far more important in the long range correlations case, as expected from Fig. 7. Further, table I shows that, for fixed m + p, AMI evolves differently depending on the precise values of m and p separately. For a fixed m + p AMI is larger when m and p are close, which can be understood as a closer similarity between the two embedded vectors appearing as the arguments of the AMI in eq.(4). Fig. 8. Statistical test. p-value of Wilcoxon rank-sum test between the statistics ofÎ (1,1) (1) andÎ (m,p) (1) for processes with exponentially decaying correlation (top line) or with a power law decay (bottom line). Left column is for Y I , middle column for Y F 1 and right column for Y F 2 .\"\n },\n {\n \"title\": \"V. CONCLUSIONS AND PERSPECTIVES\",\n \"content\": \"We explored the evolution of AMI (I (m,p,τ ) ) and entropy rate (h (m,τ ) ) on the scale parameter τ . Even when considering only two point interactions (m = p = 1) AMI and entropy rate probe statistics of any order, and hence non-linear dependences, and as such appear as unambiguous generalizations of the correlation function and the power spectrum.\\nAs an illustration, we analyzed two log-normal processes with identical 1-point distribution and identical correlation function. AMI (I (1,1,τ ) ) and entropy rate (h (1,τ ) ) are able to discriminate these two processes, which appear as identical for classical analysis, by enlightening differences in high order dependences. The behavior of these quantities along scales reveal the existence of stronger high-order dependences at smaller scale, which allows an easier separation of processes.\\nIn addition, AMI generalizes easily to consider explicitly interactions between more than two points. We thus probe the complexity of the dependence structure above linear order, i.e., the additional information measured by AMI when considering an extra point should indicate the existence and relevance of next-order interactions. As an example, comparing the effect of increasing the order of interactions for two dependence structures showed that while exponential decay -short rangedependences do not involve next order interactions, power law -long range -dependences do.\\nThe same qualitative results were obtained when considering others one-point distribution, which supports the validity of the methodology for any distribution and dependence structure. This generality should make the methodology very interesting to perform non-Gaussian processes characterization in several different applications.\"\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 {\n \"title\": \"Fig. 7 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"the (m + p)-dimensional time-embedded vector x (m+p)","content":"the (m + p)-dimensional time-embedded vector x (m+p)"},{"id":"","caption":"Short-range dependencies. Classical analysis: histogram (a), power spectrum (b) and correlation function (c). Information theory analysis for m = 1, p = 1: entropy (d), entropy rate (e) and auto mutual information (f). The color indicates the process: blue for standard log-normal (Y F 1 ), red for Y F 2 log-normal and black for Gaussian (Y I ). The continuous lines in (e,d,f) correspond to analytical expressions. We used N = 216 and k = 5.","content":"Fig. 1 .\n1Short-range dependencies. Classical analysis: histogram (a), power spectrum (b) and correlation function (c). Information theory analysis for m = 1, p = 1: entropy (d), entropy rate (e) and auto mutual information (f). The color indicates the process: blue for standard log-normal (Y F 1 ), red for Y F 2 log-normal and black for Gaussian (Y I ). The continuous lines in (e,d,f) correspond to analytical expressions. We used N = 216 and k = 5."},{"id":"","caption":"Long-range dependencies. Classical analysis: histogram (a), power spectrum (b) and correlation function (c). Information theory analysis for m = 1, p = 1: entropy (d), entropy rate (e) and auto mutual information (f). The color indicates the process: blue for standard log-normal (Y F 1 ), red for Y F 2 log-normal and black for Gaussian (Y I ). The continuous lines in (e,d,f) correspond to analytical expressions. We used N = 216 and k = 5.","content":"Fig. 2 .\n2Long-range dependencies. Classical analysis: histogram (a), power spectrum (b) and correlation function (c). Information theory analysis for m = 1, p = 1: entropy (d), entropy rate (e) and auto mutual information (f). The color indicates the process: blue for standard log-normal (Y F 1 ), red for Y F 2 log-normal and black for Gaussian (Y I ). The continuous lines in (e,d,f) correspond to analytical expressions. We used N = 216 and k = 5."},{"id":"","caption":"Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of N , for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, c(τ ) = 0.32 and k = 5.","content":"Fig. 3 .\n3Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of N , for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, c(τ ) = 0.32 and k = 5."},{"id":"","caption":"Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of k, for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, c(τ ) = 0.32 and N = 2 16 . biases, and this irrespective of the marginal distributions and of short-range or long-range nature of the correlations. Because increasing k further implies increasing the computational load, k = 5 is used hereafter. 4) Influence of the correlation strength c Y (τ ): Fig. 5 reports estimation performance as a function of c Y (τ ) and","content":"Fig. 4 .\n4Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of k, for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, c(τ ) = 0.32 and N = 2 16 . biases, and this irrespective of the marginal distributions and of short-range or long-range nature of the correlations. Because increasing k further implies increasing the computational load, k = 5 is used hereafter. 4) Influence of the correlation strength c Y (τ ): Fig. 5 reports estimation performance as a function of c Y (τ ) and"},{"id":"","caption":"Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of the correlation strength c Y , for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, k = 5 and N = 2 12 .","content":"Fig. 5 .\n5Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of the correlation strength c Y , for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, k = 5 and N = 2 12 ."},{"id":"","caption":"Statistical test. Histogram of the estimation ofĥ(1,1) (1) (a) and I (1,1) (1) (b) for the scale invariant processes (blue for Y F 1 and red for Y F 2 of size N = 2 16 . (c) p-value of the Wilcoxon rank-sum test between the two log-nomal processes Y F 1 and Y F 2 as fonction of the scale for the statistics ofĥ (1,1) (τ ) (dotted line) andÎ (1,1) (τ ) (full line) with N = 2 16 . (d) same quantities as (c) for scale τ = 1 as functions of the process size N . The statistics has been done on 100 realizations of both process. In (c) and (d) the gray curves correspond to the results obtained on the Exponential Decay processes. Mann-Whitney U-test ranksum.","content":"Fig. 6 .\n6Statistical test. Histogram of the estimation ofĥ(1,1) (1) (a) and I (1,1) (1) (b) for the scale invariant processes (blue for Y F 1 and red for Y F 2 of size N = 2 16 . (c) p-value of the Wilcoxon rank-sum test between the two log-nomal processes Y F 1 and Y F 2 as fonction of the scale for the statistics ofĥ (1,1) (τ ) (dotted line) andÎ (1,1) (τ ) (full line) with N = 2 16 . (d) same quantities as (c) for scale τ = 1 as functions of the process size N . The statistics has been done on 100 realizations of both process. In (c) and (d) the gray curves correspond to the results obtained on the Exponential Decay processes. Mann-Whitney U-test ranksum."},{"id":"","caption":"Evolution of AMI with embedding.Î (m,p) (τ = 1) for processes with exponentially decaying correlation (left column) and with power law decay (right column). Black is for Y I , blue is for Y F 1 and red for Y F 2 . First three lines are for fixed m and increasing p:m = 1 (a,b), m = 2 (c,d) and m = 3 (e,f). Last line is for m = p. Other parameters are N = 2 16 , k = 5,","content":"Fig. 7 .\n7Evolution of AMI with embedding.Î (m,p) (τ = 1) for processes with exponentially decaying correlation (left column) and with power law decay (right column). Black is for Y I , blue is for Y F 1 and red for Y F 2 . First three lines are for fixed m and increasing p:m = 1 (a,b), m = 2 (c,d) and m = 3 (e,f). Last line is for m = p. Other parameters are N = 2 16 , k = 5,"},{"id":"","caption":"Statistical test. p-VALUE OF WILCOXON TEST BETWEEN THE STATISTICS OFÎ (1,3) (1) ANDÎ (2,2) (1).","content":"Statistical test. p-VALUE OF WILCOXON TEST BETWEEN THE STATISTICS OFÎ (1,3) (1) ANDÎ (2,2) (1).range correlations \n0.34 \n0.96 \n0.72 \nLong range correlations \n6e −15 \n5e −20 \n2e −3 \nTABLE I \nLong range correlationŝ \n\nI (2,p,1) \n\n(a) \n\nI (1,p,1) \n\nI (p,p,1) \n\n(c) \n\n(b) \n\n(e) \n\nShort range correlations \n\n(h) \n(g) \n\n(f) \n\np \n\n(d) \n\np \n\nI (3,p,1)"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"the (m + p)-dimensional time-embedded vector x (m+p)\",\n \"content\": \"the (m + p)-dimensional time-embedded vector x (m+p)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Short-range dependencies. Classical analysis: histogram (a), power spectrum (b) and correlation function (c). Information theory analysis for m = 1, p = 1: entropy (d), entropy rate (e) and auto mutual information (f). The color indicates the process: blue for standard log-normal (Y F 1 ), red for Y F 2 log-normal and black for Gaussian (Y I ). The continuous lines in (e,d,f) correspond to analytical expressions. We used N = 216 and k = 5.\",\n \"content\": \"Fig. 1 .\\n1Short-range dependencies. Classical analysis: histogram (a), power spectrum (b) and correlation function (c). Information theory analysis for m = 1, p = 1: entropy (d), entropy rate (e) and auto mutual information (f). The color indicates the process: blue for standard log-normal (Y F 1 ), red for Y F 2 log-normal and black for Gaussian (Y I ). The continuous lines in (e,d,f) correspond to analytical expressions. We used N = 216 and k = 5.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Long-range dependencies. Classical analysis: histogram (a), power spectrum (b) and correlation function (c). Information theory analysis for m = 1, p = 1: entropy (d), entropy rate (e) and auto mutual information (f). The color indicates the process: blue for standard log-normal (Y F 1 ), red for Y F 2 log-normal and black for Gaussian (Y I ). The continuous lines in (e,d,f) correspond to analytical expressions. We used N = 216 and k = 5.\",\n \"content\": \"Fig. 2 .\\n2Long-range dependencies. Classical analysis: histogram (a), power spectrum (b) and correlation function (c). Information theory analysis for m = 1, p = 1: entropy (d), entropy rate (e) and auto mutual information (f). The color indicates the process: blue for standard log-normal (Y F 1 ), red for Y F 2 log-normal and black for Gaussian (Y I ). The continuous lines in (e,d,f) correspond to analytical expressions. We used N = 216 and k = 5.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of N , for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, c(τ ) = 0.32 and k = 5.\",\n \"content\": \"Fig. 3 .\\n3Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of N , for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, c(τ ) = 0.32 and k = 5.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of k, for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, c(τ ) = 0.32 and N = 2 16 . biases, and this irrespective of the marginal distributions and of short-range or long-range nature of the correlations. Because increasing k further implies increasing the computational load, k = 5 is used hereafter. 4) Influence of the correlation strength c Y (τ ): Fig. 5 reports estimation performance as a function of c Y (τ ) and\",\n \"content\": \"Fig. 4 .\\n4Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of k, for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, c(τ ) = 0.32 and N = 2 16 . biases, and this irrespective of the marginal distributions and of short-range or long-range nature of the correlations. Because increasing k further implies increasing the computational load, k = 5 is used hereafter. 4) Influence of the correlation strength c Y (τ ): Fig. 5 reports estimation performance as a function of c Y (τ ) and\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of the correlation strength c Y , for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, k = 5 and N = 2 12 .\",\n \"content\": \"Fig. 5 .\\n5Estimation performance forĤ (entropy),Î (1,1,τ ) (auto mutual information) andĥ (1,τ ) (entropy rate), as functions of the correlation strength c Y , for log-normal (blue) and Gaussian (black) processes, with short-range (left) and long range (right) correlations. fc = 0.1, H = 0.7, k = 5 and N = 2 12 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Statistical test. Histogram of the estimation ofĥ(1,1) (1) (a) and I (1,1) (1) (b) for the scale invariant processes (blue for Y F 1 and red for Y F 2 of size N = 2 16 . (c) p-value of the Wilcoxon rank-sum test between the two log-nomal processes Y F 1 and Y F 2 as fonction of the scale for the statistics ofĥ (1,1) (τ ) (dotted line) andÎ (1,1) (τ ) (full line) with N = 2 16 . (d) same quantities as (c) for scale τ = 1 as functions of the process size N . The statistics has been done on 100 realizations of both process. In (c) and (d) the gray curves correspond to the results obtained on the Exponential Decay processes. Mann-Whitney U-test ranksum.\",\n \"content\": \"Fig. 6 .\\n6Statistical test. Histogram of the estimation ofĥ(1,1) (1) (a) and I (1,1) (1) (b) for the scale invariant processes (blue for Y F 1 and red for Y F 2 of size N = 2 16 . (c) p-value of the Wilcoxon rank-sum test between the two log-nomal processes Y F 1 and Y F 2 as fonction of the scale for the statistics ofĥ (1,1) (τ ) (dotted line) andÎ (1,1) (τ ) (full line) with N = 2 16 . (d) same quantities as (c) for scale τ = 1 as functions of the process size N . The statistics has been done on 100 realizations of both process. In (c) and (d) the gray curves correspond to the results obtained on the Exponential Decay processes. Mann-Whitney U-test ranksum.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Evolution of AMI with embedding.Î (m,p) (τ = 1) for processes with exponentially decaying correlation (left column) and with power law decay (right column). Black is for Y I , blue is for Y F 1 and red for Y F 2 . First three lines are for fixed m and increasing p:m = 1 (a,b), m = 2 (c,d) and m = 3 (e,f). Last line is for m = p. Other parameters are N = 2 16 , k = 5,\",\n \"content\": \"Fig. 7 .\\n7Evolution of AMI with embedding.Î (m,p) (τ = 1) for processes with exponentially decaying correlation (left column) and with power law decay (right column). Black is for Y I , blue is for Y F 1 and red for Y F 2 . First three lines are for fixed m and increasing p:m = 1 (a,b), m = 2 (c,d) and m = 3 (e,f). Last line is for m = p. Other parameters are N = 2 16 , k = 5,\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Statistical test. p-VALUE OF WILCOXON TEST BETWEEN THE STATISTICS OFÎ (1,3) (1) ANDÎ (2,2) (1).\",\n \"content\": \"Statistical test. p-VALUE OF WILCOXON TEST BETWEEN THE STATISTICS OFÎ (1,3) (1) ANDÎ (2,2) (1).range correlations \\n0.34 \\n0.96 \\n0.72 \\nLong range correlations \\n6e −15 \\n5e −20 \\n2e −3 \\nTABLE I \\nLong range correlationŝ \\n\\nI (2,p,1) \\n\\n(a) \\n\\nI (1,p,1) \\n\\nI (p,p,1) \\n\\n(c) \\n\\n(b) \\n\\n(e) \\n\\nShort range correlations \\n\\n(h) \\n(g) \\n\\n(f) \\n\\np \\n\\n(d) \\n\\np \\n\\nI (3,p,1)\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Statistical signs of synaptic interaction in neurons. G P Moore, J P Segundo, D H Perkel, H Levitan, Biophysical Journal. 109G. P. Moore, J. P. Segundo, D. H. Perkel, and H. 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M Cohen, A Kohn, Nature Neuroscience. 147M. Cohen and A. Kohn, \\\"Measuring and interpreting neuronal correla- tions,\\\" Nature Neuroscience, vol. 14(7), pp. 811-819, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Chattering cells: Superficial pyramidal neurons contributing to the generation of synchronous oscillations in the visual cortex. C Gray, D Mccormick, Science. 274C. Gray and D. McCormick, \\\"Chattering cells: Superficial pyramidal neurons contributing to the generation of synchronous oscillations in the visual cortex.\\\" Science, vol. 274, pp. 109-113, 1996.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Decoupling the cortical power spectrum reveals real-time representation of individual finger movements in humans. K J Miller, S Zanos, E E Fetz, M Nijs, J G Ojemann, Journal of Neuroscience. 2910K. J. Miller, S. Zanos, E. E. Fetz, M. den Nijs, and J. G. Ojemann, \\\"Decoupling the cortical power spectrum reveals real-time representation of individual finger movements in humans,\\\" Journal of Neuroscience, vol. 29, no. 10, pp. 3132-3137, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The local structure of turbulence incompressible viscous fluid for very large reynolds number. A Kolmogorov, Dokl. Akad. Nauk SSSR. 30A. Kolmogorov, \\\"The local structure of turbulence incompressible viscous fluid for very large reynolds number.\\\" Dokl. Akad. Nauk SSSR, vol. 30, pp. 299-303, 1941.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"High-order velocity structure functions in turbulent shear flow. F Anselmet, Y Gagne, E Hopfinger, R Antonia, Journal of Fluid Mechanics. 140F. Anselmet, Y. Gagne, E. Hopfinger, and R. Antonia, \\\"High-order velocity structure functions in turbulent shear flow,\\\" Journal of Fluid Mechanics, vol. 140, pp. 63-89, 1984.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Constructing ecological interaction networks by correlation analysis: hints from community sampling. W Zhang, Network Biology. 12W. Zhang, \\\"Constructing ecological interaction networks by correlation analysis: hints from community sampling,\\\" Network Biology, vol. 1(2), pp. 81-98, 2011.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Climate networks around the globe are significantly affected by el niño. K Yamasaki, A Gozolchiani, S Havlin, Phys. Rev. Lett. 100228501K. Yamasaki, A. Gozolchiani, and S. Havlin, \\\"Climate networks around the globe are significantly affected by el niño,\\\" Phys. Rev. Lett., vol. 100, p. 228501, Jun 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Power-law strength-degree correlation from resource-allocation dynamics on weighted networks. Q Ou, Y Jin, T Zhou, B Wang, B Yin, Phys. Rev. E. 7521102Q. Ou, Y. Jin, T. Zhou, B. Wang, and B. Yin, \\\"Power-law strength-degree correlation from resource-allocation dynamics on weighted networks,\\\" Phys. Rev. E, vol. 75, p. 021102, Feb 2007.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The three-point correlation function in cosmology. M Takada, B Jain, Monthly Notices of the Royal Astronomical Society. 3402M. Takada and B. Jain, \\\"The three-point correlation function in cosmol- ogy,\\\" Monthly Notices of the Royal Astronomical Society, vol. 340(2), 2002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Computing the three-point correlation function of galaxies in O. Z Slepian, D Eisenstein, Monthly Notices of the Royal Astronomical Society. 4542Z. Slepian and D. Eisenstein, \\\"Computing the three-point correlation function of galaxies in O(n 2 ),\\\" Monthly Notices of the Royal Astro- nomical Society, vol. 454(4), pp. 4142-4158, 2015.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Signal processing with higher-order spectra. C L Nikias, J M Mendel, IEEE signal processing magazine. 103C. L. Nikias and J. M. Mendel, \\\"Signal processing with higher-order spectra,\\\" IEEE signal processing magazine, vol. 10, no. 3, pp. 10-37, 1993.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Statistiques d'ordre supérieur pour le traitement du signal. J.-L Lacoume, P.-O Amblard, P Comon, iSBN = 2-225- 83118-1290J.-L. Lacoume, P.-O. Amblard, and P. Comon, \\\"Statistiques d'ordre supérieur pour le traitement du signal,\\\" p. 290, 1997, iSBN = 2-225- 83118-1.\",\n \"doi\": \"iSBN = 2-225- 83118-1\"\n },\n {\n \"text\": \"Higher-order spectra: The bispectrum and trispectrum. W Collis, P White, J Hammond, Mechanical Systems and Signal Processing. 123W. Collis, P. White, and J. Hammond, \\\"Higher-order spectra: The bis- pectrum and trispectrum,\\\" Mechanical Systems and Signal Processing, vol. 12, no. 3, pp. 375 -394, 1998.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Statistics of tricoherence. V Chandran, S Elgar, B Vanhoff, IEEE Transactions on Signal Processing. 42V. Chandran, S. Elgar, and B. Vanhoff, \\\"Statistics of tricoherence,\\\" IEEE Transactions on Signal Processing, vol. 42, pp. 3430-3440, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Analytic minkowski functionals of the cosmic microwave background: Second-order non-gaussianity with bispectrum and trispectrum. T Matsubara, Phys. Rev. D. 8183505T. Matsubara, \\\"Analytic minkowski functionals of the cosmic microwave background: Second-order non-gaussianity with bispectrum and trispec- trum,\\\" Phys. Rev. D, vol. 81, p. 083505, Apr 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Cosmic microwave background trispectrum and primordial magnetic field limits. 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The widely accepted, post-Apollo paradigm of early lunar impact cratering (ca. 1975-2014) proposed very low or negligible impact cratering in the period from accretion (>4.4 Ga) to ~4.0 Ga ago, followed by a ~170-million-yearlong spike of cataclysmic cratering, during which most prominent multi-ring impact basins formed at age ~3.9 Ga. More recent dynamical models suggest very early intense impact rates, declining throughout the period from accretion until an age of ~3.0 Ga. These models remove the basin-forming spike. This shift has important consequences vis-à-vis megaregolith evolution and properties of rock samples that can be collected on the lunar surface today. We adopt the Morbidelli et al.(2018)\"accretion tail\" model of early intense bombardment, declining as a function of time. We find effects differing from the previous models: early crater saturation and supersaturation; disturbance of magma ocean solidification; deep early megaregolith; and erosive destruction of the earliest multi-ring basins, their impact melts, and their ejecta blankets. Our results explain observations such as differences in numbers of early lunar impact melts vs. numbers of early igneous crustal rocks, highland breccias containing impact melts as old as 4.35 Ga, absence of a 170 Myr-long spike in impact melt ages at 3.9 Ga among lunar and asteroidal meteorites, and GRAIL observations of lunar crustal structure.\n\nContrary data began to appear but were not always recognized as such. For example, KREEP-poor impact melt clasts in lunar meteorites showed no spike at 3.9 Ga, and also showed a paucity of impact melts before ~4.0 Ga, but these valuable data were said to support the terminal cataclysm model in spite of the missing spike (Cohen et al. 2000). Dynamical models, such as the original \"Nice model\" (Gomes et al. 2005) and models by Bottke et al. (2010Bottke et al. ( , 2012 were presented as support or confirmation for the \"classic\" terminal cataclysm paradigm, because they provided a conceptual explanation for the possible existence of a sudden surge of impactors at 3.9 Ga, associated with a phase of dynamical instability of the giant planets. The timing of the spike at 3.9 Ga in those models was not a direct consequence of the dynamical evolution and (as stated in the models) was assumed. These models implied that the sharp spike in cratering applied throughout the solar system at 3.9 Ga, but this was not confirmed in asteroidal meteorites (e.g., see data related to Vesta, Cohen 2013).\n\nDynamicists continued to use the term \"LHB\" but changed the definition from a sharp spike lasting ~170 million years (Ryder 1990) to a gentle surge lasting as much as 1600 million years (Bottke et al. 2012). This maintained the \"LHB\" terminology in spite of modifying the phenomenon. More recently, it has been realized that lack of early impact melts may not require lack of impacts, but rather may simply mean lack of survival of impact melts -due either to lack of impacts or destruction of the early specimens (Hartmann 1975(Hartmann , 2003(Hartmann , 2019Chapman et al. 2007). Morbidelli et al. (2018) compared an updated \"classic\" paradigm dynamical model (which they refer to as a \"cataclysm scenario\") to a second, early intense bombardment model with a monotonically decreasing impact rate, which they call the \"accretion tail scenario.\" The accretion tail model is similar to some of the early early intense bombardment models with decreasing impact rate, based on planetary accretion theory and crater counts (Safronov 1972;Hartmann 1966Hartmann , 1970aHartmann ,b, 1972Hartmann , 1975Turner 1979;Neukum 1983. Those models, however, lacked detailed dynamical modeling.\n\nOne problem with the early versions of the accretion tail scenario is that, in order to cause the proposed impact rates around 4 Ga, the total amount of mass accreted by the Moon since its formation would have exceeded by one order of magnitude that inferred by some researchers from the concentration of highly siderophile elements (HSEs) in the lunar mantle (Day et al. 2007;Day and Walker 2015). However, Morbidelli et al. (2018) suggested that HSEs should have been sequestered into the lunar core during mantle crystallization and overturn. If the latter happened around 4.35 Gy ago, as predicted by lunar thermal evolution models (Elkins-Tanton et al. 2011) and supported by the ages of the oldest lunar rocks (Borg et al. 2015), an intense early bombardment followed by the accretion tail scenario could also have been an influence on HSE concentration in the current lunar mantle and is consistent with the cratering rate observed on terrains younger than 4 Ga. Zhu et al. (2019) supported this result by computing the amount of accreted material implanted in the lunar crust and mantle in each impact of the accretion tail scenario. (See also Joy et al. 2016 for discussion of the types of meteorites striking the Moon at different times.) Based on such work, Morbidelli et al. (2018) concluded that the accretion tail scenario is preferable to the terminal cataclysm scenario, because it avoids the need to delay a giant planet orbit instability until 3.9 Ga -a delay that has been suggested to be inconsistent with the probable dynamics of giant planet formation (Ribeiro de Sousa et al. 2020). Similarly, Nesvorny et al. (2019) found evidence that the trans-Neptunian disk of planetesimals (scattered in order to produce an impact spike in the inner solar system) should have been dynamically depleted within 100 Ma of solar system origin, consistent with the growing evidence that a sharp spike at 3.9 Ga did not occur. According to such studies, the maximum cratering rates in the inner solar system should have occurred much earlier than 3.9 Ga (Clement et al. 2018), possibly helping to explain the giant Moon-forming impact at a very early date >4.4 Ga. In short, the present paper will examine the lunar geological consequence of the early intense bombardment/accretion tail scenario that we infer from the apparent collapse of the \"classic\" cataclysm/LHB paradigm.\n\nThe need for our present work is shown by the fact that, as mentioned above, numbers of authors have continued to use the term \"LHB\" in spite of the changing definitions (sometimes from year to year), while other authors hold to the original definition of a spike at 3.9 Ga. (See Hartmann 2019 for more detailed discussion.) To emphasize the importance of the definitions, we note that in spite of the shift away from the terminal cataclysm/LHB model, the scientific and popular literature in various fields has continued to appeal to the \"classic\" version of the terminal cataclysm or LHB as well established. For example:\n\n• Perkins (2014, writing in Science, reviewing how cosmic impacts could have transformed simple precursor materials into nucleobases in RNA): \"During a period aptly dubbed the Late Heavy Bombardment, which began about 4 billion years ago and lasted some 150 million years….\" • Jolliff and Robinson (2019, in Physics Today, reviewing the legacy of Apollo): \"The Apollo samples provided the first evidence of the so-called late, heavy bombardment…thought to have spiked around 3.9-4.0 billion years ago. Models of the early solar system's orbital dynamics suggest that shifts in the orbits of Jupiter and Saturn may have destabilized early asteroid and cometary belts and led to that cataclysm some 500 million years after the solar system formed.\" • Cartier (2019, in Eos, writing about two zircon grains found in a lunar meteorite): \"Past studies have shown that these two lunar zircons are about 4.1 billion years old and formed during a tumultuous time in the solar system's history called the Late Heavy Bombardment.\" • Shigeru Ida (2019, in Science, writing about Saturn's rings' origin possibly involving breakup of an incoming Kuiper Belt object): \"Such an encounter would have been very rare except during the Solar System formation stage…and the Late Heavy Bombardment era at ~4 billion years ago.\"\n\nThese continued applications of the cataclysm/LHB concept in fields as diverse as biology, lunar history, meteorite evolution, and Saturn rings origin, show the need to reassess the inner solar system cratering from ages of ~ 4.4 Ga to ~ 3.8 Ga.\n\nA note about our terminology: The present paper will deal with various kinds of intervals of years, not only conventional \"age,\" (measured back in time from today), but also intervals measured forward in time from the formation of the Moon, and simple intervals such as the duration of magma ocean solidification, or the ~170 million year interval centered around 3.9 billion years ago that was the reported duration of the terminal cataclysm or LHB, based on impact melt ages (Ryder 1990). In the interest of clarity and brevity, we adopt a current convention (e.g., Christie-Blick 2012) in which conventional age is designated with \"a\" (Latin annum for year, hence Ga and Ma), but intervals, which are not ages, are designated with \"yr.\" (Hence in this paper the reader fill find use both Ma and Myr).\n\nAnother clarification of the present paper may be helpful. Various readers of earlier drafts suggested that we add discussions of certain current issues along with references to various papers regarding the history of the terminal cataclysm/LHB paradigm. These issues included discussions of whether \"paradigm\" is an appropriate term, and more discussion of certain papers. Examples of such issues are more discussion of the useful work of Haskin (1998), Petro and Pieters (2008), and others about Imbrium as a source of 3.9 Ga impact melts and widely distributed ejecta; the role of Orientale basin as \"abruptly\" ending big-basin formation; alterations of the original Nice model with redefinitions of \"LHB;\" more discussion of various papers in the first decades of lunar sample analysis; etc. Because all of these issues were treated at length by one of us in a recent, open-access study of the origins, acceptance, and eventual rejection of the terminal cataclysm/LHB scenarios (78 pages with 181 references, Hartmann 2019), we beg the readers' indulgence in our referencing of that paper, rather than lengthening this paper by repeating those discussions here. We focus here on our main question: If the quantitative Morbidelli et al. (2018) \"accretion tail\" model of lunar history is correct, how does that affect the evolution of megaregolith and how does megaregolith evolution affect rock samples on the lunar surface today and, by extension, other planetary surfaces?\n\n\nANALYSIS AND APPLICATION OF ACCRETION TAIL SCENARIO\n\n\nVIS-À-VIS IMPACT RATE VS. TIME\n\nAs pointed out by Morbidelli et al. (2018), the \"accretion tail\" dynamical model creates a curve of declining impact rate vs. time, similar to -but independent of -early pre-and post-Apollo suggestions about the nature of that curve, based mainly on cratering statistics. Hartmann (1965Hartmann ( , 1966 estimated the characteristic age of lunar mare lava plains at ~3.6 Ga, using cratering rates derived from the Canadian shield. The second paper pointed out that since the pre-mare lunar highlands had 32 × the crater density of the maria, the brief pre-3.6 Ga cratering rate had to average at least ~160× the post-mare crater rate. The uncertainty in that number was estimated to be a factor 2. The number 160 was considered a lower limit, because the factor 32 crater density was found on a number of solar system surfaces, indicating that it is represents the crater saturation equilibrium level found on many bodies (Hartmann 1984). Craters formed in excess of the saturation level are destroyed by later cratering, and this means that the total impact crater numbers in the highlands may be much larger than what we can see today. The saturation equilibrium level is important in the present paper because, as we will show, the total number of impacts during very early intense bombardment, based on Morbidelli et al. (2018), was well beyond the number required to reach saturation. Thus, the saturation curve found in most areas today does not represent the actual number of craters that were formed, but only a lower limit. Once saturation was exceeded on a surface, we lose information on the impact numbers and the shape of the earlier size-frequency distribution curve.\n\nAfter investigators dated Apollo mission samples collected from various landing sites, crater density measurements as a function of measured surface age indicated that as we go backward in time from ages ~3.2 to ~3.8 Ga, the cratering rate reached a value of order ≥200 times the current rate (e.g., Hartmann 1972;Neukum 1983;Neukum et al. 2001). In the \"accretion tail\" scenario, this decline in cratering can be seen as the tail end of the more or less monotonic decline in impact rate from >4.4 Ga ago to 3.8 Ga ago, when the cratering record does not reach saturation, and is more clearly measurable. We stress that the time evolution of the impact rate in this paper not a backward extrapolation of the observed impact rate evolution observed in the 3.2-3.8 Ga interval, but is the result of the numerical modelling of the dynamical evolution of the planetesimals remaining in the terrestrial planet region at the end of the planets' formation process (Morbidelli et al. 2018). Table 1 shows an application of this model. Column 1 lists dates at 100 Myr intervals. Columns 2 and 3 illustrate two ways of normalizing the bombardment model data to give cratering rates relative to the modern cratering rate (defined as 1.0). In column 2 (model A) we normalize to an average value of 1.0 for the last 400 Ma (averaging over the Morbidelli et al. 2018 model values for the last four 100-Ma bins, including T=0=today). Column 3 shows a second, different normalization of the curve (model B). It applies suggestions that the cratering rate at 3.0 Ga was probably several times that observed today, and was still declining from 3.0 Ga till today (Quantin et al. 2004Hartmann et al. 2007). Thus, column 3 normalizes the data to a value of 3.0 at 3.0 Ga. The two normalizations give similar results, as plotted in Fig. 1.\n\nIn this context, it is important to note that the Morbidelli et al. (2018) data set, in order to extend the timeline from 3.0 Ga until today, involves a conceptual discontinuity at ~3.0 Ga ago. At age >3 Ga it follows the calculated decay of asteroids and leftover planetesimals assuming their dynamics were dominated by planetary perturbations (the main subject of Morbidelli et al. 2018). For age <3 Ga, the model shifts to dominance of near-Earth asteroids escaping from the asteroid belt under Yarkovsky forces, giving a relatively constant bombardment in Table 1 (similar to the normalization in column 2 of Table 1). In reality (especially with an asteroid belt a bit more massive 3 Ga ago than now, giving more NEAs) the two curves would merge more smoothly. For this reason, Morbidelli et al. (2018, their Fig. 5) originally plotted their results from 4.5 Ga only to 3.0 Ga ago, but here we use additional data from their model to extend the curve.\n\nThe fourth column in Table 1 shows a few examples of earlier cratering rate estimates (still normalized to current rate = 1) based on extrapolation of crater count data backward to ~4.3 or 4.4 Ga, assuming the early estimates of decline in planetesimal numbers (Hartmann 1972;Neukum et al. 2001;Quantin et al. 2007). In Table 1, fluctuations in cratering rate and half-lives of asteroids against collisions with other bodies, at ages around 3.0 Ga may be due mostly to the merger of the two models. The final (fifth) column of Table 1 lists some estimates of half-lives of the projectile population published during earlier crater-count studies (Hartmann 1972;Neukum et al. 2001;Quantin et al. 2007). We note that these are similar to the results from Morbidelli et al. (2018). Figure 1 shows the Morbidelli at al. (2018) cratering rate curve as a function of time (on a logarithmic scale), using the two normalization methods from Table 1, allowing either no decline since 3.0 Ga or decline by a factor 3. We infer that the most likely cratering rate behavior occurs between these two curves, but with a more gradual transition at about 3.5 to 2.5 Ga, ending (by definition) with today's rate of 1.0.\n\nTo summarize, the \"accretion tail\" scenario favored by Morbidelli et al. (2018) indicates that the early intense bombardment rates in the first few hundred million years (surface formation until ~4.15 Ga) exceeded 1000 times the present rate. At dates before 4.4 Ga, it may have exceeded 10,000 the current rate, although, as we probe back toward the beginning, the lunar surface may have been in magma-ocean state. Note the contrast with the \"classic\" terminal cataclysm paradigm, which called for more or less negligible cratering in the ~4.4 Ga to ~4.0 age interval, with few if any of the major front-side multi-ring basins forming in that period. Hartmann 1972, Fig. 3, p. 55;Q = Quantin et al. 2007). Note: Morbidelli et al. (2018) are given precedence in this paper over the earlier estimates, which are cited merely to demonstrate similarity with early post-Apollo models based on crater statistics.\n\nAs Morbidelli et al. (2018) reported, the crater count results (and their extrapolation to before 4.0 Ga) come strikingly close to the results from the Apollo-era models in both shape and absolute numbers. For example, Hartmann (1972), using early Apollo sample data stated that prior to 4.1 [Ga] ago, the cratering rate on the Moon was at least 10 3 times the present rate and the rate declined with a half-life less than 8 × 10 7 yr. During the interval from 4.1 to 3.2 [Ga] ago, the number of planetesimals showed an exponential decay with a half-life about 3 × 10 8 yr…. A more constant cratering rate applied in the last three aeons. Neukum (1983, as reviewed in Neukum et al. 2001) developed an analytic interpretation of the crater-count data, an equation fitting their observations of crater cumulative crater density as a function of time. The differential values of that equation (i.e., difference in craters/km 2 in a specified size range from beginning to end of 100 Myr intervals), give the crater formation rate (i.e., craters/km 2 in a specified size range) formed per 100 Myr interval). As an example of their results, Neukum et al. (2001) stated that \"…the cratering rate 4 Gyr ago was 500 times higher than the constant rate during the last 3 Gyr,\" a statement close to the curves in our Fig. 1.\n\n\nFigure 1.\n\nCurve showing cratering rate as a function of time (relative to today's cratering rate) derived from Morbidelli et al. (2018) accretion tail model. Lower curve assumes cratering rate asymptotically arrives at today's rate (i.e., 10 • ). Upper curve assumes cratering rate at 3.0 Ga equals 3 × today's rate. \"N\" symbols are independent data from crater counts, taken from equation fitted by Neukum to crater count data (see text for further discussion).\n\nThe impact rate data points derived from the Neukum equation, based entirely on crater counts (not on a dynamical model) are included with symbol \"N\" for comparison in our Fig. 1. The agreement between the dynamical models A and B, and the cratering rate analysis, is remarkable for three distinct reasons. (1) The research teams are entirely independent. The Morbidelli et al. (2018) team made no attempt to match their dynamical model to the earlier crater chronometry results. (2) The techniques used are entirely independent; mathematical results from orbital dynamical calculations are unrelated in practice to counts of lunar craters.\n\n(3) Perhaps most striking, Morbidelli et al. (2018) model predictions match not only the bestestablished parts of the Neukum curve at ages ≲3.8 Ga, but also an extrapolation of that curve to earlier dates. To repeat, the best data used for the Neukum curve come from dated samples returned from Apollo and Luna landing sites, but surfaces for crater counts dating from before ~3.8 Ga (a) are less well documented (if available at all), and (b) have crater densities near the saturation equilibrium line so that super-saturation densities on older surface cannot be directly measured. All the more noteworthy, then, that the Neukum equation for cratering rates fits so well the results of the dynamical model.\n\nThe conclusion of this section is that the accretion tail model, along with certain Apollo era suggestions of early intense cratering, imply that the cratering rate prior to 3.8 or 3.9 Ga likely exceeded hundreds and even thousands of times the rate from the last 1-3 Gyr (and could plausibly have exceeded ten thousand times higher than it is today, prior to ~4.35 Ga), as shown in Table 1, columns 2 and 3, from the Morbidelli et al. (2018) data. This is contrary to \"classic\" terminal cataclysm/LHB paradigm, where cratering rates prior to 3.9-4.0 Ga were considered to be comparable to the present rate. It is significant that, half a century after the Apollo and Luna samples were returned, the accuracy of first significant figures is an issue in understanding important lunar phenomena.\n\n\nDEVELOPING TOOLS TO STUDY EFFECTS OF INTENSE EARLY BOMBARDMENT\n\nA useful tool in understanding the lunar cratering record is the \"isochron diagram,\" which shows the observed size-frequency distribution (SFD) of impact craters as a function of surface age. In the template version, shown here in Fig. 2a, the crater density (craters/km 2 ) is shown incrementally, for each individual size bin, and the size bins are plotted on a log√2 scale, for example from 1.0 km to 1.4 km, from 1.4 km to 2 km, and so on. This plot has an advantage over widely used cumulative plots, because it shows exactly what is happening in each individual size bin, rather than smoothing the curve's structure over many sizes, as cumulative curves do.\n\nThe isochron diagram has a number of additional useful features. The upper diagonal boundary, marked \"Saturation\" shows an empirical curve (mentioned earlier), which has been repeatedly observed in the most heavily cratered surfaces on the Moon, various asteroids, and outer planet satellites throughout the solar system (Hartmann 1984). The observed saturation curve is not a \"geometric\" saturation, since more small craters could, geometrically, be squeezed into existing empty spaces on these surfaces. However, nature does not allow the addition of craters in only a narrow size range; the whole SFD must be added all at once, which requires occasional, large, basin-scale impacts, whose excavation and ejecta blankets completely resurface some areas. Furthermore, \"equilibrium\" connotes stability, whereas empirical saturation in a given region does not represent a static situation. Not only do the distant (often \"off-stage\") basins' ejecta occasionally wipe out shallower craters (at, say, D <100 m diameter, depending on distance and size of the basin), but also the continual rain of meteorites creates craters smaller than any specified diameter D, and erodes rims of craters larger than size D by a simple sandblasting effect. An inexorably increasing supersaturation would produce many big, deep impact craters (some no longer visible) and thus pulverize increasingly deep layers, including the earliest impact melt lenses formed in the near-surface kilometers under the earliest large impact basin floors. Thus, different parts of the saturation line can thus oscillate over time by factors about 2 to 4 (Hartmann and Gaskell 1997). The dashed lines on either side of the saturation curve thus show the typical range of oscillation of visible saturation crater densities. The important larger lesson is that a surface characterized by the saturation line may have been impacted far more times than is evident from the numbers of visible craters, preventing a clear measurement of the SFD of the earliest impactors.\n\n(a) (b) Figure 2. (a) Template for lunar isochron diagram. Each curve shows the size-frequency distribution of craters observed on lunar surfaces of specified ages. The diagram assumes constant crater production rate going back in time until about 3 Ga but higher rates before that, as based on Apollo data (Hartmann 1970a(Hartmann ,b, 1972Neukum et al. 2001). (b) Crater accumulation diagram under conditions of impact rate 10,000 times the current rate, as suggested by Morbidelli et al. (2018) for the period around 4.35 Ga. Given the half-life estimates in Table 2, a rate of ~ 10,000× current rate lasts only tens of millions of years, so the graph is extended only to 10 Myr (see text for further discussion). Because crater chronometry is constantly improving with new data, fine print above diagrams refers to date of our latest iteration of the lunar isochron plot, and the date of our data set. Now suppose the cratering rate ~4.35 Ga was ~10,000 times the present rate, as the accretion tail model indicates for an interval of about 50-100 Myr, centered on that date (see Table 1 and Fig. 1). To create an isochron diagram to match conditions under that elevated impact rate, we simply relabel the isochrons, stopping at 10 Myr, because the half-life of decline in cratering rate at 4.35 would have been on the order of 50-60 Myr, according to Table 1. The result is shown in Fig. 2b. The diagram suggests that any surface formed during cratering at that rate would have reached supersaturation equilibrium conditions on a timescale ~50 Ma, and we can surmise that multi-ring basins of D >300 km would have been forming in stochastic intervals averaging 50-100 Myr. If the cratering rates were higher than 10,000 times the present rate before 4.35 Ga (as indicated in Table 1 and Fig. 1), and if the surface were solid, supersaturation would have been reached in <50 Ma.\n\n\nCRATER ACCUMULATION ON SURFACES FORMED AT DIFFERENT TIMES\n\nAs suggested by Fig. 2b and the discussion in the last section, the accretion tail model offers an approach to understanding how the earliest cratering conditions affected the early lunar crust and megaregolith, and even the kinds of samples picked up on the surface today. This investigation is detailed in Table 2. 0.78 ** *Average mare N(1) density is assumed to be 3.2(10 -3 )craters/km 2 , larger than 1 km, based on Hartmann (1966Hartmann ( , 2005 crater counts on a variety of lunar mare surfaces, which were found to have virtually the same crater density, independent of the Morbidelli et al. (2018) In Table 2, column 1 defines 100 Myr-long time bins, and lists dates of surface formation. In column 2, N(1) refers to the cumulative number of craters/km 2 that have diameter >1 km. N(1) is often used as a proxy for the entire size-frequency distribution curve for all impact craters, from microscopic pits to 1000-km multi-ring basin structures. We adopt it as a first-order proxy, but we recommend caution, because the N(1) values fall just in the diameter zone where the -2 slope SFD of larger craters intersects the steeper -3 to -4 slope at smaller diameters, as shown in Fig. 2. Our work concentrates on the rate of change in the entire crater population as a function of time, but N(1) still has first-order value in our discussions. Thus, in column 2, ∆N(1) values from the accretion tail model are defined as the total number of craters with diameter D >1 km added per km 2 in the 100 Ma intervals shown in column 1. For example, from 3.05 to 2.95 Ga, centered on 3.0 Ga, the model shows ∆N(1) addition of ~2.3 (10 -4 ) craters/km 2 larger than 1 km.\n\nColumn 3 repeats column 2 by listing the ∆N(1) number of craters in terms of average mare crater density. This gives a more direct connection to the appearance of the lunar surface, making the table more accessible to readers not familiar with N(1). Entries of ~1.0 in column 2 thus represent a surface with crater densities typical of average lunar mare on the front side of the Moon, based partly on data from the University of Arizona Lunar and Planetary Laboratory crater catalogs (for a summary of those catalogs, see Wood 1972) and additional counts over the years, by Hartmann, treating the mare surface as one giant mare. The mare surfaces in the major basins have remarkably similar densities, and \"average lunar mare,\" as mentioned in a footnote to Table 2, thus refers to a summation of counts in each diameter bin from surfaces of various major basins, divided by the total area counted in each bin.\n\nThe remaining columns, 4 to 11, give crater densities relative to average mare crater density, as they accumulate on surfaces of eight difference starting dates. Thus for example, if we look at a surface (perhaps a lava flow) formed at 4.35 Ga (column 6), it has 0 crater density at 4.35, but adds ∆N(1) of 103 craters/km 2 by 4.25 Ga. And so on, down the table until we see it leveling off at roughly 170 craters/km 2 larger than 1 km, which would be observed today. In columns 4 to 11, entries of ~32 represent lunar highland surfaces, matching saturation equilibrium as measured by Hartmann (1984). Table 2. Three vertical scales have been added to the diagram to demonstrate its range of important information. At the left side we see the crater densities in terms of average lunar mare and also in terms of N(1) as used in Neukum (1983) cumulative crater densities. At the right side we see a listing of the estimated megaregolith depths, which are a direct function of accumulated crater densities, as will be discussed below. The shallower depth estimates come from Apollo observations and GRAIL data (Wieczorek et al. 2013), and the deeper depths, in the regions dating to the first 500 Myr, come from our calculations. Figure 3 also contains three horizontal shaded bands, marking average lunar mare crater densities, saturation density (32× average mare density), and 10× saturation density. The vertical shaded band indicates a plausible time span in which a putative magma ocean might have solidified, i.e., forming a stable solid surface; accumulation of craters on solid surfaces would occur only after that. Table 2, showing accumulation of crater density as a function of time, during the putative early intense bombardment. Bottom of each curve (on abscissa) shows formation dates for surfaces, and the curves show subsequent crater accumulation on those surfaces. Data were initially plotted at 100 Ma intervals, leading to a sharp kink in each upper curve, ~100 Ma after the start date, due to short half-life of early impact rates; those kinks have been smoothed out by adding artificial data points to indicate a more realistic transition. (See text for discussion of the shaded bands and the three scales provided for the ordinate axis on left and right sides. See Table 2 footnote for definition of \"average mare.\" The curve for 4.55 Ga is included to indicate the extreme impact rate if solid bodies existed at that time; the Moon probably did not yet exist.) Figure 3 gives insights into a number of lunar properties and aspects of lunar history. For example:\n\n• Crater SFD numbers formed on the earliest inner solar system surfaces, as represented by the 4.55 Ga and 4.45 Ga curves, are no longer measurable on surfaces today, because of extreme oversaturation. The earliest craters have been overlapped by later large craters, multi-ring basin impact structures, and basaltic mare lava flows.\n\n• Surfaces formed prior to ~4.2 to ~4.3 Ga were intensely bombarded and super-saturated with craters, due to the very high early cratering rates, but are getting closer to saturation levels. Near saturation levels affect the survival of impact melts from upper kilometers that time.\n\n• The youngest crater-saturated surfaces visible today should date from ~4.1 to ~4.2 Ga, consistent with Fig. 2a. The reduced saturation effects by this time allow for increasing survival of brecciated materials and impact melt inclusions from that time.\n\n• Crater densities matching average mare surface appear around 3.5 to 3.6 as predicted from pre-Apollo crater counts and supported with dated Apollo samples.\n\n\nINTENSE EARLY BOMBARDMENT: CONSEQUENCES FOR MEGAREGOLITH EVOLUTION AND MEGAREGOLITH DEPTHS\n\nApplying the information discussed so far, we investigate the evolution of megaregolith in the first 600 Myr of lunar history. Figure 4 introduces a technique for using the SFD curve of crater populations to calculate depth of pulverization, based on an earlier technique utilizing the isochron diagram (Hartmann 1980(Hartmann , 2005. The technique works for under-saturated, saturated, and over-saturated surfaces. For various SFD crater densities formed on such surfaces, we calculate the area covered by craters (per km 2 ) in each diameter bin, then add up the cumulative area starting with the largest bin and working toward smaller bins. As shown in Fig. 4, at each diameter D, this gives the total area covered by craters larger than D (and thus ejecting material deeper than some effective depth d excavated by such craters). At some sizes we reach a point size where the total area of larger craters is 100% of the area available. At 100% coverage, however, not all areas have been impacted and excavated, because of crater overlap. Thus, the 200% parameter is included to allow more realistically for such overlap. We will define our quantitative definition of d in a moment. We assume, as a first-order approximation, that when 100% to 200% of the surface is covered by craters larger than critical diameters designated as \"D100%\" and \"D200%,\" respectively, then a characteristic depth of megaregolith is at least of the order ~d100% to ~d200%. The effective depth at any diameter D must be less than the deepest part of the transient cavity, but deeper than the observed floor depth in the existing crater. Here we assume also that nearly all of the \"effectively\" excavated material is distributed elsewhere on the Moon. The depths affected by craters exceeding D100% and D200% are somewhat in question, since the transient cavity depths near the center of a crater are considerably greater than the final crater depths. What, then, is the depth actually affected by saturation coverage among craters of these sizes? Taking into account the effects mentioned so far, we assume conservatively that craters of the relevant diameters (some kilometers) churned and pulverized materials to effective depths equaling 1/3 of their diameter. The depth of megaregolith, while averaging around the order of magnitude of ~d100% to ~d200%, will vary from region to region, because stochastically-produced largest basin-forming impacts will blow away early surfaces and create new, localized, surfaces -which then begin to accumulate mare lava flows, new impacts, and new regolith. For example, the large, relatively late Imbrium basin must have blown away most earlier megaregolith (not to mention overlapping, obliterating, and ejecting material from earlier large craters and possible earlier, small multi-ring basins with their impact melts). At the same time, Imbrium's ejecta delivered such earlier megaregolith materials onto other regions (McGetchin et al. 1973;Haskin 1998;Hartmann 2003;Petro and Pieters 2008;Norman and Nemchin 2012;Norman et al. 2015). The giant basins thus create relatively thin spots in the megaregolith cover (possibly zero kilometers immediately after impact in some cases), while depositing the previous megaregolith material in a surrounding ejecta blanket. Another example of these effects may be present at South Pole-Aitken (SP-A) basin. Using GRAIL results, Besserer et al. (2014) state that \"The SP-A region appears to possess a thinner low-density (porous) layer than the rest of the farside.\" This result suggests that that SP-A blew away a fair amount of megaregolith which had formed before the SP-A impact. Even cursory inspection of lunar photo images reveals that SP-A has a higher crater density and higher albedo than the major front-side maria. It has evidence of mare basaltic lava deposits, but those mare surfaces themselves have been heavily cratered, with the lava apparently pulverized and brightened by ray deposits. These statements suggest that SP-A and its lava flooding date from before most of better-preserved mare lava production dating from ~3.8 Ga to ~3.2 Ga, but nonetheless that a substantial thickness of megaregolith existed before the impact, to be blown away. This is consistent with our model, but inconsistent with the LHB paradigm, with its sparse cratering before 4.0 Ga. Here, 100% of area is covered by craters larger than ~4 km, and 200% by craters larger than ~350 m. (b) The same but for a surface that is supersaturated by a factor 2. In this case, 100% of area is covered by craters larger than ~60 km and 200% covered by craters larger than ~5 km. The diagram illustrates how the sizes of craters excavating 100% and 200% of total area increases dramatically as super-saturation is reached. Table 3 and Fig. 4 illustrate the technique described here. We apply the technique starting with the saturation equilibrium curve, because it reflects the highest observable crater densities and thus suggests a minimal megaregolith depth in lunar highland saturated regions. In Table 3, column 1 shows crater diameter and column 2 shows the calculated fractional area per km 2 covered by observed craters in each bin, as follows:\n\nFractional area covered in each bin = π(D/2) 2 N = 0.785ND 2 (1) where D = crater diameter near mid-point in each bin N = no. craters/km 2 observed in each bin This fractional area is then expressed as a percentage of the area in column 2 of Table 3.\n\nThe formula does not take into account that some craters overlap earlier craters. This is a valid approach if we are interested in the total volume being excavated and ejected by observable craters. But if we are interested in the total surface area that has been excavated, the 200% figure is better, since most of the area has been impacted at least once by craters larger than D200%.\n\nColumn 3 in Table 3 shows the cumulative percentage of area covered by all craters larger than D, starting with the largest craters at the bottom of the table. At D >128 km there are poorer statistics because of the smaller number of large craters and basins, and the isochron curves are not plotted at still larger sizes for this reason. Among the giant multi-ring basins at the largestdiameter end of the size-frequency distribution (SFD), at D >~360 km, there appears to be a continued turn-down at largest sizes, and we have added modest provisional percentage numbers in column 3 at D >360 km. The areas covered, however, are not large enough to affect our conclusions significantly. Columns 4, 5, and 6 add the same results for 2× saturation and 2.5× saturation In Table 3, the important transitions at 100% and 200% coverage by impact craters are shown in bold face in columns 3, 5, and 6. In this calculation, at D <1.4 km, we follow the wellknown steep branch of the SFD upward into the region above the saturation line, not stopping at the saturation curve per se. This is in order to include the effects of all impacts called for by the SFD at small sizes, whether we can see the resulting craters or not. This steep branch (and presumably the larger craters of the large-D branch at D >1.4 km), include an uncertain fraction of secondary craters, which add to the churning and pulverization involved in megaregolith evolution. [Our system of crater counts and crater chronometry includes all visible craters (excluding obvious clusters with rays), without trying to distinguish primaries from the remaining \"field secondaries,\" since attempts to distinguish them visually are likely to be in error. This system is practical, since nature can't produce large craters without producing secondaries, and can't produce secondaries without primaries, and on old surfaces with high crater densities, the addition of unrecognized scattered field secondaries in a single shower from a fresh distant primary will not have a drastic effect on the total crater density.] Note that, as seen in Fig. 4a and mentioned above, the small-D end of the steep branch of the SFD is first to hit the saturation line. What appears to be sparse cratering as in lunar maria at large D thus produces supersaturation at small D creating a shallow regolith of roughly 5 to 15 meters depth, as detected in Apollo landings on lunar maria. But once saturation is approached at D ≳1.4 km, as indicated in Fig. 4a, the saturation line is reached virtually simultaneously at all larger crater diameters, causing an explosive growth in megaregolith.\n\nHere we pause to ask whether these last statements apply to the earliest crater populations in the first few hundred million years. The problem is that if the early intense bombardment/accretion tail scenario is correct, we must ask if the SFD of crater populations at that time was the same as that observed in post-mare time, when the observed crater density is low, so that the SFD can be accurately measured. On earlier surfaces, dating from ≳4.2-4.3 Ga, the SFD is uncertain because supersaturation occurred, and details of the SFD shape are masked by supersaturation. However, Delbo et al. (2017) found that the collisional evolution of the asteroidal population (which produces the observed size distribution of asteroids) was already well developed prior to the dynamical instability among the giant planets, and possibly by the time of the formation of the Moon itself -and they concluded that its SFD at that time was already consistent with that of current asteroids (which, in turn, is at least roughly consistent with current craters). We thus see no firm observational or theoretical evidence that the slope of the SFD among the larger impactors in impactor populations before ~4.1-4.2 Ga departed from what we see recorded on unsaturated lunar surfaces today. Changes in typical velocity of craterforming impactors are not expected to have an overwhelming effect on the shape of the SFD. The main point here is that the near-simultaneous approach to saturation by all craters of D ≳1.4 km caused a rapid transition from shallow regolith to deep megaregolith. What are the consequences of saturation and super-saturation for megaregolith evolution? At saturation level, as shown in Fig. 4a and Table 3, column 3, we see that D200% is about 350 m, and D100% is about 4 km. This would suggest effective regolith depths (allowing for the moment two significant figures) in the range of d ~120 m to ~1.3 km -somewhat deeper in some areas, and shallower in other areas, since craters larger than 4 km have more stochastic effects. These estimated regolith depths correspond to surfaces observed today but formed ~4.1 to ~4.2 Ga. Now consider still older surfaces that are supersaturated with craters. Although the supersaturation cratering SFDs are hidden from the view of modern observers, Fig. 4b and Table 3 show how the accretion tail model would produce at least twice the level of the saturation curve on surfaces that formed as early as ~4.2-4.3 Ga -a date from which we have impact melt samples. In this case, as shown in column 4 of Table 3, craters of D >60 km would have covered 100% the surface, and craters of D >5 km would have covered 200% of the surface. These figures suggest that the megaregolith depth produced by 2× saturation would be (by our calculation) in the range of ~2 km to ~20 km. Note the explosive regolith growth within an interval of only ~100 Myr: While crater densities have increased by only factor 2 above saturation, the d100% figure has increased from a depth of ~120 m to 2 km.\n\nBecause of the \"explosive\" growth effect, megaregolith depths increase rapidly as we consider slightly higher degrees of supersaturation on earlier (older) surfaces. On surfaces formed during a period earlier than ~4.3 Ga, when Table 3 indicates 2.5× saturation at D100% = ~16 km to D200% = ~90 km, megaregolith depth, as observed today, would be in the range 5 to 30 km (in areas not excavated by subsequent large impact basins -an effect mentioned above and discussed in the next section).\n\nInterestingly, the megaregolith gravels appear frequently to have consolidated into coherent breccia material. Many of the Apollo highland breccias samples, perhaps excavated from depth, are strong, coherent rocks. Coherent materials may have arisen as a result of several different effects. First, impacts in the first hundred Myr or so may have penetrated primarily into molten materials, regionally stirring them and mixing them with shattered crustal materials, so that molten matrix materials may have helped solidify some breccias. Second, pressure effects may have helped cement the heated fragmented materials below some depth. Third, hypervelocity meteorite impacts at various scales produce glassy matrix material that helps to cement rocky fragments into coherent breccias. Wieczorek et al. (2013) and Spray (2016) have discussed such processes in more detail. To summarize, the depths of tens of kilometers, calculated here for impact-affected materials from the geometry of craters, should not necessarily be construed as occupied by loose material; the material may, however, be more porous and less dense than deeper plutonic layers unfractured by impacts. These results are consistent with GRAIL gravitational findings of low densities and higher porosities in the upper few kilometers of the lunar surface. For example, Wieczorek et al. (2013, p. 671) found densities of 2550 kg/m 3 and average crustal porosity of 12% \"to depths of at least a few kilometers.\" We suggest that these measurements refer to loosely consolidated and/or heavily fractured megaregolith, grading at depth into more coherent breccias and fractured crustal bedrock with very early plutonic igneous intrusions along some of the fractures.\n\nTo summarize this section so far, our discussions depart considerably from the \"classic\" terminal cataclysm/LHB models. To start with, a whole series of impacts at all scales, including a few 10 3 -km-scale basins, would have been forming as the putative magma ocean solidified. The end of magma ocean crystallization is believed to have happened by about 4.45 to 4.35 Ga (Elkins-Tanton et al. 2011;Zhu et al. 2019). Borg et al. (2011Borg et al. ( , 2015 describe an age of 4.360±0.003 Ga for a ferroan anorthosite sample 60025 and discuss it as marking a possible magma ocean crystallization age. However, they also refer to three ferroan anorthosites dated by other authors at 4.43±0.03, 4.47±0.07, and 4.53±0.12 Ga, which could record earlier impact and recrystallization events. Numbers of impact melt clasts, dated at ~4.35 Ga, offer vestiges of one or more large impact structures during that period (Nemchin et al. 2008, Grange et al. 2013, and White et al. 2020; see also Hartmann 2019 for review). Figures 1, 2a , and 3 of our present paper imply very intense bombardment, including multiple impacts of basin-forming magnitude, along with extreme crater oversaturation, during the suggested periods of magma ocean presence. In contrast, the LHB paradigm, from its original version (Wetherill 1975(Wetherill , 1977Ryder 1990) with its period of several hundred million years of negligible impacts before 4.0 Gyr, to the re-defined versions (e.g., Bottke 2012, with LHB reduced in intensity to a surge from 4.1 to 2.5 Ga) all involved a period of relatively low bombardment during the suggested periods of magma ocean presence.\n\nOur accretion tail scenario, with early intense bombardment during the putative magma ocean's lifetime and solidification, would produce radically different petrologic results than the LHB models. Some early discussions of the solidification of the putative lunar magma ocean, under conditions pictured in LHB models, assumed the process starting with smooth-layered stratigraphy evolving in the absence of large impacts, with density-controlled layering according to the Bowen reaction series. For example, Walker (2009, p. 109) stated that \"…the lunar mantle is likely stratified resulting from magma ocean crystallization and little subsequent mixing.\" The accretion tail model of impact history, together with early work on planetary accretion (e.g., Safronov 1972) and our analysis here, indicate that during accretion of the Moon and the final solidification of the magma ocean, the uppermost mantle of the Moon and/or magma ocean could not have solidified in a smoothly \"stratified\" way with \"little subsequent mixing.\" Addressing the date of solidification for the magma ocean, Borg et al. (2011) considered their zircon-derived 4.36 Ga date to be surprisingly young and concluded \"much of the lunar crust may have been produced by non-magma-ocean processes, such as serial magmatism.\" In contrast to LHB scenarios, our proposed early intense bombardment allows an explanation for complex crustal evolution. serial magmatism, and HSE anomalies. Large-scale basin impacts, would have broken up newly forming surface crustal layers in many regions and ejected masses of deeper material into widespread megaregolith layers, as well as triggering magmatic eruptions. Impactfracturing and impact-stirring of deep magmas may have been a factor in allowing HSE to collect in the lunar interior. Magma at depth from would have been stirred and redistributed during the period from, say, 4.35 Ga to ~4.1 Ga. Figure 2b, representing the intense impact regime about 4.35 Ga, suggests impact basins of >300 km diameter forming stochastically at intervals on the order of ten million years in random lunar locations at that time. Each one would have excavated to depths of many tens of km and dumped ejected fragmental material over wide areas of the Moon. Consistent with this, Wieczorek and Phillips (1999) suggested that large impacts may have disturbed the crustal structure being formed by magma ocean crystallization. Gross and Joy (2016) and Pernet-Fisher and Joy (2016) also discussed effects of impacts on magma ocean evolution and crustal formation. Rolf et al. (2017) suggested that accumulating ejecta deposits (megaregolith) from impact basins scattered over time may have an insulating effect, influencing the rate of heat flow from the interior.\n\n\nCONSEQUENCES OF LARGEST EARLY IMPACTS VIS À VIS HETEROGENEITY OF THE MEGAREGOLITH\n\nThe formation of multi-ring basins approaching 1000 km diameters, scattered in space and time during the first 600 Myr, created localized effects on the depth of megaregolith (which included mixed layers of local and distant basin ejecta). Such giant basins also affect the observable survival of earlier, nearby basins, by obliterating parts of rim structures and/or burying parts of them in ejecta. The accretion tail model with early intense bombardment thus casts the South Polar Aitken Basin and the controversial Oceanus Procellarum in a new light. Figures 1, 2a, and Table 1 suggest that under our proposed cratering rates before ~4.3 Ga, several giant impact basins, in the 10 3 km diameter (outer ring) range would probably have formed.\n\nThe Procellarum area was once considered to be a candidate for such a primordial impact basin, and this hypothesis gains plausibility in the context of our discussion. The GRAIL team discovered a 3000-km diameter \"ring\" of linear gravitational anomalies under the outer part of Oceanus Procellarum, interpreted as intrusion-filled fractures -which might seem to support that Procellarum is a form of a multi-ring impact feature. They suggested, however, that since these features comprise about five or six linear segments, they are not related to the \"rings\" of faults around multi-ring impact basins (Andrews-Hanna et al. 2014). Thus they argued against an impact origin for Procellarum. Figure 5, however, shows ~30-40-degree linear segments in some rings around the Orientale impact basin. Figure 6 shows similar linear segments in the ring structures around the Humboldtianum impact basin. Note that if a truly circular mass of material along a crater rim is trying to slide downward into a basin interior along a conical fault, it is inhibited from sliding because it is buttressed by its own material. Linear normal fault segments make it easier for such material to slip, in segments, down into the crater interior. Furthermore, a Procellarum-scale impact involves restructuring the spherical curvature of the Moon, and we have little knowledge of how a spherical planetary body's brittle and/or plastic lithospheric layers adjust to world-scale impact craters. Our conclusion is that linear subsurface features at Procellarum do not argue against an impact origin for that large lunar feature. Zhu et al. (2017) also argue in favor of a collisional origin of Procellarum because this event would help explain the dichotomy in crustal thickness between the near-side and far-side of the Moon. Figure 6. Humboldtianum multi-ring basin. Photo (a) and topographic map (b) reveal that the southern half of the outer ring (boundary between orange highlands surrounding and green, depressed interior), shows three clear linear segments. Diameter of outer ring is ~620 km. NE half portion of that ring and the basin are overlain by a large crater. (a) Discovery rectified photo by WKH, from Hartmann and Kuiper (1962). This early image is shown in order to illustrate that these linear features have been apparent since the early 1960s. (b) NASA; \"LOLA\" topographic map: NASA/Goddard Space Flight Center.\n\nAnother consequence of the accretion tail model with early intense bombardment is that the earliest, extremely high impact rates such as 10,000 times the present rate, go beyond the rates typically contemplated in the terminal cataclysm/LHB model. We have already mentioned that the earliest basins would have been beaten into presently undetectable states. Figure 7 illustrates the effectiveness of that process. It shows a vivid example of an older basin, Serenitatis, whose rim has been broken into linear ridges and partly demolished by ejecta or faulting associated with the younger, neighboring Imbrium basin. The mountains of the Serenitatis rim are laced by ridges and valleys structured not radial to Serenitatis but radial to Imbrium, whose center is ~980 km away, beyond the lower edge of the frame. The destruction here is also ~270 to ~390 km beyond the Apennine rim of the Imbrium basin. The precise mechanism for this reshaping of the Serenitatis rim by the Imbrium impact is still not certain. Tectonic fractures caused by the Imbrium impact, with collapses during lava flooding of the area, seem a possibility. A second possibility might be gouging by local grazing, low-angle rock masses in ejecta ejected from Imbrium, possibly embedded in a somewhat volatilized mass like a base surge. The second mechanism seems perhaps less preposterous when one considers that the \"mountain ranges\" creating basin rims are, to some extent, piled-up, ejected megaregolith gravels. Whatever the cause, the degree of Serenitatis destruction by Imbrium is remarkableand all the more so since (as discussed in more detail by Hartmann 2019, pp. 19-20, Table 1) there is at least one case where researchers relying on radiometric dating of Apollo samples listed Serenitatis as about one error bar younger than Imbrium. (Most proposed dating of Serenitatis put it within an error bar to two older than Imbrium.) Such young dates reported for Serenitatis may, however, have involved the Imbrium ejecta that modified the Serenitatis rim (see Hartmann 2019 for more detailed discussion). \n\n\nCONSEQUENCES OF MEGAREGOLITH EVOLUTION VIS-À-VIS ROCK SAMPLES COLLECTED ON PRESENT-DAY LUNAR AND PLANETARY SURFACES\n\nThe evolution of the megaregolith, and its current properties, have important effects on the rocks and soil samples collected on the present-day surface of the Moon, not to mention other planetary surfaces. To take impact melt samples first, the high formation rate of early basins and large craters had important effects on any impact melts formed in the first few hundred Myr. The large, multi-ring basins are believed to contribute most of the volume of lunar impact melts (Artemieva and Shuvalov 2008), but the exact disposition and depth of impact melts within basins appear to be uncertain. For example, based on terrestrial observations and theoretical modeling, Cintala and Grieve (1998) depict impact melt sheets as layers much thinner than the transient cavity basin depth, spread across the basin interior. Artemieva and Shuvalov (2008) and others suggest more concentration in deeper plugs under the middle of the crater. Impact melt plugs as deep as 20 km have been suggested under Imbrium. It is implicit in either case that impact melts are created and distributed in more or less surficial layers having to do with impacts and ejecta blankets. We have shown, however, that in the accretion tail model, creation of the initial megaregolith in the first few hundred Myr involved pulverization to depths of tens of kilometers, probably deeper than most impact melts. As also shown in Fig. 3, surfaces and near-surface structures formed around 4.15 Ga are only slightly supersaturated. By about 4.0 or 3.9, near-surface structures such as impact melt lenses in the upper few kilometers are more likely to survive. Furthermore, as indicated in Fig. 3 and Fig. 2a, structures formed around 4.0 Ga no longer become saturated with large craters. In short, the accretion tail scenario, contrary to terminal cataclysm/LHB models, explains why impact melts older than ~4.1 Ga are hard to find, while increasing numbers of impact melts are found from ~4.0 Ga to 3.9 Ga. The number of impact melts decreases after that because the impact rate itself was declining. In this view, the unique aspect of 3.9 Ga is not a sudden narrow spike in impact rate, but rather that this date marks the beginning of the period when massive formations on the scale of impact melt lenses will survive -not to mention that the Imbrium and Orientale basins formed around that time.\n\nBut what about the fact that primordial crustal samples date back as far as 4.44-4.45 Ga? Why weren't they also destroyed? The answer lies in the fact that the base of the megaregolith, by definition, lies as deep as the deepest pulverization and fragmentation, reaching into more intact primordial rock layers. According to our scenario of early intense bombardment and accretion tail, the megaregolith reached tens of kilometers depth. Richardson and Abramov (2020) emphasize different layers in the megaregolith, with a similar conclusion. They picture a surface dusty regolith layer typically about 20 m deep, underlain by an \"upper megaregolith\" from ~20 m to a depth of 1 to 3 km, consisting of depositional layers of \"brecciated and/or melted material…characterized by the transport and deposition of material via either transient crater gravitational collapse or impact ejecta.\" Under the \"upper regolith,\" they picture a \"lower megaregolith\" layer, running from ~1-3 km at the top to \"about 20-25 km,\" consisting of bedrock that has been fractured in place. In all such models there is a semi-infinite supply of primordial, lower crustal or upper mantle plutonic rocks below the bottom of the megaregolith. Thus, the largest modern craters, whose transient cavities exceed roughly 20 km depth, excavate samples of primordial crustal or upper mantle rocks and scatter them on the surface. This excavation process is aided by additional aspect of megaregolith evolution: It is heterogeneous in the sense that the biggest basin impacts blow away much of the accumulated megaregolith under their floors, creating \"thin spots.\" Those areas were then coated with lavas, but even modest-sized craters on floors of basins can penetrate the lavas and eject ancient, deepseated rocks. Craters with bright ray systems, associated with Imbrium and postdating mare lava plains, such as Autolycus (diameter D = 81 km) and Aristillus (D = 55 km) (both inside the Imbrium's Apennine ring) are good examples. Copernicus (D = 93 km) and Eratosthenes (D = 59 km), somewhat outside that ring, are also candidates for ejecting early crustal samples.\n\nOnce a rock has been expelled onto the lunar surface, how long does it last? This question was not much considered when the Apollo missions were planned and the first samples were collected, because the Moon was thought to lack erosive processes. However, a number of diverse studies show that lunar rock samples typically have cosmic ray exposure ages of tens of Ma to a few hundred Ma, and meter-scale rocks typically erode due to the \"sandblasting\" flux of small meteorites with a few hundred Ma. Moreover, craters of D <12 km (depth ≲2 km) typically produce rocky ejecta blankets in lunar maria (from lava layers) but not so much in lunar highlands (where the subsurface is mostly pulverized). For discussion of such effects, see Basilevsky et al. 2013;Ghent et al. 2014;Costello et al. 2018;Hartmann 2019. Thus, in collecting lunar rock samples, we are entirely at the mercy of excavation processes in the megaregolith; megaregolith properties \"filter\" what we can collect on the surface.\n\n\nFUTURE WORK\n\nWe regard our current work as a demonstration of concept. In the past there has been some separation of sub-disciplines in planetary science. Research in sample petrology, radiometric dating, impact crater studies, numerical modeling of interplanetary impact fluxes have typically involved different research groups, different meetings, different sessions at large annual meetings, and different journals. One area of fruitful future research may attempt to combine better the insights from (1) dynamical models of accretion, planetesimal collisions, and planetesimal fragmentation events, (2) the resulting geological consequences for lunar and planetary cratering rates as a function of time, (3) studies of the processes of excavation and deposition of impact debris. We note that such work will apply to more or less airless bodies throughout at least the asteroid belt and inner solar system, including Mars. At the same time, we caution that to do more detailed modeling of megaregolith evolution we need better quantitative information on still-vague parameters such as (1) extent and depth of fracturing and pulverization of rock during impacts of various sizes; (2) velocity distributions for the different kinds of meteorites hitting the Moon at any given time; (3) cross-section and volume of transient cavities during impacts of various sizes, especially multi-ring basins; (4) cross-section and depth of impact melt lenses on or under the floors of multi-ring impact basins; (5) deposition depths, compositions, and homogeneity of structure in ejecta blankets as a function or distance from crater center for craters of various sizes (especially looking at sources of ejected KREEP materials and uniqueness of KREEP material to the Imbrium region); (6) degree and causes of cementation of fractured and pulverized megaregolith materials as a function of depth; (7) nature of rocks excavated from the base of the megaregolith by the largest recent craters in terms of our access to the original lunar crust or the upper mantle; (8) survival times of excavated rocks as a function of sizes on the lunar (or planetary) surfaces (i.e., erosion histories of samples accessible on the surface.\n\n\nCONCLUSIONS\n\nThe concepts of \"early intense bombardment\" and the \"accretion tail\" dynamical model (Morbidelli et al. 2018) provide a coherent view of age distributions of radiometric dates of lunar primordial crustal rocks vs. impact melts, GRAIL data on crustal structures, and early evolution of lunar features and multi-ring basin morphologies. They also clarify phenomena not adequately explained by the terminal cataclysm/LHB paradigm. Among the implications of our study:\n\n• The solidification of the magma ocean was affected by extremely high cratering rates during that early era. This prevented \"smooth,\" layered deposits based on Bowen reaction series in at least some areas, and may help explain complex petrology among the most ancient samples from various regions of the present-day Moon. • If a surface formed (or attempted to form) by solidification of a magma ocean during the extreme bombardment as early as >4.4 to 4.35, it would have experienced supersaturation of multi-kilometer diameter crater size-frequency distributions by about 4.35 to 4.3 Ga. • Surviving remnants of such early surfaces, i.e., between the largest impact basins, would have continued to accumulate even more craters, so that early crater densities would have gone into supersaturation, probably by factors at least 2.5 to 10, resulting in pulverization of the earliest impact melts. This result differs from predictions of the terminal cataclysm/LHB models, and helps explain the dearth of impact melt clasts from before ~4.35 Ga. • Continuing bombardment in the first few hundred Myr of crust existence produced reworking and redistribution of materials in upper megaregolith layers, consistent with GRAIL measurements of low density and high porosity in the upper kilometers, Richardson and Abramov's (2020) modeling, and age distribution of earliest known impact melts. • The surface today would display localized regions destroyed and resurfaced by multi-ring basin-scale impacts, typically filled with basaltic lavas, all in various stages of degradation by subsequent cratering. This scenario is consistent with the presence of the large, very degraded South Polar Aitken basin, and with Oceanus Procellarum being a very early giant impact feature. • Subsurface linear features comprising a \"ring\" around Oceanus Procellarum are not a compelling argument against Procellarum being an impact feature. • The oldest lava-plain formations, for example, in South Polar Aiken, are covered by a veneer of high-albedo ejecta and overlapping rays, and were often mapped in the pre-Apollo and early post-Apollo eras as \"highlands.\" They may, however, have quite different megaregolith composition (and other properties) than highland regions that survive from earliest crustal formation • The supersaturation-pulverization of the upper kilometers would have comminuted the impact melt lenses produced in the surface layers of the early impact basins, explaining the presence of impact melts from ~ 4.1 to 4.35 Ga as fine-scale clasts in lunar breccias. • Surviving surfaces formed around 4.1-4.2 Ga have reached at least 1.0× saturation crater density.\n\n• Surfaces formed after about 4.0-3.9 Ga are not quite saturated with craters, and experienced much lower regolith production rates than earlier surfaces. Survival times of rocks in surface layers rapidly increase at this time. This explains the survival of many impact melt rock samples after ~4.0-~3.9 Ga as contrasted with few before that time. • The blasting away of megaregolith layers by a few large, late basin scale impacts (especially Imbrium) created thin spots in the megaregolith where recent Copernicusscale and Autolycus-scale craters can eject primordial crustal rocks from the base of the local megaregolith, which in many instances is covered by mare basalt layers. This explains the presence of primordial plutonic crustal rocks in present-day surface sample collections (in spite of the paucity of impact melt rocks before ~4.0 Ga). • Many concepts in this paper, regarding megaregolith evolution and sample collection apply not only to the Moon but to all relatively airless planetary bodies in the inner solar system (Mercury, moons, asteroids, and even Mars), and at least into the main asteroid belt, and probably beyond, depending on the histories of asteroid/comet impact rates in the outer solar system. • We are still in an era where first-significant-figure issues remain to be considered, even while assessing certain processes with numerical models citing two-and three-significantfigures.\n\n\nmodel. See text. **The accumulated crater numbers in the last bins of the table should probably be higher. This involves intersection of two different processes of scattering interplanetary bodies in the Morbidelli et al. (2018) model, around 3 Ga. (See text for discussion.)\n\nFigure 3\n3shows a plot of the data in\n\nFigure 3 .\n3A plot of data from\n\nFigure 4 .\n4Estimated areas covered by craters larger than given diameters (indicated by vertical \"%\" figures. Fine-dashed SFD curves are isochrons showing observed crater SFDs as inFig. 2a. Solid saturation line shows maximum observable crater densities reached on airless planetary bodies, based on empirical measurements. Note that when a given D bin reaches the saturation line it does not mean that craters larger than that D cover 100% of the Moon (see text). Heavy dotted curve in each figure shows actual number of impacts and craters formed, eventually rising above the saturation for small enough craters and large enough ages. Along the heavy dotted curves, calculations are made of cumulative area covered by craters larger than D, starting at lower right corner. (a): Isochron diagram for area just reaching saturation.\n\nFigure 5 .\n5Orientale basin, in an orbital photo (a) and a topographic map (b) showing linear segments in the surrounding rings. Three linear segments are pronounced along the north (top) third of the outer (930 km diameter) ring, especially at top of the topographic map. The outer ring and next inner ring are formed by linear segments on ENE side. Note in 5(a) both of those segments have a narrow strip of dark lava at the inner foot of the scarp, suggesting a fault plane that allowed magma to intrude along the fault and then extrude onto the surface. An outer ring of diameter 1300 km has also been mapped, but is not easily visible here. (a) NASA Lunar Reconnaissance Orbiter Camera WAC image [NASA/Goddard Space Flight Center]. (b) Laser altimeter \"LOLA\" topographic map from NASA/Goddard Space Flight Center.)\n\nFigure 7 .\n7Two views of south rim of Serenitatis basin (rugged terrain from bottom to top along center of image), showing degradation by linear structural features radial to Imbrium basin (centered ~270 km off the bottom of the image; north at left). The pattern of ridges radial to Imbrium extends as far as the Apollo 16 site, where Imbrium-age material was collected. Bottom to top length of each image ~350 km. (Photos by WKH, during Lunar Rectified Atlas project ca. 1962, University of Arizona, Lunar and Planetary Lab.)\n\nTable 1 .\n1Cratering rate relative to present impact rate based on Morbidelli et al.(2018) \n\n\nTable 2 .\n2Calculation of early crater accumulation. (Deriving the accumulated number of craters/km 2 starting at specified date, expressed in terms of average mare density* = 1.)Age \n(Ga) \n\n∆N(1) \n1 km \ncraters \nformed \nper km 2 \nduring \n100 Myr \ninterval \nspecified \nin column \n1 \n\n∆N(1) \nrelative \nto \naverage \nmare, \nwhere \nN(1) \ncrater \ndensity \n[= 3.2 \n(10 -3 )]* \n\nStart at \n4.55 Ga \n(if Moon \nexisted) \n\nStart \nat \n4.45 \nGa \n\nStart \nat \n4.35 \nGa \n\nStart \nat \n4.25 \nGa \n\nStart \nat \n4.05 \nGa \n\nStart \nat \n3.85 \nGa \n\nStart \nat \n3.65 \nGa \n\nStart \nat \n3.45 \nGa \n\n4.55 \n0 \n4.5 \n31.7 \n9906 \n4.45 \n9906 \n0 \n4.4 \n1.21 \n378 \n4.35 \n10280 \n378 \n0 \n4.3 \n0.33 \n103 \n4.25 \n10380 \n481 103 \n0 \n4.2 \n0.12 \n37 \n4.15 \n10420 \n518 140 \n37 \n4.1 \n0.052 \n16.1 \n4.05 \n10440 \n534 156 \n53 \n0 \n4.0 \n0.024 \n7.62 \n3.95 \n10440 \n541 164 \n61 \n7.62 \n3.9 \n0.013 \n4.13 \n3.85 \n545 168 \n65 \n11.8 \n0 \n3.8 \n6.2 (-3) \n1.94 \n\n3.75 \n170 \n67 \n13.8 1.94 \n3.7 3.8 (-3) \n1.19 \n3.65 \n* \n68 \n14.8 3.13 0 \n3.6 2.3 (-3) \n0.718 \n\n\nTable 3 .\n3Cumulative area covered by craters larger than diameter D among crater populations reaching the saturation equilibrium line, and also reaching 2.0× and 2.5× saturation equilibrium levels. Bold face type in columns 3, 5, and 6 emphasize conditions where ~100% and ~200% of area are covered by craters.Crater \ndiameter D \n(km) \nmarking \nboundaries \nof \ndiameter \nbins \n\nPercentage \narea \ncovered by \ncraters \nwithin \nspecified bin \nwhen crater \nsaturation is \nreached at \nD>1.4 km \n\nCumulative % \nof area in bin \ncovered by \nall craters of \ndiameter >D \nwhen \nsaturation is \nreached \n(according to \ndiscussion in \ntext, the \nyoungest \nexamples of \nsaturated \nsurfaces \nformed at 4.1 \nto 4.2 Ga) \n\nPercentage \narea of \ncraters/km 2 \nin bin when \n2× crater \nsaturation is \nreached at \nD>1.4 km \n\nCumulative \n% of area \ncovered by \ncraters of \ndiameter >D \nat 2× \nsaturation \n(according \nto Fig. 3, the \nyoungest \nexamples \noccur on \nsurfaces \nformed at \n~4.3 Ga) \n\nCumulative % \nof area \ncovered by \ncraters of \ndiameter >D \nat 2.5× \nsaturation \n(according to \nFig. 3, the \nyoungest \nexamples \nwould be >4.3 \nGa) \n\n0.25 \n\n48% \n227% \n96% \n456% \n567% \n\n0.353 \n\n30% \n179% \n60% \n360% \n447% \n\n0.5 \n\n17% \n149% \n34% \n300% \n372% \n\n0.707 \n\n13% \n132% \n26% \n266% \n330% \n\n1 \n\n6% \n119% \n12% \n240% \n297% \n\n1.41 \n\nACKNOWLEDGMENTSThis paper was written mostly during visiting scientist invitations at the International Space Science Institute in Bern, Switzerland; thanks to ISSI for hospitality. 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Nature 571:226-229.\n"},"corpus_id":{"kind":"number","value":225075660,"string":"225,075,660"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2010.14275\",\n \"mag\": 3111441048,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1111/maps.13595\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/2010.14275v1.pdf\"\n ],\n \"pdf_sha\": \"b1fb818a282f71f8266eb8c1560ecc52345f3150\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/2010.14275\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"EFFECTS OF EARLY INTENSE BOMBARDMENT ON MEGAREGOLITH EVOLUTION AND ON LUNAR (AND PLANETARY) SURFACE SAMPLES"},"authors":{"kind":"list like","value":["William K Hartmann \nPlanetary Science Institute\nTucsonArizona","Alessandro Morbidelli \nUniversité Côte d'Azur\nCNRS-Lagrange Observatoiré de la Côte d'Azur\n34229-F 06304, Cedex 4NiceCSFrance"],"string":"[\n \"William K Hartmann \\nPlanetary Science Institute\\nTucsonArizona\",\n \"Alessandro Morbidelli \\nUniversité Côte d'Azur\\nCNRS-Lagrange Observatoiré de la Côte d'Azur\\n34229-F 06304, Cedex 4NiceCSFrance\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Impact rates in the first 500 Myr of the solar system are critical to an understanding of lunar geological history, but they have been controversial. The widely accepted, post-Apollo paradigm of early lunar impact cratering (ca. 1975-2014) proposed very low or negligible impact cratering in the period from accretion (>4.4 Ga) to ~4.0 Ga ago, followed by a ~170-million-yearlong spike of cataclysmic cratering, during which most prominent multi-ring impact basins formed at age ~3.9 Ga. More recent dynamical models suggest very early intense impact rates, declining throughout the period from accretion until an age of ~3.0 Ga. These models remove the basin-forming spike. This shift has important consequences vis-à-vis megaregolith evolution and properties of rock samples that can be collected on the lunar surface today. We adopt the Morbidelli et al.(2018)\"accretion tail\" model of early intense bombardment, declining as a function of time. We find effects differing from the previous models: early crater saturation and supersaturation; disturbance of magma ocean solidification; deep early megaregolith; and erosive destruction of the earliest multi-ring basins, their impact melts, and their ejecta blankets. Our results explain observations such as differences in numbers of early lunar impact melts vs. numbers of early igneous crustal rocks, highland breccias containing impact melts as old as 4.35 Ga, absence of a 170 Myr-long spike in impact melt ages at 3.9 Ga among lunar and asteroidal meteorites, and GRAIL observations of lunar crustal structure."},"sections":{"kind":"list like","value":[{"title":"ANALYSIS AND APPLICATION OF ACCRETION TAIL SCENARIO","content":""},{"title":"VIS-À-VIS IMPACT RATE VS. TIME","content":"As pointed out by Morbidelli et al. (2018), the \"accretion tail\" dynamical model creates a curve of declining impact rate vs. time, similar to -but independent of -early pre-and post-Apollo suggestions about the nature of that curve, based mainly on cratering statistics. Hartmann (1965Hartmann ( , 1966 estimated the characteristic age of lunar mare lava plains at ~3.6 Ga, using cratering rates derived from the Canadian shield. The second paper pointed out that since the pre-mare lunar highlands had 32 × the crater density of the maria, the brief pre-3.6 Ga cratering rate had to average at least ~160× the post-mare crater rate. The uncertainty in that number was estimated to be a factor 2. The number 160 was considered a lower limit, because the factor 32 crater density was found on a number of solar system surfaces, indicating that it is represents the crater saturation equilibrium level found on many bodies (Hartmann 1984). Craters formed in excess of the saturation level are destroyed by later cratering, and this means that the total impact crater numbers in the highlands may be much larger than what we can see today. The saturation equilibrium level is important in the present paper because, as we will show, the total number of impacts during very early intense bombardment, based on Morbidelli et al. (2018), was well beyond the number required to reach saturation. Thus, the saturation curve found in most areas today does not represent the actual number of craters that were formed, but only a lower limit. Once saturation was exceeded on a surface, we lose information on the impact numbers and the shape of the earlier size-frequency distribution curve.\nAfter investigators dated Apollo mission samples collected from various landing sites, crater density measurements as a function of measured surface age indicated that as we go backward in time from ages ~3.2 to ~3.8 Ga, the cratering rate reached a value of order ≥200 times the current rate (e.g., Hartmann 1972;Neukum 1983;Neukum et al. 2001). In the \"accretion tail\" scenario, this decline in cratering can be seen as the tail end of the more or less monotonic decline in impact rate from >4.4 Ga ago to 3.8 Ga ago, when the cratering record does not reach saturation, and is more clearly measurable. We stress that the time evolution of the impact rate in this paper not a backward extrapolation of the observed impact rate evolution observed in the 3.2-3.8 Ga interval, but is the result of the numerical modelling of the dynamical evolution of the planetesimals remaining in the terrestrial planet region at the end of the planets' formation process (Morbidelli et al. 2018). Table 1 shows an application of this model. Column 1 lists dates at 100 Myr intervals. Columns 2 and 3 illustrate two ways of normalizing the bombardment model data to give cratering rates relative to the modern cratering rate (defined as 1.0). In column 2 (model A) we normalize to an average value of 1.0 for the last 400 Ma (averaging over the Morbidelli et al. 2018 model values for the last four 100-Ma bins, including T=0=today). Column 3 shows a second, different normalization of the curve (model B). It applies suggestions that the cratering rate at 3.0 Ga was probably several times that observed today, and was still declining from 3.0 Ga till today (Quantin et al. 2004Hartmann et al. 2007). Thus, column 3 normalizes the data to a value of 3.0 at 3.0 Ga. The two normalizations give similar results, as plotted in Fig. 1.\nIn this context, it is important to note that the Morbidelli et al. (2018) data set, in order to extend the timeline from 3.0 Ga until today, involves a conceptual discontinuity at ~3.0 Ga ago. At age >3 Ga it follows the calculated decay of asteroids and leftover planetesimals assuming their dynamics were dominated by planetary perturbations (the main subject of Morbidelli et al. 2018). For age <3 Ga, the model shifts to dominance of near-Earth asteroids escaping from the asteroid belt under Yarkovsky forces, giving a relatively constant bombardment in Table 1 (similar to the normalization in column 2 of Table 1). In reality (especially with an asteroid belt a bit more massive 3 Ga ago than now, giving more NEAs) the two curves would merge more smoothly. For this reason, Morbidelli et al. (2018, their Fig. 5) originally plotted their results from 4.5 Ga only to 3.0 Ga ago, but here we use additional data from their model to extend the curve.\nThe fourth column in Table 1 shows a few examples of earlier cratering rate estimates (still normalized to current rate = 1) based on extrapolation of crater count data backward to ~4.3 or 4.4 Ga, assuming the early estimates of decline in planetesimal numbers (Hartmann 1972;Neukum et al. 2001;Quantin et al. 2007). In Table 1, fluctuations in cratering rate and half-lives of asteroids against collisions with other bodies, at ages around 3.0 Ga may be due mostly to the merger of the two models. The final (fifth) column of Table 1 lists some estimates of half-lives of the projectile population published during earlier crater-count studies (Hartmann 1972;Neukum et al. 2001;Quantin et al. 2007). We note that these are similar to the results from Morbidelli et al. (2018). Figure 1 shows the Morbidelli at al. (2018) cratering rate curve as a function of time (on a logarithmic scale), using the two normalization methods from Table 1, allowing either no decline since 3.0 Ga or decline by a factor 3. We infer that the most likely cratering rate behavior occurs between these two curves, but with a more gradual transition at about 3.5 to 2.5 Ga, ending (by definition) with today's rate of 1.0.\nTo summarize, the \"accretion tail\" scenario favored by Morbidelli et al. (2018) indicates that the early intense bombardment rates in the first few hundred million years (surface formation until ~4.15 Ga) exceeded 1000 times the present rate. At dates before 4.4 Ga, it may have exceeded 10,000 the current rate, although, as we probe back toward the beginning, the lunar surface may have been in magma-ocean state. Note the contrast with the \"classic\" terminal cataclysm paradigm, which called for more or less negligible cratering in the ~4.4 Ga to ~4.0 age interval, with few if any of the major front-side multi-ring basins forming in that period. Hartmann 1972, Fig. 3, p. 55;Q = Quantin et al. 2007). Note: Morbidelli et al. (2018) are given precedence in this paper over the earlier estimates, which are cited merely to demonstrate similarity with early post-Apollo models based on crater statistics.\nAs Morbidelli et al. (2018) reported, the crater count results (and their extrapolation to before 4.0 Ga) come strikingly close to the results from the Apollo-era models in both shape and absolute numbers. For example, Hartmann (1972), using early Apollo sample data stated that prior to 4.1 [Ga] ago, the cratering rate on the Moon was at least 10 3 times the present rate and the rate declined with a half-life less than 8 × 10 7 yr. During the interval from 4.1 to 3.2 [Ga] ago, the number of planetesimals showed an exponential decay with a half-life about 3 × 10 8 yr…. A more constant cratering rate applied in the last three aeons. Neukum (1983, as reviewed in Neukum et al. 2001) developed an analytic interpretation of the crater-count data, an equation fitting their observations of crater cumulative crater density as a function of time. The differential values of that equation (i.e., difference in craters/km 2 in a specified size range from beginning to end of 100 Myr intervals), give the crater formation rate (i.e., craters/km 2 in a specified size range) formed per 100 Myr interval). As an example of their results, Neukum et al. (2001) stated that \"…the cratering rate 4 Gyr ago was 500 times higher than the constant rate during the last 3 Gyr,\" a statement close to the curves in our Fig. 1."},{"title":"Figure 1.","content":"Curve showing cratering rate as a function of time (relative to today's cratering rate) derived from Morbidelli et al. (2018) accretion tail model. Lower curve assumes cratering rate asymptotically arrives at today's rate (i.e., 10 • ). Upper curve assumes cratering rate at 3.0 Ga equals 3 × today's rate. \"N\" symbols are independent data from crater counts, taken from equation fitted by Neukum to crater count data (see text for further discussion).\nThe impact rate data points derived from the Neukum equation, based entirely on crater counts (not on a dynamical model) are included with symbol \"N\" for comparison in our Fig. 1. The agreement between the dynamical models A and B, and the cratering rate analysis, is remarkable for three distinct reasons. (1) The research teams are entirely independent. The Morbidelli et al. (2018) team made no attempt to match their dynamical model to the earlier crater chronometry results. (2) The techniques used are entirely independent; mathematical results from orbital dynamical calculations are unrelated in practice to counts of lunar craters.\n(3) Perhaps most striking, Morbidelli et al. (2018) model predictions match not only the bestestablished parts of the Neukum curve at ages ≲3.8 Ga, but also an extrapolation of that curve to earlier dates. To repeat, the best data used for the Neukum curve come from dated samples returned from Apollo and Luna landing sites, but surfaces for crater counts dating from before ~3.8 Ga (a) are less well documented (if available at all), and (b) have crater densities near the saturation equilibrium line so that super-saturation densities on older surface cannot be directly measured. All the more noteworthy, then, that the Neukum equation for cratering rates fits so well the results of the dynamical model.\nThe conclusion of this section is that the accretion tail model, along with certain Apollo era suggestions of early intense cratering, imply that the cratering rate prior to 3.8 or 3.9 Ga likely exceeded hundreds and even thousands of times the rate from the last 1-3 Gyr (and could plausibly have exceeded ten thousand times higher than it is today, prior to ~4.35 Ga), as shown in Table 1, columns 2 and 3, from the Morbidelli et al. (2018) data. This is contrary to \"classic\" terminal cataclysm/LHB paradigm, where cratering rates prior to 3.9-4.0 Ga were considered to be comparable to the present rate. It is significant that, half a century after the Apollo and Luna samples were returned, the accuracy of first significant figures is an issue in understanding important lunar phenomena."},{"title":"DEVELOPING TOOLS TO STUDY EFFECTS OF INTENSE EARLY BOMBARDMENT","content":"A useful tool in understanding the lunar cratering record is the \"isochron diagram,\" which shows the observed size-frequency distribution (SFD) of impact craters as a function of surface age. In the template version, shown here in Fig. 2a, the crater density (craters/km 2 ) is shown incrementally, for each individual size bin, and the size bins are plotted on a log√2 scale, for example from 1.0 km to 1.4 km, from 1.4 km to 2 km, and so on. This plot has an advantage over widely used cumulative plots, because it shows exactly what is happening in each individual size bin, rather than smoothing the curve's structure over many sizes, as cumulative curves do.\nThe isochron diagram has a number of additional useful features. The upper diagonal boundary, marked \"Saturation\" shows an empirical curve (mentioned earlier), which has been repeatedly observed in the most heavily cratered surfaces on the Moon, various asteroids, and outer planet satellites throughout the solar system (Hartmann 1984). The observed saturation curve is not a \"geometric\" saturation, since more small craters could, geometrically, be squeezed into existing empty spaces on these surfaces. However, nature does not allow the addition of craters in only a narrow size range; the whole SFD must be added all at once, which requires occasional, large, basin-scale impacts, whose excavation and ejecta blankets completely resurface some areas. Furthermore, \"equilibrium\" connotes stability, whereas empirical saturation in a given region does not represent a static situation. Not only do the distant (often \"off-stage\") basins' ejecta occasionally wipe out shallower craters (at, say, D <100 m diameter, depending on distance and size of the basin), but also the continual rain of meteorites creates craters smaller than any specified diameter D, and erodes rims of craters larger than size D by a simple sandblasting effect. An inexorably increasing supersaturation would produce many big, deep impact craters (some no longer visible) and thus pulverize increasingly deep layers, including the earliest impact melt lenses formed in the near-surface kilometers under the earliest large impact basin floors. Thus, different parts of the saturation line can thus oscillate over time by factors about 2 to 4 (Hartmann and Gaskell 1997). The dashed lines on either side of the saturation curve thus show the typical range of oscillation of visible saturation crater densities. The important larger lesson is that a surface characterized by the saturation line may have been impacted far more times than is evident from the numbers of visible craters, preventing a clear measurement of the SFD of the earliest impactors.\n(a) (b) Figure 2. (a) Template for lunar isochron diagram. Each curve shows the size-frequency distribution of craters observed on lunar surfaces of specified ages. The diagram assumes constant crater production rate going back in time until about 3 Ga but higher rates before that, as based on Apollo data (Hartmann 1970a(Hartmann ,b, 1972Neukum et al. 2001). (b) Crater accumulation diagram under conditions of impact rate 10,000 times the current rate, as suggested by Morbidelli et al. (2018) for the period around 4.35 Ga. Given the half-life estimates in Table 2, a rate of ~ 10,000× current rate lasts only tens of millions of years, so the graph is extended only to 10 Myr (see text for further discussion). Because crater chronometry is constantly improving with new data, fine print above diagrams refers to date of our latest iteration of the lunar isochron plot, and the date of our data set. Now suppose the cratering rate ~4.35 Ga was ~10,000 times the present rate, as the accretion tail model indicates for an interval of about 50-100 Myr, centered on that date (see Table 1 and Fig. 1). To create an isochron diagram to match conditions under that elevated impact rate, we simply relabel the isochrons, stopping at 10 Myr, because the half-life of decline in cratering rate at 4.35 would have been on the order of 50-60 Myr, according to Table 1. The result is shown in Fig. 2b. The diagram suggests that any surface formed during cratering at that rate would have reached supersaturation equilibrium conditions on a timescale ~50 Ma, and we can surmise that multi-ring basins of D >300 km would have been forming in stochastic intervals averaging 50-100 Myr. If the cratering rates were higher than 10,000 times the present rate before 4.35 Ga (as indicated in Table 1 and Fig. 1), and if the surface were solid, supersaturation would have been reached in <50 Ma."},{"title":"CRATER ACCUMULATION ON SURFACES FORMED AT DIFFERENT TIMES","content":"As suggested by Fig. 2b and the discussion in the last section, the accretion tail model offers an approach to understanding how the earliest cratering conditions affected the early lunar crust and megaregolith, and even the kinds of samples picked up on the surface today. This investigation is detailed in Table 2. 0.78 ** *Average mare N(1) density is assumed to be 3.2(10 -3 )craters/km 2 , larger than 1 km, based on Hartmann (1966Hartmann ( , 2005 crater counts on a variety of lunar mare surfaces, which were found to have virtually the same crater density, independent of the Morbidelli et al. (2018) In Table 2, column 1 defines 100 Myr-long time bins, and lists dates of surface formation. In column 2, N(1) refers to the cumulative number of craters/km 2 that have diameter >1 km. N(1) is often used as a proxy for the entire size-frequency distribution curve for all impact craters, from microscopic pits to 1000-km multi-ring basin structures. We adopt it as a first-order proxy, but we recommend caution, because the N(1) values fall just in the diameter zone where the -2 slope SFD of larger craters intersects the steeper -3 to -4 slope at smaller diameters, as shown in Fig. 2. Our work concentrates on the rate of change in the entire crater population as a function of time, but N(1) still has first-order value in our discussions. Thus, in column 2, ∆N(1) values from the accretion tail model are defined as the total number of craters with diameter D >1 km added per km 2 in the 100 Ma intervals shown in column 1. For example, from 3.05 to 2.95 Ga, centered on 3.0 Ga, the model shows ∆N(1) addition of ~2.3 (10 -4 ) craters/km 2 larger than 1 km.\nColumn 3 repeats column 2 by listing the ∆N(1) number of craters in terms of average mare crater density. This gives a more direct connection to the appearance of the lunar surface, making the table more accessible to readers not familiar with N(1). Entries of ~1.0 in column 2 thus represent a surface with crater densities typical of average lunar mare on the front side of the Moon, based partly on data from the University of Arizona Lunar and Planetary Laboratory crater catalogs (for a summary of those catalogs, see Wood 1972) and additional counts over the years, by Hartmann, treating the mare surface as one giant mare. The mare surfaces in the major basins have remarkably similar densities, and \"average lunar mare,\" as mentioned in a footnote to Table 2, thus refers to a summation of counts in each diameter bin from surfaces of various major basins, divided by the total area counted in each bin.\nThe remaining columns, 4 to 11, give crater densities relative to average mare crater density, as they accumulate on surfaces of eight difference starting dates. Thus for example, if we look at a surface (perhaps a lava flow) formed at 4.35 Ga (column 6), it has 0 crater density at 4.35, but adds ∆N(1) of 103 craters/km 2 by 4.25 Ga. And so on, down the table until we see it leveling off at roughly 170 craters/km 2 larger than 1 km, which would be observed today. In columns 4 to 11, entries of ~32 represent lunar highland surfaces, matching saturation equilibrium as measured by Hartmann (1984). Table 2. Three vertical scales have been added to the diagram to demonstrate its range of important information. At the left side we see the crater densities in terms of average lunar mare and also in terms of N(1) as used in Neukum (1983) cumulative crater densities. At the right side we see a listing of the estimated megaregolith depths, which are a direct function of accumulated crater densities, as will be discussed below. The shallower depth estimates come from Apollo observations and GRAIL data (Wieczorek et al. 2013), and the deeper depths, in the regions dating to the first 500 Myr, come from our calculations. Figure 3 also contains three horizontal shaded bands, marking average lunar mare crater densities, saturation density (32× average mare density), and 10× saturation density. The vertical shaded band indicates a plausible time span in which a putative magma ocean might have solidified, i.e., forming a stable solid surface; accumulation of craters on solid surfaces would occur only after that. Table 2, showing accumulation of crater density as a function of time, during the putative early intense bombardment. Bottom of each curve (on abscissa) shows formation dates for surfaces, and the curves show subsequent crater accumulation on those surfaces. Data were initially plotted at 100 Ma intervals, leading to a sharp kink in each upper curve, ~100 Ma after the start date, due to short half-life of early impact rates; those kinks have been smoothed out by adding artificial data points to indicate a more realistic transition. (See text for discussion of the shaded bands and the three scales provided for the ordinate axis on left and right sides. See Table 2 footnote for definition of \"average mare.\" The curve for 4.55 Ga is included to indicate the extreme impact rate if solid bodies existed at that time; the Moon probably did not yet exist.) Figure 3 gives insights into a number of lunar properties and aspects of lunar history. For example:\n• Crater SFD numbers formed on the earliest inner solar system surfaces, as represented by the 4.55 Ga and 4.45 Ga curves, are no longer measurable on surfaces today, because of extreme oversaturation. The earliest craters have been overlapped by later large craters, multi-ring basin impact structures, and basaltic mare lava flows.\n• Surfaces formed prior to ~4.2 to ~4.3 Ga were intensely bombarded and super-saturated with craters, due to the very high early cratering rates, but are getting closer to saturation levels. Near saturation levels affect the survival of impact melts from upper kilometers that time.\n• The youngest crater-saturated surfaces visible today should date from ~4.1 to ~4.2 Ga, consistent with Fig. 2a. The reduced saturation effects by this time allow for increasing survival of brecciated materials and impact melt inclusions from that time.\n• Crater densities matching average mare surface appear around 3.5 to 3.6 as predicted from pre-Apollo crater counts and supported with dated Apollo samples."},{"title":"INTENSE EARLY BOMBARDMENT: CONSEQUENCES FOR MEGAREGOLITH EVOLUTION AND MEGAREGOLITH DEPTHS","content":"Applying the information discussed so far, we investigate the evolution of megaregolith in the first 600 Myr of lunar history. Figure 4 introduces a technique for using the SFD curve of crater populations to calculate depth of pulverization, based on an earlier technique utilizing the isochron diagram (Hartmann 1980(Hartmann , 2005. The technique works for under-saturated, saturated, and over-saturated surfaces. For various SFD crater densities formed on such surfaces, we calculate the area covered by craters (per km 2 ) in each diameter bin, then add up the cumulative area starting with the largest bin and working toward smaller bins. As shown in Fig. 4, at each diameter D, this gives the total area covered by craters larger than D (and thus ejecting material deeper than some effective depth d excavated by such craters). At some sizes we reach a point size where the total area of larger craters is 100% of the area available. At 100% coverage, however, not all areas have been impacted and excavated, because of crater overlap. Thus, the 200% parameter is included to allow more realistically for such overlap. We will define our quantitative definition of d in a moment. We assume, as a first-order approximation, that when 100% to 200% of the surface is covered by craters larger than critical diameters designated as \"D100%\" and \"D200%,\" respectively, then a characteristic depth of megaregolith is at least of the order ~d100% to ~d200%. The effective depth at any diameter D must be less than the deepest part of the transient cavity, but deeper than the observed floor depth in the existing crater. Here we assume also that nearly all of the \"effectively\" excavated material is distributed elsewhere on the Moon. The depths affected by craters exceeding D100% and D200% are somewhat in question, since the transient cavity depths near the center of a crater are considerably greater than the final crater depths. What, then, is the depth actually affected by saturation coverage among craters of these sizes? Taking into account the effects mentioned so far, we assume conservatively that craters of the relevant diameters (some kilometers) churned and pulverized materials to effective depths equaling 1/3 of their diameter. The depth of megaregolith, while averaging around the order of magnitude of ~d100% to ~d200%, will vary from region to region, because stochastically-produced largest basin-forming impacts will blow away early surfaces and create new, localized, surfaces -which then begin to accumulate mare lava flows, new impacts, and new regolith. For example, the large, relatively late Imbrium basin must have blown away most earlier megaregolith (not to mention overlapping, obliterating, and ejecting material from earlier large craters and possible earlier, small multi-ring basins with their impact melts). At the same time, Imbrium's ejecta delivered such earlier megaregolith materials onto other regions (McGetchin et al. 1973;Haskin 1998;Hartmann 2003;Petro and Pieters 2008;Norman and Nemchin 2012;Norman et al. 2015). The giant basins thus create relatively thin spots in the megaregolith cover (possibly zero kilometers immediately after impact in some cases), while depositing the previous megaregolith material in a surrounding ejecta blanket. Another example of these effects may be present at South Pole-Aitken (SP-A) basin. Using GRAIL results, Besserer et al. (2014) state that \"The SP-A region appears to possess a thinner low-density (porous) layer than the rest of the farside.\" This result suggests that that SP-A blew away a fair amount of megaregolith which had formed before the SP-A impact. Even cursory inspection of lunar photo images reveals that SP-A has a higher crater density and higher albedo than the major front-side maria. It has evidence of mare basaltic lava deposits, but those mare surfaces themselves have been heavily cratered, with the lava apparently pulverized and brightened by ray deposits. These statements suggest that SP-A and its lava flooding date from before most of better-preserved mare lava production dating from ~3.8 Ga to ~3.2 Ga, but nonetheless that a substantial thickness of megaregolith existed before the impact, to be blown away. This is consistent with our model, but inconsistent with the LHB paradigm, with its sparse cratering before 4.0 Ga. Here, 100% of area is covered by craters larger than ~4 km, and 200% by craters larger than ~350 m. (b) The same but for a surface that is supersaturated by a factor 2. In this case, 100% of area is covered by craters larger than ~60 km and 200% covered by craters larger than ~5 km. The diagram illustrates how the sizes of craters excavating 100% and 200% of total area increases dramatically as super-saturation is reached. Table 3 and Fig. 4 illustrate the technique described here. We apply the technique starting with the saturation equilibrium curve, because it reflects the highest observable crater densities and thus suggests a minimal megaregolith depth in lunar highland saturated regions. In Table 3, column 1 shows crater diameter and column 2 shows the calculated fractional area per km 2 covered by observed craters in each bin, as follows:\nFractional area covered in each bin = π(D/2) 2 N = 0.785ND 2 (1) where D = crater diameter near mid-point in each bin N = no. craters/km 2 observed in each bin This fractional area is then expressed as a percentage of the area in column 2 of Table 3.\nThe formula does not take into account that some craters overlap earlier craters. This is a valid approach if we are interested in the total volume being excavated and ejected by observable craters. But if we are interested in the total surface area that has been excavated, the 200% figure is better, since most of the area has been impacted at least once by craters larger than D200%.\nColumn 3 in Table 3 shows the cumulative percentage of area covered by all craters larger than D, starting with the largest craters at the bottom of the table. At D >128 km there are poorer statistics because of the smaller number of large craters and basins, and the isochron curves are not plotted at still larger sizes for this reason. Among the giant multi-ring basins at the largestdiameter end of the size-frequency distribution (SFD), at D >~360 km, there appears to be a continued turn-down at largest sizes, and we have added modest provisional percentage numbers in column 3 at D >360 km. The areas covered, however, are not large enough to affect our conclusions significantly. Columns 4, 5, and 6 add the same results for 2× saturation and 2.5× saturation In Table 3, the important transitions at 100% and 200% coverage by impact craters are shown in bold face in columns 3, 5, and 6. In this calculation, at D <1.4 km, we follow the wellknown steep branch of the SFD upward into the region above the saturation line, not stopping at the saturation curve per se. This is in order to include the effects of all impacts called for by the SFD at small sizes, whether we can see the resulting craters or not. This steep branch (and presumably the larger craters of the large-D branch at D >1.4 km), include an uncertain fraction of secondary craters, which add to the churning and pulverization involved in megaregolith evolution. [Our system of crater counts and crater chronometry includes all visible craters (excluding obvious clusters with rays), without trying to distinguish primaries from the remaining \"field secondaries,\" since attempts to distinguish them visually are likely to be in error. This system is practical, since nature can't produce large craters without producing secondaries, and can't produce secondaries without primaries, and on old surfaces with high crater densities, the addition of unrecognized scattered field secondaries in a single shower from a fresh distant primary will not have a drastic effect on the total crater density.] Note that, as seen in Fig. 4a and mentioned above, the small-D end of the steep branch of the SFD is first to hit the saturation line. What appears to be sparse cratering as in lunar maria at large D thus produces supersaturation at small D creating a shallow regolith of roughly 5 to 15 meters depth, as detected in Apollo landings on lunar maria. But once saturation is approached at D ≳1.4 km, as indicated in Fig. 4a, the saturation line is reached virtually simultaneously at all larger crater diameters, causing an explosive growth in megaregolith.\nHere we pause to ask whether these last statements apply to the earliest crater populations in the first few hundred million years. The problem is that if the early intense bombardment/accretion tail scenario is correct, we must ask if the SFD of crater populations at that time was the same as that observed in post-mare time, when the observed crater density is low, so that the SFD can be accurately measured. On earlier surfaces, dating from ≳4.2-4.3 Ga, the SFD is uncertain because supersaturation occurred, and details of the SFD shape are masked by supersaturation. However, Delbo et al. (2017) found that the collisional evolution of the asteroidal population (which produces the observed size distribution of asteroids) was already well developed prior to the dynamical instability among the giant planets, and possibly by the time of the formation of the Moon itself -and they concluded that its SFD at that time was already consistent with that of current asteroids (which, in turn, is at least roughly consistent with current craters). We thus see no firm observational or theoretical evidence that the slope of the SFD among the larger impactors in impactor populations before ~4.1-4.2 Ga departed from what we see recorded on unsaturated lunar surfaces today. Changes in typical velocity of craterforming impactors are not expected to have an overwhelming effect on the shape of the SFD. The main point here is that the near-simultaneous approach to saturation by all craters of D ≳1.4 km caused a rapid transition from shallow regolith to deep megaregolith. What are the consequences of saturation and super-saturation for megaregolith evolution? At saturation level, as shown in Fig. 4a and Table 3, column 3, we see that D200% is about 350 m, and D100% is about 4 km. This would suggest effective regolith depths (allowing for the moment two significant figures) in the range of d ~120 m to ~1.3 km -somewhat deeper in some areas, and shallower in other areas, since craters larger than 4 km have more stochastic effects. These estimated regolith depths correspond to surfaces observed today but formed ~4.1 to ~4.2 Ga. Now consider still older surfaces that are supersaturated with craters. Although the supersaturation cratering SFDs are hidden from the view of modern observers, Fig. 4b and Table 3 show how the accretion tail model would produce at least twice the level of the saturation curve on surfaces that formed as early as ~4.2-4.3 Ga -a date from which we have impact melt samples. In this case, as shown in column 4 of Table 3, craters of D >60 km would have covered 100% the surface, and craters of D >5 km would have covered 200% of the surface. These figures suggest that the megaregolith depth produced by 2× saturation would be (by our calculation) in the range of ~2 km to ~20 km. Note the explosive regolith growth within an interval of only ~100 Myr: While crater densities have increased by only factor 2 above saturation, the d100% figure has increased from a depth of ~120 m to 2 km.\nBecause of the \"explosive\" growth effect, megaregolith depths increase rapidly as we consider slightly higher degrees of supersaturation on earlier (older) surfaces. On surfaces formed during a period earlier than ~4.3 Ga, when Table 3 indicates 2.5× saturation at D100% = ~16 km to D200% = ~90 km, megaregolith depth, as observed today, would be in the range 5 to 30 km (in areas not excavated by subsequent large impact basins -an effect mentioned above and discussed in the next section).\nInterestingly, the megaregolith gravels appear frequently to have consolidated into coherent breccia material. Many of the Apollo highland breccias samples, perhaps excavated from depth, are strong, coherent rocks. Coherent materials may have arisen as a result of several different effects. First, impacts in the first hundred Myr or so may have penetrated primarily into molten materials, regionally stirring them and mixing them with shattered crustal materials, so that molten matrix materials may have helped solidify some breccias. Second, pressure effects may have helped cement the heated fragmented materials below some depth. Third, hypervelocity meteorite impacts at various scales produce glassy matrix material that helps to cement rocky fragments into coherent breccias. Wieczorek et al. (2013) and Spray (2016) have discussed such processes in more detail. To summarize, the depths of tens of kilometers, calculated here for impact-affected materials from the geometry of craters, should not necessarily be construed as occupied by loose material; the material may, however, be more porous and less dense than deeper plutonic layers unfractured by impacts. These results are consistent with GRAIL gravitational findings of low densities and higher porosities in the upper few kilometers of the lunar surface. For example, Wieczorek et al. (2013, p. 671) found densities of 2550 kg/m 3 and average crustal porosity of 12% \"to depths of at least a few kilometers.\" We suggest that these measurements refer to loosely consolidated and/or heavily fractured megaregolith, grading at depth into more coherent breccias and fractured crustal bedrock with very early plutonic igneous intrusions along some of the fractures.\nTo summarize this section so far, our discussions depart considerably from the \"classic\" terminal cataclysm/LHB models. To start with, a whole series of impacts at all scales, including a few 10 3 -km-scale basins, would have been forming as the putative magma ocean solidified. The end of magma ocean crystallization is believed to have happened by about 4.45 to 4.35 Ga (Elkins-Tanton et al. 2011;Zhu et al. 2019). Borg et al. (2011Borg et al. ( , 2015 describe an age of 4.360±0.003 Ga for a ferroan anorthosite sample 60025 and discuss it as marking a possible magma ocean crystallization age. However, they also refer to three ferroan anorthosites dated by other authors at 4.43±0.03, 4.47±0.07, and 4.53±0.12 Ga, which could record earlier impact and recrystallization events. Numbers of impact melt clasts, dated at ~4.35 Ga, offer vestiges of one or more large impact structures during that period (Nemchin et al. 2008, Grange et al. 2013, and White et al. 2020; see also Hartmann 2019 for review). Figures 1, 2a , and 3 of our present paper imply very intense bombardment, including multiple impacts of basin-forming magnitude, along with extreme crater oversaturation, during the suggested periods of magma ocean presence. In contrast, the LHB paradigm, from its original version (Wetherill 1975(Wetherill , 1977Ryder 1990) with its period of several hundred million years of negligible impacts before 4.0 Gyr, to the re-defined versions (e.g., Bottke 2012, with LHB reduced in intensity to a surge from 4.1 to 2.5 Ga) all involved a period of relatively low bombardment during the suggested periods of magma ocean presence.\nOur accretion tail scenario, with early intense bombardment during the putative magma ocean's lifetime and solidification, would produce radically different petrologic results than the LHB models. Some early discussions of the solidification of the putative lunar magma ocean, under conditions pictured in LHB models, assumed the process starting with smooth-layered stratigraphy evolving in the absence of large impacts, with density-controlled layering according to the Bowen reaction series. For example, Walker (2009, p. 109) stated that \"…the lunar mantle is likely stratified resulting from magma ocean crystallization and little subsequent mixing.\" The accretion tail model of impact history, together with early work on planetary accretion (e.g., Safronov 1972) and our analysis here, indicate that during accretion of the Moon and the final solidification of the magma ocean, the uppermost mantle of the Moon and/or magma ocean could not have solidified in a smoothly \"stratified\" way with \"little subsequent mixing.\" Addressing the date of solidification for the magma ocean, Borg et al. (2011) considered their zircon-derived 4.36 Ga date to be surprisingly young and concluded \"much of the lunar crust may have been produced by non-magma-ocean processes, such as serial magmatism.\" In contrast to LHB scenarios, our proposed early intense bombardment allows an explanation for complex crustal evolution. serial magmatism, and HSE anomalies. Large-scale basin impacts, would have broken up newly forming surface crustal layers in many regions and ejected masses of deeper material into widespread megaregolith layers, as well as triggering magmatic eruptions. Impactfracturing and impact-stirring of deep magmas may have been a factor in allowing HSE to collect in the lunar interior. Magma at depth from would have been stirred and redistributed during the period from, say, 4.35 Ga to ~4.1 Ga. Figure 2b, representing the intense impact regime about 4.35 Ga, suggests impact basins of >300 km diameter forming stochastically at intervals on the order of ten million years in random lunar locations at that time. Each one would have excavated to depths of many tens of km and dumped ejected fragmental material over wide areas of the Moon. Consistent with this, Wieczorek and Phillips (1999) suggested that large impacts may have disturbed the crustal structure being formed by magma ocean crystallization. Gross and Joy (2016) and Pernet-Fisher and Joy (2016) also discussed effects of impacts on magma ocean evolution and crustal formation. Rolf et al. (2017) suggested that accumulating ejecta deposits (megaregolith) from impact basins scattered over time may have an insulating effect, influencing the rate of heat flow from the interior."},{"title":"CONSEQUENCES OF LARGEST EARLY IMPACTS VIS À VIS HETEROGENEITY OF THE MEGAREGOLITH","content":"The formation of multi-ring basins approaching 1000 km diameters, scattered in space and time during the first 600 Myr, created localized effects on the depth of megaregolith (which included mixed layers of local and distant basin ejecta). Such giant basins also affect the observable survival of earlier, nearby basins, by obliterating parts of rim structures and/or burying parts of them in ejecta. The accretion tail model with early intense bombardment thus casts the South Polar Aitken Basin and the controversial Oceanus Procellarum in a new light. Figures 1, 2a, and Table 1 suggest that under our proposed cratering rates before ~4.3 Ga, several giant impact basins, in the 10 3 km diameter (outer ring) range would probably have formed.\nThe Procellarum area was once considered to be a candidate for such a primordial impact basin, and this hypothesis gains plausibility in the context of our discussion. The GRAIL team discovered a 3000-km diameter \"ring\" of linear gravitational anomalies under the outer part of Oceanus Procellarum, interpreted as intrusion-filled fractures -which might seem to support that Procellarum is a form of a multi-ring impact feature. They suggested, however, that since these features comprise about five or six linear segments, they are not related to the \"rings\" of faults around multi-ring impact basins (Andrews-Hanna et al. 2014). Thus they argued against an impact origin for Procellarum. Figure 5, however, shows ~30-40-degree linear segments in some rings around the Orientale impact basin. Figure 6 shows similar linear segments in the ring structures around the Humboldtianum impact basin. Note that if a truly circular mass of material along a crater rim is trying to slide downward into a basin interior along a conical fault, it is inhibited from sliding because it is buttressed by its own material. Linear normal fault segments make it easier for such material to slip, in segments, down into the crater interior. Furthermore, a Procellarum-scale impact involves restructuring the spherical curvature of the Moon, and we have little knowledge of how a spherical planetary body's brittle and/or plastic lithospheric layers adjust to world-scale impact craters. Our conclusion is that linear subsurface features at Procellarum do not argue against an impact origin for that large lunar feature. Zhu et al. (2017) also argue in favor of a collisional origin of Procellarum because this event would help explain the dichotomy in crustal thickness between the near-side and far-side of the Moon. Figure 6. Humboldtianum multi-ring basin. Photo (a) and topographic map (b) reveal that the southern half of the outer ring (boundary between orange highlands surrounding and green, depressed interior), shows three clear linear segments. Diameter of outer ring is ~620 km. NE half portion of that ring and the basin are overlain by a large crater. (a) Discovery rectified photo by WKH, from Hartmann and Kuiper (1962). This early image is shown in order to illustrate that these linear features have been apparent since the early 1960s. (b) NASA; \"LOLA\" topographic map: NASA/Goddard Space Flight Center.\nAnother consequence of the accretion tail model with early intense bombardment is that the earliest, extremely high impact rates such as 10,000 times the present rate, go beyond the rates typically contemplated in the terminal cataclysm/LHB model. We have already mentioned that the earliest basins would have been beaten into presently undetectable states. Figure 7 illustrates the effectiveness of that process. It shows a vivid example of an older basin, Serenitatis, whose rim has been broken into linear ridges and partly demolished by ejecta or faulting associated with the younger, neighboring Imbrium basin. The mountains of the Serenitatis rim are laced by ridges and valleys structured not radial to Serenitatis but radial to Imbrium, whose center is ~980 km away, beyond the lower edge of the frame. The destruction here is also ~270 to ~390 km beyond the Apennine rim of the Imbrium basin. The precise mechanism for this reshaping of the Serenitatis rim by the Imbrium impact is still not certain. Tectonic fractures caused by the Imbrium impact, with collapses during lava flooding of the area, seem a possibility. A second possibility might be gouging by local grazing, low-angle rock masses in ejecta ejected from Imbrium, possibly embedded in a somewhat volatilized mass like a base surge. The second mechanism seems perhaps less preposterous when one considers that the \"mountain ranges\" creating basin rims are, to some extent, piled-up, ejected megaregolith gravels. Whatever the cause, the degree of Serenitatis destruction by Imbrium is remarkableand all the more so since (as discussed in more detail by Hartmann 2019, pp. 19-20, Table 1) there is at least one case where researchers relying on radiometric dating of Apollo samples listed Serenitatis as about one error bar younger than Imbrium. (Most proposed dating of Serenitatis put it within an error bar to two older than Imbrium.) Such young dates reported for Serenitatis may, however, have involved the Imbrium ejecta that modified the Serenitatis rim (see Hartmann 2019 for more detailed discussion)."},{"title":"CONSEQUENCES OF MEGAREGOLITH EVOLUTION VIS-À-VIS ROCK SAMPLES COLLECTED ON PRESENT-DAY LUNAR AND PLANETARY SURFACES","content":"The evolution of the megaregolith, and its current properties, have important effects on the rocks and soil samples collected on the present-day surface of the Moon, not to mention other planetary surfaces. To take impact melt samples first, the high formation rate of early basins and large craters had important effects on any impact melts formed in the first few hundred Myr. The large, multi-ring basins are believed to contribute most of the volume of lunar impact melts (Artemieva and Shuvalov 2008), but the exact disposition and depth of impact melts within basins appear to be uncertain. For example, based on terrestrial observations and theoretical modeling, Cintala and Grieve (1998) depict impact melt sheets as layers much thinner than the transient cavity basin depth, spread across the basin interior. Artemieva and Shuvalov (2008) and others suggest more concentration in deeper plugs under the middle of the crater. Impact melt plugs as deep as 20 km have been suggested under Imbrium. It is implicit in either case that impact melts are created and distributed in more or less surficial layers having to do with impacts and ejecta blankets. We have shown, however, that in the accretion tail model, creation of the initial megaregolith in the first few hundred Myr involved pulverization to depths of tens of kilometers, probably deeper than most impact melts. As also shown in Fig. 3, surfaces and near-surface structures formed around 4.15 Ga are only slightly supersaturated. By about 4.0 or 3.9, near-surface structures such as impact melt lenses in the upper few kilometers are more likely to survive. Furthermore, as indicated in Fig. 3 and Fig. 2a, structures formed around 4.0 Ga no longer become saturated with large craters. In short, the accretion tail scenario, contrary to terminal cataclysm/LHB models, explains why impact melts older than ~4.1 Ga are hard to find, while increasing numbers of impact melts are found from ~4.0 Ga to 3.9 Ga. The number of impact melts decreases after that because the impact rate itself was declining. In this view, the unique aspect of 3.9 Ga is not a sudden narrow spike in impact rate, but rather that this date marks the beginning of the period when massive formations on the scale of impact melt lenses will survive -not to mention that the Imbrium and Orientale basins formed around that time.\nBut what about the fact that primordial crustal samples date back as far as 4.44-4.45 Ga? Why weren't they also destroyed? The answer lies in the fact that the base of the megaregolith, by definition, lies as deep as the deepest pulverization and fragmentation, reaching into more intact primordial rock layers. According to our scenario of early intense bombardment and accretion tail, the megaregolith reached tens of kilometers depth. Richardson and Abramov (2020) emphasize different layers in the megaregolith, with a similar conclusion. They picture a surface dusty regolith layer typically about 20 m deep, underlain by an \"upper megaregolith\" from ~20 m to a depth of 1 to 3 km, consisting of depositional layers of \"brecciated and/or melted material…characterized by the transport and deposition of material via either transient crater gravitational collapse or impact ejecta.\" Under the \"upper regolith,\" they picture a \"lower megaregolith\" layer, running from ~1-3 km at the top to \"about 20-25 km,\" consisting of bedrock that has been fractured in place. In all such models there is a semi-infinite supply of primordial, lower crustal or upper mantle plutonic rocks below the bottom of the megaregolith. Thus, the largest modern craters, whose transient cavities exceed roughly 20 km depth, excavate samples of primordial crustal or upper mantle rocks and scatter them on the surface. This excavation process is aided by additional aspect of megaregolith evolution: It is heterogeneous in the sense that the biggest basin impacts blow away much of the accumulated megaregolith under their floors, creating \"thin spots.\" Those areas were then coated with lavas, but even modest-sized craters on floors of basins can penetrate the lavas and eject ancient, deepseated rocks. Craters with bright ray systems, associated with Imbrium and postdating mare lava plains, such as Autolycus (diameter D = 81 km) and Aristillus (D = 55 km) (both inside the Imbrium's Apennine ring) are good examples. Copernicus (D = 93 km) and Eratosthenes (D = 59 km), somewhat outside that ring, are also candidates for ejecting early crustal samples.\nOnce a rock has been expelled onto the lunar surface, how long does it last? This question was not much considered when the Apollo missions were planned and the first samples were collected, because the Moon was thought to lack erosive processes. However, a number of diverse studies show that lunar rock samples typically have cosmic ray exposure ages of tens of Ma to a few hundred Ma, and meter-scale rocks typically erode due to the \"sandblasting\" flux of small meteorites with a few hundred Ma. Moreover, craters of D <12 km (depth ≲2 km) typically produce rocky ejecta blankets in lunar maria (from lava layers) but not so much in lunar highlands (where the subsurface is mostly pulverized). For discussion of such effects, see Basilevsky et al. 2013;Ghent et al. 2014;Costello et al. 2018;Hartmann 2019. Thus, in collecting lunar rock samples, we are entirely at the mercy of excavation processes in the megaregolith; megaregolith properties \"filter\" what we can collect on the surface."},{"title":"FUTURE WORK","content":"We regard our current work as a demonstration of concept. In the past there has been some separation of sub-disciplines in planetary science. Research in sample petrology, radiometric dating, impact crater studies, numerical modeling of interplanetary impact fluxes have typically involved different research groups, different meetings, different sessions at large annual meetings, and different journals. One area of fruitful future research may attempt to combine better the insights from (1) dynamical models of accretion, planetesimal collisions, and planetesimal fragmentation events, (2) the resulting geological consequences for lunar and planetary cratering rates as a function of time, (3) studies of the processes of excavation and deposition of impact debris. We note that such work will apply to more or less airless bodies throughout at least the asteroid belt and inner solar system, including Mars. At the same time, we caution that to do more detailed modeling of megaregolith evolution we need better quantitative information on still-vague parameters such as (1) extent and depth of fracturing and pulverization of rock during impacts of various sizes; (2) velocity distributions for the different kinds of meteorites hitting the Moon at any given time; (3) cross-section and volume of transient cavities during impacts of various sizes, especially multi-ring basins; (4) cross-section and depth of impact melt lenses on or under the floors of multi-ring impact basins; (5) deposition depths, compositions, and homogeneity of structure in ejecta blankets as a function or distance from crater center for craters of various sizes (especially looking at sources of ejected KREEP materials and uniqueness of KREEP material to the Imbrium region); (6) degree and causes of cementation of fractured and pulverized megaregolith materials as a function of depth; (7) nature of rocks excavated from the base of the megaregolith by the largest recent craters in terms of our access to the original lunar crust or the upper mantle; (8) survival times of excavated rocks as a function of sizes on the lunar (or planetary) surfaces (i.e., erosion histories of samples accessible on the surface."},{"title":"CONCLUSIONS","content":"The concepts of \"early intense bombardment\" and the \"accretion tail\" dynamical model (Morbidelli et al. 2018) provide a coherent view of age distributions of radiometric dates of lunar primordial crustal rocks vs. impact melts, GRAIL data on crustal structures, and early evolution of lunar features and multi-ring basin morphologies. They also clarify phenomena not adequately explained by the terminal cataclysm/LHB paradigm. Among the implications of our study:\n• The solidification of the magma ocean was affected by extremely high cratering rates during that early era. This prevented \"smooth,\" layered deposits based on Bowen reaction series in at least some areas, and may help explain complex petrology among the most ancient samples from various regions of the present-day Moon. • If a surface formed (or attempted to form) by solidification of a magma ocean during the extreme bombardment as early as >4.4 to 4.35, it would have experienced supersaturation of multi-kilometer diameter crater size-frequency distributions by about 4.35 to 4.3 Ga. • Surviving remnants of such early surfaces, i.e., between the largest impact basins, would have continued to accumulate even more craters, so that early crater densities would have gone into supersaturation, probably by factors at least 2.5 to 10, resulting in pulverization of the earliest impact melts. This result differs from predictions of the terminal cataclysm/LHB models, and helps explain the dearth of impact melt clasts from before ~4.35 Ga. • Continuing bombardment in the first few hundred Myr of crust existence produced reworking and redistribution of materials in upper megaregolith layers, consistent with GRAIL measurements of low density and high porosity in the upper kilometers, Richardson and Abramov's (2020) modeling, and age distribution of earliest known impact melts. • The surface today would display localized regions destroyed and resurfaced by multi-ring basin-scale impacts, typically filled with basaltic lavas, all in various stages of degradation by subsequent cratering. This scenario is consistent with the presence of the large, very degraded South Polar Aitken basin, and with Oceanus Procellarum being a very early giant impact feature. • Subsurface linear features comprising a \"ring\" around Oceanus Procellarum are not a compelling argument against Procellarum being an impact feature. • The oldest lava-plain formations, for example, in South Polar Aiken, are covered by a veneer of high-albedo ejecta and overlapping rays, and were often mapped in the pre-Apollo and early post-Apollo eras as \"highlands.\" They may, however, have quite different megaregolith composition (and other properties) than highland regions that survive from earliest crustal formation • The supersaturation-pulverization of the upper kilometers would have comminuted the impact melt lenses produced in the surface layers of the early impact basins, explaining the presence of impact melts from ~ 4.1 to 4.35 Ga as fine-scale clasts in lunar breccias. • Surviving surfaces formed around 4.1-4.2 Ga have reached at least 1.0× saturation crater density.\n• Surfaces formed after about 4.0-3.9 Ga are not quite saturated with craters, and experienced much lower regolith production rates than earlier surfaces. Survival times of rocks in surface layers rapidly increase at this time. This explains the survival of many impact melt rock samples after ~4.0-~3.9 Ga as contrasted with few before that time. • The blasting away of megaregolith layers by a few large, late basin scale impacts (especially Imbrium) created thin spots in the megaregolith where recent Copernicusscale and Autolycus-scale craters can eject primordial crustal rocks from the base of the local megaregolith, which in many instances is covered by mare basalt layers. This explains the presence of primordial plutonic crustal rocks in present-day surface sample collections (in spite of the paucity of impact melt rocks before ~4.0 Ga). • Many concepts in this paper, regarding megaregolith evolution and sample collection apply not only to the Moon but to all relatively airless planetary bodies in the inner solar system (Mercury, moons, asteroids, and even Mars), and at least into the main asteroid belt, and probably beyond, depending on the histories of asteroid/comet impact rates in the outer solar system. • We are still in an era where first-significant-figure issues remain to be considered, even while assessing certain processes with numerical models citing two-and three-significantfigures."},{"title":"Figure 3","content":""},{"title":"Figure 3 .","content":""},{"title":"Figure 4 .","content":""},{"title":"Figure 5 .","content":""},{"title":"Figure 7 .","content":""},{"title":"Table 1 .","content":""},{"title":"Table 2 .","content":""},{"title":"Table 3 .","content":""}],"string":"[\n {\n \"title\": \"ANALYSIS AND APPLICATION OF ACCRETION TAIL SCENARIO\",\n \"content\": \"\"\n },\n {\n \"title\": \"VIS-À-VIS IMPACT RATE VS. TIME\",\n \"content\": \"As pointed out by Morbidelli et al. (2018), the \\\"accretion tail\\\" dynamical model creates a curve of declining impact rate vs. time, similar to -but independent of -early pre-and post-Apollo suggestions about the nature of that curve, based mainly on cratering statistics. Hartmann (1965Hartmann ( , 1966 estimated the characteristic age of lunar mare lava plains at ~3.6 Ga, using cratering rates derived from the Canadian shield. The second paper pointed out that since the pre-mare lunar highlands had 32 × the crater density of the maria, the brief pre-3.6 Ga cratering rate had to average at least ~160× the post-mare crater rate. The uncertainty in that number was estimated to be a factor 2. The number 160 was considered a lower limit, because the factor 32 crater density was found on a number of solar system surfaces, indicating that it is represents the crater saturation equilibrium level found on many bodies (Hartmann 1984). Craters formed in excess of the saturation level are destroyed by later cratering, and this means that the total impact crater numbers in the highlands may be much larger than what we can see today. The saturation equilibrium level is important in the present paper because, as we will show, the total number of impacts during very early intense bombardment, based on Morbidelli et al. (2018), was well beyond the number required to reach saturation. Thus, the saturation curve found in most areas today does not represent the actual number of craters that were formed, but only a lower limit. Once saturation was exceeded on a surface, we lose information on the impact numbers and the shape of the earlier size-frequency distribution curve.\\nAfter investigators dated Apollo mission samples collected from various landing sites, crater density measurements as a function of measured surface age indicated that as we go backward in time from ages ~3.2 to ~3.8 Ga, the cratering rate reached a value of order ≥200 times the current rate (e.g., Hartmann 1972;Neukum 1983;Neukum et al. 2001). In the \\\"accretion tail\\\" scenario, this decline in cratering can be seen as the tail end of the more or less monotonic decline in impact rate from >4.4 Ga ago to 3.8 Ga ago, when the cratering record does not reach saturation, and is more clearly measurable. We stress that the time evolution of the impact rate in this paper not a backward extrapolation of the observed impact rate evolution observed in the 3.2-3.8 Ga interval, but is the result of the numerical modelling of the dynamical evolution of the planetesimals remaining in the terrestrial planet region at the end of the planets' formation process (Morbidelli et al. 2018). Table 1 shows an application of this model. Column 1 lists dates at 100 Myr intervals. Columns 2 and 3 illustrate two ways of normalizing the bombardment model data to give cratering rates relative to the modern cratering rate (defined as 1.0). In column 2 (model A) we normalize to an average value of 1.0 for the last 400 Ma (averaging over the Morbidelli et al. 2018 model values for the last four 100-Ma bins, including T=0=today). Column 3 shows a second, different normalization of the curve (model B). It applies suggestions that the cratering rate at 3.0 Ga was probably several times that observed today, and was still declining from 3.0 Ga till today (Quantin et al. 2004Hartmann et al. 2007). Thus, column 3 normalizes the data to a value of 3.0 at 3.0 Ga. The two normalizations give similar results, as plotted in Fig. 1.\\nIn this context, it is important to note that the Morbidelli et al. (2018) data set, in order to extend the timeline from 3.0 Ga until today, involves a conceptual discontinuity at ~3.0 Ga ago. At age >3 Ga it follows the calculated decay of asteroids and leftover planetesimals assuming their dynamics were dominated by planetary perturbations (the main subject of Morbidelli et al. 2018). For age <3 Ga, the model shifts to dominance of near-Earth asteroids escaping from the asteroid belt under Yarkovsky forces, giving a relatively constant bombardment in Table 1 (similar to the normalization in column 2 of Table 1). In reality (especially with an asteroid belt a bit more massive 3 Ga ago than now, giving more NEAs) the two curves would merge more smoothly. For this reason, Morbidelli et al. (2018, their Fig. 5) originally plotted their results from 4.5 Ga only to 3.0 Ga ago, but here we use additional data from their model to extend the curve.\\nThe fourth column in Table 1 shows a few examples of earlier cratering rate estimates (still normalized to current rate = 1) based on extrapolation of crater count data backward to ~4.3 or 4.4 Ga, assuming the early estimates of decline in planetesimal numbers (Hartmann 1972;Neukum et al. 2001;Quantin et al. 2007). In Table 1, fluctuations in cratering rate and half-lives of asteroids against collisions with other bodies, at ages around 3.0 Ga may be due mostly to the merger of the two models. The final (fifth) column of Table 1 lists some estimates of half-lives of the projectile population published during earlier crater-count studies (Hartmann 1972;Neukum et al. 2001;Quantin et al. 2007). We note that these are similar to the results from Morbidelli et al. (2018). Figure 1 shows the Morbidelli at al. (2018) cratering rate curve as a function of time (on a logarithmic scale), using the two normalization methods from Table 1, allowing either no decline since 3.0 Ga or decline by a factor 3. We infer that the most likely cratering rate behavior occurs between these two curves, but with a more gradual transition at about 3.5 to 2.5 Ga, ending (by definition) with today's rate of 1.0.\\nTo summarize, the \\\"accretion tail\\\" scenario favored by Morbidelli et al. (2018) indicates that the early intense bombardment rates in the first few hundred million years (surface formation until ~4.15 Ga) exceeded 1000 times the present rate. At dates before 4.4 Ga, it may have exceeded 10,000 the current rate, although, as we probe back toward the beginning, the lunar surface may have been in magma-ocean state. Note the contrast with the \\\"classic\\\" terminal cataclysm paradigm, which called for more or less negligible cratering in the ~4.4 Ga to ~4.0 age interval, with few if any of the major front-side multi-ring basins forming in that period. Hartmann 1972, Fig. 3, p. 55;Q = Quantin et al. 2007). Note: Morbidelli et al. (2018) are given precedence in this paper over the earlier estimates, which are cited merely to demonstrate similarity with early post-Apollo models based on crater statistics.\\nAs Morbidelli et al. (2018) reported, the crater count results (and their extrapolation to before 4.0 Ga) come strikingly close to the results from the Apollo-era models in both shape and absolute numbers. For example, Hartmann (1972), using early Apollo sample data stated that prior to 4.1 [Ga] ago, the cratering rate on the Moon was at least 10 3 times the present rate and the rate declined with a half-life less than 8 × 10 7 yr. During the interval from 4.1 to 3.2 [Ga] ago, the number of planetesimals showed an exponential decay with a half-life about 3 × 10 8 yr…. A more constant cratering rate applied in the last three aeons. Neukum (1983, as reviewed in Neukum et al. 2001) developed an analytic interpretation of the crater-count data, an equation fitting their observations of crater cumulative crater density as a function of time. The differential values of that equation (i.e., difference in craters/km 2 in a specified size range from beginning to end of 100 Myr intervals), give the crater formation rate (i.e., craters/km 2 in a specified size range) formed per 100 Myr interval). As an example of their results, Neukum et al. (2001) stated that \\\"…the cratering rate 4 Gyr ago was 500 times higher than the constant rate during the last 3 Gyr,\\\" a statement close to the curves in our Fig. 1.\"\n },\n {\n \"title\": \"Figure 1.\",\n \"content\": \"Curve showing cratering rate as a function of time (relative to today's cratering rate) derived from Morbidelli et al. (2018) accretion tail model. Lower curve assumes cratering rate asymptotically arrives at today's rate (i.e., 10 • ). Upper curve assumes cratering rate at 3.0 Ga equals 3 × today's rate. \\\"N\\\" symbols are independent data from crater counts, taken from equation fitted by Neukum to crater count data (see text for further discussion).\\nThe impact rate data points derived from the Neukum equation, based entirely on crater counts (not on a dynamical model) are included with symbol \\\"N\\\" for comparison in our Fig. 1. The agreement between the dynamical models A and B, and the cratering rate analysis, is remarkable for three distinct reasons. (1) The research teams are entirely independent. The Morbidelli et al. (2018) team made no attempt to match their dynamical model to the earlier crater chronometry results. (2) The techniques used are entirely independent; mathematical results from orbital dynamical calculations are unrelated in practice to counts of lunar craters.\\n(3) Perhaps most striking, Morbidelli et al. (2018) model predictions match not only the bestestablished parts of the Neukum curve at ages ≲3.8 Ga, but also an extrapolation of that curve to earlier dates. To repeat, the best data used for the Neukum curve come from dated samples returned from Apollo and Luna landing sites, but surfaces for crater counts dating from before ~3.8 Ga (a) are less well documented (if available at all), and (b) have crater densities near the saturation equilibrium line so that super-saturation densities on older surface cannot be directly measured. All the more noteworthy, then, that the Neukum equation for cratering rates fits so well the results of the dynamical model.\\nThe conclusion of this section is that the accretion tail model, along with certain Apollo era suggestions of early intense cratering, imply that the cratering rate prior to 3.8 or 3.9 Ga likely exceeded hundreds and even thousands of times the rate from the last 1-3 Gyr (and could plausibly have exceeded ten thousand times higher than it is today, prior to ~4.35 Ga), as shown in Table 1, columns 2 and 3, from the Morbidelli et al. (2018) data. This is contrary to \\\"classic\\\" terminal cataclysm/LHB paradigm, where cratering rates prior to 3.9-4.0 Ga were considered to be comparable to the present rate. It is significant that, half a century after the Apollo and Luna samples were returned, the accuracy of first significant figures is an issue in understanding important lunar phenomena.\"\n },\n {\n \"title\": \"DEVELOPING TOOLS TO STUDY EFFECTS OF INTENSE EARLY BOMBARDMENT\",\n \"content\": \"A useful tool in understanding the lunar cratering record is the \\\"isochron diagram,\\\" which shows the observed size-frequency distribution (SFD) of impact craters as a function of surface age. In the template version, shown here in Fig. 2a, the crater density (craters/km 2 ) is shown incrementally, for each individual size bin, and the size bins are plotted on a log√2 scale, for example from 1.0 km to 1.4 km, from 1.4 km to 2 km, and so on. This plot has an advantage over widely used cumulative plots, because it shows exactly what is happening in each individual size bin, rather than smoothing the curve's structure over many sizes, as cumulative curves do.\\nThe isochron diagram has a number of additional useful features. The upper diagonal boundary, marked \\\"Saturation\\\" shows an empirical curve (mentioned earlier), which has been repeatedly observed in the most heavily cratered surfaces on the Moon, various asteroids, and outer planet satellites throughout the solar system (Hartmann 1984). The observed saturation curve is not a \\\"geometric\\\" saturation, since more small craters could, geometrically, be squeezed into existing empty spaces on these surfaces. However, nature does not allow the addition of craters in only a narrow size range; the whole SFD must be added all at once, which requires occasional, large, basin-scale impacts, whose excavation and ejecta blankets completely resurface some areas. Furthermore, \\\"equilibrium\\\" connotes stability, whereas empirical saturation in a given region does not represent a static situation. Not only do the distant (often \\\"off-stage\\\") basins' ejecta occasionally wipe out shallower craters (at, say, D <100 m diameter, depending on distance and size of the basin), but also the continual rain of meteorites creates craters smaller than any specified diameter D, and erodes rims of craters larger than size D by a simple sandblasting effect. An inexorably increasing supersaturation would produce many big, deep impact craters (some no longer visible) and thus pulverize increasingly deep layers, including the earliest impact melt lenses formed in the near-surface kilometers under the earliest large impact basin floors. Thus, different parts of the saturation line can thus oscillate over time by factors about 2 to 4 (Hartmann and Gaskell 1997). The dashed lines on either side of the saturation curve thus show the typical range of oscillation of visible saturation crater densities. The important larger lesson is that a surface characterized by the saturation line may have been impacted far more times than is evident from the numbers of visible craters, preventing a clear measurement of the SFD of the earliest impactors.\\n(a) (b) Figure 2. (a) Template for lunar isochron diagram. Each curve shows the size-frequency distribution of craters observed on lunar surfaces of specified ages. The diagram assumes constant crater production rate going back in time until about 3 Ga but higher rates before that, as based on Apollo data (Hartmann 1970a(Hartmann ,b, 1972Neukum et al. 2001). (b) Crater accumulation diagram under conditions of impact rate 10,000 times the current rate, as suggested by Morbidelli et al. (2018) for the period around 4.35 Ga. Given the half-life estimates in Table 2, a rate of ~ 10,000× current rate lasts only tens of millions of years, so the graph is extended only to 10 Myr (see text for further discussion). Because crater chronometry is constantly improving with new data, fine print above diagrams refers to date of our latest iteration of the lunar isochron plot, and the date of our data set. Now suppose the cratering rate ~4.35 Ga was ~10,000 times the present rate, as the accretion tail model indicates for an interval of about 50-100 Myr, centered on that date (see Table 1 and Fig. 1). To create an isochron diagram to match conditions under that elevated impact rate, we simply relabel the isochrons, stopping at 10 Myr, because the half-life of decline in cratering rate at 4.35 would have been on the order of 50-60 Myr, according to Table 1. The result is shown in Fig. 2b. The diagram suggests that any surface formed during cratering at that rate would have reached supersaturation equilibrium conditions on a timescale ~50 Ma, and we can surmise that multi-ring basins of D >300 km would have been forming in stochastic intervals averaging 50-100 Myr. If the cratering rates were higher than 10,000 times the present rate before 4.35 Ga (as indicated in Table 1 and Fig. 1), and if the surface were solid, supersaturation would have been reached in <50 Ma.\"\n },\n {\n \"title\": \"CRATER ACCUMULATION ON SURFACES FORMED AT DIFFERENT TIMES\",\n \"content\": \"As suggested by Fig. 2b and the discussion in the last section, the accretion tail model offers an approach to understanding how the earliest cratering conditions affected the early lunar crust and megaregolith, and even the kinds of samples picked up on the surface today. This investigation is detailed in Table 2. 0.78 ** *Average mare N(1) density is assumed to be 3.2(10 -3 )craters/km 2 , larger than 1 km, based on Hartmann (1966Hartmann ( , 2005 crater counts on a variety of lunar mare surfaces, which were found to have virtually the same crater density, independent of the Morbidelli et al. (2018) In Table 2, column 1 defines 100 Myr-long time bins, and lists dates of surface formation. In column 2, N(1) refers to the cumulative number of craters/km 2 that have diameter >1 km. N(1) is often used as a proxy for the entire size-frequency distribution curve for all impact craters, from microscopic pits to 1000-km multi-ring basin structures. We adopt it as a first-order proxy, but we recommend caution, because the N(1) values fall just in the diameter zone where the -2 slope SFD of larger craters intersects the steeper -3 to -4 slope at smaller diameters, as shown in Fig. 2. Our work concentrates on the rate of change in the entire crater population as a function of time, but N(1) still has first-order value in our discussions. Thus, in column 2, ∆N(1) values from the accretion tail model are defined as the total number of craters with diameter D >1 km added per km 2 in the 100 Ma intervals shown in column 1. For example, from 3.05 to 2.95 Ga, centered on 3.0 Ga, the model shows ∆N(1) addition of ~2.3 (10 -4 ) craters/km 2 larger than 1 km.\\nColumn 3 repeats column 2 by listing the ∆N(1) number of craters in terms of average mare crater density. This gives a more direct connection to the appearance of the lunar surface, making the table more accessible to readers not familiar with N(1). Entries of ~1.0 in column 2 thus represent a surface with crater densities typical of average lunar mare on the front side of the Moon, based partly on data from the University of Arizona Lunar and Planetary Laboratory crater catalogs (for a summary of those catalogs, see Wood 1972) and additional counts over the years, by Hartmann, treating the mare surface as one giant mare. The mare surfaces in the major basins have remarkably similar densities, and \\\"average lunar mare,\\\" as mentioned in a footnote to Table 2, thus refers to a summation of counts in each diameter bin from surfaces of various major basins, divided by the total area counted in each bin.\\nThe remaining columns, 4 to 11, give crater densities relative to average mare crater density, as they accumulate on surfaces of eight difference starting dates. Thus for example, if we look at a surface (perhaps a lava flow) formed at 4.35 Ga (column 6), it has 0 crater density at 4.35, but adds ∆N(1) of 103 craters/km 2 by 4.25 Ga. And so on, down the table until we see it leveling off at roughly 170 craters/km 2 larger than 1 km, which would be observed today. In columns 4 to 11, entries of ~32 represent lunar highland surfaces, matching saturation equilibrium as measured by Hartmann (1984). Table 2. Three vertical scales have been added to the diagram to demonstrate its range of important information. At the left side we see the crater densities in terms of average lunar mare and also in terms of N(1) as used in Neukum (1983) cumulative crater densities. At the right side we see a listing of the estimated megaregolith depths, which are a direct function of accumulated crater densities, as will be discussed below. The shallower depth estimates come from Apollo observations and GRAIL data (Wieczorek et al. 2013), and the deeper depths, in the regions dating to the first 500 Myr, come from our calculations. Figure 3 also contains three horizontal shaded bands, marking average lunar mare crater densities, saturation density (32× average mare density), and 10× saturation density. The vertical shaded band indicates a plausible time span in which a putative magma ocean might have solidified, i.e., forming a stable solid surface; accumulation of craters on solid surfaces would occur only after that. Table 2, showing accumulation of crater density as a function of time, during the putative early intense bombardment. Bottom of each curve (on abscissa) shows formation dates for surfaces, and the curves show subsequent crater accumulation on those surfaces. Data were initially plotted at 100 Ma intervals, leading to a sharp kink in each upper curve, ~100 Ma after the start date, due to short half-life of early impact rates; those kinks have been smoothed out by adding artificial data points to indicate a more realistic transition. (See text for discussion of the shaded bands and the three scales provided for the ordinate axis on left and right sides. See Table 2 footnote for definition of \\\"average mare.\\\" The curve for 4.55 Ga is included to indicate the extreme impact rate if solid bodies existed at that time; the Moon probably did not yet exist.) Figure 3 gives insights into a number of lunar properties and aspects of lunar history. For example:\\n• Crater SFD numbers formed on the earliest inner solar system surfaces, as represented by the 4.55 Ga and 4.45 Ga curves, are no longer measurable on surfaces today, because of extreme oversaturation. The earliest craters have been overlapped by later large craters, multi-ring basin impact structures, and basaltic mare lava flows.\\n• Surfaces formed prior to ~4.2 to ~4.3 Ga were intensely bombarded and super-saturated with craters, due to the very high early cratering rates, but are getting closer to saturation levels. Near saturation levels affect the survival of impact melts from upper kilometers that time.\\n• The youngest crater-saturated surfaces visible today should date from ~4.1 to ~4.2 Ga, consistent with Fig. 2a. The reduced saturation effects by this time allow for increasing survival of brecciated materials and impact melt inclusions from that time.\\n• Crater densities matching average mare surface appear around 3.5 to 3.6 as predicted from pre-Apollo crater counts and supported with dated Apollo samples.\"\n },\n {\n \"title\": \"INTENSE EARLY BOMBARDMENT: CONSEQUENCES FOR MEGAREGOLITH EVOLUTION AND MEGAREGOLITH DEPTHS\",\n \"content\": \"Applying the information discussed so far, we investigate the evolution of megaregolith in the first 600 Myr of lunar history. Figure 4 introduces a technique for using the SFD curve of crater populations to calculate depth of pulverization, based on an earlier technique utilizing the isochron diagram (Hartmann 1980(Hartmann , 2005. The technique works for under-saturated, saturated, and over-saturated surfaces. For various SFD crater densities formed on such surfaces, we calculate the area covered by craters (per km 2 ) in each diameter bin, then add up the cumulative area starting with the largest bin and working toward smaller bins. As shown in Fig. 4, at each diameter D, this gives the total area covered by craters larger than D (and thus ejecting material deeper than some effective depth d excavated by such craters). At some sizes we reach a point size where the total area of larger craters is 100% of the area available. At 100% coverage, however, not all areas have been impacted and excavated, because of crater overlap. Thus, the 200% parameter is included to allow more realistically for such overlap. We will define our quantitative definition of d in a moment. We assume, as a first-order approximation, that when 100% to 200% of the surface is covered by craters larger than critical diameters designated as \\\"D100%\\\" and \\\"D200%,\\\" respectively, then a characteristic depth of megaregolith is at least of the order ~d100% to ~d200%. The effective depth at any diameter D must be less than the deepest part of the transient cavity, but deeper than the observed floor depth in the existing crater. Here we assume also that nearly all of the \\\"effectively\\\" excavated material is distributed elsewhere on the Moon. The depths affected by craters exceeding D100% and D200% are somewhat in question, since the transient cavity depths near the center of a crater are considerably greater than the final crater depths. What, then, is the depth actually affected by saturation coverage among craters of these sizes? Taking into account the effects mentioned so far, we assume conservatively that craters of the relevant diameters (some kilometers) churned and pulverized materials to effective depths equaling 1/3 of their diameter. The depth of megaregolith, while averaging around the order of magnitude of ~d100% to ~d200%, will vary from region to region, because stochastically-produced largest basin-forming impacts will blow away early surfaces and create new, localized, surfaces -which then begin to accumulate mare lava flows, new impacts, and new regolith. For example, the large, relatively late Imbrium basin must have blown away most earlier megaregolith (not to mention overlapping, obliterating, and ejecting material from earlier large craters and possible earlier, small multi-ring basins with their impact melts). At the same time, Imbrium's ejecta delivered such earlier megaregolith materials onto other regions (McGetchin et al. 1973;Haskin 1998;Hartmann 2003;Petro and Pieters 2008;Norman and Nemchin 2012;Norman et al. 2015). The giant basins thus create relatively thin spots in the megaregolith cover (possibly zero kilometers immediately after impact in some cases), while depositing the previous megaregolith material in a surrounding ejecta blanket. Another example of these effects may be present at South Pole-Aitken (SP-A) basin. Using GRAIL results, Besserer et al. (2014) state that \\\"The SP-A region appears to possess a thinner low-density (porous) layer than the rest of the farside.\\\" This result suggests that that SP-A blew away a fair amount of megaregolith which had formed before the SP-A impact. Even cursory inspection of lunar photo images reveals that SP-A has a higher crater density and higher albedo than the major front-side maria. It has evidence of mare basaltic lava deposits, but those mare surfaces themselves have been heavily cratered, with the lava apparently pulverized and brightened by ray deposits. These statements suggest that SP-A and its lava flooding date from before most of better-preserved mare lava production dating from ~3.8 Ga to ~3.2 Ga, but nonetheless that a substantial thickness of megaregolith existed before the impact, to be blown away. This is consistent with our model, but inconsistent with the LHB paradigm, with its sparse cratering before 4.0 Ga. Here, 100% of area is covered by craters larger than ~4 km, and 200% by craters larger than ~350 m. (b) The same but for a surface that is supersaturated by a factor 2. In this case, 100% of area is covered by craters larger than ~60 km and 200% covered by craters larger than ~5 km. The diagram illustrates how the sizes of craters excavating 100% and 200% of total area increases dramatically as super-saturation is reached. Table 3 and Fig. 4 illustrate the technique described here. We apply the technique starting with the saturation equilibrium curve, because it reflects the highest observable crater densities and thus suggests a minimal megaregolith depth in lunar highland saturated regions. In Table 3, column 1 shows crater diameter and column 2 shows the calculated fractional area per km 2 covered by observed craters in each bin, as follows:\\nFractional area covered in each bin = π(D/2) 2 N = 0.785ND 2 (1) where D = crater diameter near mid-point in each bin N = no. craters/km 2 observed in each bin This fractional area is then expressed as a percentage of the area in column 2 of Table 3.\\nThe formula does not take into account that some craters overlap earlier craters. This is a valid approach if we are interested in the total volume being excavated and ejected by observable craters. But if we are interested in the total surface area that has been excavated, the 200% figure is better, since most of the area has been impacted at least once by craters larger than D200%.\\nColumn 3 in Table 3 shows the cumulative percentage of area covered by all craters larger than D, starting with the largest craters at the bottom of the table. At D >128 km there are poorer statistics because of the smaller number of large craters and basins, and the isochron curves are not plotted at still larger sizes for this reason. Among the giant multi-ring basins at the largestdiameter end of the size-frequency distribution (SFD), at D >~360 km, there appears to be a continued turn-down at largest sizes, and we have added modest provisional percentage numbers in column 3 at D >360 km. The areas covered, however, are not large enough to affect our conclusions significantly. Columns 4, 5, and 6 add the same results for 2× saturation and 2.5× saturation In Table 3, the important transitions at 100% and 200% coverage by impact craters are shown in bold face in columns 3, 5, and 6. In this calculation, at D <1.4 km, we follow the wellknown steep branch of the SFD upward into the region above the saturation line, not stopping at the saturation curve per se. This is in order to include the effects of all impacts called for by the SFD at small sizes, whether we can see the resulting craters or not. This steep branch (and presumably the larger craters of the large-D branch at D >1.4 km), include an uncertain fraction of secondary craters, which add to the churning and pulverization involved in megaregolith evolution. [Our system of crater counts and crater chronometry includes all visible craters (excluding obvious clusters with rays), without trying to distinguish primaries from the remaining \\\"field secondaries,\\\" since attempts to distinguish them visually are likely to be in error. This system is practical, since nature can't produce large craters without producing secondaries, and can't produce secondaries without primaries, and on old surfaces with high crater densities, the addition of unrecognized scattered field secondaries in a single shower from a fresh distant primary will not have a drastic effect on the total crater density.] Note that, as seen in Fig. 4a and mentioned above, the small-D end of the steep branch of the SFD is first to hit the saturation line. What appears to be sparse cratering as in lunar maria at large D thus produces supersaturation at small D creating a shallow regolith of roughly 5 to 15 meters depth, as detected in Apollo landings on lunar maria. But once saturation is approached at D ≳1.4 km, as indicated in Fig. 4a, the saturation line is reached virtually simultaneously at all larger crater diameters, causing an explosive growth in megaregolith.\\nHere we pause to ask whether these last statements apply to the earliest crater populations in the first few hundred million years. The problem is that if the early intense bombardment/accretion tail scenario is correct, we must ask if the SFD of crater populations at that time was the same as that observed in post-mare time, when the observed crater density is low, so that the SFD can be accurately measured. On earlier surfaces, dating from ≳4.2-4.3 Ga, the SFD is uncertain because supersaturation occurred, and details of the SFD shape are masked by supersaturation. However, Delbo et al. (2017) found that the collisional evolution of the asteroidal population (which produces the observed size distribution of asteroids) was already well developed prior to the dynamical instability among the giant planets, and possibly by the time of the formation of the Moon itself -and they concluded that its SFD at that time was already consistent with that of current asteroids (which, in turn, is at least roughly consistent with current craters). We thus see no firm observational or theoretical evidence that the slope of the SFD among the larger impactors in impactor populations before ~4.1-4.2 Ga departed from what we see recorded on unsaturated lunar surfaces today. Changes in typical velocity of craterforming impactors are not expected to have an overwhelming effect on the shape of the SFD. The main point here is that the near-simultaneous approach to saturation by all craters of D ≳1.4 km caused a rapid transition from shallow regolith to deep megaregolith. What are the consequences of saturation and super-saturation for megaregolith evolution? At saturation level, as shown in Fig. 4a and Table 3, column 3, we see that D200% is about 350 m, and D100% is about 4 km. This would suggest effective regolith depths (allowing for the moment two significant figures) in the range of d ~120 m to ~1.3 km -somewhat deeper in some areas, and shallower in other areas, since craters larger than 4 km have more stochastic effects. These estimated regolith depths correspond to surfaces observed today but formed ~4.1 to ~4.2 Ga. Now consider still older surfaces that are supersaturated with craters. Although the supersaturation cratering SFDs are hidden from the view of modern observers, Fig. 4b and Table 3 show how the accretion tail model would produce at least twice the level of the saturation curve on surfaces that formed as early as ~4.2-4.3 Ga -a date from which we have impact melt samples. In this case, as shown in column 4 of Table 3, craters of D >60 km would have covered 100% the surface, and craters of D >5 km would have covered 200% of the surface. These figures suggest that the megaregolith depth produced by 2× saturation would be (by our calculation) in the range of ~2 km to ~20 km. Note the explosive regolith growth within an interval of only ~100 Myr: While crater densities have increased by only factor 2 above saturation, the d100% figure has increased from a depth of ~120 m to 2 km.\\nBecause of the \\\"explosive\\\" growth effect, megaregolith depths increase rapidly as we consider slightly higher degrees of supersaturation on earlier (older) surfaces. On surfaces formed during a period earlier than ~4.3 Ga, when Table 3 indicates 2.5× saturation at D100% = ~16 km to D200% = ~90 km, megaregolith depth, as observed today, would be in the range 5 to 30 km (in areas not excavated by subsequent large impact basins -an effect mentioned above and discussed in the next section).\\nInterestingly, the megaregolith gravels appear frequently to have consolidated into coherent breccia material. Many of the Apollo highland breccias samples, perhaps excavated from depth, are strong, coherent rocks. Coherent materials may have arisen as a result of several different effects. First, impacts in the first hundred Myr or so may have penetrated primarily into molten materials, regionally stirring them and mixing them with shattered crustal materials, so that molten matrix materials may have helped solidify some breccias. Second, pressure effects may have helped cement the heated fragmented materials below some depth. Third, hypervelocity meteorite impacts at various scales produce glassy matrix material that helps to cement rocky fragments into coherent breccias. Wieczorek et al. (2013) and Spray (2016) have discussed such processes in more detail. To summarize, the depths of tens of kilometers, calculated here for impact-affected materials from the geometry of craters, should not necessarily be construed as occupied by loose material; the material may, however, be more porous and less dense than deeper plutonic layers unfractured by impacts. These results are consistent with GRAIL gravitational findings of low densities and higher porosities in the upper few kilometers of the lunar surface. For example, Wieczorek et al. (2013, p. 671) found densities of 2550 kg/m 3 and average crustal porosity of 12% \\\"to depths of at least a few kilometers.\\\" We suggest that these measurements refer to loosely consolidated and/or heavily fractured megaregolith, grading at depth into more coherent breccias and fractured crustal bedrock with very early plutonic igneous intrusions along some of the fractures.\\nTo summarize this section so far, our discussions depart considerably from the \\\"classic\\\" terminal cataclysm/LHB models. To start with, a whole series of impacts at all scales, including a few 10 3 -km-scale basins, would have been forming as the putative magma ocean solidified. The end of magma ocean crystallization is believed to have happened by about 4.45 to 4.35 Ga (Elkins-Tanton et al. 2011;Zhu et al. 2019). Borg et al. (2011Borg et al. ( , 2015 describe an age of 4.360±0.003 Ga for a ferroan anorthosite sample 60025 and discuss it as marking a possible magma ocean crystallization age. However, they also refer to three ferroan anorthosites dated by other authors at 4.43±0.03, 4.47±0.07, and 4.53±0.12 Ga, which could record earlier impact and recrystallization events. Numbers of impact melt clasts, dated at ~4.35 Ga, offer vestiges of one or more large impact structures during that period (Nemchin et al. 2008, Grange et al. 2013, and White et al. 2020; see also Hartmann 2019 for review). Figures 1, 2a , and 3 of our present paper imply very intense bombardment, including multiple impacts of basin-forming magnitude, along with extreme crater oversaturation, during the suggested periods of magma ocean presence. In contrast, the LHB paradigm, from its original version (Wetherill 1975(Wetherill , 1977Ryder 1990) with its period of several hundred million years of negligible impacts before 4.0 Gyr, to the re-defined versions (e.g., Bottke 2012, with LHB reduced in intensity to a surge from 4.1 to 2.5 Ga) all involved a period of relatively low bombardment during the suggested periods of magma ocean presence.\\nOur accretion tail scenario, with early intense bombardment during the putative magma ocean's lifetime and solidification, would produce radically different petrologic results than the LHB models. Some early discussions of the solidification of the putative lunar magma ocean, under conditions pictured in LHB models, assumed the process starting with smooth-layered stratigraphy evolving in the absence of large impacts, with density-controlled layering according to the Bowen reaction series. For example, Walker (2009, p. 109) stated that \\\"…the lunar mantle is likely stratified resulting from magma ocean crystallization and little subsequent mixing.\\\" The accretion tail model of impact history, together with early work on planetary accretion (e.g., Safronov 1972) and our analysis here, indicate that during accretion of the Moon and the final solidification of the magma ocean, the uppermost mantle of the Moon and/or magma ocean could not have solidified in a smoothly \\\"stratified\\\" way with \\\"little subsequent mixing.\\\" Addressing the date of solidification for the magma ocean, Borg et al. (2011) considered their zircon-derived 4.36 Ga date to be surprisingly young and concluded \\\"much of the lunar crust may have been produced by non-magma-ocean processes, such as serial magmatism.\\\" In contrast to LHB scenarios, our proposed early intense bombardment allows an explanation for complex crustal evolution. serial magmatism, and HSE anomalies. Large-scale basin impacts, would have broken up newly forming surface crustal layers in many regions and ejected masses of deeper material into widespread megaregolith layers, as well as triggering magmatic eruptions. Impactfracturing and impact-stirring of deep magmas may have been a factor in allowing HSE to collect in the lunar interior. Magma at depth from would have been stirred and redistributed during the period from, say, 4.35 Ga to ~4.1 Ga. Figure 2b, representing the intense impact regime about 4.35 Ga, suggests impact basins of >300 km diameter forming stochastically at intervals on the order of ten million years in random lunar locations at that time. Each one would have excavated to depths of many tens of km and dumped ejected fragmental material over wide areas of the Moon. Consistent with this, Wieczorek and Phillips (1999) suggested that large impacts may have disturbed the crustal structure being formed by magma ocean crystallization. Gross and Joy (2016) and Pernet-Fisher and Joy (2016) also discussed effects of impacts on magma ocean evolution and crustal formation. Rolf et al. (2017) suggested that accumulating ejecta deposits (megaregolith) from impact basins scattered over time may have an insulating effect, influencing the rate of heat flow from the interior.\"\n },\n {\n \"title\": \"CONSEQUENCES OF LARGEST EARLY IMPACTS VIS À VIS HETEROGENEITY OF THE MEGAREGOLITH\",\n \"content\": \"The formation of multi-ring basins approaching 1000 km diameters, scattered in space and time during the first 600 Myr, created localized effects on the depth of megaregolith (which included mixed layers of local and distant basin ejecta). Such giant basins also affect the observable survival of earlier, nearby basins, by obliterating parts of rim structures and/or burying parts of them in ejecta. The accretion tail model with early intense bombardment thus casts the South Polar Aitken Basin and the controversial Oceanus Procellarum in a new light. Figures 1, 2a, and Table 1 suggest that under our proposed cratering rates before ~4.3 Ga, several giant impact basins, in the 10 3 km diameter (outer ring) range would probably have formed.\\nThe Procellarum area was once considered to be a candidate for such a primordial impact basin, and this hypothesis gains plausibility in the context of our discussion. The GRAIL team discovered a 3000-km diameter \\\"ring\\\" of linear gravitational anomalies under the outer part of Oceanus Procellarum, interpreted as intrusion-filled fractures -which might seem to support that Procellarum is a form of a multi-ring impact feature. They suggested, however, that since these features comprise about five or six linear segments, they are not related to the \\\"rings\\\" of faults around multi-ring impact basins (Andrews-Hanna et al. 2014). Thus they argued against an impact origin for Procellarum. Figure 5, however, shows ~30-40-degree linear segments in some rings around the Orientale impact basin. Figure 6 shows similar linear segments in the ring structures around the Humboldtianum impact basin. Note that if a truly circular mass of material along a crater rim is trying to slide downward into a basin interior along a conical fault, it is inhibited from sliding because it is buttressed by its own material. Linear normal fault segments make it easier for such material to slip, in segments, down into the crater interior. Furthermore, a Procellarum-scale impact involves restructuring the spherical curvature of the Moon, and we have little knowledge of how a spherical planetary body's brittle and/or plastic lithospheric layers adjust to world-scale impact craters. Our conclusion is that linear subsurface features at Procellarum do not argue against an impact origin for that large lunar feature. Zhu et al. (2017) also argue in favor of a collisional origin of Procellarum because this event would help explain the dichotomy in crustal thickness between the near-side and far-side of the Moon. Figure 6. Humboldtianum multi-ring basin. Photo (a) and topographic map (b) reveal that the southern half of the outer ring (boundary between orange highlands surrounding and green, depressed interior), shows three clear linear segments. Diameter of outer ring is ~620 km. NE half portion of that ring and the basin are overlain by a large crater. (a) Discovery rectified photo by WKH, from Hartmann and Kuiper (1962). This early image is shown in order to illustrate that these linear features have been apparent since the early 1960s. (b) NASA; \\\"LOLA\\\" topographic map: NASA/Goddard Space Flight Center.\\nAnother consequence of the accretion tail model with early intense bombardment is that the earliest, extremely high impact rates such as 10,000 times the present rate, go beyond the rates typically contemplated in the terminal cataclysm/LHB model. We have already mentioned that the earliest basins would have been beaten into presently undetectable states. Figure 7 illustrates the effectiveness of that process. It shows a vivid example of an older basin, Serenitatis, whose rim has been broken into linear ridges and partly demolished by ejecta or faulting associated with the younger, neighboring Imbrium basin. The mountains of the Serenitatis rim are laced by ridges and valleys structured not radial to Serenitatis but radial to Imbrium, whose center is ~980 km away, beyond the lower edge of the frame. The destruction here is also ~270 to ~390 km beyond the Apennine rim of the Imbrium basin. The precise mechanism for this reshaping of the Serenitatis rim by the Imbrium impact is still not certain. Tectonic fractures caused by the Imbrium impact, with collapses during lava flooding of the area, seem a possibility. A second possibility might be gouging by local grazing, low-angle rock masses in ejecta ejected from Imbrium, possibly embedded in a somewhat volatilized mass like a base surge. The second mechanism seems perhaps less preposterous when one considers that the \\\"mountain ranges\\\" creating basin rims are, to some extent, piled-up, ejected megaregolith gravels. Whatever the cause, the degree of Serenitatis destruction by Imbrium is remarkableand all the more so since (as discussed in more detail by Hartmann 2019, pp. 19-20, Table 1) there is at least one case where researchers relying on radiometric dating of Apollo samples listed Serenitatis as about one error bar younger than Imbrium. (Most proposed dating of Serenitatis put it within an error bar to two older than Imbrium.) Such young dates reported for Serenitatis may, however, have involved the Imbrium ejecta that modified the Serenitatis rim (see Hartmann 2019 for more detailed discussion).\"\n },\n {\n \"title\": \"CONSEQUENCES OF MEGAREGOLITH EVOLUTION VIS-À-VIS ROCK SAMPLES COLLECTED ON PRESENT-DAY LUNAR AND PLANETARY SURFACES\",\n \"content\": \"The evolution of the megaregolith, and its current properties, have important effects on the rocks and soil samples collected on the present-day surface of the Moon, not to mention other planetary surfaces. To take impact melt samples first, the high formation rate of early basins and large craters had important effects on any impact melts formed in the first few hundred Myr. The large, multi-ring basins are believed to contribute most of the volume of lunar impact melts (Artemieva and Shuvalov 2008), but the exact disposition and depth of impact melts within basins appear to be uncertain. For example, based on terrestrial observations and theoretical modeling, Cintala and Grieve (1998) depict impact melt sheets as layers much thinner than the transient cavity basin depth, spread across the basin interior. Artemieva and Shuvalov (2008) and others suggest more concentration in deeper plugs under the middle of the crater. Impact melt plugs as deep as 20 km have been suggested under Imbrium. It is implicit in either case that impact melts are created and distributed in more or less surficial layers having to do with impacts and ejecta blankets. We have shown, however, that in the accretion tail model, creation of the initial megaregolith in the first few hundred Myr involved pulverization to depths of tens of kilometers, probably deeper than most impact melts. As also shown in Fig. 3, surfaces and near-surface structures formed around 4.15 Ga are only slightly supersaturated. By about 4.0 or 3.9, near-surface structures such as impact melt lenses in the upper few kilometers are more likely to survive. Furthermore, as indicated in Fig. 3 and Fig. 2a, structures formed around 4.0 Ga no longer become saturated with large craters. In short, the accretion tail scenario, contrary to terminal cataclysm/LHB models, explains why impact melts older than ~4.1 Ga are hard to find, while increasing numbers of impact melts are found from ~4.0 Ga to 3.9 Ga. The number of impact melts decreases after that because the impact rate itself was declining. In this view, the unique aspect of 3.9 Ga is not a sudden narrow spike in impact rate, but rather that this date marks the beginning of the period when massive formations on the scale of impact melt lenses will survive -not to mention that the Imbrium and Orientale basins formed around that time.\\nBut what about the fact that primordial crustal samples date back as far as 4.44-4.45 Ga? Why weren't they also destroyed? The answer lies in the fact that the base of the megaregolith, by definition, lies as deep as the deepest pulverization and fragmentation, reaching into more intact primordial rock layers. According to our scenario of early intense bombardment and accretion tail, the megaregolith reached tens of kilometers depth. Richardson and Abramov (2020) emphasize different layers in the megaregolith, with a similar conclusion. They picture a surface dusty regolith layer typically about 20 m deep, underlain by an \\\"upper megaregolith\\\" from ~20 m to a depth of 1 to 3 km, consisting of depositional layers of \\\"brecciated and/or melted material…characterized by the transport and deposition of material via either transient crater gravitational collapse or impact ejecta.\\\" Under the \\\"upper regolith,\\\" they picture a \\\"lower megaregolith\\\" layer, running from ~1-3 km at the top to \\\"about 20-25 km,\\\" consisting of bedrock that has been fractured in place. In all such models there is a semi-infinite supply of primordial, lower crustal or upper mantle plutonic rocks below the bottom of the megaregolith. Thus, the largest modern craters, whose transient cavities exceed roughly 20 km depth, excavate samples of primordial crustal or upper mantle rocks and scatter them on the surface. This excavation process is aided by additional aspect of megaregolith evolution: It is heterogeneous in the sense that the biggest basin impacts blow away much of the accumulated megaregolith under their floors, creating \\\"thin spots.\\\" Those areas were then coated with lavas, but even modest-sized craters on floors of basins can penetrate the lavas and eject ancient, deepseated rocks. Craters with bright ray systems, associated with Imbrium and postdating mare lava plains, such as Autolycus (diameter D = 81 km) and Aristillus (D = 55 km) (both inside the Imbrium's Apennine ring) are good examples. Copernicus (D = 93 km) and Eratosthenes (D = 59 km), somewhat outside that ring, are also candidates for ejecting early crustal samples.\\nOnce a rock has been expelled onto the lunar surface, how long does it last? This question was not much considered when the Apollo missions were planned and the first samples were collected, because the Moon was thought to lack erosive processes. However, a number of diverse studies show that lunar rock samples typically have cosmic ray exposure ages of tens of Ma to a few hundred Ma, and meter-scale rocks typically erode due to the \\\"sandblasting\\\" flux of small meteorites with a few hundred Ma. Moreover, craters of D <12 km (depth ≲2 km) typically produce rocky ejecta blankets in lunar maria (from lava layers) but not so much in lunar highlands (where the subsurface is mostly pulverized). For discussion of such effects, see Basilevsky et al. 2013;Ghent et al. 2014;Costello et al. 2018;Hartmann 2019. Thus, in collecting lunar rock samples, we are entirely at the mercy of excavation processes in the megaregolith; megaregolith properties \\\"filter\\\" what we can collect on the surface.\"\n },\n {\n \"title\": \"FUTURE WORK\",\n \"content\": \"We regard our current work as a demonstration of concept. In the past there has been some separation of sub-disciplines in planetary science. Research in sample petrology, radiometric dating, impact crater studies, numerical modeling of interplanetary impact fluxes have typically involved different research groups, different meetings, different sessions at large annual meetings, and different journals. One area of fruitful future research may attempt to combine better the insights from (1) dynamical models of accretion, planetesimal collisions, and planetesimal fragmentation events, (2) the resulting geological consequences for lunar and planetary cratering rates as a function of time, (3) studies of the processes of excavation and deposition of impact debris. We note that such work will apply to more or less airless bodies throughout at least the asteroid belt and inner solar system, including Mars. At the same time, we caution that to do more detailed modeling of megaregolith evolution we need better quantitative information on still-vague parameters such as (1) extent and depth of fracturing and pulverization of rock during impacts of various sizes; (2) velocity distributions for the different kinds of meteorites hitting the Moon at any given time; (3) cross-section and volume of transient cavities during impacts of various sizes, especially multi-ring basins; (4) cross-section and depth of impact melt lenses on or under the floors of multi-ring impact basins; (5) deposition depths, compositions, and homogeneity of structure in ejecta blankets as a function or distance from crater center for craters of various sizes (especially looking at sources of ejected KREEP materials and uniqueness of KREEP material to the Imbrium region); (6) degree and causes of cementation of fractured and pulverized megaregolith materials as a function of depth; (7) nature of rocks excavated from the base of the megaregolith by the largest recent craters in terms of our access to the original lunar crust or the upper mantle; (8) survival times of excavated rocks as a function of sizes on the lunar (or planetary) surfaces (i.e., erosion histories of samples accessible on the surface.\"\n },\n {\n \"title\": \"CONCLUSIONS\",\n \"content\": \"The concepts of \\\"early intense bombardment\\\" and the \\\"accretion tail\\\" dynamical model (Morbidelli et al. 2018) provide a coherent view of age distributions of radiometric dates of lunar primordial crustal rocks vs. impact melts, GRAIL data on crustal structures, and early evolution of lunar features and multi-ring basin morphologies. They also clarify phenomena not adequately explained by the terminal cataclysm/LHB paradigm. Among the implications of our study:\\n• The solidification of the magma ocean was affected by extremely high cratering rates during that early era. This prevented \\\"smooth,\\\" layered deposits based on Bowen reaction series in at least some areas, and may help explain complex petrology among the most ancient samples from various regions of the present-day Moon. • If a surface formed (or attempted to form) by solidification of a magma ocean during the extreme bombardment as early as >4.4 to 4.35, it would have experienced supersaturation of multi-kilometer diameter crater size-frequency distributions by about 4.35 to 4.3 Ga. • Surviving remnants of such early surfaces, i.e., between the largest impact basins, would have continued to accumulate even more craters, so that early crater densities would have gone into supersaturation, probably by factors at least 2.5 to 10, resulting in pulverization of the earliest impact melts. This result differs from predictions of the terminal cataclysm/LHB models, and helps explain the dearth of impact melt clasts from before ~4.35 Ga. • Continuing bombardment in the first few hundred Myr of crust existence produced reworking and redistribution of materials in upper megaregolith layers, consistent with GRAIL measurements of low density and high porosity in the upper kilometers, Richardson and Abramov's (2020) modeling, and age distribution of earliest known impact melts. • The surface today would display localized regions destroyed and resurfaced by multi-ring basin-scale impacts, typically filled with basaltic lavas, all in various stages of degradation by subsequent cratering. This scenario is consistent with the presence of the large, very degraded South Polar Aitken basin, and with Oceanus Procellarum being a very early giant impact feature. • Subsurface linear features comprising a \\\"ring\\\" around Oceanus Procellarum are not a compelling argument against Procellarum being an impact feature. • The oldest lava-plain formations, for example, in South Polar Aiken, are covered by a veneer of high-albedo ejecta and overlapping rays, and were often mapped in the pre-Apollo and early post-Apollo eras as \\\"highlands.\\\" They may, however, have quite different megaregolith composition (and other properties) than highland regions that survive from earliest crustal formation • The supersaturation-pulverization of the upper kilometers would have comminuted the impact melt lenses produced in the surface layers of the early impact basins, explaining the presence of impact melts from ~ 4.1 to 4.35 Ga as fine-scale clasts in lunar breccias. • Surviving surfaces formed around 4.1-4.2 Ga have reached at least 1.0× saturation crater density.\\n• Surfaces formed after about 4.0-3.9 Ga are not quite saturated with craters, and experienced much lower regolith production rates than earlier surfaces. Survival times of rocks in surface layers rapidly increase at this time. This explains the survival of many impact melt rock samples after ~4.0-~3.9 Ga as contrasted with few before that time. • The blasting away of megaregolith layers by a few large, late basin scale impacts (especially Imbrium) created thin spots in the megaregolith where recent Copernicusscale and Autolycus-scale craters can eject primordial crustal rocks from the base of the local megaregolith, which in many instances is covered by mare basalt layers. This explains the presence of primordial plutonic crustal rocks in present-day surface sample collections (in spite of the paucity of impact melt rocks before ~4.0 Ga). • Many concepts in this paper, regarding megaregolith evolution and sample collection apply not only to the Moon but to all relatively airless planetary bodies in the inner solar system (Mercury, moons, asteroids, and even Mars), and at least into the main asteroid belt, and probably beyond, depending on the histories of asteroid/comet impact rates in the outer solar system. • We are still in an era where first-significant-figure issues remain to be considered, even while assessing certain processes with numerical models citing two-and three-significantfigures.\"\n },\n {\n \"title\": \"Figure 3\",\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 7 .\",\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":"model. See text. **The accumulated crater numbers in the last bins of the table should probably be higher. This involves intersection of two different processes of scattering interplanetary bodies in the Morbidelli et al. (2018) model, around 3 Ga. (See text for discussion.)","content":"model. See text. **The accumulated crater numbers in the last bins of the table should probably be higher. This involves intersection of two different processes of scattering interplanetary bodies in the Morbidelli et al. (2018) model, around 3 Ga. (See text for discussion.)"},{"id":"","caption":"shows a plot of the data in","content":"Figure 3\n3shows a plot of the data in"},{"id":"","caption":"A plot of data from","content":"Figure 3 .\n3A plot of data from"},{"id":"","caption":"Estimated areas covered by craters larger than given diameters (indicated by vertical \"%\" figures. Fine-dashed SFD curves are isochrons showing observed crater SFDs as inFig. 2a. Solid saturation line shows maximum observable crater densities reached on airless planetary bodies, based on empirical measurements. Note that when a given D bin reaches the saturation line it does not mean that craters larger than that D cover 100% of the Moon (see text). Heavy dotted curve in each figure shows actual number of impacts and craters formed, eventually rising above the saturation for small enough craters and large enough ages. Along the heavy dotted curves, calculations are made of cumulative area covered by craters larger than D, starting at lower right corner. (a): Isochron diagram for area just reaching saturation.","content":"Figure 4 .\n4Estimated areas covered by craters larger than given diameters (indicated by vertical \"%\" figures. Fine-dashed SFD curves are isochrons showing observed crater SFDs as inFig. 2a. Solid saturation line shows maximum observable crater densities reached on airless planetary bodies, based on empirical measurements. Note that when a given D bin reaches the saturation line it does not mean that craters larger than that D cover 100% of the Moon (see text). Heavy dotted curve in each figure shows actual number of impacts and craters formed, eventually rising above the saturation for small enough craters and large enough ages. Along the heavy dotted curves, calculations are made of cumulative area covered by craters larger than D, starting at lower right corner. (a): Isochron diagram for area just reaching saturation."},{"id":"","caption":"Orientale basin, in an orbital photo (a) and a topographic map (b) showing linear segments in the surrounding rings. Three linear segments are pronounced along the north (top) third of the outer (930 km diameter) ring, especially at top of the topographic map. The outer ring and next inner ring are formed by linear segments on ENE side. Note in 5(a) both of those segments have a narrow strip of dark lava at the inner foot of the scarp, suggesting a fault plane that allowed magma to intrude along the fault and then extrude onto the surface. An outer ring of diameter 1300 km has also been mapped, but is not easily visible here. (a) NASA Lunar Reconnaissance Orbiter Camera WAC image [NASA/Goddard Space Flight Center]. (b) Laser altimeter \"LOLA\" topographic map from NASA/Goddard Space Flight Center.)","content":"Figure 5 .\n5Orientale basin, in an orbital photo (a) and a topographic map (b) showing linear segments in the surrounding rings. Three linear segments are pronounced along the north (top) third of the outer (930 km diameter) ring, especially at top of the topographic map. The outer ring and next inner ring are formed by linear segments on ENE side. Note in 5(a) both of those segments have a narrow strip of dark lava at the inner foot of the scarp, suggesting a fault plane that allowed magma to intrude along the fault and then extrude onto the surface. An outer ring of diameter 1300 km has also been mapped, but is not easily visible here. (a) NASA Lunar Reconnaissance Orbiter Camera WAC image [NASA/Goddard Space Flight Center]. (b) Laser altimeter \"LOLA\" topographic map from NASA/Goddard Space Flight Center.)"},{"id":"","caption":"Two views of south rim of Serenitatis basin (rugged terrain from bottom to top along center of image), showing degradation by linear structural features radial to Imbrium basin (centered ~270 km off the bottom of the image; north at left). The pattern of ridges radial to Imbrium extends as far as the Apollo 16 site, where Imbrium-age material was collected. Bottom to top length of each image ~350 km. (Photos by WKH, during Lunar Rectified Atlas project ca. 1962, University of Arizona, Lunar and Planetary Lab.)","content":"Figure 7 .\n7Two views of south rim of Serenitatis basin (rugged terrain from bottom to top along center of image), showing degradation by linear structural features radial to Imbrium basin (centered ~270 km off the bottom of the image; north at left). The pattern of ridges radial to Imbrium extends as far as the Apollo 16 site, where Imbrium-age material was collected. Bottom to top length of each image ~350 km. (Photos by WKH, during Lunar Rectified Atlas project ca. 1962, University of Arizona, Lunar and Planetary Lab.)"},{"id":"","caption":"Cratering rate relative to present impact rate based on Morbidelli et al.","content":"Table 1 .\n1Cratering rate relative to present impact rate based on Morbidelli et al.(2018)"},{"id":"","caption":"Calculation of early crater accumulation. (Deriving the accumulated number of craters/km 2 starting at specified date, expressed in terms of average mare density* = 1.)","content":"Table 2 .\n2Calculation of early crater accumulation. (Deriving the accumulated number of craters/km 2 starting at specified date, expressed in terms of average mare density* = 1.)Age \n(Ga) \n\n∆N(1) \n1 km \ncraters \nformed \nper km 2 \nduring \n100 Myr \ninterval \nspecified \nin column \n1 \n\n∆N(1) \nrelative \nto \naverage \nmare, \nwhere \nN(1) \ncrater \ndensity \n[= 3.2 \n(10 -3 )]* \n\nStart at \n4.55 Ga \n(if Moon \nexisted) \n\nStart \nat \n4.45 \nGa \n\nStart \nat \n4.35 \nGa \n\nStart \nat \n4.25 \nGa \n\nStart \nat \n4.05 \nGa \n\nStart \nat \n3.85 \nGa \n\nStart \nat \n3.65 \nGa \n\nStart \nat \n3.45 \nGa \n\n4.55 \n0 \n4.5 \n31.7 \n9906 \n4.45 \n9906 \n0 \n4.4 \n1.21 \n378 \n4.35 \n10280 \n378 \n0 \n4.3 \n0.33 \n103 \n4.25 \n10380 \n481 103 \n0 \n4.2 \n0.12 \n37 \n4.15 \n10420 \n518 140 \n37 \n4.1 \n0.052 \n16.1 \n4.05 \n10440 \n534 156 \n53 \n0 \n4.0 \n0.024 \n7.62 \n3.95 \n10440 \n541 164 \n61 \n7.62 \n3.9 \n0.013 \n4.13 \n3.85 \n545 168 \n65 \n11.8 \n0 \n3.8 \n6.2 (-3) \n1.94 \n\n3.75 \n170 \n67 \n13.8 1.94 \n3.7 3.8 (-3) \n1.19 \n3.65 \n* \n68 \n14.8 3.13 0 \n3.6 2.3 (-3) \n0.718"},{"id":"","caption":"Cumulative area covered by craters larger than diameter D among crater populations reaching the saturation equilibrium line, and also reaching 2.0× and 2.5× saturation equilibrium levels. Bold face type in columns 3, 5, and 6 emphasize conditions where ~100% and ~200% of area are covered by craters.","content":"Table 3 .\n3Cumulative area covered by craters larger than diameter D among crater populations reaching the saturation equilibrium line, and also reaching 2.0× and 2.5× saturation equilibrium levels. Bold face type in columns 3, 5, and 6 emphasize conditions where ~100% and ~200% of area are covered by craters.Crater \ndiameter D \n(km) \nmarking \nboundaries \nof \ndiameter \nbins \n\nPercentage \narea \ncovered by \ncraters \nwithin \nspecified bin \nwhen crater \nsaturation is \nreached at \nD>1.4 km \n\nCumulative % \nof area in bin \ncovered by \nall craters of \ndiameter >D \nwhen \nsaturation is \nreached \n(according to \ndiscussion in \ntext, the \nyoungest \nexamples of \nsaturated \nsurfaces \nformed at 4.1 \nto 4.2 Ga) \n\nPercentage \narea of \ncraters/km 2 \nin bin when \n2× crater \nsaturation is \nreached at \nD>1.4 km \n\nCumulative \n% of area \ncovered by \ncraters of \ndiameter >D \nat 2× \nsaturation \n(according \nto Fig. 3, the \nyoungest \nexamples \noccur on \nsurfaces \nformed at \n~4.3 Ga) \n\nCumulative % \nof area \ncovered by \ncraters of \ndiameter >D \nat 2.5× \nsaturation \n(according to \nFig. 3, the \nyoungest \nexamples \nwould be >4.3 \nGa) \n\n0.25 \n\n48% \n227% \n96% \n456% \n567% \n\n0.353 \n\n30% \n179% \n60% \n360% \n447% \n\n0.5 \n\n17% \n149% \n34% \n300% \n372% \n\n0.707 \n\n13% \n132% \n26% \n266% \n330% \n\n1 \n\n6% \n119% \n12% \n240% \n297% \n\n1.41"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"model. See text. **The accumulated crater numbers in the last bins of the table should probably be higher. This involves intersection of two different processes of scattering interplanetary bodies in the Morbidelli et al. (2018) model, around 3 Ga. (See text for discussion.)\",\n \"content\": \"model. See text. **The accumulated crater numbers in the last bins of the table should probably be higher. This involves intersection of two different processes of scattering interplanetary bodies in the Morbidelli et al. (2018) model, around 3 Ga. (See text for discussion.)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"shows a plot of the data in\",\n \"content\": \"Figure 3\\n3shows a plot of the data in\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A plot of data from\",\n \"content\": \"Figure 3 .\\n3A plot of data from\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Estimated areas covered by craters larger than given diameters (indicated by vertical \\\"%\\\" figures. Fine-dashed SFD curves are isochrons showing observed crater SFDs as inFig. 2a. Solid saturation line shows maximum observable crater densities reached on airless planetary bodies, based on empirical measurements. Note that when a given D bin reaches the saturation line it does not mean that craters larger than that D cover 100% of the Moon (see text). Heavy dotted curve in each figure shows actual number of impacts and craters formed, eventually rising above the saturation for small enough craters and large enough ages. Along the heavy dotted curves, calculations are made of cumulative area covered by craters larger than D, starting at lower right corner. (a): Isochron diagram for area just reaching saturation.\",\n \"content\": \"Figure 4 .\\n4Estimated areas covered by craters larger than given diameters (indicated by vertical \\\"%\\\" figures. Fine-dashed SFD curves are isochrons showing observed crater SFDs as inFig. 2a. Solid saturation line shows maximum observable crater densities reached on airless planetary bodies, based on empirical measurements. Note that when a given D bin reaches the saturation line it does not mean that craters larger than that D cover 100% of the Moon (see text). Heavy dotted curve in each figure shows actual number of impacts and craters formed, eventually rising above the saturation for small enough craters and large enough ages. Along the heavy dotted curves, calculations are made of cumulative area covered by craters larger than D, starting at lower right corner. (a): Isochron diagram for area just reaching saturation.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Orientale basin, in an orbital photo (a) and a topographic map (b) showing linear segments in the surrounding rings. Three linear segments are pronounced along the north (top) third of the outer (930 km diameter) ring, especially at top of the topographic map. The outer ring and next inner ring are formed by linear segments on ENE side. Note in 5(a) both of those segments have a narrow strip of dark lava at the inner foot of the scarp, suggesting a fault plane that allowed magma to intrude along the fault and then extrude onto the surface. An outer ring of diameter 1300 km has also been mapped, but is not easily visible here. (a) NASA Lunar Reconnaissance Orbiter Camera WAC image [NASA/Goddard Space Flight Center]. (b) Laser altimeter \\\"LOLA\\\" topographic map from NASA/Goddard Space Flight Center.)\",\n \"content\": \"Figure 5 .\\n5Orientale basin, in an orbital photo (a) and a topographic map (b) showing linear segments in the surrounding rings. Three linear segments are pronounced along the north (top) third of the outer (930 km diameter) ring, especially at top of the topographic map. The outer ring and next inner ring are formed by linear segments on ENE side. Note in 5(a) both of those segments have a narrow strip of dark lava at the inner foot of the scarp, suggesting a fault plane that allowed magma to intrude along the fault and then extrude onto the surface. An outer ring of diameter 1300 km has also been mapped, but is not easily visible here. (a) NASA Lunar Reconnaissance Orbiter Camera WAC image [NASA/Goddard Space Flight Center]. (b) Laser altimeter \\\"LOLA\\\" topographic map from NASA/Goddard Space Flight Center.)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Two views of south rim of Serenitatis basin (rugged terrain from bottom to top along center of image), showing degradation by linear structural features radial to Imbrium basin (centered ~270 km off the bottom of the image; north at left). The pattern of ridges radial to Imbrium extends as far as the Apollo 16 site, where Imbrium-age material was collected. Bottom to top length of each image ~350 km. (Photos by WKH, during Lunar Rectified Atlas project ca. 1962, University of Arizona, Lunar and Planetary Lab.)\",\n \"content\": \"Figure 7 .\\n7Two views of south rim of Serenitatis basin (rugged terrain from bottom to top along center of image), showing degradation by linear structural features radial to Imbrium basin (centered ~270 km off the bottom of the image; north at left). The pattern of ridges radial to Imbrium extends as far as the Apollo 16 site, where Imbrium-age material was collected. Bottom to top length of each image ~350 km. (Photos by WKH, during Lunar Rectified Atlas project ca. 1962, University of Arizona, Lunar and Planetary Lab.)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Cratering rate relative to present impact rate based on Morbidelli et al.\",\n \"content\": \"Table 1 .\\n1Cratering rate relative to present impact rate based on Morbidelli et al.(2018)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Calculation of early crater accumulation. (Deriving the accumulated number of craters/km 2 starting at specified date, expressed in terms of average mare density* = 1.)\",\n \"content\": \"Table 2 .\\n2Calculation of early crater accumulation. (Deriving the accumulated number of craters/km 2 starting at specified date, expressed in terms of average mare density* = 1.)Age \\n(Ga) \\n\\n∆N(1) \\n1 km \\ncraters \\nformed \\nper km 2 \\nduring \\n100 Myr \\ninterval \\nspecified \\nin column \\n1 \\n\\n∆N(1) \\nrelative \\nto \\naverage \\nmare, \\nwhere \\nN(1) \\ncrater \\ndensity \\n[= 3.2 \\n(10 -3 )]* \\n\\nStart at \\n4.55 Ga \\n(if Moon \\nexisted) \\n\\nStart \\nat \\n4.45 \\nGa \\n\\nStart \\nat \\n4.35 \\nGa \\n\\nStart \\nat \\n4.25 \\nGa \\n\\nStart \\nat \\n4.05 \\nGa \\n\\nStart \\nat \\n3.85 \\nGa \\n\\nStart \\nat \\n3.65 \\nGa \\n\\nStart \\nat \\n3.45 \\nGa \\n\\n4.55 \\n0 \\n4.5 \\n31.7 \\n9906 \\n4.45 \\n9906 \\n0 \\n4.4 \\n1.21 \\n378 \\n4.35 \\n10280 \\n378 \\n0 \\n4.3 \\n0.33 \\n103 \\n4.25 \\n10380 \\n481 103 \\n0 \\n4.2 \\n0.12 \\n37 \\n4.15 \\n10420 \\n518 140 \\n37 \\n4.1 \\n0.052 \\n16.1 \\n4.05 \\n10440 \\n534 156 \\n53 \\n0 \\n4.0 \\n0.024 \\n7.62 \\n3.95 \\n10440 \\n541 164 \\n61 \\n7.62 \\n3.9 \\n0.013 \\n4.13 \\n3.85 \\n545 168 \\n65 \\n11.8 \\n0 \\n3.8 \\n6.2 (-3) \\n1.94 \\n\\n3.75 \\n170 \\n67 \\n13.8 1.94 \\n3.7 3.8 (-3) \\n1.19 \\n3.65 \\n* \\n68 \\n14.8 3.13 0 \\n3.6 2.3 (-3) \\n0.718\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Cumulative area covered by craters larger than diameter D among crater populations reaching the saturation equilibrium line, and also reaching 2.0× and 2.5× saturation equilibrium levels. Bold face type in columns 3, 5, and 6 emphasize conditions where ~100% and ~200% of area are covered by craters.\",\n \"content\": \"Table 3 .\\n3Cumulative area covered by craters larger than diameter D among crater populations reaching the saturation equilibrium line, and also reaching 2.0× and 2.5× saturation equilibrium levels. Bold face type in columns 3, 5, and 6 emphasize conditions where ~100% and ~200% of area are covered by craters.Crater \\ndiameter D \\n(km) \\nmarking \\nboundaries \\nof \\ndiameter \\nbins \\n\\nPercentage \\narea \\ncovered by \\ncraters \\nwithin \\nspecified bin \\nwhen crater \\nsaturation is \\nreached at \\nD>1.4 km \\n\\nCumulative % \\nof area in bin \\ncovered by \\nall craters of \\ndiameter >D \\nwhen \\nsaturation is \\nreached \\n(according to \\ndiscussion in \\ntext, the \\nyoungest \\nexamples of \\nsaturated \\nsurfaces \\nformed at 4.1 \\nto 4.2 Ga) \\n\\nPercentage \\narea of \\ncraters/km 2 \\nin bin when \\n2× crater \\nsaturation is \\nreached at \\nD>1.4 km \\n\\nCumulative \\n% of area \\ncovered by \\ncraters of \\ndiameter >D \\nat 2× \\nsaturation \\n(according \\nto Fig. 3, the \\nyoungest \\nexamples \\noccur on \\nsurfaces \\nformed at \\n~4.3 Ga) \\n\\nCumulative % \\nof area \\ncovered by \\ncraters of \\ndiameter >D \\nat 2.5× \\nsaturation \\n(according to \\nFig. 3, the \\nyoungest \\nexamples \\nwould be >4.3 \\nGa) \\n\\n0.25 \\n\\n48% \\n227% \\n96% \\n456% \\n567% \\n\\n0.353 \\n\\n30% \\n179% \\n60% \\n360% \\n447% \\n\\n0.5 \\n\\n17% \\n149% \\n34% \\n300% \\n372% \\n\\n0.707 \\n\\n13% \\n132% \\n26% \\n266% \\n330% \\n\\n1 \\n\\n6% \\n119% \\n12% \\n240% \\n297% \\n\\n1.41\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Structure and evolution of the lunar Procellarum region as revealed by GRAIL gravity data. 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Received 2019 October 22; in original form 2019 July 25Preprint 18 December 2019 Compiled using MNRAS L A T E X style file v3.0supernovae -cosmological parameters -large-scale structure of Universe\nIn this work we utilise the most recent publicly available type Ia supernova (SN Ia) compilations and implement a well formulated cosmological model based on Lemaître-Tolman-Bondi metric in presence of cosmological constant Λ (ΛLTB) to test for signatures of large local inhomogeneities at z ≤ 0.15. Local underdensities in this redshift range have been previously found based on luminosity density (LD) data and galaxy number counts. Our main constraints on the possible local void using the Pantheon SN Ia dataset are: redshift size of z size = 0.068 +0.021 −0.030 ; density contrast of δΩ 0 /Ω 0 = −10.5 +9.3 −7.4 % between 16th and 84th percentiles. Investigating the possibility to alleviate the ∼ 9% disagreement between measurements of present expansion rate H 0 coming from calibrated local SN Ia and high-z cosmic microwave background data, we find large local void to be a very unlikely explanation alone, consistently with previous studies. However, the level of matter inhomogeneity at a scale of ∼100Mpc that is allowed by SN Ia data, although not expected from cosmic variance calculations in standard model of cosmology, could be the origin of additonal systematic error in distance ladder measurements based on SN Ia. Fitting low-redshift Pantheon data with a cut 0.023 < z < 0.15 to the ΛLTB model and to the Taylor expanded luminosity distance formula we estimate that this systematic error amounts to 1.1% towards the lower H 0 value. A test for local anisotropy in Pantheon SN Ia data yields null evidence. Analysis of LD data provides a constraint on contrast of large isotropic void δΩ 0 /Ω 0 = −51.9% ± 6.3%, which is in ∼ 4σ tension with SN Ia results. More data are necessary to better constrain the local matter density profile and understand the disagreement between SN and LD samples.\n\nINTRODUCTION\n\nThe most reliable standard candles and one of the fundamental low-redshift astrophysical observables (Betoule et al. 2014;Scolnic et al. 2018), type Ia supernovae (SN Ia), also provide a cosmologyindependent measurement of present expansion rate through the local distance ladder (Riess et al. 2016) (hereafter R16). In concert with observations of baryon acoustic oscillation (BAO) peak from galaxy redshift surveys (Eisenstein et al. 2005;Aubourg et al. 2015;Alam et al. 2017;Bautista et al. 2017;du Mas des Bourboux et al. 2017;Zhao et al. 2019a) and cosmic microwave background (CMB) radiation data (Hinshaw et al. 2013;Planck Collaboration et al. 2018a), SN Ia contributed to establishment of the standard model of cosmology. Astonishing concordance between different cosmological observables brought by technical advancements in the last decades was followed by a number of inconsistent and/or unexpected results, e.g. estimated values of matter clustering parameter, S 8 , from low and high redshifts (Planck Collabo-E-mail: vladimir.lukovic@roma2.infn.it † E-mail: haridasu@roma2.infn.it ‡ E-mail: nicola.vittorio@roma2.infn.it ration et al. 2016;Joudaki et al. 2019;Lange et al. 2019;Martinelli & Tutusaus 2019), CMB parameter constraints in curvature extension of base Λ cold dark matter (ΛCDM) model (Planck Collaboration et al. 2018b) (hereafter P18), as well as several unexplained CMB anomalies (Schwarz et al. 2016), etc. The best-known amongst ΛCDM tensions and one of the biggest challenges in modern cosmology is related to the H 0 value. Defining the scale of extra-galactic distance through the Hubble radius, as well as the present expansion rate, H 0 is a fundamental cosmological parameter both for CMB and low-redshift data (Freedman et al. 2001;Freedman & Madore 2010). The most stringent ΛCDM estimate coming from P18, H 0 = 67.36 ± 0.54 km s −1 /Mpc, is in 4.4σ tension with the most recent local measurement based on calibrated SN Ia, H 0 = 74.03 ± 1.42 km s −1 /Mpc, by Riess et al. (2019) (hereafter R19). In comparison, H 0 measurements from the time delay of strongly lensed quasars support a higher value (Birrer et al. 2019;Wong et al. 2019); using differential age method with cosmic chronometers (Jimenez & Loeb 2002) provides an estimate in the middle of two results in tension (Jimenez et al. 2019;Chen et al. 2017;Luković et al. 2016); while the constraints coming from gravitational waves are still fairly weak (Liao et al. 2017). Although the quality of BAO data has substantially improved over the last decade, providing some of the most stringent cosmological constraints on expansion dynamics (Haridasu et al. 2018a;Ramanah et al. 2019;Zhang et al. 2019), their H 0 estimate is degenerated with the sound horizon parameter, whose value relies on CMB observations and early Universe physics (Macaulay et al. 2019;Lemos et al. 2019;Aubourg et al. 2015;Cuceu et al. 2019).\n\nThe intercept of SN Ia magnitude-redshift relation is determined by their absolute magnitude (M B ) and the Hubble radius, resulting in two degenerate parameters in SN Ia analyses. While the highredshift SN Ia are necessary for assessing the underlying physics of dark energy, the low-redshift end (z ≤ 0.15) of SN Hubble diagram can be analysed with minimal assumptions about properties of the cosmic fluid and expansion dynamics related only to the well-accepted cosmic acceleration (Haridasu et al. 2017). Indeed, calibrating SN Ia absolute magnitude with the use of Cepheid variable stars and fixing the local distance-redshift relation (at z ≤ 0.15) leads to a remarkable cosmology-independent measurement of H 0 (Riess et al. 2005(Riess et al. , 2009(Riess et al. , 2011(Riess et al. , 2016(Riess et al. , 2018b. Different calibration techniques consider the usage of strong gravitational lensing (Taubenberger et al. 2019), tip of the red giant branch (Beaton et al. 2016;Freedman et al. 2019), HII regions (Fernández Arenas et al. 2018), etc. Due to the strong correlation of cosmic expansion dynamics and H 0 value, the model-independent local estimates remain crucial for the study of late-time cosmic evolution [see also Tutusaus et al. (2019); Haridasu et al. (2018b)].\n\nAs is well-known, a large inhomogeneity in local matter density distribution can affect the geometry of local Universe and the distance measures, consequently biasing the resulting model inferences and the H 0 estimate. A few independent groups used galaxy survey catalogues to test the local matter distribution, reporting hints of a large local void (Keenan et al. 2012(Keenan et al. , 2013Whitbourn & Shanks 2014Boehringer et al. 2019). Concretely, Keenan et al. (2013) (hereafter KBC13) used UKIDSS Large Area Survey (Lawrence et al. 2007), Galaxy and Mass Assembly (GAMA) (Driver & Robotham 2010;Driver et al. 2011) and 2M++ catalogue (Lavaux & Hudson 2011) to construct luminosity density sample (hereafter LD) over the redshift range 0.01 < z < 0.2. Relating it to the total matter density in the local Universe, they find evidence for an underdense region and suggest a void of about 300h 70 −1 Mpc in size and −30% contrast with respect to the background. Whitbourn & Shanks (2014) (hereafter WS14) used 6dFGS, SDSS, and GAMA surveys to examine the local galaxy density field over 20% of the sky and up to 450h 70 −1 Mpc in depth. Both number counts and peculiar velocity fields show evidence for at least 200h 70 −1 Mpc large local underdensity with −15% ± 3% average contrast, ranging between −40% and −5% in different directions. Analysing a catalogue of 1653 galaxy clusters detected by their X-ray emission Boehringer et al. (2019) report evidence for ∼ 140 Mpc large local underdensity with density contrast of 20% ± 10%.\n\nThe results of KBC13, WS14 and Boehringer et al. (2019) go against the standard cosmological paradigm and the assumption of large-scale homogeneity. In fact, indications for a few hundred Mpc large local underdensity encouraged renewed popularity of void models based on Lemaître-Tolman-Bondi (LTB) metric, also motivated as a remedy for the H 0 tension (Moffat 2016;Tokutake et al. 2018;Hoscheit & Barger 2017;Shanks et al. 2019;Kenworthy et al. 2019) and even as a possible explanation for the CMB cold spot (Szapudi et al. 2015). Contrasting the homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) models, the LTB metric (Lemaître 1927(Lemaître , 1933Tolman 1934;Bondi 1947) describes an isotropic but inhomogeneous dust system, which on cosmological scales can be used for studying cold dark matter den-sity distribution. Since the discovery of cosmic acceleration, LTB metric was used to construct toy models that challenge the cosmological constant paradigm with an alternative scenario in which the apparent accelerated expansion is a result of the strongly underdense local Universe that smoothly converges to Einstein de Sitter solution on higher redshifts (Célérier 2000;Alnes et al. 2006;Clifton et al. 2008;February et al. 2010;Nadathur & Sarkar 2011;Bolejko & Sussman 2011;Zhang et al. 2015;Stahl 2016). Since the deceleration-acceleration transition redshift is well constrained at z t ≈ 0.6 [see e.g. Haridasu et al. (2018b); Gómez-Valent (2019); Mukherjee et al. (2019) for model-independent estimates], this alternative explanation requires a giant ( ≈ 3Gpc) isotropic void that proved to be very unlikely compared to the standard cosmological model (Zibin 2011;Vargas et al. 2017;Amendola et al. 2013;Luković et al. 2016). However, extending the LTB model with the addition of cosmological constant Λ, one can describe large (order of 100Mpc) local inhomogeneous matter distribution inside a ΛCDM background (Valkenburg 2012;Valkenburg et al. 2014;Rigopoulos & Valkenburg 2012).\n\nWhile the level of cosmic variance in matter distribution expected for standard model can only partially relieve the Hubble constant problem (Marra et al. 2013;Wu & Huterer 2017;Camarena & Marra 2018), Tokutake et al. (2018) found that a strongly underdense local inhomogeneity with non-vanishing cosmological constant, modeled with an approximate numerical solution for LTB metric, can fully coincide the CMB constraints with the local H 0 measurement. Similarly, Hoscheit & Barger (2017) (hereafter HB17) extended the work of KBC13 by testing the consistency of their earlier findings with constraints coming from SN Ia and linear kinematic Sunyaev-Zel'dovich effect and showed that local void found by KBC13 could reduce the H 0 tension. A recent work of Shanks et al. (2019) argues that the combination of local inhomogeneity effect together with the re-calibration of Cepheids and local SN Ia distances using the recent parallax measurements from Gaia mission, is sufficient to completely remove the tension [see also Riess et al. (2018a); Shanks et al. (2018) for follow-up discussions]. Kenworthy et al. (2019) (hereafter KSR19) find that any inhomogeneity with contrast > 20% is strongly inconsistent with SN Ia data, consequently reassuring the ability to measure the distance and Hubble constant with locally calibrated SN to a 1% precision.\n\nThe goal of this work is to probe the local matter density using the SN Ia as one of the most reliable low-redshift astrophysical observables, constraining size and amplitude of possible matter inhomogeneity at our local position. We investigate the findings of KBC13 and KSR19 by using the complete analytic description for treating the LTB metric in presence of cosmological constant (hereafter ΛLTB) introduced in Valkenburg (2012). Extending the earlier works, we directly fit the ΛLTB model to two biggest publicly available SN Ia samples -Joint light curve analysis Betoule et al. (2014) (hereafter JLA) and Pantheon compilation Scolnic et al. (2018) (hereafter Pan), consisting of 740 and 1048 SN Ia, respectively. We also utilise the luminosity density data from KBC13 within the given formalism to assess the (dis)agreement. In section 2 we review the theoretical description of LTB metric in presence of cosmological constant and proceed by presenting the effect that a local void can have on observable physical quantities in section 3. Constraints on the void contrast and size coming from fitting JLA, Pantheon and KBC13 datasets are reported in section 4, together with the revision of their consistency. Additionally, we discuss on the possible effect of local isotropic inhomogeneity on the H 0 measurement, and explore the level of anisotropy allowed by the data. Summarising our results in section 5, we examine the dif-ferences compared to earlier findings by other teams. Finally, in the appendix A we provide a simplified void model capable of correctly reproducing our results based on complete ΛLTB description to a good approximation.\n\nThroughout the paper we use the dimensionless variable h 70 = H 0 /70 km s −1 /Mpc for distance measures; acronym w.r.t. stands for 'with respect to'.\n\n\nLOCAL MATTER DENSITY IN ΛLTB MODEL\n\nThe simplest extension of standard homogeneous cosmological model in presence of large matter inhomogeneity is done using the LTB metric (Lemaître 1933;Krasiński 1997) that describes spherically symmetric pressureless cosmic fluid\nds 2 = c 2 dt 2 − R r (t, r) 2 1 − k(r)r 2 dr 2 − R(t, r) 2 dΩ 2 .(1)\nHere R(t, r) has the units of length and defines all physical distance measures as well as the expansion history in this model. The second free function of the metric, k(r) is dimensionless and defines the curvature within the shell of a radius r. We note that radial coordinate r is also dimensionless, with no physical meaning, and it should be considered only as a flag coordinate of different shells all centred at r = 0. Using the above metric and the Einstein field equations (FE) for pressureless inhomogeneous matter fluid in presence of cosmological constant Λ, one can derive the generalised Friedmann-Lemaître equation as\nR t R 2 = 2GM R 3 − c 2 k r 2 R 2 + c 2 Λ 3 .(2)\nHere M(r) is an integration constant, whose physical meaning is the total mass enclosed inside a sphere with radial coordinate r,\nM(r) = r 0 4πR(t, r) 2 ρ m (t, r)R r (t, r) dr .(3)\nInhomogeneous cosmic matter density profile, ρ m (t, r), satisfies the conservation law coming from FE, ensuring that integrated mass M(r) remains constant in time. In inhomogeneous cosmology we differ two expansion rates, namely transverse and radial:\nH ⊥ (t, r) = R t R ,(4)H || (t, r) = R tr R r .(5)\nHence, the one appearing in Friedmann-Lemaître eq. (2) is the transverse expansion rate. As usual, we denote t = t 0 as present age of the Universe and all the physical quantities with subscript \"0\" represent their values today, while H 0 (r) = H ⊥ (t 0 , r) is used for the transverse expansion rate. Introducing scale factor, the Friedmann-Lemaître equation can be rewritten in a simpler form, although the definition of scale factor a(t, r) is not trivial nor customary as in the homogeneous cosmology. The value of flag coordinate r is commonly defined through a gauge with some other physical variable. The choice of this gauge is arbitrary and has no implications on the model predictions nor on the results of data analysis. The most often used gauge (Alnes et al. 2006;Alfedeel & Hellaby 2010;Nadathur & Sarkar 2011) in LTB framework is:\nR 0 (r) = ct 0 r ⇒ r ≡ R 0 (r) ct 0 .(6)\nHowever, in this work we use an alternative definition which turns out to be more convenient choice for the latter calculations of elliptical integrals present in the geodesic equations and distance variables. Following Valkenburg (2012), we define the dimensionless radial coordinate r through the relation with the total mass enclosed in a given shell as\nM(r) = 4π 3 C M r 3 =⇒ r ≡ 3 3 4π M(r) C M ,(7)\nwhere C M is a normalisation constant in units of mass whose value will be chosen at a later stage. Commonly, in both gauges the scale factor can be defined as\na(t, r) ≡ R(t, r) ct 0 r .(8)\nThis function clearly converges to the usual scale factor definition and the spatially uniform values in the limit of homogeneous FLRW metric. Generalised Friedmann-Lemaître eq. (2) can be rewritten in terms of three dimensionless density parameters normalised to the present time,\nH ⊥ (t, r) 2 = H 0 (r) 2 Ω m (r) R 0 (r) 3 R(t, r) 3 + Ω k (r) R 0 (r) 2 R(t, r) 2 + Ω Λ (r) ,(9)\nThese can be evaluated combining eqs.\n\n(2), (4) and (7) to (9) as,\nΩ m = 1 H 0 2 2GM R 0 3 = 1 H 0 2 8πGC M r 3 /3 c 3 t 0 3 a 0 3 r 3 =M t 0 2 H 0 2 a 0 3 ,(10)Ω k = − 1 H 0 2 c 2 k r 2 R 0 2 = − k t 0 2 H 0 2 a 0 2 ,(11)Ω Λ = 1 H 0 2 c 2 Λ 3 =Λ t 0 2 H 0 2 .(12)\nHereM = 8πGC M /(3c 3 t 0 ) andΛ = 1 3 Λc 2 t 0 2 are dimensionless constants, whereas k(r), H 0 (r) and a 0 (r) are not uniform in LTB metric. The introduced dimensionless density parameters are different from one shell to another, but do not depend on time.\n\nSince Ω m (r) is proportional to the total mass enclosed in a shell, M(r), it can be represented as the ratio of average and critical matter densities of each shell. Starting from eq. (3) and then using eqs. (7) and (8) one can define average matter density:\nρ m (t, r) = M(r) 4πR(t, r) 3 /3 = C M c 3 t 0 3 1 a(t, r) 3 .(13)\nThen, definition of critical matter density together with eqs. (10) and (13) straightforwardly confirm:\nρ crit ≡ 3H ⊥ 2 8πG =⇒ Ω m (r) = ρ m (t 0 , r) ρ crit (t 0 , r) .(14)\nFriedmann-Lemaître eq. (2) can also be rewritten in dimensionless form using eqs. (10) to (12)\nt 0 2 H ⊥ (t, r) 2 =M a(t, r) 3 − k(r) a(t, r) 2 +Λ .(15)\nIn terms of the Birkhoff theorem, eqs.\n\n(2), (9) and (15) dictate every shell in ΛLTB model to evolve as an independent homogeneous FLRW model with a specific curvature k(r), a total mass enclosed M(r) and the same value of cosmological constant Λ for all shells, determining its transverse expansion rate H ⊥ (t, r). Each shell is described with relative density parameters Ω i (r), present expansion rate H 0 (r), and evolving scale factor a(t, r)/a 0 (r). Cosmological constant, included in the Einstein field equations, is uniform over space and constant in time due to its physical nature, which is also true for the dimensionless constantΛ that we introduced. The density parameter corresponding to the cosmological constant, Ω Λ (r), can not be uniform in ΛLTB model, as this would imply that Λ is inhomogeneous and coupled to the cold dark matter density profile. This approach is reflected in a number of recent studies Valkenburg (2012) 2019), while not fully followed by HB17 who assume cosmic density parameter Ω Λ to be constant throughout the underdense region and at the background. Similarly, the methods of inverse reconstruction of matter density profile in presence of cosmological constant must be constructed such that FE are satisfied (c.f. Tokutake & Yoo (2016); Wojtak & Prada (2017)).\n\nThe gauge choice for the flag coordinate r, which is set by eq. (7), enabled us to rewrite the Friedmann-Lemaître eq. (2) in a more elegant analytic form, i.e. eq. (9). The remaining degree of freedom in the scaling constant C M , or equivalentlyM, can be used to normalise the scale factor a(t, r). We setM such that a * = a (t 0 , r * ) = 1 for a chosen shell r = r * and at the present time t 0 . For example, one can choose r * = 0 to normalise the scale factor at the observer's position or r * = +∞ to normalise at the background, instead. We follow the normalisation of scale factor on the background, setting the numerical value ofM such that a * = a (t 0 , r * = +∞) = 1 for the background shells. In particular, normalisation gives the physical meaning to the scaling constant, i.e. eq. (13) yields\nρ bg m (t) = ρ bg m (t 0 ) a bg (t) 3 =⇒ ρ bg m (t 0 ) = C M c 3 t 0 3 .(16)\nTherefore, normalising far from the inhomogeneity, which is for any large enough r * , implies that background scale factor and eq. (15) converge to the usual forms that we have in homogeneous FLRW model. It is easy to see that at present time t = t 0 the background quantities satisfy\nt 0 H bg 0 = M − k bg +Λ .(17)\nPartial differential eq. (15) can be used to express the derivative ∂t/∂a and integrate from the Big Bang up to scale a:\nt(a, r) − t BB (r) t 0 = a 0 √ α Λ α 3 − k(r) α +M dα .(18)\nAll variables on the right hand side are dimensionless and this elliptical integral is analytically solvable (for details see Valkenburg (2012)). The Big Bang time t BB (r) is an integration constant that may vary in space, affecting the age difference between the shells. Nevertheless, one does not expect large variation of t BB (r) in the case of a small local inhomogeneity. Non-uniform Big Bang time is also related to instabilities of matter perturbations (Zibin 2008). Hence, in this work we consider only the models with homogeneous Big Bang time t BB (r) ≡ const = 0. Integrating eq. (18) on the background (r = r * ) from the Big Bang a = 0 up to today a(t = t 0 , r * ) = 1 we have\n1 = t(a = 1, r * ) t 0 = 1 0 √ α √Λ α 3 − k bg α +M dα,(19)\nwhich is the numerical equation defining a value ofM, necessary for the normalisation. At any other point of interest eq. (18) can be seen as the solution for t(a, r) or, after invertion, as a(t, r). Hence, calculating numerically the scale factor at any spatial position and time gives us the angular diameter distance R(t, r) and other physical observables of interest.\n\n\nEFFECT ON ASTROPHYSICAL OBSERVABLES\n\nThe high-redshift background observables are not affected by the local isotropic inhomogeneity. Nevertheless, the analysis of local observables can yield different constraints on cosmological parameters from those obtained with FLRW models. Examples presented in this section form the basis for analysing observable data, but do not immediately represent the effect on cosmological inferences. We use a specific analytic form of the curvature, parametrised with k bg , k loc , r 0 , ∆r and given as García-Bellido-Haugbølle (GBH) profile (Garcia-Bellido & Haugbølle 2008;February et al. 2010):\nk(r) = k bg − k bg − k loc 1 − tanh r−r 0 2∆r 1 + tanh r 0 2∆r .(20)\nOur focus is the case of ΛCDM background, i.e. k bg ≡ 0, which we call ΛLTB model. The analytic form of k(r) together with the value of dimensionless cosmological constantΛ define the evolution of expansion and all physical quantities in LTB metric. In the limit of ∆r → 0, a simpler top-hat profile (TH) is recovered,\nk(r) = k loc for r < r 0 , k(r) = k bg for r ≥ r 0 .(21)\nWhile GBH form describes a smooth matter density profile, TH profile has one parameter less and is trivial to use for data fitting as two joint homogeneous models. Care must be given to relation between the physical quantities in two homogeneous regions.\n\n\nReconstruction of geodesic\n\nThe LTB metric describes an inhomogeneous dust cloud with spherical symmetry, indicated by the angular term in eq. (1). Hence, the observer located at the centre of this symmetry has an isotropic view. The assumption of isotropy gives us only the first approximation for treating the large cosmological inhomogeneities. To quantify the overall effect of large local void on cosmological inferences, we limit ourselves to the on-centre model that considers an observer located at the centre of symmetry in the local matter inhomogeneity. The assumption of this very special position can be avoided, although the geodesic equations for an off-centre observer become increasingly complex, affecting the feasibility of data analysis. We find the on-centre model as adequate for presenting all the relevant points in this work, but we extend the discussion about observer's spatial position later on. Geodesic equations can be derived following Célérier (2000):\ndr dz = 1 1 + z c √ 1 − k r 2 R tr ,(22)dt dz = − 1 1 + z R r R tr .(23)\nwhere z is the cosmological redshift. This system of differential equations must be solved numerically along the geodesic with help of eqs. (4), (8), (15) and (18), in the entire redshift range. In order to derive distance-redshift relations one needs to numerically reconstruct the function R(z), or equivalently a(z), which can be obtained by numerical inversion of eq. (18) at every z along the geodesic (Valkenburg 2012). However, we find it easier and computationally faster to use the geodesic equations and all physical variables of interest rewritten with (a, r) as independent fundamental coordinates, after the change of variables from (t, r) da dz = a t dt dz + a r dr dz .\n\nAdding the assumption of homogeneity, geodesic equations reduce to the standard form; using eqs. (8) and (23) we have: We note that this relation is valid on all shells where spatial derivative is null, i.e. in every homogeneous subregion along the matter density profile.\ndt dz = − 1 1 + z a + r a r a t + r a tr = − 1 1 + z a a t ⇒ da a = − dz 1 + z .(25)\n\nDensity contrasts\n\nIt is useful to define the relative contrasts for quantities of interest, such as physical matter density, dimensionless density parameter Ω m , and expansion rates. The contrast can be defined at any time and any spatial point inside inhomogeneous region. Consequently, for physical matter density we have\nδρ m ρ m = ρ m (t, r) ρ bg m (t) − 1(26)\nThe evolution of matter density contrast is shown in Fig. 1 for a ΛLTB void model with specific parameters determining cosmological constantΛ, central curvature k loc , size and steep transition zone, given with r 0 and ∆r. As shown in this case, the physical matter density ρ m will have compensated radial profile in the case of small ∆r and/or large central contrast. Since we use the comoving radial distance, the physical expansion of void size can not seen on Fig. 1. Most often we consider the present (t = t 0 ) contrast between observer's position (r = 0) and background (r = + inf) and we denote it with subscript 0. The contrast of dimesionless matter density parameter Ω m (r) and both expansion rates are defined in the same manner as for physical matter density in eq. (26). We will focus on present central contrasts which can be derived using eqs. (3), (7), (8), (10) and (16):\nδρ 0 ρ 0 = ρ loc m (t 0 ) ρ bg m (t 0 ) − 1 = 1 a loc 0 3 − 1 ,(27)δΩ 0 Ω 0 = Ω loc m Ω bg m − 1 = H bg 0 H loc 0 2 1 a loc 0 3 − 1 .(28)\nNotably, these two contrast are different due to the induced spatial inhomogeneity of the expansion rates, In conclusion, relation |δρ 0 /ρ 0 | < |δΩ 0 /Ω 0 | holds both in the case of local underdensity as well as overdensity. Writing the equivalent of eq.\nδH 0 H 0 = H loc 0 − H bg 0 H bg 0 .(29)\n(3) at a(t = t 0 , r = 0) = a loc 0 and after some simple calculations, one can derive analytic expressions that relate these three contrasts.\n\nWe would like to note that the current ΛLTB modelling can be equivalently parametrised with Ω bg m , δρ 0 /ρ 0 , z size and ∆z, which are derived fromΛ, k loc , r 0 and ∆r. Approximate series expansion formulae, valid for ΛCDM background, are provided in appendix A.\n\n\nScale factor\n\nThe radial dependence of scale factor in LTB models can be observed only along the geodesic as a(z) = a(t(z), r(z)) (see Fig. 2). Moreover, modelling a(z) can be used a posteriori to inversely reconstruct the matter density profile (Wojtak & Prada 2017).\n\nInside homogeneous region close to the centre of GBH profile and at the background, scale factor satisfies eq. (25) and, hence:\na(z) = a loc 0 1 + z for z << z size ,(30)a(z) = 1 1 + z for z >> z size .(31)\nIn fact, redshift size of an inhomogeneous matter profile can be defined through the equality 2 (1 + z size ) a(z size ) = a loc 0 + 1 .\n\n\nLocal expansion rate\n\nAs explained in previous section, LTB metric is characterised with two expansion rates, shown in Fig. 2. The transverse expansion rate given by Friedmann-Lemaître eq. (9) can be used together with density parameters Ω i to construct the equivalent FLRW model of each shell. On the contrary, radial expansion rate is related to the spatial derivative along the line of sight (cf. eq. (5)). Certainly, all physical quantities defined in ΛLTB model converge to their ΛCDM form at high redshift limit as a consequence of the Birkhoff theorem for the local spherical inhomogeneity (see Fig. 2). The intercept of SN Ia m(z) relation is determined by the absolute Figure 3. 1σ confidence stripes of SN Ia magnitude parameter M obtained with standard (grey) and binned: z ≤ 0.08 (green), 0.08 < z ≤ 0.15 (violet) and z > 0.15 (orange) fits to Pan+P data.\n\nmagnitude of SN and the local expansion rate. Calibrated SN Ia are able to measure the H 0 only after fixing the slope of distanceredshift relation as presented in Riess et al. (2016). However, if the inhomogeneity extends up to z ∼ 0.1, not accounting for its effect properly can represent an important systematic in H 0 measurement.\n\n\nRESULTS AND DISCUSSION\n\nIn this section we present our main findings in the analysis of LD data from KBC13 and the SN Ia data from JLA and Pantheon 1 samples (Betoule et al. 2014;Scolnic et al. 2018). The likelihoods for SN Ia datasets are implemented as in their respective releases. Results are presented for the derived parameters: relative matter density at the background Ω bg m , central contrast δΩ 0 /Ω 0 , inhomogeneity size z size and density profile shape ∆z. We use a prior of Ω bg m = 0.3153 ± 0.0073 taken from CMB TT,TE,EE+lowE+lensing data analysis for the ΛCDM background (Planck Collaboration et al. 2018b), which appropriately aids low-redshift SNe to constrain the local matter density profile (hereafter indicated as JLA+P and Pan+P). Other parameters are sampled from wide flat prior ranges: −0.4 ≤ δΩ 0 /Ω 0 ≤ 0.2, z size ≤ 0.13 (≈ 0.5Gpc in comoving distance) and ∆z ≤ 0.6 z size , for which we verify a posteriori that relaxing the priors does not affect our results.\n\nAs elaborated in section 3, the large local void is expected to affect the magnitude-redshift relation of SN Ia inside the void, due to the higher transverse expansion rate and lower values of relative density parameters Ω m (z) and Ω Λ (z) w.r.t. ΛCDM constraints. We start our analysis by testing for redshift dependence of SN Ia magnitude parameter M = M B + 5Log 10 (c/H 0 ).\n\nTo this end, we use Pantheon SN Ia whose apparent magnitudes have already been corrected with a cosmology-independent method which marginalises the effect of all magnitude correction parameters and leaves only M as a free parameter, besides the cosmological model (see Scolnic et al. (2018)). Full Pantheon dataset, along with the CMB prior, is fitted to standard ΛCDM model, allowing for three binned values of M in redshift ranges z ≤ 0.08 The grey area is a 1σ reconstruction zone obtained using Gaussian process. The blue curve is our best-fit GBH profile to 10 LD points, which shows δρ 0 /ρ 0 = −43% contrast. The green curve has a weaker contrast δρ 0 /ρ 0 = −30% suggested by KBC13.\n\n(green), 0.08 < z ≤ 0.15 (purple) and z > 0.15 (orange), with 194, 103 and 751 SNe, respectively. The resulting 1σ confidence interval for M bg = −1.184 ± 0.007 is in excellent agreement with M tot = −1.185 ± 0.005 from the standard non binned fit shown as grey stripe in Fig. 3. The visible inconsistency in the first two bins, M 1 = −1.210 ± 0.015, M 2 = −1.153 ± 0.012 might be hinting to possible local geometry effects. Fitting the Pantheon SN Ia in the redshift range 0.0233 < z < 0.15 to the FLRW series expansion (SE) formula for luminosity distance with q 0 = −0.55, as presented in R16, we obtain M(SE) = −1.182 ± 0.010, which is also different from values obtained in the first two bins. Various works in the literature, including R16 and their previous articles, introduce a parameter a B that can be derived from the intercept magnitude M as\na B = −0.2M − 5 + log 10 c ,(34)\nwhere c is the speed of light in km/s. Results from the simple binned analysis already provide a strong motivation to study a more physical, inhomogeneous cosmological model, characterised by varying H 0 (r) and matter density profile Ω m (r). Although an extended analysis of the M evolution at higher redshifts could have important cosmological implications (see e.g. Tutusaus et al. (2019) for effect on cosmic acceleration), here we looked for possible local variation in M as a hint of local matter inhomogeneity, while our background cosmology is assumed to be flat ΛCDM.\n\n\nΛLTB analysis with LD data:\n\nKBC13 summarises 22 LD data points obtained from several surveys, of which 10 are their original data points. Due to the unknown correlations that could arise from the overlap between different redshift surveys we opt to utilise only these 10 data points to constrain local matter density profile. They were obtained using data from 35,000 galaxies divided in 10 redshift bins in the range 0.005 < z < 0.20. Using LD points, KBC13 report the existence of a large local void with a luminosity density contrast of about −30% in the inner region, which is expected to be proportional to Table 1. Primary results from the ΛLTB analysis performed to: luminosity density using all 10 original data points presented in KBC13 (LD) and after removal of z = 0.034 data point from 2MASS galaxy catalogue (LD * ); 740 SN Ia from JLA dataset alone (JLA) and with Planck constraint on Ω bg m = 0.3153 ± 0.0073 (JLA+P); 1048 SN Ia from Pantheon dataset (Pan) and with Planck constraint (Pan+P); 237 low-redshift Pantheon SN Ia from the range 0.0233 < z < 0.15 with the Ω bg m = 0.3 constraint (Pan * ). The best-fit values for parameters of interest (noted as b. f. ) are provided together with confidence interval between 16th and 84th percentiles (noted as c.i. ) of their respective 1D marginalised likelihoods. Three contrast presented here are defined by Eqs. (27) to (29). The comparison to ΛCDM model is given using the relative ∆AIC values. the contrast of total matter density δρ 0 /ρ 0 (see Fig. 4). We initially perform a simple reconstruction of the density profile using the Gaussian process -a model-independent technique for data analysis (see e.g. is obtained by imposing homogeneity both inside and outside the expected void size. As such, it is allowed to predict no evidence of a void, but our resulting posterior of the reconstructed region shows definite indications of the local underdensity with δρ 0 /ρ 0 ∼ −40%.\ndata z b. f. size z c.i. size δΩ 0 Ω 0 b. f. [%] δΩ 0 Ω 0 c.i. [%] δρ 0 ρ 0 b. f. [%] δρ 0 ρ 0 c.i. [%] δH 0 H 0 b. f. [%] δH 0 H 0 c.i\nWe proceed by fitting LD data points to the full ΛLTB model and confirm that the conservative density contrast of δρ 0 /ρ 0 ∼ −30%, suggested by KBC13 and later utilised in HB17 and KSR19, appears to be an underestimate w.r.t. our best-fit shown in Table 1.\n\nDue to the small amount of data, we test our findings by performing a leave one out (LOO) analysis, implemented as random removal of each of the 10 points from the fit. It clearly shows the stability of the estimated redshift size, while the value of the profile contrast δρ 0 /ρ 0 ∼ −44.0±6.0 is found to be dominated by one stringent measurement at z = 0.034. Eliminating this data point coming from 2MASS survey, the remaining 9 points from UKIDSS and GAMA surveys estimate the contrast to be δρ 0 /ρ 0 ∼ −32.0 ± 9.5, in consistence with the conservative value suggested by KBC13. A direct comparison of our best-fit profile (blue) and the profile with δρ 0 /ρ 0 = −30% (green) in Fig. 4 clearly shows the importance of 2MASS data point at z = 0.034. Nevertheless, the induced contrast of fractional matter density δΩ 0 /Ω 0 coming from the fit to 9 points (quoted as LD * in Table 1) is still higher than the one used in previous works where the difference between δΩ/Ω and δρ/ρ contrasts is not considered (c.f. KSR19, HB17, Shanks et al. (2019)).\n\nThe primary inferences of this analysis are verified to remain unaltered changing the assumed analytic form of matter density profile. Taking the result in Table 1 at face value, would indicate an extreme tension with a homogeneous matter density scenario. While the standard model of cosmology expects variation of the local matter density and velocity fields, a more pressing problem arising from this result is the size of the estimated inhomogeneity. Based on ΛCDM matter power spectrum constrained by CMB, density perturbations of size z size ≈ 0.08 are expected to have the variance of δρ 0 /ρ 0 ≈ 1.5−2% (corresponds to δΩ 0 /Ω 0 ≈ 2−3%) at 68% confidence level (Marra et al. 2013;Camarena & Marra 2018). Hence, this expectation is at ∼ 8σ tension with the result of the fit to complete LD data, while the exclusion of 2MASS data point at z = 0.034 relaxes deviation to a however large ∼ 4σ tension.\n\nThe best-fit ΛLTB (blue) curve in Fig. 4 is closely following the predicted model-independent reconstruction (grey), except for the short overdense region, which is an expected compensation of a large and steep underdense matter profile. In fact, the data point at z = 0.081 might be a signature of the same. Although not statistically significant to draw conclusions for an overdensity, it aids to constrain the void size. The specific interplay between the model description and the fact that only 10 data points are available for constraining 4 profile parameters, results in over-fitting difficulties. For this reason we perform a Gaussian approximation of the posterior likelihood distribution before estimating the confidence regions presented in Fig. 6, which does not alter our primary inferences.\n\nGiven the lack of consistency in reported results based on LD data, limited sky coverage of the data, reports about angular variation (WS14), and possible effects of binning we also deem it important to estimate the level of (dis)agreement with the SN Ia data.\n\n\nΛLTB analysis with SN Ia datasets:\n\nProbing for signatures of large local inhomogeneity (underdense or overdense) in the SN Ia datasets, we fit the local matter density profile simultaneously with the background cosmology (Ω bg m ) which is, nevertheless, dominantly guided by the Planck CMB constraint. Since the variation of H 0 (r) is modelled, in ΛLTB we define the SN Ia intercept magnitude parameter using background expansion rate\nM = M B + 5Log 10 (c/H bg 0 ).(35)\nIn the case of JLA dataset, fitting χ 2 also depends on SN Ia magnitude correction parameters for lightcurve stretch, colour and host galaxy mass, which are later marginalised as nuisance parameters (Betoule et al. 2014).\n\nThe SN Ia constraints on local matter density profile presented in the form of confidence regions in Fiqs. 5 and 6 and Table 1 show high consistency with homogeneous ΛCDM fit, characterised by δΩ 0 /Ω 0 = 0. The analysis of JLA dataset yields two distinguished local χ 2 minima, emphasised with dashed lines in Fig. 5. The underdensity reported by WS14, with the size of ≈ 215h 70 −1 Mpc and the average contrast δρ 0 /ρ 0 = −15 ± 3% (corresponds to δΩ 0 /Ω 0 = −19 ± 4%) is in between the two minima we find for JLA dataset. Interestingly, the second local minimum specifies a void with the physical size in line with the findings of KBC13, d A (c) = 300h 70 −1 Mpc, but with a much weaker contrast. Indeed, the large local void reported by KBC13 is rejected at 4σ confidence by JLA dataset. However, our analysis of reduced LD * data points gives less stringent constraint on the void contrast which is no longer in tension with our JLA result (cf. Table 1).\n\nEven though the analysis of Pantheon dataset does not recover the primary minimum obtained for JLA supernovae, the resulting confidence regions are of high similarity. Our Pantheon and Pan+P constraints for ΛLTB model shown in Fig. 6 include inside 1σ confidence region voids with contrast δΩ 0 /Ω 0 ≈ −20%, such as the representative void of WS14. The constraints from LD data of KBC13 (blue shaded ellipses in Fig. 6) are pointing to much deeper void, although the redshift size of the two best-fits are aligned. In fact, the estimated contrasts from Pan+P and LD data are at ∼ 4σ tension (see Table 1). The nature of inferences resulting from LD data, their disagreement with the expected cosmic variance and the SN Ia constraints, further strengthens our motive to perform LOO analysis to test this data sample. LOO analysis led us to consider the fit to the reduced LD * dataset, which is shown with blue dashed contours in Fig. 6. As mentioned earlier, the LD * constraint on void size remains unaltered, while the estimated void contrast shifts towards a shallow inhomogeneity that has no tension with the Pantheon results.\n\nOur ΛLTB inferences coming from Pantheon dataset are contrasting the recent result of KSR19 which reports a very high significance of ∼ 5σ against any large local void with contrast |δρ 0 /ρ 0 | > 20%. The higher amount of SN Ia data points that they used, 1295 compared to 1048 available in Pantheon, are additionally populating the low-redshift range and are expected to improve the error bars of profile parameters by 50%. However, we sus- Figure 6. 1σ and 2σ confidence contours of local matter density profile parameters are shown for Pan+P (green regions), Pantheon (light green solid curves), full LD data from KBC13 (blue regions) and reduced LD * data (dashed blue curves). The isotropic voids proposed by WS14 and KBC13 are marked with orange crosses. Contrasts presented on vertical axes are defined by Eqs. (28) and (29). pect that theoretical modelling is also contributing to the difference w.r.t. our result. While the newer data is not publicly available at this moment, we intend to extend the current analysis in a future communication. In appendix A we present the effects of simplifying the density profile from GBH to TH solution and fixing the model parameters such as Ω bg m and z size , instead of marginalising over them to obtain the constraints on void contrast.\n\nWe chose to report the results of SN Ia analysis with and without the Planck CMB constraint on background matter density. Both SN Ia datasets provide excellent fits in the case of a local underdensity, but do not exclude the possibility for a z size > 0.1 overdensity (see Fiqs. 5 and 6). The Planck prior on background cosmology reduces the degeneracy between Ω bg m and δΩ 0 /Ω 0 parameters in SN Ia likelihood, providing tighter constraints. Nevertheless, the main inferences presented here are not strongly affected by this prior due to the excellent agreement between the Planck value and our SN Ia best-fits for Ω bg m . Examples presented in appendix A also explore the case of using a fixed value for the relative matter density at the background, as contrary to sampling this likelihood parameter. Compared to homogeneous model, the additional three parameters that characterise the void profile afford better fits to the SN Ia data, namely χ 2 JLA = 678.30 and χ 2 Pan = 1020.72 for ΛLTB model in contrast to χ 2 JLA = 682.94 and χ 2 Pan = 1026.05 for ΛCDM model. We quantify this comparison with Akaike information criteria Akaike (1974) that penalises the usage of three extra parameters as ∆AIC = ∆χ 2 + 2 * 3. All fits reported in Table 1 have ∆AIC < 2 implying no preference between the two models.\n\n\nAnisotropy considerations\n\nUsing galaxy redshift distribution and number counts up to the depth of z < 0.1, WS14 observed 215h 70 −1 Mpc local underden-sity and an angular variation of its estimated density contrast in the range −4% to −40%, with the mean of −15% (corresponds to δΩ 0 /Ω 0 = −19% shown in Fiqs. 5 and 6). Likewise, the Fig. 11a in KBC13 clearly shows discrepancy between LD points observed in different regions on the sky, again pointing towards a possibly anisotropic local density profile. Hence, an important extension to the analysis presented here can be done by taking into account the anisotropy of the data, as well as the matter density profile and the observer's position inside of the considered inhomogeneity (see e.g. Blomqvist & Mörtsell (2010)). If there was any anisotropy present in the matter distribution, it would be averaged out in the analysis that uses isotropic model like ours, leaving us with the expectation to find angular variation of density contrast around this average when modelling for anisotropic matter profiles. Be that as it may, the statistical significance in favour of the observed anisotropy in LD data is limited by the sky coverage of the used surveys, which amounts to ∼ 15% for KBC13 and ∼ 22% for WS14, spread over three distant patches on the celestial plane. Several groups performed tests on SN Ia data looking for features of local and/or global anisotropy Bengaly et al. 2015;Kalus et al. 2013;Wang & Wang 2014;Sun & Wang 2018;Andrade et al. 2018), also in relation to the local matter distribution (Colin et al. 2011). These anisotropy estimates are also impaired by the inhomogeneous angular distribution of the observed SN on the sky, which is particularly visible on medium and high redshifts due to the narrow fields of the surveys, such as SDSS. Probing low-redshift SN Ia for signatures of anisotropy, we estimate the statistical variance of model parameters over angular directions. Concretely, the two parameters of interest, M and δΩ 0 /Ω 0 in ΛLTB model, are tested separately for dipole anisotropy (see also e.g. Mariano & Perivolaropoulos (2012)). The magnitude dipole is modelled as a vector of intensity M d and direction n d . Then, the magnitude parameter M (i) of each SN Ia inside the void becomes\nM (i) =M + M d cos(n (i) , n d ) for z (i) ≤ z size , M (i) =M for z (i) > z size .(36)\nwhereM is the average intercept magnitude, while the normalised vector n (i) represents the angular position of each SN Ia. Although we fitted the full Pantheon dataset, the dipole is constrained mainly by 190 SN Ia under z ≤ 0.08, which are also more homogeneously distributed on the celestial plane compared to the full dataset. Here we used a TH matter density profile for simplicity (see section 2). Finally our estimate for the magnitude dipole\nM d = 0.011 +0.026 −0.031(37)\nwould produce the angular variation\n∆M r.m.s. = Var(∆M) = √ 2 2 M d = 0.008 ± 0.018 .(38)\nThis angular variance of magnitude inside the void can be related to the anisotropy of local expansion rate as \n\nOur result for the level of local anisotropy in Pantheon SN Ia is consistent with previous works that find null evidence against assumption of the isotropy (Andrade et al. 2018;Zhao et al. 2019b), although less stringent compared to global anisotropy constraint found by Soltis et al. (2019). Analogously to eq. (36) we statistically probed the dipole anisotropy of the estimated void contrast and found that it could account for the angular variance ∆ δΩ 0 Ω 0 r.m.s. = 3.7 ± 4.1% .\n\nWhile this result at face value shows that Pantheon dataset does not exclude the angular variation of estimated void contrast ∆(δΩ 0 /Ω 0 ) ∼ 10% at 2σ confidence level, we still find it difficult to coincide our constraints on matter density profile from Pantheon SN Ia and LD data, or explain the level of anisotropy observed by WS14. We would like to note that two former examples should be considered only as rough estimates of anisotropy level since: the dipole model does not represent a complete relativistic solution of FE, the variation of profile size is not considered, and the Pantheon SN Ia have had the apparent magnitudes corrected using the BBC method which relies on the assumption of isotropy (Scolnic et al. 2018). Although crucial in future studies, the present data is not quantitatively sufficing to benefit from the full anisotropic analysis given as an extension of the cosmological metric and theoretical model for the anisotropic density profile.\n\n\nDistance ladder measurements\n\nOne of the most important implications of a large local void is the effect on direct measurement of present cosmic expansion rate by the means of low-redshift SN Ia. Consider a spherical matter inhomogeneity, characterised with a density contrast of e.g. δΩ 0 /Ω 0 = −25% which is inside the 1σ confidence region of Pan+P constraints, presented in Table 1 and Fig. 6. This matter contrast will also induce a higher local present expansion rate by δH 0 /H 0 ≈ 3.6% w.r.t. the background expansion rate (see Eqs. (28) and (29). Hence, the luminosity distance of a SN located on any shell inside the local void is higher than what one would estimate using the background metric. This can affect the analysis of SN Ia data in the redshift range inside the local void as well as the resulting model inferences.\n\nIn light of everything said above, let's assume our local matter distribution has indeed an isotropic underdense profile. Analysing SN Ia data from the redshift range inside this local void with a simpler homogeneous model would provide a different constraint on H bg 0 than the LTB model. In fact, the assumption of locally homogeneous cosmic expansion (H loc 0 = H bg 0 ) can overestimate the value of background expansion rate, while underestimating H loc 0 , by ∼ 1 2 δH 0 /H 0 and affect the inferences on other background cosmological parameters.\n\nAs discussed in Section 4.1 theoretical expectations for the large local void based on ΛCDM model, and its effect on H 0 measurement are small, which is reflected in results presented by recent studies. On one side, the level of expected cosmic variance in local matter density for the Planck ΛCDM model is estimated to impact the systematic error of H bg 0 at 2% when using SN Ia from z ≥ 0.01 and at 1% when using SN Ia from z ≥ 0.0233 (Marra et al. 2013). On the other side, the large-volume cosmological Nbody simulations based on the Planck ΛCDM model, such as e.g. in Skillman et al. (2014), were used to quantify effects of both the cosmic variance and the SN Ia sample selection on H bg 0 value -Wu & Huterer (2017); Odderskov et al. (2016) estimate the expected variance level of measured H bg 0 to be ∼ 0.5% for 0.0233 < z < 0.15 cut (c.f. an earlier work by Wojtak et al. (2014)). In order to minimise the effect of local matter distribution, R16 (including their previous works) use only the SN Ia above z > 0.0233 for the modelindependent estimate of H bg 0 (Jha et al. 2007). In consistence with\n\nWu & Huterer (2017), KSR19 report no evidence for strong variation of matter density profile in the range z < 0.15 from SN Ia data, securing the ability to measure H bg 0 value to a percent value. The measurement of background expansion rate based on lowredshift SN Ia is obtained from the intercept magnitude parameter M (see Section 4.2). While the exact value of H bg 0 relies on local SN Ia luminosity calibration, we are mainly interested at the relative change of this estimate due to the effect of a large local void, which can be seen as the shift in estimated value of M. For this purpose, and only in this subsection, we analyse 237 Pantheon SN Ia from 0.0233 < z < 0.15 redshift cut. Fitting them to FLRW series expansion (SE) formula for luminosity distance and using a fixed value of q 0 = −0.55, as presented in R16, we obtain M(SE) = −1.1825 ± 0.0094. In low redshift range, this fit is equivalent to ΛCDM fit with Ω m = 0.3, which we will use for comparison to underdense scenario. The same Pantheon cut is fitted to ΛLTB model with background cosmology prior Ω bg m = 0.3 (q bg 0 = −0.55), density contrast prior range −50% ≤ δΩ 0 /Ω 0 ≤ 5%, void size 0.025 ≤ z size ≤ 0.12, and shape parameter 0 ≤ ∆z ≤ z size . In order to ensure enough SN Ia that constrain the background, we consider only profiles that reach the background regime up to z ≤ 0.13 by additionally removing the region of parameter space with such combination of large z size and ∆z values. After marginalising the matter density profile out of the low-redshift Pantheon likelihood, we estimate the intercept SN magnitude parameter 2 to be M(ΛLTB) = −1.1577 +0.0158 −0.0148 . The remaining parameter constraints are presented in the last row of Table 1. Regardless the three additional parameters that model the local matter underdesnity, the ΛLTB fit to low-redshift Pantheon SN Ia reveals an improvement of the best-fit by ∆AIC=-3.3.\n\nThe local matter underdensity affects the estimated value of M due to its degenearcy with density constrast. This can be seen on Fig. 7, where we show the low-redshift Pantheon constraints in δΩ 0 /Ω 0 −M plane. The best-fit of ΛCDM model (equivalent to SE) can be found on vertical axis for zero void contrast and is marked as a red point in Fig. 7, which is 1.7σ away from our constraints on ΛLTB model. Both values for the intercept magnitude parameter M have been estimated fitting the same 237 Pantheon SN Ia to two different models and are, hence, fully correlated. Taking into account this correlation, the difference between two values is evalueted to be ∆M = 0.0247 +0.0065 −0.0055 . As the ΛLTB model has more parameters in fitting, the error bar of resulting M(ΛLTB) is bigger than in the case of M(ΛCDM), and elseways, we note that if they were equivalent then the uncertainty on estimated shift ∆M would have been zero. Since this change of intercept magnitude value is coming only from the theoretical modelling and the two models have comperable AIC values (see Table 1), it should be seen as a possible source of systematic error on M estimate. The positive shift of ∆M = 0.0247 corresponds to 1.14% lower H bg 0 obtained with ΛLTB model than with SE formula. Compared to the total error of M(SE), this value of ∆M is significant. However, the major contribution to the uncertainty of H 0 measurement is presently constricted to calibration of standard SN Ia absolute magnitude, M B , that together with much smaller uncertainty on intercept magnitude M(SE) reaches the total error bar of 2.4% reported by R16. Adding the effect of large local matter inhomogeneity (z size > 0.025) on estimate of SN Ia intercept magnitude could increase this uncertainty level to 2.7% on the lower end of H 0 measurement. The improvement in Cepheid and SN Ia calibation techniques (see R19), followed by reduction 2 Confidence intervals are given between 16th and 84th percentiles of uncertainty on M B , can lead to a situation in which the effect of local matter inhomogeneity becomes more relevant for the measurement of background cosmic expansion rate.\n\n\nCONCLUSIONS\n\nIn the present work we have investigated the evidence for a large local void using a well formulated ΛLTB model against the most recent publicly available SN Ia compilations. In order to reduce the correlation with background cosmology, we have used the CMB constraint on Ω bg m (Planck Collaboration et al. 2018b), which aids to provide a slightly tighter constraints on the local void profile. The analysis on SN Ia datasets is also complimented with the luminosity density data obtained from galaxy surveys by Keenan et al. (2013), which together with Whitbourn & Shanks (2014) played a key role in propelling the recent discussion about possible existence of a large local void and its effect towards resolving one of the most prominent discordances in modern cosmology -the H 0 tension. We summarise our primary results in the following. All reported confidence intervals (c.i.) are between 16th and 84th percentiles.\n\n• Fitting ΛLTB model to LD data from KBC13, we find a void with density contrast of (δρ 0 /ρ 0 ) c.i. = −43.8 ± 6.0%, deeper than originally proposed δρ 0 /ρ 0 = −30%.\n\n• From SN Ia we do not find a strong evidence for a void or otherwise -a fit to Pan+P yields (δρ 0 /ρ 0 ) c.i. = −7.9 +7.0 −5.8 % which corresponds to (δΩ 0 /Ω 0 ) c.i. = −10.5 +9.3 −7.4 %, and a wide range for the redshift size z c.i. size = 0.068 +0.021 −0.030 . However, Pantheon constraints do not exclude a void of e.g. z size ≈ 0.075 and δΩ 0 /Ω 0 ≈ −25% at 1σ confidence level (c.f. Fig. 6).\n\n• JLA likelihood is in a good overall agreement with Pantheon, except in the range z ≤ 0.04 (c.f. Fiqs. 5 and 6). Both SN Ia datasets are at 3σ tension with our result obtained from KBC13 data, but in excellent agreement with parameters of the isotropic void proposed by WS14.\n\n• Leave one out analysis of KBC13 data reveals that the corresponding estimated contrast is dominantly constrained by one stringent point. Removing this only data point from 2MASS survey and using the remaining 9 points from UKIDSS and GAMA surveys, relaxes their tension with SN Ia data.\n\n• The model comparison with Akaike information criteria shows no preference between the tested models (c.f. Table 1).\n\n• Probing the level of statistical variance in ΛLTB model parameters over angular direction, we find null evidence for the dipole anisotropy. Its contribution to the angular variance is small: ∆M r.m.s. = 0.008 ± 0.018 for the SN intercept magnitude parameter or ∆ δΩ 0 Ω 0 r.m.s. = 3.7 ± 4.1% for the void contrast.\n\n• In our analysis of SN Ia data we do not find any evidence for a large isotropic void that could resolve the 9% discrepancy between two H 0 estimates in tension, leaving this local effect alone to be a highly unlikely explanation.\n\n• We fitted low-redshift range of Pantheon SN Ia data from 0.0233 < z < 0.15 to the FLRW series expansion formula for luminosity distance and to the ΛLTB model, comparing their estimates of SN Ia intercept magnitude, which is used for H 0 measurement. Adding the possibility of a large local void, we observe a small improvement in the fit by ∆AIC = −3.3 and a shift of the inferred intercept magnitude parameter ∆M = 0.0247 +0.065 −0.055 that could be a source of additional systematic error in H bg 0 of 1.14%. While the simpler model, which is based on series expansion formula of luminosity distance in FLRW metric, is only 1.7σ away from our ΛLTB constraints (see Fig. 7), this result is questioning whether the total error budget of distance ladder measurements based on low-redshift SN Ia should be reconsidered.\n\nIncrease of presently available data, followed by more complex theoretical modelling, is necessary to better understand the observed disagreement between LD and SN Ia samples. As already mentioned, considering off-centre position of the observer would extend the present isotropic ΛLTB formalism, also allowing for an anisotropic point of view by construction. With increasing number of observations in future surveys, SN Ia may prove as effective tracers of local matter density distribution. Figure A1. 1σ and 2σ confidence regions for ΛLTB model with TH density profile, fitted to Pantheon dataset are shown as coloured contours, while the solid lines represent the confidence regions obtained from the smoothed χ 2 function. In the first panel we used Planck prior for Ω bg m , whereas the second and third panels are obtained with fixed Ω bg m = 0.3. Moreover, in the third case z size parameter was a priori fixed to a range of z split values, followed by χ 2 sampling over δΩ 0 /Ω 0 contrast values.\n\n\n; Tokutake et al. (2018); Kenworthy et al. (2019); Boehringer et al. (\n\nFigure 1 .\n1The evolution of matter density contrast δρ/ρ is shown for ΛLTB model with present local contrast δρ 0 = −30% and redshift size z size = 0.07. The profile is shown in comoving angular diameter distance, at time lapses t = t/t 0 running from 0.01 up to 1.00 (present time).\n\nFigure 2 .\n2Redshift dependence of the cosmic expansion rates (red for radial and orange for transverse) and the scale factor (blue) are shown for ΛLTB model with contrast δρ 0 = −30% and redshift size z size = 0.07 (solid lines) in comparison with the ΛCDM model (dashed lines).\n\nFigure 4 .\n4Luminosity density data from KBC13: solid orange data points are their original data, while dashed points are from additional surveys.\n\n\nHolsclaw et al. (2010); Seikel et al. (2012); Haridasu et al. (2018b)). The reconstructed 1σ region shown in grey on Fig. 4\n\nFigure 5 .\n5Constraints on the local matter density profile from JLA (solid curves) and JLA+P (coloured regions) are shown at 1σ and 2σ confidence level. The positions of the only two local χ 2 minima are emphasised with the dashed lines. The isotropic voids proposed by WS14 and KBC13 are marked with orange crosses.\n\n\n.m.s. ≈ 0.2 ln(10) ∆M r.m.s. = 0.4 ± 0.8%\n\nFigure 7 .\n71σ and 2σ confidence regions for ΛLTB model with GBH density profile, fitted to 237 low-redshift (0.0233 < z < 0.15) SN Ia from Pantheon dataset. The best fit (b.f.) for ΛCDM model, shown as red point is almost 2σ away from ΛLTB constraints.\n\n\n.[%] \nχ 2 \n∆AIC ΛCDM \n\nLD \n0.082 0.079 +0.012 \n\n−0.012 \n\n−51.1 \n−51.9 +6.3 \n\n−6.3 \n\n−42.9 \n−43.8 +6.0 \n\n−6.1 \n\n8.1 \n8.2 +1.3 \n\n−1.2 \n\n2.62 \n\nLD * \n0.082 0.075 +0.015 \n\n−0.015 \n\n−39.7 \n−39.4 +10.3 \n\n−10.3 \n\n−32.4 \n−32.1 +9.0 \n\n−9.5 \n\n5.9 \n5.9 +1.9 \n\n−1.7 \n\n0.29 \n\nJLA \n0.025 0.039 +0.062 \n\n−0.018 \n\n−19.5 \n−9.9 +17.3 \n\n−13.9 \n\n−15.1 \n−7.5 +12.9 \n\n−11.0 \n\n2.7 \n1.3 +2.0 \n\n−2.3 \n\n678.30 \n1.35 \n\nJLA+P 0.025 0.032 +0.056 \n\n−0.011 \n\n−19.6 \n−12.5 +13.8 \n\n−12.4 \n\n−15.2 \n−9.5 +10.4 \n\n−9.9 \n\n2.7 \n1.7 +1.9 \n\n−1.9 \n\n678.34 \n1.07 \n\nPan \n0.075 0.070 +0.023 \n\n−0.031 \n\n−16.2 \n−9.8 +14.0 \n\n−8.9 \n\n−12.4 \n−7.4 +10.5 \n\n−7.0 \n\n2.2 \n1.3 +1.3 \n\n−1.9 \n\n1020.72 \n0.67 \n\nPan+P \n0.075 0.068 +0.021 \n\n−0.030 \n\n−14.4 \n−10.5 +9.3 \n\n−7.4 \n\n−11.0 \n−7.9 +7.0 \n\n−5.8 \n\n2.0 \n1.4 +1.1 \n\n−1.3 \n\n1021.01 \n0.40 \n\nPan * \n0.075 0.075 +0.013 \n\n−0.022 \n\n−23.1 \n−21.5 +8.9 \n\n−9.5 \n\n−18.1 \n−16.8 +7.1 \n\n−7.9 \n\n3.2 \n3.0 +1.5 \n\n−1.3 \n\n225.63 \n−3.30 \n\n\nMNRAS 000, 1-12(2019)\nWe use the latest release of Pantheon dataset from Nov 2018, while verifying that using the SN Ia redshifts published earlier has minimal to no effect on the likelihood in all fits performed here (see alsoRameez (2019))\nACKNOWLEDGEMENTSThe authors would like to thank Amy J. Barger, Benjamin L. Hoscheit, Adam Riess and Tom Shanks for useful discussions, and Caterina Traficante for contribution to this work during her thesis. Authors acknowledge financial support by ASI Grant No. 2016-24-H.0.APPENDIX A: TOP-HAT DENSITY PROFILEOur results for ΛLTB model can be reproduced to a good extent using a simpler TH density profile constructed as a two-step homogeneous kΛCDM model. The following approximate formulae are derived from ΛLTB model as third order series expansion in terms of present matter density contrast x = δρ 0 /ρ 0 . Given that Λ is con-stant, its dimensionless density parameter is not (see section 2).In the range 0.2 < Ω bg m < 0.4 and −30% < δρ 0 < 30% these eqs. are correct with less than 1% error on the value. The constructed TH model can be characterised with three cosmological parameters: Ω bg m , δΩ 0 /Ω 0 and z size , as well as the intercept SN Ia magnitude parameter M. As usual, using a prior on Ω bg m ensures tighter constraints on void parameters. Using eqs. (A1) to (A3) together with two-step kΛCDM formalism, it is easy to evaluate all the distances of SN Ia inside and outside the void.Sampling Pantheon χ 2 for different model parameters' values, we immediately notice it can be very irregular along the redshift axis, since the top-hat profile does not have a smooth transition from local to background geometry (seeFig. A1). We find that a TH void with a contrast δΩ 0 /Ω 0 ≈ −25% has comparable χ 2 to the bestfit of ΛCDM model for specifically chosen values of z size ≈ 0.075, although this fitting method has no way of accounting for the systematic error arousing with fixing the physical size of the void. To overcome this, one can perform smoothing of the sampled TH likelihood over redshift bins of size ∆z = 0.01. Although short in size, these bins have ∼ 20 low-redshift SN each, regularising the average likelihood dependence on z size . The 1σ and 2σ confidence regions resulting from smoothed likelihoods are shown as solid lines inFig. A1. We note that contours obtained in this way for Pan+P data are substantially similar to what we got using the GBH profile (c.f.Fig. 6). After properly marginalising out other parameters in the smoothed likelihood, we get a Pan+P constraint on contrast as (δΩ 0 /Ω 0 ) c.i. = −9.0 +7.6 −6.8 %. The corresponding result inTable 1has somewhat wider error bars since the smooth GBH function allows for more freedom in density profile than the TH form.The second and third panels inFig. A1show the importance of fitting the model parameters in this scheme as opposed to fixing them. Specifically, the last panel is obtained by sampling Pantheon χ 2 for contrast δΩ 0 /Ω 0 and M parameters, while fixing the void size to z split and Ω bg 0 = 0.3. Hence, this method is unable to provide a constraint on void size, just as the constraint on density contrast depends on the prior choice of z split . 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D Zhao, Y Zhou, Z Chang, 10.1093/mnras/stz1259MNRAS. 4865679Zhao D., Zhou Y., Chang Z., 2019b, MNRAS, 486, 5679\n\n. J P Zibin, 10.1103/PhysRevD.78.043504Phys. Rev. D. 7843504Zibin J. P., 2008, Phys. Rev. D, 78, 043504\n\n. J P Zibin, 10.1103/PhysRevD.84.123508Phys. Rev. D. 84123508Zibin J. P., 2011, Phys. Rev. D, 84, 123508\n\n. H Du Mas Des Bourboux, 10.1051/0004-6361/201731731A&A. 608130du Mas des Bourboux H., et al., 2017, A&A, 608, A130\n"},"corpus_id":{"kind":"number","value":198897423,"string":"198,897,423"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1907.11219\",\n \"mag\": 2986977326,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1093/mnras/stz3070\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1907.11219v2.pdf\"\n ],\n \"pdf_sha\": \"a00da862e84d207e8fcfb492e15e3abf0d9159e6\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1907.11219\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Exploring the evidence for a large local void with supernovae Ia data"},"authors":{"kind":"list like","value":["Vladimir V Luković \nDipartimento di Fisica\nUniversità di Roma \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomaItaly\n\nSezione INFN\nUniversità di Roma \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomaItaly","Balakrishna S Haridasu \nDipartimento di Fisica\nUniversità di Roma \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomaItaly\n\nSezione INFN\nUniversità di Roma \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomaItaly","Nicola Vittorio \nDipartimento di Fisica\nUniversità di Roma \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomaItaly\n\nSezione INFN\nUniversità di Roma \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomaItaly"],"string":"[\n \"Vladimir V Luković \\nDipartimento di Fisica\\nUniversità di Roma \\\"Tor Vergata\\\"\\nVia della Ricerca Scientifica 1I-00133RomaItaly\\n\\nSezione INFN\\nUniversità di Roma \\\"Tor Vergata\\\"\\nVia della Ricerca Scientifica 1I-00133RomaItaly\",\n \"Balakrishna S Haridasu \\nDipartimento di Fisica\\nUniversità di Roma \\\"Tor Vergata\\\"\\nVia della Ricerca Scientifica 1I-00133RomaItaly\\n\\nSezione INFN\\nUniversità di Roma \\\"Tor Vergata\\\"\\nVia della Ricerca Scientifica 1I-00133RomaItaly\",\n \"Nicola Vittorio \\nDipartimento di Fisica\\nUniversità di Roma \\\"Tor Vergata\\\"\\nVia della Ricerca Scientifica 1I-00133RomaItaly\\n\\nSezione INFN\\nUniversità di Roma \\\"Tor Vergata\\\"\\nVia della Ricerca Scientifica 1I-00133RomaItaly\"\n]","isConversational":false},"abstract":{"kind":"string","value":"In this work we utilise the most recent publicly available type Ia supernova (SN Ia) compilations and implement a well formulated cosmological model based on Lemaître-Tolman-Bondi metric in presence of cosmological constant Λ (ΛLTB) to test for signatures of large local inhomogeneities at z ≤ 0.15. Local underdensities in this redshift range have been previously found based on luminosity density (LD) data and galaxy number counts. Our main constraints on the possible local void using the Pantheon SN Ia dataset are: redshift size of z size = 0.068 +0.021 −0.030 ; density contrast of δΩ 0 /Ω 0 = −10.5 +9.3 −7.4 % between 16th and 84th percentiles. Investigating the possibility to alleviate the ∼ 9% disagreement between measurements of present expansion rate H 0 coming from calibrated local SN Ia and high-z cosmic microwave background data, we find large local void to be a very unlikely explanation alone, consistently with previous studies. However, the level of matter inhomogeneity at a scale of ∼100Mpc that is allowed by SN Ia data, although not expected from cosmic variance calculations in standard model of cosmology, could be the origin of additonal systematic error in distance ladder measurements based on SN Ia. Fitting low-redshift Pantheon data with a cut 0.023 < z < 0.15 to the ΛLTB model and to the Taylor expanded luminosity distance formula we estimate that this systematic error amounts to 1.1% towards the lower H 0 value. A test for local anisotropy in Pantheon SN Ia data yields null evidence. Analysis of LD data provides a constraint on contrast of large isotropic void δΩ 0 /Ω 0 = −51.9% ± 6.3%, which is in ∼ 4σ tension with SN Ia results. More data are necessary to better constrain the local matter density profile and understand the disagreement between SN and LD samples."},"sections":{"kind":"list like","value":[{"title":"INTRODUCTION","content":"The most reliable standard candles and one of the fundamental low-redshift astrophysical observables (Betoule et al. 2014;Scolnic et al. 2018), type Ia supernovae (SN Ia), also provide a cosmologyindependent measurement of present expansion rate through the local distance ladder (Riess et al. 2016) (hereafter R16). In concert with observations of baryon acoustic oscillation (BAO) peak from galaxy redshift surveys (Eisenstein et al. 2005;Aubourg et al. 2015;Alam et al. 2017;Bautista et al. 2017;du Mas des Bourboux et al. 2017;Zhao et al. 2019a) and cosmic microwave background (CMB) radiation data (Hinshaw et al. 2013;Planck Collaboration et al. 2018a), SN Ia contributed to establishment of the standard model of cosmology. Astonishing concordance between different cosmological observables brought by technical advancements in the last decades was followed by a number of inconsistent and/or unexpected results, e.g. estimated values of matter clustering parameter, S 8 , from low and high redshifts (Planck Collabo-E-mail: vladimir.lukovic@roma2.infn.it † E-mail: haridasu@roma2.infn.it ‡ E-mail: nicola.vittorio@roma2.infn.it ration et al. 2016;Joudaki et al. 2019;Lange et al. 2019;Martinelli & Tutusaus 2019), CMB parameter constraints in curvature extension of base Λ cold dark matter (ΛCDM) model (Planck Collaboration et al. 2018b) (hereafter P18), as well as several unexplained CMB anomalies (Schwarz et al. 2016), etc. The best-known amongst ΛCDM tensions and one of the biggest challenges in modern cosmology is related to the H 0 value. Defining the scale of extra-galactic distance through the Hubble radius, as well as the present expansion rate, H 0 is a fundamental cosmological parameter both for CMB and low-redshift data (Freedman et al. 2001;Freedman & Madore 2010). The most stringent ΛCDM estimate coming from P18, H 0 = 67.36 ± 0.54 km s −1 /Mpc, is in 4.4σ tension with the most recent local measurement based on calibrated SN Ia, H 0 = 74.03 ± 1.42 km s −1 /Mpc, by Riess et al. (2019) (hereafter R19). In comparison, H 0 measurements from the time delay of strongly lensed quasars support a higher value (Birrer et al. 2019;Wong et al. 2019); using differential age method with cosmic chronometers (Jimenez & Loeb 2002) provides an estimate in the middle of two results in tension (Jimenez et al. 2019;Chen et al. 2017;Luković et al. 2016); while the constraints coming from gravitational waves are still fairly weak (Liao et al. 2017). Although the quality of BAO data has substantially improved over the last decade, providing some of the most stringent cosmological constraints on expansion dynamics (Haridasu et al. 2018a;Ramanah et al. 2019;Zhang et al. 2019), their H 0 estimate is degenerated with the sound horizon parameter, whose value relies on CMB observations and early Universe physics (Macaulay et al. 2019;Lemos et al. 2019;Aubourg et al. 2015;Cuceu et al. 2019).\nThe intercept of SN Ia magnitude-redshift relation is determined by their absolute magnitude (M B ) and the Hubble radius, resulting in two degenerate parameters in SN Ia analyses. While the highredshift SN Ia are necessary for assessing the underlying physics of dark energy, the low-redshift end (z ≤ 0.15) of SN Hubble diagram can be analysed with minimal assumptions about properties of the cosmic fluid and expansion dynamics related only to the well-accepted cosmic acceleration (Haridasu et al. 2017). Indeed, calibrating SN Ia absolute magnitude with the use of Cepheid variable stars and fixing the local distance-redshift relation (at z ≤ 0.15) leads to a remarkable cosmology-independent measurement of H 0 (Riess et al. 2005(Riess et al. , 2009(Riess et al. , 2011(Riess et al. , 2016(Riess et al. , 2018b. Different calibration techniques consider the usage of strong gravitational lensing (Taubenberger et al. 2019), tip of the red giant branch (Beaton et al. 2016;Freedman et al. 2019), HII regions (Fernández Arenas et al. 2018), etc. Due to the strong correlation of cosmic expansion dynamics and H 0 value, the model-independent local estimates remain crucial for the study of late-time cosmic evolution [see also Tutusaus et al. (2019); Haridasu et al. (2018b)].\nAs is well-known, a large inhomogeneity in local matter density distribution can affect the geometry of local Universe and the distance measures, consequently biasing the resulting model inferences and the H 0 estimate. A few independent groups used galaxy survey catalogues to test the local matter distribution, reporting hints of a large local void (Keenan et al. 2012(Keenan et al. , 2013Whitbourn & Shanks 2014Boehringer et al. 2019). Concretely, Keenan et al. (2013) (hereafter KBC13) used UKIDSS Large Area Survey (Lawrence et al. 2007), Galaxy and Mass Assembly (GAMA) (Driver & Robotham 2010;Driver et al. 2011) and 2M++ catalogue (Lavaux & Hudson 2011) to construct luminosity density sample (hereafter LD) over the redshift range 0.01 < z < 0.2. Relating it to the total matter density in the local Universe, they find evidence for an underdense region and suggest a void of about 300h 70 −1 Mpc in size and −30% contrast with respect to the background. Whitbourn & Shanks (2014) (hereafter WS14) used 6dFGS, SDSS, and GAMA surveys to examine the local galaxy density field over 20% of the sky and up to 450h 70 −1 Mpc in depth. Both number counts and peculiar velocity fields show evidence for at least 200h 70 −1 Mpc large local underdensity with −15% ± 3% average contrast, ranging between −40% and −5% in different directions. Analysing a catalogue of 1653 galaxy clusters detected by their X-ray emission Boehringer et al. (2019) report evidence for ∼ 140 Mpc large local underdensity with density contrast of 20% ± 10%.\nThe results of KBC13, WS14 and Boehringer et al. (2019) go against the standard cosmological paradigm and the assumption of large-scale homogeneity. In fact, indications for a few hundred Mpc large local underdensity encouraged renewed popularity of void models based on Lemaître-Tolman-Bondi (LTB) metric, also motivated as a remedy for the H 0 tension (Moffat 2016;Tokutake et al. 2018;Hoscheit & Barger 2017;Shanks et al. 2019;Kenworthy et al. 2019) and even as a possible explanation for the CMB cold spot (Szapudi et al. 2015). Contrasting the homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) models, the LTB metric (Lemaître 1927(Lemaître , 1933Tolman 1934;Bondi 1947) describes an isotropic but inhomogeneous dust system, which on cosmological scales can be used for studying cold dark matter den-sity distribution. Since the discovery of cosmic acceleration, LTB metric was used to construct toy models that challenge the cosmological constant paradigm with an alternative scenario in which the apparent accelerated expansion is a result of the strongly underdense local Universe that smoothly converges to Einstein de Sitter solution on higher redshifts (Célérier 2000;Alnes et al. 2006;Clifton et al. 2008;February et al. 2010;Nadathur & Sarkar 2011;Bolejko & Sussman 2011;Zhang et al. 2015;Stahl 2016). Since the deceleration-acceleration transition redshift is well constrained at z t ≈ 0.6 [see e.g. Haridasu et al. (2018b); Gómez-Valent (2019); Mukherjee et al. (2019) for model-independent estimates], this alternative explanation requires a giant ( ≈ 3Gpc) isotropic void that proved to be very unlikely compared to the standard cosmological model (Zibin 2011;Vargas et al. 2017;Amendola et al. 2013;Luković et al. 2016). However, extending the LTB model with the addition of cosmological constant Λ, one can describe large (order of 100Mpc) local inhomogeneous matter distribution inside a ΛCDM background (Valkenburg 2012;Valkenburg et al. 2014;Rigopoulos & Valkenburg 2012).\nWhile the level of cosmic variance in matter distribution expected for standard model can only partially relieve the Hubble constant problem (Marra et al. 2013;Wu & Huterer 2017;Camarena & Marra 2018), Tokutake et al. (2018) found that a strongly underdense local inhomogeneity with non-vanishing cosmological constant, modeled with an approximate numerical solution for LTB metric, can fully coincide the CMB constraints with the local H 0 measurement. Similarly, Hoscheit & Barger (2017) (hereafter HB17) extended the work of KBC13 by testing the consistency of their earlier findings with constraints coming from SN Ia and linear kinematic Sunyaev-Zel'dovich effect and showed that local void found by KBC13 could reduce the H 0 tension. A recent work of Shanks et al. (2019) argues that the combination of local inhomogeneity effect together with the re-calibration of Cepheids and local SN Ia distances using the recent parallax measurements from Gaia mission, is sufficient to completely remove the tension [see also Riess et al. (2018a); Shanks et al. (2018) for follow-up discussions]. Kenworthy et al. (2019) (hereafter KSR19) find that any inhomogeneity with contrast > 20% is strongly inconsistent with SN Ia data, consequently reassuring the ability to measure the distance and Hubble constant with locally calibrated SN to a 1% precision.\nThe goal of this work is to probe the local matter density using the SN Ia as one of the most reliable low-redshift astrophysical observables, constraining size and amplitude of possible matter inhomogeneity at our local position. We investigate the findings of KBC13 and KSR19 by using the complete analytic description for treating the LTB metric in presence of cosmological constant (hereafter ΛLTB) introduced in Valkenburg (2012). Extending the earlier works, we directly fit the ΛLTB model to two biggest publicly available SN Ia samples -Joint light curve analysis Betoule et al. (2014) (hereafter JLA) and Pantheon compilation Scolnic et al. (2018) (hereafter Pan), consisting of 740 and 1048 SN Ia, respectively. We also utilise the luminosity density data from KBC13 within the given formalism to assess the (dis)agreement. In section 2 we review the theoretical description of LTB metric in presence of cosmological constant and proceed by presenting the effect that a local void can have on observable physical quantities in section 3. Constraints on the void contrast and size coming from fitting JLA, Pantheon and KBC13 datasets are reported in section 4, together with the revision of their consistency. Additionally, we discuss on the possible effect of local isotropic inhomogeneity on the H 0 measurement, and explore the level of anisotropy allowed by the data. Summarising our results in section 5, we examine the dif-ferences compared to earlier findings by other teams. Finally, in the appendix A we provide a simplified void model capable of correctly reproducing our results based on complete ΛLTB description to a good approximation.\nThroughout the paper we use the dimensionless variable h 70 = H 0 /70 km s −1 /Mpc for distance measures; acronym w.r.t. stands for 'with respect to'."},{"title":"LOCAL MATTER DENSITY IN ΛLTB MODEL","content":"The simplest extension of standard homogeneous cosmological model in presence of large matter inhomogeneity is done using the LTB metric (Lemaître 1933;Krasiński 1997) that describes spherically symmetric pressureless cosmic fluid\nHere R(t, r) has the units of length and defines all physical distance measures as well as the expansion history in this model. The second free function of the metric, k(r) is dimensionless and defines the curvature within the shell of a radius r. We note that radial coordinate r is also dimensionless, with no physical meaning, and it should be considered only as a flag coordinate of different shells all centred at r = 0. Using the above metric and the Einstein field equations (FE) for pressureless inhomogeneous matter fluid in presence of cosmological constant Λ, one can derive the generalised Friedmann-Lemaître equation as\nHere M(r) is an integration constant, whose physical meaning is the total mass enclosed inside a sphere with radial coordinate r,\nInhomogeneous cosmic matter density profile, ρ m (t, r), satisfies the conservation law coming from FE, ensuring that integrated mass M(r) remains constant in time. In inhomogeneous cosmology we differ two expansion rates, namely transverse and radial:\nHence, the one appearing in Friedmann-Lemaître eq. (2) is the transverse expansion rate. As usual, we denote t = t 0 as present age of the Universe and all the physical quantities with subscript \"0\" represent their values today, while H 0 (r) = H ⊥ (t 0 , r) is used for the transverse expansion rate. Introducing scale factor, the Friedmann-Lemaître equation can be rewritten in a simpler form, although the definition of scale factor a(t, r) is not trivial nor customary as in the homogeneous cosmology. The value of flag coordinate r is commonly defined through a gauge with some other physical variable. The choice of this gauge is arbitrary and has no implications on the model predictions nor on the results of data analysis. The most often used gauge (Alnes et al. 2006;Alfedeel & Hellaby 2010;Nadathur & Sarkar 2011) in LTB framework is:\nHowever, in this work we use an alternative definition which turns out to be more convenient choice for the latter calculations of elliptical integrals present in the geodesic equations and distance variables. Following Valkenburg (2012), we define the dimensionless radial coordinate r through the relation with the total mass enclosed in a given shell as\nwhere C M is a normalisation constant in units of mass whose value will be chosen at a later stage. Commonly, in both gauges the scale factor can be defined as\nThis function clearly converges to the usual scale factor definition and the spatially uniform values in the limit of homogeneous FLRW metric. Generalised Friedmann-Lemaître eq. (2) can be rewritten in terms of three dimensionless density parameters normalised to the present time,\nThese can be evaluated combining eqs.\n(2), (4) and (7) to (9) as,\nHereM = 8πGC M /(3c 3 t 0 ) andΛ = 1 3 Λc 2 t 0 2 are dimensionless constants, whereas k(r), H 0 (r) and a 0 (r) are not uniform in LTB metric. The introduced dimensionless density parameters are different from one shell to another, but do not depend on time.\nSince Ω m (r) is proportional to the total mass enclosed in a shell, M(r), it can be represented as the ratio of average and critical matter densities of each shell. Starting from eq. (3) and then using eqs. (7) and (8) one can define average matter density:\nThen, definition of critical matter density together with eqs. (10) and (13) straightforwardly confirm:\nFriedmann-Lemaître eq. (2) can also be rewritten in dimensionless form using eqs. (10) to (12)\nIn terms of the Birkhoff theorem, eqs.\n(2), (9) and (15) dictate every shell in ΛLTB model to evolve as an independent homogeneous FLRW model with a specific curvature k(r), a total mass enclosed M(r) and the same value of cosmological constant Λ for all shells, determining its transverse expansion rate H ⊥ (t, r). Each shell is described with relative density parameters Ω i (r), present expansion rate H 0 (r), and evolving scale factor a(t, r)/a 0 (r). Cosmological constant, included in the Einstein field equations, is uniform over space and constant in time due to its physical nature, which is also true for the dimensionless constantΛ that we introduced. The density parameter corresponding to the cosmological constant, Ω Λ (r), can not be uniform in ΛLTB model, as this would imply that Λ is inhomogeneous and coupled to the cold dark matter density profile. This approach is reflected in a number of recent studies Valkenburg (2012) 2019), while not fully followed by HB17 who assume cosmic density parameter Ω Λ to be constant throughout the underdense region and at the background. Similarly, the methods of inverse reconstruction of matter density profile in presence of cosmological constant must be constructed such that FE are satisfied (c.f. Tokutake & Yoo (2016); Wojtak & Prada (2017)).\nThe gauge choice for the flag coordinate r, which is set by eq. (7), enabled us to rewrite the Friedmann-Lemaître eq. (2) in a more elegant analytic form, i.e. eq. (9). The remaining degree of freedom in the scaling constant C M , or equivalentlyM, can be used to normalise the scale factor a(t, r). We setM such that a * = a (t 0 , r * ) = 1 for a chosen shell r = r * and at the present time t 0 . For example, one can choose r * = 0 to normalise the scale factor at the observer's position or r * = +∞ to normalise at the background, instead. We follow the normalisation of scale factor on the background, setting the numerical value ofM such that a * = a (t 0 , r * = +∞) = 1 for the background shells. In particular, normalisation gives the physical meaning to the scaling constant, i.e. eq. (13) yields\nTherefore, normalising far from the inhomogeneity, which is for any large enough r * , implies that background scale factor and eq. (15) converge to the usual forms that we have in homogeneous FLRW model. It is easy to see that at present time t = t 0 the background quantities satisfy\nPartial differential eq. (15) can be used to express the derivative ∂t/∂a and integrate from the Big Bang up to scale a:\nAll variables on the right hand side are dimensionless and this elliptical integral is analytically solvable (for details see Valkenburg (2012)). The Big Bang time t BB (r) is an integration constant that may vary in space, affecting the age difference between the shells. Nevertheless, one does not expect large variation of t BB (r) in the case of a small local inhomogeneity. Non-uniform Big Bang time is also related to instabilities of matter perturbations (Zibin 2008). Hence, in this work we consider only the models with homogeneous Big Bang time t BB (r) ≡ const = 0. Integrating eq. (18) on the background (r = r * ) from the Big Bang a = 0 up to today a(t = t 0 , r * ) = 1 we have\nwhich is the numerical equation defining a value ofM, necessary for the normalisation. At any other point of interest eq. (18) can be seen as the solution for t(a, r) or, after invertion, as a(t, r). Hence, calculating numerically the scale factor at any spatial position and time gives us the angular diameter distance R(t, r) and other physical observables of interest."},{"title":"EFFECT ON ASTROPHYSICAL OBSERVABLES","content":"The high-redshift background observables are not affected by the local isotropic inhomogeneity. Nevertheless, the analysis of local observables can yield different constraints on cosmological parameters from those obtained with FLRW models. Examples presented in this section form the basis for analysing observable data, but do not immediately represent the effect on cosmological inferences. We use a specific analytic form of the curvature, parametrised with k bg , k loc , r 0 , ∆r and given as García-Bellido-Haugbølle (GBH) profile (Garcia-Bellido & Haugbølle 2008;February et al. 2010):\nOur focus is the case of ΛCDM background, i.e. k bg ≡ 0, which we call ΛLTB model. The analytic form of k(r) together with the value of dimensionless cosmological constantΛ define the evolution of expansion and all physical quantities in LTB metric. In the limit of ∆r → 0, a simpler top-hat profile (TH) is recovered,\nWhile GBH form describes a smooth matter density profile, TH profile has one parameter less and is trivial to use for data fitting as two joint homogeneous models. Care must be given to relation between the physical quantities in two homogeneous regions."},{"title":"Reconstruction of geodesic","content":"The LTB metric describes an inhomogeneous dust cloud with spherical symmetry, indicated by the angular term in eq. (1). Hence, the observer located at the centre of this symmetry has an isotropic view. The assumption of isotropy gives us only the first approximation for treating the large cosmological inhomogeneities. To quantify the overall effect of large local void on cosmological inferences, we limit ourselves to the on-centre model that considers an observer located at the centre of symmetry in the local matter inhomogeneity. The assumption of this very special position can be avoided, although the geodesic equations for an off-centre observer become increasingly complex, affecting the feasibility of data analysis. We find the on-centre model as adequate for presenting all the relevant points in this work, but we extend the discussion about observer's spatial position later on. Geodesic equations can be derived following Célérier (2000):\nwhere z is the cosmological redshift. This system of differential equations must be solved numerically along the geodesic with help of eqs. (4), (8), (15) and (18), in the entire redshift range. In order to derive distance-redshift relations one needs to numerically reconstruct the function R(z), or equivalently a(z), which can be obtained by numerical inversion of eq. (18) at every z along the geodesic (Valkenburg 2012). However, we find it easier and computationally faster to use the geodesic equations and all physical variables of interest rewritten with (a, r) as independent fundamental coordinates, after the change of variables from (t, r) da dz = a t dt dz + a r dr dz .\nAdding the assumption of homogeneity, geodesic equations reduce to the standard form; using eqs. (8) and (23) we have: We note that this relation is valid on all shells where spatial derivative is null, i.e. in every homogeneous subregion along the matter density profile."},{"title":"Density contrasts","content":"It is useful to define the relative contrasts for quantities of interest, such as physical matter density, dimensionless density parameter Ω m , and expansion rates. The contrast can be defined at any time and any spatial point inside inhomogeneous region. Consequently, for physical matter density we have\nThe evolution of matter density contrast is shown in Fig. 1 for a ΛLTB void model with specific parameters determining cosmological constantΛ, central curvature k loc , size and steep transition zone, given with r 0 and ∆r. As shown in this case, the physical matter density ρ m will have compensated radial profile in the case of small ∆r and/or large central contrast. Since we use the comoving radial distance, the physical expansion of void size can not seen on Fig. 1. Most often we consider the present (t = t 0 ) contrast between observer's position (r = 0) and background (r = + inf) and we denote it with subscript 0. The contrast of dimesionless matter density parameter Ω m (r) and both expansion rates are defined in the same manner as for physical matter density in eq. (26). We will focus on present central contrasts which can be derived using eqs. (3), (7), (8), (10) and (16):\nNotably, these two contrast are different due to the induced spatial inhomogeneity of the expansion rates, In conclusion, relation |δρ 0 /ρ 0 | < |δΩ 0 /Ω 0 | holds both in the case of local underdensity as well as overdensity. Writing the equivalent of eq.\n(3) at a(t = t 0 , r = 0) = a loc 0 and after some simple calculations, one can derive analytic expressions that relate these three contrasts.\nWe would like to note that the current ΛLTB modelling can be equivalently parametrised with Ω bg m , δρ 0 /ρ 0 , z size and ∆z, which are derived fromΛ, k loc , r 0 and ∆r. Approximate series expansion formulae, valid for ΛCDM background, are provided in appendix A."},{"title":"Scale factor","content":"The radial dependence of scale factor in LTB models can be observed only along the geodesic as a(z) = a(t(z), r(z)) (see Fig. 2). Moreover, modelling a(z) can be used a posteriori to inversely reconstruct the matter density profile (Wojtak & Prada 2017).\nInside homogeneous region close to the centre of GBH profile and at the background, scale factor satisfies eq. (25) and, hence:\nIn fact, redshift size of an inhomogeneous matter profile can be defined through the equality 2 (1 + z size ) a(z size ) = a loc 0 + 1 ."},{"title":"Local expansion rate","content":"As explained in previous section, LTB metric is characterised with two expansion rates, shown in Fig. 2. The transverse expansion rate given by Friedmann-Lemaître eq. (9) can be used together with density parameters Ω i to construct the equivalent FLRW model of each shell. On the contrary, radial expansion rate is related to the spatial derivative along the line of sight (cf. eq. (5)). Certainly, all physical quantities defined in ΛLTB model converge to their ΛCDM form at high redshift limit as a consequence of the Birkhoff theorem for the local spherical inhomogeneity (see Fig. 2). The intercept of SN Ia m(z) relation is determined by the absolute Figure 3. 1σ confidence stripes of SN Ia magnitude parameter M obtained with standard (grey) and binned: z ≤ 0.08 (green), 0.08 < z ≤ 0.15 (violet) and z > 0.15 (orange) fits to Pan+P data.\nmagnitude of SN and the local expansion rate. Calibrated SN Ia are able to measure the H 0 only after fixing the slope of distanceredshift relation as presented in Riess et al. (2016). However, if the inhomogeneity extends up to z ∼ 0.1, not accounting for its effect properly can represent an important systematic in H 0 measurement."},{"title":"RESULTS AND DISCUSSION","content":"In this section we present our main findings in the analysis of LD data from KBC13 and the SN Ia data from JLA and Pantheon 1 samples (Betoule et al. 2014;Scolnic et al. 2018). The likelihoods for SN Ia datasets are implemented as in their respective releases. Results are presented for the derived parameters: relative matter density at the background Ω bg m , central contrast δΩ 0 /Ω 0 , inhomogeneity size z size and density profile shape ∆z. We use a prior of Ω bg m = 0.3153 ± 0.0073 taken from CMB TT,TE,EE+lowE+lensing data analysis for the ΛCDM background (Planck Collaboration et al. 2018b), which appropriately aids low-redshift SNe to constrain the local matter density profile (hereafter indicated as JLA+P and Pan+P). Other parameters are sampled from wide flat prior ranges: −0.4 ≤ δΩ 0 /Ω 0 ≤ 0.2, z size ≤ 0.13 (≈ 0.5Gpc in comoving distance) and ∆z ≤ 0.6 z size , for which we verify a posteriori that relaxing the priors does not affect our results.\nAs elaborated in section 3, the large local void is expected to affect the magnitude-redshift relation of SN Ia inside the void, due to the higher transverse expansion rate and lower values of relative density parameters Ω m (z) and Ω Λ (z) w.r.t. ΛCDM constraints. We start our analysis by testing for redshift dependence of SN Ia magnitude parameter M = M B + 5Log 10 (c/H 0 ).\nTo this end, we use Pantheon SN Ia whose apparent magnitudes have already been corrected with a cosmology-independent method which marginalises the effect of all magnitude correction parameters and leaves only M as a free parameter, besides the cosmological model (see Scolnic et al. (2018)). Full Pantheon dataset, along with the CMB prior, is fitted to standard ΛCDM model, allowing for three binned values of M in redshift ranges z ≤ 0.08 The grey area is a 1σ reconstruction zone obtained using Gaussian process. The blue curve is our best-fit GBH profile to 10 LD points, which shows δρ 0 /ρ 0 = −43% contrast. The green curve has a weaker contrast δρ 0 /ρ 0 = −30% suggested by KBC13.\n(green), 0.08 < z ≤ 0.15 (purple) and z > 0.15 (orange), with 194, 103 and 751 SNe, respectively. The resulting 1σ confidence interval for M bg = −1.184 ± 0.007 is in excellent agreement with M tot = −1.185 ± 0.005 from the standard non binned fit shown as grey stripe in Fig. 3. The visible inconsistency in the first two bins, M 1 = −1.210 ± 0.015, M 2 = −1.153 ± 0.012 might be hinting to possible local geometry effects. Fitting the Pantheon SN Ia in the redshift range 0.0233 < z < 0.15 to the FLRW series expansion (SE) formula for luminosity distance with q 0 = −0.55, as presented in R16, we obtain M(SE) = −1.182 ± 0.010, which is also different from values obtained in the first two bins. Various works in the literature, including R16 and their previous articles, introduce a parameter a B that can be derived from the intercept magnitude M as\nwhere c is the speed of light in km/s. Results from the simple binned analysis already provide a strong motivation to study a more physical, inhomogeneous cosmological model, characterised by varying H 0 (r) and matter density profile Ω m (r). Although an extended analysis of the M evolution at higher redshifts could have important cosmological implications (see e.g. Tutusaus et al. (2019) for effect on cosmic acceleration), here we looked for possible local variation in M as a hint of local matter inhomogeneity, while our background cosmology is assumed to be flat ΛCDM."},{"title":"ΛLTB analysis with LD data:","content":"KBC13 summarises 22 LD data points obtained from several surveys, of which 10 are their original data points. Due to the unknown correlations that could arise from the overlap between different redshift surveys we opt to utilise only these 10 data points to constrain local matter density profile. They were obtained using data from 35,000 galaxies divided in 10 redshift bins in the range 0.005 < z < 0.20. Using LD points, KBC13 report the existence of a large local void with a luminosity density contrast of about −30% in the inner region, which is expected to be proportional to Table 1. Primary results from the ΛLTB analysis performed to: luminosity density using all 10 original data points presented in KBC13 (LD) and after removal of z = 0.034 data point from 2MASS galaxy catalogue (LD * ); 740 SN Ia from JLA dataset alone (JLA) and with Planck constraint on Ω bg m = 0.3153 ± 0.0073 (JLA+P); 1048 SN Ia from Pantheon dataset (Pan) and with Planck constraint (Pan+P); 237 low-redshift Pantheon SN Ia from the range 0.0233 < z < 0.15 with the Ω bg m = 0.3 constraint (Pan * ). The best-fit values for parameters of interest (noted as b. f. ) are provided together with confidence interval between 16th and 84th percentiles (noted as c.i. ) of their respective 1D marginalised likelihoods. Three contrast presented here are defined by Eqs. (27) to (29). The comparison to ΛCDM model is given using the relative ∆AIC values. the contrast of total matter density δρ 0 /ρ 0 (see Fig. 4). We initially perform a simple reconstruction of the density profile using the Gaussian process -a model-independent technique for data analysis (see e.g. is obtained by imposing homogeneity both inside and outside the expected void size. As such, it is allowed to predict no evidence of a void, but our resulting posterior of the reconstructed region shows definite indications of the local underdensity with δρ 0 /ρ 0 ∼ −40%.\nWe proceed by fitting LD data points to the full ΛLTB model and confirm that the conservative density contrast of δρ 0 /ρ 0 ∼ −30%, suggested by KBC13 and later utilised in HB17 and KSR19, appears to be an underestimate w.r.t. our best-fit shown in Table 1.\nDue to the small amount of data, we test our findings by performing a leave one out (LOO) analysis, implemented as random removal of each of the 10 points from the fit. It clearly shows the stability of the estimated redshift size, while the value of the profile contrast δρ 0 /ρ 0 ∼ −44.0±6.0 is found to be dominated by one stringent measurement at z = 0.034. Eliminating this data point coming from 2MASS survey, the remaining 9 points from UKIDSS and GAMA surveys estimate the contrast to be δρ 0 /ρ 0 ∼ −32.0 ± 9.5, in consistence with the conservative value suggested by KBC13. A direct comparison of our best-fit profile (blue) and the profile with δρ 0 /ρ 0 = −30% (green) in Fig. 4 clearly shows the importance of 2MASS data point at z = 0.034. Nevertheless, the induced contrast of fractional matter density δΩ 0 /Ω 0 coming from the fit to 9 points (quoted as LD * in Table 1) is still higher than the one used in previous works where the difference between δΩ/Ω and δρ/ρ contrasts is not considered (c.f. KSR19, HB17, Shanks et al. (2019)).\nThe primary inferences of this analysis are verified to remain unaltered changing the assumed analytic form of matter density profile. Taking the result in Table 1 at face value, would indicate an extreme tension with a homogeneous matter density scenario. While the standard model of cosmology expects variation of the local matter density and velocity fields, a more pressing problem arising from this result is the size of the estimated inhomogeneity. Based on ΛCDM matter power spectrum constrained by CMB, density perturbations of size z size ≈ 0.08 are expected to have the variance of δρ 0 /ρ 0 ≈ 1.5−2% (corresponds to δΩ 0 /Ω 0 ≈ 2−3%) at 68% confidence level (Marra et al. 2013;Camarena & Marra 2018). Hence, this expectation is at ∼ 8σ tension with the result of the fit to complete LD data, while the exclusion of 2MASS data point at z = 0.034 relaxes deviation to a however large ∼ 4σ tension.\nThe best-fit ΛLTB (blue) curve in Fig. 4 is closely following the predicted model-independent reconstruction (grey), except for the short overdense region, which is an expected compensation of a large and steep underdense matter profile. In fact, the data point at z = 0.081 might be a signature of the same. Although not statistically significant to draw conclusions for an overdensity, it aids to constrain the void size. The specific interplay between the model description and the fact that only 10 data points are available for constraining 4 profile parameters, results in over-fitting difficulties. For this reason we perform a Gaussian approximation of the posterior likelihood distribution before estimating the confidence regions presented in Fig. 6, which does not alter our primary inferences.\nGiven the lack of consistency in reported results based on LD data, limited sky coverage of the data, reports about angular variation (WS14), and possible effects of binning we also deem it important to estimate the level of (dis)agreement with the SN Ia data."},{"title":"ΛLTB analysis with SN Ia datasets:","content":"Probing for signatures of large local inhomogeneity (underdense or overdense) in the SN Ia datasets, we fit the local matter density profile simultaneously with the background cosmology (Ω bg m ) which is, nevertheless, dominantly guided by the Planck CMB constraint. Since the variation of H 0 (r) is modelled, in ΛLTB we define the SN Ia intercept magnitude parameter using background expansion rate\nIn the case of JLA dataset, fitting χ 2 also depends on SN Ia magnitude correction parameters for lightcurve stretch, colour and host galaxy mass, which are later marginalised as nuisance parameters (Betoule et al. 2014).\nThe SN Ia constraints on local matter density profile presented in the form of confidence regions in Fiqs. 5 and 6 and Table 1 show high consistency with homogeneous ΛCDM fit, characterised by δΩ 0 /Ω 0 = 0. The analysis of JLA dataset yields two distinguished local χ 2 minima, emphasised with dashed lines in Fig. 5. The underdensity reported by WS14, with the size of ≈ 215h 70 −1 Mpc and the average contrast δρ 0 /ρ 0 = −15 ± 3% (corresponds to δΩ 0 /Ω 0 = −19 ± 4%) is in between the two minima we find for JLA dataset. Interestingly, the second local minimum specifies a void with the physical size in line with the findings of KBC13, d A (c) = 300h 70 −1 Mpc, but with a much weaker contrast. Indeed, the large local void reported by KBC13 is rejected at 4σ confidence by JLA dataset. However, our analysis of reduced LD * data points gives less stringent constraint on the void contrast which is no longer in tension with our JLA result (cf. Table 1).\nEven though the analysis of Pantheon dataset does not recover the primary minimum obtained for JLA supernovae, the resulting confidence regions are of high similarity. Our Pantheon and Pan+P constraints for ΛLTB model shown in Fig. 6 include inside 1σ confidence region voids with contrast δΩ 0 /Ω 0 ≈ −20%, such as the representative void of WS14. The constraints from LD data of KBC13 (blue shaded ellipses in Fig. 6) are pointing to much deeper void, although the redshift size of the two best-fits are aligned. In fact, the estimated contrasts from Pan+P and LD data are at ∼ 4σ tension (see Table 1). The nature of inferences resulting from LD data, their disagreement with the expected cosmic variance and the SN Ia constraints, further strengthens our motive to perform LOO analysis to test this data sample. LOO analysis led us to consider the fit to the reduced LD * dataset, which is shown with blue dashed contours in Fig. 6. As mentioned earlier, the LD * constraint on void size remains unaltered, while the estimated void contrast shifts towards a shallow inhomogeneity that has no tension with the Pantheon results.\nOur ΛLTB inferences coming from Pantheon dataset are contrasting the recent result of KSR19 which reports a very high significance of ∼ 5σ against any large local void with contrast |δρ 0 /ρ 0 | > 20%. The higher amount of SN Ia data points that they used, 1295 compared to 1048 available in Pantheon, are additionally populating the low-redshift range and are expected to improve the error bars of profile parameters by 50%. However, we sus- Figure 6. 1σ and 2σ confidence contours of local matter density profile parameters are shown for Pan+P (green regions), Pantheon (light green solid curves), full LD data from KBC13 (blue regions) and reduced LD * data (dashed blue curves). The isotropic voids proposed by WS14 and KBC13 are marked with orange crosses. Contrasts presented on vertical axes are defined by Eqs. (28) and (29). pect that theoretical modelling is also contributing to the difference w.r.t. our result. While the newer data is not publicly available at this moment, we intend to extend the current analysis in a future communication. In appendix A we present the effects of simplifying the density profile from GBH to TH solution and fixing the model parameters such as Ω bg m and z size , instead of marginalising over them to obtain the constraints on void contrast.\nWe chose to report the results of SN Ia analysis with and without the Planck CMB constraint on background matter density. Both SN Ia datasets provide excellent fits in the case of a local underdensity, but do not exclude the possibility for a z size > 0.1 overdensity (see Fiqs. 5 and 6). The Planck prior on background cosmology reduces the degeneracy between Ω bg m and δΩ 0 /Ω 0 parameters in SN Ia likelihood, providing tighter constraints. Nevertheless, the main inferences presented here are not strongly affected by this prior due to the excellent agreement between the Planck value and our SN Ia best-fits for Ω bg m . Examples presented in appendix A also explore the case of using a fixed value for the relative matter density at the background, as contrary to sampling this likelihood parameter. Compared to homogeneous model, the additional three parameters that characterise the void profile afford better fits to the SN Ia data, namely χ 2 JLA = 678.30 and χ 2 Pan = 1020.72 for ΛLTB model in contrast to χ 2 JLA = 682.94 and χ 2 Pan = 1026.05 for ΛCDM model. We quantify this comparison with Akaike information criteria Akaike (1974) that penalises the usage of three extra parameters as ∆AIC = ∆χ 2 + 2 * 3. All fits reported in Table 1 have ∆AIC < 2 implying no preference between the two models."},{"title":"Anisotropy considerations","content":"Using galaxy redshift distribution and number counts up to the depth of z < 0.1, WS14 observed 215h 70 −1 Mpc local underden-sity and an angular variation of its estimated density contrast in the range −4% to −40%, with the mean of −15% (corresponds to δΩ 0 /Ω 0 = −19% shown in Fiqs. 5 and 6). Likewise, the Fig. 11a in KBC13 clearly shows discrepancy between LD points observed in different regions on the sky, again pointing towards a possibly anisotropic local density profile. Hence, an important extension to the analysis presented here can be done by taking into account the anisotropy of the data, as well as the matter density profile and the observer's position inside of the considered inhomogeneity (see e.g. Blomqvist & Mörtsell (2010)). If there was any anisotropy present in the matter distribution, it would be averaged out in the analysis that uses isotropic model like ours, leaving us with the expectation to find angular variation of density contrast around this average when modelling for anisotropic matter profiles. Be that as it may, the statistical significance in favour of the observed anisotropy in LD data is limited by the sky coverage of the used surveys, which amounts to ∼ 15% for KBC13 and ∼ 22% for WS14, spread over three distant patches on the celestial plane. Several groups performed tests on SN Ia data looking for features of local and/or global anisotropy Bengaly et al. 2015;Kalus et al. 2013;Wang & Wang 2014;Sun & Wang 2018;Andrade et al. 2018), also in relation to the local matter distribution (Colin et al. 2011). These anisotropy estimates are also impaired by the inhomogeneous angular distribution of the observed SN on the sky, which is particularly visible on medium and high redshifts due to the narrow fields of the surveys, such as SDSS. Probing low-redshift SN Ia for signatures of anisotropy, we estimate the statistical variance of model parameters over angular directions. Concretely, the two parameters of interest, M and δΩ 0 /Ω 0 in ΛLTB model, are tested separately for dipole anisotropy (see also e.g. Mariano & Perivolaropoulos (2012)). The magnitude dipole is modelled as a vector of intensity M d and direction n d . Then, the magnitude parameter M (i) of each SN Ia inside the void becomes\nwhereM is the average intercept magnitude, while the normalised vector n (i) represents the angular position of each SN Ia. Although we fitted the full Pantheon dataset, the dipole is constrained mainly by 190 SN Ia under z ≤ 0.08, which are also more homogeneously distributed on the celestial plane compared to the full dataset. Here we used a TH matter density profile for simplicity (see section 2). Finally our estimate for the magnitude dipole\nwould produce the angular variation\nThis angular variance of magnitude inside the void can be related to the anisotropy of local expansion rate as\nOur result for the level of local anisotropy in Pantheon SN Ia is consistent with previous works that find null evidence against assumption of the isotropy (Andrade et al. 2018;Zhao et al. 2019b), although less stringent compared to global anisotropy constraint found by Soltis et al. (2019). Analogously to eq. (36) we statistically probed the dipole anisotropy of the estimated void contrast and found that it could account for the angular variance ∆ δΩ 0 Ω 0 r.m.s. = 3.7 ± 4.1% .\nWhile this result at face value shows that Pantheon dataset does not exclude the angular variation of estimated void contrast ∆(δΩ 0 /Ω 0 ) ∼ 10% at 2σ confidence level, we still find it difficult to coincide our constraints on matter density profile from Pantheon SN Ia and LD data, or explain the level of anisotropy observed by WS14. We would like to note that two former examples should be considered only as rough estimates of anisotropy level since: the dipole model does not represent a complete relativistic solution of FE, the variation of profile size is not considered, and the Pantheon SN Ia have had the apparent magnitudes corrected using the BBC method which relies on the assumption of isotropy (Scolnic et al. 2018). Although crucial in future studies, the present data is not quantitatively sufficing to benefit from the full anisotropic analysis given as an extension of the cosmological metric and theoretical model for the anisotropic density profile."},{"title":"Distance ladder measurements","content":"One of the most important implications of a large local void is the effect on direct measurement of present cosmic expansion rate by the means of low-redshift SN Ia. Consider a spherical matter inhomogeneity, characterised with a density contrast of e.g. δΩ 0 /Ω 0 = −25% which is inside the 1σ confidence region of Pan+P constraints, presented in Table 1 and Fig. 6. This matter contrast will also induce a higher local present expansion rate by δH 0 /H 0 ≈ 3.6% w.r.t. the background expansion rate (see Eqs. (28) and (29). Hence, the luminosity distance of a SN located on any shell inside the local void is higher than what one would estimate using the background metric. This can affect the analysis of SN Ia data in the redshift range inside the local void as well as the resulting model inferences.\nIn light of everything said above, let's assume our local matter distribution has indeed an isotropic underdense profile. Analysing SN Ia data from the redshift range inside this local void with a simpler homogeneous model would provide a different constraint on H bg 0 than the LTB model. In fact, the assumption of locally homogeneous cosmic expansion (H loc 0 = H bg 0 ) can overestimate the value of background expansion rate, while underestimating H loc 0 , by ∼ 1 2 δH 0 /H 0 and affect the inferences on other background cosmological parameters.\nAs discussed in Section 4.1 theoretical expectations for the large local void based on ΛCDM model, and its effect on H 0 measurement are small, which is reflected in results presented by recent studies. On one side, the level of expected cosmic variance in local matter density for the Planck ΛCDM model is estimated to impact the systematic error of H bg 0 at 2% when using SN Ia from z ≥ 0.01 and at 1% when using SN Ia from z ≥ 0.0233 (Marra et al. 2013). On the other side, the large-volume cosmological Nbody simulations based on the Planck ΛCDM model, such as e.g. in Skillman et al. (2014), were used to quantify effects of both the cosmic variance and the SN Ia sample selection on H bg 0 value -Wu & Huterer (2017); Odderskov et al. (2016) estimate the expected variance level of measured H bg 0 to be ∼ 0.5% for 0.0233 < z < 0.15 cut (c.f. an earlier work by Wojtak et al. (2014)). In order to minimise the effect of local matter distribution, R16 (including their previous works) use only the SN Ia above z > 0.0233 for the modelindependent estimate of H bg 0 (Jha et al. 2007). In consistence with\nWu & Huterer (2017), KSR19 report no evidence for strong variation of matter density profile in the range z < 0.15 from SN Ia data, securing the ability to measure H bg 0 value to a percent value. The measurement of background expansion rate based on lowredshift SN Ia is obtained from the intercept magnitude parameter M (see Section 4.2). While the exact value of H bg 0 relies on local SN Ia luminosity calibration, we are mainly interested at the relative change of this estimate due to the effect of a large local void, which can be seen as the shift in estimated value of M. For this purpose, and only in this subsection, we analyse 237 Pantheon SN Ia from 0.0233 < z < 0.15 redshift cut. Fitting them to FLRW series expansion (SE) formula for luminosity distance and using a fixed value of q 0 = −0.55, as presented in R16, we obtain M(SE) = −1.1825 ± 0.0094. In low redshift range, this fit is equivalent to ΛCDM fit with Ω m = 0.3, which we will use for comparison to underdense scenario. The same Pantheon cut is fitted to ΛLTB model with background cosmology prior Ω bg m = 0.3 (q bg 0 = −0.55), density contrast prior range −50% ≤ δΩ 0 /Ω 0 ≤ 5%, void size 0.025 ≤ z size ≤ 0.12, and shape parameter 0 ≤ ∆z ≤ z size . In order to ensure enough SN Ia that constrain the background, we consider only profiles that reach the background regime up to z ≤ 0.13 by additionally removing the region of parameter space with such combination of large z size and ∆z values. After marginalising the matter density profile out of the low-redshift Pantheon likelihood, we estimate the intercept SN magnitude parameter 2 to be M(ΛLTB) = −1.1577 +0.0158 −0.0148 . The remaining parameter constraints are presented in the last row of Table 1. Regardless the three additional parameters that model the local matter underdesnity, the ΛLTB fit to low-redshift Pantheon SN Ia reveals an improvement of the best-fit by ∆AIC=-3.3.\nThe local matter underdensity affects the estimated value of M due to its degenearcy with density constrast. This can be seen on Fig. 7, where we show the low-redshift Pantheon constraints in δΩ 0 /Ω 0 −M plane. The best-fit of ΛCDM model (equivalent to SE) can be found on vertical axis for zero void contrast and is marked as a red point in Fig. 7, which is 1.7σ away from our constraints on ΛLTB model. Both values for the intercept magnitude parameter M have been estimated fitting the same 237 Pantheon SN Ia to two different models and are, hence, fully correlated. Taking into account this correlation, the difference between two values is evalueted to be ∆M = 0.0247 +0.0065 −0.0055 . As the ΛLTB model has more parameters in fitting, the error bar of resulting M(ΛLTB) is bigger than in the case of M(ΛCDM), and elseways, we note that if they were equivalent then the uncertainty on estimated shift ∆M would have been zero. Since this change of intercept magnitude value is coming only from the theoretical modelling and the two models have comperable AIC values (see Table 1), it should be seen as a possible source of systematic error on M estimate. The positive shift of ∆M = 0.0247 corresponds to 1.14% lower H bg 0 obtained with ΛLTB model than with SE formula. Compared to the total error of M(SE), this value of ∆M is significant. However, the major contribution to the uncertainty of H 0 measurement is presently constricted to calibration of standard SN Ia absolute magnitude, M B , that together with much smaller uncertainty on intercept magnitude M(SE) reaches the total error bar of 2.4% reported by R16. Adding the effect of large local matter inhomogeneity (z size > 0.025) on estimate of SN Ia intercept magnitude could increase this uncertainty level to 2.7% on the lower end of H 0 measurement. The improvement in Cepheid and SN Ia calibation techniques (see R19), followed by reduction 2 Confidence intervals are given between 16th and 84th percentiles of uncertainty on M B , can lead to a situation in which the effect of local matter inhomogeneity becomes more relevant for the measurement of background cosmic expansion rate."},{"title":"CONCLUSIONS","content":"In the present work we have investigated the evidence for a large local void using a well formulated ΛLTB model against the most recent publicly available SN Ia compilations. In order to reduce the correlation with background cosmology, we have used the CMB constraint on Ω bg m (Planck Collaboration et al. 2018b), which aids to provide a slightly tighter constraints on the local void profile. The analysis on SN Ia datasets is also complimented with the luminosity density data obtained from galaxy surveys by Keenan et al. (2013), which together with Whitbourn & Shanks (2014) played a key role in propelling the recent discussion about possible existence of a large local void and its effect towards resolving one of the most prominent discordances in modern cosmology -the H 0 tension. We summarise our primary results in the following. All reported confidence intervals (c.i.) are between 16th and 84th percentiles.\n• Fitting ΛLTB model to LD data from KBC13, we find a void with density contrast of (δρ 0 /ρ 0 ) c.i. = −43.8 ± 6.0%, deeper than originally proposed δρ 0 /ρ 0 = −30%.\n• From SN Ia we do not find a strong evidence for a void or otherwise -a fit to Pan+P yields (δρ 0 /ρ 0 ) c.i. = −7.9 +7.0 −5.8 % which corresponds to (δΩ 0 /Ω 0 ) c.i. = −10.5 +9.3 −7.4 %, and a wide range for the redshift size z c.i. size = 0.068 +0.021 −0.030 . However, Pantheon constraints do not exclude a void of e.g. z size ≈ 0.075 and δΩ 0 /Ω 0 ≈ −25% at 1σ confidence level (c.f. Fig. 6).\n• JLA likelihood is in a good overall agreement with Pantheon, except in the range z ≤ 0.04 (c.f. Fiqs. 5 and 6). Both SN Ia datasets are at 3σ tension with our result obtained from KBC13 data, but in excellent agreement with parameters of the isotropic void proposed by WS14.\n• Leave one out analysis of KBC13 data reveals that the corresponding estimated contrast is dominantly constrained by one stringent point. Removing this only data point from 2MASS survey and using the remaining 9 points from UKIDSS and GAMA surveys, relaxes their tension with SN Ia data.\n• The model comparison with Akaike information criteria shows no preference between the tested models (c.f. Table 1).\n• Probing the level of statistical variance in ΛLTB model parameters over angular direction, we find null evidence for the dipole anisotropy. Its contribution to the angular variance is small: ∆M r.m.s. = 0.008 ± 0.018 for the SN intercept magnitude parameter or ∆ δΩ 0 Ω 0 r.m.s. = 3.7 ± 4.1% for the void contrast.\n• In our analysis of SN Ia data we do not find any evidence for a large isotropic void that could resolve the 9% discrepancy between two H 0 estimates in tension, leaving this local effect alone to be a highly unlikely explanation.\n• We fitted low-redshift range of Pantheon SN Ia data from 0.0233 < z < 0.15 to the FLRW series expansion formula for luminosity distance and to the ΛLTB model, comparing their estimates of SN Ia intercept magnitude, which is used for H 0 measurement. Adding the possibility of a large local void, we observe a small improvement in the fit by ∆AIC = −3.3 and a shift of the inferred intercept magnitude parameter ∆M = 0.0247 +0.065 −0.055 that could be a source of additional systematic error in H bg 0 of 1.14%. While the simpler model, which is based on series expansion formula of luminosity distance in FLRW metric, is only 1.7σ away from our ΛLTB constraints (see Fig. 7), this result is questioning whether the total error budget of distance ladder measurements based on low-redshift SN Ia should be reconsidered.\nIncrease of presently available data, followed by more complex theoretical modelling, is necessary to better understand the observed disagreement between LD and SN Ia samples. As already mentioned, considering off-centre position of the observer would extend the present isotropic ΛLTB formalism, also allowing for an anisotropic point of view by construction. With increasing number of observations in future surveys, SN Ia may prove as effective tracers of local matter density distribution. Figure A1. 1σ and 2σ confidence regions for ΛLTB model with TH density profile, fitted to Pantheon dataset are shown as coloured contours, while the solid lines represent the confidence regions obtained from the smoothed χ 2 function. In the first panel we used Planck prior for Ω bg m , whereas the second and third panels are obtained with fixed Ω bg m = 0.3. Moreover, in the third case z size parameter was a priori fixed to a range of z split values, followed by χ 2 sampling over δΩ 0 /Ω 0 contrast values."},{"title":"Figure 1 .","content":""},{"title":"Figure 2 .","content":""},{"title":"Figure 4 .","content":""},{"title":"Figure 5 .","content":""},{"title":"Figure 7 .","content":""}],"string":"[\n {\n \"title\": \"INTRODUCTION\",\n \"content\": \"The most reliable standard candles and one of the fundamental low-redshift astrophysical observables (Betoule et al. 2014;Scolnic et al. 2018), type Ia supernovae (SN Ia), also provide a cosmologyindependent measurement of present expansion rate through the local distance ladder (Riess et al. 2016) (hereafter R16). In concert with observations of baryon acoustic oscillation (BAO) peak from galaxy redshift surveys (Eisenstein et al. 2005;Aubourg et al. 2015;Alam et al. 2017;Bautista et al. 2017;du Mas des Bourboux et al. 2017;Zhao et al. 2019a) and cosmic microwave background (CMB) radiation data (Hinshaw et al. 2013;Planck Collaboration et al. 2018a), SN Ia contributed to establishment of the standard model of cosmology. Astonishing concordance between different cosmological observables brought by technical advancements in the last decades was followed by a number of inconsistent and/or unexpected results, e.g. estimated values of matter clustering parameter, S 8 , from low and high redshifts (Planck Collabo-E-mail: vladimir.lukovic@roma2.infn.it † E-mail: haridasu@roma2.infn.it ‡ E-mail: nicola.vittorio@roma2.infn.it ration et al. 2016;Joudaki et al. 2019;Lange et al. 2019;Martinelli & Tutusaus 2019), CMB parameter constraints in curvature extension of base Λ cold dark matter (ΛCDM) model (Planck Collaboration et al. 2018b) (hereafter P18), as well as several unexplained CMB anomalies (Schwarz et al. 2016), etc. The best-known amongst ΛCDM tensions and one of the biggest challenges in modern cosmology is related to the H 0 value. Defining the scale of extra-galactic distance through the Hubble radius, as well as the present expansion rate, H 0 is a fundamental cosmological parameter both for CMB and low-redshift data (Freedman et al. 2001;Freedman & Madore 2010). The most stringent ΛCDM estimate coming from P18, H 0 = 67.36 ± 0.54 km s −1 /Mpc, is in 4.4σ tension with the most recent local measurement based on calibrated SN Ia, H 0 = 74.03 ± 1.42 km s −1 /Mpc, by Riess et al. (2019) (hereafter R19). In comparison, H 0 measurements from the time delay of strongly lensed quasars support a higher value (Birrer et al. 2019;Wong et al. 2019); using differential age method with cosmic chronometers (Jimenez & Loeb 2002) provides an estimate in the middle of two results in tension (Jimenez et al. 2019;Chen et al. 2017;Luković et al. 2016); while the constraints coming from gravitational waves are still fairly weak (Liao et al. 2017). Although the quality of BAO data has substantially improved over the last decade, providing some of the most stringent cosmological constraints on expansion dynamics (Haridasu et al. 2018a;Ramanah et al. 2019;Zhang et al. 2019), their H 0 estimate is degenerated with the sound horizon parameter, whose value relies on CMB observations and early Universe physics (Macaulay et al. 2019;Lemos et al. 2019;Aubourg et al. 2015;Cuceu et al. 2019).\\nThe intercept of SN Ia magnitude-redshift relation is determined by their absolute magnitude (M B ) and the Hubble radius, resulting in two degenerate parameters in SN Ia analyses. While the highredshift SN Ia are necessary for assessing the underlying physics of dark energy, the low-redshift end (z ≤ 0.15) of SN Hubble diagram can be analysed with minimal assumptions about properties of the cosmic fluid and expansion dynamics related only to the well-accepted cosmic acceleration (Haridasu et al. 2017). Indeed, calibrating SN Ia absolute magnitude with the use of Cepheid variable stars and fixing the local distance-redshift relation (at z ≤ 0.15) leads to a remarkable cosmology-independent measurement of H 0 (Riess et al. 2005(Riess et al. , 2009(Riess et al. , 2011(Riess et al. , 2016(Riess et al. , 2018b. Different calibration techniques consider the usage of strong gravitational lensing (Taubenberger et al. 2019), tip of the red giant branch (Beaton et al. 2016;Freedman et al. 2019), HII regions (Fernández Arenas et al. 2018), etc. Due to the strong correlation of cosmic expansion dynamics and H 0 value, the model-independent local estimates remain crucial for the study of late-time cosmic evolution [see also Tutusaus et al. (2019); Haridasu et al. (2018b)].\\nAs is well-known, a large inhomogeneity in local matter density distribution can affect the geometry of local Universe and the distance measures, consequently biasing the resulting model inferences and the H 0 estimate. A few independent groups used galaxy survey catalogues to test the local matter distribution, reporting hints of a large local void (Keenan et al. 2012(Keenan et al. , 2013Whitbourn & Shanks 2014Boehringer et al. 2019). Concretely, Keenan et al. (2013) (hereafter KBC13) used UKIDSS Large Area Survey (Lawrence et al. 2007), Galaxy and Mass Assembly (GAMA) (Driver & Robotham 2010;Driver et al. 2011) and 2M++ catalogue (Lavaux & Hudson 2011) to construct luminosity density sample (hereafter LD) over the redshift range 0.01 < z < 0.2. Relating it to the total matter density in the local Universe, they find evidence for an underdense region and suggest a void of about 300h 70 −1 Mpc in size and −30% contrast with respect to the background. Whitbourn & Shanks (2014) (hereafter WS14) used 6dFGS, SDSS, and GAMA surveys to examine the local galaxy density field over 20% of the sky and up to 450h 70 −1 Mpc in depth. Both number counts and peculiar velocity fields show evidence for at least 200h 70 −1 Mpc large local underdensity with −15% ± 3% average contrast, ranging between −40% and −5% in different directions. Analysing a catalogue of 1653 galaxy clusters detected by their X-ray emission Boehringer et al. (2019) report evidence for ∼ 140 Mpc large local underdensity with density contrast of 20% ± 10%.\\nThe results of KBC13, WS14 and Boehringer et al. (2019) go against the standard cosmological paradigm and the assumption of large-scale homogeneity. In fact, indications for a few hundred Mpc large local underdensity encouraged renewed popularity of void models based on Lemaître-Tolman-Bondi (LTB) metric, also motivated as a remedy for the H 0 tension (Moffat 2016;Tokutake et al. 2018;Hoscheit & Barger 2017;Shanks et al. 2019;Kenworthy et al. 2019) and even as a possible explanation for the CMB cold spot (Szapudi et al. 2015). Contrasting the homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) models, the LTB metric (Lemaître 1927(Lemaître , 1933Tolman 1934;Bondi 1947) describes an isotropic but inhomogeneous dust system, which on cosmological scales can be used for studying cold dark matter den-sity distribution. Since the discovery of cosmic acceleration, LTB metric was used to construct toy models that challenge the cosmological constant paradigm with an alternative scenario in which the apparent accelerated expansion is a result of the strongly underdense local Universe that smoothly converges to Einstein de Sitter solution on higher redshifts (Célérier 2000;Alnes et al. 2006;Clifton et al. 2008;February et al. 2010;Nadathur & Sarkar 2011;Bolejko & Sussman 2011;Zhang et al. 2015;Stahl 2016). Since the deceleration-acceleration transition redshift is well constrained at z t ≈ 0.6 [see e.g. Haridasu et al. (2018b); Gómez-Valent (2019); Mukherjee et al. (2019) for model-independent estimates], this alternative explanation requires a giant ( ≈ 3Gpc) isotropic void that proved to be very unlikely compared to the standard cosmological model (Zibin 2011;Vargas et al. 2017;Amendola et al. 2013;Luković et al. 2016). However, extending the LTB model with the addition of cosmological constant Λ, one can describe large (order of 100Mpc) local inhomogeneous matter distribution inside a ΛCDM background (Valkenburg 2012;Valkenburg et al. 2014;Rigopoulos & Valkenburg 2012).\\nWhile the level of cosmic variance in matter distribution expected for standard model can only partially relieve the Hubble constant problem (Marra et al. 2013;Wu & Huterer 2017;Camarena & Marra 2018), Tokutake et al. (2018) found that a strongly underdense local inhomogeneity with non-vanishing cosmological constant, modeled with an approximate numerical solution for LTB metric, can fully coincide the CMB constraints with the local H 0 measurement. Similarly, Hoscheit & Barger (2017) (hereafter HB17) extended the work of KBC13 by testing the consistency of their earlier findings with constraints coming from SN Ia and linear kinematic Sunyaev-Zel'dovich effect and showed that local void found by KBC13 could reduce the H 0 tension. A recent work of Shanks et al. (2019) argues that the combination of local inhomogeneity effect together with the re-calibration of Cepheids and local SN Ia distances using the recent parallax measurements from Gaia mission, is sufficient to completely remove the tension [see also Riess et al. (2018a); Shanks et al. (2018) for follow-up discussions]. Kenworthy et al. (2019) (hereafter KSR19) find that any inhomogeneity with contrast > 20% is strongly inconsistent with SN Ia data, consequently reassuring the ability to measure the distance and Hubble constant with locally calibrated SN to a 1% precision.\\nThe goal of this work is to probe the local matter density using the SN Ia as one of the most reliable low-redshift astrophysical observables, constraining size and amplitude of possible matter inhomogeneity at our local position. We investigate the findings of KBC13 and KSR19 by using the complete analytic description for treating the LTB metric in presence of cosmological constant (hereafter ΛLTB) introduced in Valkenburg (2012). Extending the earlier works, we directly fit the ΛLTB model to two biggest publicly available SN Ia samples -Joint light curve analysis Betoule et al. (2014) (hereafter JLA) and Pantheon compilation Scolnic et al. (2018) (hereafter Pan), consisting of 740 and 1048 SN Ia, respectively. We also utilise the luminosity density data from KBC13 within the given formalism to assess the (dis)agreement. In section 2 we review the theoretical description of LTB metric in presence of cosmological constant and proceed by presenting the effect that a local void can have on observable physical quantities in section 3. Constraints on the void contrast and size coming from fitting JLA, Pantheon and KBC13 datasets are reported in section 4, together with the revision of their consistency. Additionally, we discuss on the possible effect of local isotropic inhomogeneity on the H 0 measurement, and explore the level of anisotropy allowed by the data. Summarising our results in section 5, we examine the dif-ferences compared to earlier findings by other teams. Finally, in the appendix A we provide a simplified void model capable of correctly reproducing our results based on complete ΛLTB description to a good approximation.\\nThroughout the paper we use the dimensionless variable h 70 = H 0 /70 km s −1 /Mpc for distance measures; acronym w.r.t. stands for 'with respect to'.\"\n },\n {\n \"title\": \"LOCAL MATTER DENSITY IN ΛLTB MODEL\",\n \"content\": \"The simplest extension of standard homogeneous cosmological model in presence of large matter inhomogeneity is done using the LTB metric (Lemaître 1933;Krasiński 1997) that describes spherically symmetric pressureless cosmic fluid\\nHere R(t, r) has the units of length and defines all physical distance measures as well as the expansion history in this model. The second free function of the metric, k(r) is dimensionless and defines the curvature within the shell of a radius r. We note that radial coordinate r is also dimensionless, with no physical meaning, and it should be considered only as a flag coordinate of different shells all centred at r = 0. Using the above metric and the Einstein field equations (FE) for pressureless inhomogeneous matter fluid in presence of cosmological constant Λ, one can derive the generalised Friedmann-Lemaître equation as\\nHere M(r) is an integration constant, whose physical meaning is the total mass enclosed inside a sphere with radial coordinate r,\\nInhomogeneous cosmic matter density profile, ρ m (t, r), satisfies the conservation law coming from FE, ensuring that integrated mass M(r) remains constant in time. In inhomogeneous cosmology we differ two expansion rates, namely transverse and radial:\\nHence, the one appearing in Friedmann-Lemaître eq. (2) is the transverse expansion rate. As usual, we denote t = t 0 as present age of the Universe and all the physical quantities with subscript \\\"0\\\" represent their values today, while H 0 (r) = H ⊥ (t 0 , r) is used for the transverse expansion rate. Introducing scale factor, the Friedmann-Lemaître equation can be rewritten in a simpler form, although the definition of scale factor a(t, r) is not trivial nor customary as in the homogeneous cosmology. The value of flag coordinate r is commonly defined through a gauge with some other physical variable. The choice of this gauge is arbitrary and has no implications on the model predictions nor on the results of data analysis. The most often used gauge (Alnes et al. 2006;Alfedeel & Hellaby 2010;Nadathur & Sarkar 2011) in LTB framework is:\\nHowever, in this work we use an alternative definition which turns out to be more convenient choice for the latter calculations of elliptical integrals present in the geodesic equations and distance variables. Following Valkenburg (2012), we define the dimensionless radial coordinate r through the relation with the total mass enclosed in a given shell as\\nwhere C M is a normalisation constant in units of mass whose value will be chosen at a later stage. Commonly, in both gauges the scale factor can be defined as\\nThis function clearly converges to the usual scale factor definition and the spatially uniform values in the limit of homogeneous FLRW metric. Generalised Friedmann-Lemaître eq. (2) can be rewritten in terms of three dimensionless density parameters normalised to the present time,\\nThese can be evaluated combining eqs.\\n(2), (4) and (7) to (9) as,\\nHereM = 8πGC M /(3c 3 t 0 ) andΛ = 1 3 Λc 2 t 0 2 are dimensionless constants, whereas k(r), H 0 (r) and a 0 (r) are not uniform in LTB metric. The introduced dimensionless density parameters are different from one shell to another, but do not depend on time.\\nSince Ω m (r) is proportional to the total mass enclosed in a shell, M(r), it can be represented as the ratio of average and critical matter densities of each shell. Starting from eq. (3) and then using eqs. (7) and (8) one can define average matter density:\\nThen, definition of critical matter density together with eqs. (10) and (13) straightforwardly confirm:\\nFriedmann-Lemaître eq. (2) can also be rewritten in dimensionless form using eqs. (10) to (12)\\nIn terms of the Birkhoff theorem, eqs.\\n(2), (9) and (15) dictate every shell in ΛLTB model to evolve as an independent homogeneous FLRW model with a specific curvature k(r), a total mass enclosed M(r) and the same value of cosmological constant Λ for all shells, determining its transverse expansion rate H ⊥ (t, r). Each shell is described with relative density parameters Ω i (r), present expansion rate H 0 (r), and evolving scale factor a(t, r)/a 0 (r). Cosmological constant, included in the Einstein field equations, is uniform over space and constant in time due to its physical nature, which is also true for the dimensionless constantΛ that we introduced. The density parameter corresponding to the cosmological constant, Ω Λ (r), can not be uniform in ΛLTB model, as this would imply that Λ is inhomogeneous and coupled to the cold dark matter density profile. This approach is reflected in a number of recent studies Valkenburg (2012) 2019), while not fully followed by HB17 who assume cosmic density parameter Ω Λ to be constant throughout the underdense region and at the background. Similarly, the methods of inverse reconstruction of matter density profile in presence of cosmological constant must be constructed such that FE are satisfied (c.f. Tokutake & Yoo (2016); Wojtak & Prada (2017)).\\nThe gauge choice for the flag coordinate r, which is set by eq. (7), enabled us to rewrite the Friedmann-Lemaître eq. (2) in a more elegant analytic form, i.e. eq. (9). The remaining degree of freedom in the scaling constant C M , or equivalentlyM, can be used to normalise the scale factor a(t, r). We setM such that a * = a (t 0 , r * ) = 1 for a chosen shell r = r * and at the present time t 0 . For example, one can choose r * = 0 to normalise the scale factor at the observer's position or r * = +∞ to normalise at the background, instead. We follow the normalisation of scale factor on the background, setting the numerical value ofM such that a * = a (t 0 , r * = +∞) = 1 for the background shells. In particular, normalisation gives the physical meaning to the scaling constant, i.e. eq. (13) yields\\nTherefore, normalising far from the inhomogeneity, which is for any large enough r * , implies that background scale factor and eq. (15) converge to the usual forms that we have in homogeneous FLRW model. It is easy to see that at present time t = t 0 the background quantities satisfy\\nPartial differential eq. (15) can be used to express the derivative ∂t/∂a and integrate from the Big Bang up to scale a:\\nAll variables on the right hand side are dimensionless and this elliptical integral is analytically solvable (for details see Valkenburg (2012)). The Big Bang time t BB (r) is an integration constant that may vary in space, affecting the age difference between the shells. Nevertheless, one does not expect large variation of t BB (r) in the case of a small local inhomogeneity. Non-uniform Big Bang time is also related to instabilities of matter perturbations (Zibin 2008). Hence, in this work we consider only the models with homogeneous Big Bang time t BB (r) ≡ const = 0. Integrating eq. (18) on the background (r = r * ) from the Big Bang a = 0 up to today a(t = t 0 , r * ) = 1 we have\\nwhich is the numerical equation defining a value ofM, necessary for the normalisation. At any other point of interest eq. (18) can be seen as the solution for t(a, r) or, after invertion, as a(t, r). Hence, calculating numerically the scale factor at any spatial position and time gives us the angular diameter distance R(t, r) and other physical observables of interest.\"\n },\n {\n \"title\": \"EFFECT ON ASTROPHYSICAL OBSERVABLES\",\n \"content\": \"The high-redshift background observables are not affected by the local isotropic inhomogeneity. Nevertheless, the analysis of local observables can yield different constraints on cosmological parameters from those obtained with FLRW models. Examples presented in this section form the basis for analysing observable data, but do not immediately represent the effect on cosmological inferences. We use a specific analytic form of the curvature, parametrised with k bg , k loc , r 0 , ∆r and given as García-Bellido-Haugbølle (GBH) profile (Garcia-Bellido & Haugbølle 2008;February et al. 2010):\\nOur focus is the case of ΛCDM background, i.e. k bg ≡ 0, which we call ΛLTB model. The analytic form of k(r) together with the value of dimensionless cosmological constantΛ define the evolution of expansion and all physical quantities in LTB metric. In the limit of ∆r → 0, a simpler top-hat profile (TH) is recovered,\\nWhile GBH form describes a smooth matter density profile, TH profile has one parameter less and is trivial to use for data fitting as two joint homogeneous models. Care must be given to relation between the physical quantities in two homogeneous regions.\"\n },\n {\n \"title\": \"Reconstruction of geodesic\",\n \"content\": \"The LTB metric describes an inhomogeneous dust cloud with spherical symmetry, indicated by the angular term in eq. (1). Hence, the observer located at the centre of this symmetry has an isotropic view. The assumption of isotropy gives us only the first approximation for treating the large cosmological inhomogeneities. To quantify the overall effect of large local void on cosmological inferences, we limit ourselves to the on-centre model that considers an observer located at the centre of symmetry in the local matter inhomogeneity. The assumption of this very special position can be avoided, although the geodesic equations for an off-centre observer become increasingly complex, affecting the feasibility of data analysis. We find the on-centre model as adequate for presenting all the relevant points in this work, but we extend the discussion about observer's spatial position later on. Geodesic equations can be derived following Célérier (2000):\\nwhere z is the cosmological redshift. This system of differential equations must be solved numerically along the geodesic with help of eqs. (4), (8), (15) and (18), in the entire redshift range. In order to derive distance-redshift relations one needs to numerically reconstruct the function R(z), or equivalently a(z), which can be obtained by numerical inversion of eq. (18) at every z along the geodesic (Valkenburg 2012). However, we find it easier and computationally faster to use the geodesic equations and all physical variables of interest rewritten with (a, r) as independent fundamental coordinates, after the change of variables from (t, r) da dz = a t dt dz + a r dr dz .\\nAdding the assumption of homogeneity, geodesic equations reduce to the standard form; using eqs. (8) and (23) we have: We note that this relation is valid on all shells where spatial derivative is null, i.e. in every homogeneous subregion along the matter density profile.\"\n },\n {\n \"title\": \"Density contrasts\",\n \"content\": \"It is useful to define the relative contrasts for quantities of interest, such as physical matter density, dimensionless density parameter Ω m , and expansion rates. The contrast can be defined at any time and any spatial point inside inhomogeneous region. Consequently, for physical matter density we have\\nThe evolution of matter density contrast is shown in Fig. 1 for a ΛLTB void model with specific parameters determining cosmological constantΛ, central curvature k loc , size and steep transition zone, given with r 0 and ∆r. As shown in this case, the physical matter density ρ m will have compensated radial profile in the case of small ∆r and/or large central contrast. Since we use the comoving radial distance, the physical expansion of void size can not seen on Fig. 1. Most often we consider the present (t = t 0 ) contrast between observer's position (r = 0) and background (r = + inf) and we denote it with subscript 0. The contrast of dimesionless matter density parameter Ω m (r) and both expansion rates are defined in the same manner as for physical matter density in eq. (26). We will focus on present central contrasts which can be derived using eqs. (3), (7), (8), (10) and (16):\\nNotably, these two contrast are different due to the induced spatial inhomogeneity of the expansion rates, In conclusion, relation |δρ 0 /ρ 0 | < |δΩ 0 /Ω 0 | holds both in the case of local underdensity as well as overdensity. Writing the equivalent of eq.\\n(3) at a(t = t 0 , r = 0) = a loc 0 and after some simple calculations, one can derive analytic expressions that relate these three contrasts.\\nWe would like to note that the current ΛLTB modelling can be equivalently parametrised with Ω bg m , δρ 0 /ρ 0 , z size and ∆z, which are derived fromΛ, k loc , r 0 and ∆r. Approximate series expansion formulae, valid for ΛCDM background, are provided in appendix A.\"\n },\n {\n \"title\": \"Scale factor\",\n \"content\": \"The radial dependence of scale factor in LTB models can be observed only along the geodesic as a(z) = a(t(z), r(z)) (see Fig. 2). Moreover, modelling a(z) can be used a posteriori to inversely reconstruct the matter density profile (Wojtak & Prada 2017).\\nInside homogeneous region close to the centre of GBH profile and at the background, scale factor satisfies eq. (25) and, hence:\\nIn fact, redshift size of an inhomogeneous matter profile can be defined through the equality 2 (1 + z size ) a(z size ) = a loc 0 + 1 .\"\n },\n {\n \"title\": \"Local expansion rate\",\n \"content\": \"As explained in previous section, LTB metric is characterised with two expansion rates, shown in Fig. 2. The transverse expansion rate given by Friedmann-Lemaître eq. (9) can be used together with density parameters Ω i to construct the equivalent FLRW model of each shell. On the contrary, radial expansion rate is related to the spatial derivative along the line of sight (cf. eq. (5)). Certainly, all physical quantities defined in ΛLTB model converge to their ΛCDM form at high redshift limit as a consequence of the Birkhoff theorem for the local spherical inhomogeneity (see Fig. 2). The intercept of SN Ia m(z) relation is determined by the absolute Figure 3. 1σ confidence stripes of SN Ia magnitude parameter M obtained with standard (grey) and binned: z ≤ 0.08 (green), 0.08 < z ≤ 0.15 (violet) and z > 0.15 (orange) fits to Pan+P data.\\nmagnitude of SN and the local expansion rate. Calibrated SN Ia are able to measure the H 0 only after fixing the slope of distanceredshift relation as presented in Riess et al. (2016). However, if the inhomogeneity extends up to z ∼ 0.1, not accounting for its effect properly can represent an important systematic in H 0 measurement.\"\n },\n {\n \"title\": \"RESULTS AND DISCUSSION\",\n \"content\": \"In this section we present our main findings in the analysis of LD data from KBC13 and the SN Ia data from JLA and Pantheon 1 samples (Betoule et al. 2014;Scolnic et al. 2018). The likelihoods for SN Ia datasets are implemented as in their respective releases. Results are presented for the derived parameters: relative matter density at the background Ω bg m , central contrast δΩ 0 /Ω 0 , inhomogeneity size z size and density profile shape ∆z. We use a prior of Ω bg m = 0.3153 ± 0.0073 taken from CMB TT,TE,EE+lowE+lensing data analysis for the ΛCDM background (Planck Collaboration et al. 2018b), which appropriately aids low-redshift SNe to constrain the local matter density profile (hereafter indicated as JLA+P and Pan+P). Other parameters are sampled from wide flat prior ranges: −0.4 ≤ δΩ 0 /Ω 0 ≤ 0.2, z size ≤ 0.13 (≈ 0.5Gpc in comoving distance) and ∆z ≤ 0.6 z size , for which we verify a posteriori that relaxing the priors does not affect our results.\\nAs elaborated in section 3, the large local void is expected to affect the magnitude-redshift relation of SN Ia inside the void, due to the higher transverse expansion rate and lower values of relative density parameters Ω m (z) and Ω Λ (z) w.r.t. ΛCDM constraints. We start our analysis by testing for redshift dependence of SN Ia magnitude parameter M = M B + 5Log 10 (c/H 0 ).\\nTo this end, we use Pantheon SN Ia whose apparent magnitudes have already been corrected with a cosmology-independent method which marginalises the effect of all magnitude correction parameters and leaves only M as a free parameter, besides the cosmological model (see Scolnic et al. (2018)). Full Pantheon dataset, along with the CMB prior, is fitted to standard ΛCDM model, allowing for three binned values of M in redshift ranges z ≤ 0.08 The grey area is a 1σ reconstruction zone obtained using Gaussian process. The blue curve is our best-fit GBH profile to 10 LD points, which shows δρ 0 /ρ 0 = −43% contrast. The green curve has a weaker contrast δρ 0 /ρ 0 = −30% suggested by KBC13.\\n(green), 0.08 < z ≤ 0.15 (purple) and z > 0.15 (orange), with 194, 103 and 751 SNe, respectively. The resulting 1σ confidence interval for M bg = −1.184 ± 0.007 is in excellent agreement with M tot = −1.185 ± 0.005 from the standard non binned fit shown as grey stripe in Fig. 3. The visible inconsistency in the first two bins, M 1 = −1.210 ± 0.015, M 2 = −1.153 ± 0.012 might be hinting to possible local geometry effects. Fitting the Pantheon SN Ia in the redshift range 0.0233 < z < 0.15 to the FLRW series expansion (SE) formula for luminosity distance with q 0 = −0.55, as presented in R16, we obtain M(SE) = −1.182 ± 0.010, which is also different from values obtained in the first two bins. Various works in the literature, including R16 and their previous articles, introduce a parameter a B that can be derived from the intercept magnitude M as\\nwhere c is the speed of light in km/s. Results from the simple binned analysis already provide a strong motivation to study a more physical, inhomogeneous cosmological model, characterised by varying H 0 (r) and matter density profile Ω m (r). Although an extended analysis of the M evolution at higher redshifts could have important cosmological implications (see e.g. Tutusaus et al. (2019) for effect on cosmic acceleration), here we looked for possible local variation in M as a hint of local matter inhomogeneity, while our background cosmology is assumed to be flat ΛCDM.\"\n },\n {\n \"title\": \"ΛLTB analysis with LD data:\",\n \"content\": \"KBC13 summarises 22 LD data points obtained from several surveys, of which 10 are their original data points. Due to the unknown correlations that could arise from the overlap between different redshift surveys we opt to utilise only these 10 data points to constrain local matter density profile. They were obtained using data from 35,000 galaxies divided in 10 redshift bins in the range 0.005 < z < 0.20. Using LD points, KBC13 report the existence of a large local void with a luminosity density contrast of about −30% in the inner region, which is expected to be proportional to Table 1. Primary results from the ΛLTB analysis performed to: luminosity density using all 10 original data points presented in KBC13 (LD) and after removal of z = 0.034 data point from 2MASS galaxy catalogue (LD * ); 740 SN Ia from JLA dataset alone (JLA) and with Planck constraint on Ω bg m = 0.3153 ± 0.0073 (JLA+P); 1048 SN Ia from Pantheon dataset (Pan) and with Planck constraint (Pan+P); 237 low-redshift Pantheon SN Ia from the range 0.0233 < z < 0.15 with the Ω bg m = 0.3 constraint (Pan * ). The best-fit values for parameters of interest (noted as b. f. ) are provided together with confidence interval between 16th and 84th percentiles (noted as c.i. ) of their respective 1D marginalised likelihoods. Three contrast presented here are defined by Eqs. (27) to (29). The comparison to ΛCDM model is given using the relative ∆AIC values. the contrast of total matter density δρ 0 /ρ 0 (see Fig. 4). We initially perform a simple reconstruction of the density profile using the Gaussian process -a model-independent technique for data analysis (see e.g. is obtained by imposing homogeneity both inside and outside the expected void size. As such, it is allowed to predict no evidence of a void, but our resulting posterior of the reconstructed region shows definite indications of the local underdensity with δρ 0 /ρ 0 ∼ −40%.\\nWe proceed by fitting LD data points to the full ΛLTB model and confirm that the conservative density contrast of δρ 0 /ρ 0 ∼ −30%, suggested by KBC13 and later utilised in HB17 and KSR19, appears to be an underestimate w.r.t. our best-fit shown in Table 1.\\nDue to the small amount of data, we test our findings by performing a leave one out (LOO) analysis, implemented as random removal of each of the 10 points from the fit. It clearly shows the stability of the estimated redshift size, while the value of the profile contrast δρ 0 /ρ 0 ∼ −44.0±6.0 is found to be dominated by one stringent measurement at z = 0.034. Eliminating this data point coming from 2MASS survey, the remaining 9 points from UKIDSS and GAMA surveys estimate the contrast to be δρ 0 /ρ 0 ∼ −32.0 ± 9.5, in consistence with the conservative value suggested by KBC13. A direct comparison of our best-fit profile (blue) and the profile with δρ 0 /ρ 0 = −30% (green) in Fig. 4 clearly shows the importance of 2MASS data point at z = 0.034. Nevertheless, the induced contrast of fractional matter density δΩ 0 /Ω 0 coming from the fit to 9 points (quoted as LD * in Table 1) is still higher than the one used in previous works where the difference between δΩ/Ω and δρ/ρ contrasts is not considered (c.f. KSR19, HB17, Shanks et al. (2019)).\\nThe primary inferences of this analysis are verified to remain unaltered changing the assumed analytic form of matter density profile. Taking the result in Table 1 at face value, would indicate an extreme tension with a homogeneous matter density scenario. While the standard model of cosmology expects variation of the local matter density and velocity fields, a more pressing problem arising from this result is the size of the estimated inhomogeneity. Based on ΛCDM matter power spectrum constrained by CMB, density perturbations of size z size ≈ 0.08 are expected to have the variance of δρ 0 /ρ 0 ≈ 1.5−2% (corresponds to δΩ 0 /Ω 0 ≈ 2−3%) at 68% confidence level (Marra et al. 2013;Camarena & Marra 2018). Hence, this expectation is at ∼ 8σ tension with the result of the fit to complete LD data, while the exclusion of 2MASS data point at z = 0.034 relaxes deviation to a however large ∼ 4σ tension.\\nThe best-fit ΛLTB (blue) curve in Fig. 4 is closely following the predicted model-independent reconstruction (grey), except for the short overdense region, which is an expected compensation of a large and steep underdense matter profile. In fact, the data point at z = 0.081 might be a signature of the same. Although not statistically significant to draw conclusions for an overdensity, it aids to constrain the void size. The specific interplay between the model description and the fact that only 10 data points are available for constraining 4 profile parameters, results in over-fitting difficulties. For this reason we perform a Gaussian approximation of the posterior likelihood distribution before estimating the confidence regions presented in Fig. 6, which does not alter our primary inferences.\\nGiven the lack of consistency in reported results based on LD data, limited sky coverage of the data, reports about angular variation (WS14), and possible effects of binning we also deem it important to estimate the level of (dis)agreement with the SN Ia data.\"\n },\n {\n \"title\": \"ΛLTB analysis with SN Ia datasets:\",\n \"content\": \"Probing for signatures of large local inhomogeneity (underdense or overdense) in the SN Ia datasets, we fit the local matter density profile simultaneously with the background cosmology (Ω bg m ) which is, nevertheless, dominantly guided by the Planck CMB constraint. Since the variation of H 0 (r) is modelled, in ΛLTB we define the SN Ia intercept magnitude parameter using background expansion rate\\nIn the case of JLA dataset, fitting χ 2 also depends on SN Ia magnitude correction parameters for lightcurve stretch, colour and host galaxy mass, which are later marginalised as nuisance parameters (Betoule et al. 2014).\\nThe SN Ia constraints on local matter density profile presented in the form of confidence regions in Fiqs. 5 and 6 and Table 1 show high consistency with homogeneous ΛCDM fit, characterised by δΩ 0 /Ω 0 = 0. The analysis of JLA dataset yields two distinguished local χ 2 minima, emphasised with dashed lines in Fig. 5. The underdensity reported by WS14, with the size of ≈ 215h 70 −1 Mpc and the average contrast δρ 0 /ρ 0 = −15 ± 3% (corresponds to δΩ 0 /Ω 0 = −19 ± 4%) is in between the two minima we find for JLA dataset. Interestingly, the second local minimum specifies a void with the physical size in line with the findings of KBC13, d A (c) = 300h 70 −1 Mpc, but with a much weaker contrast. Indeed, the large local void reported by KBC13 is rejected at 4σ confidence by JLA dataset. However, our analysis of reduced LD * data points gives less stringent constraint on the void contrast which is no longer in tension with our JLA result (cf. Table 1).\\nEven though the analysis of Pantheon dataset does not recover the primary minimum obtained for JLA supernovae, the resulting confidence regions are of high similarity. Our Pantheon and Pan+P constraints for ΛLTB model shown in Fig. 6 include inside 1σ confidence region voids with contrast δΩ 0 /Ω 0 ≈ −20%, such as the representative void of WS14. The constraints from LD data of KBC13 (blue shaded ellipses in Fig. 6) are pointing to much deeper void, although the redshift size of the two best-fits are aligned. In fact, the estimated contrasts from Pan+P and LD data are at ∼ 4σ tension (see Table 1). The nature of inferences resulting from LD data, their disagreement with the expected cosmic variance and the SN Ia constraints, further strengthens our motive to perform LOO analysis to test this data sample. LOO analysis led us to consider the fit to the reduced LD * dataset, which is shown with blue dashed contours in Fig. 6. As mentioned earlier, the LD * constraint on void size remains unaltered, while the estimated void contrast shifts towards a shallow inhomogeneity that has no tension with the Pantheon results.\\nOur ΛLTB inferences coming from Pantheon dataset are contrasting the recent result of KSR19 which reports a very high significance of ∼ 5σ against any large local void with contrast |δρ 0 /ρ 0 | > 20%. The higher amount of SN Ia data points that they used, 1295 compared to 1048 available in Pantheon, are additionally populating the low-redshift range and are expected to improve the error bars of profile parameters by 50%. However, we sus- Figure 6. 1σ and 2σ confidence contours of local matter density profile parameters are shown for Pan+P (green regions), Pantheon (light green solid curves), full LD data from KBC13 (blue regions) and reduced LD * data (dashed blue curves). The isotropic voids proposed by WS14 and KBC13 are marked with orange crosses. Contrasts presented on vertical axes are defined by Eqs. (28) and (29). pect that theoretical modelling is also contributing to the difference w.r.t. our result. While the newer data is not publicly available at this moment, we intend to extend the current analysis in a future communication. In appendix A we present the effects of simplifying the density profile from GBH to TH solution and fixing the model parameters such as Ω bg m and z size , instead of marginalising over them to obtain the constraints on void contrast.\\nWe chose to report the results of SN Ia analysis with and without the Planck CMB constraint on background matter density. Both SN Ia datasets provide excellent fits in the case of a local underdensity, but do not exclude the possibility for a z size > 0.1 overdensity (see Fiqs. 5 and 6). The Planck prior on background cosmology reduces the degeneracy between Ω bg m and δΩ 0 /Ω 0 parameters in SN Ia likelihood, providing tighter constraints. Nevertheless, the main inferences presented here are not strongly affected by this prior due to the excellent agreement between the Planck value and our SN Ia best-fits for Ω bg m . Examples presented in appendix A also explore the case of using a fixed value for the relative matter density at the background, as contrary to sampling this likelihood parameter. Compared to homogeneous model, the additional three parameters that characterise the void profile afford better fits to the SN Ia data, namely χ 2 JLA = 678.30 and χ 2 Pan = 1020.72 for ΛLTB model in contrast to χ 2 JLA = 682.94 and χ 2 Pan = 1026.05 for ΛCDM model. We quantify this comparison with Akaike information criteria Akaike (1974) that penalises the usage of three extra parameters as ∆AIC = ∆χ 2 + 2 * 3. All fits reported in Table 1 have ∆AIC < 2 implying no preference between the two models.\"\n },\n {\n \"title\": \"Anisotropy considerations\",\n \"content\": \"Using galaxy redshift distribution and number counts up to the depth of z < 0.1, WS14 observed 215h 70 −1 Mpc local underden-sity and an angular variation of its estimated density contrast in the range −4% to −40%, with the mean of −15% (corresponds to δΩ 0 /Ω 0 = −19% shown in Fiqs. 5 and 6). Likewise, the Fig. 11a in KBC13 clearly shows discrepancy between LD points observed in different regions on the sky, again pointing towards a possibly anisotropic local density profile. Hence, an important extension to the analysis presented here can be done by taking into account the anisotropy of the data, as well as the matter density profile and the observer's position inside of the considered inhomogeneity (see e.g. Blomqvist & Mörtsell (2010)). If there was any anisotropy present in the matter distribution, it would be averaged out in the analysis that uses isotropic model like ours, leaving us with the expectation to find angular variation of density contrast around this average when modelling for anisotropic matter profiles. Be that as it may, the statistical significance in favour of the observed anisotropy in LD data is limited by the sky coverage of the used surveys, which amounts to ∼ 15% for KBC13 and ∼ 22% for WS14, spread over three distant patches on the celestial plane. Several groups performed tests on SN Ia data looking for features of local and/or global anisotropy Bengaly et al. 2015;Kalus et al. 2013;Wang & Wang 2014;Sun & Wang 2018;Andrade et al. 2018), also in relation to the local matter distribution (Colin et al. 2011). These anisotropy estimates are also impaired by the inhomogeneous angular distribution of the observed SN on the sky, which is particularly visible on medium and high redshifts due to the narrow fields of the surveys, such as SDSS. Probing low-redshift SN Ia for signatures of anisotropy, we estimate the statistical variance of model parameters over angular directions. Concretely, the two parameters of interest, M and δΩ 0 /Ω 0 in ΛLTB model, are tested separately for dipole anisotropy (see also e.g. Mariano & Perivolaropoulos (2012)). The magnitude dipole is modelled as a vector of intensity M d and direction n d . Then, the magnitude parameter M (i) of each SN Ia inside the void becomes\\nwhereM is the average intercept magnitude, while the normalised vector n (i) represents the angular position of each SN Ia. Although we fitted the full Pantheon dataset, the dipole is constrained mainly by 190 SN Ia under z ≤ 0.08, which are also more homogeneously distributed on the celestial plane compared to the full dataset. Here we used a TH matter density profile for simplicity (see section 2). Finally our estimate for the magnitude dipole\\nwould produce the angular variation\\nThis angular variance of magnitude inside the void can be related to the anisotropy of local expansion rate as\\nOur result for the level of local anisotropy in Pantheon SN Ia is consistent with previous works that find null evidence against assumption of the isotropy (Andrade et al. 2018;Zhao et al. 2019b), although less stringent compared to global anisotropy constraint found by Soltis et al. (2019). Analogously to eq. (36) we statistically probed the dipole anisotropy of the estimated void contrast and found that it could account for the angular variance ∆ δΩ 0 Ω 0 r.m.s. = 3.7 ± 4.1% .\\nWhile this result at face value shows that Pantheon dataset does not exclude the angular variation of estimated void contrast ∆(δΩ 0 /Ω 0 ) ∼ 10% at 2σ confidence level, we still find it difficult to coincide our constraints on matter density profile from Pantheon SN Ia and LD data, or explain the level of anisotropy observed by WS14. We would like to note that two former examples should be considered only as rough estimates of anisotropy level since: the dipole model does not represent a complete relativistic solution of FE, the variation of profile size is not considered, and the Pantheon SN Ia have had the apparent magnitudes corrected using the BBC method which relies on the assumption of isotropy (Scolnic et al. 2018). Although crucial in future studies, the present data is not quantitatively sufficing to benefit from the full anisotropic analysis given as an extension of the cosmological metric and theoretical model for the anisotropic density profile.\"\n },\n {\n \"title\": \"Distance ladder measurements\",\n \"content\": \"One of the most important implications of a large local void is the effect on direct measurement of present cosmic expansion rate by the means of low-redshift SN Ia. Consider a spherical matter inhomogeneity, characterised with a density contrast of e.g. δΩ 0 /Ω 0 = −25% which is inside the 1σ confidence region of Pan+P constraints, presented in Table 1 and Fig. 6. This matter contrast will also induce a higher local present expansion rate by δH 0 /H 0 ≈ 3.6% w.r.t. the background expansion rate (see Eqs. (28) and (29). Hence, the luminosity distance of a SN located on any shell inside the local void is higher than what one would estimate using the background metric. This can affect the analysis of SN Ia data in the redshift range inside the local void as well as the resulting model inferences.\\nIn light of everything said above, let's assume our local matter distribution has indeed an isotropic underdense profile. Analysing SN Ia data from the redshift range inside this local void with a simpler homogeneous model would provide a different constraint on H bg 0 than the LTB model. In fact, the assumption of locally homogeneous cosmic expansion (H loc 0 = H bg 0 ) can overestimate the value of background expansion rate, while underestimating H loc 0 , by ∼ 1 2 δH 0 /H 0 and affect the inferences on other background cosmological parameters.\\nAs discussed in Section 4.1 theoretical expectations for the large local void based on ΛCDM model, and its effect on H 0 measurement are small, which is reflected in results presented by recent studies. On one side, the level of expected cosmic variance in local matter density for the Planck ΛCDM model is estimated to impact the systematic error of H bg 0 at 2% when using SN Ia from z ≥ 0.01 and at 1% when using SN Ia from z ≥ 0.0233 (Marra et al. 2013). On the other side, the large-volume cosmological Nbody simulations based on the Planck ΛCDM model, such as e.g. in Skillman et al. (2014), were used to quantify effects of both the cosmic variance and the SN Ia sample selection on H bg 0 value -Wu & Huterer (2017); Odderskov et al. (2016) estimate the expected variance level of measured H bg 0 to be ∼ 0.5% for 0.0233 < z < 0.15 cut (c.f. an earlier work by Wojtak et al. (2014)). In order to minimise the effect of local matter distribution, R16 (including their previous works) use only the SN Ia above z > 0.0233 for the modelindependent estimate of H bg 0 (Jha et al. 2007). In consistence with\\nWu & Huterer (2017), KSR19 report no evidence for strong variation of matter density profile in the range z < 0.15 from SN Ia data, securing the ability to measure H bg 0 value to a percent value. The measurement of background expansion rate based on lowredshift SN Ia is obtained from the intercept magnitude parameter M (see Section 4.2). While the exact value of H bg 0 relies on local SN Ia luminosity calibration, we are mainly interested at the relative change of this estimate due to the effect of a large local void, which can be seen as the shift in estimated value of M. For this purpose, and only in this subsection, we analyse 237 Pantheon SN Ia from 0.0233 < z < 0.15 redshift cut. Fitting them to FLRW series expansion (SE) formula for luminosity distance and using a fixed value of q 0 = −0.55, as presented in R16, we obtain M(SE) = −1.1825 ± 0.0094. In low redshift range, this fit is equivalent to ΛCDM fit with Ω m = 0.3, which we will use for comparison to underdense scenario. The same Pantheon cut is fitted to ΛLTB model with background cosmology prior Ω bg m = 0.3 (q bg 0 = −0.55), density contrast prior range −50% ≤ δΩ 0 /Ω 0 ≤ 5%, void size 0.025 ≤ z size ≤ 0.12, and shape parameter 0 ≤ ∆z ≤ z size . In order to ensure enough SN Ia that constrain the background, we consider only profiles that reach the background regime up to z ≤ 0.13 by additionally removing the region of parameter space with such combination of large z size and ∆z values. After marginalising the matter density profile out of the low-redshift Pantheon likelihood, we estimate the intercept SN magnitude parameter 2 to be M(ΛLTB) = −1.1577 +0.0158 −0.0148 . The remaining parameter constraints are presented in the last row of Table 1. Regardless the three additional parameters that model the local matter underdesnity, the ΛLTB fit to low-redshift Pantheon SN Ia reveals an improvement of the best-fit by ∆AIC=-3.3.\\nThe local matter underdensity affects the estimated value of M due to its degenearcy with density constrast. This can be seen on Fig. 7, where we show the low-redshift Pantheon constraints in δΩ 0 /Ω 0 −M plane. The best-fit of ΛCDM model (equivalent to SE) can be found on vertical axis for zero void contrast and is marked as a red point in Fig. 7, which is 1.7σ away from our constraints on ΛLTB model. Both values for the intercept magnitude parameter M have been estimated fitting the same 237 Pantheon SN Ia to two different models and are, hence, fully correlated. Taking into account this correlation, the difference between two values is evalueted to be ∆M = 0.0247 +0.0065 −0.0055 . As the ΛLTB model has more parameters in fitting, the error bar of resulting M(ΛLTB) is bigger than in the case of M(ΛCDM), and elseways, we note that if they were equivalent then the uncertainty on estimated shift ∆M would have been zero. Since this change of intercept magnitude value is coming only from the theoretical modelling and the two models have comperable AIC values (see Table 1), it should be seen as a possible source of systematic error on M estimate. The positive shift of ∆M = 0.0247 corresponds to 1.14% lower H bg 0 obtained with ΛLTB model than with SE formula. Compared to the total error of M(SE), this value of ∆M is significant. However, the major contribution to the uncertainty of H 0 measurement is presently constricted to calibration of standard SN Ia absolute magnitude, M B , that together with much smaller uncertainty on intercept magnitude M(SE) reaches the total error bar of 2.4% reported by R16. Adding the effect of large local matter inhomogeneity (z size > 0.025) on estimate of SN Ia intercept magnitude could increase this uncertainty level to 2.7% on the lower end of H 0 measurement. The improvement in Cepheid and SN Ia calibation techniques (see R19), followed by reduction 2 Confidence intervals are given between 16th and 84th percentiles of uncertainty on M B , can lead to a situation in which the effect of local matter inhomogeneity becomes more relevant for the measurement of background cosmic expansion rate.\"\n },\n {\n \"title\": \"CONCLUSIONS\",\n \"content\": \"In the present work we have investigated the evidence for a large local void using a well formulated ΛLTB model against the most recent publicly available SN Ia compilations. In order to reduce the correlation with background cosmology, we have used the CMB constraint on Ω bg m (Planck Collaboration et al. 2018b), which aids to provide a slightly tighter constraints on the local void profile. The analysis on SN Ia datasets is also complimented with the luminosity density data obtained from galaxy surveys by Keenan et al. (2013), which together with Whitbourn & Shanks (2014) played a key role in propelling the recent discussion about possible existence of a large local void and its effect towards resolving one of the most prominent discordances in modern cosmology -the H 0 tension. We summarise our primary results in the following. All reported confidence intervals (c.i.) are between 16th and 84th percentiles.\\n• Fitting ΛLTB model to LD data from KBC13, we find a void with density contrast of (δρ 0 /ρ 0 ) c.i. = −43.8 ± 6.0%, deeper than originally proposed δρ 0 /ρ 0 = −30%.\\n• From SN Ia we do not find a strong evidence for a void or otherwise -a fit to Pan+P yields (δρ 0 /ρ 0 ) c.i. = −7.9 +7.0 −5.8 % which corresponds to (δΩ 0 /Ω 0 ) c.i. = −10.5 +9.3 −7.4 %, and a wide range for the redshift size z c.i. size = 0.068 +0.021 −0.030 . However, Pantheon constraints do not exclude a void of e.g. z size ≈ 0.075 and δΩ 0 /Ω 0 ≈ −25% at 1σ confidence level (c.f. Fig. 6).\\n• JLA likelihood is in a good overall agreement with Pantheon, except in the range z ≤ 0.04 (c.f. Fiqs. 5 and 6). Both SN Ia datasets are at 3σ tension with our result obtained from KBC13 data, but in excellent agreement with parameters of the isotropic void proposed by WS14.\\n• Leave one out analysis of KBC13 data reveals that the corresponding estimated contrast is dominantly constrained by one stringent point. Removing this only data point from 2MASS survey and using the remaining 9 points from UKIDSS and GAMA surveys, relaxes their tension with SN Ia data.\\n• The model comparison with Akaike information criteria shows no preference between the tested models (c.f. Table 1).\\n• Probing the level of statistical variance in ΛLTB model parameters over angular direction, we find null evidence for the dipole anisotropy. Its contribution to the angular variance is small: ∆M r.m.s. = 0.008 ± 0.018 for the SN intercept magnitude parameter or ∆ δΩ 0 Ω 0 r.m.s. = 3.7 ± 4.1% for the void contrast.\\n• In our analysis of SN Ia data we do not find any evidence for a large isotropic void that could resolve the 9% discrepancy between two H 0 estimates in tension, leaving this local effect alone to be a highly unlikely explanation.\\n• We fitted low-redshift range of Pantheon SN Ia data from 0.0233 < z < 0.15 to the FLRW series expansion formula for luminosity distance and to the ΛLTB model, comparing their estimates of SN Ia intercept magnitude, which is used for H 0 measurement. Adding the possibility of a large local void, we observe a small improvement in the fit by ∆AIC = −3.3 and a shift of the inferred intercept magnitude parameter ∆M = 0.0247 +0.065 −0.055 that could be a source of additional systematic error in H bg 0 of 1.14%. While the simpler model, which is based on series expansion formula of luminosity distance in FLRW metric, is only 1.7σ away from our ΛLTB constraints (see Fig. 7), this result is questioning whether the total error budget of distance ladder measurements based on low-redshift SN Ia should be reconsidered.\\nIncrease of presently available data, followed by more complex theoretical modelling, is necessary to better understand the observed disagreement between LD and SN Ia samples. As already mentioned, considering off-centre position of the observer would extend the present isotropic ΛLTB formalism, also allowing for an anisotropic point of view by construction. With increasing number of observations in future surveys, SN Ia may prove as effective tracers of local matter density distribution. Figure A1. 1σ and 2σ confidence regions for ΛLTB model with TH density profile, fitted to Pantheon dataset are shown as coloured contours, while the solid lines represent the confidence regions obtained from the smoothed χ 2 function. In the first panel we used Planck prior for Ω bg m , whereas the second and third panels are obtained with fixed Ω bg m = 0.3. Moreover, in the third case z size parameter was a priori fixed to a range of z split values, followed by χ 2 sampling over δΩ 0 /Ω 0 contrast values.\"\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 7 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"; Tokutake et al. (2018); Kenworthy et al. (2019); Boehringer et al. (","content":"; Tokutake et al. (2018); Kenworthy et al. (2019); Boehringer et al. ("},{"id":"","caption":"The evolution of matter density contrast δρ/ρ is shown for ΛLTB model with present local contrast δρ 0 = −30% and redshift size z size = 0.07. The profile is shown in comoving angular diameter distance, at time lapses t = t/t 0 running from 0.01 up to 1.00 (present time).","content":"Figure 1 .\n1The evolution of matter density contrast δρ/ρ is shown for ΛLTB model with present local contrast δρ 0 = −30% and redshift size z size = 0.07. The profile is shown in comoving angular diameter distance, at time lapses t = t/t 0 running from 0.01 up to 1.00 (present time)."},{"id":"","caption":"Redshift dependence of the cosmic expansion rates (red for radial and orange for transverse) and the scale factor (blue) are shown for ΛLTB model with contrast δρ 0 = −30% and redshift size z size = 0.07 (solid lines) in comparison with the ΛCDM model (dashed lines).","content":"Figure 2 .\n2Redshift dependence of the cosmic expansion rates (red for radial and orange for transverse) and the scale factor (blue) are shown for ΛLTB model with contrast δρ 0 = −30% and redshift size z size = 0.07 (solid lines) in comparison with the ΛCDM model (dashed lines)."},{"id":"","caption":"Luminosity density data from KBC13: solid orange data points are their original data, while dashed points are from additional surveys.","content":"Figure 4 .\n4Luminosity density data from KBC13: solid orange data points are their original data, while dashed points are from additional surveys."},{"id":"","caption":"Holsclaw et al. (2010); Seikel et al. (2012); Haridasu et al. (2018b)). The reconstructed 1σ region shown in grey on Fig. 4","content":"Holsclaw et al. (2010); Seikel et al. (2012); Haridasu et al. (2018b)). The reconstructed 1σ region shown in grey on Fig. 4"},{"id":"","caption":"Constraints on the local matter density profile from JLA (solid curves) and JLA+P (coloured regions) are shown at 1σ and 2σ confidence level. The positions of the only two local χ 2 minima are emphasised with the dashed lines. The isotropic voids proposed by WS14 and KBC13 are marked with orange crosses.","content":"Figure 5 .\n5Constraints on the local matter density profile from JLA (solid curves) and JLA+P (coloured regions) are shown at 1σ and 2σ confidence level. The positions of the only two local χ 2 minima are emphasised with the dashed lines. The isotropic voids proposed by WS14 and KBC13 are marked with orange crosses."},{"id":"","caption":".m.s. ≈ 0.2 ln(10) ∆M r.m.s. = 0.4 ± 0.8%","content":".m.s. ≈ 0.2 ln(10) ∆M r.m.s. = 0.4 ± 0.8%"},{"id":"","caption":"1σ and 2σ confidence regions for ΛLTB model with GBH density profile, fitted to 237 low-redshift (0.0233 < z < 0.15) SN Ia from Pantheon dataset. The best fit (b.f.) for ΛCDM model, shown as red point is almost 2σ away from ΛLTB constraints.","content":"Figure 7 .\n71σ and 2σ confidence regions for ΛLTB model with GBH density profile, fitted to 237 low-redshift (0.0233 < z < 0.15) SN Ia from Pantheon dataset. The best fit (b.f.) for ΛCDM model, shown as red point is almost 2σ away from ΛLTB constraints."},{"id":"","caption":".","content":".[%] \nχ 2 \n∆AIC ΛCDM \n\nLD \n0.082 0.079 +0.012 \n\n−0.012 \n\n−51.1 \n−51.9 +6.3 \n\n−6.3 \n\n−42.9 \n−43.8 +6.0 \n\n−6.1 \n\n8.1 \n8.2 +1.3 \n\n−1.2 \n\n2.62 \n\nLD * \n0.082 0.075 +0.015 \n\n−0.015 \n\n−39.7 \n−39.4 +10.3 \n\n−10.3 \n\n−32.4 \n−32.1 +9.0 \n\n−9.5 \n\n5.9 \n5.9 +1.9 \n\n−1.7 \n\n0.29 \n\nJLA \n0.025 0.039 +0.062 \n\n−0.018 \n\n−19.5 \n−9.9 +17.3 \n\n−13.9 \n\n−15.1 \n−7.5 +12.9 \n\n−11.0 \n\n2.7 \n1.3 +2.0 \n\n−2.3 \n\n678.30 \n1.35 \n\nJLA+P 0.025 0.032 +0.056 \n\n−0.011 \n\n−19.6 \n−12.5 +13.8 \n\n−12.4 \n\n−15.2 \n−9.5 +10.4 \n\n−9.9 \n\n2.7 \n1.7 +1.9 \n\n−1.9 \n\n678.34 \n1.07 \n\nPan \n0.075 0.070 +0.023 \n\n−0.031 \n\n−16.2 \n−9.8 +14.0 \n\n−8.9 \n\n−12.4 \n−7.4 +10.5 \n\n−7.0 \n\n2.2 \n1.3 +1.3 \n\n−1.9 \n\n1020.72 \n0.67 \n\nPan+P \n0.075 0.068 +0.021 \n\n−0.030 \n\n−14.4 \n−10.5 +9.3 \n\n−7.4 \n\n−11.0 \n−7.9 +7.0 \n\n−5.8 \n\n2.0 \n1.4 +1.1 \n\n−1.3 \n\n1021.01 \n0.40 \n\nPan * \n0.075 0.075 +0.013 \n\n−0.022 \n\n−23.1 \n−21.5 +8.9 \n\n−9.5 \n\n−18.1 \n−16.8 +7.1 \n\n−7.9 \n\n3.2 \n3.0 +1.5 \n\n−1.3 \n\n225.63 \n−3.30"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"; Tokutake et al. (2018); Kenworthy et al. (2019); Boehringer et al. (\",\n \"content\": \"; Tokutake et al. (2018); Kenworthy et al. (2019); Boehringer et al. (\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The evolution of matter density contrast δρ/ρ is shown for ΛLTB model with present local contrast δρ 0 = −30% and redshift size z size = 0.07. The profile is shown in comoving angular diameter distance, at time lapses t = t/t 0 running from 0.01 up to 1.00 (present time).\",\n \"content\": \"Figure 1 .\\n1The evolution of matter density contrast δρ/ρ is shown for ΛLTB model with present local contrast δρ 0 = −30% and redshift size z size = 0.07. The profile is shown in comoving angular diameter distance, at time lapses t = t/t 0 running from 0.01 up to 1.00 (present time).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Redshift dependence of the cosmic expansion rates (red for radial and orange for transverse) and the scale factor (blue) are shown for ΛLTB model with contrast δρ 0 = −30% and redshift size z size = 0.07 (solid lines) in comparison with the ΛCDM model (dashed lines).\",\n \"content\": \"Figure 2 .\\n2Redshift dependence of the cosmic expansion rates (red for radial and orange for transverse) and the scale factor (blue) are shown for ΛLTB model with contrast δρ 0 = −30% and redshift size z size = 0.07 (solid lines) in comparison with the ΛCDM model (dashed lines).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Luminosity density data from KBC13: solid orange data points are their original data, while dashed points are from additional surveys.\",\n \"content\": \"Figure 4 .\\n4Luminosity density data from KBC13: solid orange data points are their original data, while dashed points are from additional surveys.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Holsclaw et al. (2010); Seikel et al. (2012); Haridasu et al. (2018b)). The reconstructed 1σ region shown in grey on Fig. 4\",\n \"content\": \"Holsclaw et al. (2010); Seikel et al. (2012); Haridasu et al. (2018b)). The reconstructed 1σ region shown in grey on Fig. 4\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Constraints on the local matter density profile from JLA (solid curves) and JLA+P (coloured regions) are shown at 1σ and 2σ confidence level. The positions of the only two local χ 2 minima are emphasised with the dashed lines. The isotropic voids proposed by WS14 and KBC13 are marked with orange crosses.\",\n \"content\": \"Figure 5 .\\n5Constraints on the local matter density profile from JLA (solid curves) and JLA+P (coloured regions) are shown at 1σ and 2σ confidence level. The positions of the only two local χ 2 minima are emphasised with the dashed lines. The isotropic voids proposed by WS14 and KBC13 are marked with orange crosses.\"\n },\n {\n \"id\": \"\",\n \"caption\": \".m.s. ≈ 0.2 ln(10) ∆M r.m.s. = 0.4 ± 0.8%\",\n \"content\": \".m.s. ≈ 0.2 ln(10) ∆M r.m.s. = 0.4 ± 0.8%\"\n },\n {\n \"id\": \"\",\n \"caption\": \"1σ and 2σ confidence regions for ΛLTB model with GBH density profile, fitted to 237 low-redshift (0.0233 < z < 0.15) SN Ia from Pantheon dataset. The best fit (b.f.) for ΛCDM model, shown as red point is almost 2σ away from ΛLTB constraints.\",\n \"content\": \"Figure 7 .\\n71σ and 2σ confidence regions for ΛLTB model with GBH density profile, fitted to 237 low-redshift (0.0233 < z < 0.15) SN Ia from Pantheon dataset. The best fit (b.f.) for ΛCDM model, shown as red point is almost 2σ away from ΛLTB constraints.\"\n },\n {\n \"id\": \"\",\n \"caption\": \".\",\n \"content\": \".[%] \\nχ 2 \\n∆AIC ΛCDM \\n\\nLD \\n0.082 0.079 +0.012 \\n\\n−0.012 \\n\\n−51.1 \\n−51.9 +6.3 \\n\\n−6.3 \\n\\n−42.9 \\n−43.8 +6.0 \\n\\n−6.1 \\n\\n8.1 \\n8.2 +1.3 \\n\\n−1.2 \\n\\n2.62 \\n\\nLD * \\n0.082 0.075 +0.015 \\n\\n−0.015 \\n\\n−39.7 \\n−39.4 +10.3 \\n\\n−10.3 \\n\\n−32.4 \\n−32.1 +9.0 \\n\\n−9.5 \\n\\n5.9 \\n5.9 +1.9 \\n\\n−1.7 \\n\\n0.29 \\n\\nJLA \\n0.025 0.039 +0.062 \\n\\n−0.018 \\n\\n−19.5 \\n−9.9 +17.3 \\n\\n−13.9 \\n\\n−15.1 \\n−7.5 +12.9 \\n\\n−11.0 \\n\\n2.7 \\n1.3 +2.0 \\n\\n−2.3 \\n\\n678.30 \\n1.35 \\n\\nJLA+P 0.025 0.032 +0.056 \\n\\n−0.011 \\n\\n−19.6 \\n−12.5 +13.8 \\n\\n−12.4 \\n\\n−15.2 \\n−9.5 +10.4 \\n\\n−9.9 \\n\\n2.7 \\n1.7 +1.9 \\n\\n−1.9 \\n\\n678.34 \\n1.07 \\n\\nPan \\n0.075 0.070 +0.023 \\n\\n−0.031 \\n\\n−16.2 \\n−9.8 +14.0 \\n\\n−8.9 \\n\\n−12.4 \\n−7.4 +10.5 \\n\\n−7.0 \\n\\n2.2 \\n1.3 +1.3 \\n\\n−1.9 \\n\\n1020.72 \\n0.67 \\n\\nPan+P \\n0.075 0.068 +0.021 \\n\\n−0.030 \\n\\n−14.4 \\n−10.5 +9.3 \\n\\n−7.4 \\n\\n−11.0 \\n−7.9 +7.0 \\n\\n−5.8 \\n\\n2.0 \\n1.4 +1.1 \\n\\n−1.3 \\n\\n1021.01 \\n0.40 \\n\\nPan * \\n0.075 0.075 +0.013 \\n\\n−0.022 \\n\\n−23.1 \\n−21.5 +8.9 \\n\\n−9.5 \\n\\n−18.1 \\n−16.8 +7.1 \\n\\n−7.9 \\n\\n3.2 \\n3.0 +1.5 \\n\\n−1.3 \\n\\n225.63 \\n−3.30\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". 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H Du Mas Des Bourboux, 10.1051/0004-6361/201731731A&A. 608130du Mas des Bourboux H., et al., 2017, A&A, 608, A130\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":590,"cells":{"text":{"kind":"string","value":"\nPhotoacoustic imaging taking into account thermodynamic attenuation\n\n\nSebastián Acosta sacosta@bcm.edu \nDepartment of Pediatrics -Cardiology\nBaylor College of Medicine and Texas Children's Hospital\nHoustonTXUSA\n\nCarlos Montalto \nDepartment of Mathematics\nUniversity of Washington\nSeattleWAUSA\n\nPhotoacoustic imaging taking into account thermodynamic attenuation\nSubmitted to: Inverse ProblemsThermoacoustic and photoacoustic imagingcontrol theoryobservability estimatesmedical imaginghybrid and multiwave methodsattenuationdissipation\nIn this paper we consider a mathematical model for photoacoustic imaging which takes into account attenuation due to thermodynamic dissipation. The propagation of acoustic (compressional) waves is governed by a scalar wave equation coupled to the heat equation for the excess temperature. We seek to recover the initial acoustic profile from knowledge of acoustic measurements at the boundary.We recognize that this inverse problem is a special case of boundary observability for a thermoelastic system. This leads to the use of control/observability tools to prove the unique and stable recovery of the initial acoustic profile in the weak thermoelastic coupling regime. This approach is constructive, yielding a solvable equation for the unknown acoustic profile. Moreover, the solution to this reconstruction equation can be approximated numerically using the conjugate gradient method. If certain geometrical conditions for the wave speed are satisfied, this approach is well-suited for variable media and for measurements on a subset of the boundary. We also present a numerical implementation of the proposed reconstruction algorithm.\n\nIntroduction\n\nPhotoacoustic tomography is an imaging technique that takes advantage of the highcontrast exhibited by optical absorption and the high-resolution properties of wideband acoustic waves in soft biological tissue. Details concerning this type of imaging modalities are found in [1,2,3,4,5,6,7,8,9,10,11]. Qualitative photoacoustic imaging consists of recovering an initial pressure profile from acoustic measurements acquired on the boundary of a region of interest. The successful transformation of boundary measurements into the sought interior pressure profile requires mathematical algorithms arXiv:1602.01872v1 [math.AP] 4 Feb 2016 that have been studied by numerous researchers [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28].\n\nThere have been recent efforts to incorporate acoustic attenuation in the modeling of photoacoustic tomography (PAT) and thermoacoustic tomography (TAT). See [29,30,31,32,33,34,35,36,37,38,18,39] and references therein. Most of these results aim at modeling attenuation in the frequency domain to account for dissipation and dispersion. In this paper, however, we adopt a model where the propagation of acoustic waves is thermodynamically coupled to the diffusion of heat. The photoacoustic effect, on which PAT is physically based, consists of two transformations of energy. First, electromagnetic energy is absorbed and transformed into heat. Second, there is a conversion of heat into mechanical energy due to thermal expansivity of the tissues. Concerning this second step, due to the thermodynamic interaction between temperature and pressure, the reverse transformation of energy also takes place. Since heat diffuses, this process attenuates the mechanical energy of the acoustic waves. See [40,Ch. 8] for an introduction to thermoelasticity in biomechanics. We claim that this type of acoustic attenuation should be naturally considered in PAT because PAT itself is based on the thermo-elastic interaction.\n\nWe realize that, in mathematical terms, the PAT problem coincides with a problem of boundary observability, which is one of the central concepts of control theory for partial differential equations [41,42,43,44]. We have already employed similar tools to address the PAT problem in an enclosure [26] and other related problems [45,46]. The objective of this paper is to constructively employ the tools of observability for hyperbolic equations together with certain regularity properties of parabolic equations to solve the PAT problem in the presence of thermodynamic attenuation. For the thermoelastic system, there is a series of works on establishing exact, approximate and null controllability or observability estimates [47,48,49,50,51,52,53,54,43]. In these works, however, either the boundary condition or the distribution of control/observation is not of the type we need to model the PAT problem. Therefore, we modify some ideas provided by these references to seek a solution for PAT in the weak coupling regime. Although it might be possible to use Carleman estimates from [53,54,43] to treat the strongly coupled system, we refrain from doing so because the thermoelastic coupling in PAT is known to be relatively weak. Hence, we claim that the results of this paper are sufficient for the nature of PAT in biological tissues.\n\n\nMathematical Formulation and Main Results\n\nIn this paper we study the photoacoustic tomography problem in the presence thermodynamic attenuation. This is modeled by the linear equations of elasticity coupled with thermal diffusivity [40]. Let Ω ⊂ R n be a smooth bounded connected domain with boundary ∂Ω where n ≥ 2. The propagation of thermoelastic waves in isotropic media is governed by the following system [40,Ch. 8],\nρ∂ 2 t u − ∇ (λ div u) − div µ ∇u + ∇u T + βK ∇θ = 0, in (0, τ ) × Ω ∂ t θ − α∆θ + θ ref βK ρc p div ∂ t u = 0, in (0, τ ) × Ω\nfor the displacement u and where θ denotes the deviation from the reference temperature θ ref . Also, ρ is the mass density, λ and µ are the Lamé coefficients, α denotes the thermal diffusivity and c p is the specific heat at constant pressure. The thermoelastic coupling is given by βK where β is the coefficient of volumetric thermal expansion and K is a constant reference bulk modulus. In soft biological tissue, λ µ which implies that K/(λ + 2µ) ≈ 1. For the moment let us assume that λ, µ and ρ are constants in Ω. Later, we will drop this assumption.\n\nSince photoacoustic imaging is primarily concerned with the compressional waves, we define the pressure p = −(λ + 2µ) div u and the square of the compressional wave speed c 2 = (λ + 2µ)/ρ and proceed to obtain a scalar model for the thermoacoustic waves. Simultaneously, we seek to reveal the strength of the thermoelastic coupling by writing the governing equations in unitless form. Let L be a characteristic length scale of the domain Ω (such as its diameter). Let the characteristic time scale be given by\nT = L/c ref where c 2 ref = K/ρ\nis the square of a reference wave speed. We define the following unitless variables and parameters: Pressurep = p/K and temperatureθ = θ/θ ref .\n\nLengthx = x/L and timet = t/T . Parameters:ĉ 2 = c 2 T 2 /L 2 ,α = αT /L 2 , σ = K/(θ ref ρc p ) and = βθ ref .\n\nThe unitless coupling parameter > 0 is introduced to analyze the case where is sufficiently small. This is valid for small coefficient of thermal expansion β. The unitless product G = σ is known as the Grüneisen coefficient. Table 1 displays rough estimates for the values of these physical parameters for soft biological tissues. We obtain that 0.05 0.1 and 0.5 σ 10. For notational convenience, we assume that σ = 1. This presents no impediment to the theory as we could easily treat the case σ > 0. At this point, to alleviate the notation, we also drop the caret to denote the unitless quantities. The unitless scalar governing system then becomes,\n∂ 2 t p − c 2 ∆p − c 2 ∆θ = 0 in (0, τ ) × Ω,(1)∂ t θ − α∆θ − ∂ t p = 0 in (0, τ ) × Ω.(2)\nWe take (1)-(2) as the starting point for the modeling of this problem. While the equivalence of the thermoelastic system and equations (1)-(2) only holds when λ, µ and ρ are constants, we will consider c(x) to be a sufficiently smooth function in Ω. This follows the common practice of considering a variable wave speed to model heterogeneous media, even when the wave equation is not in divergence form. The system (1)- (2) is augmented by the following initial and boundary conditions,\np = p 0 , ∂ t p = p 1 and θ = θ 0 on {t = 0} × Ω,(3)∂ ν p + γ∂ t p = 0 and ∂ ν θ = 0 on (0, τ ) × ∂Ω.(4)\nHere, γ : ∂Ω → [0, ∞) denotes the acoustic impedance coefficient at the boundary ∂Ω. Physically, γ = 0 models an acoustically hard surface (such as reflectors) and γ → ∞ approximates an acoustically soft boundary. In general we allow γ to vary on the boundary ∂Ω to model the inhomogeneous nature of an enclosing surface and the interface with sensors or air. The length of the observation window of time is given by τ < ∞ which is defined below. In (4), the symbol ∂ ν denotes the outward normal derivative at the boundary ∂Ω. All the coefficients in (1)-(4) are assumed to be sufficiently smooth.\n\nConcerning the initial conditions, it is common in the modeling of photoacoustic tomography to assume the following. These two expressions are respectively valid when pressure relaxation and thermal diffusion are negligible in the very short lapse of heat deposition from the optical source. These conditions can be achieved in biological tissues by using nanosecond optical pulses [1,2,3,12,5,6,4,55,56]. It turns out that the assumption p 1 = 0 is not needed from the mathematical point of view because it is possible to recover both terms of the initial Cauchy data for the acoustic field. On the other hand, the assumption θ 0 = p 0 is mathematically crucial because it removes an important degree of freedom in the analysis. It would not be possible to recover, in stable manner, an independent initial condition for the thermal field. This is a well-known consequence of the smoothing effect of the heat equation.\n\nIn order to consider partial measurements and the presence of acoustic reflectors, we divide the boundary as the disjoint union ∂Ω = Γ ∪ (∂Ω \\ Γ) where Γ is the portion where we make observations of the acoustic field. As reviewed in the next section, the forward problem (1)-(4) has a unique solution, and we can define the measurement map given by\nMp 0 = p| (0,τ )×Γ (5)\nwhere p is the solution of (1)-(4) with initial conditions satisfying Assumption 2.1. The goal of the photoacoustic tomography problem is to find the initial profile p 0 from knowledge of Mp 0 . This is a challenging problem with intricate dependencies between the domain Ω, the partial boundary Γ, the wave speed c and the time interval (0, τ ). The admissible dependencies are made precise by a sophisticated assumption of geometric character. Following Bardos, Lebeau and Rauch [42], we assume that our problem enjoys the geometric control condition for the Riemannian manifold (Ω, c −2 δ ij ) with only a portion Γ of the boundary ∂Ω being accessible for observation. We assume that Γ is a smooth open domain relative to ∂Ω. This geometric assumption is the following.\n\nAssumption 2.2 (Geometric Condition). There exists τ < ∞ such that any geodesic ray, originating from any point in Ω at t = 0, eventually reaches Γ in a non-diffractive manner (after possible geometrical reflections on ∂Ω \\ Γ) before time t = τ . We refer to τ as the controllability or observability time.\n\nThe main theoretical result of this paper is the following theorem. \np 0 H 1 (Ω) ≤ C Mp 0 H 1 ((0,τ );H 0 (Γ)) ,\nfor all p 0 ∈ H 1 (Ω), and some constant C > 0.\n\nIn addition to this theoretical result, we also propose a convergent iterative reconstruction algorithm. This algorithm is described in Section 4.\n\n\nProof of the Main Result\n\nIn order to properly analyze the inverse problem we must first state some mathematical properties of the initial boundary value problem (1)-(4). Our guiding references are [57,58,59,60]. We use the following definition of energy for the thermoacoustic system,\nE(t) = 1 2 Ω |∇p(t, x)| 2 + c −2 (x)|∂ t p(t, x)| 2 + |∇θ(t, x)| 2 dV(x).\nIt is convenient to seek a space that is complete under the energy norm. Since p = const. and θ = const. is a non-trivial solution to the governing equations, the sought space should rule out nonzero constant solutions. This is achieved by considering only solutions such that\nΩ c −2 (x)∂ t p(t, x) dV(x) + ∂Ω γ(x)p(t, x) dS(x) = 0,(6)Ω (θ(t, x) − p(t, x)) dV(x) = 0.(7)\nThis is motivated by the fact that the left-hand sides of (6)- (7) are indeed independent of time provided that p and θ solve the governing equations. Therefore, it is only needed to require (6)-(7) at time t = 0. Then the appropriate energy space is given by\nH := (p 0 , p 1 , θ 0 ) ∈ H 1 (Ω) × H 0 (Ω) × H 1 (Ω) : (6)-(7) are satisfied .\nNotice that H is a closed subspace of H 1 (Ω) × H 0 (Ω) × H 1 (Ω). So it is complete under the norm of H 1 (Ω) × H 0 (Ω) × H 1 (Ω) as well as under the energy norm. The problem (1)-(4) for (p, ∂ t p, θ) is well-posed on the space H and the energy is non-increasing. We state this in the form of a lemma. The proof follows from the standard analysis of partial differential equations and semigroup theory [57,58,59,60].\n\nLemma 3.1. Given (p 0 , p 1 , θ 0 ) ∈ H, the unique weak solution of (1)-(4) satisfies (p, ∂ t p, θ) ∈ H for t ≥ 0, and p ∈ C k ([0, τ ]; H 1−k (Ω)) for k = 0, 1 and θ ∈ C([0, τ ]; H 1 (Ω)). Moreover, E(t) ≤ E(0) for all t ≥ 0 and all ≥ 0. In fact (due to parabolic regularity) the energy E ∈ H 1 (0, τ ) and\ndE dt (t) = − Ω α|∆θ(t, x)| 2 dV(x) − ∂Ω γ(x)|∂ t p(t, x)| 2 dS(x).\nThe well-posedness and regularity implications of Theorem 3.2 also apply to arbitrary initial conditions (p 0 , p 1 , θ 0 ) ∈ H 1 (Ω) × H 0 (Ω) × H 1 (Ω). By virtue of linearity, we can decompose the initial conditions as follows,\np 0 = (p 0 − p 0,const ) + p 0,const θ 0 = (θ 0 − θ 0,const ) + θ 0,const where p 0,const = Γ γ(x)dS(x) −1 Ω c −2 (x)p 1 (x)dS(x) + ∂Ω γ(x)p 0 (x)dS(x) θ 0,const = 1 |Ω| Ω (θ 0 (x) − p 0 (x)) dV(x) + p 0,const .\nTherefore, the evolution of the triplet (p 0 , p 1 , θ 0 ) ∈ H 1 (Ω) × H 0 (Ω) × H 1 (Ω) can be decomposed into the evolution of the initial condition (p 0 − p 0,const , p 1 , θ 0 − θ 0,const ) in H plus a time-independent solution given by (p 0,const , 0, θ 0,const ) for t ≥ 0. Notice that the energy of this particular solution is zero because it is constant both in space and time. This leads to the following result using Lemma 3.1 and the decomposition described above.\n\nTheorem 3.2 (Forward well-posedness). Given (p 0 , p 1 , θ 0 ) ∈ H 1 (Ω) × H 0 (Ω) × H 1 (Ω), the unique weak solution of (1)-(4) satisfies p ∈ C k ([0, τ ]; H 1−k (Ω)) for k = 0, 1 and θ ∈ C([0, τ ]; H 1 (Ω)). Moreover, E(t) ≤ E(0) for all t ≥ 0 and all ≥ 0.\n\nNow we proceed to prove the main result.\n\nProof of Theorem 2.3. Consider the problem (1)-(4) for p 0 ∈ H 1 (Ω) and p 1 = 0 and θ 0 = p 0 (Assumption 2.1). From Theorem 3.2 we have that the solution to this problem satisfies p ∈ C k ([0, τ ]; H 1−k (Ω)) for k = 0, 1 and θ ∈ C([0, τ ]; H 1 (Ω)). Now notice that θ has initial condition in H 1 (Ω), homogeneous Neumann boundary condition, and a forcing term ∂ t p ∈ C([0, τ ]; H 0 (Ω)) ⊂ H 0 ((0, τ ); H 0 (Ω)). Hence, from regularity theory for the parabolic equation [57,Thm 4.3] (see also [58]), we obtain that θ ∈ H 0 ((0, τ ); H 2 (Ω)) ∩ H 1 ((0, τ ); H 0 (Ω)) and an estimate of the form\n∆θ 2 H 0 ((0,τ );H 0 (Ω)) ≤ 2 C ∂ t p 2 H 0 ((0,τ );H 0 (Ω)) + ∇p 0 2 H 0 (Ω) ,(8)\nfor some constant C > 0 independent of > 0. From the energy estimate in Theorem 3.2, we also have that\n∂ t p 2 H 0 ((0,τ );H 0 (Ω)) ≤ C(1 + 2 ) ∇p 0 2 H 0 (Ω) .(9)\nNow, under the geometric condition in Assumption 2.2, the acoustic problem for p enjoys the following observability property [41,42,43,44],\np 0 2 H 1 (Ω) ≤ C 2 ∆θ 2 H 0 ((0,τ );H 0 (Ω)) + p 2 H 1 ((0,τ );H 0 (Γ)) ,(10)\nwhere C > 0 is also independent of > 0. Therefore, combining the above three inequalities, for sufficiently small > 0 we conclude that there is C > 0 such that p 0 2 H 1 (Ω) ≤ C p 2 H 1 ((0,τ );H 0 (Γ)) , This concludes the proof.\n\n\nReconstruction Algorithm\n\nIn this section we explicitly recover the initial acoustic profile p 0 in terms of the boundary measurements Mp 0 = p| (0,τ )×Γ . This is accomplished by using the observability properties obtained in the previous section leading to the invertibility of the normal operator (M * M).\n\nIn order to obtain an applicable expression for the operator M * , in this section we state the dual or adjoint problem associated with (1)-(4). This is equivalent to constructing the well-known Hilbert Uniqueness Method (HUM) for control of partial differential equations. See [61,41] for details. Throughout, we assume that > 0 is sufficiently small for Theorem 2.3 to apply. This adjoint problem is to find a weak solution (ψ, ξ) for the following IBVP,\n∂ 2 t ψ − c 2 ∆ψ − c 2 α −1 ∂ t ξ = 0 in (0, τ ) × Ω,(11)∂ t ξ + α∆ξ − α∆ψ = 0 in (0, τ ) × Ω,(12)ψ = 0, ∂ t ψ = 0 and ξ = 0 on {t = τ } × Ω,(13)∂ ν ψ − γ∂ t ψ = η and ∂ ν ξ − ∂ ν ψ = 0 on (0, τ ) × ∂Ω.(14)\nfor a given η ∈ H 0 ((0, τ ) × Γ) (extended as zero on (0, τ ) × ∂Ω \\ Γ). Notice that this problem is solved backwards in time with vanishing Cauchy data at time t = τ . Since this problem is well-posed [57], we obtain the following definition.\n\nDefinition 4.1. Let S be the mapping η → −∂ t ψ| t=0 , where ψ is the solution of (11)- (14) for the provided η.\n\nIntegrating by parts the terms of equations (1)-(2) against (ψ, ξ), where (ψ, ξ) is the solution of (11)- (14), we easily obtain that\np 0 , Sη Ω = Mp 0 , η (0,τ )×Γ , for all η ∈ H 0 ((0, τ ) × Γ).\nHence, invoking duality and the Riesz representation, we have that M * = S : ((0, τ ); H 0 (Γ)) → H 1 (Ω) given in Definition 4.1 is surjective. Therefore, for any φ ∈ H 1 (Ω), there exists a boundary control η ∈ H 1 ((0, τ ); H 0 (Γ)) such that the solution (ψ, ξ) of (11)- (14) satisfies\nH 1 ((0, τ ); H 0 (Γ)) → H 1 (Ω). Now,∂ t ψ = −φ, at time t = 0.\nAmong all such boundary controls, there exists η min which is uniquely determined by φ as the minimum norm control and satisfies the following stability condition\nη min H 1 ((0,τ );H 0 (Γ)) ≤ C φ H 1 (Ω)\nfor some constant C > 0. As a consequence, the mapping φ → η min defines a bounded control operator C : H 1 (Ω) → H 1 ((0, τ ); H 0 (Γ)), and this operator is given by\nC = S * (SS * ) −1 = M(M * M) −1 .\nLet (ψ, ξ) be the solution of (11)- (14) with η = Cφ and φ ∈ H 1 (Ω) arbitrary. Then by construction, Cφ H 1 ((0,τ );H 0 (Γ)) , for all φ ∈ H 1 (Ω), which implies that the unknown initial condition p 0 is explicitly recovered as follows,\np 0 , φ H 1 (Ω) = Mp 0 ,p 0 = C * Mp 0 ,(15)\nwhere C * is the adjoint of the control operator C : H 1 (Ω) → H 1 ((0, τ ); H 0 (Γ)) defined in Theorem 4.2. The reconstruction algorithm is based on the identity (15) and an iterative algorithm to approximate the action of C * . This algorithm is based on the following points provided by Theorem 4.2 (cf. [61,41]): (2) For ζ ∈ H 1 (Ω), the solution to (M * M)φ = ζ can be approximated using the conjugate gradient method.\n\nNow we proceed to describe how the action of C * can be approximated using the conjugate gradient method. See [62,Sect. 4.6] for a standard description of the conjugate gradient method in a Hilbert space setting. For sake of completeness, we describe the inversion of a generic equation (M * M)φ = ζ. Let φ 0 be an initial guess for the true solution φ * . Define r 0 = ζ − (M * M)φ 0 as the initial residue and s 0 = r 0 . For k ≥ 0, define\nφ k+1 = φ k + α k s k , where α k = r k 2 H 1 (Ω) s k , (M * M)s k H 1 (Ω) r k+1 = ζ − (M * M)φ k+1 s k+1 = r k+1 + β k s k , where β k = r k+1 2 H 1 (Ω) r k 2 H 1 (Ω)\n.\n\nSince the operator (M * M) : H 1 (Ω) → H 1 (Ω) is bounded and coercive, then there are positive constants m and M such that\nm φ 2 H 1 (Ω) ≤ φ, (M * M)φ H 1 (Ω) ≤ M φ 2 H 1 (Ω)\n. The conjugate gradient iterates can be shown to converge as follows (see [62,Sect. 4.6] and references therein),\nφ * − φ k H 1 (Ω) ≤ e −σk φ * − φ 0 H 1 (Ω) , for k ≥ 0, where σ = ln M + m M − m .(16)\nNotice that at each iteration, one must apply the operator (M * M) which amounts to solve the problem (1)-(4) (under Assumption 2.1) followed by solving the adjoint problem (11)- (14). In practice, this can be approximated using numerical methods for PDEs. However, depending on the method of choice, there are intrinsic complications that may prevent a convergence estimate such as (16) from being satisfied in the limit as the discretization is refined. We shall not elaborate any further on these complications as they lie outside of the scope of this paper. For details on these numerical issues we refer to [41,63,64,65,66] and references therein. In this paper, we adopted the two-grid approach described in [65,41] using second order finite difference methods.\n\n\nNumerical Results\n\nNow we present some numerical results to illustrate the performance of the reconstruction algorithm described in Section 4. We implemented a numerical solver for the governing system (1)-(4) and its adjoint (11)-(14) based on second order finite differences. To avoid spurious numerical instabilities, we adopted the two-grid approach described in [65,41]. We worked in R 2 where the domain Ω was taken as the unit-square. The initial profile p 0 corresponds the Shepp-Logan phantom.\n\nWe present two examples. One with constant wave speed c(x) ≡ 1, and the other with variable wave speed c = c(x) defined below. In both cases, we used the following parameters: impedance γ(x) = c −1 (x) over the boundary of Ω, thermal diffusivity α = 0.01, and coupling parameter = 0.1. The observability time was chosen to be τ = 2 which is enough for more than 99% of the energy contained in the initial profile to dissipate or leave the domain through the boundary when the wave speed is constant.\n\nWe shall compare the results from the proposed algorithm against the results from purely acoustic time-reversal. The latter is accomplished by producing measurements using the thermoacoustic forward solver M, and then back-propagating the boundary measurements in a purely acoustic medium ( = 0), that is, by ignoring the thermodynamic attenuation. See details in [26,16] for the purely acoustic time-reversal approach. The acoustic time-reversal is approximated using the same finite difference method. The initial guess for the conjugate gradient algorithm is the approximate solution obtained from the purely acoustic time-reversal algorithm. Although the proposed reconstruction algorithm has been described in the H 1 (Ω) setting, a similar study could be performed in the H 0 (Ω) setting where the inner-products in the conjugate gradient algorithm would need to be understood appropriately. In this section, we present results from the implementation both in the H 1 (Ω) and H 0 (Ω) formulations.\n\n\nConstant wave speed\n\nFor the first example where c ≡ 1, Figure 1 displays the exact initial profile and the reconstructions. The relative errors in the H 1 (Ω) and H 0 (Ω) formulations are reported in Table 2 for the first few iterations of the conjugate gradient algorithm. We notice that by ignoring the thermodynamic attenuation in the purely acoustic timereversal reconstruction (Iter = 0), the edges in the Shepp-Logan phantom are blurred considerably. Some of the sharpness is recovered by accounting for the attenuation in the proposed algorithm even after a single iteration. Table 2.\n\nConstant wave speed example. Relative error at each iteration of the conjugate gradient method described in Section 4. Iter = 0 corresponds to the initial guess given by a purely acoustic time-reversal algorithm. \nIter H 1 (Ω)-norm H 0 (Ω)-\n\nVariable wave speed\n\nFor the second example we have selected a variable wave speed defined as a layer of higher speed surrounding the smaller ellipses in the Shepp-Logan phantom. The actual profile is illustrated in the top-right panel of Figure 2. The relative errors in the H 1 (Ω) and H 0 (Ω) formulations are reported in Table 3 for the first few iterations of the conjugate gradient algorithm. Again, as shown in the lower panels of Figure 2, we see great improvements over the purely acoustic time-reversal reconstruction. We highlight the ability in capturing the jump discontinuities and the reduction of the artifacts introduced by ignoring the attenuation. \n\n\nConclusion\n\nWe have presented a PAT/TAT model based on thermoelasticity. The thermoelastic coupling accounts for how pressure changes can induce temperture changes in a body and vice versa. The coupling between temperture and deformation is a fundamental feature of PAT/TAT. The current literature dealing with PAT/TAT only considers one side of the thermoelastic interaction (the photoacoustic effect). By considering both effects simultaneously we account for a natural attenuation phenomenon of the acoustic signal. We related the thermoelastic model of PAT/TAT with boundary observavility for the thermoelastic system. We showed uniqueness and stability of recovering the initial profile of the pressure from boundary data. The initial profile was reconstructed using a conjugate gradient method. The recovery analysis of the inital wave profile works under a geometric control assumption on the wave speed. The performance of the reconstruction method is very similar in both the constant and variable wave speed scenarios.\n\nThe coupling uniteless parameter of the thermoelastic model is approximately between 0.05 and 0.1 as obtained from Table 1. Theorem 2.3 requires to be sufficiently small. Given that in PAT/TAT the thermodynamic interaction is small, such a condition on is reasonable. Nonethelesss, it might be possible to remove this condition by using Carleman estimates for the coupled thermoelastic system (e.g., [53,54,43]).\n\n\nAssumption 2.1 (Rapid Deposition of Heat). The initial conditions (3) satisfy, p 1 = 0 and θ 0 = p 0 .\n\nTheorem 2. 3 (\n3Main Result). Suppose that Assumptions 2.1 and 2.2 hold. For sufficiently small > 0, the map p 0 → Mp 0 is injective and satisfies a stability estimate of the form\n\n\nif we choose η = Mp 0 and use the estimate from Theorem 2.3, we obtain that, p 0 , (M * M)p 0 H 1 (Ω) (Ω) , for all p 0 ∈ H 1 (Ω) and some constant C > 0. Therefore, the operator (M * M) : H 1 (Ω) → H 1 (Ω) is coercive, and S : H 1((0, τ ); H 0 (Γ)) → H 1 (Ω) is surjective. With these results, we can establish the following controllability theorem.\n\n\nTheorem 4.2 (Acoustic Control). Let the Geometric Condition 2.2 hold. For sufficiently small > 0, the operator S : H 1\n\n( 1 )\n1The observability operator C * = (M * M) −1 M * , where (M * M) : H 1 (Ω) → H 1 (Ω) is coercive.\n\nFigure 1 .\n1Exact initial acoustic profile (top-left), the reconstruction from purely acoustic time-reversal (top-right), and the reconstruction from the proposed algorithm described in Section 4 using 1 iteration (bottom-left) and 5 iterations (bottom-right).\n\nFigure 2 .\n2Exact initial acoustic profile (top-left), wave speed profile (top-right), the reconstruction from purely acoustic time-reversal (bottom-left) and the reconstruction from the proposed algorithm described in Section 4 using 5 iterations (bottom-right).\n\nTable 1 .\n1Physical parameters for the thermoacoustic coupling in typical soft biological tissues.Physical Parameter \nSymbol Range \nUnits \n\nBulk modulus \nK \n2000 -2500 10 6 Pa \nDensity \nρ \n900 -1100 \nKg / m 3 \nRef. Temperature \nθ ref \n290 -310 \nK \nCoeff. thermal expansion β \n200 -300 \n10 −6 / K \nSpecific heat \nc p \n500 -5000 \nJ / (Kg K) \n\n\nTable 3 .\n3Variable wave speed example. Relative error at each iteration of the conjugate gradient method described in Section 4. Iter = 0 corresponds to the initial guess given by a purely acoustic time-reversal algorithm.Iter H 1 (Ω)-norm H 0 (Ω)-norm0 \n55.4 % \n34.3 % \n1 \n24.2 % \n16.6 % \n2 \n15.5 % \n8.0 % \n3 \n11.8 % \n4.7 % \n4 \n10.2 % \n4.0 % \n5 \n9.7 % \n3.7 % \n\nAcknowledgmentsThe authors would like to thank Plamen Stefanov for recommendations on the first draft of the paper. We also want to thank Benjamín Palacios for his ideas on a Neumann series approach for this problem.\nBiomedical optics : principles and imaging. L V Wang, H Wu, Wiley-Interscience14Hoboken, N.J.L. V. Wang and H. Wu. Biomedical optics : principles and imaging. Hoboken, N.J. : Wiley- Interscience, 2007. 1, 4\n\nPhotoacoustic imaging and spectroscopy. 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Differential Equations. 13L. de Teresa and E. Zuazua. Controllability of the linear system of thermoelastic plates. Adv. Differential Equations, 1(3):369-402, 1996. 2\n\nNullcontrollability of a system of linear thermoelasticity. Archive for Rational Mechanics and Analysis. G Lebeau, E Zuazua, 141G. Lebeau and E. Zuazua. Nullcontrollability of a system of linear thermoelasticity. Archive for Rational Mechanics and Analysis, 141(4):297-329, 1998. 2\n\nPartial exact controllability and exponential stability in higher-dimensional linear thermoelasticity. ESAIM: Control, Optimisation and Calculus of Variations. W Liu, 3W. Liu. Partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity. ESAIM: Control, Optimisation and Calculus of Variations, 3:23-48, 1998. 2\n\nCorrection to partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity. ESAIM: Control, Optimisation and Calculus of Variations. W Liu, 3W. Liu. Correction to partial exact controllability and exponential stability in higher-dimensional linear thermoelasticity. ESAIM: Control, Optimisation and Calculus of Variations, 3:323-327, 1998. 2\n\nPartial exact controllability for the linear thermo-viscoelastic model. W.-J Liu, G H Williams, Electronic Journal of Differential Equations. 17electronic onlyW.-J. Liu and G. H. Williams. Partial exact controllability for the linear thermo-viscoelastic model. Electronic Journal of Differential Equations [electronic only], 1998:No. 17, pp. 1-11, 1998. 2\n\nSimultaneous exact/approximate boundary controllability of thermo-elastic plates with variable thermal coefficient and moment control. M Eller, I Lasiecka, R Triggiani, Journal of Mathematical Analysis and Applications. 25114M. Eller, I. Lasiecka, and R. Triggiani. Simultaneous exact/approximate boundary controllability of thermo-elastic plates with variable thermal coefficient and moment control. Journal of Mathematical Analysis and Applications, 251:452-478, 2000. 2, 14\n\nCarleman estimates and boundary observability for a coupled parabolichyperbolic system. P Albano, D Tataru, Electronic Journal of Differential Equations. 214electronic onlyP. Albano and D. Tataru. Carleman estimates and boundary observability for a coupled parabolic- hyperbolic system. Electronic Journal of Differential Equations [electronic only], 2000:No. 22, pp. 1-15, 2000. 2, 14\n\nThermoacoustic computed tomography -technical considerations. R Kruger, D Reinecke, G Kruger, Med. Phys. 269R. Kruger, D. Reinecke, and G. Kruger. Thermoacoustic computed tomography -technical considerations. Med. Phys., 26(9):1832-1837, 1999. 4\n\nReal-time optoacoustic monitoring of temperature in tissues. I Larina, K Larin, R Esenaliev, Journal of Physics D: Applied Physics. 38152633I. Larina, K. Larin, and R. Esenaliev. Real-time optoacoustic monitoring of temperature in tissues. Journal of Physics D: Applied Physics, 38(15):2633, 2005. 4\n\nNon-homogeneous boundary value problems and applications. Vol. I-II-III. Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. J.-L Lions, E Magenes, J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications. Vol. I-II-III. Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd. 181-183.\n\nL C Evans, Partial Differential Equations. Providence, RIAmerican Mathematical Society197L.C. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998. 5, 6, 7\n\nAn introduction to partial differential equations. M Renardy, R Rogers, Texts in applied mathematics. 136SpringerM. Renardy and R. Rogers. An introduction to partial differential equations, volume 13 of Texts in applied mathematics. New York : Springer, 2004. 5, 6\n\nOne-parameter semigroups for linear evolution equations. K.-J Engel, R Nagel, Springer1946New Yorkof Graduate texts in mathematicsK.-J. Engel and R. Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate texts in mathematics. New York : Springer, 2000. 5, 6\n\nExact controllability, stabilization and perturbations for distributed systems. J.-L Lions, SIAM Review. 3019J.-L. Lions. Exact controllability, stabilization and perturbations for distributed systems. SIAM Review, 30(1):1-68, 1988. 7, 9\n\nTheoretical Numerical Analysis: A Functional Analysis Framework. K Atkinson, W Han, Texts in Applied Mathematics. 399Springer-VerlagK. Atkinson and W. Han. Theoretical Numerical Analysis: A Functional Analysis Framework, volume 39 of Texts in Applied Mathematics. New York : Springer-Verlag, 2001. 9\n\nThe wave equation: Control and numerics. S Ervedoza, E Zuazua, Control of Partial Differential Equations. Berlin HeidelbergSpringer2048S. Ervedoza and E. Zuazua. The wave equation: Control and numerics. In Control of Partial Differential Equations, volume 2048 of Lecture Notes in Mathematics, pages 245-339. Springer Berlin Heidelberg, 2012. 10\n\nSpectral conditions for admissibility and observability of wave systems: applications to finite element schemes. S Ervedoza, Numerische Mathematik. 1133S. Ervedoza. Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes. Numerische Mathematik, 113(3):377-415, 2009. 10\n\nGeometrical aspects of exact boundary controllability for the wave equation -a numerical study. M Asch, G Lebeau, 3ESAIM: Control, Optimisation and Calculus of VariationsM. Asch and G. Lebeau. Geometrical aspects of exact boundary controllability for the wave equation -a numerical study. ESAIM: Control, Optimisation and Calculus of Variations, 3:163- 212, 1 1998. 10\n\nNumerical Approximation of Exact Controls for Waves. S Ervedoza, E Zuazua, SpringerBriefs in Mathematics. 10SpringerS. Ervedoza and E. Zuazua. Numerical Approximation of Exact Controls for Waves. SpringerBriefs in Mathematics. Springer : New York, 2013. 10\n"},"corpus_id":{"kind":"number","value":16564308,"string":"16,564,308"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1602.01872\",\n \"mag\": 3105025489,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/0266-5611/32/11/115001\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1602.01872v1.pdf\"\n ],\n \"pdf_sha\": \"d93a690d4e2c22aeff6d7ae51322744facd12f27\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1602.01872\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Photoacoustic imaging taking into account thermodynamic attenuation"},"authors":{"kind":"list like","value":["Sebastián Acosta sacosta@bcm.edu \nDepartment of Pediatrics -Cardiology\nBaylor College of Medicine and Texas Children's Hospital\nHoustonTXUSA","Carlos Montalto \nDepartment of Mathematics\nUniversity of Washington\nSeattleWAUSA"],"string":"[\n \"Sebastián Acosta sacosta@bcm.edu \\nDepartment of Pediatrics -Cardiology\\nBaylor College of Medicine and Texas Children's Hospital\\nHoustonTXUSA\",\n \"Carlos Montalto \\nDepartment of Mathematics\\nUniversity of Washington\\nSeattleWAUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"In this paper we consider a mathematical model for photoacoustic imaging which takes into account attenuation due to thermodynamic dissipation. The propagation of acoustic (compressional) waves is governed by a scalar wave equation coupled to the heat equation for the excess temperature. We seek to recover the initial acoustic profile from knowledge of acoustic measurements at the boundary.We recognize that this inverse problem is a special case of boundary observability for a thermoelastic system. This leads to the use of control/observability tools to prove the unique and stable recovery of the initial acoustic profile in the weak thermoelastic coupling regime. This approach is constructive, yielding a solvable equation for the unknown acoustic profile. Moreover, the solution to this reconstruction equation can be approximated numerically using the conjugate gradient method. If certain geometrical conditions for the wave speed are satisfied, this approach is well-suited for variable media and for measurements on a subset of the boundary. We also present a numerical implementation of the proposed reconstruction algorithm."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Photoacoustic tomography is an imaging technique that takes advantage of the highcontrast exhibited by optical absorption and the high-resolution properties of wideband acoustic waves in soft biological tissue. Details concerning this type of imaging modalities are found in [1,2,3,4,5,6,7,8,9,10,11]. Qualitative photoacoustic imaging consists of recovering an initial pressure profile from acoustic measurements acquired on the boundary of a region of interest. The successful transformation of boundary measurements into the sought interior pressure profile requires mathematical algorithms arXiv:1602.01872v1 [math.AP] 4 Feb 2016 that have been studied by numerous researchers [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28].\nThere have been recent efforts to incorporate acoustic attenuation in the modeling of photoacoustic tomography (PAT) and thermoacoustic tomography (TAT). See [29,30,31,32,33,34,35,36,37,38,18,39] and references therein. Most of these results aim at modeling attenuation in the frequency domain to account for dissipation and dispersion. In this paper, however, we adopt a model where the propagation of acoustic waves is thermodynamically coupled to the diffusion of heat. The photoacoustic effect, on which PAT is physically based, consists of two transformations of energy. First, electromagnetic energy is absorbed and transformed into heat. Second, there is a conversion of heat into mechanical energy due to thermal expansivity of the tissues. Concerning this second step, due to the thermodynamic interaction between temperature and pressure, the reverse transformation of energy also takes place. Since heat diffuses, this process attenuates the mechanical energy of the acoustic waves. See [40,Ch. 8] for an introduction to thermoelasticity in biomechanics. We claim that this type of acoustic attenuation should be naturally considered in PAT because PAT itself is based on the thermo-elastic interaction.\nWe realize that, in mathematical terms, the PAT problem coincides with a problem of boundary observability, which is one of the central concepts of control theory for partial differential equations [41,42,43,44]. We have already employed similar tools to address the PAT problem in an enclosure [26] and other related problems [45,46]. The objective of this paper is to constructively employ the tools of observability for hyperbolic equations together with certain regularity properties of parabolic equations to solve the PAT problem in the presence of thermodynamic attenuation. For the thermoelastic system, there is a series of works on establishing exact, approximate and null controllability or observability estimates [47,48,49,50,51,52,53,54,43]. In these works, however, either the boundary condition or the distribution of control/observation is not of the type we need to model the PAT problem. Therefore, we modify some ideas provided by these references to seek a solution for PAT in the weak coupling regime. Although it might be possible to use Carleman estimates from [53,54,43] to treat the strongly coupled system, we refrain from doing so because the thermoelastic coupling in PAT is known to be relatively weak. Hence, we claim that the results of this paper are sufficient for the nature of PAT in biological tissues."},{"title":"Mathematical Formulation and Main Results","content":"In this paper we study the photoacoustic tomography problem in the presence thermodynamic attenuation. This is modeled by the linear equations of elasticity coupled with thermal diffusivity [40]. Let Ω ⊂ R n be a smooth bounded connected domain with boundary ∂Ω where n ≥ 2. The propagation of thermoelastic waves in isotropic media is governed by the following system [40,Ch. 8],\nfor the displacement u and where θ denotes the deviation from the reference temperature θ ref . Also, ρ is the mass density, λ and µ are the Lamé coefficients, α denotes the thermal diffusivity and c p is the specific heat at constant pressure. The thermoelastic coupling is given by βK where β is the coefficient of volumetric thermal expansion and K is a constant reference bulk modulus. In soft biological tissue, λ µ which implies that K/(λ + 2µ) ≈ 1. For the moment let us assume that λ, µ and ρ are constants in Ω. Later, we will drop this assumption.\nSince photoacoustic imaging is primarily concerned with the compressional waves, we define the pressure p = −(λ + 2µ) div u and the square of the compressional wave speed c 2 = (λ + 2µ)/ρ and proceed to obtain a scalar model for the thermoacoustic waves. Simultaneously, we seek to reveal the strength of the thermoelastic coupling by writing the governing equations in unitless form. Let L be a characteristic length scale of the domain Ω (such as its diameter). Let the characteristic time scale be given by\nis the square of a reference wave speed. We define the following unitless variables and parameters: Pressurep = p/K and temperatureθ = θ/θ ref .\nLengthx = x/L and timet = t/T . Parameters:ĉ 2 = c 2 T 2 /L 2 ,α = αT /L 2 , σ = K/(θ ref ρc p ) and = βθ ref .\nThe unitless coupling parameter > 0 is introduced to analyze the case where is sufficiently small. This is valid for small coefficient of thermal expansion β. The unitless product G = σ is known as the Grüneisen coefficient. Table 1 displays rough estimates for the values of these physical parameters for soft biological tissues. We obtain that 0.05 0.1 and 0.5 σ 10. For notational convenience, we assume that σ = 1. This presents no impediment to the theory as we could easily treat the case σ > 0. At this point, to alleviate the notation, we also drop the caret to denote the unitless quantities. The unitless scalar governing system then becomes,\nWe take (1)-(2) as the starting point for the modeling of this problem. While the equivalence of the thermoelastic system and equations (1)-(2) only holds when λ, µ and ρ are constants, we will consider c(x) to be a sufficiently smooth function in Ω. This follows the common practice of considering a variable wave speed to model heterogeneous media, even when the wave equation is not in divergence form. The system (1)- (2) is augmented by the following initial and boundary conditions,\nHere, γ : ∂Ω → [0, ∞) denotes the acoustic impedance coefficient at the boundary ∂Ω. Physically, γ = 0 models an acoustically hard surface (such as reflectors) and γ → ∞ approximates an acoustically soft boundary. In general we allow γ to vary on the boundary ∂Ω to model the inhomogeneous nature of an enclosing surface and the interface with sensors or air. The length of the observation window of time is given by τ < ∞ which is defined below. In (4), the symbol ∂ ν denotes the outward normal derivative at the boundary ∂Ω. All the coefficients in (1)-(4) are assumed to be sufficiently smooth.\nConcerning the initial conditions, it is common in the modeling of photoacoustic tomography to assume the following. These two expressions are respectively valid when pressure relaxation and thermal diffusion are negligible in the very short lapse of heat deposition from the optical source. These conditions can be achieved in biological tissues by using nanosecond optical pulses [1,2,3,12,5,6,4,55,56]. It turns out that the assumption p 1 = 0 is not needed from the mathematical point of view because it is possible to recover both terms of the initial Cauchy data for the acoustic field. On the other hand, the assumption θ 0 = p 0 is mathematically crucial because it removes an important degree of freedom in the analysis. It would not be possible to recover, in stable manner, an independent initial condition for the thermal field. This is a well-known consequence of the smoothing effect of the heat equation.\nIn order to consider partial measurements and the presence of acoustic reflectors, we divide the boundary as the disjoint union ∂Ω = Γ ∪ (∂Ω \\ Γ) where Γ is the portion where we make observations of the acoustic field. As reviewed in the next section, the forward problem (1)-(4) has a unique solution, and we can define the measurement map given by\nwhere p is the solution of (1)-(4) with initial conditions satisfying Assumption 2.1. The goal of the photoacoustic tomography problem is to find the initial profile p 0 from knowledge of Mp 0 . This is a challenging problem with intricate dependencies between the domain Ω, the partial boundary Γ, the wave speed c and the time interval (0, τ ). The admissible dependencies are made precise by a sophisticated assumption of geometric character. Following Bardos, Lebeau and Rauch [42], we assume that our problem enjoys the geometric control condition for the Riemannian manifold (Ω, c −2 δ ij ) with only a portion Γ of the boundary ∂Ω being accessible for observation. We assume that Γ is a smooth open domain relative to ∂Ω. This geometric assumption is the following.\nAssumption 2.2 (Geometric Condition). There exists τ < ∞ such that any geodesic ray, originating from any point in Ω at t = 0, eventually reaches Γ in a non-diffractive manner (after possible geometrical reflections on ∂Ω \\ Γ) before time t = τ . We refer to τ as the controllability or observability time.\nThe main theoretical result of this paper is the following theorem.\nfor all p 0 ∈ H 1 (Ω), and some constant C > 0.\nIn addition to this theoretical result, we also propose a convergent iterative reconstruction algorithm. This algorithm is described in Section 4."},{"title":"Proof of the Main Result","content":"In order to properly analyze the inverse problem we must first state some mathematical properties of the initial boundary value problem (1)-(4). Our guiding references are [57,58,59,60]. We use the following definition of energy for the thermoacoustic system,\nIt is convenient to seek a space that is complete under the energy norm. Since p = const. and θ = const. is a non-trivial solution to the governing equations, the sought space should rule out nonzero constant solutions. This is achieved by considering only solutions such that\nThis is motivated by the fact that the left-hand sides of (6)- (7) are indeed independent of time provided that p and θ solve the governing equations. Therefore, it is only needed to require (6)-(7) at time t = 0. Then the appropriate energy space is given by\nNotice that H is a closed subspace of H 1 (Ω) × H 0 (Ω) × H 1 (Ω). So it is complete under the norm of H 1 (Ω) × H 0 (Ω) × H 1 (Ω) as well as under the energy norm. The problem (1)-(4) for (p, ∂ t p, θ) is well-posed on the space H and the energy is non-increasing. We state this in the form of a lemma. The proof follows from the standard analysis of partial differential equations and semigroup theory [57,58,59,60].\nLemma 3.1. Given (p 0 , p 1 , θ 0 ) ∈ H, the unique weak solution of (1)-(4) satisfies (p, ∂ t p, θ) ∈ H for t ≥ 0, and p ∈ C k ([0, τ ]; H 1−k (Ω)) for k = 0, 1 and θ ∈ C([0, τ ]; H 1 (Ω)). Moreover, E(t) ≤ E(0) for all t ≥ 0 and all ≥ 0. In fact (due to parabolic regularity) the energy E ∈ H 1 (0, τ ) and\nThe well-posedness and regularity implications of Theorem 3.2 also apply to arbitrary initial conditions (p 0 , p 1 , θ 0 ) ∈ H 1 (Ω) × H 0 (Ω) × H 1 (Ω). By virtue of linearity, we can decompose the initial conditions as follows,\nTherefore, the evolution of the triplet (p 0 , p 1 , θ 0 ) ∈ H 1 (Ω) × H 0 (Ω) × H 1 (Ω) can be decomposed into the evolution of the initial condition (p 0 − p 0,const , p 1 , θ 0 − θ 0,const ) in H plus a time-independent solution given by (p 0,const , 0, θ 0,const ) for t ≥ 0. Notice that the energy of this particular solution is zero because it is constant both in space and time. This leads to the following result using Lemma 3.1 and the decomposition described above.\nTheorem 3.2 (Forward well-posedness). Given (p 0 , p 1 , θ 0 ) ∈ H 1 (Ω) × H 0 (Ω) × H 1 (Ω), the unique weak solution of (1)-(4) satisfies p ∈ C k ([0, τ ]; H 1−k (Ω)) for k = 0, 1 and θ ∈ C([0, τ ]; H 1 (Ω)). Moreover, E(t) ≤ E(0) for all t ≥ 0 and all ≥ 0.\nNow we proceed to prove the main result.\nProof of Theorem 2.3. Consider the problem (1)-(4) for p 0 ∈ H 1 (Ω) and p 1 = 0 and θ 0 = p 0 (Assumption 2.1). From Theorem 3.2 we have that the solution to this problem satisfies p ∈ C k ([0, τ ]; H 1−k (Ω)) for k = 0, 1 and θ ∈ C([0, τ ]; H 1 (Ω)). Now notice that θ has initial condition in H 1 (Ω), homogeneous Neumann boundary condition, and a forcing term ∂ t p ∈ C([0, τ ]; H 0 (Ω)) ⊂ H 0 ((0, τ ); H 0 (Ω)). Hence, from regularity theory for the parabolic equation [57,Thm 4.3] (see also [58]), we obtain that θ ∈ H 0 ((0, τ ); H 2 (Ω)) ∩ H 1 ((0, τ ); H 0 (Ω)) and an estimate of the form\nfor some constant C > 0 independent of > 0. From the energy estimate in Theorem 3.2, we also have that\nNow, under the geometric condition in Assumption 2.2, the acoustic problem for p enjoys the following observability property [41,42,43,44],\nwhere C > 0 is also independent of > 0. Therefore, combining the above three inequalities, for sufficiently small > 0 we conclude that there is C > 0 such that p 0 2 H 1 (Ω) ≤ C p 2 H 1 ((0,τ );H 0 (Γ)) , This concludes the proof."},{"title":"Reconstruction Algorithm","content":"In this section we explicitly recover the initial acoustic profile p 0 in terms of the boundary measurements Mp 0 = p| (0,τ )×Γ . This is accomplished by using the observability properties obtained in the previous section leading to the invertibility of the normal operator (M * M).\nIn order to obtain an applicable expression for the operator M * , in this section we state the dual or adjoint problem associated with (1)-(4). This is equivalent to constructing the well-known Hilbert Uniqueness Method (HUM) for control of partial differential equations. See [61,41] for details. Throughout, we assume that > 0 is sufficiently small for Theorem 2.3 to apply. This adjoint problem is to find a weak solution (ψ, ξ) for the following IBVP,\nfor a given η ∈ H 0 ((0, τ ) × Γ) (extended as zero on (0, τ ) × ∂Ω \\ Γ). Notice that this problem is solved backwards in time with vanishing Cauchy data at time t = τ . Since this problem is well-posed [57], we obtain the following definition.\nDefinition 4.1. Let S be the mapping η → −∂ t ψ| t=0 , where ψ is the solution of (11)- (14) for the provided η.\nIntegrating by parts the terms of equations (1)-(2) against (ψ, ξ), where (ψ, ξ) is the solution of (11)- (14), we easily obtain that\nHence, invoking duality and the Riesz representation, we have that M * = S : ((0, τ ); H 0 (Γ)) → H 1 (Ω) given in Definition 4.1 is surjective. Therefore, for any φ ∈ H 1 (Ω), there exists a boundary control η ∈ H 1 ((0, τ ); H 0 (Γ)) such that the solution (ψ, ξ) of (11)- (14) satisfies\nAmong all such boundary controls, there exists η min which is uniquely determined by φ as the minimum norm control and satisfies the following stability condition\nfor some constant C > 0. As a consequence, the mapping φ → η min defines a bounded control operator C : H 1 (Ω) → H 1 ((0, τ ); H 0 (Γ)), and this operator is given by\nLet (ψ, ξ) be the solution of (11)- (14) with η = Cφ and φ ∈ H 1 (Ω) arbitrary. Then by construction, Cφ H 1 ((0,τ );H 0 (Γ)) , for all φ ∈ H 1 (Ω), which implies that the unknown initial condition p 0 is explicitly recovered as follows,\nwhere C * is the adjoint of the control operator C : H 1 (Ω) → H 1 ((0, τ ); H 0 (Γ)) defined in Theorem 4.2. The reconstruction algorithm is based on the identity (15) and an iterative algorithm to approximate the action of C * . This algorithm is based on the following points provided by Theorem 4.2 (cf. [61,41]): (2) For ζ ∈ H 1 (Ω), the solution to (M * M)φ = ζ can be approximated using the conjugate gradient method.\nNow we proceed to describe how the action of C * can be approximated using the conjugate gradient method. See [62,Sect. 4.6] for a standard description of the conjugate gradient method in a Hilbert space setting. For sake of completeness, we describe the inversion of a generic equation (M * M)φ = ζ. Let φ 0 be an initial guess for the true solution φ * . Define r 0 = ζ − (M * M)φ 0 as the initial residue and s 0 = r 0 . For k ≥ 0, define\n.\nSince the operator (M * M) : H 1 (Ω) → H 1 (Ω) is bounded and coercive, then there are positive constants m and M such that\n. The conjugate gradient iterates can be shown to converge as follows (see [62,Sect. 4.6] and references therein),\nNotice that at each iteration, one must apply the operator (M * M) which amounts to solve the problem (1)-(4) (under Assumption 2.1) followed by solving the adjoint problem (11)- (14). In practice, this can be approximated using numerical methods for PDEs. However, depending on the method of choice, there are intrinsic complications that may prevent a convergence estimate such as (16) from being satisfied in the limit as the discretization is refined. We shall not elaborate any further on these complications as they lie outside of the scope of this paper. For details on these numerical issues we refer to [41,63,64,65,66] and references therein. In this paper, we adopted the two-grid approach described in [65,41] using second order finite difference methods."},{"title":"Numerical Results","content":"Now we present some numerical results to illustrate the performance of the reconstruction algorithm described in Section 4. We implemented a numerical solver for the governing system (1)-(4) and its adjoint (11)-(14) based on second order finite differences. To avoid spurious numerical instabilities, we adopted the two-grid approach described in [65,41]. We worked in R 2 where the domain Ω was taken as the unit-square. The initial profile p 0 corresponds the Shepp-Logan phantom.\nWe present two examples. One with constant wave speed c(x) ≡ 1, and the other with variable wave speed c = c(x) defined below. In both cases, we used the following parameters: impedance γ(x) = c −1 (x) over the boundary of Ω, thermal diffusivity α = 0.01, and coupling parameter = 0.1. The observability time was chosen to be τ = 2 which is enough for more than 99% of the energy contained in the initial profile to dissipate or leave the domain through the boundary when the wave speed is constant.\nWe shall compare the results from the proposed algorithm against the results from purely acoustic time-reversal. The latter is accomplished by producing measurements using the thermoacoustic forward solver M, and then back-propagating the boundary measurements in a purely acoustic medium ( = 0), that is, by ignoring the thermodynamic attenuation. See details in [26,16] for the purely acoustic time-reversal approach. The acoustic time-reversal is approximated using the same finite difference method. The initial guess for the conjugate gradient algorithm is the approximate solution obtained from the purely acoustic time-reversal algorithm. Although the proposed reconstruction algorithm has been described in the H 1 (Ω) setting, a similar study could be performed in the H 0 (Ω) setting where the inner-products in the conjugate gradient algorithm would need to be understood appropriately. In this section, we present results from the implementation both in the H 1 (Ω) and H 0 (Ω) formulations."},{"title":"Constant wave speed","content":"For the first example where c ≡ 1, Figure 1 displays the exact initial profile and the reconstructions. The relative errors in the H 1 (Ω) and H 0 (Ω) formulations are reported in Table 2 for the first few iterations of the conjugate gradient algorithm. We notice that by ignoring the thermodynamic attenuation in the purely acoustic timereversal reconstruction (Iter = 0), the edges in the Shepp-Logan phantom are blurred considerably. Some of the sharpness is recovered by accounting for the attenuation in the proposed algorithm even after a single iteration. Table 2.\nConstant wave speed example. Relative error at each iteration of the conjugate gradient method described in Section 4. Iter = 0 corresponds to the initial guess given by a purely acoustic time-reversal algorithm."},{"title":"Variable wave speed","content":"For the second example we have selected a variable wave speed defined as a layer of higher speed surrounding the smaller ellipses in the Shepp-Logan phantom. The actual profile is illustrated in the top-right panel of Figure 2. The relative errors in the H 1 (Ω) and H 0 (Ω) formulations are reported in Table 3 for the first few iterations of the conjugate gradient algorithm. Again, as shown in the lower panels of Figure 2, we see great improvements over the purely acoustic time-reversal reconstruction. We highlight the ability in capturing the jump discontinuities and the reduction of the artifacts introduced by ignoring the attenuation."},{"title":"Conclusion","content":"We have presented a PAT/TAT model based on thermoelasticity. The thermoelastic coupling accounts for how pressure changes can induce temperture changes in a body and vice versa. The coupling between temperture and deformation is a fundamental feature of PAT/TAT. The current literature dealing with PAT/TAT only considers one side of the thermoelastic interaction (the photoacoustic effect). By considering both effects simultaneously we account for a natural attenuation phenomenon of the acoustic signal. We related the thermoelastic model of PAT/TAT with boundary observavility for the thermoelastic system. We showed uniqueness and stability of recovering the initial profile of the pressure from boundary data. The initial profile was reconstructed using a conjugate gradient method. The recovery analysis of the inital wave profile works under a geometric control assumption on the wave speed. The performance of the reconstruction method is very similar in both the constant and variable wave speed scenarios.\nThe coupling uniteless parameter of the thermoelastic model is approximately between 0.05 and 0.1 as obtained from Table 1. Theorem 2.3 requires to be sufficiently small. Given that in PAT/TAT the thermodynamic interaction is small, such a condition on is reasonable. Nonethelesss, it might be possible to remove this condition by using Carleman estimates for the coupled thermoelastic system (e.g., [53,54,43])."},{"title":"Theorem 2. 3 (","content":""},{"title":"( 1 )","content":""},{"title":"Figure 1 .","content":""},{"title":"Figure 2 .","content":""},{"title":"Table 1 .","content":""},{"title":"Table 3 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Photoacoustic tomography is an imaging technique that takes advantage of the highcontrast exhibited by optical absorption and the high-resolution properties of wideband acoustic waves in soft biological tissue. Details concerning this type of imaging modalities are found in [1,2,3,4,5,6,7,8,9,10,11]. Qualitative photoacoustic imaging consists of recovering an initial pressure profile from acoustic measurements acquired on the boundary of a region of interest. The successful transformation of boundary measurements into the sought interior pressure profile requires mathematical algorithms arXiv:1602.01872v1 [math.AP] 4 Feb 2016 that have been studied by numerous researchers [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28].\\nThere have been recent efforts to incorporate acoustic attenuation in the modeling of photoacoustic tomography (PAT) and thermoacoustic tomography (TAT). See [29,30,31,32,33,34,35,36,37,38,18,39] and references therein. Most of these results aim at modeling attenuation in the frequency domain to account for dissipation and dispersion. In this paper, however, we adopt a model where the propagation of acoustic waves is thermodynamically coupled to the diffusion of heat. The photoacoustic effect, on which PAT is physically based, consists of two transformations of energy. First, electromagnetic energy is absorbed and transformed into heat. Second, there is a conversion of heat into mechanical energy due to thermal expansivity of the tissues. Concerning this second step, due to the thermodynamic interaction between temperature and pressure, the reverse transformation of energy also takes place. Since heat diffuses, this process attenuates the mechanical energy of the acoustic waves. See [40,Ch. 8] for an introduction to thermoelasticity in biomechanics. We claim that this type of acoustic attenuation should be naturally considered in PAT because PAT itself is based on the thermo-elastic interaction.\\nWe realize that, in mathematical terms, the PAT problem coincides with a problem of boundary observability, which is one of the central concepts of control theory for partial differential equations [41,42,43,44]. We have already employed similar tools to address the PAT problem in an enclosure [26] and other related problems [45,46]. The objective of this paper is to constructively employ the tools of observability for hyperbolic equations together with certain regularity properties of parabolic equations to solve the PAT problem in the presence of thermodynamic attenuation. For the thermoelastic system, there is a series of works on establishing exact, approximate and null controllability or observability estimates [47,48,49,50,51,52,53,54,43]. In these works, however, either the boundary condition or the distribution of control/observation is not of the type we need to model the PAT problem. Therefore, we modify some ideas provided by these references to seek a solution for PAT in the weak coupling regime. Although it might be possible to use Carleman estimates from [53,54,43] to treat the strongly coupled system, we refrain from doing so because the thermoelastic coupling in PAT is known to be relatively weak. Hence, we claim that the results of this paper are sufficient for the nature of PAT in biological tissues.\"\n },\n {\n \"title\": \"Mathematical Formulation and Main Results\",\n \"content\": \"In this paper we study the photoacoustic tomography problem in the presence thermodynamic attenuation. This is modeled by the linear equations of elasticity coupled with thermal diffusivity [40]. Let Ω ⊂ R n be a smooth bounded connected domain with boundary ∂Ω where n ≥ 2. The propagation of thermoelastic waves in isotropic media is governed by the following system [40,Ch. 8],\\nfor the displacement u and where θ denotes the deviation from the reference temperature θ ref . Also, ρ is the mass density, λ and µ are the Lamé coefficients, α denotes the thermal diffusivity and c p is the specific heat at constant pressure. The thermoelastic coupling is given by βK where β is the coefficient of volumetric thermal expansion and K is a constant reference bulk modulus. In soft biological tissue, λ µ which implies that K/(λ + 2µ) ≈ 1. For the moment let us assume that λ, µ and ρ are constants in Ω. Later, we will drop this assumption.\\nSince photoacoustic imaging is primarily concerned with the compressional waves, we define the pressure p = −(λ + 2µ) div u and the square of the compressional wave speed c 2 = (λ + 2µ)/ρ and proceed to obtain a scalar model for the thermoacoustic waves. Simultaneously, we seek to reveal the strength of the thermoelastic coupling by writing the governing equations in unitless form. Let L be a characteristic length scale of the domain Ω (such as its diameter). Let the characteristic time scale be given by\\nis the square of a reference wave speed. We define the following unitless variables and parameters: Pressurep = p/K and temperatureθ = θ/θ ref .\\nLengthx = x/L and timet = t/T . Parameters:ĉ 2 = c 2 T 2 /L 2 ,α = αT /L 2 , σ = K/(θ ref ρc p ) and = βθ ref .\\nThe unitless coupling parameter > 0 is introduced to analyze the case where is sufficiently small. This is valid for small coefficient of thermal expansion β. The unitless product G = σ is known as the Grüneisen coefficient. Table 1 displays rough estimates for the values of these physical parameters for soft biological tissues. We obtain that 0.05 0.1 and 0.5 σ 10. For notational convenience, we assume that σ = 1. This presents no impediment to the theory as we could easily treat the case σ > 0. At this point, to alleviate the notation, we also drop the caret to denote the unitless quantities. The unitless scalar governing system then becomes,\\nWe take (1)-(2) as the starting point for the modeling of this problem. While the equivalence of the thermoelastic system and equations (1)-(2) only holds when λ, µ and ρ are constants, we will consider c(x) to be a sufficiently smooth function in Ω. This follows the common practice of considering a variable wave speed to model heterogeneous media, even when the wave equation is not in divergence form. The system (1)- (2) is augmented by the following initial and boundary conditions,\\nHere, γ : ∂Ω → [0, ∞) denotes the acoustic impedance coefficient at the boundary ∂Ω. Physically, γ = 0 models an acoustically hard surface (such as reflectors) and γ → ∞ approximates an acoustically soft boundary. In general we allow γ to vary on the boundary ∂Ω to model the inhomogeneous nature of an enclosing surface and the interface with sensors or air. The length of the observation window of time is given by τ < ∞ which is defined below. In (4), the symbol ∂ ν denotes the outward normal derivative at the boundary ∂Ω. All the coefficients in (1)-(4) are assumed to be sufficiently smooth.\\nConcerning the initial conditions, it is common in the modeling of photoacoustic tomography to assume the following. These two expressions are respectively valid when pressure relaxation and thermal diffusion are negligible in the very short lapse of heat deposition from the optical source. These conditions can be achieved in biological tissues by using nanosecond optical pulses [1,2,3,12,5,6,4,55,56]. It turns out that the assumption p 1 = 0 is not needed from the mathematical point of view because it is possible to recover both terms of the initial Cauchy data for the acoustic field. On the other hand, the assumption θ 0 = p 0 is mathematically crucial because it removes an important degree of freedom in the analysis. It would not be possible to recover, in stable manner, an independent initial condition for the thermal field. This is a well-known consequence of the smoothing effect of the heat equation.\\nIn order to consider partial measurements and the presence of acoustic reflectors, we divide the boundary as the disjoint union ∂Ω = Γ ∪ (∂Ω \\\\ Γ) where Γ is the portion where we make observations of the acoustic field. As reviewed in the next section, the forward problem (1)-(4) has a unique solution, and we can define the measurement map given by\\nwhere p is the solution of (1)-(4) with initial conditions satisfying Assumption 2.1. The goal of the photoacoustic tomography problem is to find the initial profile p 0 from knowledge of Mp 0 . This is a challenging problem with intricate dependencies between the domain Ω, the partial boundary Γ, the wave speed c and the time interval (0, τ ). The admissible dependencies are made precise by a sophisticated assumption of geometric character. Following Bardos, Lebeau and Rauch [42], we assume that our problem enjoys the geometric control condition for the Riemannian manifold (Ω, c −2 δ ij ) with only a portion Γ of the boundary ∂Ω being accessible for observation. We assume that Γ is a smooth open domain relative to ∂Ω. This geometric assumption is the following.\\nAssumption 2.2 (Geometric Condition). There exists τ < ∞ such that any geodesic ray, originating from any point in Ω at t = 0, eventually reaches Γ in a non-diffractive manner (after possible geometrical reflections on ∂Ω \\\\ Γ) before time t = τ . We refer to τ as the controllability or observability time.\\nThe main theoretical result of this paper is the following theorem.\\nfor all p 0 ∈ H 1 (Ω), and some constant C > 0.\\nIn addition to this theoretical result, we also propose a convergent iterative reconstruction algorithm. This algorithm is described in Section 4.\"\n },\n {\n \"title\": \"Proof of the Main Result\",\n \"content\": \"In order to properly analyze the inverse problem we must first state some mathematical properties of the initial boundary value problem (1)-(4). Our guiding references are [57,58,59,60]. We use the following definition of energy for the thermoacoustic system,\\nIt is convenient to seek a space that is complete under the energy norm. Since p = const. and θ = const. is a non-trivial solution to the governing equations, the sought space should rule out nonzero constant solutions. This is achieved by considering only solutions such that\\nThis is motivated by the fact that the left-hand sides of (6)- (7) are indeed independent of time provided that p and θ solve the governing equations. Therefore, it is only needed to require (6)-(7) at time t = 0. Then the appropriate energy space is given by\\nNotice that H is a closed subspace of H 1 (Ω) × H 0 (Ω) × H 1 (Ω). So it is complete under the norm of H 1 (Ω) × H 0 (Ω) × H 1 (Ω) as well as under the energy norm. The problem (1)-(4) for (p, ∂ t p, θ) is well-posed on the space H and the energy is non-increasing. We state this in the form of a lemma. The proof follows from the standard analysis of partial differential equations and semigroup theory [57,58,59,60].\\nLemma 3.1. Given (p 0 , p 1 , θ 0 ) ∈ H, the unique weak solution of (1)-(4) satisfies (p, ∂ t p, θ) ∈ H for t ≥ 0, and p ∈ C k ([0, τ ]; H 1−k (Ω)) for k = 0, 1 and θ ∈ C([0, τ ]; H 1 (Ω)). Moreover, E(t) ≤ E(0) for all t ≥ 0 and all ≥ 0. In fact (due to parabolic regularity) the energy E ∈ H 1 (0, τ ) and\\nThe well-posedness and regularity implications of Theorem 3.2 also apply to arbitrary initial conditions (p 0 , p 1 , θ 0 ) ∈ H 1 (Ω) × H 0 (Ω) × H 1 (Ω). By virtue of linearity, we can decompose the initial conditions as follows,\\nTherefore, the evolution of the triplet (p 0 , p 1 , θ 0 ) ∈ H 1 (Ω) × H 0 (Ω) × H 1 (Ω) can be decomposed into the evolution of the initial condition (p 0 − p 0,const , p 1 , θ 0 − θ 0,const ) in H plus a time-independent solution given by (p 0,const , 0, θ 0,const ) for t ≥ 0. Notice that the energy of this particular solution is zero because it is constant both in space and time. This leads to the following result using Lemma 3.1 and the decomposition described above.\\nTheorem 3.2 (Forward well-posedness). Given (p 0 , p 1 , θ 0 ) ∈ H 1 (Ω) × H 0 (Ω) × H 1 (Ω), the unique weak solution of (1)-(4) satisfies p ∈ C k ([0, τ ]; H 1−k (Ω)) for k = 0, 1 and θ ∈ C([0, τ ]; H 1 (Ω)). Moreover, E(t) ≤ E(0) for all t ≥ 0 and all ≥ 0.\\nNow we proceed to prove the main result.\\nProof of Theorem 2.3. Consider the problem (1)-(4) for p 0 ∈ H 1 (Ω) and p 1 = 0 and θ 0 = p 0 (Assumption 2.1). From Theorem 3.2 we have that the solution to this problem satisfies p ∈ C k ([0, τ ]; H 1−k (Ω)) for k = 0, 1 and θ ∈ C([0, τ ]; H 1 (Ω)). Now notice that θ has initial condition in H 1 (Ω), homogeneous Neumann boundary condition, and a forcing term ∂ t p ∈ C([0, τ ]; H 0 (Ω)) ⊂ H 0 ((0, τ ); H 0 (Ω)). Hence, from regularity theory for the parabolic equation [57,Thm 4.3] (see also [58]), we obtain that θ ∈ H 0 ((0, τ ); H 2 (Ω)) ∩ H 1 ((0, τ ); H 0 (Ω)) and an estimate of the form\\nfor some constant C > 0 independent of > 0. From the energy estimate in Theorem 3.2, we also have that\\nNow, under the geometric condition in Assumption 2.2, the acoustic problem for p enjoys the following observability property [41,42,43,44],\\nwhere C > 0 is also independent of > 0. Therefore, combining the above three inequalities, for sufficiently small > 0 we conclude that there is C > 0 such that p 0 2 H 1 (Ω) ≤ C p 2 H 1 ((0,τ );H 0 (Γ)) , This concludes the proof.\"\n },\n {\n \"title\": \"Reconstruction Algorithm\",\n \"content\": \"In this section we explicitly recover the initial acoustic profile p 0 in terms of the boundary measurements Mp 0 = p| (0,τ )×Γ . This is accomplished by using the observability properties obtained in the previous section leading to the invertibility of the normal operator (M * M).\\nIn order to obtain an applicable expression for the operator M * , in this section we state the dual or adjoint problem associated with (1)-(4). This is equivalent to constructing the well-known Hilbert Uniqueness Method (HUM) for control of partial differential equations. See [61,41] for details. Throughout, we assume that > 0 is sufficiently small for Theorem 2.3 to apply. This adjoint problem is to find a weak solution (ψ, ξ) for the following IBVP,\\nfor a given η ∈ H 0 ((0, τ ) × Γ) (extended as zero on (0, τ ) × ∂Ω \\\\ Γ). Notice that this problem is solved backwards in time with vanishing Cauchy data at time t = τ . Since this problem is well-posed [57], we obtain the following definition.\\nDefinition 4.1. Let S be the mapping η → −∂ t ψ| t=0 , where ψ is the solution of (11)- (14) for the provided η.\\nIntegrating by parts the terms of equations (1)-(2) against (ψ, ξ), where (ψ, ξ) is the solution of (11)- (14), we easily obtain that\\nHence, invoking duality and the Riesz representation, we have that M * = S : ((0, τ ); H 0 (Γ)) → H 1 (Ω) given in Definition 4.1 is surjective. Therefore, for any φ ∈ H 1 (Ω), there exists a boundary control η ∈ H 1 ((0, τ ); H 0 (Γ)) such that the solution (ψ, ξ) of (11)- (14) satisfies\\nAmong all such boundary controls, there exists η min which is uniquely determined by φ as the minimum norm control and satisfies the following stability condition\\nfor some constant C > 0. As a consequence, the mapping φ → η min defines a bounded control operator C : H 1 (Ω) → H 1 ((0, τ ); H 0 (Γ)), and this operator is given by\\nLet (ψ, ξ) be the solution of (11)- (14) with η = Cφ and φ ∈ H 1 (Ω) arbitrary. Then by construction, Cφ H 1 ((0,τ );H 0 (Γ)) , for all φ ∈ H 1 (Ω), which implies that the unknown initial condition p 0 is explicitly recovered as follows,\\nwhere C * is the adjoint of the control operator C : H 1 (Ω) → H 1 ((0, τ ); H 0 (Γ)) defined in Theorem 4.2. The reconstruction algorithm is based on the identity (15) and an iterative algorithm to approximate the action of C * . This algorithm is based on the following points provided by Theorem 4.2 (cf. [61,41]): (2) For ζ ∈ H 1 (Ω), the solution to (M * M)φ = ζ can be approximated using the conjugate gradient method.\\nNow we proceed to describe how the action of C * can be approximated using the conjugate gradient method. See [62,Sect. 4.6] for a standard description of the conjugate gradient method in a Hilbert space setting. For sake of completeness, we describe the inversion of a generic equation (M * M)φ = ζ. Let φ 0 be an initial guess for the true solution φ * . Define r 0 = ζ − (M * M)φ 0 as the initial residue and s 0 = r 0 . For k ≥ 0, define\\n.\\nSince the operator (M * M) : H 1 (Ω) → H 1 (Ω) is bounded and coercive, then there are positive constants m and M such that\\n. The conjugate gradient iterates can be shown to converge as follows (see [62,Sect. 4.6] and references therein),\\nNotice that at each iteration, one must apply the operator (M * M) which amounts to solve the problem (1)-(4) (under Assumption 2.1) followed by solving the adjoint problem (11)- (14). In practice, this can be approximated using numerical methods for PDEs. However, depending on the method of choice, there are intrinsic complications that may prevent a convergence estimate such as (16) from being satisfied in the limit as the discretization is refined. We shall not elaborate any further on these complications as they lie outside of the scope of this paper. For details on these numerical issues we refer to [41,63,64,65,66] and references therein. In this paper, we adopted the two-grid approach described in [65,41] using second order finite difference methods.\"\n },\n {\n \"title\": \"Numerical Results\",\n \"content\": \"Now we present some numerical results to illustrate the performance of the reconstruction algorithm described in Section 4. We implemented a numerical solver for the governing system (1)-(4) and its adjoint (11)-(14) based on second order finite differences. To avoid spurious numerical instabilities, we adopted the two-grid approach described in [65,41]. We worked in R 2 where the domain Ω was taken as the unit-square. The initial profile p 0 corresponds the Shepp-Logan phantom.\\nWe present two examples. One with constant wave speed c(x) ≡ 1, and the other with variable wave speed c = c(x) defined below. In both cases, we used the following parameters: impedance γ(x) = c −1 (x) over the boundary of Ω, thermal diffusivity α = 0.01, and coupling parameter = 0.1. The observability time was chosen to be τ = 2 which is enough for more than 99% of the energy contained in the initial profile to dissipate or leave the domain through the boundary when the wave speed is constant.\\nWe shall compare the results from the proposed algorithm against the results from purely acoustic time-reversal. The latter is accomplished by producing measurements using the thermoacoustic forward solver M, and then back-propagating the boundary measurements in a purely acoustic medium ( = 0), that is, by ignoring the thermodynamic attenuation. See details in [26,16] for the purely acoustic time-reversal approach. The acoustic time-reversal is approximated using the same finite difference method. The initial guess for the conjugate gradient algorithm is the approximate solution obtained from the purely acoustic time-reversal algorithm. Although the proposed reconstruction algorithm has been described in the H 1 (Ω) setting, a similar study could be performed in the H 0 (Ω) setting where the inner-products in the conjugate gradient algorithm would need to be understood appropriately. In this section, we present results from the implementation both in the H 1 (Ω) and H 0 (Ω) formulations.\"\n },\n {\n \"title\": \"Constant wave speed\",\n \"content\": \"For the first example where c ≡ 1, Figure 1 displays the exact initial profile and the reconstructions. The relative errors in the H 1 (Ω) and H 0 (Ω) formulations are reported in Table 2 for the first few iterations of the conjugate gradient algorithm. We notice that by ignoring the thermodynamic attenuation in the purely acoustic timereversal reconstruction (Iter = 0), the edges in the Shepp-Logan phantom are blurred considerably. Some of the sharpness is recovered by accounting for the attenuation in the proposed algorithm even after a single iteration. Table 2.\\nConstant wave speed example. Relative error at each iteration of the conjugate gradient method described in Section 4. Iter = 0 corresponds to the initial guess given by a purely acoustic time-reversal algorithm.\"\n },\n {\n \"title\": \"Variable wave speed\",\n \"content\": \"For the second example we have selected a variable wave speed defined as a layer of higher speed surrounding the smaller ellipses in the Shepp-Logan phantom. The actual profile is illustrated in the top-right panel of Figure 2. The relative errors in the H 1 (Ω) and H 0 (Ω) formulations are reported in Table 3 for the first few iterations of the conjugate gradient algorithm. Again, as shown in the lower panels of Figure 2, we see great improvements over the purely acoustic time-reversal reconstruction. We highlight the ability in capturing the jump discontinuities and the reduction of the artifacts introduced by ignoring the attenuation.\"\n },\n {\n \"title\": \"Conclusion\",\n \"content\": \"We have presented a PAT/TAT model based on thermoelasticity. The thermoelastic coupling accounts for how pressure changes can induce temperture changes in a body and vice versa. The coupling between temperture and deformation is a fundamental feature of PAT/TAT. The current literature dealing with PAT/TAT only considers one side of the thermoelastic interaction (the photoacoustic effect). By considering both effects simultaneously we account for a natural attenuation phenomenon of the acoustic signal. We related the thermoelastic model of PAT/TAT with boundary observavility for the thermoelastic system. We showed uniqueness and stability of recovering the initial profile of the pressure from boundary data. The initial profile was reconstructed using a conjugate gradient method. The recovery analysis of the inital wave profile works under a geometric control assumption on the wave speed. The performance of the reconstruction method is very similar in both the constant and variable wave speed scenarios.\\nThe coupling uniteless parameter of the thermoelastic model is approximately between 0.05 and 0.1 as obtained from Table 1. Theorem 2.3 requires to be sufficiently small. Given that in PAT/TAT the thermodynamic interaction is small, such a condition on is reasonable. Nonethelesss, it might be possible to remove this condition by using Carleman estimates for the coupled thermoelastic system (e.g., [53,54,43]).\"\n },\n {\n \"title\": \"Theorem 2. 3 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"( 1 )\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 3 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Assumption 2.1 (Rapid Deposition of Heat). The initial conditions (3) satisfy, p 1 = 0 and θ 0 = p 0 .","content":"Assumption 2.1 (Rapid Deposition of Heat). The initial conditions (3) satisfy, p 1 = 0 and θ 0 = p 0 ."},{"id":"","caption":"Main Result). Suppose that Assumptions 2.1 and 2.2 hold. For sufficiently small > 0, the map p 0 → Mp 0 is injective and satisfies a stability estimate of the form","content":"Theorem 2. 3 (\n3Main Result). Suppose that Assumptions 2.1 and 2.2 hold. For sufficiently small > 0, the map p 0 → Mp 0 is injective and satisfies a stability estimate of the form"},{"id":"","caption":"if we choose η = Mp 0 and use the estimate from Theorem 2.3, we obtain that, p 0 , (M * M)p 0 H 1 (Ω) (Ω) , for all p 0 ∈ H 1 (Ω) and some constant C > 0. Therefore, the operator (M * M) : H 1 (Ω) → H 1 (Ω) is coercive, and S : H 1((0, τ ); H 0 (Γ)) → H 1 (Ω) is surjective. With these results, we can establish the following controllability theorem.","content":"if we choose η = Mp 0 and use the estimate from Theorem 2.3, we obtain that, p 0 , (M * M)p 0 H 1 (Ω) (Ω) , for all p 0 ∈ H 1 (Ω) and some constant C > 0. Therefore, the operator (M * M) : H 1 (Ω) → H 1 (Ω) is coercive, and S : H 1((0, τ ); H 0 (Γ)) → H 1 (Ω) is surjective. With these results, we can establish the following controllability theorem."},{"id":"","caption":"Theorem 4.2 (Acoustic Control). Let the Geometric Condition 2.2 hold. For sufficiently small > 0, the operator S : H 1","content":"Theorem 4.2 (Acoustic Control). Let the Geometric Condition 2.2 hold. For sufficiently small > 0, the operator S : H 1"},{"id":"","caption":"The observability operator C * = (M * M) −1 M * , where (M * M) : H 1 (Ω) → H 1 (Ω) is coercive.","content":"( 1 )\n1The observability operator C * = (M * M) −1 M * , where (M * M) : H 1 (Ω) → H 1 (Ω) is coercive."},{"id":"","caption":"Exact initial acoustic profile (top-left), the reconstruction from purely acoustic time-reversal (top-right), and the reconstruction from the proposed algorithm described in Section 4 using 1 iteration (bottom-left) and 5 iterations (bottom-right).","content":"Figure 1 .\n1Exact initial acoustic profile (top-left), the reconstruction from purely acoustic time-reversal (top-right), and the reconstruction from the proposed algorithm described in Section 4 using 1 iteration (bottom-left) and 5 iterations (bottom-right)."},{"id":"","caption":"Exact initial acoustic profile (top-left), wave speed profile (top-right), the reconstruction from purely acoustic time-reversal (bottom-left) and the reconstruction from the proposed algorithm described in Section 4 using 5 iterations (bottom-right).","content":"Figure 2 .\n2Exact initial acoustic profile (top-left), wave speed profile (top-right), the reconstruction from purely acoustic time-reversal (bottom-left) and the reconstruction from the proposed algorithm described in Section 4 using 5 iterations (bottom-right)."},{"id":"","caption":"Physical parameters for the thermoacoustic coupling in typical soft biological tissues.","content":"Table 1 .\n1Physical parameters for the thermoacoustic coupling in typical soft biological tissues.Physical Parameter \nSymbol Range \nUnits \n\nBulk modulus \nK \n2000 -2500 10 6 Pa \nDensity \nρ \n900 -1100 \nKg / m 3 \nRef. Temperature \nθ ref \n290 -310 \nK \nCoeff. thermal expansion β \n200 -300 \n10 −6 / K \nSpecific heat \nc p \n500 -5000 \nJ / (Kg K)"},{"id":"","caption":"Variable wave speed example. Relative error at each iteration of the conjugate gradient method described in Section 4. Iter = 0 corresponds to the initial guess given by a purely acoustic time-reversal algorithm.Iter H 1 (Ω)-norm H 0 (Ω)-norm","content":"Table 3 .\n3Variable wave speed example. Relative error at each iteration of the conjugate gradient method described in Section 4. Iter = 0 corresponds to the initial guess given by a purely acoustic time-reversal algorithm.Iter H 1 (Ω)-norm H 0 (Ω)-norm0 \n55.4 % \n34.3 % \n1 \n24.2 % \n16.6 % \n2 \n15.5 % \n8.0 % \n3 \n11.8 % \n4.7 % \n4 \n10.2 % \n4.0 % \n5 \n9.7 % \n3.7 %"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Assumption 2.1 (Rapid Deposition of Heat). The initial conditions (3) satisfy, p 1 = 0 and θ 0 = p 0 .\",\n \"content\": \"Assumption 2.1 (Rapid Deposition of Heat). The initial conditions (3) satisfy, p 1 = 0 and θ 0 = p 0 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Main Result). Suppose that Assumptions 2.1 and 2.2 hold. For sufficiently small > 0, the map p 0 → Mp 0 is injective and satisfies a stability estimate of the form\",\n \"content\": \"Theorem 2. 3 (\\n3Main Result). Suppose that Assumptions 2.1 and 2.2 hold. For sufficiently small > 0, the map p 0 → Mp 0 is injective and satisfies a stability estimate of the form\"\n },\n {\n \"id\": \"\",\n \"caption\": \"if we choose η = Mp 0 and use the estimate from Theorem 2.3, we obtain that, p 0 , (M * M)p 0 H 1 (Ω) (Ω) , for all p 0 ∈ H 1 (Ω) and some constant C > 0. Therefore, the operator (M * M) : H 1 (Ω) → H 1 (Ω) is coercive, and S : H 1((0, τ ); H 0 (Γ)) → H 1 (Ω) is surjective. With these results, we can establish the following controllability theorem.\",\n \"content\": \"if we choose η = Mp 0 and use the estimate from Theorem 2.3, we obtain that, p 0 , (M * M)p 0 H 1 (Ω) (Ω) , for all p 0 ∈ H 1 (Ω) and some constant C > 0. Therefore, the operator (M * M) : H 1 (Ω) → H 1 (Ω) is coercive, and S : H 1((0, τ ); H 0 (Γ)) → H 1 (Ω) is surjective. With these results, we can establish the following controllability theorem.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Theorem 4.2 (Acoustic Control). Let the Geometric Condition 2.2 hold. For sufficiently small > 0, the operator S : H 1\",\n \"content\": \"Theorem 4.2 (Acoustic Control). Let the Geometric Condition 2.2 hold. For sufficiently small > 0, the operator S : H 1\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The observability operator C * = (M * M) −1 M * , where (M * M) : H 1 (Ω) → H 1 (Ω) is coercive.\",\n \"content\": \"( 1 )\\n1The observability operator C * = (M * M) −1 M * , where (M * M) : H 1 (Ω) → H 1 (Ω) is coercive.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Exact initial acoustic profile (top-left), the reconstruction from purely acoustic time-reversal (top-right), and the reconstruction from the proposed algorithm described in Section 4 using 1 iteration (bottom-left) and 5 iterations (bottom-right).\",\n \"content\": \"Figure 1 .\\n1Exact initial acoustic profile (top-left), the reconstruction from purely acoustic time-reversal (top-right), and the reconstruction from the proposed algorithm described in Section 4 using 1 iteration (bottom-left) and 5 iterations (bottom-right).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Exact initial acoustic profile (top-left), wave speed profile (top-right), the reconstruction from purely acoustic time-reversal (bottom-left) and the reconstruction from the proposed algorithm described in Section 4 using 5 iterations (bottom-right).\",\n \"content\": \"Figure 2 .\\n2Exact initial acoustic profile (top-left), wave speed profile (top-right), the reconstruction from purely acoustic time-reversal (bottom-left) and the reconstruction from the proposed algorithm described in Section 4 using 5 iterations (bottom-right).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Physical parameters for the thermoacoustic coupling in typical soft biological tissues.\",\n \"content\": \"Table 1 .\\n1Physical parameters for the thermoacoustic coupling in typical soft biological tissues.Physical Parameter \\nSymbol Range \\nUnits \\n\\nBulk modulus \\nK \\n2000 -2500 10 6 Pa \\nDensity \\nρ \\n900 -1100 \\nKg / m 3 \\nRef. Temperature \\nθ ref \\n290 -310 \\nK \\nCoeff. thermal expansion β \\n200 -300 \\n10 −6 / K \\nSpecific heat \\nc p \\n500 -5000 \\nJ / (Kg K)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Variable wave speed example. Relative error at each iteration of the conjugate gradient method described in Section 4. Iter = 0 corresponds to the initial guess given by a purely acoustic time-reversal algorithm.Iter H 1 (Ω)-norm H 0 (Ω)-norm\",\n \"content\": \"Table 3 .\\n3Variable wave speed example. Relative error at each iteration of the conjugate gradient method described in Section 4. Iter = 0 corresponds to the initial guess given by a purely acoustic time-reversal algorithm.Iter H 1 (Ω)-norm H 0 (Ω)-norm0 \\n55.4 % \\n34.3 % \\n1 \\n24.2 % \\n16.6 % \\n2 \\n15.5 % \\n8.0 % \\n3 \\n11.8 % \\n4.7 % \\n4 \\n10.2 % \\n4.0 % \\n5 \\n9.7 % \\n3.7 %\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Biomedical optics : principles and imaging. 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Springer New York, 2011. 1, 4\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Photoacoustic tomography: in vivo imaging from organelles to organs. L V Wang, S Hu, Science. 33560754L. V. Wang and S. Hu. Photoacoustic tomography: in vivo imaging from organelles to organs. Science, 335(6075):1458-1462, 2012. 1, 4\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Noninvasive laser-induced photoacoustic tomography for structural and functional in vivo imaging of the brain. X Wang, Y Pang, G Ku, X Xie, G Stoica, L V Wang, Nature biotechnology. 217X. Wang, Y. Pang, G. Ku, X. Xie, G. Stoica, and L. V. Wang. Noninvasive laser-induced photoacoustic tomography for structural and functional in vivo imaging of the brain. Nature biotechnology, 21(7):803-806, 2003. 1\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Imaging of hemoglobin oxygen saturation variations in single vessels in vivo using photoacoustic microscopy. H F Zhang, K Maslov, M Sivaramakrishnan, G Stoica, L V Wang, Applied Physics Letters. 905H. F. 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This dynamics works as follows: at every stage of the game a player is selected uniformly at random and she plays according to a noisy best-response where the noise level is tuned by a parameter β. Such a dynamics defines a family of ergodic Markov chains, indexed by β, over the set of strategy profiles. We believe that the stationary distribution of these Markov chains gives a meaningful description of the long-term behavior for systems whose agents are not completely rational.Our aim is twofold: On the one hand, we are interested in evaluating the performance of the game at equilibrium, i.e. the expected social welfare when the strategy profiles are random according to the stationary distribution. On the other hand, we want to estimate how long it takes, for a system starting at an arbitrary profile and running the logit dynamics, to get close to its stationary distribution; i.e., the mixing time of the chain.In this paper we study the stationary expected social welfare for the 3-player CK game [6], for 2-player coordination games, and for two simple n-player games. For all these games, we also give almost tight upper and lower bounds on the mixing time of logit dynamics. Our results show two different behaviors: in some games the mixing time depends exponentially on β, while for other games it can be upper bounded by a function independent of β.\n\nIntroduction\n\nThe evolution of a system is determined by its dynamics and complex systems are often described by looking at the equilibrium states induced by their dynamics. Once the system reaches an equilibrium state it stays there, thus equilibrium states describe the long-term behavior of the system. In this paper we are mainly interested in systems whose individual components are selfish agents. The state of a selfish system is fully described by a vector of strategies, each controlled by one agent, and each state assigns a payoff to each agent. The agents are selfish in the sense that they pick their strategy so to maximize their payoff, given the strategies of the other agents. Nash equilibrium is the classical notion of equilibrium for selfish systems and it corresponds to the equilibrium induced by the best-response dynamics. The observation that selfish systems are described by their equilibrium states (that is, by the Nash equilibria) has motivated the notions of Price of Anarchy [15] and Price of Stability [1] and the analysis of efficiency of selfish systems based on such notions.\n\nHowever, such analysis inherits some of the shortcomings of the concept of a Nash equilibrium. First of all, the best-response dynamics assumes that selfish agents have complete knowledge of the current state of the system; that is, they know the payoff associated with each of their possible choices and each of the strategies chosen by other agents. Instead, in most cases, agents have only approximate knowledge of the system state or they are not able to compute their best choice. Moreover, in presence of multiple equilibria, it is not clear which one of them will be reached by the system, as it may depend on the initial state: Price of Anarchy considers the worst-case equilibrium, whereas Price of Stability focuses on the best-case equilibrium. Finally, Nash equilibria are hard to compute [7,5] and thus for some systems it might take very long to reach a Nash equilibrium: in this case using equilibrium states to describe the system performance is not well justified. Rather, one would like to analyze the performance of a system by using a dynamics (and its related equilibrium notion) that has the following three properties:\n\n• The dynamics takes into account the fact that the system components might have a perturbed or noisy knowledge of the system;\n\n• For every system the equilibrium state exists and is unique;\n\n• The system reaches the equilibrium very quickly regardless of the starting state.\n\nIn this paper, we consider noisy best-response dynamics in which the behavior of the agents is described by a parameter β 0. The case β = 0 corresponds to agents picking their strategies completely at random (that is, the agents have no knowledge of the system) and the case β = ∞ corresponds to agents picking their strategies according to the best-response dynamics (in which the agents have full and complete knowledge of the system). The intermediate values of β correspond to agents that are roughly guided by the best-response dynamics but can make a sub-optimal choice due, for example, to bounded rationality of the agent or limited knowledge about the system: this sub-optimal behavior occurs with some probability that depends on β (and on the associated payoff).\n\nWe will study a specific noisy best-response dynamics for which the system evolves according to an ergodic Markov chain for all β 0. For these systems, it is natural to look at the stationary distribution (which is the equilibrium state of the Markov chain) and to analyze the expected social welfare (the sum of utility functions) of the system at that distribution. We stress that the noisy best-response dynamics well models agents that only have approximate or noisy knowledge of the system and that for ergodic Markov chains (such as the ones arising in our study) the stationary distribution is known to exist and to be unique. Moreover, to justify the use of the stationary distribution for analyzing the performance of the system, we will study how fast the Markov chain converges to the stationary distribution.\n\nRelated Works and Our Results. Several dynamics, besides the best-response dynamics, and several notions of equilibrium, besides Nash equilibria, have been considered to describe the evolution of a selfish system and to analyze its performance. See, for example, [11,21,20]. Equilibrium concepts based on the best-response. When the game does not possess a Pure Nash equilibrium, the best-response dynamics will eventually cycle over a set of states (in a Nash equilibrium the set is a singleton). These states are called sink equilibria [12]. Sink equilibria exist for all games and, in some contexts, they seem a better approximation of the real setting than mixed Nash equilibria. Unfortunately, sink equilibria share two undesirable properties with Nash equilibria: a game can have more that one sink equilibrium and sink equilibria seem hard to compute [9].\n\nOther notions of equilibrium state associated with best-response dynamics are the unit-recall equilibria and component-wise unit-recall equilibria (see [9]). However, we point out that the former does not always exist and that the latter imposes too strict limitations on the players. No-Regret Dynamics. Another broadly explored set of dynamics are the no-regret dynamics (see, for example, [11]). The regret of an user is the difference between the long-term average cost and the average cost of the best strategy in hindsight. In the no-regret dynamics the regret of every player after t steps is o(t) (sublinear with time). In [10,14] it is showed that the no-regret dynamics converges to the set of correlated equilibria. Note that the convergence is to the set of correlated equilibria and not to a specific correlated equilibrium. Our work. In this paper we consider a specific noisy best-response dynamics called the logit dynamics (see [3]) and we study its mixing time (that is, the time it takes to converge to the stationary distribution) and the stationary expected social welfare. Specifically,\n\n• We start by analyzing the logit dynamics for a simple 3-player linear congestion game (the CK game [6]) which exhibits the worst Price of Anarchy among linear congestion games. We show that the mixing time of the logit dynamics is upper bounded by a constant independent of β. Moreover, we show that the stationary expected social welfare is larger than the social welfare of the worst Nash equilibrium for all β;\n\n• We then analyze the 2 × 2 coordination games studied in [3]. Here we show that, under some conditions, the stationary expected social welfare is larger than the social welfare of the worst Nash equilibrium. We give upper and lower bounds on the mixing time exponential in β. We also observe that the same bounds apply to anti-coordination games;\n\n• Finally, we apply our analysis to two simple n-player games: the OR game and XOR game. We give upper and lower bounds on the mixing time: we show that the mixing time of the OR game can be upper bounded by a function independent of β, while the mixing time of the XOR game increases exponentially in β. We also prove that for β = O(log n) the mixing time is polynomial in n for both games.\n\nThe logit dynamics has been first studied by Blume [3] who showed that, for 2 × 2 coordination games, the long-term behavior of the Markov chain is concentrated in the risk dominant equilibrium (see [13]) for sufficiently large β. Ellison [8] studied different noisy best-response dynamics for coordination games assuming that interaction among players were described by a graph; that is, the utility of a player is determined only by the strategies of the adjacent players. Specifically, Ellison [8] studied interaction modeled by rings and showed that some large fraction of the players will eventually choose the risk dominant strategy. Similar results were obtained by Peyton Young [22] for the logit dynamics and for more general families of graphs. Montanari and Saberi [17] gave bounds on the hitting time (the expected time that the logit dynamics takes to reach a specific state) of the risk dominant equilibrium state in terms of some graph theoretic properties of the underlying interaction network. Asadpour and Saberi [2] studied the hitting time for a broader class of congestion games. We notice that none of [3,8,22] gave any bound on the convergence time to the risk dominant equilibrium. Montanari and Saberi [17] were the first to do so but their study focuses on the hitting time of a specific configuration and not on the convergence time to the stationary distribution. From a technical point of view, our work follows the lead of [3,8,22] and extends their technical findings by giving bounds on the mixing time of the Markov chain of the logit dynamics. We stress that previous results only proved that, for sufficiently large β, eventually the system concentrates around certain states without further quantifying the rate of convergence nor the asymptotic behaviour of the system for small values of β. Instead, we identify the stationary distribution of the logit dynamics as the global equilibrium and we evaluate the social welfare at stationarity and the time it takes the system to reach it (the mixing time) as explicit functions of β.\n\nWe choose to start our study from the class of coordination games considered in [3] and two simple n-player games (the OR game and the XOR game). We give nearly tight upper and lower bounds on the mixing time. Despite their game-theoretic simplicity, the analytical study of the mixing time of the logit dynamics for the two n-player games is far from trivial. We notice that the results in [17] cannot be used to derive upper bounds on the mixing time.\n\nFrom a more conceptual point of view, our work tries (similarly to [12,9,19]) to introduce a solution concept that well models the behavior of selfish agents, is uniquely defined for any game, and is quickly reached from any starting state. We propose the stationary distribution induced by the logit dynamics as a possible solution concept and exemplify its use in the analysis of the performance of some 2 × 2 games (as the ones considered in [3]), of games used to obtain tight bounds on the Price of Anarchy, and of two simple multi-player games.\n\nOrganization of the paper. In Section 2 we summarize some Markov chain notions that we will use throughout the paper. In Section 3 we formally describe the logit dynamics for strategic games. We also describe the coupling we will repeatedly use in the proofs of the upper bounds on mixing times. In Sections 4, 5, 6 and 7 we study the stationary expected social welfare and the mixing time of the logit dynamics for CK game, coordination games, the OR game, and the XOR game, respectively. Finally, in Section 8 we present conclusions and some open problems. Notation. We write S for the complementary set of a set S; we write |S| for its size. We use bold symbols for vectors; when x = (x 1 , . . . , x n ) ∈ {0, 1} n we write |x| for the number of 1s in x; i.e., |x| = |{i ∈ [n] :\n\nx i = 1}|. For two vectors x, y let H(x, y) = |{i ∈ [n] : x i = y i }| be their Hamming distance: we write x ∼ y if H(x, y) = 1. We use the standard game theoretic notation (x −i , y) to mean the vector obtained from x by replacing the i-th entry with y, i.e. (x −i , y) = (x 1 , . . . , x i−1 , y, x i+1 , . . . , x n ).\n\n\nMarkov chains summary and notation\n\nWe summarize the main tools we use to bound the mixing time of Markov chains (for a complete description of such tools see, for example, Chapters 5.2, 7.2, 12.2 and 14.2 of [16]. We refer the reader to [16] also for notational conventions).\n\nConsider a Markov chain M with finite state space Ω and transition matrix P . It is a classical result that for an irreducible and aperiodic Markov chain 1 there exists an unique stationary distribution π over Ω; that is, a distribution π on Ω such that π · P = π. The total variation distance µ − ν TV between two probability distributions µ and ν on Ω is defined as\nµ − ν TV = max A⊂Ω |µ(A) − ν(A)| .\nAn irreducible and aperiodic Markov chain M converges to its stationary distribution π; specifically, there exists 1 > α > 0 such that\nd(t) α t ,\nwhere\nd(t) = max x∈Ω P t (x, ·) − π TV\nand P t (x, ·) is the distribution at time t of the Markov chain starting at x. For 1/2 > ε > 0, the mixing time is defined as t mix (ε) = min{t ∈ N : d(t) ε}.\n\nIt is usual to set ε = 1/4 or ε = 1/2e. If not explicitly specified, when we write t mix we mean t mix (1/4). Observe that t mix (ǫ) ⌈log 2 ǫ −1 ⌉t mix .\n\nCoupling. A coupling of two probability distributions µ and ν on Ω is a pair of random variables (X, Y ) defined on Ω × Ω such that the marginal distribution of X is µ and the marginal distribution of Y is ν. A coupling of a Markov chain M with transition matrix P is a process (X t , Y t ) ∞ t=0 with the property that both X t and Y t are Markov chains with transition matrix P . When the two coupled chains start at (X 0 , Y 0 ) = (x, y), we write P x,y (·) and E x,y [·] for the probability and the expectation on the space where the two chains are both defined. We denote by τ couple the first time the two chains meet; that is,\nτ couple = min{t : X t = Y t } .\nWe will consider only couplings of Markov chains with the property that for s τ couple , it holds X s = Y s . The following theorem can be used to give an upper bound on t mix (see, for example, Corollary 5.3 in [16]).\n\nTheorem 1 (Coupling) Let M be a Markov chain with state space Ω and transition matrix P . For each pair of states x, y ∈ Ω consider a coupling (X t , Y t ) of M with starting states X 0 = x and Y 0 = y. Then\nd(t) max x,y∈Ω P x,y (τ couple > t) .\nSometimes it is difficult to specify a coupling and to analyze the coupling time τ couple for each pair of starting states x and y. The Path Coupling theorem says that it is sufficient to define a coupling only for pairs of Markov chains starting from adjacent states and an upper bound on the mixing time can be obtained if each of these couplings contracts their distance on average. More precisely, consider a Markov chain M with state space Ω and transition matrix P ; let G = (Ω, E) be a connected graph and let w : E → R be a function assigning weights to the edges such that w(e) 1 for every edge e ∈ E; for x, y ∈ Ω, we denote by ρ(x, y) the weight of the shortest path in G between x and y. The following theorem holds.\n\nTheorem 2 (Path Coupling [4]) Suppose that for every edge {x, y} ∈ E a coupling (X t , Y t ) of M with X 0 = x and Y 0 = y exists such that E x,y [ρ(X 1 , Y 1 )] e −α · w({x, y}) for some α > 0. Then\nt mix (ε) log(diam(G)) + log(1/ε) α where diam(G) is the (weighted) diameter of G.\nSpectral techniques. A Markov chain M with state space Ω and transition matrix P is said reversible if for all x, y ∈ Ω, it holds that π(x) · P (x, y) = π(y) · P (y, x).\n\nThe eigenvalues of the transition matrix P of a reversible Markov chain M can be used to obtain upper and lower bounds on the mixing time. Observe that all the eigenvalues of any transition matrix P have absolute value at most 1, λ = 1 is an eigenvalue, and for irreducible and aperiodic chains, −1 is not an eigenvalue. The relaxation time t rel of a Markov chain M is defined as\nt rel = 1 1 − λ ⋆\nwhere λ ⋆ is the largest absolute value among eigenvalues other than 1, λ ⋆ = max{|λ| : λ is an eigenvalue of P, λ = 1} .\n\nObserve that, for M reversible, irreducible and aperiodic, 0 λ ⋆ < 1 and thus t rel is positive and finite. We have the following theorem (see, for example, Theorems 12.3 and 12.4 in [16]).\n\n\nTheorem 3 (Relaxation time)\n\nLet P be the transition matrix of a reversible, irreducible, and aperiodic Markov chain with state space Ω and stationary distribution π. Then\n(t rel − 1) log 1 2ǫ t mix (ǫ) log 1 ǫπ min t rel\nwhere π min = min x∈Ω π(x).\n\nLower bound. We will use the following theorem to derive our lower bounds (see, for example, Theorem 7.3 in [16]).\n\nTheorem 4 (Bottleneck ratio) Let M = {X t : t ∈ N} be an irreducible and aperiodic Markov chain with finite state space Ω, transition matrix P , and stationary distribution π. Let S ⊆ Ω be any set with π(S) 1/2. Then the mixing time is\nt mix (ε) 1 − 2ǫ 2Φ(S)\nwhere Φ(S) = Q(S, S) π(S) and Q(S, S) = x∈S, y∈S π(x)P (x, y). \n\n\nThe model and the problem\nσ i (y | x) = 1 T i (x) e βui(x−i,y)(1)\nwhere x ∈ S 1 × · · · × S n is the strategy profile played at the current time step, T i (x) = z∈Si e βui(x−i,z) is the normalizing factor, and β 0.\n\nParameter β is called inverse noise of the system, indeed from (1) it is easy to see that, for β = 0 player i selects her strategy uniformly at random, for β > 0 the probability is biased toward strategies promising higher payoffs, and for β → ∞ player i chooses her best response strategy (if more than one best response is available, she chooses uniformly at random one of them). Moreover observe that probability σ i (y | x) does not depend on the strategy x i currently adopted by player i. The above dynamics defines a Markov chain with the set of strategy profiles as state space, and where the transition probability from profile x = (x 1 , . . . , x n ) to profile y = (y 1 , . . . , y n ) is zero if the H(x, y) 2 and it is 1 n σ i (y i | x) if the two profiles differ exactly at player i. More formally, we can define the logit dynamics as follows.\n\nDefinition 5 (Logit dynamics [3]) Let G = ([n], S, U) be a strategic game and let β 0. The logit dynamics for G is the Markov chain M β = {X t : t ∈ N} with state space Ω = S 1 ×· · ·×S n and transition matrix\nP (x, y) = 1 n · σ i (y i | x), if y −i = x −i and y i = x i ; n i=1 σ i (y i | x), if y = x; 0, otherwise; (2) where σ i (y i | x) is defined in (1).\n\nProperties. Logit dynamics enjoys some interesting properties:\n\nErgodicity. It is easy to see that the logit dynamics is irreducible and aperiodic. Indeed, let x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ) be two profiles and let (z 0 , . . . , z n ) be a path of profiles where z 0 = x, z n = y and z i = (y 1 , . . . , y i , x i+1 , . . . x n ) for i = 1, . . . , n − 1. The probability that the chain starting at x is in y after n steps is P n (x, y) = P n (z 0 , z n ) P n−1 (z 0 , z n−1 )P (z n−1 , z n ) and recursively\nP n (x, y) n i=1 P (z i−1 , z i ) > 0\nwhere the last inequality follows from (2) because, for all i = 1, . . . , n, the Hamming distance between z i−1 and z i is at most 1. Hence there is a unique stationary distribution π and, for every starting profile x, the distribution of the chain P t (x, ·) converges to π in total variation as t tends to infinity. \ni (x) := u i (x) + c i for all x\nwe get a new gameG = ([n], S,Ũ ) but the same logit dynamics. Indeed, according to (1), the probability player i chooses strategy y when the game is at profile x is\nσ i (y | x) = e βũi(x−i,y) z∈Si e βũi(x−i,z) = 1 z∈Si e β[ũi(x−i,z)−ũi(x−i,y)] = 1 z∈Si e β[ui(x−i,z)−ui(x−i,y)] = σ i (y | x) .\nNoise changes under utility rescaling. While translations of utilities do not affect logit dynamics, a rescaling of the utility functions for the same constant α > 0 changes the inverse noise from β to α · β. Indeed, if for every player i and every profile x we set\nu i (x) := α · u i (x) , from (1) we haveσ i (y | x) = e βũi(x−i,y) z∈Si e βũi(x−i,z) = e αβui(x−i,y)\nz∈Si e αβui(x−i,z) . Notice that, unlike translations constants, we here must have the same rescaling constant α for all utility functions.\n\nPotential Games. A game G = ([n], S, U) is said a (exact) potential game if a function Φ : S 1 × · · · × S n → R exists such that, for every player i and for every pair of profiles x and y that differ only at position i, it holds that\nu i (x) − u i (y) = Φ(x) − Φ(y). It is easy to see that, if G = ([n], S, U)\nis a potential game with potential function Φ, then the Markov chain given by (2) is reversible and its stationary distribution is the Gibbs measure\nπ(x) = 1 Z e βΦ(x)(3)\nwhere Z = y∈S1×···×Sn e βΦ(y) is the normalizing constant. Except for the Matching Pennies example in Subsection 3.1, all the games we analyze in this paper are potential games. Logit dynamics vs Glauber dynamics. When G is a potential game, the logit dynamics is equivalent to the well-studied Glauber dynamics. For state space Ω = S 1 × · · · × S n and probability distribution µ over Ω, the Glauber dynamics for µ proceeds as follows: From profile x ∈ Ω, pick a player i ∈ [n] uniformly at random and update her strategy at y ∈ S i with probability µ conditioned on the other players being at\nx −i , i.e. µ(y | x −i ) = µ(x −i , y) z∈Si µ(x −i , z)\n.\n\nIt is easy to see that the Markov chain defined by the Glauber dynamics is irreducible, aperiodic, and reversible with stationary distribution µ. When G = ([n], S, U) is a potential game with potential function Φ, the logit dynamics defines the same Markov chain as the Glauber dynamics for the Gibbs distribution π in (3). Indeed, in that case we have\nσ i (y | x) = e βui(x−i,y) z∈Si e βui(x−i,z) = 1 z∈Si e β(ui(x−i,z)−ui(x−i,y)) = 1 z∈Si e β(Φ(x−i,z)−Φ(x−i,y)) = e βΦ(x−i,y) z∈Si e βΦ(x−i,z) = π(x −i , y) z∈Si π(x −i , z)\n.\n\nHence, logit dynamics for potential games and Glauber dynamics for Gibbs distributions are two ways of looking at the same Markov chains: in the former case the dynamics is derived from the potential function, in the latter case from the stationary distribution. However, observe that, if G is not a potential game and π is the stationary distribution of the logit dynamics for G, in general the Glauber dynamics for π is different from the logit dynamics (see, for example, the Matching Pennies case in Subsection 3.1). Due to the analogies between logit and Glauber dynamics, we will sometimes adopt the terminology used by physicists to indicate the quantities involved; in particular we will call parameter β the inverse noise or inverse temperature and we will call partition function the normalizing constant Z of the Gibbs distribution (3).\n\nStationary expected social welfare and mixing time. Let W : S 1 × · · · × S n −→ R be a social welfare function (in this paper we assume that W is simply the sum of all the utility functions W (x) = n i=1 u i (x), but clearly any other function of interest can be analysed). We study the stationary expected social welfare, i.e. the expectation of W when the strategy profiles are random according to the stationary distribution π of the Markov chain,\nE π [W ] = x∈S1×···×Sn W (x)π(x)\nSince the Markov chain defined in (2) is irreducible and aperiodic, from every initial profile x the distribution P t (x, ·) of chain X t starting at x will eventually converge to π as t tends to infinity. We will be interested in bounding how long it takes to get close to the stationary distribution, that is the mixing time of the Markov chain. In the next subsection we illustrate the goals of our work with two simple examples.\n\n\nTwo simple examples: Matching Pennies and a Stairs game\n\nMatching Pennies. Consider the classical Matching Pennies game. We write the utility functions in the standard bimatrix form.\nH T H +1, −1 −1, +1 T −1, +1 +1, −1 (4)\nAccording to (1), the update probabilities for the logit dynamics are, for every x ∈ {H, T }\nσ 1 (H | (x, H)) = σ 1 (T | (x, T )) = 1 1+e −2β = σ 2 (T | (H, x)) = σ 2 (H | (T, x)) , σ 1 (T | (x, H)) = σ 1 (H | (x, T )) = 1 1+e 2β = σ 2 (H | (H, x)) = σ 2 (T | (T, x)) .\nHence the transition matrix (see (2)) is\nP = HH HT T H T T HH 1/2 b/2 (1 − b)/2 0 HT (1 − b)/2 1/2 0 b/2 T H b/2 0 1/2 (1 − b)/2 T T 0 (1 − b)/2 b/2 1/2 \nwhere, for readability sake, we named b = 1 1+e −2β . Since every column of the matrix adds up to 1, the uniform distribution π over the set of strategy profiles is the stationary distribution for the logit dynamics. The stationary expected social welfare is thus 0 for every inverse noise β.\n\nAs for the mixing time, it is easy to see that it is upper bounded by a constant independent of β. Indeed, a direct calculation shows that, for every x ∈ {HH, HT, T H, T T } and for every β 0 it holds that\nP 3 (x, ·) − π tv 7 16 < 1 2 .\nA stairs game. One of the main techniques used to give upper bounds on the mixing time of Markov chains is the coupling technique (see Theorem 1). In the following example we use it to upper bound the mixing time of the logit dynamics for a simple game. Let G be a potential game where every player has two strategies, say upstairs (or 1) and downstairs (or 0), and the potential of a profile x ∈ {0, 1} n is the number of players that are upstairs, i.e. Φ(x) = |x|.\n\nNotice that the logit dynamics (and thus the stationary distribution and the mixing time) is completely defined by the potential function, while if we wanted to evaluate the stationary expected social welfare we would need to specify the utility functions.\n\nThe partition function is\nZ(β) = x∈{0,1} n e β|x| = n k=0 n k e βk = 1 + e β n .\nSo the stationary distribution is\nπ(x) = e β|x| (1 + e β ) n .\nAs for the mixing time, we can use the coupling technique as follows: observe that the probability of playing strategy 1 (or equivalently strategy 0), for the player selected for the update, is independent of the current strategies of the other players. Indeed, according to (1), for every x it holds that\nσ i (1 | x) = e βui(x−i,1) e βui(x−i,1) + e βui(x−i,0) = 1 1 + e β(ui(x−i,0)−ui(x−i,1)) = 1 1 + e β(Φ(x−i,0)−Φ(x−i,1)) = 1 1 + e β(|x−i|−(|x−i|+1)) = 1 1 + e −β .\nWe can define a coupling of two Markov chains starting at two different profiles as follows: choose i ∈ [n] uniformly at random and perform the same update at player i in both chains 2 . When every player has been chosen at least once the two chains have coalesced. From the coupon collector's argument, it takes O(n log n) to have that, with probability at least 3/4, all players have been chosen at least once. By applying Theorem 1 we have that the mixing time is O(n log n).\n\nIn the above examples, it turned out that the mixing time of the logit dynamics can be upper bounded by functions that do not depend on the inverse noise β. As we shall see in the next sections, this is not always the case. Moreover, the analysis of the mixing time is usually far from trivial.\n\n\nDescription of the Coupling\n\nThroughout the paper we will use the coupling and path-coupling techniques (see Theorem 1 and Theorem 2) to give upper bounds on mixing times. Since we will use the same coupling idea in several proofs, we describe it here and we will refer to this description when we will need it.\n\nConsider an n-player 2-strategy game G and let us rename 0 and 1 the strategies of every player. For every pair of strategy profiles x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ {0, 1} n we define a coupling (X 1 , Y 1 ) of two copies of the Markov chain with transition matrix P defined in (2) for which X 0 = x and Y 0 = y.\n\nThe coupling proceeds as follows: first, pick a player i uniformly at random; then, update the strategies x i and y i of player i in the two chains, by setting\n(x i , y i ) = (0, 0), with probability min{σ i (0 | x), σ i (0 | y)} ; (1, 1), with probability min{σ i (1 | x), σ i (1 | y)} ; (0, 1), with probability σ i (0 | x) − min{σ i (0 | x), σ i (0 | y)} ; (1, 0), with probability σ i (1 | x) − min{σ i (1 | x), σ i (1 | y)} .\nThree easy observations are in order: if σ i (0 | x) = σ i (0 | y) and player i is chosen, then, after the update, we have x i = y i ; for every player i, at most one of the updates (x i , y i ) = (0, 1) and (x i , y i ) = (1, 0) has positive probability; if i is chosen for update, then the marginal distributions of x i and y i agree with σ i (· | x) and σ i (· | y) respectively, indeed, for b ∈ {0, 1}, the probability that\nx i = b is min {σ i (b | x), σ i (b | y)} + σ i (b | x) − min{σ i (b | x), σ i (b | y)} = σ i (b | x) ,\nand the probability that\ny i = b is min{σ i (b | x), σ i (b | y)} + σ i (1 − b | x) − min{σ i (1 − b | x), σ i (1 − b | y)} = = min{σ i (b | x), σ i (b | y) + (1 − σ i (b | x)) − (1 − max{σ i (b | x), σ i (b | y)}) = σ i (b | y) .\nWe define G = (Ω, E) as the Hamming graph of the game, where Ω = {0, 1} n is the set of strategy profiles, and two profiles x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ Ω are adjacent if they differ only for the strategy of one player, i.e.\n\n{x, y} ∈ E ⇐⇒ x ∼ y .\n\nFor the path coupling technique, the coupling described above is applied only to pairs of adjacent starting profiles.\n\n\nA 3-player congestion game\n\nIn this section we analyze the CK game, a simple 3-player linear congestion game introduced in [6]. This game is interesting because it highlights the weakness of the Price of Anarchy notion for the logit dynamics. Indeed, the CK game exhibits the worst Price of Anarchy with respect to the average social welfare among all linear congestion games with 3 or more players. But, as we shall see soon, the stationary expected social welfare of the logit dynamics is always larger than the social welfare of the worst Nash equilibrium and, for large enough β, players spend most of the time in the best Nash equilibrium. Moreover, we will show that the mixing time of the logit dynamics can be bounded independently from β: that is, the stationary distribution guarantees a good social welfare and it is quickly reached by the system. Let us now describe the CK game. We have 3 players and 6 facilities divided into two sets: G = {g 1 , g 2 , g 3 } and H = {h 1 , h 2 , h 3 }. Player i ∈ {0, 1, 2} has two strategies: Strategy \"0\" consists in selecting facilities (g i , h i ); Strategy \"1\" consists in selecting facilities (g i+1 , h i−1 , h i+1 ) (index arithmetic is modulo 3). The cost of a facility is the number of players choosing such facility, and the welfare of a player is minus the sum of the costs of the facilities she selected. It easy to see that this game has two pure Nash equilibria: the solution where every player plays strategy 0 (each player pays 2, which is optimal), and the solution where every player plays strategy 1 (each player pays 5). The game is a congestion game, and thus, by [18], it is also a potential game and its potential function is:\nΦ(x) = j∈G∪H Lx(j) i=1 i ,\nwhere L x (j) is the number of players using facility j in configuration x.\n\nStationary expected social welfare. It is easy to see that the update probabilities given by the logit dynamics for this game (see Equation (1)) only depend on the number of players playing strategy 1 and not on which player is actually playing that strategy. In particular, we have that, from a profile x, the player i, if selected for update, plays strategy 0 with the following probabilities:\nσ i (0 | |x −i | = 0) = 1 1 + e −4β , σ i (0 | |x −i | = 1) = 1 1 + e −2β , σ i (0 | |x −i | = 2) = 1 2 ,(6)\nand strategy 1 with the remaining probabilities. Next theorem evaluates the stationary expected social welfare for this game.\n\nTheorem 6 (Expected social welfare) The stationary expected social welfare E π [W ] of the logit dynamics for the CK game is\nE π [W ] = − 6 + 39e −4β + 63e −6β 1 + 3e −4β + 4e −6β .\nProof. We notice that two profiles with the same number of players playing strategy 1 have both the same potential (and, by Equation (3), the same stationary distribution) and the same social welfare. Thus, π(x) = π[k] and W (x) = W [k] for a profile x such that |x| = k, where\nπ[0] = e −6β Z(β) , π[1] = e −10β Z(β) , π[2] = π[3] = e −12β Z(β) ,\nwhere Z(β) = e −6β + 3e −10β + 4e −12β , and\nW [0] = −6 , W [1] = −13 , W [2] = −16 , W [3] = −15 .\nHence, the stationary expected social welfare is\nE π [W ] = − 6 · e −6β + 3 · 13 · e −10β + (3 · 16 + 15) · e −12β e −6β + 3e −10β + 4e −12β = − 6 + 39e −4β + 63e −6β 1 + 3e −4β + 4e −6β .\nNotice that for β = 0 we have E π [W ] = −27/2, which is better than the social welfare of the worst Nash equilibrium. This means that, even if each player selects her strategy at random, the logit dynamics drives the system to a random profile whose expectation to be better than the worst Nash equilibrium. We also observe that E π [W ] increases with β and thus the long-term behavior of the logit dynamics gives a better social welfare than the worst Nash equilibrium for any β 0. Moreover, the stationary expected social welfare approaches the optimal social welfare as β tends to ∞.\n\nMixing time. Now we study the mixing time of the logit dynamics for the CK game and we show that it is bounded by a constant for any β 0. The proof will use Coupling Theorem (see Theorem 1).\n\nTheorem 7 (Mixing time) There exists a constant τ such that the mixing time t mix of the logit dynamics of the CK game is upper bounded by τ for every β 0.\n\nProof. First, we notice that the update probabilities given in Equation (6) imply that\n∀ i, ∀ x, ∀ β, σ i (0 | x) 1/2 .(7)\nLet X t and Y t be two copies of the logit dynamics for the CK game, starting in x and y respectively, coupled as described in Section 3.2. It is easy to check that, by Equation (7), the player selected for update, chooses strategy 0 in both chain with probability at least 1/2. Finally, we bound the probability that after three steps the two coupled chains coalesce: it is at least as large as the probability that we choose three different players and all of them play strategy 0 at their turn, i.e.\nP x,y (X 3 = Y 3 ) 1 2 · 1 3 · 1 6 = 1 36 .\nSince this bound holds for every starting pair (x, y), we have that the probability the two chains have not yet coalesced after 3t steps is\nP x,y (X 3t = Y 3t ) 1 − 1 36 t e −t/36 .\nThe thesis follows from the Theorem 1.\n\n\nTwo player games\n\nIn this section we analyse the performance of the logit dynamics for 2 × 2 coordination games (the same class studied in [3]) and 2 × 2 anti-coordination games.\n\nCoordination games. Coordination Games are two-player games in which the players have an advantage in selecting the same strategy. These games are often used to model the spread of a new technology [22]: two players have to decide whether to adopt or not a new technology. We assume that the players would prefer choosing the same technology as the other one and that choosing the new technology is at most as risky as choosing the old one. We name 0 the NEW strategy and 1 the OLD strategy. The game is formally described by the following payoff matrix\n0 1 0 (a, a) (c, d) 1 (d, c) (b, b)(8)\nWe assume that a > d and b > c (meaning that they prefer to coordinate) and that a − d b − c (meaning that for each player strategy 0 is at most as risky as strategy 1). Notice that we do not make any assumption on the relation between a and b. For convenience sake we name ∆ := a − d and δ := b − c .\n\nIt is easy to see that this game is a potential game and the following function is an exact potential for it:\nΦ(0, 0) = ∆ Φ(0, 1) = Φ(1, 0) = 0 Φ(1, 1) = δ.\nThis game has two pure Nash equilibria: (0, 0), where each player has utility a, and (1, 1), where each player has utility b. As d + c < a + b, the social welfare is maximized at one of the two equilibria. We analyse the mixing time of the logit dynamics for 2 × 2 coordination games and compute its stationary expected social welfare as a function of β.\n\nStationary expected social welfare. The logit dynamics for the coordination game defined by the payoffs in Table 8 establishes that, from a profile x, player i selected for update plays according to the following probability distribution (see Equation (1)):\nσ i (0 | x −i = 0) = 1 1+e −∆β , σ i (1 | x −i = 0) = 1 1+e ∆β , σ i (0 | x −i = 1) = 1 1+e δβ , σ i (1 | x −i = 1) = 1 1+e −δβ .\nNext theorem bounds the stationary expected social welfare E π [W ] obtained by the logit dynamics and gives conditions for which E π [W ] is better than the social welfare SW N of the worst Nash equilibrium.\n\nTheorem 8 (Expected social welfare) The stationary expected social welfare E π [W ] of the logit dynamics for the coordination game in Table 8 is\nE π [W ] = 2 · a + be −(∆−δ)β + (c + d)e −∆β 1 + e −(∆−δ)β + 2e −∆β . Moreover, if a = b then E π [W ] SW N for β sufficiently large.\nProof. The stationary distribution π of the logit dynamics is\nπ(0, 0) = e ∆β Z(β) π(1, 1) = e δβ Z(β) π(0, 1) = π(1, 0) = 1 Z(β)\nwhere Z(β) = e ∆β + e δβ + 2.\n\nSince E π [W ] = 2 · E π [u i ], we compute the expected utility E π [u i ] of player i at the stationary distribution,\nE π [u i ] = x∈{0,1} 2 u i (x)π(x) = ae ∆β + be δβ + c + d e ∆β + e δβ + 2 = a + be −(∆−δ)β + (c + d)e −∆β 1 + e −(∆−δ)β + 2e −∆β .\nThus, if a > b and β max 0,\n1 ∆ log 2b−c−d a−b , we have E π [W ] − SW N = 2 · a + be −(∆−δ)β + (c + d)e −∆β 1 + e −(∆−δ)β + 2e −∆β − 2b = 2 · (a − b) − (2b − c − d)e −∆β 1 + e −(∆−δ)β + 2e −∆β 0 . Similarly, we obtain E π [W ] − SW N 0 if b > a and β max 0, 1 δ log 2a−c−d b−a .\nMixing time. Now we study the mixing time of the logit dynamics for coordination games and we show that it is exponential in β and in the minimum potential difference between adjacent profiles.\n\nTheorem 9 (Mixing Time) The mixing time of the logit dynamics for the coordination game of Table 8 is Θ e δβ for every β 0.\n\nProof. Upper bound: We apply the Path Coupling technique (see Theorem 2) with the Hamming graph defined in (5) and all the edge-weights set to 1. Let x and y be two profiles differing only for the player j and consider the coupling defined in Section 3.2 for this pair of profiles. Now we bound the expected distance of the two coupled chains after one step.\n\nWe denote by b i (x, y) the probability that both chains perform the same update given that player i has been selected for strategy update. Clearly, b i (x, y) = 1 for i = j, while for i = j, we have\nb i (x, y) = min{σ i (0 | x), σ i (0 | y)} + min{σ i (1 | x), σ i (1 | y)} = 1 1 + e ∆β + 1 1 + e δβ .\nFor sake of readability we set p = 1 1 + e ∆β and q = 1 1 + e δβ . and thus b i (x, y) = p + q. To compute E x,y [ρ(X 1 , Y 1 )], we observe that the logit dynamics chooses player j with probability 1/2. In this case, as b j (x, y) = 1, the coupling updates both chains in the same way, resulting in X 1 = Y 1 . Similarly, player i = j is chosen for strategy update with probability 1/2. In this case, with probability b i (x, y) the coupling performs the same update in both chains resulting in ρ(X 1 , Y 1 ) = 1. Instead with probability 1 − b i (x, y), the coupling performs different updates on the chains resulting in ρ(X 1 , Y 1 ) = 2. Therefore we have,\nE x,y [ρ(X 1 , Y 1 )] = 1 2 b i (x, y) + 2 · 1 2 (1 − b i (x, y)) = 1 − 1 2 b i (x, y) = 1 − 1 2 (p + q) e − 1 2 (p+q) .\nFrom Theorem 2, with α = 1 2 (p + q) and diam(Ω) = 2, it follows that It is easy to see that the second largest eigenvalue of P is λ ⋆ = (1−p)+(1−q) 2 , hence the relaxation time is t rel = 1/(1 − λ ⋆ ) = 2 p+q , and for the mixing time we have\nt mix (ε) 2 (log 2 + log(1/ε)) p + q = 1 p + q log 4 ε 2 .t mix (ε) (t rel − 1) log 1 2ε = 2 − (p + q) p + q log 1 2ε 1 p + q log 1 2ε .(9)\nIn the last inequality we used that p and q are both smaller than 1/2. Finally, the theorem follows by observing that\n1 p + q = 1 1 1+e ∆β + 1 1+e δβ = Θ e δβ .\nNotice that, if we used the relaxation time to upper bound the mixing time (see Theorem 3) we would get a non-tight bound, hence in the above proof we had to resort to the path coupling for the upper bound.\n\nAnti-coordination games. Very similar results can be obtained for anti-coordination games. These are two-player games in which the players have an advantage in selecting different strategies. They model many settings where there is a common and exclusive resource: two players have to decide whether to use the resource or to drop it. If they both try to use it, then a deadlock occurs and this is bad for both players. Usually, these games are described by a payoff matrix like the one in Table 8, where we assume that d > a and c > b and that d − a c − b. Notice that Nash Equilibria of this game are unfair, as one player has utility max{c, d} and the other min{c, d}.\n\nFor the logit dynamics, we have that, for all β, the stationary expected social welfare is worse than the one guaranteed by a Nash equilibrium. On the other hand, for sufficiently large β we have that the expected utility of a player is always better than min{c, d}: that is, in the logit dynamics each player expects to gain more than in the worst Nash equilibrium. Moreover, the stationary distribution is a fair equilibrium, since every player has the same expected utility. As for the coordination games, the mixing time is exponential in β and in the minimum potential difference between adjacent profiles.\n\n\nThe OR game\n\nIn this section we consider the following simple n-player potential game that we here call OR game. Every player has two strategies, say {0, 1}, and each player pays the OR of the strategies of all players (including herself). More formally, the utility function of player i ∈ [n] is\nu i (x) = 0, if x = 0 ; −1,\notherwise.\n\nNotice that the OR game has 2 n − n Nash equilibria. The only profiles that are not Nash equilibria are the n profiles with exactly one player playing 1. Nash equilibrium 0 has social welfare 0, while all the others have social welfare −n.\n\nIn Theorem 10 we show that the stationary expected social welfare is always better than the social welfare of the worst Nash equilibrium, and it is significantly better for large β. Unfortunately, in Theorem 11 we show that if β is large enough to guarantee a good stationary expected social welfare, then the time needed to get close to the stationary distribution is exponential in n. Finally, in Theorem 12 we give upper bounds on the mixing time showing that if β is relatively small then the mixing time is polynomial in n, while for large β the upper bound is exponential in n and it is almost-tight with the lower bound. Despite the simplicity of the game, the analysis of the mixing time is far from trivial.\n\nTheorem 10 (Expected social welfare) The stationary expected social welfare of the logit dynamics for the OR game is E π [W ] = −αn where α = α(n, β) = (2 n −1)e −β 1+(2 n −1)e −β .\n\nProof. Observe that the OR game is a potential game with exact potential Φ where Φ(0) = 0 and Φ(x) = −1 for every x = 0. Hence the stationary distribution is\nπ(x) = 1/Z, if x = 0 ; e −β /Z, if x = 0 ;\nwhere the normalizing factor is Z = 1 + (2 n − 1)e −β . The expected social welfare is thus\nE π [W ] = x∈{0,1} n W (x)π(x) = −n · (2 n − 1)e −β 1 + (2 n − 1)e −β .\nIn the next theorem we show that the mixing time can be polynomial in n only if β c log n for some constant c.\n\nTheorem 11 (Lower bound on mixing time) The mixing time of the logit dynamics for the OR game is\n1. Ω(e β ) if β < log(2 n − 1); 2. Ω(2 n ) if β > log(2 n − 1).\nProof. Consider the set S ⊆ {0, 1} n containing only the state 0 = (0, . . . , 0) and observe that π(0) 1/2 for β log(2 n − 1). The bottleneck ratio is\nΦ(0) = 1 π(0) y∈{0,1} n π(0)P (0, y) = y∈{0,1} n : |y|=1 P (0, y) = n · 1 n 1 1 + e β .\nHence, by applying Theorem 4, the mixing time is\nt mix 1 Φ(0) = 1 + e β .\nIf β > log(2 n − 1) instead we consider the set R ⊆ {0, 1} n containing all states except state 0, and observe that\nπ(R) = 1 Z (2 n − 1)e −β = (2 n − 1)e −β 1 + (2 n − 1)e −β .\nand π(R) 1/2 for β > log(2 n − 1). It holds that\nQ(R, R) = x∈R π(x)P (x, 0) = x∈{0,1} n : |x|=1 π(x)P (x, 0) = n e −β Z 1 n 1 1 + e −β .\nThe bottleneck ratio is\nΦ(R) = Q(R, R) π(R) = Z (2 n − 1)e −β e −β Z 1 1 + e −β = 1 (2 n − 1)(1 + e −β ) < 1 2 n − 1 .\nHence, by applying Theorem 4, the mixing time is\nt mix 1 Φ(R) > 2 n − 1 .\nIn Proof. We apply the path coupling technique (see Theorem 2 in Section 2) with the Hamming graph defined in (5). Let x, y ∈ {0, 1} n be two profiles differing only at player j ∈ [n] and, without loss of generality, let us assume |x| = k − 1 and |y| = k for some k = 1, . . . , n. We set the weight of edge {x, y} depending only on k, i.e. ℓ(x, y) = δ k where δ k 1 will be chosen later. Consider the coupling defined in Subsection 3.2. Now we evaluate the expected distance after one step E x,y [ρ(X 1 , Y 1 )] of the two coupled chains (X t , Y t ) starting at (x, y). Let i be the player chosen for the update. Observe that if i = j, i.e. if we update the player where x and y are different (this holds with probability 1/n), then the distance after one step is zero, otherwise we distinguish four cases depending on the value of k. Case k = 1: In this case profile x is all zeros and profile y has only one 1 and the length of edge {x, y} is ℓ(x, y) = δ 1 . When choosing a player i = j (this happens with probability (n − 1)/n), at the next step the two chains will be at distance δ 1 (if in both chains player i chooses strategy 0, and this holds with probability min{σ i (0 | x), σ i (0 | y)}), or at distance δ 2 (if in both chains player i chooses strategy 1, and this holds with probability min{σ i (1 | x), σ i (1 | y)}), or at distance δ 1 + δ 2 (if player i chooses strategy 0 in chain X 1 and strategy 1 in chain Y 1 , and this holds with the remaining probability). Notice that, from the definition of the coupling, it will never happen that player i chooses strategy 1 in chain X 1 and strategy 0 in chain Y 1 , indeed we have that\nmin{σ i (0 | x), σ i (0 | y)} = σ i (0 | y) = 1 2 and min{σ i (1 | x), σ i (1 | y)} = σ i (1 | x) = 1 1 + e β . (10)\nHence the expected distance after one step is When player h is chosen (this holds with probability 1/n) we have that σ h (s | x) and σ h (s | y) for s = 0, 1 are the same as in (10). At the next step the two chains will be at distance δ 2 (if player h stays at strategy 1 in both chains), or at distance δ 1 (if player h chooses strategy 0 in both chains), or at distance δ 1 + δ 2 (if player h stays at strategy 0 in chain X 1 and chooses strategy 1 in chain Y 1 ).\nE x,y [ρ(X 1 , Y 1 )] = n − 1 n 1 2 δ 1 + 1 1 + e β δ 2 + 1 − 1 2 − 1 1 + e β (δ 1 + δ 2 ) = n − 1 n δ 1 1 + e −β +\nWhen a player i / ∈ {h, j} is chosen (this holds with probability (n − 2)/n) we have that σ i (0, x) = σ i (1, x) = σ i (0, y) = σ i (1, y) = 1/2. Thus in this case the two coupled chains always perform the same choice at player i, and at the next step they will be at distance δ 2 (if player i stays at strategy 0 in both chains) or at distance δ 3 (if player i chooses strategy 1 in both chains). Hence the expected distance after one step is\nE x,y [ρ(X 1 , Y 1 )] = 1 n 1 2 δ 1 + 1 1 + e β δ 2 + 1 − 1 2 − 1 1 + e β (δ 1 + δ 2 ) + n − 2 n 1 2 δ 2 + 1 2 δ 3 = 1 2n 2 1 + e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 .(12)\nCase 3 k n − 1: When a player i = j is chosen such that x i = y i = 1 (this holds with probability (k − 1)/n) then at the next step the two chains will be at distance δ k (if i stays at strategy 1) or at distance δ k−1 (if i moves to strategy 0). When a player i = j is chosen such that x i = y i = 0 (this holds with probability (n − k)/n) then at the next step the two chains will be at distance δ k (if i chooses to stay at strategy 0) or at distance δ k+1 (if i chooses to move to strategy 0). Hence the expected distance after one step is\nE x,y [ρ(X 1 , Y 1 )] = k − 1 n 1 2 δ k + 1 2 δ k−1 + n − k n 1 2 δ k + 1 2 δ k+1 = 1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) .(13)\nCase k = n: When a player i = j is chosen, then at the next step the two chains will be at distance δ n or at distance δ n−1 . Hence the expected distance after one step is\nE x,y [ρ(X 1 , Y 1 )] = n − 1 n 1 2 δ n + 1 2 δ n−1 = n − 1 2n (δ n + δ n−1 ) .(14)\nIn order to apply Theorem 2 we now have to show that it is possible to choose the edge weights δ 1 , . . . , δ n and a parameter α > 0 such that\nn−1 n δ1 1+e −β + δ2 2 δ 1 e −α ,1 2n 2\n1+e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 δ 2 e −α , 1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) δ k e −α , for k = 3, . . . , n − 1 ,\nn−1 2n (δ n + δ n−1 ) δ n e −α .(15)\nFor different values of β, we make different choices for α and for the weights δ k . For clarity's sake we split the proof in three different lemmas. We denote by δ max the largest δ k . In Lemma 13 we show that Inequalities (15) are satisfied for every value of β by choosing the weights as follows\nδ k = 1 2 [(n − 1)δ 2 + 1], if k = 1; n−k k δ k+1 + 1, if 2 k n − 1; 1, if k = n;\nand by setting α = 1/(2nδ max ). From Corollary 18, we have δ max = O( √ n2 n ). Observe that the diameter of the Hamming graph is n i=1 δ i nδ max , hence from Theorem 2 we obtain t mix = O(n 5/2 2 n ). In Lemma 14 we show that, if β < (1−ǫ) log n for an arbitrarily small constant ε > 0, Inequalities (15) are satisfied, for sufficiently large n, by choosing weights δ 1 = n 1−ε , δ 2 = 4/3, δ 3 = · · · = δ n = 1, and α = 1/n.. In this case the diameter is O(n) and, by Theorem 2, t mix = O(n log n).\n\nIn Lemma 15 we show that, Inequalities (15) are satisfied by choosing weights as follows\nδ k = 1+e −β 2 a1 b1 δ 2 + 1 , if k = 1; a k b k δ k+1 + 1, if 2 k n − 1; 1, if k = n;\nwhere a 1 = n − 1 and b 1 = ne −β + 1 and, for every k = 2, . . . , n − 1\na k = (n − k)b k−1 b k = (n + 1)b k−1 − (k − 1)a k−1 ;\nand by setting α = 1/(2nδ max ). From Corollary 21 it follows that, if β c log n for a constant c ∈ N, we have that δ max = O(n c+2 ) and the diameter of the Hamming graph is O(n c+3 ). Thus, by Theorem 2 it follows that t mix = O(n c+3 log n).\n\n\nTechnical lemmas\n\nIn this section we prove the technical lemmas needed for completing the proof of Theorem 12.\n\nLemma 13 Let δ 1 , . . . , δ n be as follows\nδ k = 1 2 [(n − 1)δ 2 + 1], if k = 1; n−k k δ k+1 + 1, if 2 k n − 1; 1, if k = n;(16)\nand let α = 1/(2nδ max ) where δ max = max{δ k : k = 1, . . . , n}. Then Inequalities (15) are satisfied for every β 0.\n\nProof. Observe that, for every k = 1, . . . , n, the right-hand side of the k-th inequality in (15) is\nδ k e −α = δ k e −1/(2nδ max ) δ k 1 − 1 2nδ max = δ k − δ k 2nδ max δ k − 1 2n\n.\n\nNow we check that the left-hand side is at most δ k − 1/(2n).\n\nFirst inequality (k = 1): n−1 n δ1\n1+e −β + δ2 2 δ 1 e −α .\nFrom the definition of δ 1 in (16) we have that\nδ 2 = 2δ 1 − 1 n − 1 .\nHence the left-hand side is\nn − 1 n δ 1 1 + e −β + δ 2 2 n − 1 n δ 1 + δ 2 2 = n − 1 n δ 1 + 2δ 1 − 1 2(n − 1) = 1 2n (2nδ 1 − 1) = δ 1 − 1 2n .\nSecond inequality (k = 2): 1\n2n 2 1+e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 δ 2 e −α .\nFrom the definition of δ 2 in (16) we have that\nδ 3 = 2 n − 2 (δ 2 − 1) .\nHence the left-hand side of the second inequality is 1 2n\n2 1 + e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 1 2n (2δ 1 + (n − 1)δ 2 + (n − 2)δ 3 ) = 1 2n ((n − 1)δ 2 + 1 + (n − 1)δ 2 + 2(δ 2 − 1)) = 1 2n (2nδ 2 − 1) = δ 2 − 1 2n .\nOther inequalities (k = 3, . . . , n − 1)\n: 1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) δ k e −α .\nFrom the definition of δ k in (16) we have that\nδ k+1 = k n − k (δ k − 1) .\nHence the left-hand side is\n1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) = 1 2n ((n − 1)δ k + (n − k + 1)δ k + (k − 1) + kδ k − k) = 1 2n (2nδ k − 1) = δ k − 1 2n .\nLast inequality (k = n): n−1 2n (δ n + δ n−1 ) δ n e −α . Since δ n = 1 and δ n−1 = 1 n−1 δ n + 1 = n n−1 , the left-hand side of the last inequality is\nn − 1 2n (δ n + δ n−1 ) = n − 1 2n (1 + n n − 1 ) = 1 − 1 2n .\nLemma 14 Let δ 1 , . . . , δ n be as follows\nδ 1 = n 1−ε , δ 2 = 4/3, δ 3 = · · · = δ n = 1\nwhere ε > 0 is an arbitrary small constant and let α = 1/n. Then Inequalities (15) are satisfied for every β (1 − ε) log n and n sufficiently large.\n\nProof. We check that all the inequalities in (15) are satisfied.\n\nFirst inequality (k = 1): n−1 n δ1 1+e −β + δ2 2 δ 1 e −α . For the left-hand side we have\nn − 1 n δ 1 1 + e −β + δ 2 2 = 1 − 1 n n 1−ε 1 + e −β + 2 3 1 − 1 n n 1−ε 1 + 1 n 1−ε + 2 3 = 1 − 1 n n 2(1−ε) n 1−ε + 1 + 2 3 = 1 − 1 n (n 1−ε + 1)(n 1−ε − 1) + 1 n 1−ε + 1 + 2 3 = 1 − 1 n n 1−ε + 1 n 1−ε + 1 − 1 3 .\nFor the right-hand side we have δ 1 e −α = n 1−ε e −1/n n 1−ε 1 − 1 n .\n\nHence the left-hand side is smaller than the right-hand one (for n sufficiently large).\n\nSecond inequality (k = 2): 1 2n 2 1+e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 δ 2 e −α . For the left-hand side we have 1 2n 2 1 + e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 = 1 2n 2 1 + e −β n 1−ε + (n − 1)\n4 3 + (n − 2) 1 2n 2n 1−ε + 7 3 n = 7 6 + 1 n ε .\nAnd for the right-hand side we have\nδ 2 e −α = 4 3 e −1/n 4 3 1 − 1 n 4 3 − 1 n .\nHence the left-hand side is smaller than the right-hand one (for n sufficiently large). Third inequality (k = 3): 1 2n ((n − 1)δ 3 + 2δ 2 + (n − 3)δ 4 ) δ 3 e −α . For the left-hand side we have\n1 2n ((n − 1)δ 3 + 2δ 2 + (n − 3)δ 4 ) = 1 2n (n − 1) + 2 4 3 + (n − 3) = 1 2n (2n − 3) 1 − 1 n .\nAnd for the right-hand side we have\nδ 3 e −α = e −1/n 1 − 1 n .\nHence the left-hand side is smaller than the right-hand one.\n\nOther inequalities (k 4): 1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) δ k e −α . Since δ k = δ k−1 = δ k+1 = 1 the left-hand side is equal to n−1 n and the right-hand side is e −1/n n−1 n .\n\n\nLemma 15\n\nLet δ 1 , . . . , δ n be as follows\nδ k = 1+e −β 2 a1 b1 δ 2 + 1 , if k = 1; a k b k δ k+1 + 1, if 2 k n − 1; 1, if k = n;(18)\nwhere a 1 = n − 1 and b 1 = ne −β + 1 and for every k = 2, . . . , n − 1\na k = (n − k)b k−1 and b k = (n + 1)b k−1 − (k − 1)a k−1 ,\nand let α = 1/(2nδ max ) where δ max = max{δ k : k = 1, . . . , n}. Then Inequalities (15) are satisfied for every β 0.\n\nBefore to prove the Lemma 15 we do the following observation.\n\nObservation 16 Let b k defined as in the Lemma 15. Then, for every k 2, it holds that b k kb k−1 .\n\nProof. We proceed by induction on k. The base case k = 2 follows from b 2 = (n + 1)(ne −β + 1) − (n − 1) = (n + 1)ne −β + 2 > 2(ne −β + 1) = 2b 1 .\n\n\nNow suppose the claim holds for\nk − 1, that is b k−1 (k − 1)b k−2 . Then b k = (n + 1)b k−1 − (k − 1)a k−1 = (n + 1)b k−1 − (k − 1)(n − k + 1)b k−2 [(n + 1) − (n − k + 1)] b k−1 = kb k−1 .\nProof ( Lemma 15). Observe that, as in Equation (17), for every k = 1, . . . , n, the right-hand side of the k-th inequality in (15) is\nδ k e −α δ k − 1 2n .\nNow we check that the left-hand side is at most δ k − 1/(2n).\n\nFirst inequality (k = 1): n−1 n δ1 1+e −β + δ2 2 δ 1 e −α . From the definition of δ 1 in (18) we have that\nδ 2 = ne −β + 1 n − 1 2δ 1 1 + e −β − 1 .\nHence the left-hand side is\nn − 1 n δ 1 1 + e −β + δ 2 2 = n − 1 n δ 1 1 + e −β + ne −β + 1 n − 1 δ 1 1 + e −β − 1 2 = n − 1 n δ 1 1 + e −β 1 + ne −β + 1 n − 1 − ne −β + 1 2n δ 1 − 1 2n . Second inequality (k = 2): 1 2n 2 1+e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 δ 2 e −α .\nFrom the definition of δ 2 in (18) we have that\nδ 3 = b 2 a 2 (δ 2 − 1) = (n + 1)b 1 − a 1 (n − 2)b 1 (δ 2 − 1) .\nHence the left-hand side is 1 2n 2 1 + e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 = 1 2n\na 1 b 1 δ 2 + 1 + (n − 1)δ 2 + (n + 1)b 1 − a 1 b 1 (δ 2 − 1) = δ 2 − 1 2n nb 1 − a 1 b 1 = δ 2 − 1 2n n − n − 1 ne −β + 1 δ 2 − 1 2n .\nOther inequalities (k = 3, . . . , n − 1): 1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) δ k e −α . From the definition of δ k in (18) we have that\nδ k+1 = b k a k (δ k − 1) = (n + 1)b k−1 − (k − 1)a k−1 (n − k)b k−1 (δ k − 1) .\nHence the left-hand side is\n1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) = 1 2n (n − 1)δ k + (k − 1) a k−1 b k−1 δ k + 1 + (n + 1)b k−1 − (k − 1)a k−1 b k−1 (δ k − 1) = δ k − 1 2n (n − k + 2)b k−1 − (k − 1)a k−1 b k−1 = δ k − 1 2n (n − k + 2) − (k − 1)(n − k + 1) b k−2 b k−1 δ 2 − 1 2n .\nwhere the inequality follows from the Observation 16.\n\nLast inequality (k = n): n−1 2n (δ n + δ n−1 ) δ n e −α . Since δ n = 1 and δ n−1 = an−1 bn−1 δ n + 1 = an−1 bn−1 + 1, the left-hand side of the last inequality is\nn − 1 2n (δ n + δ n−1 ) = n − 1 2n 2 + a n−1 b n−1 = n − 1 2n 2 + b n−2 b n−1 n − 1 2n 2 + 1 n − 1 = 1 − 1 2n .\nwhere the inequality follows from the Observation 16.\n\nIn order to apply the path coupling theorem, we need to bound δ max : the next observation will represent the main tool to achieve this goal.\n\nObservation 17 Let δ 1 , . . . , δ n be defined recursively as follows: δ n = 1 and for a suitable constant c.\nδ k = γ k δ k+1 + 1 ,\nIn order to bound δ max when δ 1 , . . . , δ n are defined as in Lemma 15 and β c log n for a constant c ∈ N, we define\nγ k = a k b k = p k e −β + l k q k e −β + r k .(19)\nYou can check that p 1 = 0, q 1 = n and\np k = (n − k)q k−1 q k = (n + 1)q k−1 − (k − 1)p k−1 ;\nwe notice that p k = (n + 1)q k−1 − (k + 1)q k−1 q k for every k. We can also prove the following simple observation about q k .\n\n\nObservation 19\n\nFor every k 1 constant, we have q k 2 −k n k .\n\nProof. We proceed by induction on k, with the base k = 1 being obvious. Suppose the claim holds for k − 1, that is q k−1 2 −(k−1) n k−1 , then\nq k = (n + 1)q k−1 − (k − 1)p k−1 n 2 q k−1 2 −k n k .\nMoreover, you can check that l 1 = n − 1, r 1 = 1 and l k = (n − k)r k−1 r k = (n + 1)r k−1 − (k − 1)l k−1 ;\n\nwe notice that above recursion gives l k = (n − k)(k − 1)! and r k = k!. Next lemma bounds γ k defined in Equation 19.\n\nLemma 20 Let δ 1 , . . . , δ n be defined as in Lemma 15, γ k defined as in Equation (19) and β c log n for a constant c ∈ N. Then, for sufficiently large n, it holds that\n γ k < n ∀ k; γ k < 1 if k > c + 2; γ c+2 = O(1).\nProof. Since p k q k , then (nq k − p k )e −β > 0; instead, l k − nr k = (k − 1)!(n − k − nk) < 0. Hence we have for every k\nγ k − n = p k e −β + l k q k e −β + r k − n = (l k − nr k ) − (nq k − p k )e −β q k e −β + r k < 0 .\nInductively, we show that for every k c + 3, we have γ k < 1. Set k = c + 3: c is a constant, thus Observation 19 holds for k − 1; hence and since e −β n −c , we have that\n(q c+3 − p c+3 )e −β = [(n + 1)q c+2 − (c + 2)p c+2 − (n − c − 3)q c+2 ]e −β 2q c+2 e −β 2 −(c+1) n 2 .\nInstead, l c+3 − r c+3 = (c + 2)!(n − 2c − 6) (c + 2)! · n. Thus,\nγ c+3 − 1 = (l c+3 − r c+3 ) − (q c+3 − p c+3 )e −β q c+3 e −β + r c+3 (c + 2)! · n − 2 −(c+1) n 2 q c+3 e −β + r c+3 < 0 ,\nfor n sufficiently large. Now, suppose that γ k−1 < 1; then, we have\nγ k − 1 = a k − b k b k = (k − 1)a k−1 − (k + 1)b k−1 b k < 0 ,\nwhere a k−1 < b k−1 is implied by the inductive hypothesis. In order to complete the proof, we need to show that γ c+2 = O(1). Similarly to the case k = c + 3, we obtain (q c+2 − p c+2 )e −β 2 −c n and l c+2 − r c+2 (c + 1)! · n. Hence, γ c+2 p c+2 + r c+2 + (c + 1)! · n p c+2 + r c+2 + 2 −c n (c + 1)! · 2 c = O(1) .\n\n\nCorollary 21\n\nLet δ 1 , . . . , δ n and c be defined as in Lemma 15. Then δ max = O(n c+2 ).\n\nProof. From Observation 17, Lemma 20 and the definition of δ 1 , . . . , δ n it follows that\nδ max n max j i=h a i b i : 1 h j n n c+2 i=1 a i b i = O(n c+2 ).\n\nThe XOR game\n\nIn this section we analyze the logit dynamics for another simple n-player game, the XOR game. The XOR game is a symmetric n-player game in which each player has two strategies, denoted by 0 and 1, and each player pays the XOR of the strategies of all players (including herself). More formally, for each i ∈ [n], the utility function u i (·) is defined as follows\nu i (x) = −1, if\nx has an odd number of 1's; 0, if x has an even number of 1's.\n\nNotice that the XOR game has 2 n−1 Nash equilibria, namely all profiles with an even number of players playing strategy 1. Nash equilibria have social welfare 0 and profiles not in equilibria have social welfare −n. Observe that the XOR game is a potential game with exact potential Φ where Φ(x) = u i (x) for every x and every i ∈ [n]. Hence, the stationary distribution is\nπ(x) = e −β /Z, if\nx has an odd number of 1's; 1/Z, if x has an even number of 1's;\n\nwhere the normalizing factor is Z = 2 n−1 (1 + e −β ). Even if this game looks similar to the OR game, it exhibits a different behavior. Theorem 22 gives the stationary expected social welfare of the XOR game and we can see that, as β increases, the expected social welfare tends from below to the social welfare at the Nash equilibria. In contrast the expected social welfare of the OR game is better than the worst Nash equilibrium for all values of β. Moreover, in Theorem 23 and Theorem 24 we show that the mixing time for the XOR game is polynomial in n and exponential in β, whereas the mixing time for the OR game can be bounded independently from β.\n\nTheorem 22 (Expected social welfare) The stationary expected social welfare of the logit dynamics for the XOR game is E π [W ] = − n 1+e β . Proof. The expected social welfare is\nE π [W ] = x∈{0,1} n W (x)π(x) = −n · 2 n−1 e −β 2 n−1 (1 + e −β ) = − n 1 + e β .\nThe next theorem shows that the mixing time is exponential in β for every β > 0.\n\nTheorem 23 (Lower bound on mixing time) The mixing time of the logit dynamics for the XOR game is Ω(e β ).\n\nProof. Consider the set S ⊆ {0, 1} n containing only the state 0 = (0, . . . , 0). Observe that π(0) 1/2. The bottleneck ratio is\nB(0) = 1 π(0)\ny∈{0,1} n π(0)P (0, y) = y∈{0,1} n : |y|=1\nP (0, y) = n · 1 n · 1 1 + e β .\nHence, by applying Theorem 4, the mixing time is\nt mix 1 B(0) = 1 + e β .\nFinally, in the next theorem we give an almost matching upper bound to the mixing time.\n\nTheorem 24 (Upper bound on mixing time) The mixing time of the logit dynamics for the OR game is O(n 3 e β ).\n\nThe theorem is proved using coupling (see Theorem 1) and proof is presented in the next sections. Specifically, we use the coupling described in Section 3.2; in Section 7.1 we show that if the coupled chains are at even distance then distance does not increase after one step of the coupling; in Section 7.2 we show that if the coupled chains are at odd distance then they get closer distance with probability independent from β; finally, in Section 7.3 we bound the expected time needed by the two chains to coalesce and use Theorem 1 to derive an upper bound for the mixing time.\n\n\nEven Hamming distance\n\nLet X t and Y t be two chains coupled as described in Section 3.2. Suppose that X t = x, Y t = y, and H(x, y) = 2ℓ, for ℓ > 0. In this case, u i (x) = u i (y) = b for all i ∈ [n] and some b ∈ {−1, 0}. Let i be the index selected for update and let us distinguish two cases. In the first case x i = y i and we have\nu i (x −i , 0) = u i (y −i , 0) and u i (x −i , 1) = u i (y −i , 1)\nand thus\nσ i (0 | x) = σ i (0 | y) and σ i (1 | x) = σ i (1 | y).\nTherefore the coupling always update the strategy of player i in the same way in the two chains and thus H(X t+1 , Y t+1 ) = 2ℓ.\n\nIn the second case we have x i = y i and we assume, without loss of generality, that x i = 0 and y i = 1. We observe that, for b ∈ {−1, 0},\nu i (x −i , 0) = u i (y −i , 1) = b and u i (y −i , 0) = u i (x −i , 1) = −(1 + b).\nTherefore we have\nσ i (0 | x) = σ i (1 | y) = 1 1 + e −(1+2b)β and σ i (1 | x) = σ i (0 | y) = 1\n1 + e (1+2b)β and thus we have three possible updates for the strategy of player i:\n\n1. both chains update to 0 (and thus H(X t+1 , Y t+1 ) = 2ℓ − 1) with probability min 1 1 + e (1+2b)β ,\n1 1 + e −(1+2b)β = 1 1 + e β ;\n2. both chains update to 1 (and thus H(X t+1 , Y t+1 ) = 2ℓ − 1) with probability min 1 1 + e (1+2b)β ,\n1 1 + e −(1+2b)β = 1 1 + e β ;\n3. chain X and Y choose two different strategies for updating the strategy of player i (and thus H(X t+1 , Y t+1 ) = 2ℓ) with probability\n1 − 2 1 + e β .\nThe following lemma summarizes the above observations. Lemma 25 Suppose that H(X t , Y t ) = 2ℓ, for ℓ > 0. Then\nH(X t+1 , Y t+1 ) = \n2ℓ − 1, with probability 2ℓ n · 2 1+e β ; 2ℓ, with probability 1 − 2ℓ n · 2 1+e β .\n\n\nOdd Hamming distance\n\nLet X t and Y t be two chains coupled as described in Section 3.2. Suppose that X t = x, Y t = y, and H(x, y) = 2ℓ − 1, for ℓ > 0. In this case we have u i (x) = b and u i (y) = −(1 + b) for some b ∈ {−1, 0}. Let i be the index selected for update and let us distinguish two cases. In the case in which x i = y i = c for some c ∈ {0, 1}, we have 3. chains X and Y choose two different strategies for updating the strategy player i (and thus H(X t+1 , Y t+1 ) = 2ℓ) with probability 1 − 2 1+e β .\n\nIn the second case we have x i = y i and we assume, without loss of generality, that x i = 0 and y i = 1. We observe that\nu i (x −i , 0) = u i (y −i , 0) = b and u i (x −i , 1) = u i (y −i , 1) = −(1 + b).\nTherefore we have σ i (0 | x) = σ i (0 | y) and σ i (1 | x) = σ i (1 | y)\n\nand thus in this case H(X t+1 , Y t+1 ) = 2ℓ − 2.\n\nThe following lemma summarizes the above observations.\n\n\nLemma 26\n\nSuppose that H(X t , Y t ) = 2ℓ − 1, for ℓ > 0. Then\nH(X t+1 , Y t+1 ) = \n2ℓ − 2, with probability 2ℓ−1 n ;\n\n2ℓ − 1, with probability n−2ℓ+1 n 2 1+e β ; 2ℓ, with probability n−2ℓ+1 n 1 − 2 1+e β .\n\n\nTime to coalesce\n\nWe denote with τ k the random variable indicating the first time at which the two coupled chains have distance k. More precisely, τ k = min{t : H(X t , Y t ) = k}.\n\nTherefore, τ couple = τ 0 is the time needed for the two chains to coalesce. We next give a bound on the expected time E x,y [τ couple ] for the two chains to coalesce starting from x and y. If x and y have distance 2ℓ, we denote by µ ℓ the expected time to reach distance 2ℓ − 2. That is,\nµ ℓ = E x,y [τ 2ℓ−2 ] .\nSimilarly, if x and y have distance 2ℓ − 1, we denote by ν ℓ the expected time to reach distance 2ℓ − 2.\n\nThat is,\nν ℓ = E x,y [τ 2ℓ−2 ] .\nNotice that, if H(x, y) = H(x ′ , y ′ ) then\nE x,y [τ k ] = E x ′ ,y ′ [τ k ]\nfor all k, and thus the µ ℓ and ν ℓ are well defined. From Lemma 25 and Lemma 26, we have the following relations µ ℓ = 1 + µ ℓ · 1 − 2ℓ n · 2 1 + e β + ν ℓ · 2ℓ n · 2 1 + e β ν ℓ = 1 + ν ℓ · n − 2ℓ + 1 n · 2 1 + e β + µ ℓ · n − 2ℓ + 1 n · 1 − 2 1 + e β . \n\n\nConclusions and open problems\n\nIn this paper we studied strategic games where at every run a player is selected uniformly at random and she is assumed to choose her strategy for the next run according to the logit dynamics: a noisy best-response dynamics where the noise level is tuned by a parameter β. Such dynamics defines a family of ergodic Markov chains, indexed by β, over the set of strategy profiles. We proposed the stationary distribution of these Markov chains as solution concept for games where players have bounded rationality or limited knowledge about the system. Since this solution concept does not assume full rationality of agents, it avoids one of the main drawbacks of many classical equilibria concepts. Moreover, the stationary distribution of an ergodic Markov chain always exists, it is unique, and the chain converges to such a distribution from any starting state.\n\nIn order to evaluate the long-term performance of the system, on the one hand we analyzed the expected social welfare when the strategy profiles are random according to the stationary distribution, on the other hand we studied the mixing time, i.e. how long it takes, for a chain starting at an arbitrary profile, to get close to its stationary distribution.\n\nIn this paper we applied this approach to some simple but well-studied games with a constant number of players: the CK game, that obtains the worst Price of Anarchy bound between linear congestion games, and the 2 × 2 coordination games considered in the seminal paper about logit dynamics [3]. We also considered two simple n-player games, the OR game and the XOR game: the analysis of the mixing time turned out to be far from trivial even for such simple games. The above games highlight a twofold behavior: for some games, namely CK game and OR game, the mixing time can be upper bounded by a function independent of β, whereas the mixing time for the other games depends exponentially on the noise parameter β.\n\nThe main goal of our line of research is to investigate logit dynamics for notable classes of n-player games. It would also be interesting to consider variations of the logit dynamics where players update their strategies simultaneously or where the noise is not uniform between players.\n\nWe have seen that, for some games and for some values of β, the mixing time can be exponential in the number of players. When it takes such a long time to reach the stationary distribution, it would be interesting to investigate the evolution of the system in the transient phase of the logit dynamics.\n\nA\nstrategic game is a triple ([n], S, U), where [n] = {1, . . . , n} is a finite set of players, S = {S 1 , .. . , S n } is a family of non-empty finite sets (S i is the set of strategies for player i), and U = {u 1 , . . . , u n } is a family of utility functions (or payoffs), where u i : S 1 × · · · × S n → R is the utility function of player i.\n\n\nInvariance under utility translation. Let G = ([n], S, U) be a game. If we change the utility functions by adding a constant c i to all the utilities of player i, i.e. if we define a new familyŨ = {ũ i : i ∈ [n]} of utility functions as followsũ\n\n\nLower bound: We use the relaxation time bound (see Theorem 3). The transition matrix of the logit\n\n\nthe next theorem we give upper bounds on the mixing time depending on the value of β. The theorem shows that, if β c log n for some constant c, the mixing time is effectively polynomial in n with degree depending on c. The use of the path coupling technique in the proof of the theorem requires a careful choice of the edge-weights. Theorem 12 (Upper bound on mixing time) The mixing time of the logit dynamics for the OR game is O(n 5/2 2 n ) for every β. Moreover, for small values of β the mixing time is 1. O(n log n) if β < (1 − ε) log n, for an arbitrary small constant ε > 0; 2. O(n c+3 log n) if β c log n, where c 1 is an arbitrary constant.\n\n= 2 :\n2In this case we have x j = 0 and y j = 1, there is another player h ∈ [n] \\ {j} where x h = y h = 1, and for all the other players i ∈ [n] \\ {j, h} it holds x i = y i = 0. Hence the length of edge {x, y} is ℓ(x, y) = δ 2 .\n\n\nwhere γ k > 0 for every k = 1, . . . , n − 1. Let δ max = max{δ k : k = 1, . . . , n}. Thenδ max n max j i=h γ i : 1 h j n − 1 .Proof. The observation follows from the fact that, for k = 1, . . . , n − 1, we have Corollary 18 Let δ 1 , . . . , δ n be defined as in Lemma 13. Then δ max c √ n2 n for a suitable constant c.Proof. From Observation 17 and the definition of δ 1 , . . . , δ n , it holds that\n\n\nu i (x −i , c) = u i (y −i , 1 − c) = b and u i (x −i , 1 − c) = u i (y −i , c) = −(1 + b). Therefore σ i (c | x) = σ i (1 − c | y) = 1 1 + e −(1+2b)β and σ i (1 − c | x) = σ i (c | y) = 1 1 + e (1+2b)β and thus we have three possible updates: 1. both chains update to c (and thus H(X t+1 , Y t+1 ) = 2ℓ − 1\n\n\nUpdate the strategy of player i according to the following probability distribution over the set S i of her strategies. For every y ∈ S iConsider the following noisy best-response dynamics, introduced in [3] and known as logit dynamics: at \nevery time step \n\n1. Select one player i ∈ [n] uniformly at random; \n2. \n\n\nSimple algebraic manipulations giveFrom Markov inequality we have thatP x,y (τ couple > t) E x,y [τ couple ] tand thus, by taking t 0 = 4E x,y [τ couple ], we have d(t 0 ) 1/4. Therefore, by using Theorem 1, we have that t mix = O n 3 e β .ν ℓ = \nn \n2ℓ − 1 \n1 + \nn − 2ℓ + 1 \n2ℓ \n· \ne β − 1 \n2 \n\nand \n\nµ ℓ = ν ℓ + \nn \n2ℓ \n· \n1 + e β \n2 \n\n= \nn \n2ℓ − 1 \n+ \nn \n2ℓ \n\nn \n2ℓ − 1 \n· \ne β − 1 \n2 \n+ 1 \n\nn \n2ℓ − 1 \n\nn \n2ℓ − 1 \n· \ne β − 1 \n2 \n+ 2 \n\nn n · \ne β − 1 \n2 \n+ 2 . \n\nHence, \n\nE x,y [τ couple ] 1 + \n\n2 ℓ n \n\nℓ even \n\nµ ℓ \nn 2 \n2 \nn · \ne β − 1 \n2 \n+ 2 + 1 = O n 3 e β . \n\n\nRoughly speaking, a finite-state Markov chain is irreducible and aperiodic if there is a time t such that, for all pairs of states x, y, the probability to be in y after t steps, starting from x, is positive.\nThis is the same coupling used in the analysis of the lazy random walk on the hypercube (e.g. see Section 5.3.3 in[16]), the only difference being that the probability of choosing 0 or 1 is not 1/2, 1/2 but 1/(1 + e β ), 1/(1 + e −β )\n\nTom Wexler, and Tim Roughgarden. 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Journal of the ACM, 56(3), 2009.\n\nThe price of anarchy of finite congestion games. George Christodoulou, Elias Koutsoupias, Proc. of the 37th Annual ACM Symposium on Theory of Computing (STOC'05). of the 37th Annual ACM Symposium on Theory of Computing (STOC'05)ACMGeorge Christodoulou and Elias Koutsoupias. The price of anarchy of finite congestion games. In Proc. of the 37th Annual ACM Symposium on Theory of Computing (STOC'05), pages 67-73. ACM, 2005.\n\nThe complexity of computing a Nash equilibrium. Constantinos Daskalakis, Paul W Goldberg, Christos H Papadimitriou, SIAM Journal on Computing. 391Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195-259, 2009.\n\nLearning, local interaction, and coordination. Glenn Ellison, Econometrica. 615Glenn Ellison. Learning, local interaction, and coordination. Econometrica, 61(5):1047-1071, 1993.\n\nThe complexity of game dynamics: Bgp oscillations, sink equilibria, and beyond. Alex Fabrikant, Christos H Papadimitriou, Proc. of the 19th annual ACM-SIAM symposium on Discrete algorithms (SODA'08). of the 19th annual ACM-SIAM symposium on Discrete algorithms (SODA'08)ACMAlex Fabrikant and Christos H. Papadimitriou. The complexity of game dynamics: Bgp oscilla- tions, sink equilibria, and beyond. In Proc. of the 19th annual ACM-SIAM symposium on Discrete algorithms (SODA'08), pages 844-853. ACM, 2008.\n\nCalibrated learning and correlated equilibrium. P Dean, Foster, V Rakesh, Vohra, Games and Economic Behavior. 211-2Dean P. Foster and Rakesh V. Vohra. Calibrated learning and correlated equilibrium. Games and Economic Behavior, 21(1-2):40-55, 1997.\n\nDrew Fudenberg, David K Levine, The Theory of Learning in Games. MIT. Drew Fudenberg and David K. Levine. The Theory of Learning in Games. MIT, 1998.\n\nSink equilibria and convergence. Michel Goemans, Vahab Mirrokni, Adrian Vetta, Proc. of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05). of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)IEEEMichel Goemans, Vahab Mirrokni, and Adrian Vetta. Sink equilibria and convergence. In Proc. of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pages 142-154. IEEE, 2005.\n\nA General Theory of Equilibrium Selection in Games. C John, Reinhard Harsanyi, Selton, MIT PressJohn C. Harsanyi and Reinhard Selton. A General Theory of Equilibrium Selection in Games. MIT Press, 1988.\n\nA simple adaptive procedure leading to correlated equilibrium. Sergiu Hart, Andreu Mas-Colell, Econometrica. 685Sergiu Hart and Andreu Mas-Colell. A simple adaptive procedure leading to correlated equilibrium. Econometrica, 68(5):1127-1150, 2000.\n\nWorst-case equilibria. Elias Koutsoupias, Christos H Papadimitriou, Preliminary version in STACS. 3Elias Koutsoupias and Christos H. Papadimitriou. Worst-case equilibria. Computer Science Review, 3(2):65-69, 2009. Preliminary version in STACS 1999.\n\nMarkov Chains and Mixing Times. David Levin, Yuval Peres, Elizabeth L Wilmer, American Mathematical SocietyDavid Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, 2008.\n\nConvergence to equilibrium in local interaction games. Andrea Montanari, Amin Saberi, Proc. of the 50th Annual Symposium on Foundations of Computer Science (FOCS'09. of the 50th Annual Symposium on Foundations of Computer Science (FOCS'09IEEEAndrea Montanari and Amin Saberi. Convergence to equilibrium in local interaction games. In Proc. of the 50th Annual Symposium on Foundations of Computer Science (FOCS'09). IEEE, 2009.\n\nA class of games possessing pure-strategy nash equilibria. Robert W Rosenthal, International Journal of Game Theory. 216567Robert W. Rosenthal. A class of games possessing pure-strategy nash equilibria. International Journal of Game Theory, 2(1):6567, 1973.\n\nOn the equilibria of alternating move games. Aaron Roth, Nina Balcan, Adam Kalai, Yishay Mansour, Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA'10). of ACM-SIAM Symposium on Discrete Algorithms (SODA'10)Aaron Roth, Nina Balcan, Adam Kalai, and Yishay Mansour. On the equilibria of alternating move games. In Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA'10), 2010.\n\nPopulation Games and Evolutionary Dynamics. H William, Sandholm, MIT PressWilliam H. Sandholm. Population Games and Evolutionary Dynamics. MIT Press, 2010.\n\nIndividual Strategy and Social Structure: An Evolutionary Theory of Institutions. H , Peyton Young, Princeton University PressH. Peyton Young. Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton University Press, 1998.\n\nThe diffusion of innovations in social networks. H , Peyton Young, Economics Working Paper Archive number. 437Johns Hopkins University, Department of EconomicsH. Peyton Young. The diffusion of innovations in social networks. Economics Working Paper Archive number 437, Johns Hopkins University, Department of Economics, 2000.\n"},"corpus_id":{"kind":"number","value":6772074,"string":"6,772,074"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1002.3474\",\n \"mag\": 3044044770,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"conf/sagt/AulettaFPP10\",\n \"doi\": \"10.1007/978-3-642-16170-4_6\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1002.3474v4.pdf\"\n ],\n \"pdf_sha\": \"d30d1a8bb05839eea36bd7c0d6c8db54c75d0ea1\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1002.3474\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Mixing Time and Stationary Expected Social Welfare of Logit Dynamics *"},"authors":{"kind":"list like","value":["Vincenzo Auletta auletta@dia.unisa.it \nDipartimento di Informatica \"Renato M. Capocelli\nUniversità di Salerno","Diodato Ferraioli ferraioli@dia.unisa.it \nDipartimento di Informatica \"Renato M. Capocelli\nUniversità di Salerno","Francesco Pasquale pasquale@dia.unisa.it \nDipartimento di Informatica \"Renato M. Capocelli\nUniversità di Salerno","Giuseppe Persiano \nDipartimento di Informatica \"Renato M. Capocelli\nUniversità di Salerno"],"string":"[\n \"Vincenzo Auletta auletta@dia.unisa.it \\nDipartimento di Informatica \\\"Renato M. Capocelli\\nUniversità di Salerno\",\n \"Diodato Ferraioli ferraioli@dia.unisa.it \\nDipartimento di Informatica \\\"Renato M. Capocelli\\nUniversità di Salerno\",\n \"Francesco Pasquale pasquale@dia.unisa.it \\nDipartimento di Informatica \\\"Renato M. Capocelli\\nUniversità di Salerno\",\n \"Giuseppe Persiano \\nDipartimento di Informatica \\\"Renato M. Capocelli\\nUniversità di Salerno\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We study logit dynamics [3] for strategic games. This dynamics works as follows: at every stage of the game a player is selected uniformly at random and she plays according to a noisy best-response where the noise level is tuned by a parameter β. Such a dynamics defines a family of ergodic Markov chains, indexed by β, over the set of strategy profiles. We believe that the stationary distribution of these Markov chains gives a meaningful description of the long-term behavior for systems whose agents are not completely rational.Our aim is twofold: On the one hand, we are interested in evaluating the performance of the game at equilibrium, i.e. the expected social welfare when the strategy profiles are random according to the stationary distribution. On the other hand, we want to estimate how long it takes, for a system starting at an arbitrary profile and running the logit dynamics, to get close to its stationary distribution; i.e., the mixing time of the chain.In this paper we study the stationary expected social welfare for the 3-player CK game [6], for 2-player coordination games, and for two simple n-player games. For all these games, we also give almost tight upper and lower bounds on the mixing time of logit dynamics. Our results show two different behaviors: in some games the mixing time depends exponentially on β, while for other games it can be upper bounded by a function independent of β."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"The evolution of a system is determined by its dynamics and complex systems are often described by looking at the equilibrium states induced by their dynamics. Once the system reaches an equilibrium state it stays there, thus equilibrium states describe the long-term behavior of the system. In this paper we are mainly interested in systems whose individual components are selfish agents. The state of a selfish system is fully described by a vector of strategies, each controlled by one agent, and each state assigns a payoff to each agent. The agents are selfish in the sense that they pick their strategy so to maximize their payoff, given the strategies of the other agents. Nash equilibrium is the classical notion of equilibrium for selfish systems and it corresponds to the equilibrium induced by the best-response dynamics. The observation that selfish systems are described by their equilibrium states (that is, by the Nash equilibria) has motivated the notions of Price of Anarchy [15] and Price of Stability [1] and the analysis of efficiency of selfish systems based on such notions.\nHowever, such analysis inherits some of the shortcomings of the concept of a Nash equilibrium. First of all, the best-response dynamics assumes that selfish agents have complete knowledge of the current state of the system; that is, they know the payoff associated with each of their possible choices and each of the strategies chosen by other agents. Instead, in most cases, agents have only approximate knowledge of the system state or they are not able to compute their best choice. Moreover, in presence of multiple equilibria, it is not clear which one of them will be reached by the system, as it may depend on the initial state: Price of Anarchy considers the worst-case equilibrium, whereas Price of Stability focuses on the best-case equilibrium. Finally, Nash equilibria are hard to compute [7,5] and thus for some systems it might take very long to reach a Nash equilibrium: in this case using equilibrium states to describe the system performance is not well justified. Rather, one would like to analyze the performance of a system by using a dynamics (and its related equilibrium notion) that has the following three properties:\n• The dynamics takes into account the fact that the system components might have a perturbed or noisy knowledge of the system;\n• For every system the equilibrium state exists and is unique;\n• The system reaches the equilibrium very quickly regardless of the starting state.\nIn this paper, we consider noisy best-response dynamics in which the behavior of the agents is described by a parameter β 0. The case β = 0 corresponds to agents picking their strategies completely at random (that is, the agents have no knowledge of the system) and the case β = ∞ corresponds to agents picking their strategies according to the best-response dynamics (in which the agents have full and complete knowledge of the system). The intermediate values of β correspond to agents that are roughly guided by the best-response dynamics but can make a sub-optimal choice due, for example, to bounded rationality of the agent or limited knowledge about the system: this sub-optimal behavior occurs with some probability that depends on β (and on the associated payoff).\nWe will study a specific noisy best-response dynamics for which the system evolves according to an ergodic Markov chain for all β 0. For these systems, it is natural to look at the stationary distribution (which is the equilibrium state of the Markov chain) and to analyze the expected social welfare (the sum of utility functions) of the system at that distribution. We stress that the noisy best-response dynamics well models agents that only have approximate or noisy knowledge of the system and that for ergodic Markov chains (such as the ones arising in our study) the stationary distribution is known to exist and to be unique. Moreover, to justify the use of the stationary distribution for analyzing the performance of the system, we will study how fast the Markov chain converges to the stationary distribution.\nRelated Works and Our Results. Several dynamics, besides the best-response dynamics, and several notions of equilibrium, besides Nash equilibria, have been considered to describe the evolution of a selfish system and to analyze its performance. See, for example, [11,21,20]. Equilibrium concepts based on the best-response. When the game does not possess a Pure Nash equilibrium, the best-response dynamics will eventually cycle over a set of states (in a Nash equilibrium the set is a singleton). These states are called sink equilibria [12]. Sink equilibria exist for all games and, in some contexts, they seem a better approximation of the real setting than mixed Nash equilibria. Unfortunately, sink equilibria share two undesirable properties with Nash equilibria: a game can have more that one sink equilibrium and sink equilibria seem hard to compute [9].\nOther notions of equilibrium state associated with best-response dynamics are the unit-recall equilibria and component-wise unit-recall equilibria (see [9]). However, we point out that the former does not always exist and that the latter imposes too strict limitations on the players. No-Regret Dynamics. Another broadly explored set of dynamics are the no-regret dynamics (see, for example, [11]). The regret of an user is the difference between the long-term average cost and the average cost of the best strategy in hindsight. In the no-regret dynamics the regret of every player after t steps is o(t) (sublinear with time). In [10,14] it is showed that the no-regret dynamics converges to the set of correlated equilibria. Note that the convergence is to the set of correlated equilibria and not to a specific correlated equilibrium. Our work. In this paper we consider a specific noisy best-response dynamics called the logit dynamics (see [3]) and we study its mixing time (that is, the time it takes to converge to the stationary distribution) and the stationary expected social welfare. Specifically,\n• We start by analyzing the logit dynamics for a simple 3-player linear congestion game (the CK game [6]) which exhibits the worst Price of Anarchy among linear congestion games. We show that the mixing time of the logit dynamics is upper bounded by a constant independent of β. Moreover, we show that the stationary expected social welfare is larger than the social welfare of the worst Nash equilibrium for all β;\n• We then analyze the 2 × 2 coordination games studied in [3]. Here we show that, under some conditions, the stationary expected social welfare is larger than the social welfare of the worst Nash equilibrium. We give upper and lower bounds on the mixing time exponential in β. We also observe that the same bounds apply to anti-coordination games;\n• Finally, we apply our analysis to two simple n-player games: the OR game and XOR game. We give upper and lower bounds on the mixing time: we show that the mixing time of the OR game can be upper bounded by a function independent of β, while the mixing time of the XOR game increases exponentially in β. We also prove that for β = O(log n) the mixing time is polynomial in n for both games.\nThe logit dynamics has been first studied by Blume [3] who showed that, for 2 × 2 coordination games, the long-term behavior of the Markov chain is concentrated in the risk dominant equilibrium (see [13]) for sufficiently large β. Ellison [8] studied different noisy best-response dynamics for coordination games assuming that interaction among players were described by a graph; that is, the utility of a player is determined only by the strategies of the adjacent players. Specifically, Ellison [8] studied interaction modeled by rings and showed that some large fraction of the players will eventually choose the risk dominant strategy. Similar results were obtained by Peyton Young [22] for the logit dynamics and for more general families of graphs. Montanari and Saberi [17] gave bounds on the hitting time (the expected time that the logit dynamics takes to reach a specific state) of the risk dominant equilibrium state in terms of some graph theoretic properties of the underlying interaction network. Asadpour and Saberi [2] studied the hitting time for a broader class of congestion games. We notice that none of [3,8,22] gave any bound on the convergence time to the risk dominant equilibrium. Montanari and Saberi [17] were the first to do so but their study focuses on the hitting time of a specific configuration and not on the convergence time to the stationary distribution. From a technical point of view, our work follows the lead of [3,8,22] and extends their technical findings by giving bounds on the mixing time of the Markov chain of the logit dynamics. We stress that previous results only proved that, for sufficiently large β, eventually the system concentrates around certain states without further quantifying the rate of convergence nor the asymptotic behaviour of the system for small values of β. Instead, we identify the stationary distribution of the logit dynamics as the global equilibrium and we evaluate the social welfare at stationarity and the time it takes the system to reach it (the mixing time) as explicit functions of β.\nWe choose to start our study from the class of coordination games considered in [3] and two simple n-player games (the OR game and the XOR game). We give nearly tight upper and lower bounds on the mixing time. Despite their game-theoretic simplicity, the analytical study of the mixing time of the logit dynamics for the two n-player games is far from trivial. We notice that the results in [17] cannot be used to derive upper bounds on the mixing time.\nFrom a more conceptual point of view, our work tries (similarly to [12,9,19]) to introduce a solution concept that well models the behavior of selfish agents, is uniquely defined for any game, and is quickly reached from any starting state. We propose the stationary distribution induced by the logit dynamics as a possible solution concept and exemplify its use in the analysis of the performance of some 2 × 2 games (as the ones considered in [3]), of games used to obtain tight bounds on the Price of Anarchy, and of two simple multi-player games.\nOrganization of the paper. In Section 2 we summarize some Markov chain notions that we will use throughout the paper. In Section 3 we formally describe the logit dynamics for strategic games. We also describe the coupling we will repeatedly use in the proofs of the upper bounds on mixing times. In Sections 4, 5, 6 and 7 we study the stationary expected social welfare and the mixing time of the logit dynamics for CK game, coordination games, the OR game, and the XOR game, respectively. Finally, in Section 8 we present conclusions and some open problems. Notation. We write S for the complementary set of a set S; we write |S| for its size. We use bold symbols for vectors; when x = (x 1 , . . . , x n ) ∈ {0, 1} n we write |x| for the number of 1s in x; i.e., |x| = |{i ∈ [n] :\nx i = 1}|. For two vectors x, y let H(x, y) = |{i ∈ [n] : x i = y i }| be their Hamming distance: we write x ∼ y if H(x, y) = 1. We use the standard game theoretic notation (x −i , y) to mean the vector obtained from x by replacing the i-th entry with y, i.e. (x −i , y) = (x 1 , . . . , x i−1 , y, x i+1 , . . . , x n )."},{"title":"Markov chains summary and notation","content":"We summarize the main tools we use to bound the mixing time of Markov chains (for a complete description of such tools see, for example, Chapters 5.2, 7.2, 12.2 and 14.2 of [16]. We refer the reader to [16] also for notational conventions).\nConsider a Markov chain M with finite state space Ω and transition matrix P . It is a classical result that for an irreducible and aperiodic Markov chain 1 there exists an unique stationary distribution π over Ω; that is, a distribution π on Ω such that π · P = π. The total variation distance µ − ν TV between two probability distributions µ and ν on Ω is defined as\nAn irreducible and aperiodic Markov chain M converges to its stationary distribution π; specifically, there exists 1 > α > 0 such that\nwhere\nand P t (x, ·) is the distribution at time t of the Markov chain starting at x. For 1/2 > ε > 0, the mixing time is defined as t mix (ε) = min{t ∈ N : d(t) ε}.\nIt is usual to set ε = 1/4 or ε = 1/2e. If not explicitly specified, when we write t mix we mean t mix (1/4). Observe that t mix (ǫ) ⌈log 2 ǫ −1 ⌉t mix .\nCoupling. A coupling of two probability distributions µ and ν on Ω is a pair of random variables (X, Y ) defined on Ω × Ω such that the marginal distribution of X is µ and the marginal distribution of Y is ν. A coupling of a Markov chain M with transition matrix P is a process (X t , Y t ) ∞ t=0 with the property that both X t and Y t are Markov chains with transition matrix P . When the two coupled chains start at (X 0 , Y 0 ) = (x, y), we write P x,y (·) and E x,y [·] for the probability and the expectation on the space where the two chains are both defined. We denote by τ couple the first time the two chains meet; that is,\nWe will consider only couplings of Markov chains with the property that for s τ couple , it holds X s = Y s . The following theorem can be used to give an upper bound on t mix (see, for example, Corollary 5.3 in [16]).\nTheorem 1 (Coupling) Let M be a Markov chain with state space Ω and transition matrix P . For each pair of states x, y ∈ Ω consider a coupling (X t , Y t ) of M with starting states X 0 = x and Y 0 = y. Then\nSometimes it is difficult to specify a coupling and to analyze the coupling time τ couple for each pair of starting states x and y. The Path Coupling theorem says that it is sufficient to define a coupling only for pairs of Markov chains starting from adjacent states and an upper bound on the mixing time can be obtained if each of these couplings contracts their distance on average. More precisely, consider a Markov chain M with state space Ω and transition matrix P ; let G = (Ω, E) be a connected graph and let w : E → R be a function assigning weights to the edges such that w(e) 1 for every edge e ∈ E; for x, y ∈ Ω, we denote by ρ(x, y) the weight of the shortest path in G between x and y. The following theorem holds.\nTheorem 2 (Path Coupling [4]) Suppose that for every edge {x, y} ∈ E a coupling (X t , Y t ) of M with X 0 = x and Y 0 = y exists such that E x,y [ρ(X 1 , Y 1 )] e −α · w({x, y}) for some α > 0. Then\nSpectral techniques. A Markov chain M with state space Ω and transition matrix P is said reversible if for all x, y ∈ Ω, it holds that π(x) · P (x, y) = π(y) · P (y, x).\nThe eigenvalues of the transition matrix P of a reversible Markov chain M can be used to obtain upper and lower bounds on the mixing time. Observe that all the eigenvalues of any transition matrix P have absolute value at most 1, λ = 1 is an eigenvalue, and for irreducible and aperiodic chains, −1 is not an eigenvalue. The relaxation time t rel of a Markov chain M is defined as\nwhere λ ⋆ is the largest absolute value among eigenvalues other than 1, λ ⋆ = max{|λ| : λ is an eigenvalue of P, λ = 1} .\nObserve that, for M reversible, irreducible and aperiodic, 0 λ ⋆ < 1 and thus t rel is positive and finite. We have the following theorem (see, for example, Theorems 12.3 and 12.4 in [16])."},{"title":"Theorem 3 (Relaxation time)","content":"Let P be the transition matrix of a reversible, irreducible, and aperiodic Markov chain with state space Ω and stationary distribution π. Then\nwhere π min = min x∈Ω π(x).\nLower bound. We will use the following theorem to derive our lower bounds (see, for example, Theorem 7.3 in [16]).\nTheorem 4 (Bottleneck ratio) Let M = {X t : t ∈ N} be an irreducible and aperiodic Markov chain with finite state space Ω, transition matrix P , and stationary distribution π. Let S ⊆ Ω be any set with π(S) 1/2. Then the mixing time is\nwhere Φ(S) = Q(S, S) π(S) and Q(S, S) = x∈S, y∈S π(x)P (x, y)."},{"title":"The model and the problem","content":"where x ∈ S 1 × · · · × S n is the strategy profile played at the current time step, T i (x) = z∈Si e βui(x−i,z) is the normalizing factor, and β 0.\nParameter β is called inverse noise of the system, indeed from (1) it is easy to see that, for β = 0 player i selects her strategy uniformly at random, for β > 0 the probability is biased toward strategies promising higher payoffs, and for β → ∞ player i chooses her best response strategy (if more than one best response is available, she chooses uniformly at random one of them). Moreover observe that probability σ i (y | x) does not depend on the strategy x i currently adopted by player i. The above dynamics defines a Markov chain with the set of strategy profiles as state space, and where the transition probability from profile x = (x 1 , . . . , x n ) to profile y = (y 1 , . . . , y n ) is zero if the H(x, y) 2 and it is 1 n σ i (y i | x) if the two profiles differ exactly at player i. More formally, we can define the logit dynamics as follows.\nDefinition 5 (Logit dynamics [3]) Let G = ([n], S, U) be a strategic game and let β 0. The logit dynamics for G is the Markov chain M β = {X t : t ∈ N} with state space Ω = S 1 ×· · ·×S n and transition matrix"},{"title":"Properties. Logit dynamics enjoys some interesting properties:","content":"Ergodicity. It is easy to see that the logit dynamics is irreducible and aperiodic. Indeed, let x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ) be two profiles and let (z 0 , . . . , z n ) be a path of profiles where z 0 = x, z n = y and z i = (y 1 , . . . , y i , x i+1 , . . . x n ) for i = 1, . . . , n − 1. The probability that the chain starting at x is in y after n steps is P n (x, y) = P n (z 0 , z n ) P n−1 (z 0 , z n−1 )P (z n−1 , z n ) and recursively\nwhere the last inequality follows from (2) because, for all i = 1, . . . , n, the Hamming distance between z i−1 and z i is at most 1. Hence there is a unique stationary distribution π and, for every starting profile x, the distribution of the chain P t (x, ·) converges to π in total variation as t tends to infinity.\nwe get a new gameG = ([n], S,Ũ ) but the same logit dynamics. Indeed, according to (1), the probability player i chooses strategy y when the game is at profile x is\nNoise changes under utility rescaling. While translations of utilities do not affect logit dynamics, a rescaling of the utility functions for the same constant α > 0 changes the inverse noise from β to α · β. Indeed, if for every player i and every profile x we set\nz∈Si e αβui(x−i,z) . Notice that, unlike translations constants, we here must have the same rescaling constant α for all utility functions.\nPotential Games. A game G = ([n], S, U) is said a (exact) potential game if a function Φ : S 1 × · · · × S n → R exists such that, for every player i and for every pair of profiles x and y that differ only at position i, it holds that\nis a potential game with potential function Φ, then the Markov chain given by (2) is reversible and its stationary distribution is the Gibbs measure\nwhere Z = y∈S1×···×Sn e βΦ(y) is the normalizing constant. Except for the Matching Pennies example in Subsection 3.1, all the games we analyze in this paper are potential games. Logit dynamics vs Glauber dynamics. When G is a potential game, the logit dynamics is equivalent to the well-studied Glauber dynamics. For state space Ω = S 1 × · · · × S n and probability distribution µ over Ω, the Glauber dynamics for µ proceeds as follows: From profile x ∈ Ω, pick a player i ∈ [n] uniformly at random and update her strategy at y ∈ S i with probability µ conditioned on the other players being at\n.\nIt is easy to see that the Markov chain defined by the Glauber dynamics is irreducible, aperiodic, and reversible with stationary distribution µ. When G = ([n], S, U) is a potential game with potential function Φ, the logit dynamics defines the same Markov chain as the Glauber dynamics for the Gibbs distribution π in (3). Indeed, in that case we have\n.\nHence, logit dynamics for potential games and Glauber dynamics for Gibbs distributions are two ways of looking at the same Markov chains: in the former case the dynamics is derived from the potential function, in the latter case from the stationary distribution. However, observe that, if G is not a potential game and π is the stationary distribution of the logit dynamics for G, in general the Glauber dynamics for π is different from the logit dynamics (see, for example, the Matching Pennies case in Subsection 3.1). Due to the analogies between logit and Glauber dynamics, we will sometimes adopt the terminology used by physicists to indicate the quantities involved; in particular we will call parameter β the inverse noise or inverse temperature and we will call partition function the normalizing constant Z of the Gibbs distribution (3).\nStationary expected social welfare and mixing time. Let W : S 1 × · · · × S n −→ R be a social welfare function (in this paper we assume that W is simply the sum of all the utility functions W (x) = n i=1 u i (x), but clearly any other function of interest can be analysed). We study the stationary expected social welfare, i.e. the expectation of W when the strategy profiles are random according to the stationary distribution π of the Markov chain,\nSince the Markov chain defined in (2) is irreducible and aperiodic, from every initial profile x the distribution P t (x, ·) of chain X t starting at x will eventually converge to π as t tends to infinity. We will be interested in bounding how long it takes to get close to the stationary distribution, that is the mixing time of the Markov chain. In the next subsection we illustrate the goals of our work with two simple examples."},{"title":"Two simple examples: Matching Pennies and a Stairs game","content":"Matching Pennies. Consider the classical Matching Pennies game. We write the utility functions in the standard bimatrix form.\nAccording to (1), the update probabilities for the logit dynamics are, for every x ∈ {H, T }\nHence the transition matrix (see (2)) is\nwhere, for readability sake, we named b = 1 1+e −2β . Since every column of the matrix adds up to 1, the uniform distribution π over the set of strategy profiles is the stationary distribution for the logit dynamics. The stationary expected social welfare is thus 0 for every inverse noise β.\nAs for the mixing time, it is easy to see that it is upper bounded by a constant independent of β. Indeed, a direct calculation shows that, for every x ∈ {HH, HT, T H, T T } and for every β 0 it holds that\nA stairs game. One of the main techniques used to give upper bounds on the mixing time of Markov chains is the coupling technique (see Theorem 1). In the following example we use it to upper bound the mixing time of the logit dynamics for a simple game. Let G be a potential game where every player has two strategies, say upstairs (or 1) and downstairs (or 0), and the potential of a profile x ∈ {0, 1} n is the number of players that are upstairs, i.e. Φ(x) = |x|.\nNotice that the logit dynamics (and thus the stationary distribution and the mixing time) is completely defined by the potential function, while if we wanted to evaluate the stationary expected social welfare we would need to specify the utility functions.\nThe partition function is\nSo the stationary distribution is\nAs for the mixing time, we can use the coupling technique as follows: observe that the probability of playing strategy 1 (or equivalently strategy 0), for the player selected for the update, is independent of the current strategies of the other players. Indeed, according to (1), for every x it holds that\nWe can define a coupling of two Markov chains starting at two different profiles as follows: choose i ∈ [n] uniformly at random and perform the same update at player i in both chains 2 . When every player has been chosen at least once the two chains have coalesced. From the coupon collector's argument, it takes O(n log n) to have that, with probability at least 3/4, all players have been chosen at least once. By applying Theorem 1 we have that the mixing time is O(n log n).\nIn the above examples, it turned out that the mixing time of the logit dynamics can be upper bounded by functions that do not depend on the inverse noise β. As we shall see in the next sections, this is not always the case. Moreover, the analysis of the mixing time is usually far from trivial."},{"title":"Description of the Coupling","content":"Throughout the paper we will use the coupling and path-coupling techniques (see Theorem 1 and Theorem 2) to give upper bounds on mixing times. Since we will use the same coupling idea in several proofs, we describe it here and we will refer to this description when we will need it.\nConsider an n-player 2-strategy game G and let us rename 0 and 1 the strategies of every player. For every pair of strategy profiles x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ {0, 1} n we define a coupling (X 1 , Y 1 ) of two copies of the Markov chain with transition matrix P defined in (2) for which X 0 = x and Y 0 = y.\nThe coupling proceeds as follows: first, pick a player i uniformly at random; then, update the strategies x i and y i of player i in the two chains, by setting\nThree easy observations are in order: if σ i (0 | x) = σ i (0 | y) and player i is chosen, then, after the update, we have x i = y i ; for every player i, at most one of the updates (x i , y i ) = (0, 1) and (x i , y i ) = (1, 0) has positive probability; if i is chosen for update, then the marginal distributions of x i and y i agree with σ i (· | x) and σ i (· | y) respectively, indeed, for b ∈ {0, 1}, the probability that\nand the probability that\nWe define G = (Ω, E) as the Hamming graph of the game, where Ω = {0, 1} n is the set of strategy profiles, and two profiles x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ Ω are adjacent if they differ only for the strategy of one player, i.e.\n{x, y} ∈ E ⇐⇒ x ∼ y .\nFor the path coupling technique, the coupling described above is applied only to pairs of adjacent starting profiles."},{"title":"A 3-player congestion game","content":"In this section we analyze the CK game, a simple 3-player linear congestion game introduced in [6]. This game is interesting because it highlights the weakness of the Price of Anarchy notion for the logit dynamics. Indeed, the CK game exhibits the worst Price of Anarchy with respect to the average social welfare among all linear congestion games with 3 or more players. But, as we shall see soon, the stationary expected social welfare of the logit dynamics is always larger than the social welfare of the worst Nash equilibrium and, for large enough β, players spend most of the time in the best Nash equilibrium. Moreover, we will show that the mixing time of the logit dynamics can be bounded independently from β: that is, the stationary distribution guarantees a good social welfare and it is quickly reached by the system. Let us now describe the CK game. We have 3 players and 6 facilities divided into two sets: G = {g 1 , g 2 , g 3 } and H = {h 1 , h 2 , h 3 }. Player i ∈ {0, 1, 2} has two strategies: Strategy \"0\" consists in selecting facilities (g i , h i ); Strategy \"1\" consists in selecting facilities (g i+1 , h i−1 , h i+1 ) (index arithmetic is modulo 3). The cost of a facility is the number of players choosing such facility, and the welfare of a player is minus the sum of the costs of the facilities she selected. It easy to see that this game has two pure Nash equilibria: the solution where every player plays strategy 0 (each player pays 2, which is optimal), and the solution where every player plays strategy 1 (each player pays 5). The game is a congestion game, and thus, by [18], it is also a potential game and its potential function is:\nwhere L x (j) is the number of players using facility j in configuration x.\nStationary expected social welfare. It is easy to see that the update probabilities given by the logit dynamics for this game (see Equation (1)) only depend on the number of players playing strategy 1 and not on which player is actually playing that strategy. In particular, we have that, from a profile x, the player i, if selected for update, plays strategy 0 with the following probabilities:\nand strategy 1 with the remaining probabilities. Next theorem evaluates the stationary expected social welfare for this game.\nTheorem 6 (Expected social welfare) The stationary expected social welfare E π [W ] of the logit dynamics for the CK game is\nProof. We notice that two profiles with the same number of players playing strategy 1 have both the same potential (and, by Equation (3), the same stationary distribution) and the same social welfare. Thus, π(x) = π[k] and W (x) = W [k] for a profile x such that |x| = k, where\nwhere Z(β) = e −6β + 3e −10β + 4e −12β , and\nHence, the stationary expected social welfare is\nNotice that for β = 0 we have E π [W ] = −27/2, which is better than the social welfare of the worst Nash equilibrium. This means that, even if each player selects her strategy at random, the logit dynamics drives the system to a random profile whose expectation to be better than the worst Nash equilibrium. We also observe that E π [W ] increases with β and thus the long-term behavior of the logit dynamics gives a better social welfare than the worst Nash equilibrium for any β 0. Moreover, the stationary expected social welfare approaches the optimal social welfare as β tends to ∞.\nMixing time. Now we study the mixing time of the logit dynamics for the CK game and we show that it is bounded by a constant for any β 0. The proof will use Coupling Theorem (see Theorem 1).\nTheorem 7 (Mixing time) There exists a constant τ such that the mixing time t mix of the logit dynamics of the CK game is upper bounded by τ for every β 0.\nProof. First, we notice that the update probabilities given in Equation (6) imply that\nLet X t and Y t be two copies of the logit dynamics for the CK game, starting in x and y respectively, coupled as described in Section 3.2. It is easy to check that, by Equation (7), the player selected for update, chooses strategy 0 in both chain with probability at least 1/2. Finally, we bound the probability that after three steps the two coupled chains coalesce: it is at least as large as the probability that we choose three different players and all of them play strategy 0 at their turn, i.e.\nSince this bound holds for every starting pair (x, y), we have that the probability the two chains have not yet coalesced after 3t steps is\nThe thesis follows from the Theorem 1."},{"title":"Two player games","content":"In this section we analyse the performance of the logit dynamics for 2 × 2 coordination games (the same class studied in [3]) and 2 × 2 anti-coordination games.\nCoordination games. Coordination Games are two-player games in which the players have an advantage in selecting the same strategy. These games are often used to model the spread of a new technology [22]: two players have to decide whether to adopt or not a new technology. We assume that the players would prefer choosing the same technology as the other one and that choosing the new technology is at most as risky as choosing the old one. We name 0 the NEW strategy and 1 the OLD strategy. The game is formally described by the following payoff matrix\nWe assume that a > d and b > c (meaning that they prefer to coordinate) and that a − d b − c (meaning that for each player strategy 0 is at most as risky as strategy 1). Notice that we do not make any assumption on the relation between a and b. For convenience sake we name ∆ := a − d and δ := b − c .\nIt is easy to see that this game is a potential game and the following function is an exact potential for it:\nThis game has two pure Nash equilibria: (0, 0), where each player has utility a, and (1, 1), where each player has utility b. As d + c < a + b, the social welfare is maximized at one of the two equilibria. We analyse the mixing time of the logit dynamics for 2 × 2 coordination games and compute its stationary expected social welfare as a function of β.\nStationary expected social welfare. The logit dynamics for the coordination game defined by the payoffs in Table 8 establishes that, from a profile x, player i selected for update plays according to the following probability distribution (see Equation (1)):\nNext theorem bounds the stationary expected social welfare E π [W ] obtained by the logit dynamics and gives conditions for which E π [W ] is better than the social welfare SW N of the worst Nash equilibrium.\nTheorem 8 (Expected social welfare) The stationary expected social welfare E π [W ] of the logit dynamics for the coordination game in Table 8 is\nProof. The stationary distribution π of the logit dynamics is\nwhere Z(β) = e ∆β + e δβ + 2.\nSince E π [W ] = 2 · E π [u i ], we compute the expected utility E π [u i ] of player i at the stationary distribution,\nThus, if a > b and β max 0,\nMixing time. Now we study the mixing time of the logit dynamics for coordination games and we show that it is exponential in β and in the minimum potential difference between adjacent profiles.\nTheorem 9 (Mixing Time) The mixing time of the logit dynamics for the coordination game of Table 8 is Θ e δβ for every β 0.\nProof. Upper bound: We apply the Path Coupling technique (see Theorem 2) with the Hamming graph defined in (5) and all the edge-weights set to 1. Let x and y be two profiles differing only for the player j and consider the coupling defined in Section 3.2 for this pair of profiles. Now we bound the expected distance of the two coupled chains after one step.\nWe denote by b i (x, y) the probability that both chains perform the same update given that player i has been selected for strategy update. Clearly, b i (x, y) = 1 for i = j, while for i = j, we have\nFor sake of readability we set p = 1 1 + e ∆β and q = 1 1 + e δβ . and thus b i (x, y) = p + q. To compute E x,y [ρ(X 1 , Y 1 )], we observe that the logit dynamics chooses player j with probability 1/2. In this case, as b j (x, y) = 1, the coupling updates both chains in the same way, resulting in X 1 = Y 1 . Similarly, player i = j is chosen for strategy update with probability 1/2. In this case, with probability b i (x, y) the coupling performs the same update in both chains resulting in ρ(X 1 , Y 1 ) = 1. Instead with probability 1 − b i (x, y), the coupling performs different updates on the chains resulting in ρ(X 1 , Y 1 ) = 2. Therefore we have,\nFrom Theorem 2, with α = 1 2 (p + q) and diam(Ω) = 2, it follows that It is easy to see that the second largest eigenvalue of P is λ ⋆ = (1−p)+(1−q) 2 , hence the relaxation time is t rel = 1/(1 − λ ⋆ ) = 2 p+q , and for the mixing time we have\nIn the last inequality we used that p and q are both smaller than 1/2. Finally, the theorem follows by observing that\nNotice that, if we used the relaxation time to upper bound the mixing time (see Theorem 3) we would get a non-tight bound, hence in the above proof we had to resort to the path coupling for the upper bound.\nAnti-coordination games. Very similar results can be obtained for anti-coordination games. These are two-player games in which the players have an advantage in selecting different strategies. They model many settings where there is a common and exclusive resource: two players have to decide whether to use the resource or to drop it. If they both try to use it, then a deadlock occurs and this is bad for both players. Usually, these games are described by a payoff matrix like the one in Table 8, where we assume that d > a and c > b and that d − a c − b. Notice that Nash Equilibria of this game are unfair, as one player has utility max{c, d} and the other min{c, d}.\nFor the logit dynamics, we have that, for all β, the stationary expected social welfare is worse than the one guaranteed by a Nash equilibrium. On the other hand, for sufficiently large β we have that the expected utility of a player is always better than min{c, d}: that is, in the logit dynamics each player expects to gain more than in the worst Nash equilibrium. Moreover, the stationary distribution is a fair equilibrium, since every player has the same expected utility. As for the coordination games, the mixing time is exponential in β and in the minimum potential difference between adjacent profiles."},{"title":"The OR game","content":"In this section we consider the following simple n-player potential game that we here call OR game. Every player has two strategies, say {0, 1}, and each player pays the OR of the strategies of all players (including herself). More formally, the utility function of player i ∈ [n] is\notherwise.\nNotice that the OR game has 2 n − n Nash equilibria. The only profiles that are not Nash equilibria are the n profiles with exactly one player playing 1. Nash equilibrium 0 has social welfare 0, while all the others have social welfare −n.\nIn Theorem 10 we show that the stationary expected social welfare is always better than the social welfare of the worst Nash equilibrium, and it is significantly better for large β. Unfortunately, in Theorem 11 we show that if β is large enough to guarantee a good stationary expected social welfare, then the time needed to get close to the stationary distribution is exponential in n. Finally, in Theorem 12 we give upper bounds on the mixing time showing that if β is relatively small then the mixing time is polynomial in n, while for large β the upper bound is exponential in n and it is almost-tight with the lower bound. Despite the simplicity of the game, the analysis of the mixing time is far from trivial.\nTheorem 10 (Expected social welfare) The stationary expected social welfare of the logit dynamics for the OR game is E π [W ] = −αn where α = α(n, β) = (2 n −1)e −β 1+(2 n −1)e −β .\nProof. Observe that the OR game is a potential game with exact potential Φ where Φ(0) = 0 and Φ(x) = −1 for every x = 0. Hence the stationary distribution is\nwhere the normalizing factor is Z = 1 + (2 n − 1)e −β . The expected social welfare is thus\nIn the next theorem we show that the mixing time can be polynomial in n only if β c log n for some constant c.\nTheorem 11 (Lower bound on mixing time) The mixing time of the logit dynamics for the OR game is\nProof. Consider the set S ⊆ {0, 1} n containing only the state 0 = (0, . . . , 0) and observe that π(0) 1/2 for β log(2 n − 1). The bottleneck ratio is\nHence, by applying Theorem 4, the mixing time is\nIf β > log(2 n − 1) instead we consider the set R ⊆ {0, 1} n containing all states except state 0, and observe that\nand π(R) 1/2 for β > log(2 n − 1). It holds that\nThe bottleneck ratio is\nHence, by applying Theorem 4, the mixing time is\nIn Proof. We apply the path coupling technique (see Theorem 2 in Section 2) with the Hamming graph defined in (5). Let x, y ∈ {0, 1} n be two profiles differing only at player j ∈ [n] and, without loss of generality, let us assume |x| = k − 1 and |y| = k for some k = 1, . . . , n. We set the weight of edge {x, y} depending only on k, i.e. ℓ(x, y) = δ k where δ k 1 will be chosen later. Consider the coupling defined in Subsection 3.2. Now we evaluate the expected distance after one step E x,y [ρ(X 1 , Y 1 )] of the two coupled chains (X t , Y t ) starting at (x, y). Let i be the player chosen for the update. Observe that if i = j, i.e. if we update the player where x and y are different (this holds with probability 1/n), then the distance after one step is zero, otherwise we distinguish four cases depending on the value of k. Case k = 1: In this case profile x is all zeros and profile y has only one 1 and the length of edge {x, y} is ℓ(x, y) = δ 1 . When choosing a player i = j (this happens with probability (n − 1)/n), at the next step the two chains will be at distance δ 1 (if in both chains player i chooses strategy 0, and this holds with probability min{σ i (0 | x), σ i (0 | y)}), or at distance δ 2 (if in both chains player i chooses strategy 1, and this holds with probability min{σ i (1 | x), σ i (1 | y)}), or at distance δ 1 + δ 2 (if player i chooses strategy 0 in chain X 1 and strategy 1 in chain Y 1 , and this holds with the remaining probability). Notice that, from the definition of the coupling, it will never happen that player i chooses strategy 1 in chain X 1 and strategy 0 in chain Y 1 , indeed we have that\nHence the expected distance after one step is When player h is chosen (this holds with probability 1/n) we have that σ h (s | x) and σ h (s | y) for s = 0, 1 are the same as in (10). At the next step the two chains will be at distance δ 2 (if player h stays at strategy 1 in both chains), or at distance δ 1 (if player h chooses strategy 0 in both chains), or at distance δ 1 + δ 2 (if player h stays at strategy 0 in chain X 1 and chooses strategy 1 in chain Y 1 ).\nWhen a player i / ∈ {h, j} is chosen (this holds with probability (n − 2)/n) we have that σ i (0, x) = σ i (1, x) = σ i (0, y) = σ i (1, y) = 1/2. Thus in this case the two coupled chains always perform the same choice at player i, and at the next step they will be at distance δ 2 (if player i stays at strategy 0 in both chains) or at distance δ 3 (if player i chooses strategy 1 in both chains). Hence the expected distance after one step is\nCase 3 k n − 1: When a player i = j is chosen such that x i = y i = 1 (this holds with probability (k − 1)/n) then at the next step the two chains will be at distance δ k (if i stays at strategy 1) or at distance δ k−1 (if i moves to strategy 0). When a player i = j is chosen such that x i = y i = 0 (this holds with probability (n − k)/n) then at the next step the two chains will be at distance δ k (if i chooses to stay at strategy 0) or at distance δ k+1 (if i chooses to move to strategy 0). Hence the expected distance after one step is\nCase k = n: When a player i = j is chosen, then at the next step the two chains will be at distance δ n or at distance δ n−1 . Hence the expected distance after one step is\nIn order to apply Theorem 2 we now have to show that it is possible to choose the edge weights δ 1 , . . . , δ n and a parameter α > 0 such that\n1+e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 δ 2 e −α , 1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) δ k e −α , for k = 3, . . . , n − 1 ,\nFor different values of β, we make different choices for α and for the weights δ k . For clarity's sake we split the proof in three different lemmas. We denote by δ max the largest δ k . In Lemma 13 we show that Inequalities (15) are satisfied for every value of β by choosing the weights as follows\nand by setting α = 1/(2nδ max ). From Corollary 18, we have δ max = O( √ n2 n ). Observe that the diameter of the Hamming graph is n i=1 δ i nδ max , hence from Theorem 2 we obtain t mix = O(n 5/2 2 n ). In Lemma 14 we show that, if β < (1−ǫ) log n for an arbitrarily small constant ε > 0, Inequalities (15) are satisfied, for sufficiently large n, by choosing weights δ 1 = n 1−ε , δ 2 = 4/3, δ 3 = · · · = δ n = 1, and α = 1/n.. In this case the diameter is O(n) and, by Theorem 2, t mix = O(n log n).\nIn Lemma 15 we show that, Inequalities (15) are satisfied by choosing weights as follows\nwhere a 1 = n − 1 and b 1 = ne −β + 1 and, for every k = 2, . . . , n − 1\nand by setting α = 1/(2nδ max ). From Corollary 21 it follows that, if β c log n for a constant c ∈ N, we have that δ max = O(n c+2 ) and the diameter of the Hamming graph is O(n c+3 ). Thus, by Theorem 2 it follows that t mix = O(n c+3 log n)."},{"title":"Technical lemmas","content":"In this section we prove the technical lemmas needed for completing the proof of Theorem 12.\nLemma 13 Let δ 1 , . . . , δ n be as follows\nand let α = 1/(2nδ max ) where δ max = max{δ k : k = 1, . . . , n}. Then Inequalities (15) are satisfied for every β 0.\nProof. Observe that, for every k = 1, . . . , n, the right-hand side of the k-th inequality in (15) is\n.\nNow we check that the left-hand side is at most δ k − 1/(2n).\nFirst inequality (k = 1): n−1 n δ1\nFrom the definition of δ 1 in (16) we have that\nHence the left-hand side is\nSecond inequality (k = 2): 1\nFrom the definition of δ 2 in (16) we have that\nHence the left-hand side of the second inequality is 1 2n\nOther inequalities (k = 3, . . . , n − 1)\nFrom the definition of δ k in (16) we have that\nHence the left-hand side is\nLast inequality (k = n): n−1 2n (δ n + δ n−1 ) δ n e −α . Since δ n = 1 and δ n−1 = 1 n−1 δ n + 1 = n n−1 , the left-hand side of the last inequality is\nLemma 14 Let δ 1 , . . . , δ n be as follows\nwhere ε > 0 is an arbitrary small constant and let α = 1/n. Then Inequalities (15) are satisfied for every β (1 − ε) log n and n sufficiently large.\nProof. We check that all the inequalities in (15) are satisfied.\nFirst inequality (k = 1): n−1 n δ1 1+e −β + δ2 2 δ 1 e −α . For the left-hand side we have\nFor the right-hand side we have δ 1 e −α = n 1−ε e −1/n n 1−ε 1 − 1 n .\nHence the left-hand side is smaller than the right-hand one (for n sufficiently large).\nSecond inequality (k = 2): 1 2n 2 1+e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 δ 2 e −α . For the left-hand side we have 1 2n 2 1 + e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 = 1 2n 2 1 + e −β n 1−ε + (n − 1)\nAnd for the right-hand side we have\nHence the left-hand side is smaller than the right-hand one (for n sufficiently large). Third inequality (k = 3): 1 2n ((n − 1)δ 3 + 2δ 2 + (n − 3)δ 4 ) δ 3 e −α . For the left-hand side we have\nAnd for the right-hand side we have\nHence the left-hand side is smaller than the right-hand one.\nOther inequalities (k 4): 1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) δ k e −α . Since δ k = δ k−1 = δ k+1 = 1 the left-hand side is equal to n−1 n and the right-hand side is e −1/n n−1 n ."},{"title":"Lemma 15","content":"Let δ 1 , . . . , δ n be as follows\nwhere a 1 = n − 1 and b 1 = ne −β + 1 and for every k = 2, . . . , n − 1\nand let α = 1/(2nδ max ) where δ max = max{δ k : k = 1, . . . , n}. Then Inequalities (15) are satisfied for every β 0.\nBefore to prove the Lemma 15 we do the following observation.\nObservation 16 Let b k defined as in the Lemma 15. Then, for every k 2, it holds that b k kb k−1 .\nProof. We proceed by induction on k. The base case k = 2 follows from b 2 = (n + 1)(ne −β + 1) − (n − 1) = (n + 1)ne −β + 2 > 2(ne −β + 1) = 2b 1 ."},{"title":"Now suppose the claim holds for","content":"Proof ( Lemma 15). Observe that, as in Equation (17), for every k = 1, . . . , n, the right-hand side of the k-th inequality in (15) is\nNow we check that the left-hand side is at most δ k − 1/(2n).\nFirst inequality (k = 1): n−1 n δ1 1+e −β + δ2 2 δ 1 e −α . From the definition of δ 1 in (18) we have that\nHence the left-hand side is\nFrom the definition of δ 2 in (18) we have that\nHence the left-hand side is 1 2n 2 1 + e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 = 1 2n\nOther inequalities (k = 3, . . . , n − 1): 1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) δ k e −α . From the definition of δ k in (18) we have that\nHence the left-hand side is\nwhere the inequality follows from the Observation 16.\nLast inequality (k = n): n−1 2n (δ n + δ n−1 ) δ n e −α . Since δ n = 1 and δ n−1 = an−1 bn−1 δ n + 1 = an−1 bn−1 + 1, the left-hand side of the last inequality is\nwhere the inequality follows from the Observation 16.\nIn order to apply the path coupling theorem, we need to bound δ max : the next observation will represent the main tool to achieve this goal.\nObservation 17 Let δ 1 , . . . , δ n be defined recursively as follows: δ n = 1 and for a suitable constant c.\nIn order to bound δ max when δ 1 , . . . , δ n are defined as in Lemma 15 and β c log n for a constant c ∈ N, we define\nYou can check that p 1 = 0, q 1 = n and\nwe notice that p k = (n + 1)q k−1 − (k + 1)q k−1 q k for every k. We can also prove the following simple observation about q k ."},{"title":"Observation 19","content":"For every k 1 constant, we have q k 2 −k n k .\nProof. We proceed by induction on k, with the base k = 1 being obvious. Suppose the claim holds for k − 1, that is q k−1 2 −(k−1) n k−1 , then\nMoreover, you can check that l 1 = n − 1, r 1 = 1 and l k = (n − k)r k−1 r k = (n + 1)r k−1 − (k − 1)l k−1 ;\nwe notice that above recursion gives l k = (n − k)(k − 1)! and r k = k!. Next lemma bounds γ k defined in Equation 19.\nLemma 20 Let δ 1 , . . . , δ n be defined as in Lemma 15, γ k defined as in Equation (19) and β c log n for a constant c ∈ N. Then, for sufficiently large n, it holds that\nProof. Since p k q k , then (nq k − p k )e −β > 0; instead, l k − nr k = (k − 1)!(n − k − nk) < 0. Hence we have for every k\nInductively, we show that for every k c + 3, we have γ k < 1. Set k = c + 3: c is a constant, thus Observation 19 holds for k − 1; hence and since e −β n −c , we have that\nInstead, l c+3 − r c+3 = (c + 2)!(n − 2c − 6) (c + 2)! · n. Thus,\nfor n sufficiently large. Now, suppose that γ k−1 < 1; then, we have\nwhere a k−1 < b k−1 is implied by the inductive hypothesis. In order to complete the proof, we need to show that γ c+2 = O(1). Similarly to the case k = c + 3, we obtain (q c+2 − p c+2 )e −β 2 −c n and l c+2 − r c+2 (c + 1)! · n. Hence, γ c+2 p c+2 + r c+2 + (c + 1)! · n p c+2 + r c+2 + 2 −c n (c + 1)! · 2 c = O(1) ."},{"title":"Corollary 21","content":"Let δ 1 , . . . , δ n and c be defined as in Lemma 15. Then δ max = O(n c+2 ).\nProof. From Observation 17, Lemma 20 and the definition of δ 1 , . . . , δ n it follows that"},{"title":"The XOR game","content":"In this section we analyze the logit dynamics for another simple n-player game, the XOR game. The XOR game is a symmetric n-player game in which each player has two strategies, denoted by 0 and 1, and each player pays the XOR of the strategies of all players (including herself). More formally, for each i ∈ [n], the utility function u i (·) is defined as follows\nx has an odd number of 1's; 0, if x has an even number of 1's.\nNotice that the XOR game has 2 n−1 Nash equilibria, namely all profiles with an even number of players playing strategy 1. Nash equilibria have social welfare 0 and profiles not in equilibria have social welfare −n. Observe that the XOR game is a potential game with exact potential Φ where Φ(x) = u i (x) for every x and every i ∈ [n]. Hence, the stationary distribution is\nx has an odd number of 1's; 1/Z, if x has an even number of 1's;\nwhere the normalizing factor is Z = 2 n−1 (1 + e −β ). Even if this game looks similar to the OR game, it exhibits a different behavior. Theorem 22 gives the stationary expected social welfare of the XOR game and we can see that, as β increases, the expected social welfare tends from below to the social welfare at the Nash equilibria. In contrast the expected social welfare of the OR game is better than the worst Nash equilibrium for all values of β. Moreover, in Theorem 23 and Theorem 24 we show that the mixing time for the XOR game is polynomial in n and exponential in β, whereas the mixing time for the OR game can be bounded independently from β.\nTheorem 22 (Expected social welfare) The stationary expected social welfare of the logit dynamics for the XOR game is E π [W ] = − n 1+e β . Proof. The expected social welfare is\nThe next theorem shows that the mixing time is exponential in β for every β > 0.\nTheorem 23 (Lower bound on mixing time) The mixing time of the logit dynamics for the XOR game is Ω(e β ).\nProof. Consider the set S ⊆ {0, 1} n containing only the state 0 = (0, . . . , 0). Observe that π(0) 1/2. The bottleneck ratio is\ny∈{0,1} n π(0)P (0, y) = y∈{0,1} n : |y|=1\nHence, by applying Theorem 4, the mixing time is\nFinally, in the next theorem we give an almost matching upper bound to the mixing time.\nTheorem 24 (Upper bound on mixing time) The mixing time of the logit dynamics for the OR game is O(n 3 e β ).\nThe theorem is proved using coupling (see Theorem 1) and proof is presented in the next sections. Specifically, we use the coupling described in Section 3.2; in Section 7.1 we show that if the coupled chains are at even distance then distance does not increase after one step of the coupling; in Section 7.2 we show that if the coupled chains are at odd distance then they get closer distance with probability independent from β; finally, in Section 7.3 we bound the expected time needed by the two chains to coalesce and use Theorem 1 to derive an upper bound for the mixing time."},{"title":"Even Hamming distance","content":"Let X t and Y t be two chains coupled as described in Section 3.2. Suppose that X t = x, Y t = y, and H(x, y) = 2ℓ, for ℓ > 0. In this case, u i (x) = u i (y) = b for all i ∈ [n] and some b ∈ {−1, 0}. Let i be the index selected for update and let us distinguish two cases. In the first case x i = y i and we have\nand thus\nTherefore the coupling always update the strategy of player i in the same way in the two chains and thus H(X t+1 , Y t+1 ) = 2ℓ.\nIn the second case we have x i = y i and we assume, without loss of generality, that x i = 0 and y i = 1. We observe that, for b ∈ {−1, 0},\nTherefore we have\n1 + e (1+2b)β and thus we have three possible updates for the strategy of player i:\n1. both chains update to 0 (and thus H(X t+1 , Y t+1 ) = 2ℓ − 1) with probability min 1 1 + e (1+2b)β ,\n2. both chains update to 1 (and thus H(X t+1 , Y t+1 ) = 2ℓ − 1) with probability min 1 1 + e (1+2b)β ,\n3. chain X and Y choose two different strategies for updating the strategy of player i (and thus H(X t+1 , Y t+1 ) = 2ℓ) with probability\nThe following lemma summarizes the above observations. Lemma 25 Suppose that H(X t , Y t ) = 2ℓ, for ℓ > 0. Then\n2ℓ − 1, with probability 2ℓ n · 2 1+e β ; 2ℓ, with probability 1 − 2ℓ n · 2 1+e β ."},{"title":"Odd Hamming distance","content":"Let X t and Y t be two chains coupled as described in Section 3.2. Suppose that X t = x, Y t = y, and H(x, y) = 2ℓ − 1, for ℓ > 0. In this case we have u i (x) = b and u i (y) = −(1 + b) for some b ∈ {−1, 0}. Let i be the index selected for update and let us distinguish two cases. In the case in which x i = y i = c for some c ∈ {0, 1}, we have 3. chains X and Y choose two different strategies for updating the strategy player i (and thus H(X t+1 , Y t+1 ) = 2ℓ) with probability 1 − 2 1+e β .\nIn the second case we have x i = y i and we assume, without loss of generality, that x i = 0 and y i = 1. We observe that\nTherefore we have σ i (0 | x) = σ i (0 | y) and σ i (1 | x) = σ i (1 | y)\nand thus in this case H(X t+1 , Y t+1 ) = 2ℓ − 2.\nThe following lemma summarizes the above observations."},{"title":"Lemma 26","content":"Suppose that H(X t , Y t ) = 2ℓ − 1, for ℓ > 0. Then\n2ℓ − 2, with probability 2ℓ−1 n ;\n2ℓ − 1, with probability n−2ℓ+1 n 2 1+e β ; 2ℓ, with probability n−2ℓ+1 n 1 − 2 1+e β ."},{"title":"Time to coalesce","content":"We denote with τ k the random variable indicating the first time at which the two coupled chains have distance k. More precisely, τ k = min{t : H(X t , Y t ) = k}.\nTherefore, τ couple = τ 0 is the time needed for the two chains to coalesce. We next give a bound on the expected time E x,y [τ couple ] for the two chains to coalesce starting from x and y. If x and y have distance 2ℓ, we denote by µ ℓ the expected time to reach distance 2ℓ − 2. That is,\nSimilarly, if x and y have distance 2ℓ − 1, we denote by ν ℓ the expected time to reach distance 2ℓ − 2.\nThat is,\nNotice that, if H(x, y) = H(x ′ , y ′ ) then\nfor all k, and thus the µ ℓ and ν ℓ are well defined. From Lemma 25 and Lemma 26, we have the following relations µ ℓ = 1 + µ ℓ · 1 − 2ℓ n · 2 1 + e β + ν ℓ · 2ℓ n · 2 1 + e β ν ℓ = 1 + ν ℓ · n − 2ℓ + 1 n · 2 1 + e β + µ ℓ · n − 2ℓ + 1 n · 1 − 2 1 + e β ."},{"title":"Conclusions and open problems","content":"In this paper we studied strategic games where at every run a player is selected uniformly at random and she is assumed to choose her strategy for the next run according to the logit dynamics: a noisy best-response dynamics where the noise level is tuned by a parameter β. Such dynamics defines a family of ergodic Markov chains, indexed by β, over the set of strategy profiles. We proposed the stationary distribution of these Markov chains as solution concept for games where players have bounded rationality or limited knowledge about the system. Since this solution concept does not assume full rationality of agents, it avoids one of the main drawbacks of many classical equilibria concepts. Moreover, the stationary distribution of an ergodic Markov chain always exists, it is unique, and the chain converges to such a distribution from any starting state.\nIn order to evaluate the long-term performance of the system, on the one hand we analyzed the expected social welfare when the strategy profiles are random according to the stationary distribution, on the other hand we studied the mixing time, i.e. how long it takes, for a chain starting at an arbitrary profile, to get close to its stationary distribution.\nIn this paper we applied this approach to some simple but well-studied games with a constant number of players: the CK game, that obtains the worst Price of Anarchy bound between linear congestion games, and the 2 × 2 coordination games considered in the seminal paper about logit dynamics [3]. We also considered two simple n-player games, the OR game and the XOR game: the analysis of the mixing time turned out to be far from trivial even for such simple games. The above games highlight a twofold behavior: for some games, namely CK game and OR game, the mixing time can be upper bounded by a function independent of β, whereas the mixing time for the other games depends exponentially on the noise parameter β.\nThe main goal of our line of research is to investigate logit dynamics for notable classes of n-player games. It would also be interesting to consider variations of the logit dynamics where players update their strategies simultaneously or where the noise is not uniform between players.\nWe have seen that, for some games and for some values of β, the mixing time can be exponential in the number of players. When it takes such a long time to reach the stationary distribution, it would be interesting to investigate the evolution of the system in the transient phase of the logit dynamics."},{"title":"A","content":""},{"title":"= 2 :","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"The evolution of a system is determined by its dynamics and complex systems are often described by looking at the equilibrium states induced by their dynamics. Once the system reaches an equilibrium state it stays there, thus equilibrium states describe the long-term behavior of the system. In this paper we are mainly interested in systems whose individual components are selfish agents. The state of a selfish system is fully described by a vector of strategies, each controlled by one agent, and each state assigns a payoff to each agent. The agents are selfish in the sense that they pick their strategy so to maximize their payoff, given the strategies of the other agents. Nash equilibrium is the classical notion of equilibrium for selfish systems and it corresponds to the equilibrium induced by the best-response dynamics. The observation that selfish systems are described by their equilibrium states (that is, by the Nash equilibria) has motivated the notions of Price of Anarchy [15] and Price of Stability [1] and the analysis of efficiency of selfish systems based on such notions.\\nHowever, such analysis inherits some of the shortcomings of the concept of a Nash equilibrium. First of all, the best-response dynamics assumes that selfish agents have complete knowledge of the current state of the system; that is, they know the payoff associated with each of their possible choices and each of the strategies chosen by other agents. Instead, in most cases, agents have only approximate knowledge of the system state or they are not able to compute their best choice. Moreover, in presence of multiple equilibria, it is not clear which one of them will be reached by the system, as it may depend on the initial state: Price of Anarchy considers the worst-case equilibrium, whereas Price of Stability focuses on the best-case equilibrium. Finally, Nash equilibria are hard to compute [7,5] and thus for some systems it might take very long to reach a Nash equilibrium: in this case using equilibrium states to describe the system performance is not well justified. Rather, one would like to analyze the performance of a system by using a dynamics (and its related equilibrium notion) that has the following three properties:\\n• The dynamics takes into account the fact that the system components might have a perturbed or noisy knowledge of the system;\\n• For every system the equilibrium state exists and is unique;\\n• The system reaches the equilibrium very quickly regardless of the starting state.\\nIn this paper, we consider noisy best-response dynamics in which the behavior of the agents is described by a parameter β 0. The case β = 0 corresponds to agents picking their strategies completely at random (that is, the agents have no knowledge of the system) and the case β = ∞ corresponds to agents picking their strategies according to the best-response dynamics (in which the agents have full and complete knowledge of the system). The intermediate values of β correspond to agents that are roughly guided by the best-response dynamics but can make a sub-optimal choice due, for example, to bounded rationality of the agent or limited knowledge about the system: this sub-optimal behavior occurs with some probability that depends on β (and on the associated payoff).\\nWe will study a specific noisy best-response dynamics for which the system evolves according to an ergodic Markov chain for all β 0. For these systems, it is natural to look at the stationary distribution (which is the equilibrium state of the Markov chain) and to analyze the expected social welfare (the sum of utility functions) of the system at that distribution. We stress that the noisy best-response dynamics well models agents that only have approximate or noisy knowledge of the system and that for ergodic Markov chains (such as the ones arising in our study) the stationary distribution is known to exist and to be unique. Moreover, to justify the use of the stationary distribution for analyzing the performance of the system, we will study how fast the Markov chain converges to the stationary distribution.\\nRelated Works and Our Results. Several dynamics, besides the best-response dynamics, and several notions of equilibrium, besides Nash equilibria, have been considered to describe the evolution of a selfish system and to analyze its performance. See, for example, [11,21,20]. Equilibrium concepts based on the best-response. When the game does not possess a Pure Nash equilibrium, the best-response dynamics will eventually cycle over a set of states (in a Nash equilibrium the set is a singleton). These states are called sink equilibria [12]. Sink equilibria exist for all games and, in some contexts, they seem a better approximation of the real setting than mixed Nash equilibria. Unfortunately, sink equilibria share two undesirable properties with Nash equilibria: a game can have more that one sink equilibrium and sink equilibria seem hard to compute [9].\\nOther notions of equilibrium state associated with best-response dynamics are the unit-recall equilibria and component-wise unit-recall equilibria (see [9]). However, we point out that the former does not always exist and that the latter imposes too strict limitations on the players. No-Regret Dynamics. Another broadly explored set of dynamics are the no-regret dynamics (see, for example, [11]). The regret of an user is the difference between the long-term average cost and the average cost of the best strategy in hindsight. In the no-regret dynamics the regret of every player after t steps is o(t) (sublinear with time). In [10,14] it is showed that the no-regret dynamics converges to the set of correlated equilibria. Note that the convergence is to the set of correlated equilibria and not to a specific correlated equilibrium. Our work. In this paper we consider a specific noisy best-response dynamics called the logit dynamics (see [3]) and we study its mixing time (that is, the time it takes to converge to the stationary distribution) and the stationary expected social welfare. Specifically,\\n• We start by analyzing the logit dynamics for a simple 3-player linear congestion game (the CK game [6]) which exhibits the worst Price of Anarchy among linear congestion games. We show that the mixing time of the logit dynamics is upper bounded by a constant independent of β. Moreover, we show that the stationary expected social welfare is larger than the social welfare of the worst Nash equilibrium for all β;\\n• We then analyze the 2 × 2 coordination games studied in [3]. Here we show that, under some conditions, the stationary expected social welfare is larger than the social welfare of the worst Nash equilibrium. We give upper and lower bounds on the mixing time exponential in β. We also observe that the same bounds apply to anti-coordination games;\\n• Finally, we apply our analysis to two simple n-player games: the OR game and XOR game. We give upper and lower bounds on the mixing time: we show that the mixing time of the OR game can be upper bounded by a function independent of β, while the mixing time of the XOR game increases exponentially in β. We also prove that for β = O(log n) the mixing time is polynomial in n for both games.\\nThe logit dynamics has been first studied by Blume [3] who showed that, for 2 × 2 coordination games, the long-term behavior of the Markov chain is concentrated in the risk dominant equilibrium (see [13]) for sufficiently large β. Ellison [8] studied different noisy best-response dynamics for coordination games assuming that interaction among players were described by a graph; that is, the utility of a player is determined only by the strategies of the adjacent players. Specifically, Ellison [8] studied interaction modeled by rings and showed that some large fraction of the players will eventually choose the risk dominant strategy. Similar results were obtained by Peyton Young [22] for the logit dynamics and for more general families of graphs. Montanari and Saberi [17] gave bounds on the hitting time (the expected time that the logit dynamics takes to reach a specific state) of the risk dominant equilibrium state in terms of some graph theoretic properties of the underlying interaction network. Asadpour and Saberi [2] studied the hitting time for a broader class of congestion games. We notice that none of [3,8,22] gave any bound on the convergence time to the risk dominant equilibrium. Montanari and Saberi [17] were the first to do so but their study focuses on the hitting time of a specific configuration and not on the convergence time to the stationary distribution. From a technical point of view, our work follows the lead of [3,8,22] and extends their technical findings by giving bounds on the mixing time of the Markov chain of the logit dynamics. We stress that previous results only proved that, for sufficiently large β, eventually the system concentrates around certain states without further quantifying the rate of convergence nor the asymptotic behaviour of the system for small values of β. Instead, we identify the stationary distribution of the logit dynamics as the global equilibrium and we evaluate the social welfare at stationarity and the time it takes the system to reach it (the mixing time) as explicit functions of β.\\nWe choose to start our study from the class of coordination games considered in [3] and two simple n-player games (the OR game and the XOR game). We give nearly tight upper and lower bounds on the mixing time. Despite their game-theoretic simplicity, the analytical study of the mixing time of the logit dynamics for the two n-player games is far from trivial. We notice that the results in [17] cannot be used to derive upper bounds on the mixing time.\\nFrom a more conceptual point of view, our work tries (similarly to [12,9,19]) to introduce a solution concept that well models the behavior of selfish agents, is uniquely defined for any game, and is quickly reached from any starting state. We propose the stationary distribution induced by the logit dynamics as a possible solution concept and exemplify its use in the analysis of the performance of some 2 × 2 games (as the ones considered in [3]), of games used to obtain tight bounds on the Price of Anarchy, and of two simple multi-player games.\\nOrganization of the paper. In Section 2 we summarize some Markov chain notions that we will use throughout the paper. In Section 3 we formally describe the logit dynamics for strategic games. We also describe the coupling we will repeatedly use in the proofs of the upper bounds on mixing times. In Sections 4, 5, 6 and 7 we study the stationary expected social welfare and the mixing time of the logit dynamics for CK game, coordination games, the OR game, and the XOR game, respectively. Finally, in Section 8 we present conclusions and some open problems. Notation. We write S for the complementary set of a set S; we write |S| for its size. We use bold symbols for vectors; when x = (x 1 , . . . , x n ) ∈ {0, 1} n we write |x| for the number of 1s in x; i.e., |x| = |{i ∈ [n] :\\nx i = 1}|. For two vectors x, y let H(x, y) = |{i ∈ [n] : x i = y i }| be their Hamming distance: we write x ∼ y if H(x, y) = 1. We use the standard game theoretic notation (x −i , y) to mean the vector obtained from x by replacing the i-th entry with y, i.e. (x −i , y) = (x 1 , . . . , x i−1 , y, x i+1 , . . . , x n ).\"\n },\n {\n \"title\": \"Markov chains summary and notation\",\n \"content\": \"We summarize the main tools we use to bound the mixing time of Markov chains (for a complete description of such tools see, for example, Chapters 5.2, 7.2, 12.2 and 14.2 of [16]. We refer the reader to [16] also for notational conventions).\\nConsider a Markov chain M with finite state space Ω and transition matrix P . It is a classical result that for an irreducible and aperiodic Markov chain 1 there exists an unique stationary distribution π over Ω; that is, a distribution π on Ω such that π · P = π. The total variation distance µ − ν TV between two probability distributions µ and ν on Ω is defined as\\nAn irreducible and aperiodic Markov chain M converges to its stationary distribution π; specifically, there exists 1 > α > 0 such that\\nwhere\\nand P t (x, ·) is the distribution at time t of the Markov chain starting at x. For 1/2 > ε > 0, the mixing time is defined as t mix (ε) = min{t ∈ N : d(t) ε}.\\nIt is usual to set ε = 1/4 or ε = 1/2e. If not explicitly specified, when we write t mix we mean t mix (1/4). Observe that t mix (ǫ) ⌈log 2 ǫ −1 ⌉t mix .\\nCoupling. A coupling of two probability distributions µ and ν on Ω is a pair of random variables (X, Y ) defined on Ω × Ω such that the marginal distribution of X is µ and the marginal distribution of Y is ν. A coupling of a Markov chain M with transition matrix P is a process (X t , Y t ) ∞ t=0 with the property that both X t and Y t are Markov chains with transition matrix P . When the two coupled chains start at (X 0 , Y 0 ) = (x, y), we write P x,y (·) and E x,y [·] for the probability and the expectation on the space where the two chains are both defined. We denote by τ couple the first time the two chains meet; that is,\\nWe will consider only couplings of Markov chains with the property that for s τ couple , it holds X s = Y s . The following theorem can be used to give an upper bound on t mix (see, for example, Corollary 5.3 in [16]).\\nTheorem 1 (Coupling) Let M be a Markov chain with state space Ω and transition matrix P . For each pair of states x, y ∈ Ω consider a coupling (X t , Y t ) of M with starting states X 0 = x and Y 0 = y. Then\\nSometimes it is difficult to specify a coupling and to analyze the coupling time τ couple for each pair of starting states x and y. The Path Coupling theorem says that it is sufficient to define a coupling only for pairs of Markov chains starting from adjacent states and an upper bound on the mixing time can be obtained if each of these couplings contracts their distance on average. More precisely, consider a Markov chain M with state space Ω and transition matrix P ; let G = (Ω, E) be a connected graph and let w : E → R be a function assigning weights to the edges such that w(e) 1 for every edge e ∈ E; for x, y ∈ Ω, we denote by ρ(x, y) the weight of the shortest path in G between x and y. The following theorem holds.\\nTheorem 2 (Path Coupling [4]) Suppose that for every edge {x, y} ∈ E a coupling (X t , Y t ) of M with X 0 = x and Y 0 = y exists such that E x,y [ρ(X 1 , Y 1 )] e −α · w({x, y}) for some α > 0. Then\\nSpectral techniques. A Markov chain M with state space Ω and transition matrix P is said reversible if for all x, y ∈ Ω, it holds that π(x) · P (x, y) = π(y) · P (y, x).\\nThe eigenvalues of the transition matrix P of a reversible Markov chain M can be used to obtain upper and lower bounds on the mixing time. Observe that all the eigenvalues of any transition matrix P have absolute value at most 1, λ = 1 is an eigenvalue, and for irreducible and aperiodic chains, −1 is not an eigenvalue. The relaxation time t rel of a Markov chain M is defined as\\nwhere λ ⋆ is the largest absolute value among eigenvalues other than 1, λ ⋆ = max{|λ| : λ is an eigenvalue of P, λ = 1} .\\nObserve that, for M reversible, irreducible and aperiodic, 0 λ ⋆ < 1 and thus t rel is positive and finite. We have the following theorem (see, for example, Theorems 12.3 and 12.4 in [16]).\"\n },\n {\n \"title\": \"Theorem 3 (Relaxation time)\",\n \"content\": \"Let P be the transition matrix of a reversible, irreducible, and aperiodic Markov chain with state space Ω and stationary distribution π. Then\\nwhere π min = min x∈Ω π(x).\\nLower bound. We will use the following theorem to derive our lower bounds (see, for example, Theorem 7.3 in [16]).\\nTheorem 4 (Bottleneck ratio) Let M = {X t : t ∈ N} be an irreducible and aperiodic Markov chain with finite state space Ω, transition matrix P , and stationary distribution π. Let S ⊆ Ω be any set with π(S) 1/2. Then the mixing time is\\nwhere Φ(S) = Q(S, S) π(S) and Q(S, S) = x∈S, y∈S π(x)P (x, y).\"\n },\n {\n \"title\": \"The model and the problem\",\n \"content\": \"where x ∈ S 1 × · · · × S n is the strategy profile played at the current time step, T i (x) = z∈Si e βui(x−i,z) is the normalizing factor, and β 0.\\nParameter β is called inverse noise of the system, indeed from (1) it is easy to see that, for β = 0 player i selects her strategy uniformly at random, for β > 0 the probability is biased toward strategies promising higher payoffs, and for β → ∞ player i chooses her best response strategy (if more than one best response is available, she chooses uniformly at random one of them). Moreover observe that probability σ i (y | x) does not depend on the strategy x i currently adopted by player i. The above dynamics defines a Markov chain with the set of strategy profiles as state space, and where the transition probability from profile x = (x 1 , . . . , x n ) to profile y = (y 1 , . . . , y n ) is zero if the H(x, y) 2 and it is 1 n σ i (y i | x) if the two profiles differ exactly at player i. More formally, we can define the logit dynamics as follows.\\nDefinition 5 (Logit dynamics [3]) Let G = ([n], S, U) be a strategic game and let β 0. The logit dynamics for G is the Markov chain M β = {X t : t ∈ N} with state space Ω = S 1 ×· · ·×S n and transition matrix\"\n },\n {\n \"title\": \"Properties. Logit dynamics enjoys some interesting properties:\",\n \"content\": \"Ergodicity. It is easy to see that the logit dynamics is irreducible and aperiodic. Indeed, let x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ) be two profiles and let (z 0 , . . . , z n ) be a path of profiles where z 0 = x, z n = y and z i = (y 1 , . . . , y i , x i+1 , . . . x n ) for i = 1, . . . , n − 1. The probability that the chain starting at x is in y after n steps is P n (x, y) = P n (z 0 , z n ) P n−1 (z 0 , z n−1 )P (z n−1 , z n ) and recursively\\nwhere the last inequality follows from (2) because, for all i = 1, . . . , n, the Hamming distance between z i−1 and z i is at most 1. Hence there is a unique stationary distribution π and, for every starting profile x, the distribution of the chain P t (x, ·) converges to π in total variation as t tends to infinity.\\nwe get a new gameG = ([n], S,Ũ ) but the same logit dynamics. Indeed, according to (1), the probability player i chooses strategy y when the game is at profile x is\\nNoise changes under utility rescaling. While translations of utilities do not affect logit dynamics, a rescaling of the utility functions for the same constant α > 0 changes the inverse noise from β to α · β. Indeed, if for every player i and every profile x we set\\nz∈Si e αβui(x−i,z) . Notice that, unlike translations constants, we here must have the same rescaling constant α for all utility functions.\\nPotential Games. A game G = ([n], S, U) is said a (exact) potential game if a function Φ : S 1 × · · · × S n → R exists such that, for every player i and for every pair of profiles x and y that differ only at position i, it holds that\\nis a potential game with potential function Φ, then the Markov chain given by (2) is reversible and its stationary distribution is the Gibbs measure\\nwhere Z = y∈S1×···×Sn e βΦ(y) is the normalizing constant. Except for the Matching Pennies example in Subsection 3.1, all the games we analyze in this paper are potential games. Logit dynamics vs Glauber dynamics. When G is a potential game, the logit dynamics is equivalent to the well-studied Glauber dynamics. For state space Ω = S 1 × · · · × S n and probability distribution µ over Ω, the Glauber dynamics for µ proceeds as follows: From profile x ∈ Ω, pick a player i ∈ [n] uniformly at random and update her strategy at y ∈ S i with probability µ conditioned on the other players being at\\n.\\nIt is easy to see that the Markov chain defined by the Glauber dynamics is irreducible, aperiodic, and reversible with stationary distribution µ. When G = ([n], S, U) is a potential game with potential function Φ, the logit dynamics defines the same Markov chain as the Glauber dynamics for the Gibbs distribution π in (3). Indeed, in that case we have\\n.\\nHence, logit dynamics for potential games and Glauber dynamics for Gibbs distributions are two ways of looking at the same Markov chains: in the former case the dynamics is derived from the potential function, in the latter case from the stationary distribution. However, observe that, if G is not a potential game and π is the stationary distribution of the logit dynamics for G, in general the Glauber dynamics for π is different from the logit dynamics (see, for example, the Matching Pennies case in Subsection 3.1). Due to the analogies between logit and Glauber dynamics, we will sometimes adopt the terminology used by physicists to indicate the quantities involved; in particular we will call parameter β the inverse noise or inverse temperature and we will call partition function the normalizing constant Z of the Gibbs distribution (3).\\nStationary expected social welfare and mixing time. Let W : S 1 × · · · × S n −→ R be a social welfare function (in this paper we assume that W is simply the sum of all the utility functions W (x) = n i=1 u i (x), but clearly any other function of interest can be analysed). We study the stationary expected social welfare, i.e. the expectation of W when the strategy profiles are random according to the stationary distribution π of the Markov chain,\\nSince the Markov chain defined in (2) is irreducible and aperiodic, from every initial profile x the distribution P t (x, ·) of chain X t starting at x will eventually converge to π as t tends to infinity. We will be interested in bounding how long it takes to get close to the stationary distribution, that is the mixing time of the Markov chain. In the next subsection we illustrate the goals of our work with two simple examples.\"\n },\n {\n \"title\": \"Two simple examples: Matching Pennies and a Stairs game\",\n \"content\": \"Matching Pennies. Consider the classical Matching Pennies game. We write the utility functions in the standard bimatrix form.\\nAccording to (1), the update probabilities for the logit dynamics are, for every x ∈ {H, T }\\nHence the transition matrix (see (2)) is\\nwhere, for readability sake, we named b = 1 1+e −2β . Since every column of the matrix adds up to 1, the uniform distribution π over the set of strategy profiles is the stationary distribution for the logit dynamics. The stationary expected social welfare is thus 0 for every inverse noise β.\\nAs for the mixing time, it is easy to see that it is upper bounded by a constant independent of β. Indeed, a direct calculation shows that, for every x ∈ {HH, HT, T H, T T } and for every β 0 it holds that\\nA stairs game. One of the main techniques used to give upper bounds on the mixing time of Markov chains is the coupling technique (see Theorem 1). In the following example we use it to upper bound the mixing time of the logit dynamics for a simple game. Let G be a potential game where every player has two strategies, say upstairs (or 1) and downstairs (or 0), and the potential of a profile x ∈ {0, 1} n is the number of players that are upstairs, i.e. Φ(x) = |x|.\\nNotice that the logit dynamics (and thus the stationary distribution and the mixing time) is completely defined by the potential function, while if we wanted to evaluate the stationary expected social welfare we would need to specify the utility functions.\\nThe partition function is\\nSo the stationary distribution is\\nAs for the mixing time, we can use the coupling technique as follows: observe that the probability of playing strategy 1 (or equivalently strategy 0), for the player selected for the update, is independent of the current strategies of the other players. Indeed, according to (1), for every x it holds that\\nWe can define a coupling of two Markov chains starting at two different profiles as follows: choose i ∈ [n] uniformly at random and perform the same update at player i in both chains 2 . When every player has been chosen at least once the two chains have coalesced. From the coupon collector's argument, it takes O(n log n) to have that, with probability at least 3/4, all players have been chosen at least once. By applying Theorem 1 we have that the mixing time is O(n log n).\\nIn the above examples, it turned out that the mixing time of the logit dynamics can be upper bounded by functions that do not depend on the inverse noise β. As we shall see in the next sections, this is not always the case. Moreover, the analysis of the mixing time is usually far from trivial.\"\n },\n {\n \"title\": \"Description of the Coupling\",\n \"content\": \"Throughout the paper we will use the coupling and path-coupling techniques (see Theorem 1 and Theorem 2) to give upper bounds on mixing times. Since we will use the same coupling idea in several proofs, we describe it here and we will refer to this description when we will need it.\\nConsider an n-player 2-strategy game G and let us rename 0 and 1 the strategies of every player. For every pair of strategy profiles x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ {0, 1} n we define a coupling (X 1 , Y 1 ) of two copies of the Markov chain with transition matrix P defined in (2) for which X 0 = x and Y 0 = y.\\nThe coupling proceeds as follows: first, pick a player i uniformly at random; then, update the strategies x i and y i of player i in the two chains, by setting\\nThree easy observations are in order: if σ i (0 | x) = σ i (0 | y) and player i is chosen, then, after the update, we have x i = y i ; for every player i, at most one of the updates (x i , y i ) = (0, 1) and (x i , y i ) = (1, 0) has positive probability; if i is chosen for update, then the marginal distributions of x i and y i agree with σ i (· | x) and σ i (· | y) respectively, indeed, for b ∈ {0, 1}, the probability that\\nand the probability that\\nWe define G = (Ω, E) as the Hamming graph of the game, where Ω = {0, 1} n is the set of strategy profiles, and two profiles x = (x 1 , . . . , x n ), y = (y 1 , . . . , y n ) ∈ Ω are adjacent if they differ only for the strategy of one player, i.e.\\n{x, y} ∈ E ⇐⇒ x ∼ y .\\nFor the path coupling technique, the coupling described above is applied only to pairs of adjacent starting profiles.\"\n },\n {\n \"title\": \"A 3-player congestion game\",\n \"content\": \"In this section we analyze the CK game, a simple 3-player linear congestion game introduced in [6]. This game is interesting because it highlights the weakness of the Price of Anarchy notion for the logit dynamics. Indeed, the CK game exhibits the worst Price of Anarchy with respect to the average social welfare among all linear congestion games with 3 or more players. But, as we shall see soon, the stationary expected social welfare of the logit dynamics is always larger than the social welfare of the worst Nash equilibrium and, for large enough β, players spend most of the time in the best Nash equilibrium. Moreover, we will show that the mixing time of the logit dynamics can be bounded independently from β: that is, the stationary distribution guarantees a good social welfare and it is quickly reached by the system. Let us now describe the CK game. We have 3 players and 6 facilities divided into two sets: G = {g 1 , g 2 , g 3 } and H = {h 1 , h 2 , h 3 }. Player i ∈ {0, 1, 2} has two strategies: Strategy \\\"0\\\" consists in selecting facilities (g i , h i ); Strategy \\\"1\\\" consists in selecting facilities (g i+1 , h i−1 , h i+1 ) (index arithmetic is modulo 3). The cost of a facility is the number of players choosing such facility, and the welfare of a player is minus the sum of the costs of the facilities she selected. It easy to see that this game has two pure Nash equilibria: the solution where every player plays strategy 0 (each player pays 2, which is optimal), and the solution where every player plays strategy 1 (each player pays 5). The game is a congestion game, and thus, by [18], it is also a potential game and its potential function is:\\nwhere L x (j) is the number of players using facility j in configuration x.\\nStationary expected social welfare. It is easy to see that the update probabilities given by the logit dynamics for this game (see Equation (1)) only depend on the number of players playing strategy 1 and not on which player is actually playing that strategy. In particular, we have that, from a profile x, the player i, if selected for update, plays strategy 0 with the following probabilities:\\nand strategy 1 with the remaining probabilities. Next theorem evaluates the stationary expected social welfare for this game.\\nTheorem 6 (Expected social welfare) The stationary expected social welfare E π [W ] of the logit dynamics for the CK game is\\nProof. We notice that two profiles with the same number of players playing strategy 1 have both the same potential (and, by Equation (3), the same stationary distribution) and the same social welfare. Thus, π(x) = π[k] and W (x) = W [k] for a profile x such that |x| = k, where\\nwhere Z(β) = e −6β + 3e −10β + 4e −12β , and\\nHence, the stationary expected social welfare is\\nNotice that for β = 0 we have E π [W ] = −27/2, which is better than the social welfare of the worst Nash equilibrium. This means that, even if each player selects her strategy at random, the logit dynamics drives the system to a random profile whose expectation to be better than the worst Nash equilibrium. We also observe that E π [W ] increases with β and thus the long-term behavior of the logit dynamics gives a better social welfare than the worst Nash equilibrium for any β 0. Moreover, the stationary expected social welfare approaches the optimal social welfare as β tends to ∞.\\nMixing time. Now we study the mixing time of the logit dynamics for the CK game and we show that it is bounded by a constant for any β 0. The proof will use Coupling Theorem (see Theorem 1).\\nTheorem 7 (Mixing time) There exists a constant τ such that the mixing time t mix of the logit dynamics of the CK game is upper bounded by τ for every β 0.\\nProof. First, we notice that the update probabilities given in Equation (6) imply that\\nLet X t and Y t be two copies of the logit dynamics for the CK game, starting in x and y respectively, coupled as described in Section 3.2. It is easy to check that, by Equation (7), the player selected for update, chooses strategy 0 in both chain with probability at least 1/2. Finally, we bound the probability that after three steps the two coupled chains coalesce: it is at least as large as the probability that we choose three different players and all of them play strategy 0 at their turn, i.e.\\nSince this bound holds for every starting pair (x, y), we have that the probability the two chains have not yet coalesced after 3t steps is\\nThe thesis follows from the Theorem 1.\"\n },\n {\n \"title\": \"Two player games\",\n \"content\": \"In this section we analyse the performance of the logit dynamics for 2 × 2 coordination games (the same class studied in [3]) and 2 × 2 anti-coordination games.\\nCoordination games. Coordination Games are two-player games in which the players have an advantage in selecting the same strategy. These games are often used to model the spread of a new technology [22]: two players have to decide whether to adopt or not a new technology. We assume that the players would prefer choosing the same technology as the other one and that choosing the new technology is at most as risky as choosing the old one. We name 0 the NEW strategy and 1 the OLD strategy. The game is formally described by the following payoff matrix\\nWe assume that a > d and b > c (meaning that they prefer to coordinate) and that a − d b − c (meaning that for each player strategy 0 is at most as risky as strategy 1). Notice that we do not make any assumption on the relation between a and b. For convenience sake we name ∆ := a − d and δ := b − c .\\nIt is easy to see that this game is a potential game and the following function is an exact potential for it:\\nThis game has two pure Nash equilibria: (0, 0), where each player has utility a, and (1, 1), where each player has utility b. As d + c < a + b, the social welfare is maximized at one of the two equilibria. We analyse the mixing time of the logit dynamics for 2 × 2 coordination games and compute its stationary expected social welfare as a function of β.\\nStationary expected social welfare. The logit dynamics for the coordination game defined by the payoffs in Table 8 establishes that, from a profile x, player i selected for update plays according to the following probability distribution (see Equation (1)):\\nNext theorem bounds the stationary expected social welfare E π [W ] obtained by the logit dynamics and gives conditions for which E π [W ] is better than the social welfare SW N of the worst Nash equilibrium.\\nTheorem 8 (Expected social welfare) The stationary expected social welfare E π [W ] of the logit dynamics for the coordination game in Table 8 is\\nProof. The stationary distribution π of the logit dynamics is\\nwhere Z(β) = e ∆β + e δβ + 2.\\nSince E π [W ] = 2 · E π [u i ], we compute the expected utility E π [u i ] of player i at the stationary distribution,\\nThus, if a > b and β max 0,\\nMixing time. Now we study the mixing time of the logit dynamics for coordination games and we show that it is exponential in β and in the minimum potential difference between adjacent profiles.\\nTheorem 9 (Mixing Time) The mixing time of the logit dynamics for the coordination game of Table 8 is Θ e δβ for every β 0.\\nProof. Upper bound: We apply the Path Coupling technique (see Theorem 2) with the Hamming graph defined in (5) and all the edge-weights set to 1. Let x and y be two profiles differing only for the player j and consider the coupling defined in Section 3.2 for this pair of profiles. Now we bound the expected distance of the two coupled chains after one step.\\nWe denote by b i (x, y) the probability that both chains perform the same update given that player i has been selected for strategy update. Clearly, b i (x, y) = 1 for i = j, while for i = j, we have\\nFor sake of readability we set p = 1 1 + e ∆β and q = 1 1 + e δβ . and thus b i (x, y) = p + q. To compute E x,y [ρ(X 1 , Y 1 )], we observe that the logit dynamics chooses player j with probability 1/2. In this case, as b j (x, y) = 1, the coupling updates both chains in the same way, resulting in X 1 = Y 1 . Similarly, player i = j is chosen for strategy update with probability 1/2. In this case, with probability b i (x, y) the coupling performs the same update in both chains resulting in ρ(X 1 , Y 1 ) = 1. Instead with probability 1 − b i (x, y), the coupling performs different updates on the chains resulting in ρ(X 1 , Y 1 ) = 2. Therefore we have,\\nFrom Theorem 2, with α = 1 2 (p + q) and diam(Ω) = 2, it follows that It is easy to see that the second largest eigenvalue of P is λ ⋆ = (1−p)+(1−q) 2 , hence the relaxation time is t rel = 1/(1 − λ ⋆ ) = 2 p+q , and for the mixing time we have\\nIn the last inequality we used that p and q are both smaller than 1/2. Finally, the theorem follows by observing that\\nNotice that, if we used the relaxation time to upper bound the mixing time (see Theorem 3) we would get a non-tight bound, hence in the above proof we had to resort to the path coupling for the upper bound.\\nAnti-coordination games. Very similar results can be obtained for anti-coordination games. These are two-player games in which the players have an advantage in selecting different strategies. They model many settings where there is a common and exclusive resource: two players have to decide whether to use the resource or to drop it. If they both try to use it, then a deadlock occurs and this is bad for both players. Usually, these games are described by a payoff matrix like the one in Table 8, where we assume that d > a and c > b and that d − a c − b. Notice that Nash Equilibria of this game are unfair, as one player has utility max{c, d} and the other min{c, d}.\\nFor the logit dynamics, we have that, for all β, the stationary expected social welfare is worse than the one guaranteed by a Nash equilibrium. On the other hand, for sufficiently large β we have that the expected utility of a player is always better than min{c, d}: that is, in the logit dynamics each player expects to gain more than in the worst Nash equilibrium. Moreover, the stationary distribution is a fair equilibrium, since every player has the same expected utility. As for the coordination games, the mixing time is exponential in β and in the minimum potential difference between adjacent profiles.\"\n },\n {\n \"title\": \"The OR game\",\n \"content\": \"In this section we consider the following simple n-player potential game that we here call OR game. Every player has two strategies, say {0, 1}, and each player pays the OR of the strategies of all players (including herself). More formally, the utility function of player i ∈ [n] is\\notherwise.\\nNotice that the OR game has 2 n − n Nash equilibria. The only profiles that are not Nash equilibria are the n profiles with exactly one player playing 1. Nash equilibrium 0 has social welfare 0, while all the others have social welfare −n.\\nIn Theorem 10 we show that the stationary expected social welfare is always better than the social welfare of the worst Nash equilibrium, and it is significantly better for large β. Unfortunately, in Theorem 11 we show that if β is large enough to guarantee a good stationary expected social welfare, then the time needed to get close to the stationary distribution is exponential in n. Finally, in Theorem 12 we give upper bounds on the mixing time showing that if β is relatively small then the mixing time is polynomial in n, while for large β the upper bound is exponential in n and it is almost-tight with the lower bound. Despite the simplicity of the game, the analysis of the mixing time is far from trivial.\\nTheorem 10 (Expected social welfare) The stationary expected social welfare of the logit dynamics for the OR game is E π [W ] = −αn where α = α(n, β) = (2 n −1)e −β 1+(2 n −1)e −β .\\nProof. Observe that the OR game is a potential game with exact potential Φ where Φ(0) = 0 and Φ(x) = −1 for every x = 0. Hence the stationary distribution is\\nwhere the normalizing factor is Z = 1 + (2 n − 1)e −β . The expected social welfare is thus\\nIn the next theorem we show that the mixing time can be polynomial in n only if β c log n for some constant c.\\nTheorem 11 (Lower bound on mixing time) The mixing time of the logit dynamics for the OR game is\\nProof. Consider the set S ⊆ {0, 1} n containing only the state 0 = (0, . . . , 0) and observe that π(0) 1/2 for β log(2 n − 1). The bottleneck ratio is\\nHence, by applying Theorem 4, the mixing time is\\nIf β > log(2 n − 1) instead we consider the set R ⊆ {0, 1} n containing all states except state 0, and observe that\\nand π(R) 1/2 for β > log(2 n − 1). It holds that\\nThe bottleneck ratio is\\nHence, by applying Theorem 4, the mixing time is\\nIn Proof. We apply the path coupling technique (see Theorem 2 in Section 2) with the Hamming graph defined in (5). Let x, y ∈ {0, 1} n be two profiles differing only at player j ∈ [n] and, without loss of generality, let us assume |x| = k − 1 and |y| = k for some k = 1, . . . , n. We set the weight of edge {x, y} depending only on k, i.e. ℓ(x, y) = δ k where δ k 1 will be chosen later. Consider the coupling defined in Subsection 3.2. Now we evaluate the expected distance after one step E x,y [ρ(X 1 , Y 1 )] of the two coupled chains (X t , Y t ) starting at (x, y). Let i be the player chosen for the update. Observe that if i = j, i.e. if we update the player where x and y are different (this holds with probability 1/n), then the distance after one step is zero, otherwise we distinguish four cases depending on the value of k. Case k = 1: In this case profile x is all zeros and profile y has only one 1 and the length of edge {x, y} is ℓ(x, y) = δ 1 . When choosing a player i = j (this happens with probability (n − 1)/n), at the next step the two chains will be at distance δ 1 (if in both chains player i chooses strategy 0, and this holds with probability min{σ i (0 | x), σ i (0 | y)}), or at distance δ 2 (if in both chains player i chooses strategy 1, and this holds with probability min{σ i (1 | x), σ i (1 | y)}), or at distance δ 1 + δ 2 (if player i chooses strategy 0 in chain X 1 and strategy 1 in chain Y 1 , and this holds with the remaining probability). Notice that, from the definition of the coupling, it will never happen that player i chooses strategy 1 in chain X 1 and strategy 0 in chain Y 1 , indeed we have that\\nHence the expected distance after one step is When player h is chosen (this holds with probability 1/n) we have that σ h (s | x) and σ h (s | y) for s = 0, 1 are the same as in (10). At the next step the two chains will be at distance δ 2 (if player h stays at strategy 1 in both chains), or at distance δ 1 (if player h chooses strategy 0 in both chains), or at distance δ 1 + δ 2 (if player h stays at strategy 0 in chain X 1 and chooses strategy 1 in chain Y 1 ).\\nWhen a player i / ∈ {h, j} is chosen (this holds with probability (n − 2)/n) we have that σ i (0, x) = σ i (1, x) = σ i (0, y) = σ i (1, y) = 1/2. Thus in this case the two coupled chains always perform the same choice at player i, and at the next step they will be at distance δ 2 (if player i stays at strategy 0 in both chains) or at distance δ 3 (if player i chooses strategy 1 in both chains). Hence the expected distance after one step is\\nCase 3 k n − 1: When a player i = j is chosen such that x i = y i = 1 (this holds with probability (k − 1)/n) then at the next step the two chains will be at distance δ k (if i stays at strategy 1) or at distance δ k−1 (if i moves to strategy 0). When a player i = j is chosen such that x i = y i = 0 (this holds with probability (n − k)/n) then at the next step the two chains will be at distance δ k (if i chooses to stay at strategy 0) or at distance δ k+1 (if i chooses to move to strategy 0). Hence the expected distance after one step is\\nCase k = n: When a player i = j is chosen, then at the next step the two chains will be at distance δ n or at distance δ n−1 . Hence the expected distance after one step is\\nIn order to apply Theorem 2 we now have to show that it is possible to choose the edge weights δ 1 , . . . , δ n and a parameter α > 0 such that\\n1+e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 δ 2 e −α , 1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) δ k e −α , for k = 3, . . . , n − 1 ,\\nFor different values of β, we make different choices for α and for the weights δ k . For clarity's sake we split the proof in three different lemmas. We denote by δ max the largest δ k . In Lemma 13 we show that Inequalities (15) are satisfied for every value of β by choosing the weights as follows\\nand by setting α = 1/(2nδ max ). From Corollary 18, we have δ max = O( √ n2 n ). Observe that the diameter of the Hamming graph is n i=1 δ i nδ max , hence from Theorem 2 we obtain t mix = O(n 5/2 2 n ). In Lemma 14 we show that, if β < (1−ǫ) log n for an arbitrarily small constant ε > 0, Inequalities (15) are satisfied, for sufficiently large n, by choosing weights δ 1 = n 1−ε , δ 2 = 4/3, δ 3 = · · · = δ n = 1, and α = 1/n.. In this case the diameter is O(n) and, by Theorem 2, t mix = O(n log n).\\nIn Lemma 15 we show that, Inequalities (15) are satisfied by choosing weights as follows\\nwhere a 1 = n − 1 and b 1 = ne −β + 1 and, for every k = 2, . . . , n − 1\\nand by setting α = 1/(2nδ max ). From Corollary 21 it follows that, if β c log n for a constant c ∈ N, we have that δ max = O(n c+2 ) and the diameter of the Hamming graph is O(n c+3 ). Thus, by Theorem 2 it follows that t mix = O(n c+3 log n).\"\n },\n {\n \"title\": \"Technical lemmas\",\n \"content\": \"In this section we prove the technical lemmas needed for completing the proof of Theorem 12.\\nLemma 13 Let δ 1 , . . . , δ n be as follows\\nand let α = 1/(2nδ max ) where δ max = max{δ k : k = 1, . . . , n}. Then Inequalities (15) are satisfied for every β 0.\\nProof. Observe that, for every k = 1, . . . , n, the right-hand side of the k-th inequality in (15) is\\n.\\nNow we check that the left-hand side is at most δ k − 1/(2n).\\nFirst inequality (k = 1): n−1 n δ1\\nFrom the definition of δ 1 in (16) we have that\\nHence the left-hand side is\\nSecond inequality (k = 2): 1\\nFrom the definition of δ 2 in (16) we have that\\nHence the left-hand side of the second inequality is 1 2n\\nOther inequalities (k = 3, . . . , n − 1)\\nFrom the definition of δ k in (16) we have that\\nHence the left-hand side is\\nLast inequality (k = n): n−1 2n (δ n + δ n−1 ) δ n e −α . Since δ n = 1 and δ n−1 = 1 n−1 δ n + 1 = n n−1 , the left-hand side of the last inequality is\\nLemma 14 Let δ 1 , . . . , δ n be as follows\\nwhere ε > 0 is an arbitrary small constant and let α = 1/n. Then Inequalities (15) are satisfied for every β (1 − ε) log n and n sufficiently large.\\nProof. We check that all the inequalities in (15) are satisfied.\\nFirst inequality (k = 1): n−1 n δ1 1+e −β + δ2 2 δ 1 e −α . For the left-hand side we have\\nFor the right-hand side we have δ 1 e −α = n 1−ε e −1/n n 1−ε 1 − 1 n .\\nHence the left-hand side is smaller than the right-hand one (for n sufficiently large).\\nSecond inequality (k = 2): 1 2n 2 1+e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 δ 2 e −α . For the left-hand side we have 1 2n 2 1 + e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 = 1 2n 2 1 + e −β n 1−ε + (n − 1)\\nAnd for the right-hand side we have\\nHence the left-hand side is smaller than the right-hand one (for n sufficiently large). Third inequality (k = 3): 1 2n ((n − 1)δ 3 + 2δ 2 + (n − 3)δ 4 ) δ 3 e −α . For the left-hand side we have\\nAnd for the right-hand side we have\\nHence the left-hand side is smaller than the right-hand one.\\nOther inequalities (k 4): 1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) δ k e −α . Since δ k = δ k−1 = δ k+1 = 1 the left-hand side is equal to n−1 n and the right-hand side is e −1/n n−1 n .\"\n },\n {\n \"title\": \"Lemma 15\",\n \"content\": \"Let δ 1 , . . . , δ n be as follows\\nwhere a 1 = n − 1 and b 1 = ne −β + 1 and for every k = 2, . . . , n − 1\\nand let α = 1/(2nδ max ) where δ max = max{δ k : k = 1, . . . , n}. Then Inequalities (15) are satisfied for every β 0.\\nBefore to prove the Lemma 15 we do the following observation.\\nObservation 16 Let b k defined as in the Lemma 15. Then, for every k 2, it holds that b k kb k−1 .\\nProof. We proceed by induction on k. The base case k = 2 follows from b 2 = (n + 1)(ne −β + 1) − (n − 1) = (n + 1)ne −β + 2 > 2(ne −β + 1) = 2b 1 .\"\n },\n {\n \"title\": \"Now suppose the claim holds for\",\n \"content\": \"Proof ( Lemma 15). Observe that, as in Equation (17), for every k = 1, . . . , n, the right-hand side of the k-th inequality in (15) is\\nNow we check that the left-hand side is at most δ k − 1/(2n).\\nFirst inequality (k = 1): n−1 n δ1 1+e −β + δ2 2 δ 1 e −α . From the definition of δ 1 in (18) we have that\\nHence the left-hand side is\\nFrom the definition of δ 2 in (18) we have that\\nHence the left-hand side is 1 2n 2 1 + e −β δ 1 + (n − 1)δ 2 + (n − 2)δ 3 = 1 2n\\nOther inequalities (k = 3, . . . , n − 1): 1 2n ((n − 1)δ k + (k − 1)δ k−1 + (n − k)δ k+1 ) δ k e −α . From the definition of δ k in (18) we have that\\nHence the left-hand side is\\nwhere the inequality follows from the Observation 16.\\nLast inequality (k = n): n−1 2n (δ n + δ n−1 ) δ n e −α . Since δ n = 1 and δ n−1 = an−1 bn−1 δ n + 1 = an−1 bn−1 + 1, the left-hand side of the last inequality is\\nwhere the inequality follows from the Observation 16.\\nIn order to apply the path coupling theorem, we need to bound δ max : the next observation will represent the main tool to achieve this goal.\\nObservation 17 Let δ 1 , . . . , δ n be defined recursively as follows: δ n = 1 and for a suitable constant c.\\nIn order to bound δ max when δ 1 , . . . , δ n are defined as in Lemma 15 and β c log n for a constant c ∈ N, we define\\nYou can check that p 1 = 0, q 1 = n and\\nwe notice that p k = (n + 1)q k−1 − (k + 1)q k−1 q k for every k. We can also prove the following simple observation about q k .\"\n },\n {\n \"title\": \"Observation 19\",\n \"content\": \"For every k 1 constant, we have q k 2 −k n k .\\nProof. We proceed by induction on k, with the base k = 1 being obvious. Suppose the claim holds for k − 1, that is q k−1 2 −(k−1) n k−1 , then\\nMoreover, you can check that l 1 = n − 1, r 1 = 1 and l k = (n − k)r k−1 r k = (n + 1)r k−1 − (k − 1)l k−1 ;\\nwe notice that above recursion gives l k = (n − k)(k − 1)! and r k = k!. Next lemma bounds γ k defined in Equation 19.\\nLemma 20 Let δ 1 , . . . , δ n be defined as in Lemma 15, γ k defined as in Equation (19) and β c log n for a constant c ∈ N. Then, for sufficiently large n, it holds that\\nProof. Since p k q k , then (nq k − p k )e −β > 0; instead, l k − nr k = (k − 1)!(n − k − nk) < 0. Hence we have for every k\\nInductively, we show that for every k c + 3, we have γ k < 1. Set k = c + 3: c is a constant, thus Observation 19 holds for k − 1; hence and since e −β n −c , we have that\\nInstead, l c+3 − r c+3 = (c + 2)!(n − 2c − 6) (c + 2)! · n. Thus,\\nfor n sufficiently large. Now, suppose that γ k−1 < 1; then, we have\\nwhere a k−1 < b k−1 is implied by the inductive hypothesis. In order to complete the proof, we need to show that γ c+2 = O(1). Similarly to the case k = c + 3, we obtain (q c+2 − p c+2 )e −β 2 −c n and l c+2 − r c+2 (c + 1)! · n. Hence, γ c+2 p c+2 + r c+2 + (c + 1)! · n p c+2 + r c+2 + 2 −c n (c + 1)! · 2 c = O(1) .\"\n },\n {\n \"title\": \"Corollary 21\",\n \"content\": \"Let δ 1 , . . . , δ n and c be defined as in Lemma 15. Then δ max = O(n c+2 ).\\nProof. From Observation 17, Lemma 20 and the definition of δ 1 , . . . , δ n it follows that\"\n },\n {\n \"title\": \"The XOR game\",\n \"content\": \"In this section we analyze the logit dynamics for another simple n-player game, the XOR game. The XOR game is a symmetric n-player game in which each player has two strategies, denoted by 0 and 1, and each player pays the XOR of the strategies of all players (including herself). More formally, for each i ∈ [n], the utility function u i (·) is defined as follows\\nx has an odd number of 1's; 0, if x has an even number of 1's.\\nNotice that the XOR game has 2 n−1 Nash equilibria, namely all profiles with an even number of players playing strategy 1. Nash equilibria have social welfare 0 and profiles not in equilibria have social welfare −n. Observe that the XOR game is a potential game with exact potential Φ where Φ(x) = u i (x) for every x and every i ∈ [n]. Hence, the stationary distribution is\\nx has an odd number of 1's; 1/Z, if x has an even number of 1's;\\nwhere the normalizing factor is Z = 2 n−1 (1 + e −β ). Even if this game looks similar to the OR game, it exhibits a different behavior. Theorem 22 gives the stationary expected social welfare of the XOR game and we can see that, as β increases, the expected social welfare tends from below to the social welfare at the Nash equilibria. In contrast the expected social welfare of the OR game is better than the worst Nash equilibrium for all values of β. Moreover, in Theorem 23 and Theorem 24 we show that the mixing time for the XOR game is polynomial in n and exponential in β, whereas the mixing time for the OR game can be bounded independently from β.\\nTheorem 22 (Expected social welfare) The stationary expected social welfare of the logit dynamics for the XOR game is E π [W ] = − n 1+e β . Proof. The expected social welfare is\\nThe next theorem shows that the mixing time is exponential in β for every β > 0.\\nTheorem 23 (Lower bound on mixing time) The mixing time of the logit dynamics for the XOR game is Ω(e β ).\\nProof. Consider the set S ⊆ {0, 1} n containing only the state 0 = (0, . . . , 0). Observe that π(0) 1/2. The bottleneck ratio is\\ny∈{0,1} n π(0)P (0, y) = y∈{0,1} n : |y|=1\\nHence, by applying Theorem 4, the mixing time is\\nFinally, in the next theorem we give an almost matching upper bound to the mixing time.\\nTheorem 24 (Upper bound on mixing time) The mixing time of the logit dynamics for the OR game is O(n 3 e β ).\\nThe theorem is proved using coupling (see Theorem 1) and proof is presented in the next sections. Specifically, we use the coupling described in Section 3.2; in Section 7.1 we show that if the coupled chains are at even distance then distance does not increase after one step of the coupling; in Section 7.2 we show that if the coupled chains are at odd distance then they get closer distance with probability independent from β; finally, in Section 7.3 we bound the expected time needed by the two chains to coalesce and use Theorem 1 to derive an upper bound for the mixing time.\"\n },\n {\n \"title\": \"Even Hamming distance\",\n \"content\": \"Let X t and Y t be two chains coupled as described in Section 3.2. Suppose that X t = x, Y t = y, and H(x, y) = 2ℓ, for ℓ > 0. In this case, u i (x) = u i (y) = b for all i ∈ [n] and some b ∈ {−1, 0}. Let i be the index selected for update and let us distinguish two cases. In the first case x i = y i and we have\\nand thus\\nTherefore the coupling always update the strategy of player i in the same way in the two chains and thus H(X t+1 , Y t+1 ) = 2ℓ.\\nIn the second case we have x i = y i and we assume, without loss of generality, that x i = 0 and y i = 1. We observe that, for b ∈ {−1, 0},\\nTherefore we have\\n1 + e (1+2b)β and thus we have three possible updates for the strategy of player i:\\n1. both chains update to 0 (and thus H(X t+1 , Y t+1 ) = 2ℓ − 1) with probability min 1 1 + e (1+2b)β ,\\n2. both chains update to 1 (and thus H(X t+1 , Y t+1 ) = 2ℓ − 1) with probability min 1 1 + e (1+2b)β ,\\n3. chain X and Y choose two different strategies for updating the strategy of player i (and thus H(X t+1 , Y t+1 ) = 2ℓ) with probability\\nThe following lemma summarizes the above observations. Lemma 25 Suppose that H(X t , Y t ) = 2ℓ, for ℓ > 0. Then\\n2ℓ − 1, with probability 2ℓ n · 2 1+e β ; 2ℓ, with probability 1 − 2ℓ n · 2 1+e β .\"\n },\n {\n \"title\": \"Odd Hamming distance\",\n \"content\": \"Let X t and Y t be two chains coupled as described in Section 3.2. Suppose that X t = x, Y t = y, and H(x, y) = 2ℓ − 1, for ℓ > 0. In this case we have u i (x) = b and u i (y) = −(1 + b) for some b ∈ {−1, 0}. Let i be the index selected for update and let us distinguish two cases. In the case in which x i = y i = c for some c ∈ {0, 1}, we have 3. chains X and Y choose two different strategies for updating the strategy player i (and thus H(X t+1 , Y t+1 ) = 2ℓ) with probability 1 − 2 1+e β .\\nIn the second case we have x i = y i and we assume, without loss of generality, that x i = 0 and y i = 1. We observe that\\nTherefore we have σ i (0 | x) = σ i (0 | y) and σ i (1 | x) = σ i (1 | y)\\nand thus in this case H(X t+1 , Y t+1 ) = 2ℓ − 2.\\nThe following lemma summarizes the above observations.\"\n },\n {\n \"title\": \"Lemma 26\",\n \"content\": \"Suppose that H(X t , Y t ) = 2ℓ − 1, for ℓ > 0. Then\\n2ℓ − 2, with probability 2ℓ−1 n ;\\n2ℓ − 1, with probability n−2ℓ+1 n 2 1+e β ; 2ℓ, with probability n−2ℓ+1 n 1 − 2 1+e β .\"\n },\n {\n \"title\": \"Time to coalesce\",\n \"content\": \"We denote with τ k the random variable indicating the first time at which the two coupled chains have distance k. More precisely, τ k = min{t : H(X t , Y t ) = k}.\\nTherefore, τ couple = τ 0 is the time needed for the two chains to coalesce. We next give a bound on the expected time E x,y [τ couple ] for the two chains to coalesce starting from x and y. If x and y have distance 2ℓ, we denote by µ ℓ the expected time to reach distance 2ℓ − 2. That is,\\nSimilarly, if x and y have distance 2ℓ − 1, we denote by ν ℓ the expected time to reach distance 2ℓ − 2.\\nThat is,\\nNotice that, if H(x, y) = H(x ′ , y ′ ) then\\nfor all k, and thus the µ ℓ and ν ℓ are well defined. From Lemma 25 and Lemma 26, we have the following relations µ ℓ = 1 + µ ℓ · 1 − 2ℓ n · 2 1 + e β + ν ℓ · 2ℓ n · 2 1 + e β ν ℓ = 1 + ν ℓ · n − 2ℓ + 1 n · 2 1 + e β + µ ℓ · n − 2ℓ + 1 n · 1 − 2 1 + e β .\"\n },\n {\n \"title\": \"Conclusions and open problems\",\n \"content\": \"In this paper we studied strategic games where at every run a player is selected uniformly at random and she is assumed to choose her strategy for the next run according to the logit dynamics: a noisy best-response dynamics where the noise level is tuned by a parameter β. Such dynamics defines a family of ergodic Markov chains, indexed by β, over the set of strategy profiles. We proposed the stationary distribution of these Markov chains as solution concept for games where players have bounded rationality or limited knowledge about the system. Since this solution concept does not assume full rationality of agents, it avoids one of the main drawbacks of many classical equilibria concepts. Moreover, the stationary distribution of an ergodic Markov chain always exists, it is unique, and the chain converges to such a distribution from any starting state.\\nIn order to evaluate the long-term performance of the system, on the one hand we analyzed the expected social welfare when the strategy profiles are random according to the stationary distribution, on the other hand we studied the mixing time, i.e. how long it takes, for a chain starting at an arbitrary profile, to get close to its stationary distribution.\\nIn this paper we applied this approach to some simple but well-studied games with a constant number of players: the CK game, that obtains the worst Price of Anarchy bound between linear congestion games, and the 2 × 2 coordination games considered in the seminal paper about logit dynamics [3]. We also considered two simple n-player games, the OR game and the XOR game: the analysis of the mixing time turned out to be far from trivial even for such simple games. The above games highlight a twofold behavior: for some games, namely CK game and OR game, the mixing time can be upper bounded by a function independent of β, whereas the mixing time for the other games depends exponentially on the noise parameter β.\\nThe main goal of our line of research is to investigate logit dynamics for notable classes of n-player games. It would also be interesting to consider variations of the logit dynamics where players update their strategies simultaneously or where the noise is not uniform between players.\\nWe have seen that, for some games and for some values of β, the mixing time can be exponential in the number of players. When it takes such a long time to reach the stationary distribution, it would be interesting to investigate the evolution of the system in the transient phase of the logit dynamics.\"\n },\n {\n \"title\": \"A\",\n \"content\": \"\"\n },\n {\n \"title\": \"= 2 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"strategic game is a triple ([n], S, U), where [n] = {1, . . . , n} is a finite set of players, S = {S 1 , .. . , S n } is a family of non-empty finite sets (S i is the set of strategies for player i), and U = {u 1 , . . . , u n } is a family of utility functions (or payoffs), where u i : S 1 × · · · × S n → R is the utility function of player i.","content":"A\nstrategic game is a triple ([n], S, U), where [n] = {1, . . . , n} is a finite set of players, S = {S 1 , .. . , S n } is a family of non-empty finite sets (S i is the set of strategies for player i), and U = {u 1 , . . . , u n } is a family of utility functions (or payoffs), where u i : S 1 × · · · × S n → R is the utility function of player i."},{"id":"","caption":"Invariance under utility translation. Let G = ([n], S, U) be a game. If we change the utility functions by adding a constant c i to all the utilities of player i, i.e. if we define a new familyŨ = {ũ i : i ∈ [n]} of utility functions as followsũ","content":"Invariance under utility translation. Let G = ([n], S, U) be a game. If we change the utility functions by adding a constant c i to all the utilities of player i, i.e. if we define a new familyŨ = {ũ i : i ∈ [n]} of utility functions as followsũ"},{"id":"","caption":"Lower bound: We use the relaxation time bound (see Theorem 3). The transition matrix of the logit","content":"Lower bound: We use the relaxation time bound (see Theorem 3). The transition matrix of the logit"},{"id":"","caption":"the next theorem we give upper bounds on the mixing time depending on the value of β. The theorem shows that, if β c log n for some constant c, the mixing time is effectively polynomial in n with degree depending on c. The use of the path coupling technique in the proof of the theorem requires a careful choice of the edge-weights. Theorem 12 (Upper bound on mixing time) The mixing time of the logit dynamics for the OR game is O(n 5/2 2 n ) for every β. Moreover, for small values of β the mixing time is 1. O(n log n) if β < (1 − ε) log n, for an arbitrary small constant ε > 0; 2. O(n c+3 log n) if β c log n, where c 1 is an arbitrary constant.","content":"the next theorem we give upper bounds on the mixing time depending on the value of β. The theorem shows that, if β c log n for some constant c, the mixing time is effectively polynomial in n with degree depending on c. The use of the path coupling technique in the proof of the theorem requires a careful choice of the edge-weights. Theorem 12 (Upper bound on mixing time) The mixing time of the logit dynamics for the OR game is O(n 5/2 2 n ) for every β. Moreover, for small values of β the mixing time is 1. O(n log n) if β < (1 − ε) log n, for an arbitrary small constant ε > 0; 2. O(n c+3 log n) if β c log n, where c 1 is an arbitrary constant."},{"id":"","caption":"In this case we have x j = 0 and y j = 1, there is another player h ∈ [n] \\ {j} where x h = y h = 1, and for all the other players i ∈ [n] \\ {j, h} it holds x i = y i = 0. Hence the length of edge {x, y} is ℓ(x, y) = δ 2 .","content":"= 2 :\n2In this case we have x j = 0 and y j = 1, there is another player h ∈ [n] \\ {j} where x h = y h = 1, and for all the other players i ∈ [n] \\ {j, h} it holds x i = y i = 0. Hence the length of edge {x, y} is ℓ(x, y) = δ 2 ."},{"id":"","caption":"where γ k > 0 for every k = 1, . . . , n − 1. Let δ max = max{δ k : k = 1, . . . , n}. Thenδ max n max j i=h γ i : 1 h j n − 1 .Proof. The observation follows from the fact that, for k = 1, . . . , n − 1, we have Corollary 18 Let δ 1 , . . . , δ n be defined as in Lemma 13. Then δ max c √ n2 n for a suitable constant c.Proof. From Observation 17 and the definition of δ 1 , . . . , δ n , it holds that","content":"where γ k > 0 for every k = 1, . . . , n − 1. Let δ max = max{δ k : k = 1, . . . , n}. Thenδ max n max j i=h γ i : 1 h j n − 1 .Proof. The observation follows from the fact that, for k = 1, . . . , n − 1, we have Corollary 18 Let δ 1 , . . . , δ n be defined as in Lemma 13. Then δ max c √ n2 n for a suitable constant c.Proof. From Observation 17 and the definition of δ 1 , . . . , δ n , it holds that"},{"id":"","caption":"u i (x −i , c) = u i (y −i , 1 − c) = b and u i (x −i , 1 − c) = u i (y −i , c) = −(1 + b). Therefore σ i (c | x) = σ i (1 − c | y) = 1 1 + e −(1+2b)β and σ i (1 − c | x) = σ i (c | y) = 1 1 + e (1+2b)β and thus we have three possible updates: 1. both chains update to c (and thus H(X t+1 , Y t+1 ) = 2ℓ − 1","content":"u i (x −i , c) = u i (y −i , 1 − c) = b and u i (x −i , 1 − c) = u i (y −i , c) = −(1 + b). Therefore σ i (c | x) = σ i (1 − c | y) = 1 1 + e −(1+2b)β and σ i (1 − c | x) = σ i (c | y) = 1 1 + e (1+2b)β and thus we have three possible updates: 1. both chains update to c (and thus H(X t+1 , Y t+1 ) = 2ℓ − 1"},{"id":"","caption":"Update the strategy of player i according to the following probability distribution over the set S i of her strategies. For every y ∈ S i","content":"Update the strategy of player i according to the following probability distribution over the set S i of her strategies. For every y ∈ S iConsider the following noisy best-response dynamics, introduced in [3] and known as logit dynamics: at \nevery time step \n\n1. Select one player i ∈ [n] uniformly at random; \n2."},{"id":"","caption":"Simple algebraic manipulations giveFrom Markov inequality we have thatP x,y (τ couple > t) E x,y [τ couple ] tand thus, by taking t 0 = 4E x,y [τ couple ], we have d(t 0 ) 1/4. Therefore, by using Theorem 1, we have that t mix = O n 3 e β .","content":"Simple algebraic manipulations giveFrom Markov inequality we have thatP x,y (τ couple > t) E x,y [τ couple ] tand thus, by taking t 0 = 4E x,y [τ couple ], we have d(t 0 ) 1/4. Therefore, by using Theorem 1, we have that t mix = O n 3 e β .ν ℓ = \nn \n2ℓ − 1 \n1 + \nn − 2ℓ + 1 \n2ℓ \n· \ne β − 1 \n2 \n\nand \n\nµ ℓ = ν ℓ + \nn \n2ℓ \n· \n1 + e β \n2 \n\n= \nn \n2ℓ − 1 \n+ \nn \n2ℓ \n\nn \n2ℓ − 1 \n· \ne β − 1 \n2 \n+ 1 \n\nn \n2ℓ − 1 \n\nn \n2ℓ − 1 \n· \ne β − 1 \n2 \n+ 2 \n\nn n · \ne β − 1 \n2 \n+ 2 . \n\nHence, \n\nE x,y [τ couple ] 1 + \n\n2 ℓ n \n\nℓ even \n\nµ ℓ \nn 2 \n2 \nn · \ne β − 1 \n2 \n+ 2 + 1 = O n 3 e β ."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"strategic game is a triple ([n], S, U), where [n] = {1, . . . , n} is a finite set of players, S = {S 1 , .. . , S n } is a family of non-empty finite sets (S i is the set of strategies for player i), and U = {u 1 , . . . , u n } is a family of utility functions (or payoffs), where u i : S 1 × · · · × S n → R is the utility function of player i.\",\n \"content\": \"A\\nstrategic game is a triple ([n], S, U), where [n] = {1, . . . , n} is a finite set of players, S = {S 1 , .. . , S n } is a family of non-empty finite sets (S i is the set of strategies for player i), and U = {u 1 , . . . , u n } is a family of utility functions (or payoffs), where u i : S 1 × · · · × S n → R is the utility function of player i.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Invariance under utility translation. Let G = ([n], S, U) be a game. If we change the utility functions by adding a constant c i to all the utilities of player i, i.e. if we define a new familyŨ = {ũ i : i ∈ [n]} of utility functions as followsũ\",\n \"content\": \"Invariance under utility translation. Let G = ([n], S, U) be a game. If we change the utility functions by adding a constant c i to all the utilities of player i, i.e. if we define a new familyŨ = {ũ i : i ∈ [n]} of utility functions as followsũ\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Lower bound: We use the relaxation time bound (see Theorem 3). The transition matrix of the logit\",\n \"content\": \"Lower bound: We use the relaxation time bound (see Theorem 3). The transition matrix of the logit\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the next theorem we give upper bounds on the mixing time depending on the value of β. The theorem shows that, if β c log n for some constant c, the mixing time is effectively polynomial in n with degree depending on c. The use of the path coupling technique in the proof of the theorem requires a careful choice of the edge-weights. Theorem 12 (Upper bound on mixing time) The mixing time of the logit dynamics for the OR game is O(n 5/2 2 n ) for every β. Moreover, for small values of β the mixing time is 1. O(n log n) if β < (1 − ε) log n, for an arbitrary small constant ε > 0; 2. O(n c+3 log n) if β c log n, where c 1 is an arbitrary constant.\",\n \"content\": \"the next theorem we give upper bounds on the mixing time depending on the value of β. The theorem shows that, if β c log n for some constant c, the mixing time is effectively polynomial in n with degree depending on c. The use of the path coupling technique in the proof of the theorem requires a careful choice of the edge-weights. Theorem 12 (Upper bound on mixing time) The mixing time of the logit dynamics for the OR game is O(n 5/2 2 n ) for every β. Moreover, for small values of β the mixing time is 1. O(n log n) if β < (1 − ε) log n, for an arbitrary small constant ε > 0; 2. O(n c+3 log n) if β c log n, where c 1 is an arbitrary constant.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"In this case we have x j = 0 and y j = 1, there is another player h ∈ [n] \\\\ {j} where x h = y h = 1, and for all the other players i ∈ [n] \\\\ {j, h} it holds x i = y i = 0. Hence the length of edge {x, y} is ℓ(x, y) = δ 2 .\",\n \"content\": \"= 2 :\\n2In this case we have x j = 0 and y j = 1, there is another player h ∈ [n] \\\\ {j} where x h = y h = 1, and for all the other players i ∈ [n] \\\\ {j, h} it holds x i = y i = 0. Hence the length of edge {x, y} is ℓ(x, y) = δ 2 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"where γ k > 0 for every k = 1, . . . , n − 1. Let δ max = max{δ k : k = 1, . . . , n}. Thenδ max n max j i=h γ i : 1 h j n − 1 .Proof. The observation follows from the fact that, for k = 1, . . . , n − 1, we have Corollary 18 Let δ 1 , . . . , δ n be defined as in Lemma 13. Then δ max c √ n2 n for a suitable constant c.Proof. From Observation 17 and the definition of δ 1 , . . . , δ n , it holds that\",\n \"content\": \"where γ k > 0 for every k = 1, . . . , n − 1. Let δ max = max{δ k : k = 1, . . . , n}. Thenδ max n max j i=h γ i : 1 h j n − 1 .Proof. The observation follows from the fact that, for k = 1, . . . , n − 1, we have Corollary 18 Let δ 1 , . . . , δ n be defined as in Lemma 13. Then δ max c √ n2 n for a suitable constant c.Proof. From Observation 17 and the definition of δ 1 , . . . , δ n , it holds that\"\n },\n {\n \"id\": \"\",\n \"caption\": \"u i (x −i , c) = u i (y −i , 1 − c) = b and u i (x −i , 1 − c) = u i (y −i , c) = −(1 + b). Therefore σ i (c | x) = σ i (1 − c | y) = 1 1 + e −(1+2b)β and σ i (1 − c | x) = σ i (c | y) = 1 1 + e (1+2b)β and thus we have three possible updates: 1. both chains update to c (and thus H(X t+1 , Y t+1 ) = 2ℓ − 1\",\n \"content\": \"u i (x −i , c) = u i (y −i , 1 − c) = b and u i (x −i , 1 − c) = u i (y −i , c) = −(1 + b). Therefore σ i (c | x) = σ i (1 − c | y) = 1 1 + e −(1+2b)β and σ i (1 − c | x) = σ i (c | y) = 1 1 + e (1+2b)β and thus we have three possible updates: 1. both chains update to c (and thus H(X t+1 , Y t+1 ) = 2ℓ − 1\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Update the strategy of player i according to the following probability distribution over the set S i of her strategies. For every y ∈ S i\",\n \"content\": \"Update the strategy of player i according to the following probability distribution over the set S i of her strategies. For every y ∈ S iConsider the following noisy best-response dynamics, introduced in [3] and known as logit dynamics: at \\nevery time step \\n\\n1. Select one player i ∈ [n] uniformly at random; \\n2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Simple algebraic manipulations giveFrom Markov inequality we have thatP x,y (τ couple > t) E x,y [τ couple ] tand thus, by taking t 0 = 4E x,y [τ couple ], we have d(t 0 ) 1/4. Therefore, by using Theorem 1, we have that t mix = O n 3 e β .\",\n \"content\": \"Simple algebraic manipulations giveFrom Markov inequality we have thatP x,y (τ couple > t) E x,y [τ couple ] tand thus, by taking t 0 = 4E x,y [τ couple ], we have d(t 0 ) 1/4. Therefore, by using Theorem 1, we have that t mix = O n 3 e β .ν ℓ = \\nn \\n2ℓ − 1 \\n1 + \\nn − 2ℓ + 1 \\n2ℓ \\n· \\ne β − 1 \\n2 \\n\\nand \\n\\nµ ℓ = ν ℓ + \\nn \\n2ℓ \\n· \\n1 + e β \\n2 \\n\\n= \\nn \\n2ℓ − 1 \\n+ \\nn \\n2ℓ \\n\\nn \\n2ℓ − 1 \\n· \\ne β − 1 \\n2 \\n+ 1 \\n\\nn \\n2ℓ − 1 \\n\\nn \\n2ℓ − 1 \\n· \\ne β − 1 \\n2 \\n+ 2 \\n\\nn n · \\ne β − 1 \\n2 \\n+ 2 . \\n\\nHence, \\n\\nE x,y [τ couple ] 1 + \\n\\n2 ℓ n \\n\\nℓ even \\n\\nµ ℓ \\nn 2 \\n2 \\nn · \\ne β − 1 \\n2 \\n+ 2 + 1 = O n 3 e β .\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Tom Wexler, and Tim Roughgarden. The price of stability for network design with fair cost allocation. Elliot Anshelevich, Anirban Dasgupta, Jon M Kleinberg, Éva Tardos, SIAM Journal on Computing. 384Preliminary version in FOCS'04Elliot Anshelevich, Anirban Dasgupta, Jon M. Kleinberg,Éva Tardos, Tom Wexler, and Tim Rough- garden. The price of stability for network design with fair cost allocation. SIAM Journal on Com- puting, 38(4):1602-1623, 2008. Preliminary version in FOCS'04.","doi":""},{"text":"On the inefficiency ratio of stable equilibria in congestion games. Arash Asadpour, Amin Saberi, Proc. of the 5th International Workshop on Internet and Network Economics (WINE'09). of the 5th International Workshop on Internet and Network Economics (WINE'09)Springer5929Arash Asadpour and Amin Saberi. On the inefficiency ratio of stable equilibria in congestion games. In Proc. of the 5th International Workshop on Internet and Network Economics (WINE'09), volume 5929 of Lecture Notes in Computer Science, pages 545-552. Springer, 2009.","doi":""},{"text":"The statistical mechanics of strategic interaction. Lawrence E Blume, Games and Economic Behavior. 5Lawrence E. Blume. The statistical mechanics of strategic interaction. Games and Economic Behavior, 5:387-424, 1993.","doi":""},{"text":"Path Coupling: A technique for proving rapid mixing in Markov chains. R Bubley, M Dyer, Proceedings of the 38th Annual Symposium on Foundations of Computer Science. the 38th Annual Symposium on Foundations of Computer ScienceWashington, DC, USAIEEE Computer Society223R. Bubley and M. Dyer. Path Coupling: A technique for proving rapid mixing in Markov chains. In Proceedings of the 38th Annual Symposium on Foundations of Computer Science, pages 223-, Washington, DC, USA, 1997. IEEE Computer Society.","doi":""},{"text":"Settling the complexity of computing two-player Nash equilibria. Xi Chen, Xiaotie Deng, Shang-Hua Teng, Journal of the ACM. 563Xi Chen, Xiaotie Deng, and Shang-Hua Teng. Settling the complexity of computing two-player Nash equilibria. Journal of the ACM, 56(3), 2009.","doi":""},{"text":"The price of anarchy of finite congestion games. George Christodoulou, Elias Koutsoupias, Proc. of the 37th Annual ACM Symposium on Theory of Computing (STOC'05). of the 37th Annual ACM Symposium on Theory of Computing (STOC'05)ACMGeorge Christodoulou and Elias Koutsoupias. The price of anarchy of finite congestion games. In Proc. of the 37th Annual ACM Symposium on Theory of Computing (STOC'05), pages 67-73. ACM, 2005.","doi":""},{"text":"The complexity of computing a Nash equilibrium. Constantinos Daskalakis, Paul W Goldberg, Christos H Papadimitriou, SIAM Journal on Computing. 391Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195-259, 2009.","doi":""},{"text":"Learning, local interaction, and coordination. Glenn Ellison, Econometrica. 615Glenn Ellison. Learning, local interaction, and coordination. Econometrica, 61(5):1047-1071, 1993.","doi":""},{"text":"The complexity of game dynamics: Bgp oscillations, sink equilibria, and beyond. Alex Fabrikant, Christos H Papadimitriou, Proc. of the 19th annual ACM-SIAM symposium on Discrete algorithms (SODA'08). of the 19th annual ACM-SIAM symposium on Discrete algorithms (SODA'08)ACMAlex Fabrikant and Christos H. Papadimitriou. The complexity of game dynamics: Bgp oscilla- tions, sink equilibria, and beyond. In Proc. of the 19th annual ACM-SIAM symposium on Discrete algorithms (SODA'08), pages 844-853. ACM, 2008.","doi":""},{"text":"Calibrated learning and correlated equilibrium. P Dean, Foster, V Rakesh, Vohra, Games and Economic Behavior. 211-2Dean P. Foster and Rakesh V. Vohra. Calibrated learning and correlated equilibrium. Games and Economic Behavior, 21(1-2):40-55, 1997.","doi":""},{"text":"Drew Fudenberg, David K Levine, The Theory of Learning in Games. MIT. Drew Fudenberg and David K. Levine. The Theory of Learning in Games. MIT, 1998.","doi":""},{"text":"Sink equilibria and convergence. Michel Goemans, Vahab Mirrokni, Adrian Vetta, Proc. of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05). of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)IEEEMichel Goemans, Vahab Mirrokni, and Adrian Vetta. Sink equilibria and convergence. In Proc. of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pages 142-154. IEEE, 2005.","doi":""},{"text":"A General Theory of Equilibrium Selection in Games. C John, Reinhard Harsanyi, Selton, MIT PressJohn C. Harsanyi and Reinhard Selton. A General Theory of Equilibrium Selection in Games. MIT Press, 1988.","doi":""},{"text":"A simple adaptive procedure leading to correlated equilibrium. Sergiu Hart, Andreu Mas-Colell, Econometrica. 685Sergiu Hart and Andreu Mas-Colell. A simple adaptive procedure leading to correlated equilibrium. Econometrica, 68(5):1127-1150, 2000.","doi":""},{"text":"Worst-case equilibria. Elias Koutsoupias, Christos H Papadimitriou, Preliminary version in STACS. 3Elias Koutsoupias and Christos H. Papadimitriou. Worst-case equilibria. Computer Science Review, 3(2):65-69, 2009. Preliminary version in STACS 1999.","doi":""},{"text":"Markov Chains and Mixing Times. David Levin, Yuval Peres, Elizabeth L Wilmer, American Mathematical SocietyDavid Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, 2008.","doi":""},{"text":"Convergence to equilibrium in local interaction games. Andrea Montanari, Amin Saberi, Proc. of the 50th Annual Symposium on Foundations of Computer Science (FOCS'09. of the 50th Annual Symposium on Foundations of Computer Science (FOCS'09IEEEAndrea Montanari and Amin Saberi. Convergence to equilibrium in local interaction games. In Proc. of the 50th Annual Symposium on Foundations of Computer Science (FOCS'09). IEEE, 2009.","doi":""},{"text":"A class of games possessing pure-strategy nash equilibria. Robert W Rosenthal, International Journal of Game Theory. 216567Robert W. Rosenthal. A class of games possessing pure-strategy nash equilibria. International Journal of Game Theory, 2(1):6567, 1973.","doi":""},{"text":"On the equilibria of alternating move games. Aaron Roth, Nina Balcan, Adam Kalai, Yishay Mansour, Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA'10). of ACM-SIAM Symposium on Discrete Algorithms (SODA'10)Aaron Roth, Nina Balcan, Adam Kalai, and Yishay Mansour. On the equilibria of alternating move games. In Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA'10), 2010.","doi":""},{"text":"Population Games and Evolutionary Dynamics. H William, Sandholm, MIT PressWilliam H. Sandholm. Population Games and Evolutionary Dynamics. MIT Press, 2010.","doi":""},{"text":"Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. H , Peyton Young, Princeton University PressH. Peyton Young. Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton University Press, 1998.","doi":""},{"text":"The diffusion of innovations in social networks. H , Peyton Young, Economics Working Paper Archive number. 437Johns Hopkins University, Department of EconomicsH. Peyton Young. The diffusion of innovations in social networks. Economics Working Paper Archive number 437, Johns Hopkins University, Department of Economics, 2000.","doi":""}],"string":"[\n {\n \"text\": \"Tom Wexler, and Tim Roughgarden. The price of stability for network design with fair cost allocation. Elliot Anshelevich, Anirban Dasgupta, Jon M Kleinberg, Éva Tardos, SIAM Journal on Computing. 384Preliminary version in FOCS'04Elliot Anshelevich, Anirban Dasgupta, Jon M. Kleinberg,Éva Tardos, Tom Wexler, and Tim Rough- garden. The price of stability for network design with fair cost allocation. SIAM Journal on Com- puting, 38(4):1602-1623, 2008. Preliminary version in FOCS'04.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the inefficiency ratio of stable equilibria in congestion games. Arash Asadpour, Amin Saberi, Proc. of the 5th International Workshop on Internet and Network Economics (WINE'09). of the 5th International Workshop on Internet and Network Economics (WINE'09)Springer5929Arash Asadpour and Amin Saberi. On the inefficiency ratio of stable equilibria in congestion games. In Proc. of the 5th International Workshop on Internet and Network Economics (WINE'09), volume 5929 of Lecture Notes in Computer Science, pages 545-552. Springer, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The statistical mechanics of strategic interaction. Lawrence E Blume, Games and Economic Behavior. 5Lawrence E. Blume. The statistical mechanics of strategic interaction. Games and Economic Behavior, 5:387-424, 1993.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Path Coupling: A technique for proving rapid mixing in Markov chains. R Bubley, M Dyer, Proceedings of the 38th Annual Symposium on Foundations of Computer Science. the 38th Annual Symposium on Foundations of Computer ScienceWashington, DC, USAIEEE Computer Society223R. Bubley and M. Dyer. Path Coupling: A technique for proving rapid mixing in Markov chains. In Proceedings of the 38th Annual Symposium on Foundations of Computer Science, pages 223-, Washington, DC, USA, 1997. IEEE Computer Society.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Settling the complexity of computing two-player Nash equilibria. Xi Chen, Xiaotie Deng, Shang-Hua Teng, Journal of the ACM. 563Xi Chen, Xiaotie Deng, and Shang-Hua Teng. Settling the complexity of computing two-player Nash equilibria. Journal of the ACM, 56(3), 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The price of anarchy of finite congestion games. George Christodoulou, Elias Koutsoupias, Proc. of the 37th Annual ACM Symposium on Theory of Computing (STOC'05). of the 37th Annual ACM Symposium on Theory of Computing (STOC'05)ACMGeorge Christodoulou and Elias Koutsoupias. The price of anarchy of finite congestion games. In Proc. of the 37th Annual ACM Symposium on Theory of Computing (STOC'05), pages 67-73. ACM, 2005.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The complexity of computing a Nash equilibrium. Constantinos Daskalakis, Paul W Goldberg, Christos H Papadimitriou, SIAM Journal on Computing. 391Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195-259, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Learning, local interaction, and coordination. Glenn Ellison, Econometrica. 615Glenn Ellison. Learning, local interaction, and coordination. Econometrica, 61(5):1047-1071, 1993.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The complexity of game dynamics: Bgp oscillations, sink equilibria, and beyond. Alex Fabrikant, Christos H Papadimitriou, Proc. of the 19th annual ACM-SIAM symposium on Discrete algorithms (SODA'08). of the 19th annual ACM-SIAM symposium on Discrete algorithms (SODA'08)ACMAlex Fabrikant and Christos H. Papadimitriou. The complexity of game dynamics: Bgp oscilla- tions, sink equilibria, and beyond. In Proc. of the 19th annual ACM-SIAM symposium on Discrete algorithms (SODA'08), pages 844-853. ACM, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Calibrated learning and correlated equilibrium. P Dean, Foster, V Rakesh, Vohra, Games and Economic Behavior. 211-2Dean P. Foster and Rakesh V. Vohra. Calibrated learning and correlated equilibrium. Games and Economic Behavior, 21(1-2):40-55, 1997.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Drew Fudenberg, David K Levine, The Theory of Learning in Games. MIT. Drew Fudenberg and David K. Levine. The Theory of Learning in Games. MIT, 1998.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Sink equilibria and convergence. Michel Goemans, Vahab Mirrokni, Adrian Vetta, Proc. of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05). of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)IEEEMichel Goemans, Vahab Mirrokni, and Adrian Vetta. Sink equilibria and convergence. In Proc. of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pages 142-154. IEEE, 2005.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A General Theory of Equilibrium Selection in Games. C John, Reinhard Harsanyi, Selton, MIT PressJohn C. Harsanyi and Reinhard Selton. A General Theory of Equilibrium Selection in Games. MIT Press, 1988.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A simple adaptive procedure leading to correlated equilibrium. Sergiu Hart, Andreu Mas-Colell, Econometrica. 685Sergiu Hart and Andreu Mas-Colell. A simple adaptive procedure leading to correlated equilibrium. Econometrica, 68(5):1127-1150, 2000.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Worst-case equilibria. Elias Koutsoupias, Christos H Papadimitriou, Preliminary version in STACS. 3Elias Koutsoupias and Christos H. Papadimitriou. Worst-case equilibria. Computer Science Review, 3(2):65-69, 2009. Preliminary version in STACS 1999.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Markov Chains and Mixing Times. David Levin, Yuval Peres, Elizabeth L Wilmer, American Mathematical SocietyDavid Levin, Yuval Peres, and Elizabeth L. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, 2008.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convergence to equilibrium in local interaction games. Andrea Montanari, Amin Saberi, Proc. of the 50th Annual Symposium on Foundations of Computer Science (FOCS'09. of the 50th Annual Symposium on Foundations of Computer Science (FOCS'09IEEEAndrea Montanari and Amin Saberi. Convergence to equilibrium in local interaction games. In Proc. of the 50th Annual Symposium on Foundations of Computer Science (FOCS'09). IEEE, 2009.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A class of games possessing pure-strategy nash equilibria. Robert W Rosenthal, International Journal of Game Theory. 216567Robert W. Rosenthal. A class of games possessing pure-strategy nash equilibria. International Journal of Game Theory, 2(1):6567, 1973.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the equilibria of alternating move games. Aaron Roth, Nina Balcan, Adam Kalai, Yishay Mansour, Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA'10). of ACM-SIAM Symposium on Discrete Algorithms (SODA'10)Aaron Roth, Nina Balcan, Adam Kalai, and Yishay Mansour. On the equilibria of alternating move games. In Proc. of ACM-SIAM Symposium on Discrete Algorithms (SODA'10), 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Population Games and Evolutionary Dynamics. H William, Sandholm, MIT PressWilliam H. Sandholm. Population Games and Evolutionary Dynamics. MIT Press, 2010.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. H , Peyton Young, Princeton University PressH. Peyton Young. Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton University Press, 1998.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The diffusion of innovations in social networks. H , Peyton Young, Economics Working Paper Archive number. 437Johns Hopkins University, Department of EconomicsH. Peyton Young. The diffusion of innovations in social networks. Economics Working Paper Archive number 437, Johns Hopkins University, Department of Economics, 2000.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":592,"cells":{"text":{"kind":"string","value":"\nLR Scalar Mixings and One-loop Neutrino Masses\n\n\nOtto C W Kong \nInstitute of Physics\nAcademia Sinica\n11529Nankang, TaipeiTAIWAN\n\nLR Scalar Mixings and One-loop Neutrino Masses\narXiv:hep-ph/0004107v4 19 Oct 2000 HYPER VERSIONSolar and Atmospheric Neutrinos Supersymmetric Standard Model Neutrino Physics\nWithin the framework of the complete theory of supersymmetry without R-parity, where all possible R-parity violating terms are admitted, we perform a systematic analytical study of all sources of neutrino masses up to \"direct one-loop\" (defined explicitly below) level. In the passing, we present the full result for squark and slepton masses. In particular, there are interesting LR squark and slepton mixings, which involve both bilinear and trilinear R-parity violating parameters. The existence and important phenomenological implications of such terms have been largely overlooked in previous studies. In particular, in the studies under which either one type of the couplings is assumed to vanish or neglected, the terms would not show up. The LR mixings play a central role in neutrino mass generation. Our results look straight forward to be obtained, which, in our opinion, is an illustration of the effectiveness of our formulation adopted.\n\nIntroduction\n\nThe minimal supersymmetric standard model (MSSM) is no doubt the most popular candidate theory for physics beyond the Standard Model (SM). The alternative theory with a discrete symmetry called R-parity not imposed deserves no less attention. In particular, the latter admits neutrino masses, without the need for any extra matter field beyond the minimal spectrum. At the present time, experimental results from neutrino physics [1] is actually the only data we have demanding physics beyond the SM, while signals from supersymmetry (SUSY) are still absent. The neutrino data provides strong hints for the existence of Majorana type masses. The latter means lepton number violation, which is suggestive of R-parity violation. Hence, it is easy to appreciate the interest in R-parity violating (RPV) contributions to neutrino masses. The study of this topic has a long history, starting from Ref. [2]. Two of the notable papers on different aspects of the topic are given in Ref. [3] and Ref. [4], to which readers are also referred for references to earlier works. More recent works in the subject area [5,6,7,8,9,10] mainly focus on the fitting of the neutrino oscillation data under different scenarios while a comprehensive analysis of all the RPV contributions is still missing. This paper aims at providing such a picture.\n\nLike most of the other recent studies, we will focus on the sub-eV neutrino mass scale suggested by the Super-Kamiokande atmospheric neutrino data [11], though most of our results are actually valid for a much larger range of neutrino masses.\n\nAs illustrated below, there is a tree-level but seesaw suppressed contribution and some direct 1-loop contributions. Our level of treatment in this paper stops there, i.e. we will, in general, not go into contributions that are expected to be further suppressed. There are direct 1-loop contributions which involve a further seesaw type suppression hidden inside the loop. These are 1-loop diagrams that would suggest a null result if electroweak (EW) states are used for the particles running inside the loop and only a minimal number of mass insertions is admitted. When one thinks about the exact result to be obtained from using mass eigenstates instead, a nonzero result could emerge. It is diffcult to give analytical expressions for the mass eigenstate results. An approximation to the latter could be obtained by considering the EW state diagrams with extra mass insertions. In the case that these mass insertions are RPV, it typically means extra seesaw type suppression. We refer to contributions from such diagrams as pseudo-direct 1-loop contributions, which we will discuss without giving explicit formulae. We list also the well-known results. The idea here is to perform a systematic analysis and present the exhaustive list of all contributions up to the level of treatment.\n\nA similar comprehensive listing of neutrino mass contributions up to the 1-loop level (direct or indirect) has been presented in Ref. [12]. However, the latter analysis is limited to a scenario where the \"third generation couplings dominate\". This amounts to admitting only non-zero λ ′ i33 's and λ i33 's among the trilinear RPV couplings, though all nonzero bilinear RPV are indeed included by the authors. In our opinion, the maximal mixing result from Super-Kamiokande [11] brings the wisdom of \"third generation domination\" under question. Refs. [7] and [9], for example illustrate how no (family) hierarchy, or even an anti-hierarchy, among the RPV couplings may be preferred. The present analysis handles the complete theory of supersymmetry (SUSY) without R-parity, where all kind of RPV terms are admitted without bias. We present complete tree-level mass matrices for the scalars in this generic scenario. There is another major difference between the two studies. Ref. [12] is interested in performing some numerical calculation. While the latter is important for explicit fitting of experimental numbers, much of the physical origin of the neutrino mass contributions are hidden under elements of mixing matrices parametrizing the effective couplings of the neutrinos to squark or slepton mass eigenstates. We are interested here in illustrating the explicit origin of each contribution. Hence, we stay with electroweak (EW) state notation and give diagrammatic as well as analytical expressions of each individual contribution. Of particular interest here is a new type of contribution involving a RPV LR scalar (squark or slepton) mixings, which has been larger overlooked by previous authors. We hope that results here will be useful for a better understanding the role of each RPV parameter and identifying interesting regions of the extensive parameter space.\n\nTo study all the RPV contributions in a single consistent framework, one needs an effective formulation of the complete theory of SUSY without R-parity. The latter theory is generally better motivated than ad hoc versions of RPV theories. The large number of new parameters involved, however, makes the theory difficult to analyze. It has been illustrated [13] that an optimal parametrization, called the single-VEV parametrization, can be of great help in making the task manageable. The effectiveness of the SVP has been explored to perform an extensive study on the resultant leptonic phenomenology [13], to identify new type of neutrino mass contributions [9], and to study a new contribution to neutron electric dipole moment at 1-loop level [14,15], as well as new sources of contribution to flavor changing neutral current processes such as b → s γ [16] and µ → e γ [17]. Studies of neutrino masses and mixings under the formulation also include Refs. [7,8]. In fact, neutrino masses contribution is a central aspect of RPV effects and is likely to provide the most stringent bounds on the couplings, though many of the bounds obtained depend on assumptions on interpretation of neutrino data and could be relaxed or removed by simple extensions of the theory allowing extra sterile neutrino(s).\n\nOne-loop neutrino mass generation in SUSY without R-parity typically involves LR mixings of squarks or slepton. We want to emphasize again that squark and slepton mass matrices presented here are complete, with all source of R-parity violation included. Such results are explicitly presented for the first time. We consider the results to be interesting in their own right.\n\nIn the appendix, we give also an explicit illustration that all the VEV's under the SVP may be taken as real, despite the existence of complex parameters in the scalar potential; and give some important consistence relationships among some of the parameters involved. These results have not been published before, and serve as important background for clarifying some issues on the scalar masses and neutrino mass contributions discussed.\n\n\nFormulation and Notation\n\nWe summarize our formulation and notation below. The most general renormalizable superpotential for the supersymmetric SM (without R-parity) can be written as\nW= ε ab µ αĤ a uL b α + h u ikQ a iĤ b uÛ C k + λ ′ αjkL a αQ b jD C k + 1 2 λ αβkL a αL b βÊ C k + 1 2 λ ′′ ijkÛ C iD C jD C k , (2.1)\nwhere (a, b) are SU(2) indices, (i, j, k) are the usual family (flavor) indices, and (α, β) are extended flavor index going from 0 to 3. In the limit where λ ijk , λ ′ ijk , λ ′′ ijk and µ i all vanish, one recovers the expression for the R-parity preserving case, withL 0 identified asĤ d . Without R-parity imposed, the latter is not a priori distinguishable from theL i 's. Note that λ is antisymmetric in the first two indices, as required by the SU(2) product rules, as shown explicitly here with ε 12 = −ε 21 = 1. Similarly, λ ′′ is antisymmetric in the last two indices from SU(3) C . R-parity is exactly an ad hoc symmetry put in to makeL 0 , stand out from the otherL i 's as the candidate forĤ d . It is defined in terms of baryon number, lepton number, and spin as, explicitly, R = (−1) 3B+L+2S . The consequence is that the accidental symmetries of baryon number and lepton number in the SM are preserved, at the expense of making particles and superparticles having a categorically different quantum number, R-parity. The latter is actually not the most effective discrete symmetry to control superparticle mediated proton decay [18], but is most restrictive in terms of what is admitted in the Lagrangian, or the superpotential alone.\n\nA naive look at the scenario suggests that the large number of new couplings makes the task formidable. However, it becomes quite manageable with an optimal choice of flavor bases, the SVP [13]. In fact, doing phenomenological studies without specifying a choice of flavor bases is ambiguous. It is like doing SM quark physics with 18 complex Yukawa couplings instead of the 10 real physical parameters. In SUSY without R-parity, the choice of an optimal parametrization mainly concerns the 4 L α flavors. Under the SVP, flavor bases are chosen such that : 1/ among theL α 's,\nonlyL 0 , bears a VEV i.e. L i ≡ 0; 2/ h e jk (≡ λ 0jk = −λ j0k ) = √ 2 v 0 diag{m 1 , m 2 , m 3 }; 3/ h d jk (≡ λ ′ 0jk ) = √ 2 v 0 diag{m d , m s , m b }; 4/ h u ik = vu √ 2 V † CKM diag{m u , m c , m t }, where v 0 ≡ √ 2 L 0 and v u ≡ √ 2 Ĥ u .\nThe big advantage here is that the (tree-level) mass matrices for all the fermions do not involve any of the trilinear RPV couplings, even though the approach makes no assumption on any RPV coupling, including those from soft SUSY breaking. Moreover, and all the parameters used are uniquely defined, with the exception of some removable phases. In fact, the (color-singlet) charged fermion mass matrix reduces to the simple form :\nM C = M 2 g 2 v 0 √ 2 0 0 0 g 2 vu √ 2 µ 0 µ 1 µ 2 µ 3 0 0 m 1 0 0 0 0 0 m 2 0 0 0 0 0 m 3 .\n(2.2)\n\nEach µ i parameter here characterizes directly the RPV effect on the corresponding charged lepton (ℓ i = e, µ, and τ ). For any set of other parameter inputs, the m i 's can then be determined, through a simple numerical procedure, to guarantee that the correct mass eigenvalues of m e , m µ , and m τ are obtained -an issue first addressed and solved in Ref. [13]. The latter issue is especially important when µ i 's not substantially smaller than µ 0 are considered. Such an odd scenario is not definitely ruled out [13]. However, we would concentrate here on the more popular scenario with only sub-eV neutrino masses and hence small µ i 's. Here deviations of the m i 's from the mass eigenvalues are negligible.\n\n\nGauginos, Higgsinos, and Neutrinos\n\nThe tree-level mixings among the gauginos, higgsinos, and neutrinos gives rise to a 7 × 7 neutral fermion mass matrix M N :\nM N = M 1 0 g 1 v u /2 −g 1 v 0 /2 0 0 0 0 M 2 −g 2 v u /2 g 2 v 0 /2 0 0 0 g 1 v u /2 −g 2 v u /2 0 −µ 0 −µ 1 −µ 2 −µ 3 −g 1 v 0 /2 g 2 v 0 /2 −µ 0 0 0 0 0 0 0 −µ 1 0 (m o ν ) 11 (m o ν ) 12 (m o ν ) 13 0 0 −µ 2 0 (m o ν ) 21 (m o ν ) 22 (m o ν ) 23 0 0 −µ 3 0 (m o ν ) 31 (m o ν ) 32 (m o ν ) 33 , (3.1) whose basis is (−iB, −iW ,h 0 u ,h 0 d , ν L 1 , ν L 2 , ν L 3 )\n, withh 0 d being the neutral fermion from L 0 . The latter is guaranteed to be predominately a neutralino rather than neutrino, as the mass matrix clearly illustrates. As pointed out above, for small µ i 's the charged fermion states in theL i 's are essentially the physical states of e, µ and τ . Hence, (ν L 1 , ν L 2 , ν L 3 ) are essentially ν e , ν µ , ν τ . All entires in the lower-right 3 × 3 block (m o ν ) are, of course, zero at tree level. They are induced via 1-loop contributions to be discussed below. Such contributions are the focus of the present study. They are referred to here as direct 1-loop contributions.\n\nWe can write the general mass matrix in the form of block submatrices:\nM N = M n ξ T ξ m o ν , (3.2)\nwhere M n is the upper-left 4 × 4 neutralino mass matrix, ξ is the 3 × 4 block, and m o ν is the lower-right 3 × 3 neutrino block in the 7 × 7 matrix. The resulting (effective) neutrino mass matrix after block diagonalization is given by\n(m ν ) = −ξM -1 n ξ T + (m o ν ) . (3.3)\nContributions to the first term here starts at tree level, which are, however, seesaw suppressed. The second term is the direct contribution, which, however, enters in only at 1-loop level. We are interested in the small µ i scenario, where the tree-level contribution is not necessarily expected to be stronger than such direct 1-loop effects. The 1-loop contributions to the ξ and M n blocks are likely to have only a secondary effect on (m ν ). The latter, to be called indirect 1-loop contributions, are not included in the present analysis.\n\n\nLR-mixings for Squarks and Sleptons\n\nThe soft SUSY breaking part of the Lagrangian can be written as\nV soft = ǫ ab B α H a uL b α + ǫ ab A U ijQ a i H b uŨ C j + A D ij H a dQ b iD C j + A E ij H a dL b iẼ C j + h.c. + ǫ ab A λ ′ ijkL a iQ b jD C k + 1 2 A λ ijkL a iL b jẼ C k + 1 2 A λ ′′ ijkŨ C iD C jD C k + h.c. +Q †m2 QQ +Ũ †m2 UŨ +D †m2 DD +L †m2 LL +Ẽ †m2 EẼ +m 2 Hu |H u | 2 + M 1 2BB + M 2 2WW + M 3 2gg + h.c. , (4.1)\nwhere we have separated the R-parity conserving A-terms from the RPV ones (recall H d ≡L 0 ). Note thatL †m2 LL , unlike the other soft mass terms, is given by a 4 × 4 matrix. Explicitly,m 2 L 00 corresponds tom 2 H d of the MSSM case whilem 2 L 0k 's give RPV mass mixings.\n\nThe SVP also simplifies much the otherwise complicated expressions for the masssquared matrix of the scalar sectors. Firstly, we will look at the squark sectors. The masses of up-squarks obviously have no RPV contribution. The down-squark sector, however, has an interesting result. We have the mass-squared matrix as follows :\nM 2 D = M 2 LL M 2 † RL M 2 RL M 2 RR , (4.2) where M 2 LL =m 2 Q + m † D m D + M 2 Z cos2β − 1 2 + 1 3 sin 2 θ W , M 2 RR =m 2 D + m D m † D + M 2 Z cos2β − 1 3 sin 2 θ W ,(4.3)\nand\n(M 2 RL ) T = A D v 0 √ 2 − m D µ * 0 tanβ − ( µ * i λ ′ ijk ) v u √ 2 . (4.4)\nHere, m D is the down-quark mass matrix, which is diagonal under the parametrization adopted; ( µ * i λ ′ ijk ) denotes the 3 × 3 matrix ( ) jk with elements listed; and tanβ = vu v 0 . Note that all the VEV's can be taken as real, so long as the tree level scalar potential is considered (see the appendix). Apart from the first A D term, the remaining terms in (M 2 RL ) T are F -term contributions; in particular, the last term gives \"SUSY conserving\" but R-parity violating contributions; note that the existence of nonzero F -terms or electroweak symmetry breaking VEV's can be interpreted as a consequence of SUSY breaking though. The full F -term part in the above equation can actually be written together as\n( µ * α λ ′ αjk ) vu √ 2\nwhere the α = 0 term, which vanishes for j = k, gives the second term in the RHS. The latter, of course, is just the usual µ-term contribution in the MSSM case.\n\nNext we move on to the slepton sector. From Eq.(4.1) above, we can see that the \"charged Higgs\" should be considered together with the sleptons. We have hence an 8 × 8 mass-squared matrix of the following 1 + 4 + 3 form :\nM 2 E = M 2 Hu M 2 † LH M 2 † RH M 2 LH M 2 LL M 2 † RL M 2 RH M 2 RL M 2 RR ; (4.5) where M 2 LL =m 2 L + m † L m L + (µ * α µ β ) + M 2 Z cos2β − 1 2 + sin 2 θ W , + M 2 Z cos 2 β [1 − sin 2 θ W ] 0 1×3 0 3×1 0 3×3 , M 2 RR =m 2 E + m E m † E + M 2 Z cos2β − sin 2 θ W , M 2 Hu =m 2 Hu + µ * α µ α + M 2 Z cos2β 1 2 − sin 2 θ W + M 2 Z sin 2 β [1 − sin 2 θ W ] ; (4.6)\nand\n( M 2 RL ) T = 0 A E v 0 √ 2 − 0 m E µ * 0 tanβ − ( µ * i λ iβk ) v u √ 2 , (4.7) M 2 RH = − ( µ * i λ i0k ) v 0 √ 2 , (4.8) M 2 LH = (B * α ) + 1 2 M 2 Z sin2β [1 − sin 2 θ W ] 0 3×1 . (4.9) Here, m L = diag{0, m E } ≡ diag{0, m 1 , m 2 , m 3 },\nwhere the three m i 's are masses from leptonic Yukawa terms as discussed above in relation to Eq.(2.2); and, again, ( µ * i λ iβk ) denotes a matrix (4 × 3) with elements given by ( ) βk . Recall that for the small µ i domain we focused on here in this paper, we have m E ≃ diag{m e , m µ , m τ }. In fact, the k-th element in the 3-column-vector M 2 RH in Eq.(4.8) can be written as simply as µ * k m k (no sum). Similarly, the k-th element in the first row of the 4 × 3 matrix ( M 2 RL ) T in Eq.(4.7) can be written as µ * k m k tanβ (no sum). The former is ã ℓ c R hu type, while the latter al c R hd type (hd ≡l L 0 ), mass-squared term. Or, to better illustrate the common flavor structure, one can put the full F -term part of Eq.(4.7) as − ( µ * α λ αβk ) vu √ 2 . For the sake of completeness, we also give explicitly the neutral scalar, (or sneutrino-Higgs) mass-squared matrix. The neutral scalar mass terms, in terms of the (1 + 4) complex scalar fields, φ n 's, can be written in two parts -a simple (M 2 φ ) mn φ † m φ n part, and a Majorana-like part in the form 1 2 (M 2 φφ ) mn φ m φ n + h.c.. As the neutral scalars are originated from chiral doublet superfields, the existence of the Majorana-like part is a direct consequence of the electroweak symmetry breaking VEV's, hence restricted to the scalars playing the Higgs role only. They come from the quartic terms of the Higgs fields in the scalar potential. We have explicitly\nM 2 φφ = 1 2 M 2 Z sin 2 β − cosβ sinβ 0 1×3 − cosβ sinβ cos 2 β 0 1×3 0 3×1 0 3×1 0 3×3 ;(4.10)\nand\nM 2 φ = m 2 Hu + µ * α µ α + M 2 Z cos2β − 1 2 −(B α ) −(B * α )m 2 L + (µ * α µ β ) + M 2 Z cos2β 1 2 + M 2 φφ ,(4.11)\nNote that M 2 φφ here is real (see the appendix), while M 2 φ does have complex entries. The full 10 × 10 (real and symmetric) mass-squared matrix for the real scalars is then given by\nM 2 S = M 2 SS M 2 SP (M 2 SP ) T M 2 P P ,(4.12)\nwhere the scalar, pseudo-scalar, and mixing parts are\nM 2 SS = Re(M 2 φ ) + M 2 φφ , M 2 P P = Re(M 2 φ ) − M 2 φφ , M 2 SP = −2 Im(M 2 φ ) ,(4.13)\nrespectively. If Im(M 2 φ ) vanishes, the scalars and pseudo-scalars decouple from one another and the unphysical Goldstone mode would be found among the latter. Finally, we note that the B α entries may also be considered as a kind of LR mixings. The RPV B i 's do in fact contribute to neutrino mass, as discussed below.\n\nWe would like to emphasize that the above scalar mass results are completeall RPV contributions, SUSY breaking or otherwise, are included without theoretical bias. The simplicity of the result is a consequence of the SVP. Explicitly, there are no RPV A-term contributions due to the vanishing of VEV's v i ≡ √ 2 L i . However, such new contributions, as well as their roles in the physics of neutrino masses and phenomena like fermion EDM's and b → s γ, are genuine. For instance, rotating to a basis among theL α superfields under which the µ i 's are zero would restore theL i VEV's and show e.g. the RPV ( µ * λ ) term as a term involving the latter VEV's and some A-term parameters. The Higgs-slepton results given as in Eqs.(4.5) and (4.11) are admittedly not very useful for doing scalar physics. They contain a redundancy of parameters and hide the unphysical Goldstone state. However, for the purpose of analyzing the neutrino mass contributions as done below, they serve their purpose. Hence, we will refrain from further laboring on the algebra here.\n\nBefore ending the section, we want to emphasize the following. While it should be straight forward to write down the scalar mass matrices by the complete theory of SUSY without R-parity, under any formulation or parametrization, to the best of our knowledge, this has not been published before. After completing the work, we checked the literature and found no explicitly written down complete results for M 2 D and M 2 E as given here. Especially the existence of the interesting new RPV contributions, of the type given by the ( µ * λ ′ ) term in M 2 D , and the ( µ * λ ) term in Eq.(4.7), and the µ * µ terms in Eqs.(4.5) and (4.11), if noticed, have not been much appreciated. Explicit discussion of terms of the type ( µ * λ ) and their contribution to neutrino masses was first given in a recent paper by K. Cheung and the present author [9], in a different context. Their existence and important phenomenological implication seems, otherwise, to have been overlooked. We are not aware of any other explicit discussion of the ( µ * λ ′ ) and ( µ * λ ) terms in the literature. What follows is a check into the literature for the scalar mass results. Before doing that, however, we would like to emphasize that the complete theory of SUSY without R-parity is not a very popular subject, compared to various versions of more specific (assumed) forms of R-parity violation. We also warn the readers that most of the previous authors were not working under the parametrization we used here. In our opinion, there are many other subtle complications when a different parametrization is used, which have not been explicitly addressed. The interested readers are referred to a forthcoming review by the author on the subject [19]. The complete scalar mass results in a genericL α flavor basis would look more complicated than what we have here too. In particular, there would be contributions involving the trilinear RPV A-terms. Having said that, let us take a look at some works on a more or less complete version of R-parity violation and the scalar mass expression given therein. Ref. [20] gives \"sfermion mass matrix\" exactly as in the MSSM without RPV terms at all. Ref. [12] is a more careful and detailed study. However, as mentioned above, the paper considers only one λ and one λ ′ and hence does not give result in the complete theory anyway. An admissible ( µ * λ ′ ) term is still missing in the down-squark mass-squared matrix given. An 8 × 8 matrix corresponding to M 2 E is indeed presented. The matrix seems correct, under the starting assumption, though the admissible ( µ * λ ) term is not explicitly shown. Ref. [3] is the first to give the matrix in the 8 × 8 form, but we find no clear sign of the ( µ * λ ) term. The squark mass-squared matrix is not explicitly given there. In fact, the paper is based on a specific high energy scenario, which is hence not totally generic. For instance, early in the paper, the soft SUSY breaking part of the Lagrangian is already given in a simplified form with the high energy assumptions put in, hence not for the generic complete theory we discussed here. The neutral scalar mass, corresponding to M 2 S above, is better known [21]. In particular, Ref. [10] gives the result essentially under the SVP, though truncated to one lepton family.\n\nAnother important point to note is that we have pay special attention to the fact there the RPV parameters are generally complex, while this phase information has not been explicitly given previously. As mentioned above, the λ ′ term, for example, contributes to neutron EDM through its imaginary part. The other example is the existence of scalar-pseudo-scalar mixing as a result of complex RPV parameters, as given above explicitly in M 2 SP . The significant phenomenological implication of the latter has been well illustrated in Ref. [22], in the case of MSSM, for instance.\n\nFinally, we note that there is another group that has done quite elaborated works on their RPV model (see for example Ref. [6]), which however includes no trilinear RPV parameters and hence would not have our results for the complete theory. Moreover, in the soft SUSY breaking part given and used in Ref. [6] actually hasm 2 L 0k term missing. We have not actually done a detailed term by term checking to see if there are other discrepancies between ours results given in this section and others in the papers mentioned. The above brief comparison is supposed to serve as an illustration of the point we want to make here -that perhaps insufficient care and attention have been given to the complete results for the scalar masses in SUSY without R-parity. We will report more on the various phenomenological implications of some of the RPV mass entries in some other publications.\n\n\nNeutrino Mass Contributions\n\nLet us return to RPV contribution to neutrino mass. From Eq. \n(m ν ) tree ij ∼ −v 2 cos 2 β (g 2 2 M 1 + g 2 1 M 2 ) 2µ 0 [2µ 0 M 1 M 2 − v 2 sinβ cosβ (g 2 2 M 1 + g 2 1 M 2 )] µ i µ j . (5.1)\nThis is illustrated diagrammatically in Fig. 1. Next we come to the direct 1-loop contributions. A typical 1-loop neutrino mass diagram has two couplings of scalar-fermion-neutrino type. With the two couplings being λ ′ -type, we have a quark-squark loop as shown in Fig. 2. Here, a LR squark mixing is needed. From Eqs. parts : firstly the familiar one\n(m ν ) sqA ij ∼ 3 16π 2 m D h m D k M 2 d λ ′ ihk λ ′ jkh [A d − µ * 0 tanβ] ( i ←→ j ) , (5.2)\nwhere Md denote an average down-squark mass, and A d being a constant (mass) parameter representing the \"proportional\" part of the A-term, namely\nA D v 0 √ 2 = A d m D + δA D v 0 √ 2\n, and m D h is the h-th diagonal element of the matrix m D (i.e. the quark mass); next, the \"proportionality\" violating part\n(m ν ) sqδA ij ∼ 3 16π 2 m D h M 2 d λ ′ ihl λ ′ jkh δA D kl v 0 √ 2 ( i ←→ j ) ,(5.3)\nwhich is typically expected to be suppressed in many SUSY breaking scenarios and neglected; and, finally, the part due to the new RPV LR mixings,\n(m ν ) sq R ij ∼ − 3 16π 2 m D h M 2 d λ ′ ihl λ ′ jkh µ * g λ ′ gkl v u √ 2 ( i ←→ j ) . (5.4)\nThe ( i ←→ j ) expression denote symmetrization with respect to i and j. It is interesting to note that the last result contains no SUSY breaking parameter in the LR mixings. In particular, the flavor changing parts of the latter could not be suppressed through any SUSY breaking mechanism. Similar to the quark-squark loop, a lepton-slepton loop with two λ-type coupling, as shown in Fig. 3, generates neutrino mass, in the presence of LR slepton mixings. Using Eqs.(4.5) and (4.7), again we split the result into the different parts : the familiar one from the \"proportional\" part of the A-term, where Ml denote an average charged slepton mass, and A e the constant (mass)\n(m ν ) slA ij ∼ 1 16π 2 m h m k M 2 ℓ λ ihk λ jkh [A e − µ * 0 tanβ] ( i ←→ j ) ,(5.parameter with A E v 0 √ 2 = A e m E + δA E v 0 √ 2\n, (recall that m h 's are diagonal element of m E and essentially the mass of the charged lepton); the \"proportionality\" violating part\n(m ν ) slδA ij ∼ 1 16π 2 m h M 2 ℓ λ ihl λ jkh δA E kl v 0 √ 2 ( i ←→ j ) ; (5.6)\nand the part due to the new RPV LR mixings,\n(m ν ) sl R ij ∼ − 1 16π 2 m h M 2 ℓ λ ihl λ jkh µ * g λ gkl v u √ 2 ( i ←→ j ) . (5.7)\nHowever, the above is not yet the full result for the type of contributions. We have emphasized throughout the paper the systematic treatment of making no a priori distinction between theL i 's andĤ d . The latter is denoted asL 0 and treated as a 4-th leptonic flavor. In Eq.(4.7), the last term admitted a β = 0 part the neutrino mass contribution of which has not been included in the above analysis of the lepton-slepton loop parallel to the quark-squark loop. The corresponding result is simply given by setting k to 0 in Eq.(5.7), which may then be simplified to\n(m ν ) slZ ij ∼ − 1 16π 2 √ 2 v 0 m 2 j M 2 ℓ λ ijl [ µ * l m l tanβ ] ( i ←→ j ) . (5.8)\nThe contribution corresponds to the SUSY analog of the Zee neutrino mass diagram [23], as discussed in Ref. [9]. We illustrate the contribution and its Zee model analog in Fig. 4. A careful examination of Fig. 3 shows that one cannot get any more new neutrino mass diagram by replacing some other λ ijk flavor indices with a 0. Hence, we have completed the listing of the two-λ-loop contributions. The above has exhausted all the possibilities from using from the trilinear superpotential couplings as the only source for the loop vertices. The only other couplings involving a neutrino are the gauge couplings and the bilinear µ i 's. The effect of the latter has been considered in the tree-level seesaw. Putting two gauge couplings together, we do have a 1-loop neutrino mass diagram, with scalars and gauginos running in the loop. The charged loop does not work, while a neutral loop could do (see Fig. 5) when there is a Majorana-like sneutrino mass term. The latter contribution was first pointed out in Ref. [10]. In fact, Majorana-like sneutrino mass is where the required two units of lepton number violation come in. The former may be interpreted as a result of splitting in mass of the sneutrino and anti-sneutrino due to R-parity violation. Following our general approach here, we illustrate this in Fig. 6. It is clear from the figure that it involves the SUSY breaking and RPV parameters B i 's, as shown is Eq.(4.11) above, and is seesaw suppressed (cf. Fig 1), unless the B i 's happen to be at the SUSY scale despite small µ i 's. Note that the B i 's and the µ i 's are not totally independent parameters, as illustrated in the appendix. It is very unlikely that B i B 0 would be much larger than µ i µ 0 . The gauge loop contribution discussed serves as an illustrative example of what we call pseudo-direct 1-loop. The EW state diagram as given in Fig. 5 reads zero, as the required mass insertion on the sneutrino line does not exist [cf. Eq.(4. 10)]. If we admit extra mass insertions and extend the sneutrino line as shown in Fig. 6, we obtain the nonzero result. However, one show bear in mind that there is no definite hierarchy between this gauge loop contribution and the other direct 1-loop ones discussed above, as they arise from different RPV parameters, the magnitude of which we have no exact information.\n\nFinally, it is not difficult to see that there is no contribution from 1-loop diagrams with one gauge coupling and one Yukawa or λ-coupling vertices, up to the direct 1- loop level. In fact, before we put in non-minimal number of mass insertions in the internal lines, there is only one such EW state diagram, as given in Fig. 7. The diagram requires aW --ℓ c R k mass insertion, which is zero under the SVP. When we admit extra mass insertions along the internal fermion line and go to the pseudodirect 1-loop level, there are apparent nonzero contribution. Obviously we need at least one lepton number violating mass insertion. Fig. 8 illustrates the minimal extra mass insertions along the fermion line that could complete a diagram.\n\nHowever, let us look more closely into the implications of putting in extra mass insertions along the internal fermion line. The scalar-fermion loop neutrino mass diagram result always has a mass factor of the internal fermion in it; explicit examples are m D h in Fig. 2 and m h in Fig. 3. To obtain the exact result, one should use the mass eigenstates, and of course sum over the latter (see, for example, formulae in Ref. [12]). However, neglecting the fermion mass dependence in the propagator integral part, the mass and mixing matrix element dependence of the full sum is of course nothing other than the tree-level mass entry of single insertion case, i.e. the vanishingW -ℓ c R k term for the case at hand. If one consider only the contribution from one of the mass eigenstates, the result would not be zero. But we know that it is going to be canceled by that from the other mass eigenstates. This is like the GIM mechanism, violated here only to the extent of the non-universal mass effect from the propagator integral part.\n\nIn terms of EW state diagrams, the exact mass eigenstate result would correspond to summing over all possible diagrams with any (up to infinite) number of admissible mass insertions. The contribution from putting Fig. 8 into the internal fermion line of Fig. 7 is of course just one among the nontrivial result. Hence, taking this as an independent contribution is more or less equivalent to taking one term from the summation over mass eigenstates. The result of the latter is expected to be canceled in the overall sum.\n\nFurthermore, even if that GIM-like cancellation is rendered ineffective by the propagator integral part, the dependence of the contribution on a single λ jik implies that upon symmetrization with respect to i and j, there would be another cancellation as from λ ijk = −λ jik to the extent thatl -L j andl -L i has the same mass. So, even the naive pseudo-direct 1-loop contribution has suppression from the expected degeneracy of slepton masses, and is only proportional to the violation of the latter.\n\nThis, together with the gauge loop discussed above, has exhausted our discussion of the pseudo-direct 1-loop contributions. One could certainly get other EW state diagrams by putting in extra mass insertion, into the scalar line of Figs. 5 or 7. While direct 1-loop results from an EW state diagram roughly represent the corresponding exact mass eigenstate results, pseudo-direct 1-loop diagrams may be just pieces of an otherwise small or vanishing overall sum with a GIM-like cancellation mechanism at work. We do have to go to exact numerical calculations to know the extent to which the latter is violated and extract the correct result. Perhaps it should also be said that we have not considered diagrams with mass insertion(s) on the external lines because such diagrams really correpond the indirect 1-loop contributions. A diagram with one mass insertion in one of the ν external line, for example, should correspond to something in the ξ block of Eq.(3.1). We want to emphasize that while we classified contributions into direct 1-loop, pseudo-direct 1-loop and indirect 1-loop, there is no definite hierarchy among them, when one is comparing contributions involving different RPV parameters. A clear example is given by the fact that we do not know if the tree-level contribution which involves the µ i 's is really larger than, for example a λ ′ (quark-squark) loop contribution, though one may naively expect so.\n\n\nConcluding Remarks\n\nFrom the above systematic analysis, it is clear that we have discussed and given explicit formulae for all neutrino mass contributions up to the level of direct 1loop contribution, for the complete theory of SUSY without R-parity. Psuedo-direct 1-loop contributions are also discussed. We have also given a description of the full squark and slepton masses. The latter is useful for analyzing other aspects of phenomenology, particularly those related to LR mixings such as fermion electric dipole moment and flavor changing neutral current processes. The successful simple description here illustrates well the effectiveness of the formulation (SVP) adopted.\n\nNote Added : After we posted the first version of this paper, a paper from Davidson and Losada [24] on the subject appeared. The paper does list results under our formulation (SVP) here and goes beyond direct or pseudo-direct 1-loop level. In particular, a nice discussion of the gauge loop result is included. However, the basic approach is very different from that of this paper, and the contributions from the RPV LR scalar mixing are not included. Comparing our results with theirs, there seems to be some apparent disagreements, which we hope to address in detail in a future publication.\n\nwhere we have put in a complex phase in the VEV v 0 , for generality.\n\nThe equations from the vanishing derivatives of V s along φ -1 and φ 0 give\n1 8 (g 2 1 + g 2 2 )(v 2 u − v 2 d ) +m 2 -1 v u = B 0 v d e iθv , 1 8 (g 2 1 + g 2 2 )(v 2 d − v 2 u ) +m 2 0 v d = B 0 v u e iθv . (A.4)\nHence, B 0 e iθv is real. In fact, the part of V s that is relevant to obtaining the tadpole equations is no different from that of MSSM apart from the fact thatm 2 Hu andm 2 H d of the latter are replaced bym 2 -1 andm 2 0 respectively. As in MSSM, the B 0 parameter can be taken as real. The conclusion here is therefore that θ v vanishes, or all VEV's are real, despite the existence of complex parameters in the scalar potential. Results from the other tadpole equations, in a φ i direction, are quite simple. They can be written as complex equations of the form which is equivalent to\nB i tanβ =m 2 L 0i + µ * 0 µ i , (A.6)\nwhere we have used v u = v sinβ and v d = v cosβ. Note that our tanβ has the same physical meaning as that in the R-parity conserving case. For instance, tanβ, together with the corresponding Yukawa coupling ratio, gives the mass ratio between the top and the bottom quark. The three complex equations for the B i 's reflect the redundance of parameters in a genericL α flavor basis. The equations also suggest that the B i 's are expected to be suppressed, with respect to the R-parity conserving B 0 , as the µ i 's are, with respect to µ 0 . They give consistence relationships among the involved RPV parameters (under the SVP) that should not be overlooked.\n\n\n(3.1), one neutrino state get a tree-level mass. The seesaw suppressed contribution [see Eqs.(3.2) and (3.3)] is given by\n\nFigure 1 :\n1Neutrino mass from tree-level seesaw.\n\nFigure 2 :\n2(4.2) and(4.4), we have the result, here written in three Neutrino mass from quark-squark loop.\n\nFigure 3 :\n3Neutrino mass from lepton-slepton loop.\n\nFigure 4 :\n4SUSY Zee diagram for neutrino mass. The Zee analog interpretation noted in the brackets.\n\nFigure 5 :igure 6 :\n56Gaugino-sneutrino loop requiring a Majorana-like sneutrino mass insertion. Seesaw diagram for Majorana-like sneutrino masses.\n\nFigure 7 :\n7A direct charged loop contribution requiring however a fermion mass insertion that is vanishing.\n\nFigure 8 :\n8A possible set of mass insertions apparently producing the required internal fermion line in the previous figure.\n\nm 2 -\n21i e iθ -1i tanβ = −e iθvm2 0i e iθ 0i , (A.5)\nAcknowledgmentsThe author is in debt to S.K. Kang and K. Cheung for discussions and for suggestions on improving the present manuscript, and to A. Akeroyd for proof-reading the last version and helps to improve the language. K. Cheung is especially to be thanked for pointing out some erroneous numerical factors in a couple of equations now fixed. S. Davidson and M. Losada are to be thanked for communications concerning their recent paper[24]. The Korea Institute for Advanced Study is to be thanked for their hospitality during the early phase of our work. The Yukawa Institute for Theoretical Phyiscs, Kyoto University, and KEK, where the last revision of the manuscript is done, is to be thanked likewise.A. Note on the scalar potentialIn terms of the five, plausibly electroweak symmetry breaking, neutral scalars fields φ n , the generic (tree-level) scalar potential, as constrained by SUSY, can be written as :Here, we count the φ n 's from −1 to 3 and identify a φ α (recall α = 0 to 3) asl 0 α and φ -1 as h 0 u . Parameters in the above expression for V s (all real) are then given bŷUnder the SVP, we write the VEV's as follows :v\nFor an updated review, see V. hep-ph/0003212For an updated review, see V. Barger, hep-ph/0003212.\n\n. L J Hall, M Suzuki, Nucl. Phys. 231419L.J. Hall and M. Suzuki, Nucl. Phys. B231 (1984) 419.\n\n. R Hempfling, Nucl. Phys. 4783R. Hempfling, Nucl. Phys. B478 (1996) 3.\n\n. H P Nilles, N Polonsky, Nucl. Phys. 48433H.P. Nilles and N. Polonsky, Nucl. Phys. B484 (1997) 33.\n\n. M Drees, S Pakvasa, X Tata, T Ter Veldhuis, Phys. Rev. 575335M. Drees, S. Pakvasa, X. Tata, and T. ter Veldhuis, Phys. Rev. D57 (1998) 5335;\n\n. V Bednyakov, A Faessler, S Kovalenko, Phys. Lett. B. 442203V. Bednyakov, A. Faessler, and S. Kovalenko, Phys. Lett. B 442 (1998) 203; B.\n\n. S Mukhopadhyaya, F Roy, Vissani, Phys. Lett. B. 443191Mukhopadhyaya, S. Roy, and F. Vissani, Phys. Lett. B 443 (1998) 191;\n\n. S Rakshit, G Bhattacharyya, M Raychaudhuri, Phys. Rev. 5991701S. Rakshit, G. Bhattacharyya, and M. Raychaudhuri, Phys. Rev. D59 (1999) 091701 ;\n\n. A S Joshipura, S K Vempati, Phys. Rev. 6095009A.S. Joshipura and S.K. Vempati, Phys. Rev. D60 (1999) 095009 ;\n\n. E Chun, S Kang, C Kim, U Lee, Nucl. Phys. 54489E. Chun, S. Kang, C. Kim, and U. Lee, Nucl. Phys. B544 (1999) 89;\n\n. F Takayama, M Yamaguchi, hep-ph/9910320F. Takayama and M. Yamaguchi, hep-ph/9910320.\n\n. M Hirsch, M Diaz, W Porod, J Romao, J Valle, hep-ph/0004115M. Hirsch, M. Diaz, W. Porod, J. Romao, and J. Valle, hep-ph/0004115.\n\n. O C W Kong, Mod. Phys. Lett. 14903O.C.W. Kong, Mod. Phys. Lett. A14 (1999) 903.\n\n. A Abada, M Losada, hep-ph/9908352A. Abada and M. Losada, hep-ph/9908352.\n\n. K Cheung, O C W Kong, Phys. Rev. 61113012K. Cheung and O.C.W. Kong, Phys. Rev. D61 (2000) 113012.\n\nhep-ph/9906310; see also. Y Grossman, H E Haber, ; M Hirsch, H V Klapdor-Kleingrothaus, S G Kovalenko, Phys. Lett. B. 398311Y. Grossman and H.E. Haber, hep-ph/9906310; see also M. Hirsch, H.V. Klapdor- Kleingrothaus S.G. Kovalenko, Phys. Lett. B 398 (1997) 311.\n\n. Y Fukuda, Super-Kamiokande CollaborationPhys. Rev. Lett. 811562Super-Kamiokande Collaboration, Y.Fukuda et al., Phys. Rev. Lett. 81 (1998) 1562;\n\n. P L Lipari ; G, E Fogli, A Lisi, G Marrone, Scioscia, Phys. Rev. 5933001P. Lipari, hep-ph/9904443; G.L. Fogli, E. Lisi, A. Marrone, and G. Scioscia, Phys. Rev. D59 (1999) 033001.\n\n. E J Chun, S K Kang, Phys. Rev. 6175012E.J. Chun and S.K. Kang, Phys. Rev. D61 (2000) 075012.\n\n. M Bisset, O C W Kong, C Macesanu, L H Orr, Phys. Lett. 430274M. Bisset, O.C.W. Kong, C. Macesanu, and L.H. Orr, Phys. Lett. B430 (1998) 274\n\n. Phys. Rev. 6235001Phys. Rev. D62 (2000) 035001.\n\n. Y.-Y Keum, O C W Kong, hep-ph/0004110Y.-Y. Keum and O.C.W. Kong, hep-ph/0004110.\n\nY.-Y Keum, O C W Kong, IPAS-HEP-k006. manuscript in preparationY.-Y. Keum and O.C.W. Kong, IPAS-HEP-k006, manuscript in preparation.\n\n. Y.-Y Keum, O C W Kong, work in progressY.-Y. Keum and O.C.W. Kong, work in progress.\n\nK Cheung, O C W Kong, IPAS-HEP-k007. manuscript in preparationK. Cheung and O.C.W. Kong, IPAS-HEP-k007, manuscript in preparation.\n\n. L E Ibáñez, G G Ross, Nucl. Phys. 3683L.E. Ibáñez and G.G. Ross, Nucl. Phys. B368 (1992) 3.\n\nIPAS-HEP-k008. O C W Kong, manuscript in preparationO.C.W. Kong, IPAS-HEP-k008, manuscript in preparation.\n\n. S Y Choi, E J Chun, S K Kang, J S Lee, Phys. Rev. 6075002S.Y. Choi, E.J. Chun, S.K. Kang, and J.S. Lee, Phys. Rev. D60 (1999) 075002.\n\n. Y See, H E Grossman, Haber, Phys. Rev. 5993008for exampleSee Y. Grossman and H.E. Haber, Phys. Rev. D59 (1999) 093008, for example.\n\n. A Pilaftsis, Phys. Lett. 43588A. Pilaftsis, Phys. Lett. B435 (1998) 88;\n\n. A Pilaftsis, C E M Wagner, Nucl. Phys. 5533A. Pilaftsis and C.E.M. Wagner, Nucl. Phys. B553 (1999) 3.\n\n. A Zee, Phys. Lett. 93389A. Zee, Phys. Lett. 93B (1980) 389.\n\n. S Davidson, M Losada, JHEP. 000521S. Davidson and M. Losada, JHEP 0005 (2000) 021.\n"},"corpus_id":{"kind":"number","value":1522414,"string":"1,522,414"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"hep-ph/0004107\",\n \"mag\": 2030303578,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/1126-6708/2000/09/037\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/hep-ph/0004107v4.pdf\"\n ],\n \"pdf_sha\": \"c27c3157118ba74c2f6c012dfb39da193ecf6262\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/hep-ph/0004107\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"LR Scalar Mixings and One-loop Neutrino Masses"},"authors":{"kind":"list like","value":["Otto C W Kong \nInstitute of Physics\nAcademia Sinica\n11529Nankang, TaipeiTAIWAN"],"string":"[\n \"Otto C W Kong \\nInstitute of Physics\\nAcademia Sinica\\n11529Nankang, TaipeiTAIWAN\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Within the framework of the complete theory of supersymmetry without R-parity, where all possible R-parity violating terms are admitted, we perform a systematic analytical study of all sources of neutrino masses up to \"direct one-loop\" (defined explicitly below) level. In the passing, we present the full result for squark and slepton masses. In particular, there are interesting LR squark and slepton mixings, which involve both bilinear and trilinear R-parity violating parameters. The existence and important phenomenological implications of such terms have been largely overlooked in previous studies. In particular, in the studies under which either one type of the couplings is assumed to vanish or neglected, the terms would not show up. The LR mixings play a central role in neutrino mass generation. Our results look straight forward to be obtained, which, in our opinion, is an illustration of the effectiveness of our formulation adopted."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"The minimal supersymmetric standard model (MSSM) is no doubt the most popular candidate theory for physics beyond the Standard Model (SM). The alternative theory with a discrete symmetry called R-parity not imposed deserves no less attention. In particular, the latter admits neutrino masses, without the need for any extra matter field beyond the minimal spectrum. At the present time, experimental results from neutrino physics [1] is actually the only data we have demanding physics beyond the SM, while signals from supersymmetry (SUSY) are still absent. The neutrino data provides strong hints for the existence of Majorana type masses. The latter means lepton number violation, which is suggestive of R-parity violation. Hence, it is easy to appreciate the interest in R-parity violating (RPV) contributions to neutrino masses. The study of this topic has a long history, starting from Ref. [2]. Two of the notable papers on different aspects of the topic are given in Ref. [3] and Ref. [4], to which readers are also referred for references to earlier works. More recent works in the subject area [5,6,7,8,9,10] mainly focus on the fitting of the neutrino oscillation data under different scenarios while a comprehensive analysis of all the RPV contributions is still missing. This paper aims at providing such a picture.\nLike most of the other recent studies, we will focus on the sub-eV neutrino mass scale suggested by the Super-Kamiokande atmospheric neutrino data [11], though most of our results are actually valid for a much larger range of neutrino masses.\nAs illustrated below, there is a tree-level but seesaw suppressed contribution and some direct 1-loop contributions. Our level of treatment in this paper stops there, i.e. we will, in general, not go into contributions that are expected to be further suppressed. There are direct 1-loop contributions which involve a further seesaw type suppression hidden inside the loop. These are 1-loop diagrams that would suggest a null result if electroweak (EW) states are used for the particles running inside the loop and only a minimal number of mass insertions is admitted. When one thinks about the exact result to be obtained from using mass eigenstates instead, a nonzero result could emerge. It is diffcult to give analytical expressions for the mass eigenstate results. An approximation to the latter could be obtained by considering the EW state diagrams with extra mass insertions. In the case that these mass insertions are RPV, it typically means extra seesaw type suppression. We refer to contributions from such diagrams as pseudo-direct 1-loop contributions, which we will discuss without giving explicit formulae. We list also the well-known results. The idea here is to perform a systematic analysis and present the exhaustive list of all contributions up to the level of treatment.\nA similar comprehensive listing of neutrino mass contributions up to the 1-loop level (direct or indirect) has been presented in Ref. [12]. However, the latter analysis is limited to a scenario where the \"third generation couplings dominate\". This amounts to admitting only non-zero λ ′ i33 's and λ i33 's among the trilinear RPV couplings, though all nonzero bilinear RPV are indeed included by the authors. In our opinion, the maximal mixing result from Super-Kamiokande [11] brings the wisdom of \"third generation domination\" under question. Refs. [7] and [9], for example illustrate how no (family) hierarchy, or even an anti-hierarchy, among the RPV couplings may be preferred. The present analysis handles the complete theory of supersymmetry (SUSY) without R-parity, where all kind of RPV terms are admitted without bias. We present complete tree-level mass matrices for the scalars in this generic scenario. There is another major difference between the two studies. Ref. [12] is interested in performing some numerical calculation. While the latter is important for explicit fitting of experimental numbers, much of the physical origin of the neutrino mass contributions are hidden under elements of mixing matrices parametrizing the effective couplings of the neutrinos to squark or slepton mass eigenstates. We are interested here in illustrating the explicit origin of each contribution. Hence, we stay with electroweak (EW) state notation and give diagrammatic as well as analytical expressions of each individual contribution. Of particular interest here is a new type of contribution involving a RPV LR scalar (squark or slepton) mixings, which has been larger overlooked by previous authors. We hope that results here will be useful for a better understanding the role of each RPV parameter and identifying interesting regions of the extensive parameter space.\nTo study all the RPV contributions in a single consistent framework, one needs an effective formulation of the complete theory of SUSY without R-parity. The latter theory is generally better motivated than ad hoc versions of RPV theories. The large number of new parameters involved, however, makes the theory difficult to analyze. It has been illustrated [13] that an optimal parametrization, called the single-VEV parametrization, can be of great help in making the task manageable. The effectiveness of the SVP has been explored to perform an extensive study on the resultant leptonic phenomenology [13], to identify new type of neutrino mass contributions [9], and to study a new contribution to neutron electric dipole moment at 1-loop level [14,15], as well as new sources of contribution to flavor changing neutral current processes such as b → s γ [16] and µ → e γ [17]. Studies of neutrino masses and mixings under the formulation also include Refs. [7,8]. In fact, neutrino masses contribution is a central aspect of RPV effects and is likely to provide the most stringent bounds on the couplings, though many of the bounds obtained depend on assumptions on interpretation of neutrino data and could be relaxed or removed by simple extensions of the theory allowing extra sterile neutrino(s).\nOne-loop neutrino mass generation in SUSY without R-parity typically involves LR mixings of squarks or slepton. We want to emphasize again that squark and slepton mass matrices presented here are complete, with all source of R-parity violation included. Such results are explicitly presented for the first time. We consider the results to be interesting in their own right.\nIn the appendix, we give also an explicit illustration that all the VEV's under the SVP may be taken as real, despite the existence of complex parameters in the scalar potential; and give some important consistence relationships among some of the parameters involved. These results have not been published before, and serve as important background for clarifying some issues on the scalar masses and neutrino mass contributions discussed."},{"title":"Formulation and Notation","content":"We summarize our formulation and notation below. The most general renormalizable superpotential for the supersymmetric SM (without R-parity) can be written as\nwhere (a, b) are SU(2) indices, (i, j, k) are the usual family (flavor) indices, and (α, β) are extended flavor index going from 0 to 3. In the limit where λ ijk , λ ′ ijk , λ ′′ ijk and µ i all vanish, one recovers the expression for the R-parity preserving case, withL 0 identified asĤ d . Without R-parity imposed, the latter is not a priori distinguishable from theL i 's. Note that λ is antisymmetric in the first two indices, as required by the SU(2) product rules, as shown explicitly here with ε 12 = −ε 21 = 1. Similarly, λ ′′ is antisymmetric in the last two indices from SU(3) C . R-parity is exactly an ad hoc symmetry put in to makeL 0 , stand out from the otherL i 's as the candidate forĤ d . It is defined in terms of baryon number, lepton number, and spin as, explicitly, R = (−1) 3B+L+2S . The consequence is that the accidental symmetries of baryon number and lepton number in the SM are preserved, at the expense of making particles and superparticles having a categorically different quantum number, R-parity. The latter is actually not the most effective discrete symmetry to control superparticle mediated proton decay [18], but is most restrictive in terms of what is admitted in the Lagrangian, or the superpotential alone.\nA naive look at the scenario suggests that the large number of new couplings makes the task formidable. However, it becomes quite manageable with an optimal choice of flavor bases, the SVP [13]. In fact, doing phenomenological studies without specifying a choice of flavor bases is ambiguous. It is like doing SM quark physics with 18 complex Yukawa couplings instead of the 10 real physical parameters. In SUSY without R-parity, the choice of an optimal parametrization mainly concerns the 4 L α flavors. Under the SVP, flavor bases are chosen such that : 1/ among theL α 's,\nThe big advantage here is that the (tree-level) mass matrices for all the fermions do not involve any of the trilinear RPV couplings, even though the approach makes no assumption on any RPV coupling, including those from soft SUSY breaking. Moreover, and all the parameters used are uniquely defined, with the exception of some removable phases. In fact, the (color-singlet) charged fermion mass matrix reduces to the simple form :\n(2.2)\nEach µ i parameter here characterizes directly the RPV effect on the corresponding charged lepton (ℓ i = e, µ, and τ ). For any set of other parameter inputs, the m i 's can then be determined, through a simple numerical procedure, to guarantee that the correct mass eigenvalues of m e , m µ , and m τ are obtained -an issue first addressed and solved in Ref. [13]. The latter issue is especially important when µ i 's not substantially smaller than µ 0 are considered. Such an odd scenario is not definitely ruled out [13]. However, we would concentrate here on the more popular scenario with only sub-eV neutrino masses and hence small µ i 's. Here deviations of the m i 's from the mass eigenvalues are negligible."},{"title":"Gauginos, Higgsinos, and Neutrinos","content":"The tree-level mixings among the gauginos, higgsinos, and neutrinos gives rise to a 7 × 7 neutral fermion mass matrix M N :\n, withh 0 d being the neutral fermion from L 0 . The latter is guaranteed to be predominately a neutralino rather than neutrino, as the mass matrix clearly illustrates. As pointed out above, for small µ i 's the charged fermion states in theL i 's are essentially the physical states of e, µ and τ . Hence, (ν L 1 , ν L 2 , ν L 3 ) are essentially ν e , ν µ , ν τ . All entires in the lower-right 3 × 3 block (m o ν ) are, of course, zero at tree level. They are induced via 1-loop contributions to be discussed below. Such contributions are the focus of the present study. They are referred to here as direct 1-loop contributions.\nWe can write the general mass matrix in the form of block submatrices:\nwhere M n is the upper-left 4 × 4 neutralino mass matrix, ξ is the 3 × 4 block, and m o ν is the lower-right 3 × 3 neutrino block in the 7 × 7 matrix. The resulting (effective) neutrino mass matrix after block diagonalization is given by\nContributions to the first term here starts at tree level, which are, however, seesaw suppressed. The second term is the direct contribution, which, however, enters in only at 1-loop level. We are interested in the small µ i scenario, where the tree-level contribution is not necessarily expected to be stronger than such direct 1-loop effects. The 1-loop contributions to the ξ and M n blocks are likely to have only a secondary effect on (m ν ). The latter, to be called indirect 1-loop contributions, are not included in the present analysis."},{"title":"LR-mixings for Squarks and Sleptons","content":"The soft SUSY breaking part of the Lagrangian can be written as\nwhere we have separated the R-parity conserving A-terms from the RPV ones (recall H d ≡L 0 ). Note thatL †m2 LL , unlike the other soft mass terms, is given by a 4 × 4 matrix. Explicitly,m 2 L 00 corresponds tom 2 H d of the MSSM case whilem 2 L 0k 's give RPV mass mixings.\nThe SVP also simplifies much the otherwise complicated expressions for the masssquared matrix of the scalar sectors. Firstly, we will look at the squark sectors. The masses of up-squarks obviously have no RPV contribution. The down-squark sector, however, has an interesting result. We have the mass-squared matrix as follows :\nand\nHere, m D is the down-quark mass matrix, which is diagonal under the parametrization adopted; ( µ * i λ ′ ijk ) denotes the 3 × 3 matrix ( ) jk with elements listed; and tanβ = vu v 0 . Note that all the VEV's can be taken as real, so long as the tree level scalar potential is considered (see the appendix). Apart from the first A D term, the remaining terms in (M 2 RL ) T are F -term contributions; in particular, the last term gives \"SUSY conserving\" but R-parity violating contributions; note that the existence of nonzero F -terms or electroweak symmetry breaking VEV's can be interpreted as a consequence of SUSY breaking though. The full F -term part in the above equation can actually be written together as\nwhere the α = 0 term, which vanishes for j = k, gives the second term in the RHS. The latter, of course, is just the usual µ-term contribution in the MSSM case.\nNext we move on to the slepton sector. From Eq.(4.1) above, we can see that the \"charged Higgs\" should be considered together with the sleptons. We have hence an 8 × 8 mass-squared matrix of the following 1 + 4 + 3 form :\nand\nwhere the three m i 's are masses from leptonic Yukawa terms as discussed above in relation to Eq.(2.2); and, again, ( µ * i λ iβk ) denotes a matrix (4 × 3) with elements given by ( ) βk . Recall that for the small µ i domain we focused on here in this paper, we have m E ≃ diag{m e , m µ , m τ }. In fact, the k-th element in the 3-column-vector M 2 RH in Eq.(4.8) can be written as simply as µ * k m k (no sum). Similarly, the k-th element in the first row of the 4 × 3 matrix ( M 2 RL ) T in Eq.(4.7) can be written as µ * k m k tanβ (no sum). The former is ã ℓ c R hu type, while the latter al c R hd type (hd ≡l L 0 ), mass-squared term. Or, to better illustrate the common flavor structure, one can put the full F -term part of Eq.(4.7) as − ( µ * α λ αβk ) vu √ 2 . For the sake of completeness, we also give explicitly the neutral scalar, (or sneutrino-Higgs) mass-squared matrix. The neutral scalar mass terms, in terms of the (1 + 4) complex scalar fields, φ n 's, can be written in two parts -a simple (M 2 φ ) mn φ † m φ n part, and a Majorana-like part in the form 1 2 (M 2 φφ ) mn φ m φ n + h.c.. As the neutral scalars are originated from chiral doublet superfields, the existence of the Majorana-like part is a direct consequence of the electroweak symmetry breaking VEV's, hence restricted to the scalars playing the Higgs role only. They come from the quartic terms of the Higgs fields in the scalar potential. We have explicitly\nand\nNote that M 2 φφ here is real (see the appendix), while M 2 φ does have complex entries. The full 10 × 10 (real and symmetric) mass-squared matrix for the real scalars is then given by\nwhere the scalar, pseudo-scalar, and mixing parts are\nrespectively. If Im(M 2 φ ) vanishes, the scalars and pseudo-scalars decouple from one another and the unphysical Goldstone mode would be found among the latter. Finally, we note that the B α entries may also be considered as a kind of LR mixings. The RPV B i 's do in fact contribute to neutrino mass, as discussed below.\nWe would like to emphasize that the above scalar mass results are completeall RPV contributions, SUSY breaking or otherwise, are included without theoretical bias. The simplicity of the result is a consequence of the SVP. Explicitly, there are no RPV A-term contributions due to the vanishing of VEV's v i ≡ √ 2 L i . However, such new contributions, as well as their roles in the physics of neutrino masses and phenomena like fermion EDM's and b → s γ, are genuine. For instance, rotating to a basis among theL α superfields under which the µ i 's are zero would restore theL i VEV's and show e.g. the RPV ( µ * λ ) term as a term involving the latter VEV's and some A-term parameters. The Higgs-slepton results given as in Eqs.(4.5) and (4.11) are admittedly not very useful for doing scalar physics. They contain a redundancy of parameters and hide the unphysical Goldstone state. However, for the purpose of analyzing the neutrino mass contributions as done below, they serve their purpose. Hence, we will refrain from further laboring on the algebra here.\nBefore ending the section, we want to emphasize the following. While it should be straight forward to write down the scalar mass matrices by the complete theory of SUSY without R-parity, under any formulation or parametrization, to the best of our knowledge, this has not been published before. After completing the work, we checked the literature and found no explicitly written down complete results for M 2 D and M 2 E as given here. Especially the existence of the interesting new RPV contributions, of the type given by the ( µ * λ ′ ) term in M 2 D , and the ( µ * λ ) term in Eq.(4.7), and the µ * µ terms in Eqs.(4.5) and (4.11), if noticed, have not been much appreciated. Explicit discussion of terms of the type ( µ * λ ) and their contribution to neutrino masses was first given in a recent paper by K. Cheung and the present author [9], in a different context. Their existence and important phenomenological implication seems, otherwise, to have been overlooked. We are not aware of any other explicit discussion of the ( µ * λ ′ ) and ( µ * λ ) terms in the literature. What follows is a check into the literature for the scalar mass results. Before doing that, however, we would like to emphasize that the complete theory of SUSY without R-parity is not a very popular subject, compared to various versions of more specific (assumed) forms of R-parity violation. We also warn the readers that most of the previous authors were not working under the parametrization we used here. In our opinion, there are many other subtle complications when a different parametrization is used, which have not been explicitly addressed. The interested readers are referred to a forthcoming review by the author on the subject [19]. The complete scalar mass results in a genericL α flavor basis would look more complicated than what we have here too. In particular, there would be contributions involving the trilinear RPV A-terms. Having said that, let us take a look at some works on a more or less complete version of R-parity violation and the scalar mass expression given therein. Ref. [20] gives \"sfermion mass matrix\" exactly as in the MSSM without RPV terms at all. Ref. [12] is a more careful and detailed study. However, as mentioned above, the paper considers only one λ and one λ ′ and hence does not give result in the complete theory anyway. An admissible ( µ * λ ′ ) term is still missing in the down-squark mass-squared matrix given. An 8 × 8 matrix corresponding to M 2 E is indeed presented. The matrix seems correct, under the starting assumption, though the admissible ( µ * λ ) term is not explicitly shown. Ref. [3] is the first to give the matrix in the 8 × 8 form, but we find no clear sign of the ( µ * λ ) term. The squark mass-squared matrix is not explicitly given there. In fact, the paper is based on a specific high energy scenario, which is hence not totally generic. For instance, early in the paper, the soft SUSY breaking part of the Lagrangian is already given in a simplified form with the high energy assumptions put in, hence not for the generic complete theory we discussed here. The neutral scalar mass, corresponding to M 2 S above, is better known [21]. In particular, Ref. [10] gives the result essentially under the SVP, though truncated to one lepton family.\nAnother important point to note is that we have pay special attention to the fact there the RPV parameters are generally complex, while this phase information has not been explicitly given previously. As mentioned above, the λ ′ term, for example, contributes to neutron EDM through its imaginary part. The other example is the existence of scalar-pseudo-scalar mixing as a result of complex RPV parameters, as given above explicitly in M 2 SP . The significant phenomenological implication of the latter has been well illustrated in Ref. [22], in the case of MSSM, for instance.\nFinally, we note that there is another group that has done quite elaborated works on their RPV model (see for example Ref. [6]), which however includes no trilinear RPV parameters and hence would not have our results for the complete theory. Moreover, in the soft SUSY breaking part given and used in Ref. [6] actually hasm 2 L 0k term missing. We have not actually done a detailed term by term checking to see if there are other discrepancies between ours results given in this section and others in the papers mentioned. The above brief comparison is supposed to serve as an illustration of the point we want to make here -that perhaps insufficient care and attention have been given to the complete results for the scalar masses in SUSY without R-parity. We will report more on the various phenomenological implications of some of the RPV mass entries in some other publications."},{"title":"Neutrino Mass Contributions","content":"Let us return to RPV contribution to neutrino mass. From Eq.\nThis is illustrated diagrammatically in Fig. 1. Next we come to the direct 1-loop contributions. A typical 1-loop neutrino mass diagram has two couplings of scalar-fermion-neutrino type. With the two couplings being λ ′ -type, we have a quark-squark loop as shown in Fig. 2. Here, a LR squark mixing is needed. From Eqs. parts : firstly the familiar one\nwhere Md denote an average down-squark mass, and A d being a constant (mass) parameter representing the \"proportional\" part of the A-term, namely\n, and m D h is the h-th diagonal element of the matrix m D (i.e. the quark mass); next, the \"proportionality\" violating part\nwhich is typically expected to be suppressed in many SUSY breaking scenarios and neglected; and, finally, the part due to the new RPV LR mixings,\nThe ( i ←→ j ) expression denote symmetrization with respect to i and j. It is interesting to note that the last result contains no SUSY breaking parameter in the LR mixings. In particular, the flavor changing parts of the latter could not be suppressed through any SUSY breaking mechanism. Similar to the quark-squark loop, a lepton-slepton loop with two λ-type coupling, as shown in Fig. 3, generates neutrino mass, in the presence of LR slepton mixings. Using Eqs.(4.5) and (4.7), again we split the result into the different parts : the familiar one from the \"proportional\" part of the A-term, where Ml denote an average charged slepton mass, and A e the constant (mass)\n, (recall that m h 's are diagonal element of m E and essentially the mass of the charged lepton); the \"proportionality\" violating part\nand the part due to the new RPV LR mixings,\nHowever, the above is not yet the full result for the type of contributions. We have emphasized throughout the paper the systematic treatment of making no a priori distinction between theL i 's andĤ d . The latter is denoted asL 0 and treated as a 4-th leptonic flavor. In Eq.(4.7), the last term admitted a β = 0 part the neutrino mass contribution of which has not been included in the above analysis of the lepton-slepton loop parallel to the quark-squark loop. The corresponding result is simply given by setting k to 0 in Eq.(5.7), which may then be simplified to\nThe contribution corresponds to the SUSY analog of the Zee neutrino mass diagram [23], as discussed in Ref. [9]. We illustrate the contribution and its Zee model analog in Fig. 4. A careful examination of Fig. 3 shows that one cannot get any more new neutrino mass diagram by replacing some other λ ijk flavor indices with a 0. Hence, we have completed the listing of the two-λ-loop contributions. The above has exhausted all the possibilities from using from the trilinear superpotential couplings as the only source for the loop vertices. The only other couplings involving a neutrino are the gauge couplings and the bilinear µ i 's. The effect of the latter has been considered in the tree-level seesaw. Putting two gauge couplings together, we do have a 1-loop neutrino mass diagram, with scalars and gauginos running in the loop. The charged loop does not work, while a neutral loop could do (see Fig. 5) when there is a Majorana-like sneutrino mass term. The latter contribution was first pointed out in Ref. [10]. In fact, Majorana-like sneutrino mass is where the required two units of lepton number violation come in. The former may be interpreted as a result of splitting in mass of the sneutrino and anti-sneutrino due to R-parity violation. Following our general approach here, we illustrate this in Fig. 6. It is clear from the figure that it involves the SUSY breaking and RPV parameters B i 's, as shown is Eq.(4.11) above, and is seesaw suppressed (cf. Fig 1), unless the B i 's happen to be at the SUSY scale despite small µ i 's. Note that the B i 's and the µ i 's are not totally independent parameters, as illustrated in the appendix. It is very unlikely that B i B 0 would be much larger than µ i µ 0 . The gauge loop contribution discussed serves as an illustrative example of what we call pseudo-direct 1-loop. The EW state diagram as given in Fig. 5 reads zero, as the required mass insertion on the sneutrino line does not exist [cf. Eq.(4. 10)]. If we admit extra mass insertions and extend the sneutrino line as shown in Fig. 6, we obtain the nonzero result. However, one show bear in mind that there is no definite hierarchy between this gauge loop contribution and the other direct 1-loop ones discussed above, as they arise from different RPV parameters, the magnitude of which we have no exact information.\nFinally, it is not difficult to see that there is no contribution from 1-loop diagrams with one gauge coupling and one Yukawa or λ-coupling vertices, up to the direct 1- loop level. In fact, before we put in non-minimal number of mass insertions in the internal lines, there is only one such EW state diagram, as given in Fig. 7. The diagram requires aW --ℓ c R k mass insertion, which is zero under the SVP. When we admit extra mass insertions along the internal fermion line and go to the pseudodirect 1-loop level, there are apparent nonzero contribution. Obviously we need at least one lepton number violating mass insertion. Fig. 8 illustrates the minimal extra mass insertions along the fermion line that could complete a diagram.\nHowever, let us look more closely into the implications of putting in extra mass insertions along the internal fermion line. The scalar-fermion loop neutrino mass diagram result always has a mass factor of the internal fermion in it; explicit examples are m D h in Fig. 2 and m h in Fig. 3. To obtain the exact result, one should use the mass eigenstates, and of course sum over the latter (see, for example, formulae in Ref. [12]). However, neglecting the fermion mass dependence in the propagator integral part, the mass and mixing matrix element dependence of the full sum is of course nothing other than the tree-level mass entry of single insertion case, i.e. the vanishingW -ℓ c R k term for the case at hand. If one consider only the contribution from one of the mass eigenstates, the result would not be zero. But we know that it is going to be canceled by that from the other mass eigenstates. This is like the GIM mechanism, violated here only to the extent of the non-universal mass effect from the propagator integral part.\nIn terms of EW state diagrams, the exact mass eigenstate result would correspond to summing over all possible diagrams with any (up to infinite) number of admissible mass insertions. The contribution from putting Fig. 8 into the internal fermion line of Fig. 7 is of course just one among the nontrivial result. Hence, taking this as an independent contribution is more or less equivalent to taking one term from the summation over mass eigenstates. The result of the latter is expected to be canceled in the overall sum.\nFurthermore, even if that GIM-like cancellation is rendered ineffective by the propagator integral part, the dependence of the contribution on a single λ jik implies that upon symmetrization with respect to i and j, there would be another cancellation as from λ ijk = −λ jik to the extent thatl -L j andl -L i has the same mass. So, even the naive pseudo-direct 1-loop contribution has suppression from the expected degeneracy of slepton masses, and is only proportional to the violation of the latter.\nThis, together with the gauge loop discussed above, has exhausted our discussion of the pseudo-direct 1-loop contributions. One could certainly get other EW state diagrams by putting in extra mass insertion, into the scalar line of Figs. 5 or 7. While direct 1-loop results from an EW state diagram roughly represent the corresponding exact mass eigenstate results, pseudo-direct 1-loop diagrams may be just pieces of an otherwise small or vanishing overall sum with a GIM-like cancellation mechanism at work. We do have to go to exact numerical calculations to know the extent to which the latter is violated and extract the correct result. Perhaps it should also be said that we have not considered diagrams with mass insertion(s) on the external lines because such diagrams really correpond the indirect 1-loop contributions. A diagram with one mass insertion in one of the ν external line, for example, should correspond to something in the ξ block of Eq.(3.1). We want to emphasize that while we classified contributions into direct 1-loop, pseudo-direct 1-loop and indirect 1-loop, there is no definite hierarchy among them, when one is comparing contributions involving different RPV parameters. A clear example is given by the fact that we do not know if the tree-level contribution which involves the µ i 's is really larger than, for example a λ ′ (quark-squark) loop contribution, though one may naively expect so."},{"title":"Concluding Remarks","content":"From the above systematic analysis, it is clear that we have discussed and given explicit formulae for all neutrino mass contributions up to the level of direct 1loop contribution, for the complete theory of SUSY without R-parity. Psuedo-direct 1-loop contributions are also discussed. We have also given a description of the full squark and slepton masses. The latter is useful for analyzing other aspects of phenomenology, particularly those related to LR mixings such as fermion electric dipole moment and flavor changing neutral current processes. The successful simple description here illustrates well the effectiveness of the formulation (SVP) adopted.\nNote Added : After we posted the first version of this paper, a paper from Davidson and Losada [24] on the subject appeared. The paper does list results under our formulation (SVP) here and goes beyond direct or pseudo-direct 1-loop level. In particular, a nice discussion of the gauge loop result is included. However, the basic approach is very different from that of this paper, and the contributions from the RPV LR scalar mixing are not included. Comparing our results with theirs, there seems to be some apparent disagreements, which we hope to address in detail in a future publication.\nwhere we have put in a complex phase in the VEV v 0 , for generality.\nThe equations from the vanishing derivatives of V s along φ -1 and φ 0 give\nHence, B 0 e iθv is real. In fact, the part of V s that is relevant to obtaining the tadpole equations is no different from that of MSSM apart from the fact thatm 2 Hu andm 2 H d of the latter are replaced bym 2 -1 andm 2 0 respectively. As in MSSM, the B 0 parameter can be taken as real. The conclusion here is therefore that θ v vanishes, or all VEV's are real, despite the existence of complex parameters in the scalar potential. Results from the other tadpole equations, in a φ i direction, are quite simple. They can be written as complex equations of the form which is equivalent to\nwhere we have used v u = v sinβ and v d = v cosβ. Note that our tanβ has the same physical meaning as that in the R-parity conserving case. For instance, tanβ, together with the corresponding Yukawa coupling ratio, gives the mass ratio between the top and the bottom quark. The three complex equations for the B i 's reflect the redundance of parameters in a genericL α flavor basis. The equations also suggest that the B i 's are expected to be suppressed, with respect to the R-parity conserving B 0 , as the µ i 's are, with respect to µ 0 . They give consistence relationships among the involved RPV parameters (under the SVP) that should not be overlooked."},{"title":"Figure 1 :","content":""},{"title":"Figure 2 :","content":""},{"title":"Figure 3 :","content":""},{"title":"Figure 4 :","content":""},{"title":"Figure 5 :igure 6 :","content":""},{"title":"Figure 7 :","content":""},{"title":"Figure 8 :","content":""},{"title":"m 2 -","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"The minimal supersymmetric standard model (MSSM) is no doubt the most popular candidate theory for physics beyond the Standard Model (SM). The alternative theory with a discrete symmetry called R-parity not imposed deserves no less attention. In particular, the latter admits neutrino masses, without the need for any extra matter field beyond the minimal spectrum. At the present time, experimental results from neutrino physics [1] is actually the only data we have demanding physics beyond the SM, while signals from supersymmetry (SUSY) are still absent. The neutrino data provides strong hints for the existence of Majorana type masses. The latter means lepton number violation, which is suggestive of R-parity violation. Hence, it is easy to appreciate the interest in R-parity violating (RPV) contributions to neutrino masses. The study of this topic has a long history, starting from Ref. [2]. Two of the notable papers on different aspects of the topic are given in Ref. [3] and Ref. [4], to which readers are also referred for references to earlier works. More recent works in the subject area [5,6,7,8,9,10] mainly focus on the fitting of the neutrino oscillation data under different scenarios while a comprehensive analysis of all the RPV contributions is still missing. This paper aims at providing such a picture.\\nLike most of the other recent studies, we will focus on the sub-eV neutrino mass scale suggested by the Super-Kamiokande atmospheric neutrino data [11], though most of our results are actually valid for a much larger range of neutrino masses.\\nAs illustrated below, there is a tree-level but seesaw suppressed contribution and some direct 1-loop contributions. Our level of treatment in this paper stops there, i.e. we will, in general, not go into contributions that are expected to be further suppressed. There are direct 1-loop contributions which involve a further seesaw type suppression hidden inside the loop. These are 1-loop diagrams that would suggest a null result if electroweak (EW) states are used for the particles running inside the loop and only a minimal number of mass insertions is admitted. When one thinks about the exact result to be obtained from using mass eigenstates instead, a nonzero result could emerge. It is diffcult to give analytical expressions for the mass eigenstate results. An approximation to the latter could be obtained by considering the EW state diagrams with extra mass insertions. In the case that these mass insertions are RPV, it typically means extra seesaw type suppression. We refer to contributions from such diagrams as pseudo-direct 1-loop contributions, which we will discuss without giving explicit formulae. We list also the well-known results. The idea here is to perform a systematic analysis and present the exhaustive list of all contributions up to the level of treatment.\\nA similar comprehensive listing of neutrino mass contributions up to the 1-loop level (direct or indirect) has been presented in Ref. [12]. However, the latter analysis is limited to a scenario where the \\\"third generation couplings dominate\\\". This amounts to admitting only non-zero λ ′ i33 's and λ i33 's among the trilinear RPV couplings, though all nonzero bilinear RPV are indeed included by the authors. In our opinion, the maximal mixing result from Super-Kamiokande [11] brings the wisdom of \\\"third generation domination\\\" under question. Refs. [7] and [9], for example illustrate how no (family) hierarchy, or even an anti-hierarchy, among the RPV couplings may be preferred. The present analysis handles the complete theory of supersymmetry (SUSY) without R-parity, where all kind of RPV terms are admitted without bias. We present complete tree-level mass matrices for the scalars in this generic scenario. There is another major difference between the two studies. Ref. [12] is interested in performing some numerical calculation. While the latter is important for explicit fitting of experimental numbers, much of the physical origin of the neutrino mass contributions are hidden under elements of mixing matrices parametrizing the effective couplings of the neutrinos to squark or slepton mass eigenstates. We are interested here in illustrating the explicit origin of each contribution. Hence, we stay with electroweak (EW) state notation and give diagrammatic as well as analytical expressions of each individual contribution. Of particular interest here is a new type of contribution involving a RPV LR scalar (squark or slepton) mixings, which has been larger overlooked by previous authors. We hope that results here will be useful for a better understanding the role of each RPV parameter and identifying interesting regions of the extensive parameter space.\\nTo study all the RPV contributions in a single consistent framework, one needs an effective formulation of the complete theory of SUSY without R-parity. The latter theory is generally better motivated than ad hoc versions of RPV theories. The large number of new parameters involved, however, makes the theory difficult to analyze. It has been illustrated [13] that an optimal parametrization, called the single-VEV parametrization, can be of great help in making the task manageable. The effectiveness of the SVP has been explored to perform an extensive study on the resultant leptonic phenomenology [13], to identify new type of neutrino mass contributions [9], and to study a new contribution to neutron electric dipole moment at 1-loop level [14,15], as well as new sources of contribution to flavor changing neutral current processes such as b → s γ [16] and µ → e γ [17]. Studies of neutrino masses and mixings under the formulation also include Refs. [7,8]. In fact, neutrino masses contribution is a central aspect of RPV effects and is likely to provide the most stringent bounds on the couplings, though many of the bounds obtained depend on assumptions on interpretation of neutrino data and could be relaxed or removed by simple extensions of the theory allowing extra sterile neutrino(s).\\nOne-loop neutrino mass generation in SUSY without R-parity typically involves LR mixings of squarks or slepton. We want to emphasize again that squark and slepton mass matrices presented here are complete, with all source of R-parity violation included. Such results are explicitly presented for the first time. We consider the results to be interesting in their own right.\\nIn the appendix, we give also an explicit illustration that all the VEV's under the SVP may be taken as real, despite the existence of complex parameters in the scalar potential; and give some important consistence relationships among some of the parameters involved. These results have not been published before, and serve as important background for clarifying some issues on the scalar masses and neutrino mass contributions discussed.\"\n },\n {\n \"title\": \"Formulation and Notation\",\n \"content\": \"We summarize our formulation and notation below. The most general renormalizable superpotential for the supersymmetric SM (without R-parity) can be written as\\nwhere (a, b) are SU(2) indices, (i, j, k) are the usual family (flavor) indices, and (α, β) are extended flavor index going from 0 to 3. In the limit where λ ijk , λ ′ ijk , λ ′′ ijk and µ i all vanish, one recovers the expression for the R-parity preserving case, withL 0 identified asĤ d . Without R-parity imposed, the latter is not a priori distinguishable from theL i 's. Note that λ is antisymmetric in the first two indices, as required by the SU(2) product rules, as shown explicitly here with ε 12 = −ε 21 = 1. Similarly, λ ′′ is antisymmetric in the last two indices from SU(3) C . R-parity is exactly an ad hoc symmetry put in to makeL 0 , stand out from the otherL i 's as the candidate forĤ d . It is defined in terms of baryon number, lepton number, and spin as, explicitly, R = (−1) 3B+L+2S . The consequence is that the accidental symmetries of baryon number and lepton number in the SM are preserved, at the expense of making particles and superparticles having a categorically different quantum number, R-parity. The latter is actually not the most effective discrete symmetry to control superparticle mediated proton decay [18], but is most restrictive in terms of what is admitted in the Lagrangian, or the superpotential alone.\\nA naive look at the scenario suggests that the large number of new couplings makes the task formidable. However, it becomes quite manageable with an optimal choice of flavor bases, the SVP [13]. In fact, doing phenomenological studies without specifying a choice of flavor bases is ambiguous. It is like doing SM quark physics with 18 complex Yukawa couplings instead of the 10 real physical parameters. In SUSY without R-parity, the choice of an optimal parametrization mainly concerns the 4 L α flavors. Under the SVP, flavor bases are chosen such that : 1/ among theL α 's,\\nThe big advantage here is that the (tree-level) mass matrices for all the fermions do not involve any of the trilinear RPV couplings, even though the approach makes no assumption on any RPV coupling, including those from soft SUSY breaking. Moreover, and all the parameters used are uniquely defined, with the exception of some removable phases. In fact, the (color-singlet) charged fermion mass matrix reduces to the simple form :\\n(2.2)\\nEach µ i parameter here characterizes directly the RPV effect on the corresponding charged lepton (ℓ i = e, µ, and τ ). For any set of other parameter inputs, the m i 's can then be determined, through a simple numerical procedure, to guarantee that the correct mass eigenvalues of m e , m µ , and m τ are obtained -an issue first addressed and solved in Ref. [13]. The latter issue is especially important when µ i 's not substantially smaller than µ 0 are considered. Such an odd scenario is not definitely ruled out [13]. However, we would concentrate here on the more popular scenario with only sub-eV neutrino masses and hence small µ i 's. Here deviations of the m i 's from the mass eigenvalues are negligible.\"\n },\n {\n \"title\": \"Gauginos, Higgsinos, and Neutrinos\",\n \"content\": \"The tree-level mixings among the gauginos, higgsinos, and neutrinos gives rise to a 7 × 7 neutral fermion mass matrix M N :\\n, withh 0 d being the neutral fermion from L 0 . The latter is guaranteed to be predominately a neutralino rather than neutrino, as the mass matrix clearly illustrates. As pointed out above, for small µ i 's the charged fermion states in theL i 's are essentially the physical states of e, µ and τ . Hence, (ν L 1 , ν L 2 , ν L 3 ) are essentially ν e , ν µ , ν τ . All entires in the lower-right 3 × 3 block (m o ν ) are, of course, zero at tree level. They are induced via 1-loop contributions to be discussed below. Such contributions are the focus of the present study. They are referred to here as direct 1-loop contributions.\\nWe can write the general mass matrix in the form of block submatrices:\\nwhere M n is the upper-left 4 × 4 neutralino mass matrix, ξ is the 3 × 4 block, and m o ν is the lower-right 3 × 3 neutrino block in the 7 × 7 matrix. The resulting (effective) neutrino mass matrix after block diagonalization is given by\\nContributions to the first term here starts at tree level, which are, however, seesaw suppressed. The second term is the direct contribution, which, however, enters in only at 1-loop level. We are interested in the small µ i scenario, where the tree-level contribution is not necessarily expected to be stronger than such direct 1-loop effects. The 1-loop contributions to the ξ and M n blocks are likely to have only a secondary effect on (m ν ). The latter, to be called indirect 1-loop contributions, are not included in the present analysis.\"\n },\n {\n \"title\": \"LR-mixings for Squarks and Sleptons\",\n \"content\": \"The soft SUSY breaking part of the Lagrangian can be written as\\nwhere we have separated the R-parity conserving A-terms from the RPV ones (recall H d ≡L 0 ). Note thatL †m2 LL , unlike the other soft mass terms, is given by a 4 × 4 matrix. Explicitly,m 2 L 00 corresponds tom 2 H d of the MSSM case whilem 2 L 0k 's give RPV mass mixings.\\nThe SVP also simplifies much the otherwise complicated expressions for the masssquared matrix of the scalar sectors. Firstly, we will look at the squark sectors. The masses of up-squarks obviously have no RPV contribution. The down-squark sector, however, has an interesting result. We have the mass-squared matrix as follows :\\nand\\nHere, m D is the down-quark mass matrix, which is diagonal under the parametrization adopted; ( µ * i λ ′ ijk ) denotes the 3 × 3 matrix ( ) jk with elements listed; and tanβ = vu v 0 . Note that all the VEV's can be taken as real, so long as the tree level scalar potential is considered (see the appendix). Apart from the first A D term, the remaining terms in (M 2 RL ) T are F -term contributions; in particular, the last term gives \\\"SUSY conserving\\\" but R-parity violating contributions; note that the existence of nonzero F -terms or electroweak symmetry breaking VEV's can be interpreted as a consequence of SUSY breaking though. The full F -term part in the above equation can actually be written together as\\nwhere the α = 0 term, which vanishes for j = k, gives the second term in the RHS. The latter, of course, is just the usual µ-term contribution in the MSSM case.\\nNext we move on to the slepton sector. From Eq.(4.1) above, we can see that the \\\"charged Higgs\\\" should be considered together with the sleptons. We have hence an 8 × 8 mass-squared matrix of the following 1 + 4 + 3 form :\\nand\\nwhere the three m i 's are masses from leptonic Yukawa terms as discussed above in relation to Eq.(2.2); and, again, ( µ * i λ iβk ) denotes a matrix (4 × 3) with elements given by ( ) βk . Recall that for the small µ i domain we focused on here in this paper, we have m E ≃ diag{m e , m µ , m τ }. In fact, the k-th element in the 3-column-vector M 2 RH in Eq.(4.8) can be written as simply as µ * k m k (no sum). Similarly, the k-th element in the first row of the 4 × 3 matrix ( M 2 RL ) T in Eq.(4.7) can be written as µ * k m k tanβ (no sum). The former is ã ℓ c R hu type, while the latter al c R hd type (hd ≡l L 0 ), mass-squared term. Or, to better illustrate the common flavor structure, one can put the full F -term part of Eq.(4.7) as − ( µ * α λ αβk ) vu √ 2 . For the sake of completeness, we also give explicitly the neutral scalar, (or sneutrino-Higgs) mass-squared matrix. The neutral scalar mass terms, in terms of the (1 + 4) complex scalar fields, φ n 's, can be written in two parts -a simple (M 2 φ ) mn φ † m φ n part, and a Majorana-like part in the form 1 2 (M 2 φφ ) mn φ m φ n + h.c.. As the neutral scalars are originated from chiral doublet superfields, the existence of the Majorana-like part is a direct consequence of the electroweak symmetry breaking VEV's, hence restricted to the scalars playing the Higgs role only. They come from the quartic terms of the Higgs fields in the scalar potential. We have explicitly\\nand\\nNote that M 2 φφ here is real (see the appendix), while M 2 φ does have complex entries. The full 10 × 10 (real and symmetric) mass-squared matrix for the real scalars is then given by\\nwhere the scalar, pseudo-scalar, and mixing parts are\\nrespectively. If Im(M 2 φ ) vanishes, the scalars and pseudo-scalars decouple from one another and the unphysical Goldstone mode would be found among the latter. Finally, we note that the B α entries may also be considered as a kind of LR mixings. The RPV B i 's do in fact contribute to neutrino mass, as discussed below.\\nWe would like to emphasize that the above scalar mass results are completeall RPV contributions, SUSY breaking or otherwise, are included without theoretical bias. The simplicity of the result is a consequence of the SVP. Explicitly, there are no RPV A-term contributions due to the vanishing of VEV's v i ≡ √ 2 L i . However, such new contributions, as well as their roles in the physics of neutrino masses and phenomena like fermion EDM's and b → s γ, are genuine. For instance, rotating to a basis among theL α superfields under which the µ i 's are zero would restore theL i VEV's and show e.g. the RPV ( µ * λ ) term as a term involving the latter VEV's and some A-term parameters. The Higgs-slepton results given as in Eqs.(4.5) and (4.11) are admittedly not very useful for doing scalar physics. They contain a redundancy of parameters and hide the unphysical Goldstone state. However, for the purpose of analyzing the neutrino mass contributions as done below, they serve their purpose. Hence, we will refrain from further laboring on the algebra here.\\nBefore ending the section, we want to emphasize the following. While it should be straight forward to write down the scalar mass matrices by the complete theory of SUSY without R-parity, under any formulation or parametrization, to the best of our knowledge, this has not been published before. After completing the work, we checked the literature and found no explicitly written down complete results for M 2 D and M 2 E as given here. Especially the existence of the interesting new RPV contributions, of the type given by the ( µ * λ ′ ) term in M 2 D , and the ( µ * λ ) term in Eq.(4.7), and the µ * µ terms in Eqs.(4.5) and (4.11), if noticed, have not been much appreciated. Explicit discussion of terms of the type ( µ * λ ) and their contribution to neutrino masses was first given in a recent paper by K. Cheung and the present author [9], in a different context. Their existence and important phenomenological implication seems, otherwise, to have been overlooked. We are not aware of any other explicit discussion of the ( µ * λ ′ ) and ( µ * λ ) terms in the literature. What follows is a check into the literature for the scalar mass results. Before doing that, however, we would like to emphasize that the complete theory of SUSY without R-parity is not a very popular subject, compared to various versions of more specific (assumed) forms of R-parity violation. We also warn the readers that most of the previous authors were not working under the parametrization we used here. In our opinion, there are many other subtle complications when a different parametrization is used, which have not been explicitly addressed. The interested readers are referred to a forthcoming review by the author on the subject [19]. The complete scalar mass results in a genericL α flavor basis would look more complicated than what we have here too. In particular, there would be contributions involving the trilinear RPV A-terms. Having said that, let us take a look at some works on a more or less complete version of R-parity violation and the scalar mass expression given therein. Ref. [20] gives \\\"sfermion mass matrix\\\" exactly as in the MSSM without RPV terms at all. Ref. [12] is a more careful and detailed study. However, as mentioned above, the paper considers only one λ and one λ ′ and hence does not give result in the complete theory anyway. An admissible ( µ * λ ′ ) term is still missing in the down-squark mass-squared matrix given. An 8 × 8 matrix corresponding to M 2 E is indeed presented. The matrix seems correct, under the starting assumption, though the admissible ( µ * λ ) term is not explicitly shown. Ref. [3] is the first to give the matrix in the 8 × 8 form, but we find no clear sign of the ( µ * λ ) term. The squark mass-squared matrix is not explicitly given there. In fact, the paper is based on a specific high energy scenario, which is hence not totally generic. For instance, early in the paper, the soft SUSY breaking part of the Lagrangian is already given in a simplified form with the high energy assumptions put in, hence not for the generic complete theory we discussed here. The neutral scalar mass, corresponding to M 2 S above, is better known [21]. In particular, Ref. [10] gives the result essentially under the SVP, though truncated to one lepton family.\\nAnother important point to note is that we have pay special attention to the fact there the RPV parameters are generally complex, while this phase information has not been explicitly given previously. As mentioned above, the λ ′ term, for example, contributes to neutron EDM through its imaginary part. The other example is the existence of scalar-pseudo-scalar mixing as a result of complex RPV parameters, as given above explicitly in M 2 SP . The significant phenomenological implication of the latter has been well illustrated in Ref. [22], in the case of MSSM, for instance.\\nFinally, we note that there is another group that has done quite elaborated works on their RPV model (see for example Ref. [6]), which however includes no trilinear RPV parameters and hence would not have our results for the complete theory. Moreover, in the soft SUSY breaking part given and used in Ref. [6] actually hasm 2 L 0k term missing. We have not actually done a detailed term by term checking to see if there are other discrepancies between ours results given in this section and others in the papers mentioned. The above brief comparison is supposed to serve as an illustration of the point we want to make here -that perhaps insufficient care and attention have been given to the complete results for the scalar masses in SUSY without R-parity. We will report more on the various phenomenological implications of some of the RPV mass entries in some other publications.\"\n },\n {\n \"title\": \"Neutrino Mass Contributions\",\n \"content\": \"Let us return to RPV contribution to neutrino mass. From Eq.\\nThis is illustrated diagrammatically in Fig. 1. Next we come to the direct 1-loop contributions. A typical 1-loop neutrino mass diagram has two couplings of scalar-fermion-neutrino type. With the two couplings being λ ′ -type, we have a quark-squark loop as shown in Fig. 2. Here, a LR squark mixing is needed. From Eqs. parts : firstly the familiar one\\nwhere Md denote an average down-squark mass, and A d being a constant (mass) parameter representing the \\\"proportional\\\" part of the A-term, namely\\n, and m D h is the h-th diagonal element of the matrix m D (i.e. the quark mass); next, the \\\"proportionality\\\" violating part\\nwhich is typically expected to be suppressed in many SUSY breaking scenarios and neglected; and, finally, the part due to the new RPV LR mixings,\\nThe ( i ←→ j ) expression denote symmetrization with respect to i and j. It is interesting to note that the last result contains no SUSY breaking parameter in the LR mixings. In particular, the flavor changing parts of the latter could not be suppressed through any SUSY breaking mechanism. Similar to the quark-squark loop, a lepton-slepton loop with two λ-type coupling, as shown in Fig. 3, generates neutrino mass, in the presence of LR slepton mixings. Using Eqs.(4.5) and (4.7), again we split the result into the different parts : the familiar one from the \\\"proportional\\\" part of the A-term, where Ml denote an average charged slepton mass, and A e the constant (mass)\\n, (recall that m h 's are diagonal element of m E and essentially the mass of the charged lepton); the \\\"proportionality\\\" violating part\\nand the part due to the new RPV LR mixings,\\nHowever, the above is not yet the full result for the type of contributions. We have emphasized throughout the paper the systematic treatment of making no a priori distinction between theL i 's andĤ d . The latter is denoted asL 0 and treated as a 4-th leptonic flavor. In Eq.(4.7), the last term admitted a β = 0 part the neutrino mass contribution of which has not been included in the above analysis of the lepton-slepton loop parallel to the quark-squark loop. The corresponding result is simply given by setting k to 0 in Eq.(5.7), which may then be simplified to\\nThe contribution corresponds to the SUSY analog of the Zee neutrino mass diagram [23], as discussed in Ref. [9]. We illustrate the contribution and its Zee model analog in Fig. 4. A careful examination of Fig. 3 shows that one cannot get any more new neutrino mass diagram by replacing some other λ ijk flavor indices with a 0. Hence, we have completed the listing of the two-λ-loop contributions. The above has exhausted all the possibilities from using from the trilinear superpotential couplings as the only source for the loop vertices. The only other couplings involving a neutrino are the gauge couplings and the bilinear µ i 's. The effect of the latter has been considered in the tree-level seesaw. Putting two gauge couplings together, we do have a 1-loop neutrino mass diagram, with scalars and gauginos running in the loop. The charged loop does not work, while a neutral loop could do (see Fig. 5) when there is a Majorana-like sneutrino mass term. The latter contribution was first pointed out in Ref. [10]. In fact, Majorana-like sneutrino mass is where the required two units of lepton number violation come in. The former may be interpreted as a result of splitting in mass of the sneutrino and anti-sneutrino due to R-parity violation. Following our general approach here, we illustrate this in Fig. 6. It is clear from the figure that it involves the SUSY breaking and RPV parameters B i 's, as shown is Eq.(4.11) above, and is seesaw suppressed (cf. Fig 1), unless the B i 's happen to be at the SUSY scale despite small µ i 's. Note that the B i 's and the µ i 's are not totally independent parameters, as illustrated in the appendix. It is very unlikely that B i B 0 would be much larger than µ i µ 0 . The gauge loop contribution discussed serves as an illustrative example of what we call pseudo-direct 1-loop. The EW state diagram as given in Fig. 5 reads zero, as the required mass insertion on the sneutrino line does not exist [cf. Eq.(4. 10)]. If we admit extra mass insertions and extend the sneutrino line as shown in Fig. 6, we obtain the nonzero result. However, one show bear in mind that there is no definite hierarchy between this gauge loop contribution and the other direct 1-loop ones discussed above, as they arise from different RPV parameters, the magnitude of which we have no exact information.\\nFinally, it is not difficult to see that there is no contribution from 1-loop diagrams with one gauge coupling and one Yukawa or λ-coupling vertices, up to the direct 1- loop level. In fact, before we put in non-minimal number of mass insertions in the internal lines, there is only one such EW state diagram, as given in Fig. 7. The diagram requires aW --ℓ c R k mass insertion, which is zero under the SVP. When we admit extra mass insertions along the internal fermion line and go to the pseudodirect 1-loop level, there are apparent nonzero contribution. Obviously we need at least one lepton number violating mass insertion. Fig. 8 illustrates the minimal extra mass insertions along the fermion line that could complete a diagram.\\nHowever, let us look more closely into the implications of putting in extra mass insertions along the internal fermion line. The scalar-fermion loop neutrino mass diagram result always has a mass factor of the internal fermion in it; explicit examples are m D h in Fig. 2 and m h in Fig. 3. To obtain the exact result, one should use the mass eigenstates, and of course sum over the latter (see, for example, formulae in Ref. [12]). However, neglecting the fermion mass dependence in the propagator integral part, the mass and mixing matrix element dependence of the full sum is of course nothing other than the tree-level mass entry of single insertion case, i.e. the vanishingW -ℓ c R k term for the case at hand. If one consider only the contribution from one of the mass eigenstates, the result would not be zero. But we know that it is going to be canceled by that from the other mass eigenstates. This is like the GIM mechanism, violated here only to the extent of the non-universal mass effect from the propagator integral part.\\nIn terms of EW state diagrams, the exact mass eigenstate result would correspond to summing over all possible diagrams with any (up to infinite) number of admissible mass insertions. The contribution from putting Fig. 8 into the internal fermion line of Fig. 7 is of course just one among the nontrivial result. Hence, taking this as an independent contribution is more or less equivalent to taking one term from the summation over mass eigenstates. The result of the latter is expected to be canceled in the overall sum.\\nFurthermore, even if that GIM-like cancellation is rendered ineffective by the propagator integral part, the dependence of the contribution on a single λ jik implies that upon symmetrization with respect to i and j, there would be another cancellation as from λ ijk = −λ jik to the extent thatl -L j andl -L i has the same mass. So, even the naive pseudo-direct 1-loop contribution has suppression from the expected degeneracy of slepton masses, and is only proportional to the violation of the latter.\\nThis, together with the gauge loop discussed above, has exhausted our discussion of the pseudo-direct 1-loop contributions. One could certainly get other EW state diagrams by putting in extra mass insertion, into the scalar line of Figs. 5 or 7. While direct 1-loop results from an EW state diagram roughly represent the corresponding exact mass eigenstate results, pseudo-direct 1-loop diagrams may be just pieces of an otherwise small or vanishing overall sum with a GIM-like cancellation mechanism at work. We do have to go to exact numerical calculations to know the extent to which the latter is violated and extract the correct result. Perhaps it should also be said that we have not considered diagrams with mass insertion(s) on the external lines because such diagrams really correpond the indirect 1-loop contributions. A diagram with one mass insertion in one of the ν external line, for example, should correspond to something in the ξ block of Eq.(3.1). We want to emphasize that while we classified contributions into direct 1-loop, pseudo-direct 1-loop and indirect 1-loop, there is no definite hierarchy among them, when one is comparing contributions involving different RPV parameters. A clear example is given by the fact that we do not know if the tree-level contribution which involves the µ i 's is really larger than, for example a λ ′ (quark-squark) loop contribution, though one may naively expect so.\"\n },\n {\n \"title\": \"Concluding Remarks\",\n \"content\": \"From the above systematic analysis, it is clear that we have discussed and given explicit formulae for all neutrino mass contributions up to the level of direct 1loop contribution, for the complete theory of SUSY without R-parity. Psuedo-direct 1-loop contributions are also discussed. We have also given a description of the full squark and slepton masses. The latter is useful for analyzing other aspects of phenomenology, particularly those related to LR mixings such as fermion electric dipole moment and flavor changing neutral current processes. The successful simple description here illustrates well the effectiveness of the formulation (SVP) adopted.\\nNote Added : After we posted the first version of this paper, a paper from Davidson and Losada [24] on the subject appeared. The paper does list results under our formulation (SVP) here and goes beyond direct or pseudo-direct 1-loop level. In particular, a nice discussion of the gauge loop result is included. However, the basic approach is very different from that of this paper, and the contributions from the RPV LR scalar mixing are not included. Comparing our results with theirs, there seems to be some apparent disagreements, which we hope to address in detail in a future publication.\\nwhere we have put in a complex phase in the VEV v 0 , for generality.\\nThe equations from the vanishing derivatives of V s along φ -1 and φ 0 give\\nHence, B 0 e iθv is real. In fact, the part of V s that is relevant to obtaining the tadpole equations is no different from that of MSSM apart from the fact thatm 2 Hu andm 2 H d of the latter are replaced bym 2 -1 andm 2 0 respectively. As in MSSM, the B 0 parameter can be taken as real. The conclusion here is therefore that θ v vanishes, or all VEV's are real, despite the existence of complex parameters in the scalar potential. Results from the other tadpole equations, in a φ i direction, are quite simple. They can be written as complex equations of the form which is equivalent to\\nwhere we have used v u = v sinβ and v d = v cosβ. Note that our tanβ has the same physical meaning as that in the R-parity conserving case. For instance, tanβ, together with the corresponding Yukawa coupling ratio, gives the mass ratio between the top and the bottom quark. The three complex equations for the B i 's reflect the redundance of parameters in a genericL α flavor basis. The equations also suggest that the B i 's are expected to be suppressed, with respect to the R-parity conserving B 0 , as the µ i 's are, with respect to µ 0 . They give consistence relationships among the involved RPV parameters (under the SVP) that should not be overlooked.\"\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 :igure 6 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 7 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 8 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"m 2 -\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"(3.1), one neutrino state get a tree-level mass. The seesaw suppressed contribution [see Eqs.(3.2) and (3.3)] is given by","content":"(3.1), one neutrino state get a tree-level mass. The seesaw suppressed contribution [see Eqs.(3.2) and (3.3)] is given by"},{"id":"","caption":"Neutrino mass from tree-level seesaw.","content":"Figure 1 :\n1Neutrino mass from tree-level seesaw."},{"id":"","caption":"(4.2) and(4.4), we have the result, here written in three Neutrino mass from quark-squark loop.","content":"Figure 2 :\n2(4.2) and(4.4), we have the result, here written in three Neutrino mass from quark-squark loop."},{"id":"","caption":"Neutrino mass from lepton-slepton loop.","content":"Figure 3 :\n3Neutrino mass from lepton-slepton loop."},{"id":"","caption":"SUSY Zee diagram for neutrino mass. The Zee analog interpretation noted in the brackets.","content":"Figure 4 :\n4SUSY Zee diagram for neutrino mass. The Zee analog interpretation noted in the brackets."},{"id":"","caption":"Gaugino-sneutrino loop requiring a Majorana-like sneutrino mass insertion. Seesaw diagram for Majorana-like sneutrino masses.","content":"Figure 5 :igure 6 :\n56Gaugino-sneutrino loop requiring a Majorana-like sneutrino mass insertion. Seesaw diagram for Majorana-like sneutrino masses."},{"id":"","caption":"A direct charged loop contribution requiring however a fermion mass insertion that is vanishing.","content":"Figure 7 :\n7A direct charged loop contribution requiring however a fermion mass insertion that is vanishing."},{"id":"","caption":"A possible set of mass insertions apparently producing the required internal fermion line in the previous figure.","content":"Figure 8 :\n8A possible set of mass insertions apparently producing the required internal fermion line in the previous figure."},{"id":"","caption":"1i e iθ -1i tanβ = −e iθvm2 0i e iθ 0i , (A.5)","content":"m 2 -\n21i e iθ -1i tanβ = −e iθvm2 0i e iθ 0i , (A.5)"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"(3.1), one neutrino state get a tree-level mass. The seesaw suppressed contribution [see Eqs.(3.2) and (3.3)] is given by\",\n \"content\": \"(3.1), one neutrino state get a tree-level mass. The seesaw suppressed contribution [see Eqs.(3.2) and (3.3)] is given by\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Neutrino mass from tree-level seesaw.\",\n \"content\": \"Figure 1 :\\n1Neutrino mass from tree-level seesaw.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(4.2) and(4.4), we have the result, here written in three Neutrino mass from quark-squark loop.\",\n \"content\": \"Figure 2 :\\n2(4.2) and(4.4), we have the result, here written in three Neutrino mass from quark-squark loop.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Neutrino mass from lepton-slepton loop.\",\n \"content\": \"Figure 3 :\\n3Neutrino mass from lepton-slepton loop.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"SUSY Zee diagram for neutrino mass. The Zee analog interpretation noted in the brackets.\",\n \"content\": \"Figure 4 :\\n4SUSY Zee diagram for neutrino mass. The Zee analog interpretation noted in the brackets.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Gaugino-sneutrino loop requiring a Majorana-like sneutrino mass insertion. Seesaw diagram for Majorana-like sneutrino masses.\",\n \"content\": \"Figure 5 :igure 6 :\\n56Gaugino-sneutrino loop requiring a Majorana-like sneutrino mass insertion. Seesaw diagram for Majorana-like sneutrino masses.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A direct charged loop contribution requiring however a fermion mass insertion that is vanishing.\",\n \"content\": \"Figure 7 :\\n7A direct charged loop contribution requiring however a fermion mass insertion that is vanishing.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"A possible set of mass insertions apparently producing the required internal fermion line in the previous figure.\",\n \"content\": \"Figure 8 :\\n8A possible set of mass insertions apparently producing the required internal fermion line in the previous figure.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"1i e iθ -1i tanβ = −e iθvm2 0i e iθ 0i , (A.5)\",\n \"content\": \"m 2 -\\n21i e iθ -1i tanβ = −e iθvm2 0i e iθ 0i , (A.5)\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"For an updated review, see V. hep-ph/0003212For an updated review, see V. Barger, hep-ph/0003212.","doi":"hep-ph/0003212"},{"text":". L J Hall, M Suzuki, Nucl. Phys. 231419L.J. Hall and M. Suzuki, Nucl. Phys. B231 (1984) 419.","doi":""},{"text":". R Hempfling, Nucl. Phys. 4783R. Hempfling, Nucl. Phys. B478 (1996) 3.","doi":""},{"text":". H P Nilles, N Polonsky, Nucl. Phys. 48433H.P. Nilles and N. Polonsky, Nucl. Phys. B484 (1997) 33.","doi":""},{"text":". M Drees, S Pakvasa, X Tata, T Ter Veldhuis, Phys. Rev. 575335M. Drees, S. Pakvasa, X. Tata, and T. ter Veldhuis, Phys. Rev. D57 (1998) 5335;","doi":""},{"text":". V Bednyakov, A Faessler, S Kovalenko, Phys. Lett. B. 442203V. Bednyakov, A. Faessler, and S. Kovalenko, Phys. Lett. B 442 (1998) 203; B.","doi":""},{"text":". S Mukhopadhyaya, F Roy, Vissani, Phys. Lett. B. 443191Mukhopadhyaya, S. Roy, and F. Vissani, Phys. Lett. B 443 (1998) 191;","doi":""},{"text":". S Rakshit, G Bhattacharyya, M Raychaudhuri, Phys. Rev. 5991701S. Rakshit, G. Bhattacharyya, and M. Raychaudhuri, Phys. Rev. D59 (1999) 091701 ;","doi":""},{"text":". A S Joshipura, S K Vempati, Phys. Rev. 6095009A.S. Joshipura and S.K. Vempati, Phys. Rev. D60 (1999) 095009 ;","doi":""},{"text":". E Chun, S Kang, C Kim, U Lee, Nucl. Phys. 54489E. Chun, S. Kang, C. Kim, and U. Lee, Nucl. Phys. B544 (1999) 89;","doi":""},{"text":". F Takayama, M Yamaguchi, hep-ph/9910320F. Takayama and M. Yamaguchi, hep-ph/9910320.","doi":"hep-ph/9910320"},{"text":". M Hirsch, M Diaz, W Porod, J Romao, J Valle, hep-ph/0004115M. Hirsch, M. Diaz, W. Porod, J. Romao, and J. Valle, hep-ph/0004115.","doi":"hep-ph/0004115"},{"text":". O C W Kong, Mod. Phys. Lett. 14903O.C.W. Kong, Mod. Phys. Lett. A14 (1999) 903.","doi":""},{"text":". A Abada, M Losada, hep-ph/9908352A. Abada and M. Losada, hep-ph/9908352.","doi":"hep-ph/9908352"},{"text":". K Cheung, O C W Kong, Phys. Rev. 61113012K. Cheung and O.C.W. Kong, Phys. Rev. D61 (2000) 113012.","doi":""},{"text":"hep-ph/9906310; see also. Y Grossman, H E Haber, ; M Hirsch, H V Klapdor-Kleingrothaus, S G Kovalenko, Phys. Lett. B. 398311Y. Grossman and H.E. Haber, hep-ph/9906310; see also M. Hirsch, H.V. Klapdor- Kleingrothaus S.G. Kovalenko, Phys. Lett. B 398 (1997) 311.","doi":""},{"text":". Y Fukuda, Super-Kamiokande CollaborationPhys. Rev. Lett. 811562Super-Kamiokande Collaboration, Y.Fukuda et al., Phys. Rev. Lett. 81 (1998) 1562;","doi":""},{"text":". P L Lipari ; G, E Fogli, A Lisi, G Marrone, Scioscia, Phys. Rev. 5933001P. Lipari, hep-ph/9904443; G.L. Fogli, E. Lisi, A. Marrone, and G. Scioscia, Phys. Rev. D59 (1999) 033001.","doi":""},{"text":". E J Chun, S K Kang, Phys. Rev. 6175012E.J. Chun and S.K. Kang, Phys. Rev. D61 (2000) 075012.","doi":""},{"text":". M Bisset, O C W Kong, C Macesanu, L H Orr, Phys. Lett. 430274M. Bisset, O.C.W. Kong, C. Macesanu, and L.H. Orr, Phys. Lett. B430 (1998) 274","doi":""},{"text":". Phys. Rev. 6235001Phys. Rev. D62 (2000) 035001.","doi":""},{"text":". Y.-Y Keum, O C W Kong, hep-ph/0004110Y.-Y. Keum and O.C.W. Kong, hep-ph/0004110.","doi":"hep-ph/0004110"},{"text":"Y.-Y Keum, O C W Kong, IPAS-HEP-k006. manuscript in preparationY.-Y. Keum and O.C.W. Kong, IPAS-HEP-k006, manuscript in preparation.","doi":""},{"text":". Y.-Y Keum, O C W Kong, work in progressY.-Y. Keum and O.C.W. Kong, work in progress.","doi":""},{"text":"K Cheung, O C W Kong, IPAS-HEP-k007. manuscript in preparationK. Cheung and O.C.W. Kong, IPAS-HEP-k007, manuscript in preparation.","doi":""},{"text":". L E Ibáñez, G G Ross, Nucl. Phys. 3683L.E. Ibáñez and G.G. Ross, Nucl. Phys. B368 (1992) 3.","doi":""},{"text":"IPAS-HEP-k008. O C W Kong, manuscript in preparationO.C.W. Kong, IPAS-HEP-k008, manuscript in preparation.","doi":""},{"text":". S Y Choi, E J Chun, S K Kang, J S Lee, Phys. Rev. 6075002S.Y. Choi, E.J. Chun, S.K. Kang, and J.S. Lee, Phys. Rev. D60 (1999) 075002.","doi":""},{"text":". Y See, H E Grossman, Haber, Phys. Rev. 5993008for exampleSee Y. Grossman and H.E. Haber, Phys. Rev. D59 (1999) 093008, for example.","doi":""},{"text":". A Pilaftsis, Phys. Lett. 43588A. Pilaftsis, Phys. Lett. B435 (1998) 88;","doi":""},{"text":". A Pilaftsis, C E M Wagner, Nucl. Phys. 5533A. Pilaftsis and C.E.M. Wagner, Nucl. Phys. B553 (1999) 3.","doi":""},{"text":". A Zee, Phys. Lett. 93389A. Zee, Phys. Lett. 93B (1980) 389.","doi":""},{"text":". S Davidson, M Losada, JHEP. 000521S. Davidson and M. Losada, JHEP 0005 (2000) 021.","doi":""}],"string":"[\n {\n \"text\": \"For an updated review, see V. hep-ph/0003212For an updated review, see V. Barger, hep-ph/0003212.\",\n \"doi\": \"hep-ph/0003212\"\n },\n {\n \"text\": \". L J Hall, M Suzuki, Nucl. Phys. 231419L.J. Hall and M. Suzuki, Nucl. Phys. B231 (1984) 419.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Hempfling, Nucl. Phys. 4783R. Hempfling, Nucl. Phys. B478 (1996) 3.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H P Nilles, N Polonsky, Nucl. Phys. 48433H.P. Nilles and N. Polonsky, Nucl. Phys. B484 (1997) 33.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Drees, S Pakvasa, X Tata, T Ter Veldhuis, Phys. Rev. 575335M. Drees, S. Pakvasa, X. Tata, and T. ter Veldhuis, Phys. Rev. D57 (1998) 5335;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". V Bednyakov, A Faessler, S Kovalenko, Phys. Lett. B. 442203V. Bednyakov, A. Faessler, and S. Kovalenko, Phys. Lett. B 442 (1998) 203; B.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Mukhopadhyaya, F Roy, Vissani, Phys. Lett. B. 443191Mukhopadhyaya, S. Roy, and F. Vissani, Phys. Lett. B 443 (1998) 191;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Rakshit, G Bhattacharyya, M Raychaudhuri, Phys. Rev. 5991701S. Rakshit, G. Bhattacharyya, and M. Raychaudhuri, Phys. Rev. D59 (1999) 091701 ;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A S Joshipura, S K Vempati, Phys. Rev. 6095009A.S. Joshipura and S.K. Vempati, Phys. Rev. D60 (1999) 095009 ;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Chun, S Kang, C Kim, U Lee, Nucl. Phys. 54489E. Chun, S. Kang, C. Kim, and U. Lee, Nucl. Phys. B544 (1999) 89;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F Takayama, M Yamaguchi, hep-ph/9910320F. Takayama and M. Yamaguchi, hep-ph/9910320.\",\n \"doi\": \"hep-ph/9910320\"\n },\n {\n \"text\": \". M Hirsch, M Diaz, W Porod, J Romao, J Valle, hep-ph/0004115M. Hirsch, M. Diaz, W. Porod, J. Romao, and J. Valle, hep-ph/0004115.\",\n \"doi\": \"hep-ph/0004115\"\n },\n {\n \"text\": \". O C W Kong, Mod. Phys. Lett. 14903O.C.W. Kong, Mod. Phys. Lett. A14 (1999) 903.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Abada, M Losada, hep-ph/9908352A. Abada and M. Losada, hep-ph/9908352.\",\n \"doi\": \"hep-ph/9908352\"\n },\n {\n \"text\": \". K Cheung, O C W Kong, Phys. Rev. 61113012K. Cheung and O.C.W. Kong, Phys. Rev. D61 (2000) 113012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"hep-ph/9906310; see also. Y Grossman, H E Haber, ; M Hirsch, H V Klapdor-Kleingrothaus, S G Kovalenko, Phys. Lett. B. 398311Y. Grossman and H.E. Haber, hep-ph/9906310; see also M. Hirsch, H.V. Klapdor- Kleingrothaus S.G. Kovalenko, Phys. Lett. B 398 (1997) 311.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y Fukuda, Super-Kamiokande CollaborationPhys. Rev. Lett. 811562Super-Kamiokande Collaboration, Y.Fukuda et al., Phys. Rev. Lett. 81 (1998) 1562;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P L Lipari ; G, E Fogli, A Lisi, G Marrone, Scioscia, Phys. Rev. 5933001P. Lipari, hep-ph/9904443; G.L. Fogli, E. Lisi, A. Marrone, and G. Scioscia, Phys. Rev. D59 (1999) 033001.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E J Chun, S K Kang, Phys. Rev. 6175012E.J. Chun and S.K. Kang, Phys. Rev. D61 (2000) 075012.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M Bisset, O C W Kong, C Macesanu, L H Orr, Phys. Lett. 430274M. Bisset, O.C.W. Kong, C. Macesanu, and L.H. Orr, Phys. Lett. B430 (1998) 274\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. 6235001Phys. Rev. D62 (2000) 035001.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y.-Y Keum, O C W Kong, hep-ph/0004110Y.-Y. Keum and O.C.W. Kong, hep-ph/0004110.\",\n \"doi\": \"hep-ph/0004110\"\n },\n {\n \"text\": \"Y.-Y Keum, O C W Kong, IPAS-HEP-k006. manuscript in preparationY.-Y. Keum and O.C.W. Kong, IPAS-HEP-k006, manuscript in preparation.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y.-Y Keum, O C W Kong, work in progressY.-Y. Keum and O.C.W. Kong, work in progress.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"K Cheung, O C W Kong, IPAS-HEP-k007. manuscript in preparationK. Cheung and O.C.W. Kong, IPAS-HEP-k007, manuscript in preparation.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". L E Ibáñez, G G Ross, Nucl. Phys. 3683L.E. Ibáñez and G.G. Ross, Nucl. Phys. B368 (1992) 3.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"IPAS-HEP-k008. O C W Kong, manuscript in preparationO.C.W. Kong, IPAS-HEP-k008, manuscript in preparation.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Y Choi, E J Chun, S K Kang, J S Lee, Phys. Rev. 6075002S.Y. Choi, E.J. Chun, S.K. Kang, and J.S. Lee, Phys. Rev. D60 (1999) 075002.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Y See, H E Grossman, Haber, Phys. Rev. 5993008for exampleSee Y. Grossman and H.E. Haber, Phys. Rev. D59 (1999) 093008, for example.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Pilaftsis, Phys. Lett. 43588A. Pilaftsis, Phys. Lett. B435 (1998) 88;\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Pilaftsis, C E M Wagner, Nucl. Phys. 5533A. Pilaftsis and C.E.M. Wagner, Nucl. Phys. B553 (1999) 3.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Zee, Phys. Lett. 93389A. Zee, Phys. Lett. 93B (1980) 389.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Davidson, M Losada, JHEP. 000521S. Davidson and M. Losada, JHEP 0005 (2000) 021.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":593,"cells":{"text":{"kind":"string","value":"\nOn Low Rank Classical Groups in String Theory, Gauge Theory and Matrix Models\nDec 2003\n\nKen Intriligator \nDepartment of Physics\nUniversity of California\nSan Diego, La Jolla92093-0354CAUSA\n\nDepartment of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCAUSA\n\nAnton V Ryzhov \nDepartment of Physics\nBrandeis University\n02454WalthamMAUSA\n\nMasaki Shigemori \nDepartment of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCAUSA\n\nCumrun Vafa \nJefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA\n\nOn Low Rank Classical Groups in String Theory, Gauge Theory and Matrix Models\nDec 2003arXiv:hep-th/0311181v2 9\nWe consider N = 1 supersymmetric U (N ), SO(N ), and Sp(N ) gauge theories, with two-index tensor matter and added tree-level superpotential, for general breaking patterns of the gauge group. By considering the string theory realization and geometric transitions, we clarify when glueball superfields should be included and extremized, or rather set to zero; this issue arises for unbroken group factors of low rank. The string theory results, which are equivalent to those of the matrix model, refer to a particular UV completion of the gauge theory, which could differ from conventional gauge theory results by residual instanton effects. Often, however, these effects exhibit miraculous cancellations, and the string theory or matrix model results end up agreeing with standard gauge theory. In particular, these string theory considerations explain and remove some apparent discrepancies between gauge theories and matrix models in the literature.\n\nIntroduction\n\nLarge N topological string duality [1] embedded in superstrings [2,3] has led to a new perspective on N = 1 supersymmetric gauge theories: that the exact effective superpotential can be efficiently computed by including glueball fields. For example, in a theory with gauge group G, with tree-level superpotential leading to a breaking pattern\nG(N ) → K i=1 G i (N i ),(1.1)\nthe dynamics is efficiently encoded in a superpotential W eff (S 1 , . . . S K ; g j , Λ) (g j are the parameters in W tree and Λ is the dynamical scale). Further, string theory implies [3] W eff (S i ; g j ,\nΛ) = K i=1 h i ∂F (S i ) ∂S i − 2πiτ i S i ,(1.2)\nwith h i and τ i the fluxes through A i and B i three-cycles in the geometry, as will be reviewed in sect. 3. The prepotential F (S i ) in (1.2) is computable in terms of geometric period integrals, which yields [3] ∂F (S i )\n∂S i = S i log Λ 3 i S i + 1 + ∂ ∂S i i 1 ,...,i K ≥0 c i 1 ...i K S i 1 1 · · · S i k K ,(1.3)\nwith coefficients c i 1 ...i K depending on the g j (but not on the gauge theory scale Λ). In [4] it was shown how planar diagrams of an associated matrix model can also be used to compute (1.2) and (1.3). Based on the stringy examples, this was generalized in [5] to a more general principle to gain non-perturbative information about the strong coupling dynamics of gauge theories, by extremizing the perturbatively computed glueball superpotential. There are two aspects to the above statements: first that the glueball fields S i are the 'right' variables to describe the IR physics, and second that perturbative gauge theory techniques suffice to compute the glueball superpotential. The latter statement has now been proven in two different approaches for low powers of the glueball fields S i in (1.3) [6,7]. For powers of the glueball fields S i larger than the dual Coxeter number of the group, an ambiguity sets in for the glueball computation of the coefficients c i 1 ...i K in both of these approaches. The matrix model provides a natural prescription for how to resolve this ambiguity, essentially by continuing from large N i . It was argued in [8,9] that the string geometry / matrix model result (since the string geometry and matrix model results are identical, we refer to them synonymously) has the following meaning: it computes the F -terms for different supersymmetric gauge theories, which can be expressed in terms of G(N + k|k) supergroups. The W eff (S i ) is independent of k, and the above ambiguity can be eliminated by taking k, and hence the dual Coxeter number, arbitrarily large. The G(N ) theory of interest is obtained from the G(N + k|k) theory by Higgsing; but there can be residual instanton contributions to W eff [9], which can lead to apparent discrepancies between the matrix model and gauge theory results. We will somewhat clarify here when such residual instanton effects do, or do not, lead to discrepancies with standard gauge theory results.\n\nThere is another, more non-trivial assumption in [5]: the statement that the glueball fields S i are the 'right' variables in the IR. This assumption was motivated from the string dualities [1][2][3], where geometric transition provide the explanation of why the glueball fields are the natural IR variables: heuristically, S i corresponds to confinement. However, this is not quite correct: it also applies to abelian theories, as had been noted in [3]. So the deep explanation of why we should choose certain dynamical S variables remains mysterious.\n\nIn this paper, we will uncover the precise prescription for the correct choice of IR variables. This will be done from the string theory perspective, by arguing in which cases there is a geometric transition in string theory. For the general breaking pattern (1.1), our prescription for treating the glueball field S i , corresponding to the factor G i in (1.1), is as follows: If h(G i ) > 0 we include S i and extremize W eff (S i ) with respect to it. On the other hand, if h(G i ) ≤ 0 we do not include or extremize S i , instead we just set S i → 0. Here we define the generalized dual-Coxeter numbers 1 h(U (N )) = N, h(Sp(N )) = N + 1, h(SO(N )) = N − 2, (1.4) which are generalized in that (1.4) applies for all N ≥ 0. In particular h(U (1)) = h(Sp(0)) = 1, so when some G i factor in (1.1) is U (1) or Sp(0), our prescription is to include the corresponding S i and extremize with respect to it. On the other hand, h(U (0)) = 0 and h(SO(2)) = 0, so when some G i factor in (1.1) is U (0) or SO(2), our prescription is to just set the corresponding S i = 0 from the outset. (Note that U (1) and SO(2) are treated differently here.) This investigation was motivated by trying to understand the discrepancies found in [10] for Sp(N ) theory with antisymmetric tensor matter, where the superpotentials from the matrix model and gauge theory were found to differ at order h in perturbation theory and beyond. The analysis considered the trivial breaking pattern Sp(N ) → Sp(N ) and a single glueball was introduced corresponding to the single unbroken gauge group factor. In [9], various gauge theories including this example were studied, and an explanation for the discrepancies was proposed in terms of the conjecture, mentioned above, that the string theory / matrix model actually computes the superpotential of the large k G(N + k|k) supergroup theories, rather than the ordinary G(N ) theory. In this context, the trivial breaking pattern considered in [10] should be understood as Sp(N ) → Sp(N ) × Sp(0), which is completed to Sp(N + k|k) → Sp(N + k 1 |k 1 ) × Sp(k 2 |k 2 ). In particular, Sp(0) factors, while trivial in standard gauge theory, are non-trivial in the string theory geometry / matrix model context: there can be a residual instanton contribution to the superpotential when one Higgses Sp(k 2 |k 2 ) down to Sp(0|0) = Sp(0), as explicitly seen in [9] for the case of breaking Sp(0) → Sp(0) with quadratic W tree 2 . Related aspects of \"Sp(0)\" being non-trivial in the string / matrix model context were subsequently discussed in [11,12,13]. However, it turns out that one also needs to modify the matrix model side of the computation to take into account the Sp(0) factors. This was found by Cachazo [11], who showed that the loop equations determining T (z) ≡ Tr( 1 z−Φ ) and R(z) ≡ − 1 32π 2 Tr( W α W α z−Φ ) for the Sp(N ) theory with antisymmetric tensor matter [14,15] could be related to those of a U (2N + 2K) gauge theory with adjoint matter, with Sp(N ) → Sp(N ) × Sp(0) K−1 mapped to U (2N + 2K) → U (2N + 2) × U (2) K−1 . It was thus shown in [11] that vanishing period of T (z)dz through a given cut, corresponding to an Sp(0) factor, does not imply that the cut closes up on shell (aspects of the periods in this theory were also discussed in [16]). This fits with our above prescription that the Sp(0) glueballs should be included and extremized in the string theory / matrix model picture, as would be done for U (2), rather than set to zero, as was originally done in [10]. We stress that we are not yet even discussing whether or not the string theory / matrix model result agrees with standard gauge theory. Irrespective of any comparison with standard gauge theory, the prescription to obtain the actual string theory / matrix model result is as described above (1.4). Having obtained that result, we can now discuss comparisons with standard gauge theory results. As seen in [11], by solving the U (2N + 2K) loop equations for the present case, this corrected matrix model result now agrees perfectly with standard gauge theory! This will be discussed further here, with all glueball fields S i included.\n\nThis agreement, between the matrix model result and standard gauge theory, is in a sense surprising for this particular theory, in light of the Sp(k|k) description of [9] for the unbroken Sp(0) factors, with the resulting residual instanton contributions to the 2 It was suggested in the original version of [9] that such Sp(0) residual instanton contributions could also play a role for the case of cubic and higher order W tree (where they had not yet been fully computed) and could explain the apparent matrix model vs. standard gauge theory discrepancies found in [10]. As we will discuss, we now know that this last speculation was not correct. The corrected proposal of [9] is still that the matrix model computes the superpotential of the G(N + k|k) theory, but where the matrix model side of the computation should be corrected, as we discuss in this paper, to include glueball fields for the Sp(0) factors. superpotential. As we will explain later in this paper, the agreement here between matrix models and standard gauge theory is thanks to a remarkable cancellation of the residual instanton effect terms, which could have spoiled the agreement. The cancellation occurs upon summing over the i in (1.2) from i = 1 . . . K.\n\nThere are similar remarkable cancellations of the \"residual instanton contributions\" to the superpotential in many other examples, which we will also discuss. In fact, in all cases that we know of, the only cases where the residual instantons do not cancel is when the gauge theory clearly has some ambiguity, requiring a choice of how to define the theory in the UV; string theory / matrix model gives a particular such choice. Examples of such cases is when the LHS of (1.1) is itself U (1) or Sp(0) super Yang-Mills, as discussed in [9]. Other examples where the residual instanton contributions do not cancel, is when the superpotential is of high enough order such that not all operators appearing in it are independent, e.g. terms like Tr Φ n , for a U (N ) adjoint Φ, when n > N . In standard gauge theory, there are then potential ambiguities involved in reducing such composite operators to the independent operators, since classical operator identities can receive quantum corrections. The residual instanton contributions, which do not cancel generally in these cases, imply specific quantum relations for these operators, corresponding to the specific UV completion. See [17,18] for related issues.\n\nThe organization of this paper is as follows: In section 2 we summarize the gauge theories under consideration. In section 3 we review the type IIB string theory construction of these gauge theories. We also discuss maps of the exact superpotentials of Sp and SO theories to those of U theories, generalizing observations of [19,20,21,11,13]. In section 4 we explain, from the string theory perspective in which cases we have a geometric transition. In section 5 we consider examples, where the glueball fields S i of all group factors are correctly accounted for on the matrix model side. The results thereby obtained via matrix models are found to agree with those of standard gauge theory. In many of these examples, this agreement relies on a remarkable interplay of different residual instanton contributions, which sometimes fully cancel. Residual instantons are discussed further in sect. 6, with examples illustrating cases where they do, or do not, cancel. In appendix A, a proof of a general relation between the S 2 and RP 2 contributions to the matrix model free energy is given, and also the matrix model computation of superpotential is presented. In appendix B the gauge theory computation of the superpotential is discussed.\n\n\nThe gauge theory examples\n\nThe specific examples of N = 1 supersymmetric gauge theories which we consider, with breaking patterns as in (1.1), are as follows:\nU (N ) with adjoint Φ U (N ) → K i=1 U (N i ), SO(N ) with adjoint Φ : SO(N ) → SO(N 0 ) × K i=1 U (N i ), Sp(N ) with adjoint Φ : Sp(N ) → Sp(N 0 ) × K i=1 U (N i ), SO(N ) with symmetric S : SO(N ) → K i=1 SO(N i ), Sp(N ) with antisymmetric A : Sp(N ) → K i=1 Sp(N i ), U (N ) with Φ + S + S : U (N ) → SO(N 0 ) × K i=1 U (N i ), U (N ) with Φ + A + A : U (N ) → Sp(N 0 ) × K i=1 U (N i ).\n(2.1) For U (N ) with adjoint Φ, the tree-level superpotential is taken to be\nW tree = Tr[W (Φ)], W (x) = K+1 j=1 g j j x j ,(2.2)\nwith K potential wells. In the classical vacua, with breaking pattern as in (2.1), Φ has N i eigenvalues equal to the root a i of .2), the factor of 1 2 is for convenience, because the eigenvalues of S or A appear in pairs, and the indices are contracted with δ a b for SO(N ) or J a b for Sp(N ). For SO(N ) and Sp(N ) with adjoint matter, the tree-level superpotential is\nW ′ (x) = K+1 j=1 g j x j−1 ≡ g K+1 K i=1 (x − a i ), (2.3) with K i=1 N i = N .W tree = 1 2 Tr[W (Φ)], W (x) = K+1 j=1 g 2j 2j x 2j ,(2.5)\nsince all Casimirs of the adjoint Φ are even, and the 1 2 is again for convenience because the eigenvalues appear in pairs. Φ's eigenvalues sit at the zeros of\nW ′ (x) = K+1 j=1 g 2j x 2j−1 ≡ g 2K+2 x K i=1 (x 2 − a 2 i ). (2.6)\nThe breaking pattern in (2.1) has N 0 eigenvalues of Φ equal to zero, and N i pairs at ±a i , (2)). The next to last example in (2.1) is the N = 2 U (N ) theory with a matter hypermultiplet in the two-index symmetric tensor representation, breaking N = 2 to N = 1 by a superpotential as in (2.2):\nso N = N 0 + K i=1 2N i for SO(N ) → SO(N 0 ) × K i=1 U (N i ), and N = N 0 + K i=1 N i for Sp(N ) → Sp(N 0 ) × K i=1 U (N i ) (with the convention Sp(1) ∼ = SUW = K+1 j=1 g j j TrΦ j + √ 2Tr SΦS. (2.7)\nIn addition to the possibility of Φ's eigenvalues sitting in any of the K critical points W ′ (x) analogous to (2.3), there is a vacuum where N 0 eigenvalues sits at φ = 0, with S S = 0, breaking U (N 0 ) → SO(N 0 ). The last example in (2.1) is the similar theory where the N = 2 hypermultiplet is instead in the antisymmetric tensor representation A, rather than the symmetric tensor S. These last two classes of examples were considered in [22,23,24]. In all of these theories, the low energy superpotential is of the general form\nW low (g j , Λ) = W cl (g j ) + W gc (Λ i ) + W H (g j , Λ). (2.8) W cl (g j )\nis the classical contribution (evaluating W tree in the appropriate minima). W gc (Λ i ) is the gaugino condensation contribution in the unbroken gauge groups of (1.1),\nW gc (Λ j ) = K i=1 h i e 2πin i /h i Λ 3 i ,(2.9)\nwhere h i = C 2 (G i ) is the dual Coxeter number of the group G i in (1.1), with the phase factors associated with the Z 2h i → Z 2 chiral symmetry breaking of the low-energy G i gaugino condensation. The scales Λ i are related to Λ by threshold matching for the fields which got a mass from W tree and the breaking (1.1); some examples, with breaking patterns as in (2.1), are as follows. For U (N ) with adjoint Φ:\nΛ 3N i i = Λ 2N W ′′ (a i ) N i j =i m −2N j W ij = g N i K+1 Λ 2N j =i (a j − a i ) N i −2N j . (2.10) For SO(N ) with adjoint, breaking SO(N ) → SO(N 0 ) × K i=1 U (N i ), Λ 3(N 0 −2) 0 = g N 0 −2 2K+2 Λ 2(N−2) K i=1 a 2(N 0 −2)−4N i i Λ 3N i i = 2 −N i g N i 2K+2 Λ 2(N−2) a −2(N 0 −2) i K j =i (a 2 i − a 2 j ) N i −2N j . (2.11) For Sp(N ) with adjoint Λ 3(N 0 +1) 0 = g N 0 +1 2K+2 Λ 2(N+1) K i=1 a 2(N 0 +1)−2N i i Λ 3N i i = 2 −N i g N i 2K+2 Λ 4(N+1) a −4(N 0 +1) i K j =i (a 2 i − a 2 j ) N i −2N j ;\n(2.12) the U (N i ) has index of embedding 2, which is why the U (N i ) one-instanton factor is related to the Sp(N ) two-instanton factor in (2.12). For SO(N ) with symmetric tensor,\nΛ 3(N i −2) i = g N i +2 K+1 Λ 2N−8 K j =i=1 (a i − a j ) N i +2−2N j . (2.13)\nFor Sp(N ) with antisymmetric A:\nΛ 3(N i +1) i = g N i −1 K+1 Λ 2N+4 K j =i (a i − a j ) N i −1−2N j .\n(2.14)\n\nFinally, the term W H (g j , Λ) in (2.8) are additional non-perturbative contributions, which can be regarded as coming from the massive, broken parts of the gauge group.\n\nIn the description with the glueballs S i integrated in, as in (1.2), the gaugino condensation contribution comes from the first term in (1.3):\nW gc (S i , Λ) = K i=1 h i S i log Λ 3 i S i + 1 ,(2.15)\nand W H (g i , Λ) comes from the last terms in (1.3), upon integrating out the S i . When the minima a i of the superpotential are widely separated, the contributions W H from these last terms are subleading as compared with W gc . As in (1.2), the full glueball superpotential (1.2) can be computed via the string theory geometric transition, in terms of certain period integrals [2,3], as will be reviewed in the next section, or via the matrix models. In that context, the term W gc (S i , Λ i ) comes from the integration measure (as is also natural in field theory, since it incorporates the U (1) R anomaly) and W H (S i , g j ) can be computed perturbatively [4,5]. The perturbative computation of W H (S i , g j ) can also be understood directly in the gauge theory [6], up to ambiguities in terms S n with n > h = C 2 (G). The string theory/ matrix model constructions yield a specific way of resolving these ambiguities, which correspond to a particular UV completion of the gauge theory [8,9]. As discussed in the introduction, our present interest will be in analyzing this circle of ideas when some of the gauge group factors in (1.1) are of low rank, or would naively appear to be trivial, e.g. U (1), U (0), SO(2), SO(0), and Sp(0).\n\n\nGeometric transition of U(N) and SO/Sp(N) theories\n\nIn this section we briefly review the type IIB geometric engineering of relevant U (N ) and SO/Sp(N ) theories, and their geometric transition.\n\n\nU (N ) with adjoint and W tree = Tr\nK+1 j=1 g i j Φ j\nThe Calabi-Yau geometry relevant to this theory [3] is the non-compact A 1 fibration\nW ′ (x) 2 + y 2 + u 2 + v 2 = 0. (3.1)\nThis fibration has K conifold singularities at the critical points of W (x), i.e. at W ′ (x) = 0. Near each of the singularities, the geometry (3.1) is the same as the usual conifold x ′2 + y 2 + u 2 + v 2 = 0, which is topologically a cone with base S 2 × S 3 . The singularities can be resolved by blowing up a 2-sphere S 2 = P 1 at each singularity. We can realize the U (N ) gauge theory with adjoint matter and superpotential (2.2) in type IIB superstring theory compactified on this resolved geometry, with N D5-branes partially wrapping the K P 1 's. The gauge theory degrees of freedom correspond to the open strings living on these D5-branes. The classical supersymmetric vacuum is obtained by distributing the N i D5-branes over the i-th 2-sphere P 1 i with i = 1, · · · , K. The corresponding breaking pattern of the gauge group is as in (2.1):\nU (N ) → K i=1 U (N i ).\nAt low energy, the gauge theory confines (when N i > 1), each U (N i ) factor developing nonzero vev of the glueball superfield S i . In string theory this is described by the geometric transition [1,2,3] in which the resolved conifold geometry with P 1 's wrapped by D5-branes is replaced by a deformed conifold geometry\nW ′ (x) 2 + f K−1 (x) + y 2 + u 2 + v 2 = 0, (3.2)\nwhere f K−1 (x) is a polynomial of degree K − 1 in x and parametrizes the deformation. After the geometric transition, each 2-sphere P 1 i wrapped by N i D5-branes is replaced by a 3-sphere A i with 3-form RR flux through it:\nA i H = N i ,(3.3)\nwhere H = H RR + τ H NS and τ = C (0) + ie −Φ is the complexified coupling constant of type IIB theory. We define the periods of the Calabi-Yau geometry (3.2) by\nS i ≡ 1 2πi A i Ω, Π i ≡ Λ b B i Ω, (3.4)\nwhere Ω is the holomorphic 3-form, B i is the noncompact 3-cycle dual to the 3-cycle A i , and Λ b is a cutoff needed to regulate the divergent B i integrals. The IR cutoff Λ b is to be identified with the UV cutoff of the 4d gauge theory. The set of variables S i measure the size of the blown up 3-spheres, and can be used to parametrize the deformation in place of the K coefficients of the polynomial f K−1 (x).\n\nThe dual theory after the geometric transition is described by a 4d, N = 1 U (1) K gauge theory, with K U (1) vector superfields V i and K chiral superfields S i . If not for the fluxes, this theory would be N = 2 supersymmetric U (1) K , with (V i , S i ) the N = 2 vector super-multiplets, S i being the Coulomb branch moduli. This low-energy U (1) K theory has a non-trivial prepotential F (S i ) and the dual periods Π i in (3.4) can be written as\nΠ i (S) = ∂F ∂S i . (3.5)\nWithout the fluxes, this prepotential can be understood as coming from integrating out D3 branes which wrap the A i cycles, and are charged under the low-energy U (1)'s [25]. The effect of the added fluxes is to break N = 2 supersymmetry to N = 1 by the added superpotential [26,27] \nW flux = H ∧ Ω = i A i H Λ b B i Ω − Λ b B i H A i Ω . (3.6)\nIn the present U (N ) case, (3.3) and (3.4) gives\nW flux = K i=1 (N i Π i − 2πiαS i ), (3.7) where Λ b B i H = α (3.8)\nis the 3-form NS flux through the 3-cycle B i and identified with the bare coupling constant of the gauge theory by\n2πiα = 8π 2 g 2 b = V. (3.9)\nwhere V is the complexified volume of the P 1 's. The N = 1 U (1) K vector multiplets V i remain massless, but the S i now have a superpotential, which fixes them to sit at discrete vacuum expectation values, where they are massive. The fields S i are identified with the glueballs on the gauge theory side. The superpotential W flux (S i ) is the full, exact, effective superpotential in (1.2). As can be verified by explicit calculations [2,3], the leading contribution to (3.7) is always of the form\nW flux ∼ K i=1 [N i S i (1 − ln(S i /Λ 3 i )) − 2πiαS i ], (3.10)\nwhere Λ i is related to the scale Λ b via precisely the relation (2.10). This leading term (3.10) is the gaugino condensation part of the superpotential, as in (1.3).\n\n\nSO and Sp theories\n\nThe string theory construction of SO/Sp(N ) theories can be obtained from the above U (N ) construction, by orientifolding the geometry before and after the geometric transition by a certain Z 2 action. The geometric construction of SO/Sp(N ) theory with adjoint was discussed in [28,29,20,30], and in that case the invariance of the geometry (3.2) under the Z 2 action requires that the polynomial W (x) be even. The geometric construction of SO/Sp(N ) theory with symmetric/antisymmetric tensor was studied in [31,32,13].\n\nIn the classical vacuum of the \"parent\" U (2N ) theory, the gauge group is broken into a product of U (N i ) groups. When a U (N i ) factor is identified with another U (N i ) by the Z 2 orientifold action, they lead to a single U (N i ) factor. When a U (N i ) factor is mapped to itself by the Z 2 orientifold action, it becomes an SO(N i ) or Sp( 1 2 N i ), depending on the charge of the orientifold hyperplane. As a result, the classical vacuum of the \"daughter\" SO/Sp(N ) theory has gauge group broken as in (2.1), depending on whether the theory is SO or Sp with adjoint, or SO with symmetric tensor, or Sp with antisymmetric tensor:\nSO/Sp(N ) → i G i (N i ), G i = U, SO, or Sp. (3.11)\nThe 3-form RR fluxes from orientifold hyperplanes makes an additional contribution to the superpotential (3.6), and the flux superpotential can be written as\nW flux = i [N i Π i (S i ) − 2πiη i αS i ], (3.12) wherê N i = N i G i = U (N i ), 1 2 N i ∓ 1 G i = SO(N i )/Sp( 1 2 N i ), η i = 1 G i = U , 1/2 G i = SO/Sp. (3.13) N i is the net 3-form RR flux through the A i cycle. For U (N i ) and Sp(N i ),N i in (3.13)\nis the dual Coxeter number (1.4), while for SO(N i ) it is half 3 the dual Coxeter number (1.4). The 1/2 in (3.13) is because the integration over the A i cycles should be halved due to the Z 2 identification.\n\n\nRelations between SO/Sp theories and U (N ) theories\n\nThe result (3.12), with (3.13), gives the exact superpotential of the SO/Sp theories in terms of the same periods S i and Π(S i ) as an auxiliary U theory. This was first noted in [11] at the level of the Konishi anomaly equation as a map between the resolvents of Sp theory with antisymmetric matter and U theory with adjoint matter. In [13], it was generalized to the map between the resolvents of SO/Sp theories with two-index tensor matter and U theory with adjoint matter, and string theory interpretation was discussed. In this subsection we will derive this map from the string theory perspective using the flux superpotential (3.12). Furthermore, we will clarify the relation of the superpotential and the scale of the SO/Sp theories to those of the U theory. The map between resolvents can be derived from these results. For Sp(N ) theory with an antisymmetric tensor, the scale relation was obtained in a different way in [11].\n\nAs a first example, consider SO(N ) with an adjoint, with the breaking pattern as in (2.1). The geometric transition result (3.12) and (3.13) implies that the exact superpotential is the same as for the U (N − 2) theory with adjoint, with breaking pattern map\nSO(N ) → SO(N 0 ) × K i=1 U (N i ) ⇐⇒ U (N − 2) → U (N 0 − 2) × K i=1 U (N i ) 2 . (3.\n14)\n\nThe map between the superpotential is\nW SO(N) exact = 1 2 W U(N−2) exact . (3.15)\nThe SO(N ) scale matching relation (2.11) is compatible with the map (3.14), since (2.10) for the theory (3.14) reproduces (2.11). Likewise, Sp(N ) with adjoint has the same exact superpotential as for the U (2N + 2) theory with adjoint, with\nSp(N ) → Sp(N 0 ) × K i=1 U (N i ) ⇐⇒ U (2N + 2) → U (2N 0 + 2) × K i=1 U (N i ) 2 , W Sp(N) exact = 1 2 W U(2N+2) exact .\n(3.16) The Sp(N ) scale matching relations (2.12) follow from the U (N ) matching relations (2.10) with the replacement (3.16), with the understanding that the U (2N + 2) and U (2N 0 + 2) one-instanton factors correspond to the Sp(N ) and Sp(N 0 ) two-instanton factors; this is related to the index of the imbedding mentioned after (2.12), and is accounted for by dividing the U (2N + 2) superpotential by two, as above.\n\nNext consider Sp(N ) with antisymmetric tensor A and breaking pattern as in (2.1). The geometric transition result (3.12) and (3.13) implies that the exact superpotential is the same as for the U (2N + 2K) theory with adjoint and breaking pattern\nSp(N ) → K i=1 Sp(N i ) ⇐⇒ U (2N + 2K) → K i=1 U (2N i + 2). (3.17)\nIn the present case, comparing the matching relations (2.14) for Sp(N ) with symmetric tensor with the matching relations (2.10) for U (2N + 2K) with adjoint, for the mapping as in (3.17) requires that the scales of the original Sp(N ) on the LHS of (3.17) and the U (2N + 2K) on the RHS of (3.17) be related as\nΛ 2(N+K) U(2N+2K) = g −2 K+1 Λ 2N+4 Sp(N) . (3.18)\nThen the Λ i of the unbroken groups on both sides of (3.17) coincide, with the understanding that the U (2N i + 2) one-instanton factors correspond to the Sp(N i ) two-instanton factors, as above. The map between the superpotential is\nW Sp(N) exact = 1 2 [W U(2N+2K) exact − ∆W cl ], ∆W cl = 2 K i=1 W (a i ), i.e. writing W exact = W cl + W quant , W Sp(N) quant = 1 2 W U(2N+2K) quant . (3.19)\nNote that, in order for the superpotentials on the two sides of (3.17) to fully coincide, one must compensate the classical mismatch ∆W cl , since each well is occupied by two additional eigenvalues in the theory on the RHS of (3.17). In the string theory geometric transition realization, this constant shift, which is independent of N , Λ, and the glueball fields S i , is most naturally interpreted as an additive shift of the superpotential on the Sp side, which can be regarded as coming from the orientifold planes both before and after the transitions. The classical shift of ∆W cl leads to slightly different operator expectation values (as computed via W eff (g p , Λ) as the generating function) between the Sp and U theory, as was seen in the example of [11]. Also, writing the map as in (3.17), we want the vacuum with confinement index 2 [11]. We could equivalently replace the RHS of (3.17)\nwith U (N + K) → K i=1 U (N i + 1)\n, in which case we would not have to divide by 2 in (3.19).\n\nLikewise, SO(N ) with symmetric tensor S has exact superpotential related to that of a U (N − 2K) theory with adjoint as\nSO(N ) → K i=1 SO(N i ) ⇐⇒ U (N − 2K) → K i=1 U (N i − 2), W SO(N) exact = 1 2 [W U(N−2K) exact + ∆W cl ], ∆W cl = 2 K i=1 W (a i ).\n(3.20)\n\nComparing the matching relations (2.13) for SO(N ) with symmetric tensor with those of (2.10) for U (N − 2K) with adjoint requires that the scales of the original SO(N ) on the LHS of (3.20) and those of the U (N − 2K) theory on the RHS be related as\nΛ 2(N−2K) U(N−2K) = g 4 K+1 Λ 2N−8 SO(N) . (3.21)\nThen the Λ i of the unbroken groups on both sides of (3.20) coincide. Again, in order for the superpotentials on the two sides of (3.20) to fully coincide, one must correct for the classical mismatch ∆W cl coming from the fact that the U (N − 2K) theory has two fewer eigenvalues in each well. In appendix A we will discuss these relations from the matrix model viewpoint. In this context, the relation relevant for (3.14) and (3.16) was conjectured in [19,20] based on explicit diagrammatic calculations, and it was proven for the case of unbroken gauge group N 0 = N in [21]. This will be generalized in Appendix A to all breaking patterns. Likewise, the matrix model relation relevant for (3.17) and (3.20) will be proven in the appendix; this is a generalization of the connection found in [11] for the theories in (3.17) and (3.20).\n\n\nString theory prescription for low rank\n\nThe discussion of the previous section applies for all N i ≥ 0. We now discuss under which circumstances one expects a transition in string theory, where S 3 i 's grows, and therefore an effective glueball field S i should be included in the superpotential. Whether or not there is a geometric transition in string theory is a local question, so each S 3 i can be studied independently. Near any S 3 i the local physics is just a conifold singularity, so we only need to consider the case of a conifold singularity.\n\n\nPhysics near a conifold singularity\n\nAs we saw, U (N ), SO/Sp(N ) gauge theory can be realized in type IIB theory as the open string theory living on the D5-branes partially wrapped on the exceptional P 1 of a resolved conifold geometry. There is a P 1 associated to each critical point of the polynomial W (x). By the geometric transition duality [2,3], this gauge theory is dual to the closed string theory in the deformed conifold geometry where the P 1 's have been blown down and S 3 's are blown up instead.\n\nLet us focus on one P 1 with N ≥ 0 D5-branes wrapping it. This corresponds to focusing on one critical point on the gauge theory side. We allow N = 0, which corresponds to an unoccupied critical point. In the neighborhood, the geometry after the geometric transition is approximately a deformed conifold x 2 + y 2 + z 2 + w 2 = µ with a blown up S 3 . The low energy degrees of freedom in the four-dimensional theory are the N = 1 U (1) photon vector superfield V and the N = 1 chiral superfield S. The bosonic component of S is proportional to µ and measures the size of the S 3 .\n\nFirst, consider the case without fluxes. Then the closed string theory has N = 2 and there is one N = 2 U (1) vector multiplet (V, S). It is known that as the size of the S 3 goes to zero there appears an extra massless degree of freedom [25], which corresponds to the D3-brane wrapping the S 3 . The mass of the wrapped BPS D3-brane is proportional to the area, S, of the S 3 , so the mass becomes zero as the S 3 shrinks to zero, i.e. as S → 0. This extra degree of freedom is described as an N = 2 hypermultiplet charged under the U (1) (of V ). Let us write this hypermultiplet in N = 1 language as (Q,Q), where Q andQ are both N = 1 chiral superfields with opposite U (1) charges. The N = 2 supersymmetry requires the superpotential\nW Q = √ 2 QQS, (4.1)\nwhich indeed incorporates the above situation that the Q,Q become massless as\nS → 0. The D-flatness is |Q| 2 − |Q| 2 = 0, (4.2)\nand the F -flatness is\nQS =QS = QQ = 0. (4.3)\nThe only solution to these is\nQ =Q = 0, S : any, (4.4)\nwhich just means that S ∼ µ is a modulus. Now let us come back to the case with the fluxes. As reviewed in the last section, the fluxes give rise to a superpotential (3.6) which breaks N = 2 to N = 1 [26,27]. As in (3.12), the local flux superpotential contribution is\nW flux (S) ≃N S[1 − ln(S/Λ 3 )] − 2πiηαS, (4.5)\nwhere we just keep the leading order term in (3.12), as in (3.10), witĥ\nN = N U (N ), N/2 ∓ 1 SO(N )/Sp( 1 2 N ), η = 1 U (N ), 1/2 SO(N )/Sp( 1 2 N ). (4.6)\nThe scale Λ is written in terms of the bare coupling Λ b and the coupling constants in the problem, as before, and 2πiα is related to the bare gauge coupling by (3.9). In the following, we discuss the cases withN = 0,N > 0 andN < 0 in order.\n\n\n•N=0 case\n\nIn this case, the total superpotential is simply the sum of (4.1) and (4.5):\nW = √ 2 QQS − 2πiηαS. (4.7)\nThe only solution to the equation of motion is\n|Q| 2 = |Q| 2 , QQ = 2πiηα √ 2 , S = 0. (4.8)\nThis is consistent with the fact that α is proportional to the volume of the P 1 , and the D3-brane condensation QQ corresponds to the size of the P 1 . Furthermore, since S = 0, the superpotential vanishes: W = 0. Therefore, forN = 0, i.e. for U (0) and SO (2), geometric transition does not take place and we should set the corresponding glueball field S → 0 from the beginning.\n\n•N>0 case In this case, there is a net RR flux through the A-cycle: A H =N . This means that the D3-brane hypermultiplet (Q,Q) is infinitely massive, because the RR flux will induceN units of fundamental charge on the D3-brane. Since the D3-brane is wrapping a compact space S 3 , the fundamental charge on it should be canceled byN fundamental strings attached to it. Those fundamental strings extend to infinity and thus cost infinite energy 4 . Therefore, we can forget about Q,Q in this case, and the full superpotential is given just by the flux contribution (4.5). The equation of motion gives\nSN ≃ Λ 3N e −2πiηα ,(4.9)\nwhich corresponds to the confining vacua of the gauge theory. Note that this case includes U (1) and Sp(0); these theories have a dual confining description. This may sound a little paradoxical, but is related to the fact that the string theory computes not for the standard G(N ) gauge theory but the associated G(N + k|k) higher rank gauge theory, which is confining and differs from standard U (1) and Sp(0) due to residual instanton effects [9].\n\n•N<0 case In this case, the same argument as theN > 0 case tells us that we should not include the D3-brane fields Q,Q. Hence the superpotential is just the flux part (4.5), which again leads to\nSN ≃ Λ 3N e −2πiηα .\n(4.10)\n\nHowever, now (4.10) is physically unacceptable, since S diverges in the weak coupling limit where the bare volume of\nP 1 becomes V = 2πiα → ∞ (g b → 0) -i.e.\ntaking P 1 large would lead to S 3 also being large, which does not make sense geometrically. The resolution is that S cannot be a good variable: the S 3 does not actually blow up, and S should be set to zero, S → 0, also for this case. Though S is set to zero, the non-zero flux can lead to a non-zero superpotential contribution W flux =N Π(S → 0).\n\nNote that the above result concerning the sign ofN does not mean the gauge theory prefers D5-branes to anti-D5-branes; it just means that one should choose the sign of the NS flux (i.e. the sign of 2πiα) appropriately. If one wraps the P 1 with anti-D5-branes, one should flip the sign of the NS flux in order to have a blown up S 3 (which can be viewed as a generalization of Seiberg duality to N f = 0).\n\n\nGeneral prescription\n\nAlthough we focused on the physics around just one P 1 in the above, the result is applicable to general cases where we have multiple P 1 wrapped with D5-branes, because the geometry near each P 1 is identical to the conifold geometry considered above. Therefore, if we replaceN withN i , all of the above conclusions carry over.\n\nOnce we have understood the physics, we can forget about the D3-brane hypermultiplet (Q,Q) and state the result as a general prescription for how string theory treats U (0), SO(0), SO (2), and Sp(0), U (1) groups in the geometric dual description:\n\n• U(0), SO(0), SO (2):\n\nThere are no glueball variables associated to these gauge groups, so we should take the corresponding S → 0.\n\n• All other groups, including Sp(0), and U (1):\n\nWe should consider and extremize the corresponding glueball field S.\n\nThis prescription should also be applied when using the matrix model [4,36,5] to compute the glueball superpotentials.\n\n\nExamples\n\nLet us scan over all of the examples of (2.1), considering the vacuum where the gauge group is unbroken, and ask when glueball fields S i for the apparently trivial groups in (2.1) should be set to zero, or included and extremized. For the first three cases in (2.1), U (N ), SO(N ), and Sp(N ) with adjoint, the breaking (2.1) is G → G × U (0) K−1 , and the glueball fields S i for the U (0) factors are to be set to zero. This justifies the analysis of these theories in the unbroken vacua in [37,21]. The next case is SO(N ) with a symmetric tensor S, where the vacuum with unbroken gauge group is to be understood as SO(N ) → SO(N ) × SO(0) K−1 , and again the glueball fields S i for the SO(0) factors are set to zero. This eliminates the Veneziano-Yankielowicz part of the superpotential for SO(0), but the −1 unit of flux associated with each SO(0) does contribute to flux termsN i Π i = −Π i in (3.12), even though this does not contain SO(0) glueballs any more.\n\nThe next case is Sp(N ) with an antisymmetric tensor, where the vacuum with unbroken gauge group is to be understood as Sp(N ) → Sp(N ) × Sp(0) K−1 . Unlike the above cases, here we must keep and extremize the S i for the Sp(0) factors, as will be further discussed shortly.\n\nFor the next to last example in (2.1), U (N ) with Φ+S + S, the vacuum with unbroken gauge group is to be understood as U (N ) → SO(0) × U (N ) × U (0) K−1 , and the glueball fields S i for SO(0) and U (0) are to be set to zero. Finally, for the last example in (2.1), U (N ) with Φ + A + A, the vacuum with unbroken gauge group is to be understood as\nU (N ) → Sp(0) × U (N ) × U (0) K−1 .\nThough the string engineering of these examples differs somewhat from those discussed in sect. 4 (it was obtained in [30]), the general prescription of sect. 4 is expected to carry over in general: the glueball field S 0 for the Sp(0) factor should be included and extremized, rather than set to zero. On the other hand, the S i for the U (0) factors are set to zero. These latter two theories in (2.1) were considered in [24] and it was noted there that for the case with antisymmetric one expands on the matrix model side around a different vacuum than would be naively expected; this indeed corresponds to keeping and extremizing the glueball field S 0 for the Sp(0) factor, as we have discussed.\n\nWe now illustrate some other breaking patterns in the examples of (2.1), from the matrix model perspective, for the case of K = 2. We also compare with standard gauge theory results and generally find agreement, even in cases where there was room for disagreement because of the possibility of residual instanton effects along the lines of [9]. As will be discussed in more detail in the following section, the agreement is thanks to a remarkable interplay of different residual instanton contributions.\n\n\nSO/Sp(N ) theory with adjoint\n\nConsider N = 2 SO(2N )/Sp(N ) theory broken to N = 1 by a tree level superpotential for the adjoint chiral superfield Φ:\nW tree = 1 2 Tr[W (Φ)], W (x) = m 2 x 2 + g 4 x 4 . (5.1)\nIn the SO case, we can skew-diagonalize Φ as\nΦ ∼ diag[λ 1 , · · · , λ N ] ⊗ iσ 2 . (5.2)\nThe superpotential (5.1) has critical points at λ = 0 and λ = ± m/g. The classical supersymmetric vacuum of the theory is given by distributing 2N 0 of the 2N \"eigenvalues\" λ i at the critical point λ = 0 and N 1 \"eigenvalue\" pairs at λ = ± m/g, with N 0 +N 1 = N . In this vacuum, the gauge group breaks as\nSO(2N ) → SO(2N 0 ) × U (N 1 ) or Sp(N ) → Sp(N 0 ) × U (N 1 ).\nIn the matrix model prescription, the effective superpotential in these vacua is calculated by matrix model as\nW DV (S 0 , N 0 ; S 1 , N 1 ) = (N 0 ∓ 1) S 0 [1 − ln(S 0 /Λ 3 0 )] + N 1 S 1 [1 − ln(S 1 /Λ 3 1 )] + W pert , W pert = 2N 0 ∂F S 2 ∂S 0 + N 1 ∂F S 2 ∂S 1 + 4F RP 2 . (5.3)\nwhere F S 2 and F RP 2 are the S 2 and RP 2 contributions, respectively, to the free energy of the associated SO/Sp(N ) matrix model, as defined in Appendix A. The scales Λ 0 , Λ 1 in (5.3) are the energy scales of the low energy SO(2N 0 )/Sp(N 0 ), U (N 1 ) theories with the Φ field integrated out, respectively. They are related to the high energy scale Λ by the matching conditions as in (2.11) and (2.12), which yields\n(Λ 0 ) 3(N 0 ∓1) = m N 0 −N 1 ∓1 g N 1 Λ 2(N∓1) , (Λ 1 ) 3N 1 = 2 −N 1 m −2N 0 ±2 g 2N 0 +N 1 ∓2 Λ 4(N∓1) . (5.4)\nThe matrix model free energy is computed in Appendix A, and the result is\nW pert = (N 0 ∓ 1) 3 2 S 2 0 − 8S 0 S 1 + 2S 2 1 α + − 9 2 S 3 0 + 42S 2 0 S 1 − 36S 0 S 2 1 + 4S 3 1 α 2 + 45 2 S 4 0 − 932 3 S 3 0 S 1 + 523S 2 0 S 2 1 − 608 3 S 0 S 3 1 + 40 3 S 4 1 α 3 + N 1 −2S 2 0 + 2S 0 S 1 α + 7S 3 0 − 18S 2 0 S 1 + 6S 0 S 2 1 α 2 + − 233 6 S 4 0 + 524 3 S 3 0 S 1 − 152S 2 0 S 2 1 + 80 3 S 0 S 3 1 α 3 + O(α 4 ), (5.5) where α ≡ g/m 2 .\nThe result (5.5) agrees with the one obtained in [38], where the glueball superpotential was calculated by evaluating the periods (3.6). The full result, (5.3) and (5.4), has the expected general form (3.12):\nW eff = (N 0 ∓ 1)Π 0 (S 0 , S 1 ) + N 1 Π 1 (S 0 , S 1 ) − 2πiα( 1 2 S 0 + S 1 ). (5.6)\nThe general prescription in section 3 reads in the present case as follows:\n\n• SO(2N)/Sp(N) → SO(2N)/Sp(N)×U(0) (unbroken SO/Sp): Set N 1 = 0, S 1 = 0. Then the superpotential is\nW eff (S 0 , N 0 ) = (N 0 ∓ 1) S 0 [1 − ln(S 0 /Λ 3 0 )] + 2 (N 0 ∓ 1) ∂F S 2 ∂S 0 S 1 =0 . (5.7)\nThis superpotential coincides with that of U (2N ∓2) with adjoint and breaking pattern U (2N ± 2) → U (2N ∓ 2) × U (0) × U (0), as expected from the map (3.14) or (3.16). As shown in [21] for SO(2N ), this matrix model result agrees with that of standard gauge theory, via using the corresponding Seiberg-Witten curve.\n\n• SO(2N) → SO(0)×U(N): Set N 0 = 0 and take S 0 → 0, which eliminates the Veneziano-Yankielowicz part for the SO(0). Then the superpotential is\nW eff (S 1 ) = N S 1 [1 − ln(S 1 /Λ 3 1 )] + 4 F RP 2 | S 0 =0 . = −Π 0 (S 0 , S 1 )| S 0 =0 + N Π 1 (S 0 , S 1 )| S 0 =0 − 2πiαS 1 . (5.8) Note that ∂F S 2 ∂S 1 S 0 =0\n= 0 because F S 2 does not contain terms with S 1 only (all terms are of the form S n 0 S m 1 with n > 0). And though the Veneziano-Yankielowicz part of the superpotential for SO(0) is eliminated via S 0 → 0, the −1 units of flux associated with SO(0) does make a contribution in (5.8), with the non-zero terms in −Π 0 (S 0 , S 1 )| S 0 =0 in the second line of (5.8) coming from the term 4F RP 2 | S 0 =0 .\n\n• SO(2N) → SO(2)×U(N−1):\n\nSet N 0 = 1, S 0 = 0 and remove the Veneziano-Yankielowicz part for the SO(2). Then the superpotential is\nW DV (S 1 ) = (N − 1)S 1 [1 − ln(S 1 /Λ 3 1 )],(5.9)\nwhere we used\n∂F S 2 ∂S 1 S 0 =0 = 0 again. Integrating out S 1 gives W low = (N − 1)Λ 3 1 = 1 2 (N − 1)gΛ 4 (5.10) • Sp(N) → Sp(0)×U(N):\nSet N 0 = 0 in the equation and keep both S 0 and S 1 . Then the superpotential is\nW DV (S 0 , N 0 , S 1 , N 1 ) = S 0 [1 − ln(S 0 /Λ 3 0 )] + N S 1 [1 − ln(S 1 /Λ 3 1 )] + W pert , W pert = N 1 ∂F S 2 ∂S 1 + 4F RP 2 = N 1 ∂F S 2 ∂S 1 + 2 ∂F S 2 ∂S 0 . (5.11)\nFor various breaking patterns, we integrated out the glueball superfield(s) from the glueball superpotential (5.7)-(5.11), and calculated the low energy superpotential W low as a function of coupling constants m, g, and the scale Λ. Having obtained the actual matrix model results, we can compare to the superpotential as computed via standard gauge theory methods, such as via factorizing of the Seiberg-Witten curve. This method is reviewed in Appendix B, and the results are found to agree with the matrix model results completely. The resulting W low is shown in Table 1.\n\nThe SO(2N +1) theory with adjoint in the SO(2N +1) → U (1) N vacuum was studied diagrammatically in [39].\n\n\nSp(N ) theory with antisymmetric tensor\n\nConsider Sp(N ) theory with an antisymmetric tensor chiral superfield A = −A T . Take cubic tree level superpotential Table 1: The low energy superpotential calculated from the factorization of the Seiberg-Witten curve and from matrix model. In the above, the classical contribution has been subtracted: W low = −N 1 m 2 /4g + W ′ low .\nW tree = 1 2 Tr[W (Φ)], W (x) = m 2 x 2 + g 3 x 3 ,(5.£ µ ½ ¿ ½ ¾´ £ ½ Ñ ¾ µ ½ ¿ ¿ £ Ñ ¾ · ¡ ¡ ¡ ËÇ´ µ ËÇ´ µ ¢ Í´¼µ ¿Ñ£ ¾ · ¾ £ ËÇ´ µ ËÇ´ µ ¢ Í´½µ ¾´Ñ µ ½ ¾ £ ¿ · ¾ £ Ñ ´ Ñ µ ½ ¾ £ · ¡ ¡ ¡ ËÇ´ µ ËÇ´ µ ¢ Í´¾µ ¾ ¾ £ Ñ £ ½¾ Ñ · ¿¾ £ ½ Ñ · ¡ ¡ ¡ ËÇ´ µ ËÇ´¾µ ¢ Í´¿µ ¿ ¾ £ ËÇ´ µ ËÇ´¼µ ¢ Í´ µ ¾´Ñ µ ½ ¾ £ ¿ ¾ £ ¾Ñ · ½ ´ Ñ µ ½ ¾ £ · ¡ ¡ ¡ ËÔ´¾µ ËÔ´¾µ ¢ Í´¼µ ¾Ñ£ ¾ · ¿ £ ËÔ´¾µ ËÔ´¼µ ¢ Í´½µ £ ËÔ´ µ ËÔ´ µ ¢ Í´¼µ ¿Ñ£ ¾ · ¾ £ ËÔ´ µ ËÔ´¾µ ¢ Í´½µ ¾´Ñ µ ½ ¾ £ ¿ · ¾ £ Ñ ´ Ñ µ ½ ¾ £ · ¡ ¡ ¡ ËÔ´ µ ËÔ´¼µ ¢ Í´¾µ ¾ ¾ £ Ñ £ ½¾ Ñ · ¿¾ £ ½ Ñ · ¡ ¡ ¡\nwhere Φ = AJ, and J is the invariant antisymmetric tensor J = 1 N ⊗ iσ 2 . We do not require A to be traceless, i.e. Tr[Φ] = Tr[AJ] = 0. By a complexified Sp(N ) gauge rotation, Φ can be diagonalized as [40] Φ\n∼ = diag[λ 1 , · · · , λ N ] ⊗ 1 2 , λ i ∈ C. (5.13)\nThe superpotential (5.12) has critical points at λ = 0, −m/g. The classical supersymmetric vacuum of the theory is given by distributing N 1 and N 2 \"eigenvalues\" λ i at the critical point λ = 0 and λ = −m/g, respectively, with\nN 1 + N 2 = N , breaking Sp(N ) → Sp(N 1 ) × Sp(N 2 ).\nThe glueball superpotential is calculated from the associated Sp(N ) matrix model as\nW DV (S 1 , N 1 ; S 2 , N 2 ) =(N 1 + 1)S 1 [1 − ln(S 1 /Λ 3 1 )] + (N 2 + 1)S 2 [1 − ln(S 2 /Λ 3 2 )] + W pert , W pert = 2N 1 ∂F S 2 ∂S 1 + 2N 2 ∂F S 2 ∂S 2 + 4F RP 2 .\n(5.14)\n\nThe scales Λ 1 , Λ 2 in (5.3) respectively are the energy scales of the low energy Sp(N 1 ), Sp(N 2 ) theories with the Φ field integrated out. They are related to the high energy scale Λ by the matching conditions as in (2.14), which yields\n(Λ 1 ) 3(N 1 +1) = m N 1 −2N 2 −1 g 2N 2 Λ 2N+4 , (Λ 2 ) 3(N 2 +1) = (−1) N 2 −1 m −2N 1 +N 2 −1 g 2N 1 Λ 2N+4 . (5.15)\nThe matrix model free energy is computed in Appendix A, and the result is\nW pert = 2(N 1 + 1) −S 2 1 + 5S 1 S 2 − 5 2 S 2 2 α + − 16 3 S 3 1 + 91 2 S 2 1 S 2 − 59S 1 S 2 2 + 91 6 S 3 2 α 2 + − 140 3 S 4 1 + 1742 3 S 3 1 S 2 − 1318S 2 1 S 2 2 + 2636 3 S 1 S 3 2 − 871 6 S 4 2 α 3 + 2(N 2 + 1) S 1 ↔ S 2 , α → −α ,(5.\n16) where α ≡ g 2 /m 3 . This is as expected from the map (3.17) and [11].\n\nIn particular, let us concentrate on the unbroken case, N 2 = 0. Unlike [10,15], we do not set S 2 = 0, but rather keep S 2 non-zero and extremize with respect to it, according to our general prescription, to obtain the actual matrix model result. After integrating out S 1 and S 2 from (5.16), we obtain the superpotential as a power series in Λ 1 and Λ 2 as W low = (N + 1)Λ 3 1 + Λ 3 2 + (higher order terms in Λ 1,2 ). (5.17)\n\nThe matching relation (5.15) gives\nΛ 3 2 = −(Λ 3 1 ) N+1 α N ,(5.18)\nso the terms containing Λ 2 in (5.17) starts to contribute to the superpotential at order (Λ 3 1 ) N+1 , i.e. like Sp(N ) instantons. If we use the relation (5.18) and write out all the terms in (5.17), we obtain\nN = 0 : W low = O(α 4 ), N = 1 : W low = 2Λ 3 1 + O(α 4 ), N = 2 : W low = 3Λ 3 1 − Λ 6 1 α − 2Λ 9 1 α 2 − 187 27 Λ 12 1 α 3 + O(α 4 ), N = 3 : W low = 4Λ 3 1 − 3Λ 6 1 α − 47 6 Λ 9 1 α 2 − 75 2 Λ 12 1 α 3 + O(α 4 ), N = 4 : W low = 5Λ 3 1 − 5Λ 6 1 α − 13Λ 9 1 α 2 − 65Λ 12 1 α 3 + O(α 4 ). (5.19)\nThus properly accounting for S 2 , it turns out that these matrix model results agree perfectly, up to the order presented, with the standard gauge theory results (Eq. (4.13) of [10]). (The discrepancies found in [10] set in at order Λ\n\n\n3(N+1) 1\n\n, and are cancelled e.g. by (5.18).) In (5.19), for N = 0, 1, there were rather remarkable cancellations between the instanton contributions from Sp(N 1 ) and Sp(N 2 ). This will be further discussed and generalized in the next section.\n\nThe matrix model prediction for the superpotential of the SO(2N ) theory with symmetric tensor can similarly be obtained by simply changing the N i + 1 in (5.16) to N i − 1. It should be possible to compute the superpotential from gauge theory using the duality for this theory [41]. The result is expected to be compatible with the map (3.20) to the superpotential computed for the U (N − 2K) theory with adjoint.\n\n\nResidual Instantons: String theory (matrix model) versus gauge theory\n\nA remarkable aspect of the string theory (matrix model) computation of the effective superpotential is that (1.3) can be obtained purely in terms of the dynamics of the lowenergy K i=1 G(N i ) theory on the RHS of (1.1). The only information needed about the high-energy G(N ) gauge theory is the perturbative contribution of the G(N )/ i G i (N i ) ghosts to the glueball superpotential (1.3), as discussed in [42], along with the matching relations connecting the scales Λ i of the low-energy G i (N i ) factors to the scale Λ of the highenergy G(N ) theory. This is very different from the conventional description of standard gauge theory, where there can be non-perturbative contributions to W low which are not readily seen in terms of the low-energy theory on the RHS of (1.1). An example of such an effect is instantons in the broken part of the group when π 3 (G(N )/ i G i (N i )) = 0 (see e.g. [43]). Nevertheless, the string theory/ matrix model does properly reproduce such effects, via a low-energy description.\n\nA gauge theory interpretation for the string theory/ matrix model results was given in [8,9]: the string theory / matrix model results actually refer to a particularly natural UV completion of the original G(N ) theory, where it is embedded in the supergroup G(N +k|k) with k large. This latter theory has a Higgs branch, where k can be reduced successively, eventually Higgsing the theory down to the original G(N ) theory. More generally, the theory with breaking pattern (1.1) is replaced with\nG(N + k|k) → K i=1 G i (N i + k i |k i ), (6.1)\nwhich has a Higgs branch flat direction connecting it to (1.1). Consideration of the particular matter content of the G(N + k|k) theories along the Higgs branch, which often has extended supersymmetry, suggests that no dynamically generated superpotential ever lifts this Higgs branch moduli space, i.e. that the superpotentials of these particular theories are always independent of the location of the theory on this Higgs branch [9]. Moving along the Higgs branch has the effect of reducing k, and this expected independence of the superpotential of the position on the Higgs branch fits with the fact that the G(N + k|k) matrix model results are k independent, because all k dependence cancels in the supertraces.\n\nBecause of the expected independence of the superpotential on the Higgs branch, and because we Higgs back to the original G(N ) theory, in most cases, this \"F-completion\" of the original G(N ) theory into the G(N + k|k) theory is of no consequence. There are, however, a few rare exceptions, where the superpotential of the Higgsed G(N + k|k) theory differs from that of the standard G(N ) theory. This difference comes from residual instantons in G(N + k|k)/G(N ), which need not decouple even if G (N + k|k) is Higgsed to G(N ) far in the UV. As verified in [9], these residual instanton contributions precisely account for the few differences between the string theory (matrix model) results and standard gauge theory, for example the glueball superpotentials, with coefficient h = 1, for U (1) and Sp(0), e.g. with an adjoint and quadratic superpotential.\n\nIn many cases, however, these residual instanton contributions sum up to yield precisely the result expected from standard gauge theory, including superpotential contributions which in standard gauge theory would not have had a known low-energy description. In particular, residual instanton contributions which could have lead to potential discrepancies with standard gauge theory often completely cancel. The cancellation occurs once one sums over the different terms i in (1.2), upon using the precise matching relation between the low-energy scales Λ i , and the original high-energy scale Λ.\n\nAs an example, consider U (K) with adjoint matter and breaking pattern U (K) → U (1) K . For K = 1 the string theory (matrix model) description includes a residual instanton effect, yielding W low = Λ 3 L rather than the standard gauge theory answer W sgt = 0 [9]. But for all K > 1 the string theory/ matrix model result is W low = 0, in agreement with the standard gauge theory expectation for U (K) → U (1) K . The result W low = 0 looks like a remarkable cancellation because the glueball superpotential W eff (S 1 , . . . S K ) is quite non-trivial. Nevertheless, upon solving for the S i and plugging back in, the exact result for W low = W eff ( S i ) is zero, as was proven in [44].\n\nTo illustrate this cancellation, consider the leading order gaugino condensation contribution to W eff (S i ) in the string theory (matrix model) constructions, where the unbroken U (1) factors in U (K) → U (1) K contribute as in (2.15), with h i = 1, unlike in standard gauge theory:\nW gc (S i ) = k i=1 S i log( Λ 3 i S i ) + 1 , (6.2) with Λ 3 i = g K+1 Λ 2N\nj =i (a j − a i ) −1 by using (2.10) with all N i = 1. Though this is a non-trivial superpotential, it vanishes upon integrating out the S i :\nW gc ( S i ) = K i=1 Λ 3 i = g K+1 Λ 2N K i=1 j =i (a j − a i ) −1 = g 2 k+1 Λ 2N dx 2πi 1 W ′ (x) = 0. (6.3)\nThe contour in (6.3) encloses all the zeros of W ′ (x), and we get zero for all K > 1 by pulling the contour off to infinity. We see here why K = 1 is different: we then get a residue at infinity, leading to the low energy superpotential W U(1) = Λ 3 L , as in (2.9), with h U(1) = 1 as in (1.4).\n\nTo give another example of such a cancellation of residual instanton effects, consider the string theory (matrix model) result for Sp(N ) with an antisymmetric tensor A, with W tree having K critical points, for the case N = 0. For the case of K = 1, the superpotential is just a mass term for A and the low-energy superpotential is the Sp(0) gaugino condensation superpotential, with h(Sp(0)) = 1: W = Λ 3 , unlike standard gauge theory. Again, this can be understood as a residual instanton effect in the F-completion of Sp(N ) to Sp(N + k|k), which is present precisely for the case N = 0 [9]. For a higher order superpotential, K > 1, we would write the breaking pattern as Sp(0) → Sp(0) K . For all K > 1, the residual instanton effects all cancel, precisely as in the U (K) → U (1) K example discussed above; in fact, the two theories have the same effective superpotential W eff (aside from the classical difference), as discussed in [11] and sect. 3.3. Thus, for example, (6.3) can also be interpreted as the leading gaugino condensation contributions from the Sp(0) K factors, and where we now use the matching relation (2.14) to relate the Λ 3 i to g K+1 Λ 2K\n\nj =i (a j − a i ) −1 . Again, there is complete cancellation in W eff here, except for the case K = 1.\n\nMore generally, for Sp(N ) with antisymmetric, breaking as Sp(N ) → K i=1 Sp(N i ), the results obtained via the string theory / matrix model glueball potential W eff (S 1 , . . . S K ), upon integrating out the S i , appears to always agree with standard gauge theory results for the superpotential [40], as seen in the examples of [11] and (5.19). This agreement comes about via a remarkable interplay between the different terms i in (3.12). If we treated the scales Λ i of the Sp(N i ) factors as if they were initially independent, each term N i Π( S i )−2πiη i S i in (3.12) would be a complicated function of Λ i , which does not have a known, conventional, interpretation in terms of standard gauge theory for the low-energy Sp(N i ) factor. But upon adding the different i terms and using the matching relations relating Λ i to Λ, e.g. (2.14), one nevertheless obtains the standard gauge theory results, thanks to an intricate interplay between the different terms i.\n\nBy the map of (3.17) [11], the agreement between string theory / matrix models and standard gauge theory for Sp(N ) with antisymmetric can be phrased as such an agreement for U (N + K) with adjoint and breaking pattern U (N + K) → K i=1 U (N i + 1).\n\nAs another example, consider U (N ) with adjoint Φ and superpotential having K = N − 1, in the vacuum where U (N ) → U (2) × U (1) N−2 . Factorizing the Seiberg-Witten curve yields for the exact superpotential [46] W exact = W cl (g) ± 2g N Λ N . . The fact that (6.4) will only hold if remarkable cancellations occur upon integrating out S 1 and S 2 from the non-trivial W (S 1 , S 2 ), was discussed in [3], where the cancellations were verified to indeed occur, up to order α 3 . This is checked to one higher order in (5.19), since it is related to Sp(1) → Sp(1) × Sp(0) by the map of [11] and sect. 3.3. The leading order cancellation, say in terms of U (3) → U (2) × U (1), is between U (1) gaugino condensation, Λ 3 2 , and a higher order term coming from integrating out S 1 from W pert (S i ). The residual instanton contributions associated with the UV completion (6.1), as opposed to the standard gauge theory results for (1.1) do not always cancel, however. The cases where we find non-cancellations are when the degree of the superpotential is sufficiently large, so that it contains terms which are not independent moduli. As an example, consider U (1) with W tree as in (2.2) having K minima, breaking U (1) → U (1) × U (0) K−1 . The gaugino condensation contribution to the superpotential, according to the string theory (matrix model) construction, is given by (2.15) with h 1 = 1 and all other h i = 0 and their S i set to zero. Upon integrating out S 1 , we thus obtain the superpotential\nW gc = Λ 3 1 = Λ 2 W ′′ (a 1 ) = g K+1 Λ 2 j =1 (a j − a 1 ),(6.5)\nwhere we used the matching relation (2.10) with N 1 = 1 and all other N j = 0. The full low-energy effective superpotential W low (g i , Λ) can be regarded as the generating function for the operator expectation values:\nu j = ∂W low (g i , Λ) ∂g j u i ≡ 1 j TrΦ j . (6.6)\nIn the U (1) theory, we have classical relations u j = 1 j u j 1 . But the quantum contribution (6.5) (along with additional, higher order contributions) imply quantum deformation of these classical relations, due to the residual instanton effects in the U (1 + k|k) → U (1 + k 1 |k 1 ) × U (k 2 |k 2 ) . . . U (k K |k K ) F-completion. For the simplest such example, consider U (1) with W tree = 1 2 mΦ 2 + λΦ. The low-energy superpotential is\nW low = − λ 2 2m + mΛ 2 ,(6.7)\nwith the first term the classical contribution and the second the residual instanton. Using (6.6) we then get\nu 1 = − λ m , , u 2 = λ 2 2m 2 + Λ 2 i.e. u 2 = 1 2 u 2 1 + Λ 2 ,(6.8)\nwhich can be regarded as an instanton correction to the composite operator u 2 .\n\nAs another such example, consider U (2) with an adjoint and W tree having K minima, in the vacuum where the gauge group is broken as U\n(2) → U (1) × U (1) × U (0) K−2 . The gaugino condensation contribution to W low is W gc = Λ 3 1 + Λ 3 2 = g K+1 Λ 4 K j=3 (a j − a 1 ) − K j=3 (a j − a 2 ) a 2 − a 1 . (6.9)\nFor example, for U (2) with W tree having K = 3 critical points, we break U (2) → U (1) × U (1) × U (0) and (6.9) leads to W gc = g 4 Λ 4 . (6.10)\n\nComputing expectation values as in (6.6) this leads to\nu 4 = u 4 cl + Λ 4 ,(6.11)\nwhich can be interpreted as an instanton contribution to the composite operator u 4 = TrΦ 4 in U (2) gauge theory. More generally, for U (N ) gauge theory, the independent basis of operators u j = 1 j TrΦ j are only those with j ≤ N , those with j > N can be expressed as products of these basis operators via classical relations. But these relations can be affected by instantons. In particular, for U (N ) with an adjoint, the instanton factor is Λ 2N , so operators u j with j ≥ 2N can be affected. The above residual instanton contributions of the U (N + k|k) UV completion can be interpreted as implying specific such instanton corrections to the higher Casimirs u j .\n\nA similar situation arises in the N = 1 * U (N ) theory, where the effective superpotential of the matrix model and conventional gauge theory differ by a contribution N 2 m 3 E 2 (N τ ) [17]; this was interpreted in [17] as differing operator definitions of TrΦ 2 between gauge theory and the matrix model at the level of instantons. Related issues for multi-trace operators were seen in [18].\n\n\nConclusions\n\nTo compute the correct string theory / matrix model results, we should include or not include the glueball fields S i according to the prescription of this paper. Upon doing so, in all examples that we know of, the string theory / matrix model results agree with the results of standard gauge theory, at least in those cases where the relevant gauge theory does not suffer from UV ambiguities. In the case where such ambiguities are present, for example in defining composite operators appearing in W tree , the string theory / matrix model results correspond to a particular UV definition of the theory. The agreement with standard gauge theory results is often due to a remarkable interplay between the different low-energy terms, found upon integrating out the glueball fields S i , and connecting their scales Λ i via the appropriate matching relation to the scale Λ of the original theory. In some cases, this interplay leads to complete cancellations of the residual instanton contributions to W low coming from the G(N + k|k) completion [8,9]. Perhaps there is some additional structure governing the glueball superpotentials, which would make these remarkable cancellations more manifest.\n\n\nA.1. Proof for relation between F S 2 and F RP 2\n\nHere we prove a general relation between the S 2 and RP 2 contributions to the SO(2N )/Sp(N ) matrix model free energy:\nF RP 2 = ∓ 1 2 ∂F S 2 ∂S 0 SO(2N )/Sp(N ) with adjoint, ∓ 1 2 K i=1 ∂F S 2 ∂S i SO(2N )/Sp(N )\nwith symmetric/antisymmetric tensor.\n\n(A.1) The first equation was conjectured in [19,20] based on explicit diagrammatic calculations, and proven in [21] for the case of unbroken vacua. Here we will give a general matrix model proof for arbitrary breaking pattern. These relations are equivalent to the maps (3.14), (3.16), (3.17), and (3.20), which we obtained in sect. 3.3 immediately from the string theory geometric transition construction, accounting for the orientifold contributions to the fluxes.\n\nConsider U (N) and SO(2N)/Sp(N) matrix models which correspond to U (N ) and SO(2N )/Sp(N ) gauge theories with a two-index tensor matter field. The partition function is\nZ = e − 1 g 2 F(S i ) = dΦ e − 1 g W tree (Φ) . (A.2)\nWe denote matrix model quantities by boldface letters, following the notation of [15]. Φ is an N×N (for U (N) theory) or 2N×2N (for SO(2N)/Sp(N) theory) matrix corresponding to the Φ field in gauge theory, and the \"action\" W tree is defined in (2.2), (2.4) and (2.5). The matrix integral (A.2) is evaluated perturbatively around the general broken vacua of (2.1), with N i replaced by N i . We take the double scaling limit N i → ∞, g → 0 with\ngN i ≡ S i (for U (N i ) factors) or 2gN i ≡ S i (for SO(2N i )/Sp(N i ) factors) kept finite.\nThe dependence of the free energy F(S i ) on N i are eliminated in favor of S i , and F(S i ) is expanded in the 't Hooft expansion as\nF(S i ) = M g 2−χ(M) F M (S i ) = F S 2 + gF RP 2 + · · · (A.3)\nwhere the sum is over all compact topologies M of the matrix model diagrams written in the 't Hooft double-line notation, and χ(M) is the Euler number of M.\n\nThe matrix model resolvent is defined as follows:\nR(z) ≡ g Tr 1 z − Φ = R S 2 (z) + gR RP 2 (z) + · · · . (A.4)\nFor U (N ) theory with adjoint, the expansion parameter is g 2 instead of g, and in particular, R RP 2 (z) ≡ 0. The resolvent and the free energy are related as\nR M (z) = S z δ χ(M),2 + 1 z 2 ∂F M ∂g 1 + 2 z 3 ∂F M ∂g 2 + 3 z 4 ∂F M ∂g 3 + · · · , (A.5) where S = K i=1 S i .\nThe resolvents can be determined uniquely by solving the matrix model loop equations (the loop equations for the relevant matrix models are summarized in [15]), under the condition\nA i dz 2πi R S 2 (z) = S i , A i dz 2πi R RP 2 (z) = 0. (A.6)\nA i is the contour around the i-th critical point of W (z). In general, R(z) develops a cut around each critical point in the large N i limit, and A i is taken to encircle the i-th cut. Note that the expression (A.5) should be understood as a Laurent expansion around z = ∞, and converges only if |z| is larger than r such that all the singularities (cuts) of the resolvent are inside the circle C : |z| = r.\n\nOn the other hand, gauge theory resolvents R(z), T (z) (see e.g. [7]) are determined uniquely by solving the Konishi anomaly equations (the Konishi anomaly equations for the relevant gauge theories are summarized in [15]), under the condition\nA i dz 2πi R(z) = S i , A i dz 2πi T (z) = N i U (N i ) 2N i SO(2N i )/Sp(N i ) . (A.7)\nAs was shown in [15], the matrix model resolvents R S 2 (z), R RP 2 (z) are related to the gauge theory resolvents R(z), T (z) as\nR(z) = R S 2 (z), T (z) = U(N i ) N i ∂ ∂S i R S 2 (z) + SO(2N i ) /Sp(N i ) 2N i ∂ ∂S i R S 2 (z) + 4R RP 2 (z) (A.8)\nwith S i and S i identified 5 .\n\nFirst, consider SO(2N )/Sp(N ) theory with adjoint. The general breaking pattern is\nSO(2N ) → SO(2N 0 ) × U (N 1 ) × · · · × U (N K ) or Sp(N ) → Sp(N 0 ) × U (N 1 ) × · · · × U (N K ) (Eq. (2.1)), where N = N 0 + K i=1 N i .\nNote that the eigenvalues are distributed in a symmetric manner under z ↔ −z, and hence (A.7) is It was shown in [13] that the resolvents of this SO(2N )/Sp(N ) theory are related to the resolventsR(z) andT (z) of U (Ñ ≡ 2N ∓ 2) theory with adjoint as follows:\nA 0 dz 2πi R(z) = S 0 , A i dz 2πi R(z) = A −i dz 2πi R(z) = S i , A 0 dz 2πi T (z) = 2N 0 , A i dz 2πi T (z) = A −i dz 2πi T (z) = N i ,R(z) =R(z), T (z) =T (z) ± 2 z . (A.10)\nThe tree level superpotential of the U (Ñ ) theory is related to the one for the SO(2N )/Sp(N ) theory as W U (z) = W SO/Sp (z) (see (2.2) and (2.5)), and the breaking pattern is\nU (Ñ ) → U (N −K ) × · · · × U (N −1 ) × U (2N 0 ∓ 2) × U (N 1 ) × · · · × U (N K ) with N −i = N i , 1 ≤ i ≤ K.\nNote that since there is no z ↔ −z symmetry in the U (Ñ ) theory, the U (N −i ) factors that are \"images\" for SO(2N )/Sp(N ) are \"real\" for U (Ñ ). In addition, the glue-ballS of the U (Ñ ) theory is related to the glueball S of the SO(2N )/Sp(N ) theory as\nS 0 = S 0 ,S i =S −i = S i , 1 ≤ i ≤ K.\nTherefore, e.g. the first equation in (A.10) is more precisely\nR(z, S j ) =R(z,S j ) S 0 =S 0 ,S i =S −i =S i . (A.11)\nDifferentiating (A.11) with respect to S j , we obtain\n∂R ∂S 0 = ∂R ∂S 0 S 0 =S 0 , S i =S −i =S i , ∂R ∂S j = ∂R ∂S j + ∂R ∂S −j S 0 =S 0 , S i =S −i =S i , (A.12)\nwhere 1 ≤ j ≤ K. Now, using (A.8), let us translate the relation (A.10) among gauge theory resolvents into a relation among matrix model resolvents:\nR S 2 =R S 2 , 2N 0 ∂R S 2 ∂S 0 + K i=1 N i ∂R S 2 ∂S i + 4R RP 2 = (2N 0 ∓ 2) ∂R S 2 ∂S 0 + K i=1 N i ∂R S 2 ∂S i + ∂R S 2 ∂S −i ± 2 z . (A.13)\nHereR S 2 is the matrix model resolvent associated with the U (Ñ ) theory. Using (A.12) and the relations R S 2 = R,R S 2 =R (Eq. (A.8)), we obtain\nR RP 2 (z) = ∓ 1 2 ∂ ∂S 0 R S 2 (z) ± 1 2z .\n(A.14)\n\nBy expanding the resolvents around z = ∞ using (A.5) and comparing the coefficients, we obtain a relation between matrix model free energies:\nj ∂F RP 2 ∂g j = ± 1 2 ∂ ∂S 0 Sδ j0 + j ∂F S 2 ∂g j ∓ 1 2 δ j0 . (A.15)\nwhere j = 0, 2, · · · , 2(K + 1). The j = 0 case is trivially satisfied since S = S 0 + 2 K i=1 S i here, while the j = 2, 4, . . . , 2(K + 1) cases lead to the first equation of (A.1), which we wanted to prove.\n\nNext, consider SO(2N )/Sp(N ) theory with symmetric/antisymmetric tensor. The breaking pattern is SO (2N \n) → K i=1 SO(2N i ) or Sp(N ) → K i=1 Sp(N i ) (Eq. (2.1)), where N = K i=1 N i .\nIt was shown in [11] that the resolvents of this SO/Sp theory is related to the resolventsR(z) andT (z) of U (Ñ ≡ 2N ∓ 2K) theory with adjoint as follows:\nR(z) =R(z), T (z) =T (z) ± d dz ln[W ′ (z) 2 + f K−1 (z)]. (A.16)\nThe tree level superpotential of the U (Ñ ) theory is related to the one for the SO(2N )/Sp(N ) theory as W U (z) = W SO/Sp (z) (see (2.2) and (2.4)), and the breaking pattern is\nU (Ñ ) → K i=1 U (2N i ∓ 2)\n. The glueball S i of the U (Ñ ) theory is taken to be the same as the glueball of the SO(2N )/Sp(N ) theory. In (A.16), f K−1 (z) is a polynomial of degree K − 1. Using (A.8), we can translate the relation (A.16) among gauge theory resolvents into a relation among matrix model resolvents:\nR RP 2 (z) = ∓ 1 2 K i=1 ∂ ∂S i R S 2 (z) ± 1 4 d dz ln[W ′ (z) 2 + f K−1 (z)]. (A.17)\nIn order to extract the relation between matrix model free energies, let us multiply (A.17) by z j (0 ≤ j ≤ K + 1) and integrate over z along the contour C, introduced under (A.6), which encloses all the cuts around the critical points of W (z). Taking\nW ′ (z) 2 + f K−1 (z) = g 2 K+1 K i=1 (z − a + i )(z − a − i ), (A.18)\nthe branching points of the cuts are at z = a ± i . The second term on the right hand side in (A.17) does not contribute to the contour integral unless j = 0:\n∓ 1 4 C dz 2πi K i=1 z j 1 z − a + i + 1 z − a − i = ± 1 4 C dw 2πi K i=1 1 w j+1 1 1 − a + i w + 1 1 − a − i w = ∓ K 2 δ j0 , (A.19)\nwhere w = 1/z, because all the poles w = 1/a ± i are outside of the contour C (on the w-plane). On the contour C, we can use the Laurent expansion (A.5) to evaluate the contribution from the other terms, and the final result is\nj ∂F RP 2 ∂g j = ± 1 2 K i=1 ∂ ∂S i Sδ j0 + j ∂F S 2 ∂g j ∓ K 2 δ j0 . (A.20)\nThe j = 0 case is trivially satisfied, while the 1 ≤ j ≤ K + 1 cases lead to the second equation of (A.1), which we wanted to prove.\n\n\nA.2. Computation of matrix model free energy: SO(2N )/Sp(N ) theory with adjoint\n\nLet us consider SO(2N) matrix model which corresponds to SO(2N ) gauge theory with adjoint. The tree level superpotential is taken to be quartic (5.1). The matrix variable Φ in (A.2) is a real antisymmetric matrix and can be skew-diagonalized as\nΦ ∼ = diag[λ 1 , · · · , λ N ] ⊗ iσ 2 .\n(A.21)\n\nBy changing the integration variables from Φ to λ i , we obtain\nZ ∼ N i=1 dλ i N i<j (λ 2 i − λ 2 j ) 2 e − 1 g N i=1 (− m 2 λ 2 i + g 4 λ 4 i ) , (A.22) where N i<j (λ 2 i − λ 2 j ) 2\nis the Jacobian for this change of variables [47,20]. The polynomial − m 2 λ 2 + g 4 λ 4 has critical points at λ = 0, ± m/g, around which we would like to do perturbative expansion. For this purpose, we separate λ's into two groups as \nλ i = λ (0) i 0 i 0 = 1, . . . , N 0 , m/g + λF = gf 1 (N 0 , N 1 ) + g 2 f 2 (N 0 , N 1 ) + · · · , (A.24)\nthe loop expansion tells us that f n (N 0 , N 1 ) is a polynomial of degree n + 2. Therefore, by performing the matrix integral by computer for small values of N 0 and N 1 , one can determine the polynomial f n . If we rewrite N 0,1 in favor of S 0 = 2gN 0 and S 1 = gN 1 , the expansion (A.24) arranges itself into the 't Hooft expansion (A.3), from which one can read off F S 2 , F RP 2 , etc. Following the procedure sketched above, we computed the matrix model free energy as\nF S 2 = 1 4 S 3 0 − 2S 2 0 S 1 + S 0 S 2 1 α + − 9 16 S 4 0 + 7S 3 0 S 1 − 9S 2 0 S 2 1 + 2S 0 S 3 1 α 2 + 9 4 S 5 0 − 233 6 S 4 0 S 1 + 262 3 S 3 0 S 2 1 − 152 3 S 2 0 S 3 1 + 20 3 S 0 S 4 1 α 3 + O(α 4 ), (A.25)\nwhere we defined α ≡ g/m 2 . We also checked explicitly that the relation (A.1) holds. Substituting (A.25) into the DV relation (5.3), we obtain the superpotential (5.5).\n\nThe Sp(N ) result is obtained similarly, with the result as in (A.1).\n\nA.3. Computation of matrix model free energy: Sp(N ) theory with antisymmetric tensor Consider the Sp(N) matrix model which corresponds to Sp(N ) gauge theory with an antisymmetric tensor. The superpotential is taken to be quartic (5.12). The matrix variable Φ satisfies Φ = AJ, A T = −A. The \"action\" W tree is given in (5.12). By a complexified Sp(N) gauge rotation, the matrix Φ can be brought to the form [40] Φ ∼ = diag[λ 1 , . . . , λ N ] ⊗ 1 2 , λ i ∈ C. (A.26)\n\nBy changing the integration variables from Φ to λ i , we obtain\nZ ∼ N i=1 dλ i N i<j (λ i − λ j ) 4 e − 1 g N i=1 ( m 2 λ 2 i + g 3 λ 3 i ) . (A.27)\nwhere N i<j (λ i −λ j ) 4 comes from the Jacobian for this change of variables. The polynomial m 2 λ 2 + g 3 λ 3 has two critical points z = 0, − m g , around which we would like to do perturbative expansion. For this purpose, we separate λ's into two groups as The matrix integral can be performed just the same way as for the SO/Sp(N ) theory with adjoint, as described in the last subsection. After substitution S 1,2 = 2gN 1,2 , we obtain\nλ i = λ(F S 2 = − S 3 1 3 + S 3 2 3 + 5 2 S 2 1 S 2 − 5 2 S 1 S 2 2 α + − 4 3 S 4 1 + 91 6 S 3 1 S 2 − 59 2 S 2 1 S 2 2 + 91 6 S 1 S 3 2 − 4 3 S 4 2 α 2 + − 28 3 S 5 1 + 871 6 S 4 1 S 2 − 1318 3 S 3 1 S 2 2 + 1318 3 S 2 1 S 3 2 − 871 6 S 1 S 4 2 + 28 3 S 5 2 α 3 + O(α 4 ), (A.29) where α ≡ g 2 /m 3 .\nWe also checked explicitly that the relation (A.1) holds. Substituting (A.29) into (5.14), we obtain the glueball superpotential (5.16).\n\n\nAppendix B. Gauge theory calculation of superpotential\n\nIn this appendix, we compute the exact superpotential of the N = 1 SO(2N )/Sp(N ) theory with adjoint in various vacua by considering factorization of the N = 2 curve. This factorization method was developed in [3] for U (N ), and generalized in [48] to the case with unoccupied critical points (in other words, the n < K case below). Inclusion of fundamentals was considered in [49]. The generalization to SO/Sp gauge group, which discuss below, was given in [29,38,50,51,52,12].\n\nFirst consider N = 2 SO(2N ) theory broken to N = 1 by the following polynomial tree level superpotential for the adjoint chiral superfield Φ:\nW tree = 1 2 Tr[W (Φ)], W (x) = K+1 j=1 g 2j 2j x 2j , W ′ (x) = g 2K+2 x K i=1 (x 2 − a 2 i ). (B.1)\nThe classical supersymmetric vacua are obtained by putting 2N 0 eigenvalues of Φ at x = 0 and N i pairs of eigenvalues at x = ±a i , where i = 1, · · · , K and N 0 + K i=1 N i = N . In this vacuum the gauge group breaks as SO(2N ) → SO(2N 0 ) × K i=1 U (N i ). We allow some of N i to vanish, i.e. we allow \"unoccupied\" critical points. Let the number of nonzero N i≥1 be n. Then, the N = 2 curve governing this SO(2N ) theory factorizes as [29,38,50]:\ny 2 = P 2 2N (x) − 4x 4 Λ 4N−4 = [xH 2N−2n−2 (x)] 2 F 2(2n+1) (x) N 0 > 0, H 2N−2n (x) 2 F 4n (x) N 0 = 0. (B.2)\nHere P , H and F are polynomials in x of the subscripted degree, which are invariant under x → −x, i.e. they are actually polynomials in x 2 . This factorization is required to have the appropriate number of independent, massless, monopoles and dyons, which must condense to eliminate some of the low-energy photons. The polynomial F is related to the tree level superpotential as\nF 2(2n+1) (x) = 1 g 2 2K+2 W ′ 2K+1 (x) 2 + f 2K (x) n = K, F 2(2n+1) (x)Q 2K−2n (x) 2 = 1 g 2 2K+2 W ′ 2K+1 (x) 2 + f 2K (x) n < K (B.3)\nwith some polynomial Q 2K−2n (x), f 2K (x) of the subscripted degrees, and W ′ 2K+1 (x) is as in (2.6). Equation (B.3) is for N 0 > 0, and for N 0 = 0 one must use the second equation with Q 2K−2n replaced by Q 2K−2n+2 .\n\nFor N 0 > 0 we can write the solution of (B.2) in terms of that of the corresponding U (2N − 2) breaking pattern, via:\nP SO(2N) 2N (x) = x 2 P U(2N−2) 2N−2 (x). (B.4)\nThe low energy superpotential is given by\nW low = 1 2 K+1 j=1 g 2j u 2j , (B.5)\nwhere the u 2j are constrained to satisfy (B.2). Implementing this leads to the result that [51]:\nTr 1 x − Φ = d dx ln P 2N (x) + P 2N (x) 2 − 4x 4 Λ 4N−4 . (B.6)\nPlugging back into (B.5) gives W low . Note that the superpotential takes this simple form (B.5) only after one integrates out the monopoles and dyons, whose equation of motion led to the factorization constraint (B.2) [53,54,21]. The Sp(N ) theory can be solved similarly. The N = 2 curve factorizes in the vacuum with breaking pattern Sp(N ) → Sp(N 0 ) × K i=1 U (N i ) as [29,38,50] \ny 2 = B 2N+2 (x) 2 − 4Λ 4N+4 = x 2 H 2N−2n (x) 2 F 2(2n+1) (x), B 2N+2 (x) ≡ x 2 P 2N (x) + 2Λ 2N+2 .\n(B.7)\n\nThe polynomial The Λ 2N+2 shift in (B.8) is an Sp(N ) residual instanton effect, associated with the index of the embedding of the U (N i ) factors in Sp(N ) [55,43] Again, the superpotential is given as in (B.5), subject to the constraint that u j satisfy (B.7). Implementing this, the u j can be obtained from B 2N+2 (x) by [51] Tr\n1 x − Φ = d dx ln B 2N+2 (x) + B 2N+2 (x) 2 − 4Λ 4N+4 . (B.9)\nWe now consider the exact W eff for a few SO/Sp(N ) cases, to illustrate and clarify the general features 6 . We take quartic tree level superpotential\nW tree = 1 2 TrW (Φ), W (x) = m 2 x 2 + g 4 x 4 , (B.10)\nwhich corresponds to K = 1.\n\n\nB.1. SO(2N ) unbroken\n\nBy the map (3.14), this maps to U (2N − 2) unbroken, for which P U(2N−2) (x) = 2Λ 2N−2 T 2N−2 (x/2Λ), with T N (x = 1 2 (t + t −1 )) = 1 2 (t N + t −N ) a Chebyshev polynomial [56]. Thus, using (B.4), P SO(2N) (x) = 2Λ 2N−2 x 2 T 2N−2 (x/2Λ), as found in [21]. This then leads to [21] By the map (3.14), this maps to U (2N −2) → U (0)×U (N −1)×U (N −1). Using (3.14), the multiplication map of [3] for the U (2N − 2) theory leads to a similar multiplication map for the SO(2N ) theory, which was discussed in [51]. Using this, we can construct the solution to the factorization problem for general N in terms of that of say N = 2, i.e. SO(4) → SO(2) × U (1). In this case, equation (B.2) is\ny 2 = P 2 4 − 4x 4 Λ 4 = x 2 F 6 .\n(B.14)\n\nThe solution to this factorization problem is The solution for general SO(2N ) → SO(2) × U (N − 1), the multiplication map gives the solution to the factorization problem as P 2N (x) = 2x 2 Λ 2N−2 T N−1 ((x 2 − a 2 )/2Λ 2 ), with T N−1 the Chebyshev polynomial defined above. The effect is to rescale u 2 , u 4 , and hence W low by an overall factor of N − 1:\nP 4 = x 2 (x 2 − a 2 ), F 6 = x 2 [(x 2 − a 2 ) 2 − 4Λ 2 ],W low = (N − 1) − m 2 4g + 1 2 gΛ 4 . (B.19)\nThis agrees with the result (5.10).\n\n\nB.3. SO(4) → U (2)\n\nMore generally, we could consider the breaking pattern SO(2N ) → U (N ). The map of (3.14) is less useful here, when N 0 = 0, since it suggests mapping to U (2N − 2) → U (−2) × U (N ) × U (N ) and the U (−2) needs to be interpreted. In general, this breaking pattern leads to a complicated W low (Λ). We will here illustrate the case SO(4) → U (2), corresponding to N = 2, N 0 = 0, n = 1, K = 1. Equation (B.2) is\ny 2 = P 2 4 − 4x 4 Λ 4 = H 2 2 F 4 . (B.20)\nThe solution to this factorization problem is\nP 4 = (x 2 − a 2 ) 2 + 2Λ 2 x 2 , H 2 = x 2 − a 2 , F 4 = (x 2 − a 2 ) 2 + 4Λ 2 x 2 . (B.21)\nIn the classical Λ → 0 limit, this shows P 4 → (x 2 − a 2 ) 2 , implying the breaking pattern SO(4) → U (2). (B.6) gives \ny 2 = B 2 6 − 4Λ 12 = x 2 H 2 2 F 6 , F 6 = 1 g 2 W ′ 3 2 + f 2 . (B.26)\nThis factorization problem is solved by [51]: This solution continuously connects two classically different vacua with breaking pattern Sp(2) → U (2) and Sp(2) → Sp(1) × U (1). Correspondingly there are two ways to take the classical limit: i) Λ → 0 with a fixed, and ii) Λ, a → 0 with w = 2Λ 3 /a 2 fixed. In these limits, P 4 (x) goes to i) (x 2 − a 2 ) 2 and ii) x 2 (x 2 + w 2 ), showing the aforementioned breaking pattern. In the Sp(2) → U (2) case, we solve the second equation of (B.27) with the condition a 2 → −m/g as Λ → 0. The solution is a 2 = − m g + 4g 2 Λ 6 m 2 + 32g 5 Λ 12 m 5 + 448g 8 Λ 18 m 8 + · · · . (B.29)\nP 4 = (x 2 − a 2 ) 2 + 4Λ 6 a 4 (x 2 − 2a 2 ), m g = −a 2 +\nFrom (B.28) and (B.5), one obtains the exact superpotential: W low = − m 2 4g + 2m 1/2 g 1/2 Λ 3 + g 2 Λ 6 m − g 7/2 Λ 9 m 5/2 + · · · . (B.32)\nW low = − m 2 2g − 2g 2 Λ 6 m − 4g 5 Λ 12 m 4 −\n\nof [11] and sect. 3.3 relates this to Sp(1) → Sp(1) × Sp(0) N−2 , where the exact gauge theory result agrees with (6.4), up to the classical shift, upon using the relation (3.18). A priori, one might expect the string theory / matrix model result to disagree with (6.4), due to residual instanton contributions from the U (1) N−2 or the Sp(0) N−2 in U (N ) → U (2) × U (1) N−2 and Sp(1) → Sp(1) × Sp(0) N−2 respectively. But the string theory / matrix model result nevertheless agrees with (6.4), thanks to the interplay between the different terms. Consider, in particular, the case U (3) → U (2) × U (1)\n\n\n= 1, . . . , K. The contours A i and A −i encircle counterclockwise the cuts around z = a i and z = −a i , respectively. The relation (A.8) holds as it is, with the summation understood as over SO(2N 0 )/Sp(N 0 ) and U (N i ), i = 1, . . . , K.\n\ni 1\n1= 1, . . . , N 1 , (A.23) with N 0 + N 1 = N, corresponding to the classical supersymmetric vacuum with breaking pattern SO(2N) → SO(2N 0 ) × U (N 1 ). We would like to evaluate the matrix integral (A.22) perturbatively around λ (0,1) = 0. If we expand the matrix model free energy in the coupling constant g as\n\n1) i 0 i 1\n01= 1, . . . , N 1 , −m/g + λ (2) i 1 i 2 = 1, . . . , N 2 , (A.28) with N 1 + N 2 = N. This corresponds to the classical supersymmetric vacuum with breaking pattern Sp(N) → Sp(N 1 ) × Sp(N 2 ).\n\nF 2 (\n22n+1) (x) is related to W ′ (x) by (B.3). The mapping of the Sp(N ) theory to a U (2N + 2) theory, as in (3.16), can be written as a solution of (B.7) in terms of solutions of the corresponding U (2N + 2) factorization problem:B 2n+2 (x) ≡ x 2 P Sp(N) 2N (x) + 2Λ 2N+2 = P U(2N+2) 2N+2 (x).(B.8)\n\n\nlow-energy superpotential is W low = 1 2 (m u 2 + g u 4 ):W low = (N − 1) mΛ 2 + 3 2 gΛ 4 . (B.13) B.2. SO(2N ) → SO(2) × U (N − 1)\n\n\nwe can see the breaking pattern SO(4) → SO(2) × U (1). Using (B.\n\nu 2 B. 4 .\n24= 2(a 2 − Λ 2 ), u 4 = (a 4 − 2Λ 2 ) 2 − Λ 4 . (B.22) Sp(2) → U (2), Sp(1) × U (1)This corresponds to N = 2, n = 1, K = 1. Equations (B.7) and (B.3) are\n\n32g 8 Λ 18 m 7 +\n7· · · . (B.30) In the Sp(2) → Sp(1) × U (1) case, we solve the second equation of (B.27) with the condition w 2 → m/g as Λ → 0. B.28) and (B.5), one obtains the exact superpotential:\nOur convention is such that Sp(N ) ⊂ SU (2N ), and hence Sp(1) ∼ = SU (2).\nSo we get h replaced with h/2 for SO groups in (2.9). While one could absorb the overall factor of 2 into the definition of Λ, the number of vacua should be h whereas here we apparently get h/2 for SO groups. This is because the we don't see spinors or the Z 2 part of the center which acts on them; it's analogous to U (2N ) being restricted to vacua with confinement index 2.\nThis phenomenon is the same as that observed in the context of AdS/CFT[33,34,35].\nThe relation (A.8) is an obvious generalization of the formula in[15], which was for unbroken vacua, to an arbitrary breaking pattern. The gauge theory resolvents R(z), T (z) given in (A.8) clearly satisfy the condition (A.7) provided that the matrix model resolvents R S 2 (z), R RP 2 (z) satisfy the condition (A.6).\nMore SO/Sp examples can be found in[51].\nAcknowledgmentsWe would like to thank M. Aganagic, F. Cachazo, S. Naculich and H. Schnitzer for valuable discussions.K.I. was supported in part by DOE-FG03-97ER40546 and KI would also like to thank the Rutgers theory group for their support and hospitality while he was a visitor, where part of this work was done. P.K. was supported in part by NSF grant PHY-0099570. The research of A.V.R was supported by NSF grant 99-73935. A.V.R. and C.V. thank the 2003 Simons Workshop in Mathematics and Physics for a stimulating environment. C.V. would like to thank the hospitality of theory group at Caltech, where he was a Distinguished Gordon Moore Scholar. The research of C.V. was supported in part by NSF grants PHY-9802709 and DMS-0074329.Appendix A. 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Naqvi, \"Phases of N = 1 super- symmetric gauge theories with flavors,\" arXiv:hep-th/0303065.\n\nGeometric dual and matrix theory for SO/Sp gauge theories. B Feng, arXiv:hep-th/0212010Nucl. Phys. B. 661113B. Feng, \"Geometric dual and matrix theory for SO/Sp gauge theories,\" Nucl. Phys. B 661, 113 (2003) [arXiv:hep-th/0212010].\n\nPhases of N = 1 supersymmetric SO / Sp gauge theories via matrix model. C H Ahn, Y Ookouchi, arXiv:hep-th/0302150JHEP. 030310C. h. Ahn and Y. Ookouchi, \"Phases of N = 1 supersymmetric SO / Sp gauge theories via matrix model,\" JHEP 0303, 010 (2003) [arXiv:hep-th/0302150].\n\nPhases of N = 1 SO(N(c)) gauge theories with flavors. C H Ahn, B Feng, Y Ookouchi, arXiv:hep-th/0306068C. h. Ahn, B. Feng and Y. Ookouchi, \"Phases of N = 1 SO(N(c)) gauge theories with flavors,\" arXiv:hep-th/0306068.\n\nElectric -magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory. N Seiberg, E Witten, arXiv:hep-th/9407087Nucl. Phys. 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B 447, 271 (1995) [arXiv:hep-th/9503163].\n"},"corpus_id":{"kind":"number","value":119068847,"string":"119,068,847"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"hep-th/0311181\",\n \"mag\": 2171782575,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1016/j.nuclphysb.2003.12.030\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/hep-th/0311181v2.pdf\"\n ],\n \"pdf_sha\": \"0078b4f507a921c898b1f97f6a28456164d57f21\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"https://dash.harvard.edu/bitstream/1/41385089/1/69919%200311181.pdf\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"On Low Rank Classical Groups in String Theory, Gauge Theory and Matrix Models"},"authors":{"kind":"list like","value":["Ken Intriligator \nDepartment of Physics\nUniversity of California\nSan Diego, La Jolla92093-0354CAUSA\n\nDepartment of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCAUSA","Anton V Ryzhov \nDepartment of Physics\nBrandeis University\n02454WalthamMAUSA","Masaki Shigemori \nDepartment of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCAUSA","Cumrun Vafa \nJefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA"],"string":"[\n \"Ken Intriligator \\nDepartment of Physics\\nUniversity of California\\nSan Diego, La Jolla92093-0354CAUSA\\n\\nDepartment of Physics and Astronomy\\nUCLA\\n90095-1547Los AngelesCAUSA\",\n \"Anton V Ryzhov \\nDepartment of Physics\\nBrandeis University\\n02454WalthamMAUSA\",\n \"Masaki Shigemori \\nDepartment of Physics and Astronomy\\nUCLA\\n90095-1547Los AngelesCAUSA\",\n \"Cumrun Vafa \\nJefferson Physical Laboratory\\nHarvard University\\n02138CambridgeMAUSA\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We consider N = 1 supersymmetric U (N ), SO(N ), and Sp(N ) gauge theories, with two-index tensor matter and added tree-level superpotential, for general breaking patterns of the gauge group. By considering the string theory realization and geometric transitions, we clarify when glueball superfields should be included and extremized, or rather set to zero; this issue arises for unbroken group factors of low rank. The string theory results, which are equivalent to those of the matrix model, refer to a particular UV completion of the gauge theory, which could differ from conventional gauge theory results by residual instanton effects. Often, however, these effects exhibit miraculous cancellations, and the string theory or matrix model results end up agreeing with standard gauge theory. In particular, these string theory considerations explain and remove some apparent discrepancies between gauge theories and matrix models in the literature."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Large N topological string duality [1] embedded in superstrings [2,3] has led to a new perspective on N = 1 supersymmetric gauge theories: that the exact effective superpotential can be efficiently computed by including glueball fields. For example, in a theory with gauge group G, with tree-level superpotential leading to a breaking pattern\nthe dynamics is efficiently encoded in a superpotential W eff (S 1 , . . . S K ; g j , Λ) (g j are the parameters in W tree and Λ is the dynamical scale). Further, string theory implies [3] W eff (S i ; g j ,\nwith h i and τ i the fluxes through A i and B i three-cycles in the geometry, as will be reviewed in sect. 3. The prepotential F (S i ) in (1.2) is computable in terms of geometric period integrals, which yields [3] ∂F (S i )\nwith coefficients c i 1 ...i K depending on the g j (but not on the gauge theory scale Λ). In [4] it was shown how planar diagrams of an associated matrix model can also be used to compute (1.2) and (1.3). Based on the stringy examples, this was generalized in [5] to a more general principle to gain non-perturbative information about the strong coupling dynamics of gauge theories, by extremizing the perturbatively computed glueball superpotential. There are two aspects to the above statements: first that the glueball fields S i are the 'right' variables to describe the IR physics, and second that perturbative gauge theory techniques suffice to compute the glueball superpotential. The latter statement has now been proven in two different approaches for low powers of the glueball fields S i in (1.3) [6,7]. For powers of the glueball fields S i larger than the dual Coxeter number of the group, an ambiguity sets in for the glueball computation of the coefficients c i 1 ...i K in both of these approaches. The matrix model provides a natural prescription for how to resolve this ambiguity, essentially by continuing from large N i . It was argued in [8,9] that the string geometry / matrix model result (since the string geometry and matrix model results are identical, we refer to them synonymously) has the following meaning: it computes the F -terms for different supersymmetric gauge theories, which can be expressed in terms of G(N + k|k) supergroups. The W eff (S i ) is independent of k, and the above ambiguity can be eliminated by taking k, and hence the dual Coxeter number, arbitrarily large. The G(N ) theory of interest is obtained from the G(N + k|k) theory by Higgsing; but there can be residual instanton contributions to W eff [9], which can lead to apparent discrepancies between the matrix model and gauge theory results. We will somewhat clarify here when such residual instanton effects do, or do not, lead to discrepancies with standard gauge theory results.\nThere is another, more non-trivial assumption in [5]: the statement that the glueball fields S i are the 'right' variables in the IR. This assumption was motivated from the string dualities [1][2][3], where geometric transition provide the explanation of why the glueball fields are the natural IR variables: heuristically, S i corresponds to confinement. However, this is not quite correct: it also applies to abelian theories, as had been noted in [3]. So the deep explanation of why we should choose certain dynamical S variables remains mysterious.\nIn this paper, we will uncover the precise prescription for the correct choice of IR variables. This will be done from the string theory perspective, by arguing in which cases there is a geometric transition in string theory. For the general breaking pattern (1.1), our prescription for treating the glueball field S i , corresponding to the factor G i in (1.1), is as follows: If h(G i ) > 0 we include S i and extremize W eff (S i ) with respect to it. On the other hand, if h(G i ) ≤ 0 we do not include or extremize S i , instead we just set S i → 0. Here we define the generalized dual-Coxeter numbers 1 h(U (N )) = N, h(Sp(N )) = N + 1, h(SO(N )) = N − 2, (1.4) which are generalized in that (1.4) applies for all N ≥ 0. In particular h(U (1)) = h(Sp(0)) = 1, so when some G i factor in (1.1) is U (1) or Sp(0), our prescription is to include the corresponding S i and extremize with respect to it. On the other hand, h(U (0)) = 0 and h(SO(2)) = 0, so when some G i factor in (1.1) is U (0) or SO(2), our prescription is to just set the corresponding S i = 0 from the outset. (Note that U (1) and SO(2) are treated differently here.) This investigation was motivated by trying to understand the discrepancies found in [10] for Sp(N ) theory with antisymmetric tensor matter, where the superpotentials from the matrix model and gauge theory were found to differ at order h in perturbation theory and beyond. The analysis considered the trivial breaking pattern Sp(N ) → Sp(N ) and a single glueball was introduced corresponding to the single unbroken gauge group factor. In [9], various gauge theories including this example were studied, and an explanation for the discrepancies was proposed in terms of the conjecture, mentioned above, that the string theory / matrix model actually computes the superpotential of the large k G(N + k|k) supergroup theories, rather than the ordinary G(N ) theory. In this context, the trivial breaking pattern considered in [10] should be understood as Sp(N ) → Sp(N ) × Sp(0), which is completed to Sp(N + k|k) → Sp(N + k 1 |k 1 ) × Sp(k 2 |k 2 ). In particular, Sp(0) factors, while trivial in standard gauge theory, are non-trivial in the string theory geometry / matrix model context: there can be a residual instanton contribution to the superpotential when one Higgses Sp(k 2 |k 2 ) down to Sp(0|0) = Sp(0), as explicitly seen in [9] for the case of breaking Sp(0) → Sp(0) with quadratic W tree 2 . Related aspects of \"Sp(0)\" being non-trivial in the string / matrix model context were subsequently discussed in [11,12,13]. However, it turns out that one also needs to modify the matrix model side of the computation to take into account the Sp(0) factors. This was found by Cachazo [11], who showed that the loop equations determining T (z) ≡ Tr( 1 z−Φ ) and R(z) ≡ − 1 32π 2 Tr( W α W α z−Φ ) for the Sp(N ) theory with antisymmetric tensor matter [14,15] could be related to those of a U (2N + 2K) gauge theory with adjoint matter, with Sp(N ) → Sp(N ) × Sp(0) K−1 mapped to U (2N + 2K) → U (2N + 2) × U (2) K−1 . It was thus shown in [11] that vanishing period of T (z)dz through a given cut, corresponding to an Sp(0) factor, does not imply that the cut closes up on shell (aspects of the periods in this theory were also discussed in [16]). This fits with our above prescription that the Sp(0) glueballs should be included and extremized in the string theory / matrix model picture, as would be done for U (2), rather than set to zero, as was originally done in [10]. We stress that we are not yet even discussing whether or not the string theory / matrix model result agrees with standard gauge theory. Irrespective of any comparison with standard gauge theory, the prescription to obtain the actual string theory / matrix model result is as described above (1.4). Having obtained that result, we can now discuss comparisons with standard gauge theory results. As seen in [11], by solving the U (2N + 2K) loop equations for the present case, this corrected matrix model result now agrees perfectly with standard gauge theory! This will be discussed further here, with all glueball fields S i included.\nThis agreement, between the matrix model result and standard gauge theory, is in a sense surprising for this particular theory, in light of the Sp(k|k) description of [9] for the unbroken Sp(0) factors, with the resulting residual instanton contributions to the 2 It was suggested in the original version of [9] that such Sp(0) residual instanton contributions could also play a role for the case of cubic and higher order W tree (where they had not yet been fully computed) and could explain the apparent matrix model vs. standard gauge theory discrepancies found in [10]. As we will discuss, we now know that this last speculation was not correct. The corrected proposal of [9] is still that the matrix model computes the superpotential of the G(N + k|k) theory, but where the matrix model side of the computation should be corrected, as we discuss in this paper, to include glueball fields for the Sp(0) factors. superpotential. As we will explain later in this paper, the agreement here between matrix models and standard gauge theory is thanks to a remarkable cancellation of the residual instanton effect terms, which could have spoiled the agreement. The cancellation occurs upon summing over the i in (1.2) from i = 1 . . . K.\nThere are similar remarkable cancellations of the \"residual instanton contributions\" to the superpotential in many other examples, which we will also discuss. In fact, in all cases that we know of, the only cases where the residual instantons do not cancel is when the gauge theory clearly has some ambiguity, requiring a choice of how to define the theory in the UV; string theory / matrix model gives a particular such choice. Examples of such cases is when the LHS of (1.1) is itself U (1) or Sp(0) super Yang-Mills, as discussed in [9]. Other examples where the residual instanton contributions do not cancel, is when the superpotential is of high enough order such that not all operators appearing in it are independent, e.g. terms like Tr Φ n , for a U (N ) adjoint Φ, when n > N . In standard gauge theory, there are then potential ambiguities involved in reducing such composite operators to the independent operators, since classical operator identities can receive quantum corrections. The residual instanton contributions, which do not cancel generally in these cases, imply specific quantum relations for these operators, corresponding to the specific UV completion. See [17,18] for related issues.\nThe organization of this paper is as follows: In section 2 we summarize the gauge theories under consideration. In section 3 we review the type IIB string theory construction of these gauge theories. We also discuss maps of the exact superpotentials of Sp and SO theories to those of U theories, generalizing observations of [19,20,21,11,13]. In section 4 we explain, from the string theory perspective in which cases we have a geometric transition. In section 5 we consider examples, where the glueball fields S i of all group factors are correctly accounted for on the matrix model side. The results thereby obtained via matrix models are found to agree with those of standard gauge theory. In many of these examples, this agreement relies on a remarkable interplay of different residual instanton contributions, which sometimes fully cancel. Residual instantons are discussed further in sect. 6, with examples illustrating cases where they do, or do not, cancel. In appendix A, a proof of a general relation between the S 2 and RP 2 contributions to the matrix model free energy is given, and also the matrix model computation of superpotential is presented. In appendix B the gauge theory computation of the superpotential is discussed."},{"title":"The gauge theory examples","content":"The specific examples of N = 1 supersymmetric gauge theories which we consider, with breaking patterns as in (1.1), are as follows:\n(2.1) For U (N ) with adjoint Φ, the tree-level superpotential is taken to be\nwith K potential wells. In the classical vacua, with breaking pattern as in (2.1), Φ has N i eigenvalues equal to the root a i of .2), the factor of 1 2 is for convenience, because the eigenvalues of S or A appear in pairs, and the indices are contracted with δ a b for SO(N ) or J a b for Sp(N ). For SO(N ) and Sp(N ) with adjoint matter, the tree-level superpotential is\nsince all Casimirs of the adjoint Φ are even, and the 1 2 is again for convenience because the eigenvalues appear in pairs. Φ's eigenvalues sit at the zeros of\nThe breaking pattern in (2.1) has N 0 eigenvalues of Φ equal to zero, and N i pairs at ±a i , (2)). The next to last example in (2.1) is the N = 2 U (N ) theory with a matter hypermultiplet in the two-index symmetric tensor representation, breaking N = 2 to N = 1 by a superpotential as in (2.2):\nIn addition to the possibility of Φ's eigenvalues sitting in any of the K critical points W ′ (x) analogous to (2.3), there is a vacuum where N 0 eigenvalues sits at φ = 0, with S S = 0, breaking U (N 0 ) → SO(N 0 ). The last example in (2.1) is the similar theory where the N = 2 hypermultiplet is instead in the antisymmetric tensor representation A, rather than the symmetric tensor S. These last two classes of examples were considered in [22,23,24]. In all of these theories, the low energy superpotential is of the general form\nis the classical contribution (evaluating W tree in the appropriate minima). W gc (Λ i ) is the gaugino condensation contribution in the unbroken gauge groups of (1.1),\nwhere h i = C 2 (G i ) is the dual Coxeter number of the group G i in (1.1), with the phase factors associated with the Z 2h i → Z 2 chiral symmetry breaking of the low-energy G i gaugino condensation. The scales Λ i are related to Λ by threshold matching for the fields which got a mass from W tree and the breaking (1.1); some examples, with breaking patterns as in (2.1), are as follows. For U (N ) with adjoint Φ:\n(2.12) the U (N i ) has index of embedding 2, which is why the U (N i ) one-instanton factor is related to the Sp(N ) two-instanton factor in (2.12). For SO(N ) with symmetric tensor,\nFor Sp(N ) with antisymmetric A:\n(2.14)\nFinally, the term W H (g j , Λ) in (2.8) are additional non-perturbative contributions, which can be regarded as coming from the massive, broken parts of the gauge group.\nIn the description with the glueballs S i integrated in, as in (1.2), the gaugino condensation contribution comes from the first term in (1.3):\nand W H (g i , Λ) comes from the last terms in (1.3), upon integrating out the S i . When the minima a i of the superpotential are widely separated, the contributions W H from these last terms are subleading as compared with W gc . As in (1.2), the full glueball superpotential (1.2) can be computed via the string theory geometric transition, in terms of certain period integrals [2,3], as will be reviewed in the next section, or via the matrix models. In that context, the term W gc (S i , Λ i ) comes from the integration measure (as is also natural in field theory, since it incorporates the U (1) R anomaly) and W H (S i , g j ) can be computed perturbatively [4,5]. The perturbative computation of W H (S i , g j ) can also be understood directly in the gauge theory [6], up to ambiguities in terms S n with n > h = C 2 (G). The string theory/ matrix model constructions yield a specific way of resolving these ambiguities, which correspond to a particular UV completion of the gauge theory [8,9]. As discussed in the introduction, our present interest will be in analyzing this circle of ideas when some of the gauge group factors in (1.1) are of low rank, or would naively appear to be trivial, e.g. U (1), U (0), SO(2), SO(0), and Sp(0)."},{"title":"Geometric transition of U(N) and SO/Sp(N) theories","content":"In this section we briefly review the type IIB geometric engineering of relevant U (N ) and SO/Sp(N ) theories, and their geometric transition."},{"title":"U (N ) with adjoint and W tree = Tr","content":"The Calabi-Yau geometry relevant to this theory [3] is the non-compact A 1 fibration\nThis fibration has K conifold singularities at the critical points of W (x), i.e. at W ′ (x) = 0. Near each of the singularities, the geometry (3.1) is the same as the usual conifold x ′2 + y 2 + u 2 + v 2 = 0, which is topologically a cone with base S 2 × S 3 . The singularities can be resolved by blowing up a 2-sphere S 2 = P 1 at each singularity. We can realize the U (N ) gauge theory with adjoint matter and superpotential (2.2) in type IIB superstring theory compactified on this resolved geometry, with N D5-branes partially wrapping the K P 1 's. The gauge theory degrees of freedom correspond to the open strings living on these D5-branes. The classical supersymmetric vacuum is obtained by distributing the N i D5-branes over the i-th 2-sphere P 1 i with i = 1, · · · , K. The corresponding breaking pattern of the gauge group is as in (2.1):\nAt low energy, the gauge theory confines (when N i > 1), each U (N i ) factor developing nonzero vev of the glueball superfield S i . In string theory this is described by the geometric transition [1,2,3] in which the resolved conifold geometry with P 1 's wrapped by D5-branes is replaced by a deformed conifold geometry\nwhere f K−1 (x) is a polynomial of degree K − 1 in x and parametrizes the deformation. After the geometric transition, each 2-sphere P 1 i wrapped by N i D5-branes is replaced by a 3-sphere A i with 3-form RR flux through it:\nwhere H = H RR + τ H NS and τ = C (0) + ie −Φ is the complexified coupling constant of type IIB theory. We define the periods of the Calabi-Yau geometry (3.2) by\nwhere Ω is the holomorphic 3-form, B i is the noncompact 3-cycle dual to the 3-cycle A i , and Λ b is a cutoff needed to regulate the divergent B i integrals. The IR cutoff Λ b is to be identified with the UV cutoff of the 4d gauge theory. The set of variables S i measure the size of the blown up 3-spheres, and can be used to parametrize the deformation in place of the K coefficients of the polynomial f K−1 (x).\nThe dual theory after the geometric transition is described by a 4d, N = 1 U (1) K gauge theory, with K U (1) vector superfields V i and K chiral superfields S i . If not for the fluxes, this theory would be N = 2 supersymmetric U (1) K , with (V i , S i ) the N = 2 vector super-multiplets, S i being the Coulomb branch moduli. This low-energy U (1) K theory has a non-trivial prepotential F (S i ) and the dual periods Π i in (3.4) can be written as\nWithout the fluxes, this prepotential can be understood as coming from integrating out D3 branes which wrap the A i cycles, and are charged under the low-energy U (1)'s [25]. The effect of the added fluxes is to break N = 2 supersymmetry to N = 1 by the added superpotential [26,27]\nIn the present U (N ) case, (3.3) and (3.4) gives\nis the 3-form NS flux through the 3-cycle B i and identified with the bare coupling constant of the gauge theory by\nwhere V is the complexified volume of the P 1 's. The N = 1 U (1) K vector multiplets V i remain massless, but the S i now have a superpotential, which fixes them to sit at discrete vacuum expectation values, where they are massive. The fields S i are identified with the glueballs on the gauge theory side. The superpotential W flux (S i ) is the full, exact, effective superpotential in (1.2). As can be verified by explicit calculations [2,3], the leading contribution to (3.7) is always of the form\nwhere Λ i is related to the scale Λ b via precisely the relation (2.10). This leading term (3.10) is the gaugino condensation part of the superpotential, as in (1.3)."},{"title":"SO and Sp theories","content":"The string theory construction of SO/Sp(N ) theories can be obtained from the above U (N ) construction, by orientifolding the geometry before and after the geometric transition by a certain Z 2 action. The geometric construction of SO/Sp(N ) theory with adjoint was discussed in [28,29,20,30], and in that case the invariance of the geometry (3.2) under the Z 2 action requires that the polynomial W (x) be even. The geometric construction of SO/Sp(N ) theory with symmetric/antisymmetric tensor was studied in [31,32,13].\nIn the classical vacuum of the \"parent\" U (2N ) theory, the gauge group is broken into a product of U (N i ) groups. When a U (N i ) factor is identified with another U (N i ) by the Z 2 orientifold action, they lead to a single U (N i ) factor. When a U (N i ) factor is mapped to itself by the Z 2 orientifold action, it becomes an SO(N i ) or Sp( 1 2 N i ), depending on the charge of the orientifold hyperplane. As a result, the classical vacuum of the \"daughter\" SO/Sp(N ) theory has gauge group broken as in (2.1), depending on whether the theory is SO or Sp with adjoint, or SO with symmetric tensor, or Sp with antisymmetric tensor:\nThe 3-form RR fluxes from orientifold hyperplanes makes an additional contribution to the superpotential (3.6), and the flux superpotential can be written as\nis the dual Coxeter number (1.4), while for SO(N i ) it is half 3 the dual Coxeter number (1.4). The 1/2 in (3.13) is because the integration over the A i cycles should be halved due to the Z 2 identification."},{"title":"Relations between SO/Sp theories and U (N ) theories","content":"The result (3.12), with (3.13), gives the exact superpotential of the SO/Sp theories in terms of the same periods S i and Π(S i ) as an auxiliary U theory. This was first noted in [11] at the level of the Konishi anomaly equation as a map between the resolvents of Sp theory with antisymmetric matter and U theory with adjoint matter. In [13], it was generalized to the map between the resolvents of SO/Sp theories with two-index tensor matter and U theory with adjoint matter, and string theory interpretation was discussed. In this subsection we will derive this map from the string theory perspective using the flux superpotential (3.12). Furthermore, we will clarify the relation of the superpotential and the scale of the SO/Sp theories to those of the U theory. The map between resolvents can be derived from these results. For Sp(N ) theory with an antisymmetric tensor, the scale relation was obtained in a different way in [11].\nAs a first example, consider SO(N ) with an adjoint, with the breaking pattern as in (2.1). The geometric transition result (3.12) and (3.13) implies that the exact superpotential is the same as for the U (N − 2) theory with adjoint, with breaking pattern map\n14)\nThe map between the superpotential is\nThe SO(N ) scale matching relation (2.11) is compatible with the map (3.14), since (2.10) for the theory (3.14) reproduces (2.11). Likewise, Sp(N ) with adjoint has the same exact superpotential as for the U (2N + 2) theory with adjoint, with\n(3.16) The Sp(N ) scale matching relations (2.12) follow from the U (N ) matching relations (2.10) with the replacement (3.16), with the understanding that the U (2N + 2) and U (2N 0 + 2) one-instanton factors correspond to the Sp(N ) and Sp(N 0 ) two-instanton factors; this is related to the index of the imbedding mentioned after (2.12), and is accounted for by dividing the U (2N + 2) superpotential by two, as above.\nNext consider Sp(N ) with antisymmetric tensor A and breaking pattern as in (2.1). The geometric transition result (3.12) and (3.13) implies that the exact superpotential is the same as for the U (2N + 2K) theory with adjoint and breaking pattern\nIn the present case, comparing the matching relations (2.14) for Sp(N ) with symmetric tensor with the matching relations (2.10) for U (2N + 2K) with adjoint, for the mapping as in (3.17) requires that the scales of the original Sp(N ) on the LHS of (3.17) and the U (2N + 2K) on the RHS of (3.17) be related as\nThen the Λ i of the unbroken groups on both sides of (3.17) coincide, with the understanding that the U (2N i + 2) one-instanton factors correspond to the Sp(N i ) two-instanton factors, as above. The map between the superpotential is\nNote that, in order for the superpotentials on the two sides of (3.17) to fully coincide, one must compensate the classical mismatch ∆W cl , since each well is occupied by two additional eigenvalues in the theory on the RHS of (3.17). In the string theory geometric transition realization, this constant shift, which is independent of N , Λ, and the glueball fields S i , is most naturally interpreted as an additive shift of the superpotential on the Sp side, which can be regarded as coming from the orientifold planes both before and after the transitions. The classical shift of ∆W cl leads to slightly different operator expectation values (as computed via W eff (g p , Λ) as the generating function) between the Sp and U theory, as was seen in the example of [11]. Also, writing the map as in (3.17), we want the vacuum with confinement index 2 [11]. We could equivalently replace the RHS of (3.17)\n, in which case we would not have to divide by 2 in (3.19).\nLikewise, SO(N ) with symmetric tensor S has exact superpotential related to that of a U (N − 2K) theory with adjoint as\n(3.20)\nComparing the matching relations (2.13) for SO(N ) with symmetric tensor with those of (2.10) for U (N − 2K) with adjoint requires that the scales of the original SO(N ) on the LHS of (3.20) and those of the U (N − 2K) theory on the RHS be related as\nThen the Λ i of the unbroken groups on both sides of (3.20) coincide. Again, in order for the superpotentials on the two sides of (3.20) to fully coincide, one must correct for the classical mismatch ∆W cl coming from the fact that the U (N − 2K) theory has two fewer eigenvalues in each well. In appendix A we will discuss these relations from the matrix model viewpoint. In this context, the relation relevant for (3.14) and (3.16) was conjectured in [19,20] based on explicit diagrammatic calculations, and it was proven for the case of unbroken gauge group N 0 = N in [21]. This will be generalized in Appendix A to all breaking patterns. Likewise, the matrix model relation relevant for (3.17) and (3.20) will be proven in the appendix; this is a generalization of the connection found in [11] for the theories in (3.17) and (3.20)."},{"title":"String theory prescription for low rank","content":"The discussion of the previous section applies for all N i ≥ 0. We now discuss under which circumstances one expects a transition in string theory, where S 3 i 's grows, and therefore an effective glueball field S i should be included in the superpotential. Whether or not there is a geometric transition in string theory is a local question, so each S 3 i can be studied independently. Near any S 3 i the local physics is just a conifold singularity, so we only need to consider the case of a conifold singularity."},{"title":"Physics near a conifold singularity","content":"As we saw, U (N ), SO/Sp(N ) gauge theory can be realized in type IIB theory as the open string theory living on the D5-branes partially wrapped on the exceptional P 1 of a resolved conifold geometry. There is a P 1 associated to each critical point of the polynomial W (x). By the geometric transition duality [2,3], this gauge theory is dual to the closed string theory in the deformed conifold geometry where the P 1 's have been blown down and S 3 's are blown up instead.\nLet us focus on one P 1 with N ≥ 0 D5-branes wrapping it. This corresponds to focusing on one critical point on the gauge theory side. We allow N = 0, which corresponds to an unoccupied critical point. In the neighborhood, the geometry after the geometric transition is approximately a deformed conifold x 2 + y 2 + z 2 + w 2 = µ with a blown up S 3 . The low energy degrees of freedom in the four-dimensional theory are the N = 1 U (1) photon vector superfield V and the N = 1 chiral superfield S. The bosonic component of S is proportional to µ and measures the size of the S 3 .\nFirst, consider the case without fluxes. Then the closed string theory has N = 2 and there is one N = 2 U (1) vector multiplet (V, S). It is known that as the size of the S 3 goes to zero there appears an extra massless degree of freedom [25], which corresponds to the D3-brane wrapping the S 3 . The mass of the wrapped BPS D3-brane is proportional to the area, S, of the S 3 , so the mass becomes zero as the S 3 shrinks to zero, i.e. as S → 0. This extra degree of freedom is described as an N = 2 hypermultiplet charged under the U (1) (of V ). Let us write this hypermultiplet in N = 1 language as (Q,Q), where Q andQ are both N = 1 chiral superfields with opposite U (1) charges. The N = 2 supersymmetry requires the superpotential\nwhich indeed incorporates the above situation that the Q,Q become massless as\nand the F -flatness is\nThe only solution to these is\nwhich just means that S ∼ µ is a modulus. Now let us come back to the case with the fluxes. As reviewed in the last section, the fluxes give rise to a superpotential (3.6) which breaks N = 2 to N = 1 [26,27]. As in (3.12), the local flux superpotential contribution is\nwhere we just keep the leading order term in (3.12), as in (3.10), witĥ\nThe scale Λ is written in terms of the bare coupling Λ b and the coupling constants in the problem, as before, and 2πiα is related to the bare gauge coupling by (3.9). In the following, we discuss the cases withN = 0,N > 0 andN < 0 in order."},{"title":"•N=0 case","content":"In this case, the total superpotential is simply the sum of (4.1) and (4.5):\nThe only solution to the equation of motion is\nThis is consistent with the fact that α is proportional to the volume of the P 1 , and the D3-brane condensation QQ corresponds to the size of the P 1 . Furthermore, since S = 0, the superpotential vanishes: W = 0. Therefore, forN = 0, i.e. for U (0) and SO (2), geometric transition does not take place and we should set the corresponding glueball field S → 0 from the beginning.\n•N>0 case In this case, there is a net RR flux through the A-cycle: A H =N . This means that the D3-brane hypermultiplet (Q,Q) is infinitely massive, because the RR flux will induceN units of fundamental charge on the D3-brane. Since the D3-brane is wrapping a compact space S 3 , the fundamental charge on it should be canceled byN fundamental strings attached to it. Those fundamental strings extend to infinity and thus cost infinite energy 4 . Therefore, we can forget about Q,Q in this case, and the full superpotential is given just by the flux contribution (4.5). The equation of motion gives\nwhich corresponds to the confining vacua of the gauge theory. Note that this case includes U (1) and Sp(0); these theories have a dual confining description. This may sound a little paradoxical, but is related to the fact that the string theory computes not for the standard G(N ) gauge theory but the associated G(N + k|k) higher rank gauge theory, which is confining and differs from standard U (1) and Sp(0) due to residual instanton effects [9].\n•N<0 case In this case, the same argument as theN > 0 case tells us that we should not include the D3-brane fields Q,Q. Hence the superpotential is just the flux part (4.5), which again leads to\n(4.10)\nHowever, now (4.10) is physically unacceptable, since S diverges in the weak coupling limit where the bare volume of\ntaking P 1 large would lead to S 3 also being large, which does not make sense geometrically. The resolution is that S cannot be a good variable: the S 3 does not actually blow up, and S should be set to zero, S → 0, also for this case. Though S is set to zero, the non-zero flux can lead to a non-zero superpotential contribution W flux =N Π(S → 0).\nNote that the above result concerning the sign ofN does not mean the gauge theory prefers D5-branes to anti-D5-branes; it just means that one should choose the sign of the NS flux (i.e. the sign of 2πiα) appropriately. If one wraps the P 1 with anti-D5-branes, one should flip the sign of the NS flux in order to have a blown up S 3 (which can be viewed as a generalization of Seiberg duality to N f = 0)."},{"title":"General prescription","content":"Although we focused on the physics around just one P 1 in the above, the result is applicable to general cases where we have multiple P 1 wrapped with D5-branes, because the geometry near each P 1 is identical to the conifold geometry considered above. Therefore, if we replaceN withN i , all of the above conclusions carry over.\nOnce we have understood the physics, we can forget about the D3-brane hypermultiplet (Q,Q) and state the result as a general prescription for how string theory treats U (0), SO(0), SO (2), and Sp(0), U (1) groups in the geometric dual description:\n• U(0), SO(0), SO (2):\nThere are no glueball variables associated to these gauge groups, so we should take the corresponding S → 0.\n• All other groups, including Sp(0), and U (1):\nWe should consider and extremize the corresponding glueball field S.\nThis prescription should also be applied when using the matrix model [4,36,5] to compute the glueball superpotentials."},{"title":"Examples","content":"Let us scan over all of the examples of (2.1), considering the vacuum where the gauge group is unbroken, and ask when glueball fields S i for the apparently trivial groups in (2.1) should be set to zero, or included and extremized. For the first three cases in (2.1), U (N ), SO(N ), and Sp(N ) with adjoint, the breaking (2.1) is G → G × U (0) K−1 , and the glueball fields S i for the U (0) factors are to be set to zero. This justifies the analysis of these theories in the unbroken vacua in [37,21]. The next case is SO(N ) with a symmetric tensor S, where the vacuum with unbroken gauge group is to be understood as SO(N ) → SO(N ) × SO(0) K−1 , and again the glueball fields S i for the SO(0) factors are set to zero. This eliminates the Veneziano-Yankielowicz part of the superpotential for SO(0), but the −1 unit of flux associated with each SO(0) does contribute to flux termsN i Π i = −Π i in (3.12), even though this does not contain SO(0) glueballs any more.\nThe next case is Sp(N ) with an antisymmetric tensor, where the vacuum with unbroken gauge group is to be understood as Sp(N ) → Sp(N ) × Sp(0) K−1 . Unlike the above cases, here we must keep and extremize the S i for the Sp(0) factors, as will be further discussed shortly.\nFor the next to last example in (2.1), U (N ) with Φ+S + S, the vacuum with unbroken gauge group is to be understood as U (N ) → SO(0) × U (N ) × U (0) K−1 , and the glueball fields S i for SO(0) and U (0) are to be set to zero. Finally, for the last example in (2.1), U (N ) with Φ + A + A, the vacuum with unbroken gauge group is to be understood as\nThough the string engineering of these examples differs somewhat from those discussed in sect. 4 (it was obtained in [30]), the general prescription of sect. 4 is expected to carry over in general: the glueball field S 0 for the Sp(0) factor should be included and extremized, rather than set to zero. On the other hand, the S i for the U (0) factors are set to zero. These latter two theories in (2.1) were considered in [24] and it was noted there that for the case with antisymmetric one expands on the matrix model side around a different vacuum than would be naively expected; this indeed corresponds to keeping and extremizing the glueball field S 0 for the Sp(0) factor, as we have discussed.\nWe now illustrate some other breaking patterns in the examples of (2.1), from the matrix model perspective, for the case of K = 2. We also compare with standard gauge theory results and generally find agreement, even in cases where there was room for disagreement because of the possibility of residual instanton effects along the lines of [9]. As will be discussed in more detail in the following section, the agreement is thanks to a remarkable interplay of different residual instanton contributions."},{"title":"SO/Sp(N ) theory with adjoint","content":"Consider N = 2 SO(2N )/Sp(N ) theory broken to N = 1 by a tree level superpotential for the adjoint chiral superfield Φ:\nIn the SO case, we can skew-diagonalize Φ as\nThe superpotential (5.1) has critical points at λ = 0 and λ = ± m/g. The classical supersymmetric vacuum of the theory is given by distributing 2N 0 of the 2N \"eigenvalues\" λ i at the critical point λ = 0 and N 1 \"eigenvalue\" pairs at λ = ± m/g, with N 0 +N 1 = N . In this vacuum, the gauge group breaks as\nIn the matrix model prescription, the effective superpotential in these vacua is calculated by matrix model as\nwhere F S 2 and F RP 2 are the S 2 and RP 2 contributions, respectively, to the free energy of the associated SO/Sp(N ) matrix model, as defined in Appendix A. The scales Λ 0 , Λ 1 in (5.3) are the energy scales of the low energy SO(2N 0 )/Sp(N 0 ), U (N 1 ) theories with the Φ field integrated out, respectively. They are related to the high energy scale Λ by the matching conditions as in (2.11) and (2.12), which yields\nThe matrix model free energy is computed in Appendix A, and the result is\nThe result (5.5) agrees with the one obtained in [38], where the glueball superpotential was calculated by evaluating the periods (3.6). The full result, (5.3) and (5.4), has the expected general form (3.12):\nThe general prescription in section 3 reads in the present case as follows:\n• SO(2N)/Sp(N) → SO(2N)/Sp(N)×U(0) (unbroken SO/Sp): Set N 1 = 0, S 1 = 0. Then the superpotential is\nThis superpotential coincides with that of U (2N ∓2) with adjoint and breaking pattern U (2N ± 2) → U (2N ∓ 2) × U (0) × U (0), as expected from the map (3.14) or (3.16). As shown in [21] for SO(2N ), this matrix model result agrees with that of standard gauge theory, via using the corresponding Seiberg-Witten curve.\n• SO(2N) → SO(0)×U(N): Set N 0 = 0 and take S 0 → 0, which eliminates the Veneziano-Yankielowicz part for the SO(0). Then the superpotential is\n= 0 because F S 2 does not contain terms with S 1 only (all terms are of the form S n 0 S m 1 with n > 0). And though the Veneziano-Yankielowicz part of the superpotential for SO(0) is eliminated via S 0 → 0, the −1 units of flux associated with SO(0) does make a contribution in (5.8), with the non-zero terms in −Π 0 (S 0 , S 1 )| S 0 =0 in the second line of (5.8) coming from the term 4F RP 2 | S 0 =0 .\n• SO(2N) → SO(2)×U(N−1):\nSet N 0 = 1, S 0 = 0 and remove the Veneziano-Yankielowicz part for the SO(2). Then the superpotential is\nwhere we used\nSet N 0 = 0 in the equation and keep both S 0 and S 1 . Then the superpotential is\nFor various breaking patterns, we integrated out the glueball superfield(s) from the glueball superpotential (5.7)-(5.11), and calculated the low energy superpotential W low as a function of coupling constants m, g, and the scale Λ. Having obtained the actual matrix model results, we can compare to the superpotential as computed via standard gauge theory methods, such as via factorizing of the Seiberg-Witten curve. This method is reviewed in Appendix B, and the results are found to agree with the matrix model results completely. The resulting W low is shown in Table 1.\nThe SO(2N +1) theory with adjoint in the SO(2N +1) → U (1) N vacuum was studied diagrammatically in [39]."},{"title":"Sp(N ) theory with antisymmetric tensor","content":"Consider Sp(N ) theory with an antisymmetric tensor chiral superfield A = −A T . Take cubic tree level superpotential Table 1: The low energy superpotential calculated from the factorization of the Seiberg-Witten curve and from matrix model. In the above, the classical contribution has been subtracted: W low = −N 1 m 2 /4g + W ′ low .\nwhere Φ = AJ, and J is the invariant antisymmetric tensor J = 1 N ⊗ iσ 2 . We do not require A to be traceless, i.e. Tr[Φ] = Tr[AJ] = 0. By a complexified Sp(N ) gauge rotation, Φ can be diagonalized as [40] Φ\nThe superpotential (5.12) has critical points at λ = 0, −m/g. The classical supersymmetric vacuum of the theory is given by distributing N 1 and N 2 \"eigenvalues\" λ i at the critical point λ = 0 and λ = −m/g, respectively, with\nThe glueball superpotential is calculated from the associated Sp(N ) matrix model as\n(5.14)\nThe scales Λ 1 , Λ 2 in (5.3) respectively are the energy scales of the low energy Sp(N 1 ), Sp(N 2 ) theories with the Φ field integrated out. They are related to the high energy scale Λ by the matching conditions as in (2.14), which yields\nThe matrix model free energy is computed in Appendix A, and the result is\n16) where α ≡ g 2 /m 3 . This is as expected from the map (3.17) and [11].\nIn particular, let us concentrate on the unbroken case, N 2 = 0. Unlike [10,15], we do not set S 2 = 0, but rather keep S 2 non-zero and extremize with respect to it, according to our general prescription, to obtain the actual matrix model result. After integrating out S 1 and S 2 from (5.16), we obtain the superpotential as a power series in Λ 1 and Λ 2 as W low = (N + 1)Λ 3 1 + Λ 3 2 + (higher order terms in Λ 1,2 ). (5.17)\nThe matching relation (5.15) gives\nso the terms containing Λ 2 in (5.17) starts to contribute to the superpotential at order (Λ 3 1 ) N+1 , i.e. like Sp(N ) instantons. If we use the relation (5.18) and write out all the terms in (5.17), we obtain\nThus properly accounting for S 2 , it turns out that these matrix model results agree perfectly, up to the order presented, with the standard gauge theory results (Eq. (4.13) of [10]). (The discrepancies found in [10] set in at order Λ"},{"title":"3(N+1) 1","content":", and are cancelled e.g. by (5.18).) In (5.19), for N = 0, 1, there were rather remarkable cancellations between the instanton contributions from Sp(N 1 ) and Sp(N 2 ). This will be further discussed and generalized in the next section.\nThe matrix model prediction for the superpotential of the SO(2N ) theory with symmetric tensor can similarly be obtained by simply changing the N i + 1 in (5.16) to N i − 1. It should be possible to compute the superpotential from gauge theory using the duality for this theory [41]. The result is expected to be compatible with the map (3.20) to the superpotential computed for the U (N − 2K) theory with adjoint."},{"title":"Residual Instantons: String theory (matrix model) versus gauge theory","content":"A remarkable aspect of the string theory (matrix model) computation of the effective superpotential is that (1.3) can be obtained purely in terms of the dynamics of the lowenergy K i=1 G(N i ) theory on the RHS of (1.1). The only information needed about the high-energy G(N ) gauge theory is the perturbative contribution of the G(N )/ i G i (N i ) ghosts to the glueball superpotential (1.3), as discussed in [42], along with the matching relations connecting the scales Λ i of the low-energy G i (N i ) factors to the scale Λ of the highenergy G(N ) theory. This is very different from the conventional description of standard gauge theory, where there can be non-perturbative contributions to W low which are not readily seen in terms of the low-energy theory on the RHS of (1.1). An example of such an effect is instantons in the broken part of the group when π 3 (G(N )/ i G i (N i )) = 0 (see e.g. [43]). Nevertheless, the string theory/ matrix model does properly reproduce such effects, via a low-energy description.\nA gauge theory interpretation for the string theory/ matrix model results was given in [8,9]: the string theory / matrix model results actually refer to a particularly natural UV completion of the original G(N ) theory, where it is embedded in the supergroup G(N +k|k) with k large. This latter theory has a Higgs branch, where k can be reduced successively, eventually Higgsing the theory down to the original G(N ) theory. More generally, the theory with breaking pattern (1.1) is replaced with\nwhich has a Higgs branch flat direction connecting it to (1.1). Consideration of the particular matter content of the G(N + k|k) theories along the Higgs branch, which often has extended supersymmetry, suggests that no dynamically generated superpotential ever lifts this Higgs branch moduli space, i.e. that the superpotentials of these particular theories are always independent of the location of the theory on this Higgs branch [9]. Moving along the Higgs branch has the effect of reducing k, and this expected independence of the superpotential of the position on the Higgs branch fits with the fact that the G(N + k|k) matrix model results are k independent, because all k dependence cancels in the supertraces.\nBecause of the expected independence of the superpotential on the Higgs branch, and because we Higgs back to the original G(N ) theory, in most cases, this \"F-completion\" of the original G(N ) theory into the G(N + k|k) theory is of no consequence. There are, however, a few rare exceptions, where the superpotential of the Higgsed G(N + k|k) theory differs from that of the standard G(N ) theory. This difference comes from residual instantons in G(N + k|k)/G(N ), which need not decouple even if G (N + k|k) is Higgsed to G(N ) far in the UV. As verified in [9], these residual instanton contributions precisely account for the few differences between the string theory (matrix model) results and standard gauge theory, for example the glueball superpotentials, with coefficient h = 1, for U (1) and Sp(0), e.g. with an adjoint and quadratic superpotential.\nIn many cases, however, these residual instanton contributions sum up to yield precisely the result expected from standard gauge theory, including superpotential contributions which in standard gauge theory would not have had a known low-energy description. In particular, residual instanton contributions which could have lead to potential discrepancies with standard gauge theory often completely cancel. The cancellation occurs once one sums over the different terms i in (1.2), upon using the precise matching relation between the low-energy scales Λ i , and the original high-energy scale Λ.\nAs an example, consider U (K) with adjoint matter and breaking pattern U (K) → U (1) K . For K = 1 the string theory (matrix model) description includes a residual instanton effect, yielding W low = Λ 3 L rather than the standard gauge theory answer W sgt = 0 [9]. But for all K > 1 the string theory/ matrix model result is W low = 0, in agreement with the standard gauge theory expectation for U (K) → U (1) K . The result W low = 0 looks like a remarkable cancellation because the glueball superpotential W eff (S 1 , . . . S K ) is quite non-trivial. Nevertheless, upon solving for the S i and plugging back in, the exact result for W low = W eff ( S i ) is zero, as was proven in [44].\nTo illustrate this cancellation, consider the leading order gaugino condensation contribution to W eff (S i ) in the string theory (matrix model) constructions, where the unbroken U (1) factors in U (K) → U (1) K contribute as in (2.15), with h i = 1, unlike in standard gauge theory:\nj =i (a j − a i ) −1 by using (2.10) with all N i = 1. Though this is a non-trivial superpotential, it vanishes upon integrating out the S i :\nThe contour in (6.3) encloses all the zeros of W ′ (x), and we get zero for all K > 1 by pulling the contour off to infinity. We see here why K = 1 is different: we then get a residue at infinity, leading to the low energy superpotential W U(1) = Λ 3 L , as in (2.9), with h U(1) = 1 as in (1.4).\nTo give another example of such a cancellation of residual instanton effects, consider the string theory (matrix model) result for Sp(N ) with an antisymmetric tensor A, with W tree having K critical points, for the case N = 0. For the case of K = 1, the superpotential is just a mass term for A and the low-energy superpotential is the Sp(0) gaugino condensation superpotential, with h(Sp(0)) = 1: W = Λ 3 , unlike standard gauge theory. Again, this can be understood as a residual instanton effect in the F-completion of Sp(N ) to Sp(N + k|k), which is present precisely for the case N = 0 [9]. For a higher order superpotential, K > 1, we would write the breaking pattern as Sp(0) → Sp(0) K . For all K > 1, the residual instanton effects all cancel, precisely as in the U (K) → U (1) K example discussed above; in fact, the two theories have the same effective superpotential W eff (aside from the classical difference), as discussed in [11] and sect. 3.3. Thus, for example, (6.3) can also be interpreted as the leading gaugino condensation contributions from the Sp(0) K factors, and where we now use the matching relation (2.14) to relate the Λ 3 i to g K+1 Λ 2K\nj =i (a j − a i ) −1 . Again, there is complete cancellation in W eff here, except for the case K = 1.\nMore generally, for Sp(N ) with antisymmetric, breaking as Sp(N ) → K i=1 Sp(N i ), the results obtained via the string theory / matrix model glueball potential W eff (S 1 , . . . S K ), upon integrating out the S i , appears to always agree with standard gauge theory results for the superpotential [40], as seen in the examples of [11] and (5.19). This agreement comes about via a remarkable interplay between the different terms i in (3.12). If we treated the scales Λ i of the Sp(N i ) factors as if they were initially independent, each term N i Π( S i )−2πiη i S i in (3.12) would be a complicated function of Λ i , which does not have a known, conventional, interpretation in terms of standard gauge theory for the low-energy Sp(N i ) factor. But upon adding the different i terms and using the matching relations relating Λ i to Λ, e.g. (2.14), one nevertheless obtains the standard gauge theory results, thanks to an intricate interplay between the different terms i.\nBy the map of (3.17) [11], the agreement between string theory / matrix models and standard gauge theory for Sp(N ) with antisymmetric can be phrased as such an agreement for U (N + K) with adjoint and breaking pattern U (N + K) → K i=1 U (N i + 1).\nAs another example, consider U (N ) with adjoint Φ and superpotential having K = N − 1, in the vacuum where U (N ) → U (2) × U (1) N−2 . Factorizing the Seiberg-Witten curve yields for the exact superpotential [46] W exact = W cl (g) ± 2g N Λ N . . The fact that (6.4) will only hold if remarkable cancellations occur upon integrating out S 1 and S 2 from the non-trivial W (S 1 , S 2 ), was discussed in [3], where the cancellations were verified to indeed occur, up to order α 3 . This is checked to one higher order in (5.19), since it is related to Sp(1) → Sp(1) × Sp(0) by the map of [11] and sect. 3.3. The leading order cancellation, say in terms of U (3) → U (2) × U (1), is between U (1) gaugino condensation, Λ 3 2 , and a higher order term coming from integrating out S 1 from W pert (S i ). The residual instanton contributions associated with the UV completion (6.1), as opposed to the standard gauge theory results for (1.1) do not always cancel, however. The cases where we find non-cancellations are when the degree of the superpotential is sufficiently large, so that it contains terms which are not independent moduli. As an example, consider U (1) with W tree as in (2.2) having K minima, breaking U (1) → U (1) × U (0) K−1 . The gaugino condensation contribution to the superpotential, according to the string theory (matrix model) construction, is given by (2.15) with h 1 = 1 and all other h i = 0 and their S i set to zero. Upon integrating out S 1 , we thus obtain the superpotential\nwhere we used the matching relation (2.10) with N 1 = 1 and all other N j = 0. The full low-energy effective superpotential W low (g i , Λ) can be regarded as the generating function for the operator expectation values:\nIn the U (1) theory, we have classical relations u j = 1 j u j 1 . But the quantum contribution (6.5) (along with additional, higher order contributions) imply quantum deformation of these classical relations, due to the residual instanton effects in the U (1 + k|k) → U (1 + k 1 |k 1 ) × U (k 2 |k 2 ) . . . U (k K |k K ) F-completion. For the simplest such example, consider U (1) with W tree = 1 2 mΦ 2 + λΦ. The low-energy superpotential is\nwith the first term the classical contribution and the second the residual instanton. Using (6.6) we then get\nwhich can be regarded as an instanton correction to the composite operator u 2 .\nAs another such example, consider U (2) with an adjoint and W tree having K minima, in the vacuum where the gauge group is broken as U\nFor example, for U (2) with W tree having K = 3 critical points, we break U (2) → U (1) × U (1) × U (0) and (6.9) leads to W gc = g 4 Λ 4 . (6.10)\nComputing expectation values as in (6.6) this leads to\nwhich can be interpreted as an instanton contribution to the composite operator u 4 = TrΦ 4 in U (2) gauge theory. More generally, for U (N ) gauge theory, the independent basis of operators u j = 1 j TrΦ j are only those with j ≤ N , those with j > N can be expressed as products of these basis operators via classical relations. But these relations can be affected by instantons. In particular, for U (N ) with an adjoint, the instanton factor is Λ 2N , so operators u j with j ≥ 2N can be affected. The above residual instanton contributions of the U (N + k|k) UV completion can be interpreted as implying specific such instanton corrections to the higher Casimirs u j .\nA similar situation arises in the N = 1 * U (N ) theory, where the effective superpotential of the matrix model and conventional gauge theory differ by a contribution N 2 m 3 E 2 (N τ ) [17]; this was interpreted in [17] as differing operator definitions of TrΦ 2 between gauge theory and the matrix model at the level of instantons. Related issues for multi-trace operators were seen in [18]."},{"title":"Conclusions","content":"To compute the correct string theory / matrix model results, we should include or not include the glueball fields S i according to the prescription of this paper. Upon doing so, in all examples that we know of, the string theory / matrix model results agree with the results of standard gauge theory, at least in those cases where the relevant gauge theory does not suffer from UV ambiguities. In the case where such ambiguities are present, for example in defining composite operators appearing in W tree , the string theory / matrix model results correspond to a particular UV definition of the theory. The agreement with standard gauge theory results is often due to a remarkable interplay between the different low-energy terms, found upon integrating out the glueball fields S i , and connecting their scales Λ i via the appropriate matching relation to the scale Λ of the original theory. In some cases, this interplay leads to complete cancellations of the residual instanton contributions to W low coming from the G(N + k|k) completion [8,9]. Perhaps there is some additional structure governing the glueball superpotentials, which would make these remarkable cancellations more manifest."},{"title":"A.1. Proof for relation between F S 2 and F RP 2","content":"Here we prove a general relation between the S 2 and RP 2 contributions to the SO(2N )/Sp(N ) matrix model free energy:\nwith symmetric/antisymmetric tensor.\n(A.1) The first equation was conjectured in [19,20] based on explicit diagrammatic calculations, and proven in [21] for the case of unbroken vacua. Here we will give a general matrix model proof for arbitrary breaking pattern. These relations are equivalent to the maps (3.14), (3.16), (3.17), and (3.20), which we obtained in sect. 3.3 immediately from the string theory geometric transition construction, accounting for the orientifold contributions to the fluxes.\nConsider U (N) and SO(2N)/Sp(N) matrix models which correspond to U (N ) and SO(2N )/Sp(N ) gauge theories with a two-index tensor matter field. The partition function is\nWe denote matrix model quantities by boldface letters, following the notation of [15]. Φ is an N×N (for U (N) theory) or 2N×2N (for SO(2N)/Sp(N) theory) matrix corresponding to the Φ field in gauge theory, and the \"action\" W tree is defined in (2.2), (2.4) and (2.5). The matrix integral (A.2) is evaluated perturbatively around the general broken vacua of (2.1), with N i replaced by N i . We take the double scaling limit N i → ∞, g → 0 with\nThe dependence of the free energy F(S i ) on N i are eliminated in favor of S i , and F(S i ) is expanded in the 't Hooft expansion as\nwhere the sum is over all compact topologies M of the matrix model diagrams written in the 't Hooft double-line notation, and χ(M) is the Euler number of M.\nThe matrix model resolvent is defined as follows:\nFor U (N ) theory with adjoint, the expansion parameter is g 2 instead of g, and in particular, R RP 2 (z) ≡ 0. The resolvent and the free energy are related as\nThe resolvents can be determined uniquely by solving the matrix model loop equations (the loop equations for the relevant matrix models are summarized in [15]), under the condition\nA i is the contour around the i-th critical point of W (z). In general, R(z) develops a cut around each critical point in the large N i limit, and A i is taken to encircle the i-th cut. Note that the expression (A.5) should be understood as a Laurent expansion around z = ∞, and converges only if |z| is larger than r such that all the singularities (cuts) of the resolvent are inside the circle C : |z| = r.\nOn the other hand, gauge theory resolvents R(z), T (z) (see e.g. [7]) are determined uniquely by solving the Konishi anomaly equations (the Konishi anomaly equations for the relevant gauge theories are summarized in [15]), under the condition\nAs was shown in [15], the matrix model resolvents R S 2 (z), R RP 2 (z) are related to the gauge theory resolvents R(z), T (z) as\nwith S i and S i identified 5 .\nFirst, consider SO(2N )/Sp(N ) theory with adjoint. The general breaking pattern is\nNote that the eigenvalues are distributed in a symmetric manner under z ↔ −z, and hence (A.7) is It was shown in [13] that the resolvents of this SO(2N )/Sp(N ) theory are related to the resolventsR(z) andT (z) of U (Ñ ≡ 2N ∓ 2) theory with adjoint as follows:\nThe tree level superpotential of the U (Ñ ) theory is related to the one for the SO(2N )/Sp(N ) theory as W U (z) = W SO/Sp (z) (see (2.2) and (2.5)), and the breaking pattern is\nNote that since there is no z ↔ −z symmetry in the U (Ñ ) theory, the U (N −i ) factors that are \"images\" for SO(2N )/Sp(N ) are \"real\" for U (Ñ ). In addition, the glue-ballS of the U (Ñ ) theory is related to the glueball S of the SO(2N )/Sp(N ) theory as\nTherefore, e.g. the first equation in (A.10) is more precisely\nDifferentiating (A.11) with respect to S j , we obtain\nwhere 1 ≤ j ≤ K. Now, using (A.8), let us translate the relation (A.10) among gauge theory resolvents into a relation among matrix model resolvents:\nHereR S 2 is the matrix model resolvent associated with the U (Ñ ) theory. Using (A.12) and the relations R S 2 = R,R S 2 =R (Eq. (A.8)), we obtain\n(A.14)\nBy expanding the resolvents around z = ∞ using (A.5) and comparing the coefficients, we obtain a relation between matrix model free energies:\nwhere j = 0, 2, · · · , 2(K + 1). The j = 0 case is trivially satisfied since S = S 0 + 2 K i=1 S i here, while the j = 2, 4, . . . , 2(K + 1) cases lead to the first equation of (A.1), which we wanted to prove.\nNext, consider SO(2N )/Sp(N ) theory with symmetric/antisymmetric tensor. The breaking pattern is SO (2N\nIt was shown in [11] that the resolvents of this SO/Sp theory is related to the resolventsR(z) andT (z) of U (Ñ ≡ 2N ∓ 2K) theory with adjoint as follows:\nThe tree level superpotential of the U (Ñ ) theory is related to the one for the SO(2N )/Sp(N ) theory as W U (z) = W SO/Sp (z) (see (2.2) and (2.4)), and the breaking pattern is\n. The glueball S i of the U (Ñ ) theory is taken to be the same as the glueball of the SO(2N )/Sp(N ) theory. In (A.16), f K−1 (z) is a polynomial of degree K − 1. Using (A.8), we can translate the relation (A.16) among gauge theory resolvents into a relation among matrix model resolvents:\nIn order to extract the relation between matrix model free energies, let us multiply (A.17) by z j (0 ≤ j ≤ K + 1) and integrate over z along the contour C, introduced under (A.6), which encloses all the cuts around the critical points of W (z). Taking\nthe branching points of the cuts are at z = a ± i . The second term on the right hand side in (A.17) does not contribute to the contour integral unless j = 0:\nwhere w = 1/z, because all the poles w = 1/a ± i are outside of the contour C (on the w-plane). On the contour C, we can use the Laurent expansion (A.5) to evaluate the contribution from the other terms, and the final result is\nThe j = 0 case is trivially satisfied, while the 1 ≤ j ≤ K + 1 cases lead to the second equation of (A.1), which we wanted to prove."},{"title":"A.2. Computation of matrix model free energy: SO(2N )/Sp(N ) theory with adjoint","content":"Let us consider SO(2N) matrix model which corresponds to SO(2N ) gauge theory with adjoint. The tree level superpotential is taken to be quartic (5.1). The matrix variable Φ in (A.2) is a real antisymmetric matrix and can be skew-diagonalized as\n(A.21)\nBy changing the integration variables from Φ to λ i , we obtain\nis the Jacobian for this change of variables [47,20]. The polynomial − m 2 λ 2 + g 4 λ 4 has critical points at λ = 0, ± m/g, around which we would like to do perturbative expansion. For this purpose, we separate λ's into two groups as\nthe loop expansion tells us that f n (N 0 , N 1 ) is a polynomial of degree n + 2. Therefore, by performing the matrix integral by computer for small values of N 0 and N 1 , one can determine the polynomial f n . If we rewrite N 0,1 in favor of S 0 = 2gN 0 and S 1 = gN 1 , the expansion (A.24) arranges itself into the 't Hooft expansion (A.3), from which one can read off F S 2 , F RP 2 , etc. Following the procedure sketched above, we computed the matrix model free energy as\nwhere we defined α ≡ g/m 2 . We also checked explicitly that the relation (A.1) holds. Substituting (A.25) into the DV relation (5.3), we obtain the superpotential (5.5).\nThe Sp(N ) result is obtained similarly, with the result as in (A.1).\nA.3. Computation of matrix model free energy: Sp(N ) theory with antisymmetric tensor Consider the Sp(N) matrix model which corresponds to Sp(N ) gauge theory with an antisymmetric tensor. The superpotential is taken to be quartic (5.12). The matrix variable Φ satisfies Φ = AJ, A T = −A. The \"action\" W tree is given in (5.12). By a complexified Sp(N) gauge rotation, the matrix Φ can be brought to the form [40] Φ ∼ = diag[λ 1 , . . . , λ N ] ⊗ 1 2 , λ i ∈ C. (A.26)\nBy changing the integration variables from Φ to λ i , we obtain\nwhere N i<j (λ i −λ j ) 4 comes from the Jacobian for this change of variables. The polynomial m 2 λ 2 + g 3 λ 3 has two critical points z = 0, − m g , around which we would like to do perturbative expansion. For this purpose, we separate λ's into two groups as The matrix integral can be performed just the same way as for the SO/Sp(N ) theory with adjoint, as described in the last subsection. After substitution S 1,2 = 2gN 1,2 , we obtain\nWe also checked explicitly that the relation (A.1) holds. Substituting (A.29) into (5.14), we obtain the glueball superpotential (5.16)."},{"title":"Appendix B. Gauge theory calculation of superpotential","content":"In this appendix, we compute the exact superpotential of the N = 1 SO(2N )/Sp(N ) theory with adjoint in various vacua by considering factorization of the N = 2 curve. This factorization method was developed in [3] for U (N ), and generalized in [48] to the case with unoccupied critical points (in other words, the n < K case below). Inclusion of fundamentals was considered in [49]. The generalization to SO/Sp gauge group, which discuss below, was given in [29,38,50,51,52,12].\nFirst consider N = 2 SO(2N ) theory broken to N = 1 by the following polynomial tree level superpotential for the adjoint chiral superfield Φ:\nThe classical supersymmetric vacua are obtained by putting 2N 0 eigenvalues of Φ at x = 0 and N i pairs of eigenvalues at x = ±a i , where i = 1, · · · , K and N 0 + K i=1 N i = N . In this vacuum the gauge group breaks as SO(2N ) → SO(2N 0 ) × K i=1 U (N i ). We allow some of N i to vanish, i.e. we allow \"unoccupied\" critical points. Let the number of nonzero N i≥1 be n. Then, the N = 2 curve governing this SO(2N ) theory factorizes as [29,38,50]:\nHere P , H and F are polynomials in x of the subscripted degree, which are invariant under x → −x, i.e. they are actually polynomials in x 2 . This factorization is required to have the appropriate number of independent, massless, monopoles and dyons, which must condense to eliminate some of the low-energy photons. The polynomial F is related to the tree level superpotential as\nwith some polynomial Q 2K−2n (x), f 2K (x) of the subscripted degrees, and W ′ 2K+1 (x) is as in (2.6). Equation (B.3) is for N 0 > 0, and for N 0 = 0 one must use the second equation with Q 2K−2n replaced by Q 2K−2n+2 .\nFor N 0 > 0 we can write the solution of (B.2) in terms of that of the corresponding U (2N − 2) breaking pattern, via:\nThe low energy superpotential is given by\nwhere the u 2j are constrained to satisfy (B.2). Implementing this leads to the result that [51]:\nPlugging back into (B.5) gives W low . Note that the superpotential takes this simple form (B.5) only after one integrates out the monopoles and dyons, whose equation of motion led to the factorization constraint (B.2) [53,54,21]. The Sp(N ) theory can be solved similarly. The N = 2 curve factorizes in the vacuum with breaking pattern Sp(N ) → Sp(N 0 ) × K i=1 U (N i ) as [29,38,50]\n(B.7)\nThe polynomial The Λ 2N+2 shift in (B.8) is an Sp(N ) residual instanton effect, associated with the index of the embedding of the U (N i ) factors in Sp(N ) [55,43] Again, the superpotential is given as in (B.5), subject to the constraint that u j satisfy (B.7). Implementing this, the u j can be obtained from B 2N+2 (x) by [51] Tr\nWe now consider the exact W eff for a few SO/Sp(N ) cases, to illustrate and clarify the general features 6 . We take quartic tree level superpotential\nwhich corresponds to K = 1."},{"title":"B.1. SO(2N ) unbroken","content":"By the map (3.14), this maps to U (2N − 2) unbroken, for which P U(2N−2) (x) = 2Λ 2N−2 T 2N−2 (x/2Λ), with T N (x = 1 2 (t + t −1 )) = 1 2 (t N + t −N ) a Chebyshev polynomial [56]. Thus, using (B.4), P SO(2N) (x) = 2Λ 2N−2 x 2 T 2N−2 (x/2Λ), as found in [21]. This then leads to [21] By the map (3.14), this maps to U (2N −2) → U (0)×U (N −1)×U (N −1). Using (3.14), the multiplication map of [3] for the U (2N − 2) theory leads to a similar multiplication map for the SO(2N ) theory, which was discussed in [51]. Using this, we can construct the solution to the factorization problem for general N in terms of that of say N = 2, i.e. SO(4) → SO(2) × U (1). In this case, equation (B.2) is\n(B.14)\nThe solution to this factorization problem is The solution for general SO(2N ) → SO(2) × U (N − 1), the multiplication map gives the solution to the factorization problem as P 2N (x) = 2x 2 Λ 2N−2 T N−1 ((x 2 − a 2 )/2Λ 2 ), with T N−1 the Chebyshev polynomial defined above. The effect is to rescale u 2 , u 4 , and hence W low by an overall factor of N − 1:\nThis agrees with the result (5.10)."},{"title":"B.3. SO(4) → U (2)","content":"More generally, we could consider the breaking pattern SO(2N ) → U (N ). The map of (3.14) is less useful here, when N 0 = 0, since it suggests mapping to U (2N − 2) → U (−2) × U (N ) × U (N ) and the U (−2) needs to be interpreted. In general, this breaking pattern leads to a complicated W low (Λ). We will here illustrate the case SO(4) → U (2), corresponding to N = 2, N 0 = 0, n = 1, K = 1. Equation (B.2) is\nThe solution to this factorization problem is\nIn the classical Λ → 0 limit, this shows P 4 → (x 2 − a 2 ) 2 , implying the breaking pattern SO(4) → U (2). (B.6) gives\nThis factorization problem is solved by [51]: This solution continuously connects two classically different vacua with breaking pattern Sp(2) → U (2) and Sp(2) → Sp(1) × U (1). Correspondingly there are two ways to take the classical limit: i) Λ → 0 with a fixed, and ii) Λ, a → 0 with w = 2Λ 3 /a 2 fixed. In these limits, P 4 (x) goes to i) (x 2 − a 2 ) 2 and ii) x 2 (x 2 + w 2 ), showing the aforementioned breaking pattern. In the Sp(2) → U (2) case, we solve the second equation of (B.27) with the condition a 2 → −m/g as Λ → 0. The solution is a 2 = − m g + 4g 2 Λ 6 m 2 + 32g 5 Λ 12 m 5 + 448g 8 Λ 18 m 8 + · · · . (B.29)\nFrom (B.28) and (B.5), one obtains the exact superpotential: W low = − m 2 4g + 2m 1/2 g 1/2 Λ 3 + g 2 Λ 6 m − g 7/2 Λ 9 m 5/2 + · · · . (B.32)"},{"title":"i 1","content":""},{"title":"1) i 0 i 1","content":""},{"title":"F 2 (","content":""},{"title":"u 2 B. 4 .","content":""},{"title":"32g 8 Λ 18 m 7 +","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Large N topological string duality [1] embedded in superstrings [2,3] has led to a new perspective on N = 1 supersymmetric gauge theories: that the exact effective superpotential can be efficiently computed by including glueball fields. For example, in a theory with gauge group G, with tree-level superpotential leading to a breaking pattern\\nthe dynamics is efficiently encoded in a superpotential W eff (S 1 , . . . S K ; g j , Λ) (g j are the parameters in W tree and Λ is the dynamical scale). Further, string theory implies [3] W eff (S i ; g j ,\\nwith h i and τ i the fluxes through A i and B i three-cycles in the geometry, as will be reviewed in sect. 3. The prepotential F (S i ) in (1.2) is computable in terms of geometric period integrals, which yields [3] ∂F (S i )\\nwith coefficients c i 1 ...i K depending on the g j (but not on the gauge theory scale Λ). In [4] it was shown how planar diagrams of an associated matrix model can also be used to compute (1.2) and (1.3). Based on the stringy examples, this was generalized in [5] to a more general principle to gain non-perturbative information about the strong coupling dynamics of gauge theories, by extremizing the perturbatively computed glueball superpotential. There are two aspects to the above statements: first that the glueball fields S i are the 'right' variables to describe the IR physics, and second that perturbative gauge theory techniques suffice to compute the glueball superpotential. The latter statement has now been proven in two different approaches for low powers of the glueball fields S i in (1.3) [6,7]. For powers of the glueball fields S i larger than the dual Coxeter number of the group, an ambiguity sets in for the glueball computation of the coefficients c i 1 ...i K in both of these approaches. The matrix model provides a natural prescription for how to resolve this ambiguity, essentially by continuing from large N i . It was argued in [8,9] that the string geometry / matrix model result (since the string geometry and matrix model results are identical, we refer to them synonymously) has the following meaning: it computes the F -terms for different supersymmetric gauge theories, which can be expressed in terms of G(N + k|k) supergroups. The W eff (S i ) is independent of k, and the above ambiguity can be eliminated by taking k, and hence the dual Coxeter number, arbitrarily large. The G(N ) theory of interest is obtained from the G(N + k|k) theory by Higgsing; but there can be residual instanton contributions to W eff [9], which can lead to apparent discrepancies between the matrix model and gauge theory results. We will somewhat clarify here when such residual instanton effects do, or do not, lead to discrepancies with standard gauge theory results.\\nThere is another, more non-trivial assumption in [5]: the statement that the glueball fields S i are the 'right' variables in the IR. This assumption was motivated from the string dualities [1][2][3], where geometric transition provide the explanation of why the glueball fields are the natural IR variables: heuristically, S i corresponds to confinement. However, this is not quite correct: it also applies to abelian theories, as had been noted in [3]. So the deep explanation of why we should choose certain dynamical S variables remains mysterious.\\nIn this paper, we will uncover the precise prescription for the correct choice of IR variables. This will be done from the string theory perspective, by arguing in which cases there is a geometric transition in string theory. For the general breaking pattern (1.1), our prescription for treating the glueball field S i , corresponding to the factor G i in (1.1), is as follows: If h(G i ) > 0 we include S i and extremize W eff (S i ) with respect to it. On the other hand, if h(G i ) ≤ 0 we do not include or extremize S i , instead we just set S i → 0. Here we define the generalized dual-Coxeter numbers 1 h(U (N )) = N, h(Sp(N )) = N + 1, h(SO(N )) = N − 2, (1.4) which are generalized in that (1.4) applies for all N ≥ 0. In particular h(U (1)) = h(Sp(0)) = 1, so when some G i factor in (1.1) is U (1) or Sp(0), our prescription is to include the corresponding S i and extremize with respect to it. On the other hand, h(U (0)) = 0 and h(SO(2)) = 0, so when some G i factor in (1.1) is U (0) or SO(2), our prescription is to just set the corresponding S i = 0 from the outset. (Note that U (1) and SO(2) are treated differently here.) This investigation was motivated by trying to understand the discrepancies found in [10] for Sp(N ) theory with antisymmetric tensor matter, where the superpotentials from the matrix model and gauge theory were found to differ at order h in perturbation theory and beyond. The analysis considered the trivial breaking pattern Sp(N ) → Sp(N ) and a single glueball was introduced corresponding to the single unbroken gauge group factor. In [9], various gauge theories including this example were studied, and an explanation for the discrepancies was proposed in terms of the conjecture, mentioned above, that the string theory / matrix model actually computes the superpotential of the large k G(N + k|k) supergroup theories, rather than the ordinary G(N ) theory. In this context, the trivial breaking pattern considered in [10] should be understood as Sp(N ) → Sp(N ) × Sp(0), which is completed to Sp(N + k|k) → Sp(N + k 1 |k 1 ) × Sp(k 2 |k 2 ). In particular, Sp(0) factors, while trivial in standard gauge theory, are non-trivial in the string theory geometry / matrix model context: there can be a residual instanton contribution to the superpotential when one Higgses Sp(k 2 |k 2 ) down to Sp(0|0) = Sp(0), as explicitly seen in [9] for the case of breaking Sp(0) → Sp(0) with quadratic W tree 2 . Related aspects of \\\"Sp(0)\\\" being non-trivial in the string / matrix model context were subsequently discussed in [11,12,13]. However, it turns out that one also needs to modify the matrix model side of the computation to take into account the Sp(0) factors. This was found by Cachazo [11], who showed that the loop equations determining T (z) ≡ Tr( 1 z−Φ ) and R(z) ≡ − 1 32π 2 Tr( W α W α z−Φ ) for the Sp(N ) theory with antisymmetric tensor matter [14,15] could be related to those of a U (2N + 2K) gauge theory with adjoint matter, with Sp(N ) → Sp(N ) × Sp(0) K−1 mapped to U (2N + 2K) → U (2N + 2) × U (2) K−1 . It was thus shown in [11] that vanishing period of T (z)dz through a given cut, corresponding to an Sp(0) factor, does not imply that the cut closes up on shell (aspects of the periods in this theory were also discussed in [16]). This fits with our above prescription that the Sp(0) glueballs should be included and extremized in the string theory / matrix model picture, as would be done for U (2), rather than set to zero, as was originally done in [10]. We stress that we are not yet even discussing whether or not the string theory / matrix model result agrees with standard gauge theory. Irrespective of any comparison with standard gauge theory, the prescription to obtain the actual string theory / matrix model result is as described above (1.4). Having obtained that result, we can now discuss comparisons with standard gauge theory results. As seen in [11], by solving the U (2N + 2K) loop equations for the present case, this corrected matrix model result now agrees perfectly with standard gauge theory! This will be discussed further here, with all glueball fields S i included.\\nThis agreement, between the matrix model result and standard gauge theory, is in a sense surprising for this particular theory, in light of the Sp(k|k) description of [9] for the unbroken Sp(0) factors, with the resulting residual instanton contributions to the 2 It was suggested in the original version of [9] that such Sp(0) residual instanton contributions could also play a role for the case of cubic and higher order W tree (where they had not yet been fully computed) and could explain the apparent matrix model vs. standard gauge theory discrepancies found in [10]. As we will discuss, we now know that this last speculation was not correct. The corrected proposal of [9] is still that the matrix model computes the superpotential of the G(N + k|k) theory, but where the matrix model side of the computation should be corrected, as we discuss in this paper, to include glueball fields for the Sp(0) factors. superpotential. As we will explain later in this paper, the agreement here between matrix models and standard gauge theory is thanks to a remarkable cancellation of the residual instanton effect terms, which could have spoiled the agreement. The cancellation occurs upon summing over the i in (1.2) from i = 1 . . . K.\\nThere are similar remarkable cancellations of the \\\"residual instanton contributions\\\" to the superpotential in many other examples, which we will also discuss. In fact, in all cases that we know of, the only cases where the residual instantons do not cancel is when the gauge theory clearly has some ambiguity, requiring a choice of how to define the theory in the UV; string theory / matrix model gives a particular such choice. Examples of such cases is when the LHS of (1.1) is itself U (1) or Sp(0) super Yang-Mills, as discussed in [9]. Other examples where the residual instanton contributions do not cancel, is when the superpotential is of high enough order such that not all operators appearing in it are independent, e.g. terms like Tr Φ n , for a U (N ) adjoint Φ, when n > N . In standard gauge theory, there are then potential ambiguities involved in reducing such composite operators to the independent operators, since classical operator identities can receive quantum corrections. The residual instanton contributions, which do not cancel generally in these cases, imply specific quantum relations for these operators, corresponding to the specific UV completion. See [17,18] for related issues.\\nThe organization of this paper is as follows: In section 2 we summarize the gauge theories under consideration. In section 3 we review the type IIB string theory construction of these gauge theories. We also discuss maps of the exact superpotentials of Sp and SO theories to those of U theories, generalizing observations of [19,20,21,11,13]. In section 4 we explain, from the string theory perspective in which cases we have a geometric transition. In section 5 we consider examples, where the glueball fields S i of all group factors are correctly accounted for on the matrix model side. The results thereby obtained via matrix models are found to agree with those of standard gauge theory. In many of these examples, this agreement relies on a remarkable interplay of different residual instanton contributions, which sometimes fully cancel. Residual instantons are discussed further in sect. 6, with examples illustrating cases where they do, or do not, cancel. In appendix A, a proof of a general relation between the S 2 and RP 2 contributions to the matrix model free energy is given, and also the matrix model computation of superpotential is presented. In appendix B the gauge theory computation of the superpotential is discussed.\"\n },\n {\n \"title\": \"The gauge theory examples\",\n \"content\": \"The specific examples of N = 1 supersymmetric gauge theories which we consider, with breaking patterns as in (1.1), are as follows:\\n(2.1) For U (N ) with adjoint Φ, the tree-level superpotential is taken to be\\nwith K potential wells. In the classical vacua, with breaking pattern as in (2.1), Φ has N i eigenvalues equal to the root a i of .2), the factor of 1 2 is for convenience, because the eigenvalues of S or A appear in pairs, and the indices are contracted with δ a b for SO(N ) or J a b for Sp(N ). For SO(N ) and Sp(N ) with adjoint matter, the tree-level superpotential is\\nsince all Casimirs of the adjoint Φ are even, and the 1 2 is again for convenience because the eigenvalues appear in pairs. Φ's eigenvalues sit at the zeros of\\nThe breaking pattern in (2.1) has N 0 eigenvalues of Φ equal to zero, and N i pairs at ±a i , (2)). The next to last example in (2.1) is the N = 2 U (N ) theory with a matter hypermultiplet in the two-index symmetric tensor representation, breaking N = 2 to N = 1 by a superpotential as in (2.2):\\nIn addition to the possibility of Φ's eigenvalues sitting in any of the K critical points W ′ (x) analogous to (2.3), there is a vacuum where N 0 eigenvalues sits at φ = 0, with S S = 0, breaking U (N 0 ) → SO(N 0 ). The last example in (2.1) is the similar theory where the N = 2 hypermultiplet is instead in the antisymmetric tensor representation A, rather than the symmetric tensor S. These last two classes of examples were considered in [22,23,24]. In all of these theories, the low energy superpotential is of the general form\\nis the classical contribution (evaluating W tree in the appropriate minima). W gc (Λ i ) is the gaugino condensation contribution in the unbroken gauge groups of (1.1),\\nwhere h i = C 2 (G i ) is the dual Coxeter number of the group G i in (1.1), with the phase factors associated with the Z 2h i → Z 2 chiral symmetry breaking of the low-energy G i gaugino condensation. The scales Λ i are related to Λ by threshold matching for the fields which got a mass from W tree and the breaking (1.1); some examples, with breaking patterns as in (2.1), are as follows. For U (N ) with adjoint Φ:\\n(2.12) the U (N i ) has index of embedding 2, which is why the U (N i ) one-instanton factor is related to the Sp(N ) two-instanton factor in (2.12). For SO(N ) with symmetric tensor,\\nFor Sp(N ) with antisymmetric A:\\n(2.14)\\nFinally, the term W H (g j , Λ) in (2.8) are additional non-perturbative contributions, which can be regarded as coming from the massive, broken parts of the gauge group.\\nIn the description with the glueballs S i integrated in, as in (1.2), the gaugino condensation contribution comes from the first term in (1.3):\\nand W H (g i , Λ) comes from the last terms in (1.3), upon integrating out the S i . When the minima a i of the superpotential are widely separated, the contributions W H from these last terms are subleading as compared with W gc . As in (1.2), the full glueball superpotential (1.2) can be computed via the string theory geometric transition, in terms of certain period integrals [2,3], as will be reviewed in the next section, or via the matrix models. In that context, the term W gc (S i , Λ i ) comes from the integration measure (as is also natural in field theory, since it incorporates the U (1) R anomaly) and W H (S i , g j ) can be computed perturbatively [4,5]. The perturbative computation of W H (S i , g j ) can also be understood directly in the gauge theory [6], up to ambiguities in terms S n with n > h = C 2 (G). The string theory/ matrix model constructions yield a specific way of resolving these ambiguities, which correspond to a particular UV completion of the gauge theory [8,9]. As discussed in the introduction, our present interest will be in analyzing this circle of ideas when some of the gauge group factors in (1.1) are of low rank, or would naively appear to be trivial, e.g. U (1), U (0), SO(2), SO(0), and Sp(0).\"\n },\n {\n \"title\": \"Geometric transition of U(N) and SO/Sp(N) theories\",\n \"content\": \"In this section we briefly review the type IIB geometric engineering of relevant U (N ) and SO/Sp(N ) theories, and their geometric transition.\"\n },\n {\n \"title\": \"U (N ) with adjoint and W tree = Tr\",\n \"content\": \"The Calabi-Yau geometry relevant to this theory [3] is the non-compact A 1 fibration\\nThis fibration has K conifold singularities at the critical points of W (x), i.e. at W ′ (x) = 0. Near each of the singularities, the geometry (3.1) is the same as the usual conifold x ′2 + y 2 + u 2 + v 2 = 0, which is topologically a cone with base S 2 × S 3 . The singularities can be resolved by blowing up a 2-sphere S 2 = P 1 at each singularity. We can realize the U (N ) gauge theory with adjoint matter and superpotential (2.2) in type IIB superstring theory compactified on this resolved geometry, with N D5-branes partially wrapping the K P 1 's. The gauge theory degrees of freedom correspond to the open strings living on these D5-branes. The classical supersymmetric vacuum is obtained by distributing the N i D5-branes over the i-th 2-sphere P 1 i with i = 1, · · · , K. The corresponding breaking pattern of the gauge group is as in (2.1):\\nAt low energy, the gauge theory confines (when N i > 1), each U (N i ) factor developing nonzero vev of the glueball superfield S i . In string theory this is described by the geometric transition [1,2,3] in which the resolved conifold geometry with P 1 's wrapped by D5-branes is replaced by a deformed conifold geometry\\nwhere f K−1 (x) is a polynomial of degree K − 1 in x and parametrizes the deformation. After the geometric transition, each 2-sphere P 1 i wrapped by N i D5-branes is replaced by a 3-sphere A i with 3-form RR flux through it:\\nwhere H = H RR + τ H NS and τ = C (0) + ie −Φ is the complexified coupling constant of type IIB theory. We define the periods of the Calabi-Yau geometry (3.2) by\\nwhere Ω is the holomorphic 3-form, B i is the noncompact 3-cycle dual to the 3-cycle A i , and Λ b is a cutoff needed to regulate the divergent B i integrals. The IR cutoff Λ b is to be identified with the UV cutoff of the 4d gauge theory. The set of variables S i measure the size of the blown up 3-spheres, and can be used to parametrize the deformation in place of the K coefficients of the polynomial f K−1 (x).\\nThe dual theory after the geometric transition is described by a 4d, N = 1 U (1) K gauge theory, with K U (1) vector superfields V i and K chiral superfields S i . If not for the fluxes, this theory would be N = 2 supersymmetric U (1) K , with (V i , S i ) the N = 2 vector super-multiplets, S i being the Coulomb branch moduli. This low-energy U (1) K theory has a non-trivial prepotential F (S i ) and the dual periods Π i in (3.4) can be written as\\nWithout the fluxes, this prepotential can be understood as coming from integrating out D3 branes which wrap the A i cycles, and are charged under the low-energy U (1)'s [25]. The effect of the added fluxes is to break N = 2 supersymmetry to N = 1 by the added superpotential [26,27]\\nIn the present U (N ) case, (3.3) and (3.4) gives\\nis the 3-form NS flux through the 3-cycle B i and identified with the bare coupling constant of the gauge theory by\\nwhere V is the complexified volume of the P 1 's. The N = 1 U (1) K vector multiplets V i remain massless, but the S i now have a superpotential, which fixes them to sit at discrete vacuum expectation values, where they are massive. The fields S i are identified with the glueballs on the gauge theory side. The superpotential W flux (S i ) is the full, exact, effective superpotential in (1.2). As can be verified by explicit calculations [2,3], the leading contribution to (3.7) is always of the form\\nwhere Λ i is related to the scale Λ b via precisely the relation (2.10). This leading term (3.10) is the gaugino condensation part of the superpotential, as in (1.3).\"\n },\n {\n \"title\": \"SO and Sp theories\",\n \"content\": \"The string theory construction of SO/Sp(N ) theories can be obtained from the above U (N ) construction, by orientifolding the geometry before and after the geometric transition by a certain Z 2 action. The geometric construction of SO/Sp(N ) theory with adjoint was discussed in [28,29,20,30], and in that case the invariance of the geometry (3.2) under the Z 2 action requires that the polynomial W (x) be even. The geometric construction of SO/Sp(N ) theory with symmetric/antisymmetric tensor was studied in [31,32,13].\\nIn the classical vacuum of the \\\"parent\\\" U (2N ) theory, the gauge group is broken into a product of U (N i ) groups. When a U (N i ) factor is identified with another U (N i ) by the Z 2 orientifold action, they lead to a single U (N i ) factor. When a U (N i ) factor is mapped to itself by the Z 2 orientifold action, it becomes an SO(N i ) or Sp( 1 2 N i ), depending on the charge of the orientifold hyperplane. As a result, the classical vacuum of the \\\"daughter\\\" SO/Sp(N ) theory has gauge group broken as in (2.1), depending on whether the theory is SO or Sp with adjoint, or SO with symmetric tensor, or Sp with antisymmetric tensor:\\nThe 3-form RR fluxes from orientifold hyperplanes makes an additional contribution to the superpotential (3.6), and the flux superpotential can be written as\\nis the dual Coxeter number (1.4), while for SO(N i ) it is half 3 the dual Coxeter number (1.4). The 1/2 in (3.13) is because the integration over the A i cycles should be halved due to the Z 2 identification.\"\n },\n {\n \"title\": \"Relations between SO/Sp theories and U (N ) theories\",\n \"content\": \"The result (3.12), with (3.13), gives the exact superpotential of the SO/Sp theories in terms of the same periods S i and Π(S i ) as an auxiliary U theory. This was first noted in [11] at the level of the Konishi anomaly equation as a map between the resolvents of Sp theory with antisymmetric matter and U theory with adjoint matter. In [13], it was generalized to the map between the resolvents of SO/Sp theories with two-index tensor matter and U theory with adjoint matter, and string theory interpretation was discussed. In this subsection we will derive this map from the string theory perspective using the flux superpotential (3.12). Furthermore, we will clarify the relation of the superpotential and the scale of the SO/Sp theories to those of the U theory. The map between resolvents can be derived from these results. For Sp(N ) theory with an antisymmetric tensor, the scale relation was obtained in a different way in [11].\\nAs a first example, consider SO(N ) with an adjoint, with the breaking pattern as in (2.1). The geometric transition result (3.12) and (3.13) implies that the exact superpotential is the same as for the U (N − 2) theory with adjoint, with breaking pattern map\\n14)\\nThe map between the superpotential is\\nThe SO(N ) scale matching relation (2.11) is compatible with the map (3.14), since (2.10) for the theory (3.14) reproduces (2.11). Likewise, Sp(N ) with adjoint has the same exact superpotential as for the U (2N + 2) theory with adjoint, with\\n(3.16) The Sp(N ) scale matching relations (2.12) follow from the U (N ) matching relations (2.10) with the replacement (3.16), with the understanding that the U (2N + 2) and U (2N 0 + 2) one-instanton factors correspond to the Sp(N ) and Sp(N 0 ) two-instanton factors; this is related to the index of the imbedding mentioned after (2.12), and is accounted for by dividing the U (2N + 2) superpotential by two, as above.\\nNext consider Sp(N ) with antisymmetric tensor A and breaking pattern as in (2.1). The geometric transition result (3.12) and (3.13) implies that the exact superpotential is the same as for the U (2N + 2K) theory with adjoint and breaking pattern\\nIn the present case, comparing the matching relations (2.14) for Sp(N ) with symmetric tensor with the matching relations (2.10) for U (2N + 2K) with adjoint, for the mapping as in (3.17) requires that the scales of the original Sp(N ) on the LHS of (3.17) and the U (2N + 2K) on the RHS of (3.17) be related as\\nThen the Λ i of the unbroken groups on both sides of (3.17) coincide, with the understanding that the U (2N i + 2) one-instanton factors correspond to the Sp(N i ) two-instanton factors, as above. The map between the superpotential is\\nNote that, in order for the superpotentials on the two sides of (3.17) to fully coincide, one must compensate the classical mismatch ∆W cl , since each well is occupied by two additional eigenvalues in the theory on the RHS of (3.17). In the string theory geometric transition realization, this constant shift, which is independent of N , Λ, and the glueball fields S i , is most naturally interpreted as an additive shift of the superpotential on the Sp side, which can be regarded as coming from the orientifold planes both before and after the transitions. The classical shift of ∆W cl leads to slightly different operator expectation values (as computed via W eff (g p , Λ) as the generating function) between the Sp and U theory, as was seen in the example of [11]. Also, writing the map as in (3.17), we want the vacuum with confinement index 2 [11]. We could equivalently replace the RHS of (3.17)\\n, in which case we would not have to divide by 2 in (3.19).\\nLikewise, SO(N ) with symmetric tensor S has exact superpotential related to that of a U (N − 2K) theory with adjoint as\\n(3.20)\\nComparing the matching relations (2.13) for SO(N ) with symmetric tensor with those of (2.10) for U (N − 2K) with adjoint requires that the scales of the original SO(N ) on the LHS of (3.20) and those of the U (N − 2K) theory on the RHS be related as\\nThen the Λ i of the unbroken groups on both sides of (3.20) coincide. Again, in order for the superpotentials on the two sides of (3.20) to fully coincide, one must correct for the classical mismatch ∆W cl coming from the fact that the U (N − 2K) theory has two fewer eigenvalues in each well. In appendix A we will discuss these relations from the matrix model viewpoint. In this context, the relation relevant for (3.14) and (3.16) was conjectured in [19,20] based on explicit diagrammatic calculations, and it was proven for the case of unbroken gauge group N 0 = N in [21]. This will be generalized in Appendix A to all breaking patterns. Likewise, the matrix model relation relevant for (3.17) and (3.20) will be proven in the appendix; this is a generalization of the connection found in [11] for the theories in (3.17) and (3.20).\"\n },\n {\n \"title\": \"String theory prescription for low rank\",\n \"content\": \"The discussion of the previous section applies for all N i ≥ 0. We now discuss under which circumstances one expects a transition in string theory, where S 3 i 's grows, and therefore an effective glueball field S i should be included in the superpotential. Whether or not there is a geometric transition in string theory is a local question, so each S 3 i can be studied independently. Near any S 3 i the local physics is just a conifold singularity, so we only need to consider the case of a conifold singularity.\"\n },\n {\n \"title\": \"Physics near a conifold singularity\",\n \"content\": \"As we saw, U (N ), SO/Sp(N ) gauge theory can be realized in type IIB theory as the open string theory living on the D5-branes partially wrapped on the exceptional P 1 of a resolved conifold geometry. There is a P 1 associated to each critical point of the polynomial W (x). By the geometric transition duality [2,3], this gauge theory is dual to the closed string theory in the deformed conifold geometry where the P 1 's have been blown down and S 3 's are blown up instead.\\nLet us focus on one P 1 with N ≥ 0 D5-branes wrapping it. This corresponds to focusing on one critical point on the gauge theory side. We allow N = 0, which corresponds to an unoccupied critical point. In the neighborhood, the geometry after the geometric transition is approximately a deformed conifold x 2 + y 2 + z 2 + w 2 = µ with a blown up S 3 . The low energy degrees of freedom in the four-dimensional theory are the N = 1 U (1) photon vector superfield V and the N = 1 chiral superfield S. The bosonic component of S is proportional to µ and measures the size of the S 3 .\\nFirst, consider the case without fluxes. Then the closed string theory has N = 2 and there is one N = 2 U (1) vector multiplet (V, S). It is known that as the size of the S 3 goes to zero there appears an extra massless degree of freedom [25], which corresponds to the D3-brane wrapping the S 3 . The mass of the wrapped BPS D3-brane is proportional to the area, S, of the S 3 , so the mass becomes zero as the S 3 shrinks to zero, i.e. as S → 0. This extra degree of freedom is described as an N = 2 hypermultiplet charged under the U (1) (of V ). Let us write this hypermultiplet in N = 1 language as (Q,Q), where Q andQ are both N = 1 chiral superfields with opposite U (1) charges. The N = 2 supersymmetry requires the superpotential\\nwhich indeed incorporates the above situation that the Q,Q become massless as\\nand the F -flatness is\\nThe only solution to these is\\nwhich just means that S ∼ µ is a modulus. Now let us come back to the case with the fluxes. As reviewed in the last section, the fluxes give rise to a superpotential (3.6) which breaks N = 2 to N = 1 [26,27]. As in (3.12), the local flux superpotential contribution is\\nwhere we just keep the leading order term in (3.12), as in (3.10), witĥ\\nThe scale Λ is written in terms of the bare coupling Λ b and the coupling constants in the problem, as before, and 2πiα is related to the bare gauge coupling by (3.9). In the following, we discuss the cases withN = 0,N > 0 andN < 0 in order.\"\n },\n {\n \"title\": \"•N=0 case\",\n \"content\": \"In this case, the total superpotential is simply the sum of (4.1) and (4.5):\\nThe only solution to the equation of motion is\\nThis is consistent with the fact that α is proportional to the volume of the P 1 , and the D3-brane condensation QQ corresponds to the size of the P 1 . Furthermore, since S = 0, the superpotential vanishes: W = 0. Therefore, forN = 0, i.e. for U (0) and SO (2), geometric transition does not take place and we should set the corresponding glueball field S → 0 from the beginning.\\n•N>0 case In this case, there is a net RR flux through the A-cycle: A H =N . This means that the D3-brane hypermultiplet (Q,Q) is infinitely massive, because the RR flux will induceN units of fundamental charge on the D3-brane. Since the D3-brane is wrapping a compact space S 3 , the fundamental charge on it should be canceled byN fundamental strings attached to it. Those fundamental strings extend to infinity and thus cost infinite energy 4 . Therefore, we can forget about Q,Q in this case, and the full superpotential is given just by the flux contribution (4.5). The equation of motion gives\\nwhich corresponds to the confining vacua of the gauge theory. Note that this case includes U (1) and Sp(0); these theories have a dual confining description. This may sound a little paradoxical, but is related to the fact that the string theory computes not for the standard G(N ) gauge theory but the associated G(N + k|k) higher rank gauge theory, which is confining and differs from standard U (1) and Sp(0) due to residual instanton effects [9].\\n•N<0 case In this case, the same argument as theN > 0 case tells us that we should not include the D3-brane fields Q,Q. Hence the superpotential is just the flux part (4.5), which again leads to\\n(4.10)\\nHowever, now (4.10) is physically unacceptable, since S diverges in the weak coupling limit where the bare volume of\\ntaking P 1 large would lead to S 3 also being large, which does not make sense geometrically. The resolution is that S cannot be a good variable: the S 3 does not actually blow up, and S should be set to zero, S → 0, also for this case. Though S is set to zero, the non-zero flux can lead to a non-zero superpotential contribution W flux =N Π(S → 0).\\nNote that the above result concerning the sign ofN does not mean the gauge theory prefers D5-branes to anti-D5-branes; it just means that one should choose the sign of the NS flux (i.e. the sign of 2πiα) appropriately. If one wraps the P 1 with anti-D5-branes, one should flip the sign of the NS flux in order to have a blown up S 3 (which can be viewed as a generalization of Seiberg duality to N f = 0).\"\n },\n {\n \"title\": \"General prescription\",\n \"content\": \"Although we focused on the physics around just one P 1 in the above, the result is applicable to general cases where we have multiple P 1 wrapped with D5-branes, because the geometry near each P 1 is identical to the conifold geometry considered above. Therefore, if we replaceN withN i , all of the above conclusions carry over.\\nOnce we have understood the physics, we can forget about the D3-brane hypermultiplet (Q,Q) and state the result as a general prescription for how string theory treats U (0), SO(0), SO (2), and Sp(0), U (1) groups in the geometric dual description:\\n• U(0), SO(0), SO (2):\\nThere are no glueball variables associated to these gauge groups, so we should take the corresponding S → 0.\\n• All other groups, including Sp(0), and U (1):\\nWe should consider and extremize the corresponding glueball field S.\\nThis prescription should also be applied when using the matrix model [4,36,5] to compute the glueball superpotentials.\"\n },\n {\n \"title\": \"Examples\",\n \"content\": \"Let us scan over all of the examples of (2.1), considering the vacuum where the gauge group is unbroken, and ask when glueball fields S i for the apparently trivial groups in (2.1) should be set to zero, or included and extremized. For the first three cases in (2.1), U (N ), SO(N ), and Sp(N ) with adjoint, the breaking (2.1) is G → G × U (0) K−1 , and the glueball fields S i for the U (0) factors are to be set to zero. This justifies the analysis of these theories in the unbroken vacua in [37,21]. The next case is SO(N ) with a symmetric tensor S, where the vacuum with unbroken gauge group is to be understood as SO(N ) → SO(N ) × SO(0) K−1 , and again the glueball fields S i for the SO(0) factors are set to zero. This eliminates the Veneziano-Yankielowicz part of the superpotential for SO(0), but the −1 unit of flux associated with each SO(0) does contribute to flux termsN i Π i = −Π i in (3.12), even though this does not contain SO(0) glueballs any more.\\nThe next case is Sp(N ) with an antisymmetric tensor, where the vacuum with unbroken gauge group is to be understood as Sp(N ) → Sp(N ) × Sp(0) K−1 . Unlike the above cases, here we must keep and extremize the S i for the Sp(0) factors, as will be further discussed shortly.\\nFor the next to last example in (2.1), U (N ) with Φ+S + S, the vacuum with unbroken gauge group is to be understood as U (N ) → SO(0) × U (N ) × U (0) K−1 , and the glueball fields S i for SO(0) and U (0) are to be set to zero. Finally, for the last example in (2.1), U (N ) with Φ + A + A, the vacuum with unbroken gauge group is to be understood as\\nThough the string engineering of these examples differs somewhat from those discussed in sect. 4 (it was obtained in [30]), the general prescription of sect. 4 is expected to carry over in general: the glueball field S 0 for the Sp(0) factor should be included and extremized, rather than set to zero. On the other hand, the S i for the U (0) factors are set to zero. These latter two theories in (2.1) were considered in [24] and it was noted there that for the case with antisymmetric one expands on the matrix model side around a different vacuum than would be naively expected; this indeed corresponds to keeping and extremizing the glueball field S 0 for the Sp(0) factor, as we have discussed.\\nWe now illustrate some other breaking patterns in the examples of (2.1), from the matrix model perspective, for the case of K = 2. We also compare with standard gauge theory results and generally find agreement, even in cases where there was room for disagreement because of the possibility of residual instanton effects along the lines of [9]. As will be discussed in more detail in the following section, the agreement is thanks to a remarkable interplay of different residual instanton contributions.\"\n },\n {\n \"title\": \"SO/Sp(N ) theory with adjoint\",\n \"content\": \"Consider N = 2 SO(2N )/Sp(N ) theory broken to N = 1 by a tree level superpotential for the adjoint chiral superfield Φ:\\nIn the SO case, we can skew-diagonalize Φ as\\nThe superpotential (5.1) has critical points at λ = 0 and λ = ± m/g. The classical supersymmetric vacuum of the theory is given by distributing 2N 0 of the 2N \\\"eigenvalues\\\" λ i at the critical point λ = 0 and N 1 \\\"eigenvalue\\\" pairs at λ = ± m/g, with N 0 +N 1 = N . In this vacuum, the gauge group breaks as\\nIn the matrix model prescription, the effective superpotential in these vacua is calculated by matrix model as\\nwhere F S 2 and F RP 2 are the S 2 and RP 2 contributions, respectively, to the free energy of the associated SO/Sp(N ) matrix model, as defined in Appendix A. The scales Λ 0 , Λ 1 in (5.3) are the energy scales of the low energy SO(2N 0 )/Sp(N 0 ), U (N 1 ) theories with the Φ field integrated out, respectively. They are related to the high energy scale Λ by the matching conditions as in (2.11) and (2.12), which yields\\nThe matrix model free energy is computed in Appendix A, and the result is\\nThe result (5.5) agrees with the one obtained in [38], where the glueball superpotential was calculated by evaluating the periods (3.6). The full result, (5.3) and (5.4), has the expected general form (3.12):\\nThe general prescription in section 3 reads in the present case as follows:\\n• SO(2N)/Sp(N) → SO(2N)/Sp(N)×U(0) (unbroken SO/Sp): Set N 1 = 0, S 1 = 0. Then the superpotential is\\nThis superpotential coincides with that of U (2N ∓2) with adjoint and breaking pattern U (2N ± 2) → U (2N ∓ 2) × U (0) × U (0), as expected from the map (3.14) or (3.16). As shown in [21] for SO(2N ), this matrix model result agrees with that of standard gauge theory, via using the corresponding Seiberg-Witten curve.\\n• SO(2N) → SO(0)×U(N): Set N 0 = 0 and take S 0 → 0, which eliminates the Veneziano-Yankielowicz part for the SO(0). Then the superpotential is\\n= 0 because F S 2 does not contain terms with S 1 only (all terms are of the form S n 0 S m 1 with n > 0). And though the Veneziano-Yankielowicz part of the superpotential for SO(0) is eliminated via S 0 → 0, the −1 units of flux associated with SO(0) does make a contribution in (5.8), with the non-zero terms in −Π 0 (S 0 , S 1 )| S 0 =0 in the second line of (5.8) coming from the term 4F RP 2 | S 0 =0 .\\n• SO(2N) → SO(2)×U(N−1):\\nSet N 0 = 1, S 0 = 0 and remove the Veneziano-Yankielowicz part for the SO(2). Then the superpotential is\\nwhere we used\\nSet N 0 = 0 in the equation and keep both S 0 and S 1 . Then the superpotential is\\nFor various breaking patterns, we integrated out the glueball superfield(s) from the glueball superpotential (5.7)-(5.11), and calculated the low energy superpotential W low as a function of coupling constants m, g, and the scale Λ. Having obtained the actual matrix model results, we can compare to the superpotential as computed via standard gauge theory methods, such as via factorizing of the Seiberg-Witten curve. This method is reviewed in Appendix B, and the results are found to agree with the matrix model results completely. The resulting W low is shown in Table 1.\\nThe SO(2N +1) theory with adjoint in the SO(2N +1) → U (1) N vacuum was studied diagrammatically in [39].\"\n },\n {\n \"title\": \"Sp(N ) theory with antisymmetric tensor\",\n \"content\": \"Consider Sp(N ) theory with an antisymmetric tensor chiral superfield A = −A T . Take cubic tree level superpotential Table 1: The low energy superpotential calculated from the factorization of the Seiberg-Witten curve and from matrix model. In the above, the classical contribution has been subtracted: W low = −N 1 m 2 /4g + W ′ low .\\nwhere Φ = AJ, and J is the invariant antisymmetric tensor J = 1 N ⊗ iσ 2 . We do not require A to be traceless, i.e. Tr[Φ] = Tr[AJ] = 0. By a complexified Sp(N ) gauge rotation, Φ can be diagonalized as [40] Φ\\nThe superpotential (5.12) has critical points at λ = 0, −m/g. The classical supersymmetric vacuum of the theory is given by distributing N 1 and N 2 \\\"eigenvalues\\\" λ i at the critical point λ = 0 and λ = −m/g, respectively, with\\nThe glueball superpotential is calculated from the associated Sp(N ) matrix model as\\n(5.14)\\nThe scales Λ 1 , Λ 2 in (5.3) respectively are the energy scales of the low energy Sp(N 1 ), Sp(N 2 ) theories with the Φ field integrated out. They are related to the high energy scale Λ by the matching conditions as in (2.14), which yields\\nThe matrix model free energy is computed in Appendix A, and the result is\\n16) where α ≡ g 2 /m 3 . This is as expected from the map (3.17) and [11].\\nIn particular, let us concentrate on the unbroken case, N 2 = 0. Unlike [10,15], we do not set S 2 = 0, but rather keep S 2 non-zero and extremize with respect to it, according to our general prescription, to obtain the actual matrix model result. After integrating out S 1 and S 2 from (5.16), we obtain the superpotential as a power series in Λ 1 and Λ 2 as W low = (N + 1)Λ 3 1 + Λ 3 2 + (higher order terms in Λ 1,2 ). (5.17)\\nThe matching relation (5.15) gives\\nso the terms containing Λ 2 in (5.17) starts to contribute to the superpotential at order (Λ 3 1 ) N+1 , i.e. like Sp(N ) instantons. If we use the relation (5.18) and write out all the terms in (5.17), we obtain\\nThus properly accounting for S 2 , it turns out that these matrix model results agree perfectly, up to the order presented, with the standard gauge theory results (Eq. (4.13) of [10]). (The discrepancies found in [10] set in at order Λ\"\n },\n {\n \"title\": \"3(N+1) 1\",\n \"content\": \", and are cancelled e.g. by (5.18).) In (5.19), for N = 0, 1, there were rather remarkable cancellations between the instanton contributions from Sp(N 1 ) and Sp(N 2 ). This will be further discussed and generalized in the next section.\\nThe matrix model prediction for the superpotential of the SO(2N ) theory with symmetric tensor can similarly be obtained by simply changing the N i + 1 in (5.16) to N i − 1. It should be possible to compute the superpotential from gauge theory using the duality for this theory [41]. The result is expected to be compatible with the map (3.20) to the superpotential computed for the U (N − 2K) theory with adjoint.\"\n },\n {\n \"title\": \"Residual Instantons: String theory (matrix model) versus gauge theory\",\n \"content\": \"A remarkable aspect of the string theory (matrix model) computation of the effective superpotential is that (1.3) can be obtained purely in terms of the dynamics of the lowenergy K i=1 G(N i ) theory on the RHS of (1.1). The only information needed about the high-energy G(N ) gauge theory is the perturbative contribution of the G(N )/ i G i (N i ) ghosts to the glueball superpotential (1.3), as discussed in [42], along with the matching relations connecting the scales Λ i of the low-energy G i (N i ) factors to the scale Λ of the highenergy G(N ) theory. This is very different from the conventional description of standard gauge theory, where there can be non-perturbative contributions to W low which are not readily seen in terms of the low-energy theory on the RHS of (1.1). An example of such an effect is instantons in the broken part of the group when π 3 (G(N )/ i G i (N i )) = 0 (see e.g. [43]). Nevertheless, the string theory/ matrix model does properly reproduce such effects, via a low-energy description.\\nA gauge theory interpretation for the string theory/ matrix model results was given in [8,9]: the string theory / matrix model results actually refer to a particularly natural UV completion of the original G(N ) theory, where it is embedded in the supergroup G(N +k|k) with k large. This latter theory has a Higgs branch, where k can be reduced successively, eventually Higgsing the theory down to the original G(N ) theory. More generally, the theory with breaking pattern (1.1) is replaced with\\nwhich has a Higgs branch flat direction connecting it to (1.1). Consideration of the particular matter content of the G(N + k|k) theories along the Higgs branch, which often has extended supersymmetry, suggests that no dynamically generated superpotential ever lifts this Higgs branch moduli space, i.e. that the superpotentials of these particular theories are always independent of the location of the theory on this Higgs branch [9]. Moving along the Higgs branch has the effect of reducing k, and this expected independence of the superpotential of the position on the Higgs branch fits with the fact that the G(N + k|k) matrix model results are k independent, because all k dependence cancels in the supertraces.\\nBecause of the expected independence of the superpotential on the Higgs branch, and because we Higgs back to the original G(N ) theory, in most cases, this \\\"F-completion\\\" of the original G(N ) theory into the G(N + k|k) theory is of no consequence. There are, however, a few rare exceptions, where the superpotential of the Higgsed G(N + k|k) theory differs from that of the standard G(N ) theory. This difference comes from residual instantons in G(N + k|k)/G(N ), which need not decouple even if G (N + k|k) is Higgsed to G(N ) far in the UV. As verified in [9], these residual instanton contributions precisely account for the few differences between the string theory (matrix model) results and standard gauge theory, for example the glueball superpotentials, with coefficient h = 1, for U (1) and Sp(0), e.g. with an adjoint and quadratic superpotential.\\nIn many cases, however, these residual instanton contributions sum up to yield precisely the result expected from standard gauge theory, including superpotential contributions which in standard gauge theory would not have had a known low-energy description. In particular, residual instanton contributions which could have lead to potential discrepancies with standard gauge theory often completely cancel. The cancellation occurs once one sums over the different terms i in (1.2), upon using the precise matching relation between the low-energy scales Λ i , and the original high-energy scale Λ.\\nAs an example, consider U (K) with adjoint matter and breaking pattern U (K) → U (1) K . For K = 1 the string theory (matrix model) description includes a residual instanton effect, yielding W low = Λ 3 L rather than the standard gauge theory answer W sgt = 0 [9]. But for all K > 1 the string theory/ matrix model result is W low = 0, in agreement with the standard gauge theory expectation for U (K) → U (1) K . The result W low = 0 looks like a remarkable cancellation because the glueball superpotential W eff (S 1 , . . . S K ) is quite non-trivial. Nevertheless, upon solving for the S i and plugging back in, the exact result for W low = W eff ( S i ) is zero, as was proven in [44].\\nTo illustrate this cancellation, consider the leading order gaugino condensation contribution to W eff (S i ) in the string theory (matrix model) constructions, where the unbroken U (1) factors in U (K) → U (1) K contribute as in (2.15), with h i = 1, unlike in standard gauge theory:\\nj =i (a j − a i ) −1 by using (2.10) with all N i = 1. Though this is a non-trivial superpotential, it vanishes upon integrating out the S i :\\nThe contour in (6.3) encloses all the zeros of W ′ (x), and we get zero for all K > 1 by pulling the contour off to infinity. We see here why K = 1 is different: we then get a residue at infinity, leading to the low energy superpotential W U(1) = Λ 3 L , as in (2.9), with h U(1) = 1 as in (1.4).\\nTo give another example of such a cancellation of residual instanton effects, consider the string theory (matrix model) result for Sp(N ) with an antisymmetric tensor A, with W tree having K critical points, for the case N = 0. For the case of K = 1, the superpotential is just a mass term for A and the low-energy superpotential is the Sp(0) gaugino condensation superpotential, with h(Sp(0)) = 1: W = Λ 3 , unlike standard gauge theory. Again, this can be understood as a residual instanton effect in the F-completion of Sp(N ) to Sp(N + k|k), which is present precisely for the case N = 0 [9]. For a higher order superpotential, K > 1, we would write the breaking pattern as Sp(0) → Sp(0) K . For all K > 1, the residual instanton effects all cancel, precisely as in the U (K) → U (1) K example discussed above; in fact, the two theories have the same effective superpotential W eff (aside from the classical difference), as discussed in [11] and sect. 3.3. Thus, for example, (6.3) can also be interpreted as the leading gaugino condensation contributions from the Sp(0) K factors, and where we now use the matching relation (2.14) to relate the Λ 3 i to g K+1 Λ 2K\\nj =i (a j − a i ) −1 . Again, there is complete cancellation in W eff here, except for the case K = 1.\\nMore generally, for Sp(N ) with antisymmetric, breaking as Sp(N ) → K i=1 Sp(N i ), the results obtained via the string theory / matrix model glueball potential W eff (S 1 , . . . S K ), upon integrating out the S i , appears to always agree with standard gauge theory results for the superpotential [40], as seen in the examples of [11] and (5.19). This agreement comes about via a remarkable interplay between the different terms i in (3.12). If we treated the scales Λ i of the Sp(N i ) factors as if they were initially independent, each term N i Π( S i )−2πiη i S i in (3.12) would be a complicated function of Λ i , which does not have a known, conventional, interpretation in terms of standard gauge theory for the low-energy Sp(N i ) factor. But upon adding the different i terms and using the matching relations relating Λ i to Λ, e.g. (2.14), one nevertheless obtains the standard gauge theory results, thanks to an intricate interplay between the different terms i.\\nBy the map of (3.17) [11], the agreement between string theory / matrix models and standard gauge theory for Sp(N ) with antisymmetric can be phrased as such an agreement for U (N + K) with adjoint and breaking pattern U (N + K) → K i=1 U (N i + 1).\\nAs another example, consider U (N ) with adjoint Φ and superpotential having K = N − 1, in the vacuum where U (N ) → U (2) × U (1) N−2 . Factorizing the Seiberg-Witten curve yields for the exact superpotential [46] W exact = W cl (g) ± 2g N Λ N . . The fact that (6.4) will only hold if remarkable cancellations occur upon integrating out S 1 and S 2 from the non-trivial W (S 1 , S 2 ), was discussed in [3], where the cancellations were verified to indeed occur, up to order α 3 . This is checked to one higher order in (5.19), since it is related to Sp(1) → Sp(1) × Sp(0) by the map of [11] and sect. 3.3. The leading order cancellation, say in terms of U (3) → U (2) × U (1), is between U (1) gaugino condensation, Λ 3 2 , and a higher order term coming from integrating out S 1 from W pert (S i ). The residual instanton contributions associated with the UV completion (6.1), as opposed to the standard gauge theory results for (1.1) do not always cancel, however. The cases where we find non-cancellations are when the degree of the superpotential is sufficiently large, so that it contains terms which are not independent moduli. As an example, consider U (1) with W tree as in (2.2) having K minima, breaking U (1) → U (1) × U (0) K−1 . The gaugino condensation contribution to the superpotential, according to the string theory (matrix model) construction, is given by (2.15) with h 1 = 1 and all other h i = 0 and their S i set to zero. Upon integrating out S 1 , we thus obtain the superpotential\\nwhere we used the matching relation (2.10) with N 1 = 1 and all other N j = 0. The full low-energy effective superpotential W low (g i , Λ) can be regarded as the generating function for the operator expectation values:\\nIn the U (1) theory, we have classical relations u j = 1 j u j 1 . But the quantum contribution (6.5) (along with additional, higher order contributions) imply quantum deformation of these classical relations, due to the residual instanton effects in the U (1 + k|k) → U (1 + k 1 |k 1 ) × U (k 2 |k 2 ) . . . U (k K |k K ) F-completion. For the simplest such example, consider U (1) with W tree = 1 2 mΦ 2 + λΦ. The low-energy superpotential is\\nwith the first term the classical contribution and the second the residual instanton. Using (6.6) we then get\\nwhich can be regarded as an instanton correction to the composite operator u 2 .\\nAs another such example, consider U (2) with an adjoint and W tree having K minima, in the vacuum where the gauge group is broken as U\\nFor example, for U (2) with W tree having K = 3 critical points, we break U (2) → U (1) × U (1) × U (0) and (6.9) leads to W gc = g 4 Λ 4 . (6.10)\\nComputing expectation values as in (6.6) this leads to\\nwhich can be interpreted as an instanton contribution to the composite operator u 4 = TrΦ 4 in U (2) gauge theory. More generally, for U (N ) gauge theory, the independent basis of operators u j = 1 j TrΦ j are only those with j ≤ N , those with j > N can be expressed as products of these basis operators via classical relations. But these relations can be affected by instantons. In particular, for U (N ) with an adjoint, the instanton factor is Λ 2N , so operators u j with j ≥ 2N can be affected. The above residual instanton contributions of the U (N + k|k) UV completion can be interpreted as implying specific such instanton corrections to the higher Casimirs u j .\\nA similar situation arises in the N = 1 * U (N ) theory, where the effective superpotential of the matrix model and conventional gauge theory differ by a contribution N 2 m 3 E 2 (N τ ) [17]; this was interpreted in [17] as differing operator definitions of TrΦ 2 between gauge theory and the matrix model at the level of instantons. Related issues for multi-trace operators were seen in [18].\"\n },\n {\n \"title\": \"Conclusions\",\n \"content\": \"To compute the correct string theory / matrix model results, we should include or not include the glueball fields S i according to the prescription of this paper. Upon doing so, in all examples that we know of, the string theory / matrix model results agree with the results of standard gauge theory, at least in those cases where the relevant gauge theory does not suffer from UV ambiguities. In the case where such ambiguities are present, for example in defining composite operators appearing in W tree , the string theory / matrix model results correspond to a particular UV definition of the theory. The agreement with standard gauge theory results is often due to a remarkable interplay between the different low-energy terms, found upon integrating out the glueball fields S i , and connecting their scales Λ i via the appropriate matching relation to the scale Λ of the original theory. In some cases, this interplay leads to complete cancellations of the residual instanton contributions to W low coming from the G(N + k|k) completion [8,9]. Perhaps there is some additional structure governing the glueball superpotentials, which would make these remarkable cancellations more manifest.\"\n },\n {\n \"title\": \"A.1. Proof for relation between F S 2 and F RP 2\",\n \"content\": \"Here we prove a general relation between the S 2 and RP 2 contributions to the SO(2N )/Sp(N ) matrix model free energy:\\nwith symmetric/antisymmetric tensor.\\n(A.1) The first equation was conjectured in [19,20] based on explicit diagrammatic calculations, and proven in [21] for the case of unbroken vacua. Here we will give a general matrix model proof for arbitrary breaking pattern. These relations are equivalent to the maps (3.14), (3.16), (3.17), and (3.20), which we obtained in sect. 3.3 immediately from the string theory geometric transition construction, accounting for the orientifold contributions to the fluxes.\\nConsider U (N) and SO(2N)/Sp(N) matrix models which correspond to U (N ) and SO(2N )/Sp(N ) gauge theories with a two-index tensor matter field. The partition function is\\nWe denote matrix model quantities by boldface letters, following the notation of [15]. Φ is an N×N (for U (N) theory) or 2N×2N (for SO(2N)/Sp(N) theory) matrix corresponding to the Φ field in gauge theory, and the \\\"action\\\" W tree is defined in (2.2), (2.4) and (2.5). The matrix integral (A.2) is evaluated perturbatively around the general broken vacua of (2.1), with N i replaced by N i . We take the double scaling limit N i → ∞, g → 0 with\\nThe dependence of the free energy F(S i ) on N i are eliminated in favor of S i , and F(S i ) is expanded in the 't Hooft expansion as\\nwhere the sum is over all compact topologies M of the matrix model diagrams written in the 't Hooft double-line notation, and χ(M) is the Euler number of M.\\nThe matrix model resolvent is defined as follows:\\nFor U (N ) theory with adjoint, the expansion parameter is g 2 instead of g, and in particular, R RP 2 (z) ≡ 0. The resolvent and the free energy are related as\\nThe resolvents can be determined uniquely by solving the matrix model loop equations (the loop equations for the relevant matrix models are summarized in [15]), under the condition\\nA i is the contour around the i-th critical point of W (z). In general, R(z) develops a cut around each critical point in the large N i limit, and A i is taken to encircle the i-th cut. Note that the expression (A.5) should be understood as a Laurent expansion around z = ∞, and converges only if |z| is larger than r such that all the singularities (cuts) of the resolvent are inside the circle C : |z| = r.\\nOn the other hand, gauge theory resolvents R(z), T (z) (see e.g. [7]) are determined uniquely by solving the Konishi anomaly equations (the Konishi anomaly equations for the relevant gauge theories are summarized in [15]), under the condition\\nAs was shown in [15], the matrix model resolvents R S 2 (z), R RP 2 (z) are related to the gauge theory resolvents R(z), T (z) as\\nwith S i and S i identified 5 .\\nFirst, consider SO(2N )/Sp(N ) theory with adjoint. The general breaking pattern is\\nNote that the eigenvalues are distributed in a symmetric manner under z ↔ −z, and hence (A.7) is It was shown in [13] that the resolvents of this SO(2N )/Sp(N ) theory are related to the resolventsR(z) andT (z) of U (Ñ ≡ 2N ∓ 2) theory with adjoint as follows:\\nThe tree level superpotential of the U (Ñ ) theory is related to the one for the SO(2N )/Sp(N ) theory as W U (z) = W SO/Sp (z) (see (2.2) and (2.5)), and the breaking pattern is\\nNote that since there is no z ↔ −z symmetry in the U (Ñ ) theory, the U (N −i ) factors that are \\\"images\\\" for SO(2N )/Sp(N ) are \\\"real\\\" for U (Ñ ). In addition, the glue-ballS of the U (Ñ ) theory is related to the glueball S of the SO(2N )/Sp(N ) theory as\\nTherefore, e.g. the first equation in (A.10) is more precisely\\nDifferentiating (A.11) with respect to S j , we obtain\\nwhere 1 ≤ j ≤ K. Now, using (A.8), let us translate the relation (A.10) among gauge theory resolvents into a relation among matrix model resolvents:\\nHereR S 2 is the matrix model resolvent associated with the U (Ñ ) theory. Using (A.12) and the relations R S 2 = R,R S 2 =R (Eq. (A.8)), we obtain\\n(A.14)\\nBy expanding the resolvents around z = ∞ using (A.5) and comparing the coefficients, we obtain a relation between matrix model free energies:\\nwhere j = 0, 2, · · · , 2(K + 1). The j = 0 case is trivially satisfied since S = S 0 + 2 K i=1 S i here, while the j = 2, 4, . . . , 2(K + 1) cases lead to the first equation of (A.1), which we wanted to prove.\\nNext, consider SO(2N )/Sp(N ) theory with symmetric/antisymmetric tensor. The breaking pattern is SO (2N\\nIt was shown in [11] that the resolvents of this SO/Sp theory is related to the resolventsR(z) andT (z) of U (Ñ ≡ 2N ∓ 2K) theory with adjoint as follows:\\nThe tree level superpotential of the U (Ñ ) theory is related to the one for the SO(2N )/Sp(N ) theory as W U (z) = W SO/Sp (z) (see (2.2) and (2.4)), and the breaking pattern is\\n. The glueball S i of the U (Ñ ) theory is taken to be the same as the glueball of the SO(2N )/Sp(N ) theory. In (A.16), f K−1 (z) is a polynomial of degree K − 1. Using (A.8), we can translate the relation (A.16) among gauge theory resolvents into a relation among matrix model resolvents:\\nIn order to extract the relation between matrix model free energies, let us multiply (A.17) by z j (0 ≤ j ≤ K + 1) and integrate over z along the contour C, introduced under (A.6), which encloses all the cuts around the critical points of W (z). Taking\\nthe branching points of the cuts are at z = a ± i . The second term on the right hand side in (A.17) does not contribute to the contour integral unless j = 0:\\nwhere w = 1/z, because all the poles w = 1/a ± i are outside of the contour C (on the w-plane). On the contour C, we can use the Laurent expansion (A.5) to evaluate the contribution from the other terms, and the final result is\\nThe j = 0 case is trivially satisfied, while the 1 ≤ j ≤ K + 1 cases lead to the second equation of (A.1), which we wanted to prove.\"\n },\n {\n \"title\": \"A.2. Computation of matrix model free energy: SO(2N )/Sp(N ) theory with adjoint\",\n \"content\": \"Let us consider SO(2N) matrix model which corresponds to SO(2N ) gauge theory with adjoint. The tree level superpotential is taken to be quartic (5.1). The matrix variable Φ in (A.2) is a real antisymmetric matrix and can be skew-diagonalized as\\n(A.21)\\nBy changing the integration variables from Φ to λ i , we obtain\\nis the Jacobian for this change of variables [47,20]. The polynomial − m 2 λ 2 + g 4 λ 4 has critical points at λ = 0, ± m/g, around which we would like to do perturbative expansion. For this purpose, we separate λ's into two groups as\\nthe loop expansion tells us that f n (N 0 , N 1 ) is a polynomial of degree n + 2. Therefore, by performing the matrix integral by computer for small values of N 0 and N 1 , one can determine the polynomial f n . If we rewrite N 0,1 in favor of S 0 = 2gN 0 and S 1 = gN 1 , the expansion (A.24) arranges itself into the 't Hooft expansion (A.3), from which one can read off F S 2 , F RP 2 , etc. Following the procedure sketched above, we computed the matrix model free energy as\\nwhere we defined α ≡ g/m 2 . We also checked explicitly that the relation (A.1) holds. Substituting (A.25) into the DV relation (5.3), we obtain the superpotential (5.5).\\nThe Sp(N ) result is obtained similarly, with the result as in (A.1).\\nA.3. Computation of matrix model free energy: Sp(N ) theory with antisymmetric tensor Consider the Sp(N) matrix model which corresponds to Sp(N ) gauge theory with an antisymmetric tensor. The superpotential is taken to be quartic (5.12). The matrix variable Φ satisfies Φ = AJ, A T = −A. The \\\"action\\\" W tree is given in (5.12). By a complexified Sp(N) gauge rotation, the matrix Φ can be brought to the form [40] Φ ∼ = diag[λ 1 , . . . , λ N ] ⊗ 1 2 , λ i ∈ C. (A.26)\\nBy changing the integration variables from Φ to λ i , we obtain\\nwhere N i<j (λ i −λ j ) 4 comes from the Jacobian for this change of variables. The polynomial m 2 λ 2 + g 3 λ 3 has two critical points z = 0, − m g , around which we would like to do perturbative expansion. For this purpose, we separate λ's into two groups as The matrix integral can be performed just the same way as for the SO/Sp(N ) theory with adjoint, as described in the last subsection. After substitution S 1,2 = 2gN 1,2 , we obtain\\nWe also checked explicitly that the relation (A.1) holds. Substituting (A.29) into (5.14), we obtain the glueball superpotential (5.16).\"\n },\n {\n \"title\": \"Appendix B. Gauge theory calculation of superpotential\",\n \"content\": \"In this appendix, we compute the exact superpotential of the N = 1 SO(2N )/Sp(N ) theory with adjoint in various vacua by considering factorization of the N = 2 curve. This factorization method was developed in [3] for U (N ), and generalized in [48] to the case with unoccupied critical points (in other words, the n < K case below). Inclusion of fundamentals was considered in [49]. The generalization to SO/Sp gauge group, which discuss below, was given in [29,38,50,51,52,12].\\nFirst consider N = 2 SO(2N ) theory broken to N = 1 by the following polynomial tree level superpotential for the adjoint chiral superfield Φ:\\nThe classical supersymmetric vacua are obtained by putting 2N 0 eigenvalues of Φ at x = 0 and N i pairs of eigenvalues at x = ±a i , where i = 1, · · · , K and N 0 + K i=1 N i = N . In this vacuum the gauge group breaks as SO(2N ) → SO(2N 0 ) × K i=1 U (N i ). We allow some of N i to vanish, i.e. we allow \\\"unoccupied\\\" critical points. Let the number of nonzero N i≥1 be n. Then, the N = 2 curve governing this SO(2N ) theory factorizes as [29,38,50]:\\nHere P , H and F are polynomials in x of the subscripted degree, which are invariant under x → −x, i.e. they are actually polynomials in x 2 . This factorization is required to have the appropriate number of independent, massless, monopoles and dyons, which must condense to eliminate some of the low-energy photons. The polynomial F is related to the tree level superpotential as\\nwith some polynomial Q 2K−2n (x), f 2K (x) of the subscripted degrees, and W ′ 2K+1 (x) is as in (2.6). Equation (B.3) is for N 0 > 0, and for N 0 = 0 one must use the second equation with Q 2K−2n replaced by Q 2K−2n+2 .\\nFor N 0 > 0 we can write the solution of (B.2) in terms of that of the corresponding U (2N − 2) breaking pattern, via:\\nThe low energy superpotential is given by\\nwhere the u 2j are constrained to satisfy (B.2). Implementing this leads to the result that [51]:\\nPlugging back into (B.5) gives W low . Note that the superpotential takes this simple form (B.5) only after one integrates out the monopoles and dyons, whose equation of motion led to the factorization constraint (B.2) [53,54,21]. The Sp(N ) theory can be solved similarly. The N = 2 curve factorizes in the vacuum with breaking pattern Sp(N ) → Sp(N 0 ) × K i=1 U (N i ) as [29,38,50]\\n(B.7)\\nThe polynomial The Λ 2N+2 shift in (B.8) is an Sp(N ) residual instanton effect, associated with the index of the embedding of the U (N i ) factors in Sp(N ) [55,43] Again, the superpotential is given as in (B.5), subject to the constraint that u j satisfy (B.7). Implementing this, the u j can be obtained from B 2N+2 (x) by [51] Tr\\nWe now consider the exact W eff for a few SO/Sp(N ) cases, to illustrate and clarify the general features 6 . We take quartic tree level superpotential\\nwhich corresponds to K = 1.\"\n },\n {\n \"title\": \"B.1. SO(2N ) unbroken\",\n \"content\": \"By the map (3.14), this maps to U (2N − 2) unbroken, for which P U(2N−2) (x) = 2Λ 2N−2 T 2N−2 (x/2Λ), with T N (x = 1 2 (t + t −1 )) = 1 2 (t N + t −N ) a Chebyshev polynomial [56]. Thus, using (B.4), P SO(2N) (x) = 2Λ 2N−2 x 2 T 2N−2 (x/2Λ), as found in [21]. This then leads to [21] By the map (3.14), this maps to U (2N −2) → U (0)×U (N −1)×U (N −1). Using (3.14), the multiplication map of [3] for the U (2N − 2) theory leads to a similar multiplication map for the SO(2N ) theory, which was discussed in [51]. Using this, we can construct the solution to the factorization problem for general N in terms of that of say N = 2, i.e. SO(4) → SO(2) × U (1). In this case, equation (B.2) is\\n(B.14)\\nThe solution to this factorization problem is The solution for general SO(2N ) → SO(2) × U (N − 1), the multiplication map gives the solution to the factorization problem as P 2N (x) = 2x 2 Λ 2N−2 T N−1 ((x 2 − a 2 )/2Λ 2 ), with T N−1 the Chebyshev polynomial defined above. The effect is to rescale u 2 , u 4 , and hence W low by an overall factor of N − 1:\\nThis agrees with the result (5.10).\"\n },\n {\n \"title\": \"B.3. SO(4) → U (2)\",\n \"content\": \"More generally, we could consider the breaking pattern SO(2N ) → U (N ). The map of (3.14) is less useful here, when N 0 = 0, since it suggests mapping to U (2N − 2) → U (−2) × U (N ) × U (N ) and the U (−2) needs to be interpreted. In general, this breaking pattern leads to a complicated W low (Λ). We will here illustrate the case SO(4) → U (2), corresponding to N = 2, N 0 = 0, n = 1, K = 1. Equation (B.2) is\\nThe solution to this factorization problem is\\nIn the classical Λ → 0 limit, this shows P 4 → (x 2 − a 2 ) 2 , implying the breaking pattern SO(4) → U (2). (B.6) gives\\nThis factorization problem is solved by [51]: This solution continuously connects two classically different vacua with breaking pattern Sp(2) → U (2) and Sp(2) → Sp(1) × U (1). Correspondingly there are two ways to take the classical limit: i) Λ → 0 with a fixed, and ii) Λ, a → 0 with w = 2Λ 3 /a 2 fixed. In these limits, P 4 (x) goes to i) (x 2 − a 2 ) 2 and ii) x 2 (x 2 + w 2 ), showing the aforementioned breaking pattern. In the Sp(2) → U (2) case, we solve the second equation of (B.27) with the condition a 2 → −m/g as Λ → 0. The solution is a 2 = − m g + 4g 2 Λ 6 m 2 + 32g 5 Λ 12 m 5 + 448g 8 Λ 18 m 8 + · · · . (B.29)\\nFrom (B.28) and (B.5), one obtains the exact superpotential: W low = − m 2 4g + 2m 1/2 g 1/2 Λ 3 + g 2 Λ 6 m − g 7/2 Λ 9 m 5/2 + · · · . (B.32)\"\n },\n {\n \"title\": \"i 1\",\n \"content\": \"\"\n },\n {\n \"title\": \"1) i 0 i 1\",\n \"content\": \"\"\n },\n {\n \"title\": \"F 2 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"u 2 B. 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"32g 8 Λ 18 m 7 +\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"of [11] and sect. 3.3 relates this to Sp(1) → Sp(1) × Sp(0) N−2 , where the exact gauge theory result agrees with (6.4), up to the classical shift, upon using the relation (3.18). A priori, one might expect the string theory / matrix model result to disagree with (6.4), due to residual instanton contributions from the U (1) N−2 or the Sp(0) N−2 in U (N ) → U (2) × U (1) N−2 and Sp(1) → Sp(1) × Sp(0) N−2 respectively. But the string theory / matrix model result nevertheless agrees with (6.4), thanks to the interplay between the different terms. Consider, in particular, the case U (3) → U (2) × U (1)","content":"of [11] and sect. 3.3 relates this to Sp(1) → Sp(1) × Sp(0) N−2 , where the exact gauge theory result agrees with (6.4), up to the classical shift, upon using the relation (3.18). A priori, one might expect the string theory / matrix model result to disagree with (6.4), due to residual instanton contributions from the U (1) N−2 or the Sp(0) N−2 in U (N ) → U (2) × U (1) N−2 and Sp(1) → Sp(1) × Sp(0) N−2 respectively. But the string theory / matrix model result nevertheless agrees with (6.4), thanks to the interplay between the different terms. Consider, in particular, the case U (3) → U (2) × U (1)"},{"id":"","caption":"= 1, . . . , K. The contours A i and A −i encircle counterclockwise the cuts around z = a i and z = −a i , respectively. The relation (A.8) holds as it is, with the summation understood as over SO(2N 0 )/Sp(N 0 ) and U (N i ), i = 1, . . . , K.","content":"= 1, . . . , K. The contours A i and A −i encircle counterclockwise the cuts around z = a i and z = −a i , respectively. The relation (A.8) holds as it is, with the summation understood as over SO(2N 0 )/Sp(N 0 ) and U (N i ), i = 1, . . . , K."},{"id":"","caption":"= 1, . . . , N 1 , (A.23) with N 0 + N 1 = N, corresponding to the classical supersymmetric vacuum with breaking pattern SO(2N) → SO(2N 0 ) × U (N 1 ). We would like to evaluate the matrix integral (A.22) perturbatively around λ (0,1) = 0. If we expand the matrix model free energy in the coupling constant g as","content":"i 1\n1= 1, . . . , N 1 , (A.23) with N 0 + N 1 = N, corresponding to the classical supersymmetric vacuum with breaking pattern SO(2N) → SO(2N 0 ) × U (N 1 ). We would like to evaluate the matrix integral (A.22) perturbatively around λ (0,1) = 0. If we expand the matrix model free energy in the coupling constant g as"},{"id":"","caption":"= 1, . . . , N 1 , −m/g + λ (2) i 1 i 2 = 1, . . . , N 2 , (A.28) with N 1 + N 2 = N. This corresponds to the classical supersymmetric vacuum with breaking pattern Sp(N) → Sp(N 1 ) × Sp(N 2 ).","content":"1) i 0 i 1\n01= 1, . . . , N 1 , −m/g + λ (2) i 1 i 2 = 1, . . . , N 2 , (A.28) with N 1 + N 2 = N. This corresponds to the classical supersymmetric vacuum with breaking pattern Sp(N) → Sp(N 1 ) × Sp(N 2 )."},{"id":"","caption":"2n+1) (x) is related to W ′ (x) by (B.3). The mapping of the Sp(N ) theory to a U (2N + 2) theory, as in (3.16), can be written as a solution of (B.7) in terms of solutions of the corresponding U (2N + 2) factorization problem:B 2n+2 (x) ≡ x 2 P Sp(N) 2N (x) + 2Λ 2N+2 = P U(2N+2) 2N+2 (x).(B.8)","content":"F 2 (\n22n+1) (x) is related to W ′ (x) by (B.3). The mapping of the Sp(N ) theory to a U (2N + 2) theory, as in (3.16), can be written as a solution of (B.7) in terms of solutions of the corresponding U (2N + 2) factorization problem:B 2n+2 (x) ≡ x 2 P Sp(N) 2N (x) + 2Λ 2N+2 = P U(2N+2) 2N+2 (x).(B.8)"},{"id":"","caption":"low-energy superpotential is W low = 1 2 (m u 2 + g u 4 ):W low = (N − 1) mΛ 2 + 3 2 gΛ 4 . (B.13) B.2. SO(2N ) → SO(2) × U (N − 1)","content":"low-energy superpotential is W low = 1 2 (m u 2 + g u 4 ):W low = (N − 1) mΛ 2 + 3 2 gΛ 4 . (B.13) B.2. SO(2N ) → SO(2) × U (N − 1)"},{"id":"","caption":"we can see the breaking pattern SO(4) → SO(2) × U (1). Using (B.","content":"we can see the breaking pattern SO(4) → SO(2) × U (1). Using (B."},{"id":"","caption":"= 2(a 2 − Λ 2 ), u 4 = (a 4 − 2Λ 2 ) 2 − Λ 4 . (B.22) Sp(2) → U (2), Sp(1) × U (1)This corresponds to N = 2, n = 1, K = 1. Equations (B.7) and (B.3) are","content":"u 2 B. 4 .\n24= 2(a 2 − Λ 2 ), u 4 = (a 4 − 2Λ 2 ) 2 − Λ 4 . (B.22) Sp(2) → U (2), Sp(1) × U (1)This corresponds to N = 2, n = 1, K = 1. Equations (B.7) and (B.3) are"},{"id":"","caption":"· · · . (B.30) In the Sp(2) → Sp(1) × U (1) case, we solve the second equation of (B.27) with the condition w 2 → m/g as Λ → 0. B.28) and (B.5), one obtains the exact superpotential:","content":"32g 8 Λ 18 m 7 +\n7· · · . (B.30) In the Sp(2) → Sp(1) × U (1) case, we solve the second equation of (B.27) with the condition w 2 → m/g as Λ → 0. B.28) and (B.5), one obtains the exact superpotential:"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"of [11] and sect. 3.3 relates this to Sp(1) → Sp(1) × Sp(0) N−2 , where the exact gauge theory result agrees with (6.4), up to the classical shift, upon using the relation (3.18). A priori, one might expect the string theory / matrix model result to disagree with (6.4), due to residual instanton contributions from the U (1) N−2 or the Sp(0) N−2 in U (N ) → U (2) × U (1) N−2 and Sp(1) → Sp(1) × Sp(0) N−2 respectively. But the string theory / matrix model result nevertheless agrees with (6.4), thanks to the interplay between the different terms. Consider, in particular, the case U (3) → U (2) × U (1)\",\n \"content\": \"of [11] and sect. 3.3 relates this to Sp(1) → Sp(1) × Sp(0) N−2 , where the exact gauge theory result agrees with (6.4), up to the classical shift, upon using the relation (3.18). A priori, one might expect the string theory / matrix model result to disagree with (6.4), due to residual instanton contributions from the U (1) N−2 or the Sp(0) N−2 in U (N ) → U (2) × U (1) N−2 and Sp(1) → Sp(1) × Sp(0) N−2 respectively. But the string theory / matrix model result nevertheless agrees with (6.4), thanks to the interplay between the different terms. Consider, in particular, the case U (3) → U (2) × U (1)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"= 1, . . . , K. The contours A i and A −i encircle counterclockwise the cuts around z = a i and z = −a i , respectively. The relation (A.8) holds as it is, with the summation understood as over SO(2N 0 )/Sp(N 0 ) and U (N i ), i = 1, . . . , K.\",\n \"content\": \"= 1, . . . , K. The contours A i and A −i encircle counterclockwise the cuts around z = a i and z = −a i , respectively. The relation (A.8) holds as it is, with the summation understood as over SO(2N 0 )/Sp(N 0 ) and U (N i ), i = 1, . . . , K.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"= 1, . . . , N 1 , (A.23) with N 0 + N 1 = N, corresponding to the classical supersymmetric vacuum with breaking pattern SO(2N) → SO(2N 0 ) × U (N 1 ). We would like to evaluate the matrix integral (A.22) perturbatively around λ (0,1) = 0. If we expand the matrix model free energy in the coupling constant g as\",\n \"content\": \"i 1\\n1= 1, . . . , N 1 , (A.23) with N 0 + N 1 = N, corresponding to the classical supersymmetric vacuum with breaking pattern SO(2N) → SO(2N 0 ) × U (N 1 ). We would like to evaluate the matrix integral (A.22) perturbatively around λ (0,1) = 0. If we expand the matrix model free energy in the coupling constant g as\"\n },\n {\n \"id\": \"\",\n \"caption\": \"= 1, . . . , N 1 , −m/g + λ (2) i 1 i 2 = 1, . . . , N 2 , (A.28) with N 1 + N 2 = N. This corresponds to the classical supersymmetric vacuum with breaking pattern Sp(N) → Sp(N 1 ) × Sp(N 2 ).\",\n \"content\": \"1) i 0 i 1\\n01= 1, . . . , N 1 , −m/g + λ (2) i 1 i 2 = 1, . . . , N 2 , (A.28) with N 1 + N 2 = N. This corresponds to the classical supersymmetric vacuum with breaking pattern Sp(N) → Sp(N 1 ) × Sp(N 2 ).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"2n+1) (x) is related to W ′ (x) by (B.3). The mapping of the Sp(N ) theory to a U (2N + 2) theory, as in (3.16), can be written as a solution of (B.7) in terms of solutions of the corresponding U (2N + 2) factorization problem:B 2n+2 (x) ≡ x 2 P Sp(N) 2N (x) + 2Λ 2N+2 = P U(2N+2) 2N+2 (x).(B.8)\",\n \"content\": \"F 2 (\\n22n+1) (x) is related to W ′ (x) by (B.3). The mapping of the Sp(N ) theory to a U (2N + 2) theory, as in (3.16), can be written as a solution of (B.7) in terms of solutions of the corresponding U (2N + 2) factorization problem:B 2n+2 (x) ≡ x 2 P Sp(N) 2N (x) + 2Λ 2N+2 = P U(2N+2) 2N+2 (x).(B.8)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"low-energy superpotential is W low = 1 2 (m u 2 + g u 4 ):W low = (N − 1) mΛ 2 + 3 2 gΛ 4 . (B.13) B.2. SO(2N ) → SO(2) × U (N − 1)\",\n \"content\": \"low-energy superpotential is W low = 1 2 (m u 2 + g u 4 ):W low = (N − 1) mΛ 2 + 3 2 gΛ 4 . (B.13) B.2. SO(2N ) → SO(2) × U (N − 1)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"we can see the breaking pattern SO(4) → SO(2) × U (1). Using (B.\",\n \"content\": \"we can see the breaking pattern SO(4) → SO(2) × U (1). Using (B.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"= 2(a 2 − Λ 2 ), u 4 = (a 4 − 2Λ 2 ) 2 − Λ 4 . (B.22) Sp(2) → U (2), Sp(1) × U (1)This corresponds to N = 2, n = 1, K = 1. Equations (B.7) and (B.3) are\",\n \"content\": \"u 2 B. 4 .\\n24= 2(a 2 − Λ 2 ), u 4 = (a 4 − 2Λ 2 ) 2 − Λ 4 . (B.22) Sp(2) → U (2), Sp(1) × U (1)This corresponds to N = 2, n = 1, K = 1. Equations (B.7) and (B.3) are\"\n },\n {\n \"id\": \"\",\n \"caption\": \"· · · . (B.30) In the Sp(2) → Sp(1) × U (1) case, we solve the second equation of (B.27) with the condition w 2 → m/g as Λ → 0. B.28) and (B.5), one obtains the exact superpotential:\",\n \"content\": \"32g 8 Λ 18 m 7 +\\n7· · · . (B.30) In the Sp(2) → Sp(1) × U (1) case, we solve the second equation of (B.27) with the condition w 2 → m/g as Λ → 0. B.28) and (B.5), one obtains the exact superpotential:\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"On the gauge theory/geometry correspondence. R Gopakumar, C Vafa, arXiv:hep-th/9811131Adv. Theor. Math. Phys. 31415R. Gopakumar and C. Vafa, \"On the gauge theory/geometry correspondence,\" Adv. Theor. Math. Phys. 3, 1415 (1999) [arXiv:hep-th/9811131].","doi":"arXiv:hep-th/9811131"},{"text":"Superstrings and topological strings at large N. C Vafa, arXiv:hep-th/0008142J. Math. Phys. 422798C. Vafa, \"Superstrings and topological strings at large N,\" J. Math. Phys. 42, 2798 (2001) [arXiv:hep-th/0008142].","doi":"arXiv:hep-th/0008142"},{"text":"A large N duality via a geometric transition. F Cachazo, K A Intriligator, C Vafa, arXiv:hep-th/0103067Nucl. Phys. B. 6033F. Cachazo, K. A. Intriligator and C. 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Intriligator, \\\"New RG fixed points and duality in supersymmetric SP(N(c)) and SO(N(c)) gauge theories,\\\" Nucl. Phys. B 448, 187 (1995) [arXiv:hep-th/9505051].\",\n \"doi\": \"arXiv:hep-th/9505051\"\n },\n {\n \"text\": \"Perturbative analysis of gauged matrix models. R Dijkgraaf, S Gukov, V A Kazakov, C Vafa, arXiv:hep-th/0210238R. Dijkgraaf, S. Gukov, V. A. Kazakov and C. Vafa, \\\"Perturbative analysis of gauged matrix models,\\\" arXiv:hep-th/0210238.\",\n \"doi\": \"arXiv:hep-th/0210238\"\n },\n {\n \"text\": \"Instantons in partially broken gauge groups. C Csaki, H Murayama, arXiv:hep-th/9804061Nucl. Phys. B. 532498C. Csaki and H. Murayama, \\\"Instantons in partially broken gauge groups,\\\" Nucl. Phys. B 532, 498 (1998) [arXiv:hep-th/9804061].\",\n \"doi\": \"arXiv:hep-th/9804061\"\n },\n {\n \"text\": \"N = 1 and N = 2 geometry from fluxes. F Cachazo, C Vafa, arXiv:hep-th/0206017F. Cachazo and C. Vafa,\\\"N = 1 and N = 2 geometry from fluxes,\\\" arXiv:hep- th/0206017.\",\n \"doi\": \"arXiv:hep-th/0206017\"\n },\n {\n \"text\": \"Exact results and duality for Sp(2N) SUSY gauge theories with an antisymmetric tensor. C Csaki, W Skiba, M Schmaltz, arXiv:hep-th/9607210Nucl. Phys. B. 487128C. Csaki, W. Skiba and M. Schmaltz, \\\"Exact results and duality for Sp(2N) SUSY gauge theories with an antisymmetric tensor,\\\" Nucl. Phys. B 487, 128 (1997)[arXiv:hep-th/9607210].\",\n \"doi\": \"arXiv:hep-th/9607210\"\n },\n {\n \"text\": \"Massless Monopoles Via Confining Phase Superpotentials. S Elitzur, A Forge, A Giveon, K A Intriligator, E Rabinovici, arXiv:hep-th/9603051Phys. Lett. B. 379121S. Elitzur, A. Forge, A. Giveon, K. A. Intriligator and E. Rabinovici, \\\"Massless Monopoles Via Confining Phase Superpotentials,\\\" Phys. Lett. B 379, 121 (1996) [arXiv:hep-th/9603051].\",\n \"doi\": \"arXiv:hep-th/9603051\"\n },\n {\n \"text\": \"Random matrices. M L Mehta, Academic PressM. L. Mehta, \\\"Random matrices,\\\" Academic Press (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Phases of N = 1 supersymmetric gauge theories and matrices. F Cachazo, N Seiberg, E Witten, arXiv:hep-th/0301006JHEP. 030242F. Cachazo, N. Seiberg and E. Witten, \\\"Phases of N = 1 supersymmetric gauge theories and matrices,\\\" JHEP 0302, 042 (2003) [arXiv:hep-th/0301006].\",\n \"doi\": \"arXiv:hep-th/0301006\"\n },\n {\n \"text\": \"Phases of N = 1 supersymmetric gauge theories with flavors. V Balasubramanian, B Feng, M X Huang, A Naqvi, arXiv:hep-th/0303065V. Balasubramanian, B. Feng, M. x. Huang and A. Naqvi, \\\"Phases of N = 1 super- symmetric gauge theories with flavors,\\\" arXiv:hep-th/0303065.\",\n \"doi\": \"arXiv:hep-th/0303065\"\n },\n {\n \"text\": \"Geometric dual and matrix theory for SO/Sp gauge theories. B Feng, arXiv:hep-th/0212010Nucl. Phys. B. 661113B. Feng, \\\"Geometric dual and matrix theory for SO/Sp gauge theories,\\\" Nucl. Phys. 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B 444, 125 (1995) [arXiv:hep-th/9503179].\",\n \"doi\": \"arXiv:hep-th/9503179\"\n },\n {\n \"text\": \"Dynamics of SU(N) supersymmetric gauge theory. M R Douglas, S H Shenker, arXiv:hep-th/9503163Nucl. Phys. B. 447271M. R. Douglas and S. H. Shenker, \\\"Dynamics of SU(N) supersymmetric gauge theory,\\\" Nucl. Phys. B 447, 271 (1995) [arXiv:hep-th/9503163].\",\n \"doi\": \"arXiv:hep-th/9503163\"\n }\n]","isConversational":false}}},{"rowIdx":594,"cells":{"text":{"kind":"string","value":"\nON A MIXED PROBLEM FOR THE PARABOLIC LAMÉ TYPE OPERATOR\n15 Apr 2019\n\nR Puzyrev \nA Shlapunov \nON A MIXED PROBLEM FOR THE PARABOLIC LAMÉ TYPE OPERATOR\n15 Apr 201910.1515/JIIP-2014-0043arXiv:1904.06797v1 [math.AP]\nWe consider a boundary value problem for the parabolic Lamé type operator being a linearization of the Navier-Stokes' equations for compressible flow of Newtonian fluids. It consists of recovering a vector-function, satisfying the parabolic Lamé type system in a cylindrical domain, via its values and the values of the boundary stress tensor on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural spaces of smooth functions and in the corresponding Hölder spaces; besides, additional initial data do not turn the problem to a well-posed one. Using the Integral Representation's Method we obtain the Uniqueness Theorem and solvability conditions for the problem.with ν = (ν 1 , ν 2 , ν 3 , ν 4 ) being the unit normal vector to the surface ∂D and δ i,j being the Kronecker symbol (see [1, §15, formula (15.12)]).\n\nIntroduction\n\nLet, as usual, ∆ n be the Laplace operator, ∇ n be the gradient operator and div n be the divergence operator in R n , n ≥ 2. The Navier-Stokes' equations for compressible flow of Newtonian fluids over the four-dimensional domain D ⊂ R 3\n\nx ×R t under the action of a body force F (x, t) = (F 1 (x, t), F 2 (x, t), F 3 (x, t)) can be written in the following form (see [1, §15, formulas (15.5), (15.6)]):\nρ ∂v ∂t + v · ∇ 3 v + ∇ 3 p − div 3 µ 1 ∇ 3 v − ∇ 3 µ 1 3 + µ 2 div 3 v − av = F,(1)\nwhere v(x, t) = (v 1 (x, t), v 2 (x, t), v 3 (x, t)) is the flow velocity, ρ(x, t) is the fluid density, p(x, t) is the pressure, µ j (x, t) are (positive) viscosity coefficients and av = [(∇ 3 µ 1 ) * ⊗ ∇ 3 − (∇ 3 µ 1 )div 3 ]v is the linear first order summand with M * 1 being the adjoint matrix for a matrix M 1 and M * 1 ⊗M 2 being the Kronecker product of matrices M * 1 and M 2 . If the boundary ∂D of D is piece-wise smooth then the boundary conditions for this system often involve the force νp − σ ′ v acting on the unit surface area where the force friction (or the boundary viscosity tensor) σ ′ has the components\nσ ′ i,j = δ i,j µ 1 n k=1 ν k ∂ ∂ k + µ 1 ν j ∂ ∂x i + (µ 2 − 2µ 1 /3)ν i ∂ ∂x j , 1 ≤ i, j ≤ 3,\nSince the density ρ is positive, a proper linearization of the substantial derivative term v · ∇ 3 v turns (1) into a parabolic Lamé type system related to an unknown vector u:\nL 4 u = ∂u ∂t − L 3 u − 3 j=1 a j (x, t) ∂u ∂x j − a 0 (x, t)u = f\nwhere a j (x, t), 0 ≤ j ≤ 3, are (3 × 3) matrices with functional entries and L n = div n µ∇ n + ∇ n (µ + λ)div n , n ≥ 2 is the strongly elliptic (with respect to the space variables) formally self-adjoint Lamé type operator with the Lamé coefficients satisfying µ(x, t) > 0, (µ(x, t) + λ(x, t)) ≥ 0.\n\nThe regularity (smoothness) of the Lamé coefficients and the matrices a j (x, t) depends upon the regularity of the density ρ and the viscosity coefficients µ j . Note that if µ is constant, λ + µ = 0 and a j = 0, 0 ≤ j ≤ 3, then L 4 reduces to the heat operator, though, of course, it is known that the heat equation is not ideal to model the process of the heat conduction.\n\nLet Ω be a bounded domain (i.e. bounded open connected set) in n-dimensional real space R n with the coordinates x = (x 1 , . . . , x n ). As usual we denote by Ω the closure of Ω, and we denote by ∂Ω its boundary. In the sequel we assume that ∂Ω is piece-wise smooth. As ∂Ω is piece-wise smooth, the normal vector ν = (ν 1 , ..., ν n ) is defined almost everywhere on ∂Ω and satisfies n j=1 ν 2 j = 0. Let Ω T = {x ∈ Ω, 0 < t < T } an open cylinder, having the altitude 0 < T ≤ +∞ and the base Ω, in (n + 1)-dimensional real space R n+1 = R n × {−∞ < t < +∞}. Let also Γ ⊂ ∂Ω be a non empty connected open (in the topology of ∂Ω) subset of ∂Ω and Γ T = Γ × (0, T ).\n\nIn the present paper we consider a mixed boundary problem for the parabolic system in the cylindrical domain Ω T\nL n+1 = ∂ ∂t − L n − Au, where Au = n j=1 a j (x, t) ∂ ∂x j + a 0 (x, t),\nthe Lamé coefficients and the entries of the (n × n)-matrices a j (x, t), 0 ≤ j ≤ n, are C ∞ -smooth in a neighborhood of Ω T and real analytic with respect to the space variables in a neighborhood of Ω. Instead of classical boundary value problems for parabolic equations (see, for instance, [2], [3], [4], [5]) we consider the ill-posed problem, consisting in finding a vector-function satisfying the corresponding parabolic equation in the cylinder via its values and the values of the boundary stress tensor with the components\nσ i,j = µ δ i,j n k=1 ν k ∂ ∂x k + µ ν j ∂ ∂x i + λ ν i ∂ ∂x j , 1 ≤ i, j ≤ n.(2)\non the given part Γ T of the lateral surface of the cylinder Ω T (cf. [6]). Using parabolic potentials we prove Uniqueness Theorem and obtain solvability conditions for the problem (cf. [7] related to similar results for the heat equation). Actually, the approach was invented for the investigation of the famous ill-posed Cauchy problem for elliptic equations (see, for instance, [8] for the Cauchy-Riemann operator, [9] for the elliptic Lamé operator and [10], [11], [12], for general systems with injective principal symbols).\n\n\nPreliminaries\n\nAs usual, for s ∈ Z + (here Z + = N ∪ {0}) and an open subset D ⊂ R m we denote C s (D) the set of all s times continuously differentiable functions in D. The standard topology of this metrisable space induces uniform convergence on compact subsets in D together with all the partial derivatives up to order s.\n\nFor S ⊂ ∂D we denote C s (D ∪S) the set of such functions from the space C s (D) that all their derivatives up to order s can be extended continuously onto D ∪ S. The standard topology of this metrisable space induces uniform convergence on compact subsets in D ∪ S together with all the partial derivatives up to order s. In particular, for bounded domains, C s (D ∪ ∂D) = C s (D) is a Banach space. If D is an unbounded then the Banach space C s b (D ∪ S) consists of s-times differentiable functions in D ∪ S with bounded derivatives up to order s and it is endowed with the standard sup-norm. Then C s b (D) = C s (D) for a bounded domain D. Apart from the standard functional spaces, we need also spaces taking into account the specific properties of parabolic equations in R n+1 = R n × {−∞ < t < +∞}. Namely, let C 1,0 (Ω T ) be the set of continuous functions u in Ω T , having in Ω T continuous partial derivatives u xi , and let C 2,1 (Ω T ) denote the set of continuous functions in Ω T , having in Ω T continuous partial derivatives u xi , u xixj , u t . The standard topology of this metrisable space induces uniform convergence on compact subsets in D together with all the partial derivatives used in its definition.\n\nAs before, for S ⊂ ∂Ω T we denote by C 1,0 (Ω T ∪S) the set of such functions u from the space C 1,0 (Ω T ) that their derivatives u xi can be extended continuously onto Ω T ∪S. The standard topology of this metrisable space induces uniform convergence on compact subsets of Ω T ∪S of both the functional sequences and the corresponding sequences of first partial derivatives x i . Clearly, C 1,0 (Ω T ∪ ∂Ω T ) = C 1,0 (Ω T ) is a Banach space. Similarly to the standard spaces, if D ⊂ R n+1 the Banach space C 1,0 b (D ∪ S) consisting of bounded C 0,1 (D ∪ S)-functions with bounded derivatives u xi in D ∪ S and it is endowed with the norm\nu C 1,0 b (D∪S) = sup (x,t)∈D∪S |u(x, t)| + sup (x,t)∈D∪S n j=1 | ∂u(x, t) ∂x j |.\nThen C 0,1 b (D) = C 0,1 (D) for a bounded domain D. The space of n-vector-functions u = (u 1 , . . . , u n ) of a class C will be denoted by\n[C] n .\nLet now θ be such positive constant that for all (x, t) ∈ Ω T we have\nµ(x, t) ≥ θ, (λ(x, t) + 2µ(x, t)) ≥ θ.\nThen a direct calculation shows that for all ζ ∈ R n we have det µ(x, t)|ζ| 2 ( √ −1) 2 I n + (λ(x, t) + µ(x, t))ζζ T ( √ −1) 2 − κI n = (−1) n (µ(x, t)|ζ| 2 + κ) n−1 ((2µ(x, t) + λ(x, t))|ζ| 2 + κ)\n\nwhere I n is the unit (n × n) matrix and ζ T is the transposed vector for ζ. Hence the roots of this polynomial (with respect to κ) are\nκ 1 (x, t, ζ) = −(2µ(x, t) + λ(x, t))|ζ| 2 , κ 2 (x, t, ζ) = −µ(x, t)|ζ| 2\nand, for all (x, t) ∈ Ω T , we have\nmax sup |ζ|=1 κ 1 (x, t, ζ), sup |ζ|=1 κ 2 (x, t, ζ) ≤ −θ,\ni.e. the operator L n is uniformly parabolic (according to Petrovskii) on Ω T . Now we assume that there is a n-dimensional domain U ⊃ Ω such that the Lamé coefficients µ(x, t), λ(x, t) and the entries of the (n×n)-matrices a j (x, t), 0 ≤ j ≤ n, are C ∞ -smooth in U T and real analytic with respect to the space variables in U .\n\nUnder the assumptions, the following properties hold true for parabolic operator L n+1 , which will be crucial for the approach below. Theorem 1. Each weak solution u to L n+1 u = 0 in the domain Ω T ⊂ U T belongs to C ∞ (Ω T ) and it is actually real analytic with respect to variables x in Ω.\n\nTheorem 2. The operator L n+1 has a fundamental solution in U T , i.e. a (n × n)matrix Φ(x, t, y, τ ) satisfying\n(L n+1 ) x,t Φ(x, t, y, τ ) = 0, (L * n+1 ) y,τ Φ(x, t, y, τ ) = 0, if (x, t) = (y, τ ),(3)\njj with the formal adjoint operators\n(L * n+1 ) y,τ = − ∂ ∂τ − (L n ) y − A * , A * = − n k=1 ∂ ∂y k (a * k (y, τ )·) + a * 0 (y, τ ).\nProof See, for instance, [4,Ch. 2]. We need a sort of an integral representation, similar to the famous Green Formula for the Laplace Operator, constructed with the use the fundamental solutions. More precisely, consider the cylinder type domain Ω T1,T2 = Ω T2 \\ Ω T1 and a closed measurable set S ⊂ ∂Ω.\n\nLet σ be the tensor with the components given by (2) and\nσ = σ − n k=1 a * k (x, t)ν k (x)+[(∇ n µ(x, t))ν * (x) − ν(x)(∇ n µ(x, t)) * ]\n.\nFor functions f ∈ C(Ω T1,T2 ), v ∈ C(S T ), w ∈ C(S T ), h ∈ C(Ω) we set I Ω,T1 h(x, t) = Ω Φ * (x, t, y, T 1 )h(y)dy,(4)G Ω,T1 f (x, t) = t T1 Ω Φ * (x, t, y, τ )f (y, τ )dydτ,(5)V S,T1 v(x, t) = t T1 S Φ * (x, t, y, τ )v(y, τ )ds(y)dτ,(6)W S,T1 w(x, t) = − t T1 S [σ y Φ(x, t, y, τ )] * w(y, τ )ds(y)dτ,(7)\nwhere ds is the volume form on S induced from R n . All these functions are called Parabolic Potentials with densities f , v, w and h respectively. In our situation these are convergent improper integrals depending on vector parameter (x, t) in the neighborhood U of the cylinder Ω T1,T2 in R n+1 (see, for instance, [ \nLemma 1. For all 0 ≤ T 1 < T 2 ≤ T and all u ∈ C 2,1 (Ω T1,T2 ) ∩ C 1,0 (Ω T1,T2 ) with L n+1 u ∈ C(Ω T1,T2\n) the following formula holds:\n(I Ω,T1 u + G Ω,T1 L n+1 u + V ∂Ω,T1 σu + W ∂Ω,T1 u) (x, t) = u(x, t), (x, t) ∈ Ω T1,T2 0, (x, t) ∈ U T \\ Ω T1,T2 .(8)\nProof. Indeed, it follows from Gauß-Ostrogradskii Formula that\n∂Ω v * σu = ∂Ω v * (L n u + au)dy + D Ω (u, v) (9) for all u, v ∈ C 1,0 (Ω T1,T2 ) with L n+1 u ∈ C(Ω T1,T2 ), where au = [(∇ n µ) * ⊗ ∇ n − (∇ n µ)div n ]u,(10)D Ω (u, v) = Ω µ(∇ n v) * ∇ n u + µ(∇ n v) * ⊗ ∇ n u) + λ(div n v) * div n u dy.(11)\nOn the other hand, by Gauß-Ostrogradskii Formula,\n∂Ω [(∇ n µ(x, t))ν * (x) − ν(x)(∇ n µ(x, t)) * ]ds(y) = Ω v * (au − (a * v) * u) dy. Therefore ∂Ω v * σu − (σv) * u ds(y) = Ω v * (L n u + Au) − (L n v + A * v) * u dy for all u, v ∈ C 1 (Ω) with L n u, L n v ∈ C(Ω).\nHence, again by Gauß-Ostrogradskii Formula, we obtain the (first) Green formula for the Lamé type operator:\nΩ [v * (y, T 1 )u(y, T 1 ) − v * (y, T 2 )u(y, T 2 )]dy− T2 T1 ∂Ω v * σu − (σv) * u ds(y) dt = (12) ΩT 1 ,T 2 v * L * n+1 u − (L n+1 v) * u dt dy for all u, v ∈ C 1,0 (Ω T1,T2 ) with L n+1 u, L n+1 v ∈ C(Ω T1,T2 ).\nIt follows from the definition of the fundamental solution, that\n(L n+1 ) x,t Φ(x, t, y, τ ) = δ(x − y, t − τ ), (L * n+1 ) y,τ Φ(x, t, y, τ ) = δ(x − y, t − τ ), Φ(x, t, y, τ ) = 0 for τ > t.\nThen, using the standard argument (see, for instance, [17,Ch. 6, §12] for the heat equation), we see that Green's Formula (8) follows from (12) and Fubini Theorem.\n\nTheorem 3 (Uniqueness Theorem). If Γ has at least one interior point (on ∂Ω),\nand function u ∈ C 2,1 (Ω T ) ∩ C 1,0 (Ω T ∪ Γ T ) satisfies L n+1 u ≡ 0 in Ω, u ≡ 0 on Γ T , σu ≡ 0 on Γ T , then u ≡ 0 in Ω T .\nProof. Under the hypothesis of the theorem there is an interior point x 0 on Γ.\n\nThen there is such a number\nr > 0 that B(x 0 , r) ∩ ∂Ω ⊂ Γ where B(x 0 , r) is ball in U ⊂ R n with center at x 0 and radius r. Fix an arbitrary point (x ′ , t ′ ) ∈ Ω T . It is clear that there is a domain Ω ′ ∋ x ′ satisfying Ω ′ ⊂ Ω and Ω ′ ∩ ∂Ω ⊂ Γ ∩ B(x 0 , r). Then (x ′ , t ′ ) ∈ Ω ′ T1,T2 with some 0 < T 1 < T 2 < T . But u ∈ C 2,1 (Ω ′ T1,T2 ) ∩ C 1,0 (Ω ′ T1,T2 ) and L n+1 u = 0 in Ω ′ T1,T2\nunder the hypothesis of the theorem. Hence formula (8) implies:\nI Ω ′ ,T1 u(x, t) + V ∂Ω ′ \\Γ,T1 σu(x, t) + W ∂Ω ′ \\Γ,T1 u(x, t) = u(x, t), (x, t) ∈ Ω ′ T1,T2 , 0, (x, t) ∈ U T \\ Ω ′ T1,T2 ,(13)because u ≡ σu ≡ 0 on Γ T .\nTaking into account the character of the singularity of the kernel (see [4, Theorem 2.2]) Φ(x, y, t, τ ) we conclude that the following properties are fulfilled for the integrals, depending on parameter, from the right hand side of identity (13): (3) and Leibniz rule for differentiation of integrals depending on parameter we obtain:\nI Ω ′ ,T1 (u) ∈ C 2,1 (U T1,T2 ), W ∂Ω ′ \\Γ,T1 u, V ∂Ω ′ \\Γ,T1 σu ∈ C 2,1 ((U \\ (∂Ω ′ \\ Γ)) T1,L n+1 I Ω ′ ,T1 u = 0 in U T1,T2 , L n+1 V ∂Ω ′ \\Γ,T1 σu = L n+1 W ∂Ω ′ \\Γ,T1 u = 0 in (U \\ (∂Ω ′ \\ Γ)) T1,T2 .\nHence the function\nP (x, t) = I Ω ′ ,T1 u(x, t) + V ∂Ω ′ \\Γ,T1 σu(x, t) + W ∂Ω ′ \\Γ,T1 u(x, t),\nsatisfies the Lamé type equation\n(L n+1 P )(x, t) = 0 in (U \\ (∂Ω ′ \\ Γ)) T1,T2 .\nThis implies that the function P (x, t) is real analytic with respect to the space variable x ∈ U \\ (∂Ω ′ \\ Γ) for any T 1 < t < T 2 (see, for instance, [16, Ch. VI, §1, Theorem 1]). In particular, by the construction the function P (x, t) is real analytic with respect to x in the ball B(x 0 , r) and it equals to zero for x ∈ B(x 0 , R) \\ Ω for all T 1 < t < T 2 . Therefore, the Uniqueness Theorem for real analytic functions yields P (x, t) ≡ 0 in (U \\ (∂Ω ′ \\ Γ)) T1,T2 , and in the cylinder Ω ′ T1,T2 , containing the point (x ′ , t ′ ). Now it follows from (13) that u(x ′ , t ′ ) = P (x ′ , t ′ ) = 0 and then, since the point (x ′ , t ′ ) ∈ Ω T is arbitrary we conclude that u ≡ 0 in Ω T . The proof is complete. Example 1. Let µ = 1, λ = −1 and a j = 0, 0 ≤ j ≤ n. Then L n+1 reduces to the heat operator:\nL n+1 = ∂ ∂t − ∆ n\nand corresponding fundamental solution is given by\nΦ(x, y, t, τ ) = ϕ 0 (x − y, t − τ )I n where ϕ 0 (x, t) = 1 (2 √ πt) n e − |x| 2 4t if t > 0, 0 if t ≤ 0.\nIn this caseσ = σ = ∂ ∂ν . Example 2. Let µ, λ be constant and a j = 0, 0 ≤ j ≤ n. Then L n+1 reduces to the parabolic Lamé operator\nL n+1 = ∂ ∂t − L n and corresponding fundamental solution Φ(x, y, t, τ ) is given by (n × n)-matrix with components Φ i,j (x, y, t, τ ) = ϕ i,j (x − y, t − τ ) where ϕ i,j (x, t) = ϕ 0 (x, µt)δ i,j + (2µ+λ)t µt ∂ 2 ϕ 0 (x, s) ∂x j ∂x i ds,\n(see, for instance, [4]). In this caseσ = σ = µ ∂ ∂ν + µ ν * ⊗ ∇ n + λ ν div n .\n\n\nThe boundary problem\n\nGreen formula (8) and the Uniqueness Theorem 3 suggest us to consider two kind of problems for the parabolic Lamé type equation.\n\nLet vector-functions\nu (0) (x) ∈ [C(Ω)] n , f (x, t) ∈ [C b (Ω T )] n , u (1) (x, t) ∈ [C 1,0 (Γ T ) ∩ C b (Γ T )] n , u (2) (x, t) ∈ [C b (Γ T )] n\nbe given. and boundary conditions\nu(x, t) = u (1) (x, t) on Γ T ,(15)σu(x, t) = u (2) (x, t) on Γ T .(16)\nNote that, if the surface Γ and the data of the problem are real analytic then the Cauchy-Kovalevsky Theorem implies that Problem 1 can not have more than one solution in the class of (even formal) power series. However the theorem does not imply the existence of solutions to Problem 1 because it grants the solution in a small neighborhood of the surface Γ T only (but not in a given domain Ω T !). In any case, we do not assume the real analyticity of Γ and the data u (1) , u (2) and f . Corollary 1. If Γ has at least one interior point (on ∂Ω) then Problem 1 has no more that one solution.\n\nProof. Let v(x, t) and w(x, t) be two solutions to Problem 1. Then function\nu = (v − w) ∈ C 2,1 (Ω T ) ∩ C 1,0 (Ω T ∪ Γ T ) ∩ C(Ω T \\ (∂Ω \\ Γ) T )\nis a solution to the corresponding problem with f = 0, u 1 = 0, u 2 = 0. Using 3 we conclude that u is identically zero in Ω T .\n\nThus, the Uniqueness Theorem 3 implies that the data of Problem 1 are suitable in order to uniquely define its solution.\n\nEasily, Problem 1 is ill-posed because this is the property of the Cauchy problem for elliptic systems in R n (see, for instance [13] or [16, Ch. 1, §2]). Of course, in this case the boundary data should be taken independent on t. The Uniqueness Theorem clarify why the problem is ill-posed. The reason is the redundant data. Indeed, if Γ has at least one interior point (on ∂Ω), then taking a smaller set Γ ′ ⊂ Γ we again obtain a problem with no more than one solution.\n\nAnother problem involves the initial data.\nProblem 2. Find a vector-function u(x, t) ∈ [C 2,1 (Ω T )∩C 1,0 (Ω T ∪Γ T )∩C b (Ω T )] n satisfying\nin Ω T Lamé type equation (14), boundary conditions (15), (16) and initial condition\nu(x, 0) = u (0) (x), x ∈ Ω.(17)\nOf course one should also take care on the compatibility of the data u (0) , u (1) , u (2) : at least\nu (0) (x) = u (1) (x, 0) on Γ,(18)\nand, if u (0) ∈ C 1 (Ω), even\nσu (0) (x) = u (2) (x, 0) on Γ.(19)\nThe motivation of Problems 1 and 2 is transparent. The space C b (Ω T ) is chosen because u represents the \"velocity\". The first problem describes the situation where for some reasons at each time t ≥ 0 only part Γ of the solid surface ∂Ω bounding the fluid is available for measurements. The second one describes the situation where the continuity up to ∂Ω T is postulated, the \"velocity\" u is known at every point x ∈ Ω at the initial time t = 0 but the data on ∂Ω \\ Γ were lost for t > 0.\n\nClearly, Problem 2 has no more than one solution, too, if Γ has at least one interior point (on ∂Ω).\n\nWe note that in classical theory of (initial and) boundary problems for the parabolic equation (14), initial condition (17) and boundary condition αu + βσ = u (3) on the whole lateral surface ∂Ω T of the cylinder Ω T are usually considered. As a rule, such a problem is well-posed in proper spaces (Hölder spaces, Sobolev spaces etc.), see, for instance, [2].\n\nLet us show that Problem 2 is ill-posed, too.\n\nExample 3. Let the Lamé coefficients µ, λ be constant and a j = 0, 0 ≤ j ≤ n. Take a cube Q n = {0 < x j < 1, 1 ≤ j ≤ n} as base Ω of the cylinder Ω T . Let Γ be the face {x n = 0} of the cube Q n . Then Γ T = Q n−1 × (0, T ) and the stress tensor σ is given by the diagonal matrix with the non-zero entries\nσ j,j = µ ∂ ∂x n , 1 ≤ j ≤ n − 1, σ n,n = (2µ + λ) ∂ ∂x n .\nFix N ∈ N and consider the sequence of functions u(x, t, k, r) ∈ [C ∞ (R n+1 )] n with the components: u 1 (x, t, k, r) = 0, . . . , u n−1 (x, t, k, r) = 0, u n (x, t, k, r) = e k 2 (2µ+λ)(t−r)+kxn k N , depending on a parameter 0 < r < +∞. Consider also the data f (x, t, k, r), u (0) (x, t, k, r), u (1) (x, t, k, r), u (2) (x, t, k, r) having the following components:\nf j (x, t, k, r) = 0, 1 ≤ j ≤ n, u (0) j (x, k, r) = 0, 1 ≤ j ≤ n − 1, u (0) n (x, k, r) = e −k 2 (2µ+λ)r+kxn k N , u(1)\nj (x 1 , . . . , x n−1 , t, k, r) = 0, 1 ≤ j ≤ n − 1,\nu (1) n (x 1 , . . . , x n−1 , t, k, r) = e k 2 (2µ+λ)(t−r) k N , u(2)\nj (x 1 , . . . , x n−1 , t, k, r) = 0, 1 ≤ j ≤ n − 1,\nu (2) n (x 1 , . . . , x n−1 , t, k) = (2µ + λ) e k 2 (2µ+λ)(t−T ) k N −1 .\nThen, for 0 < T < +∞, each function u(x, t, k, T ) is a solution to problem (14), (15), (16), (17) with the data f (x, t, k, T ),\nu (0) (x, t, k, T ), u (1) (x, t, k, T ), u (2) (x, t, k, T ).\n\nIt is clear, that compatibility conditions (18), (19) hold and\nf (x, t, k, T ) −→ k→∞ 0 in [C ∞ (Ω T )] n , u (0) (x, k, T ) −→ k→∞ 0 in [C ∞ (Ω)] n , u (1) (x, t, k, T ) −→ k→∞ 0 in [C s (Γ T )] n , u (2) (x, t, k, T ) −→ k→∞ 0 in [C s (Γ T )] n ,\nif N > 2s + 1. On the other hand, for all x n > 0 and all N ∈ N we have:\nu n (x, T, k) = e k 2 (2µ+λ)(T −T )+kxn k N = e kxn k N −→ k→∞ +∞.\nThus, there is no continuity with respect to the data and hence Problem 2 is ill-posed for 0 < T < +∞. Let now T = +∞. Then we may consider the data with a fixed 0 < T 0 < +∞:\nf (x, t, k, ∞) = 0 ∈ [C b (Ω T ] n , u (0) (x, k, ∞) = u (0) (x, k, T 0 ) ∈ [C(Ω)] n , u (i) j (x, t, k, ∞) = 0, 1 ≤ j ≤ n − 1, 1 ≤ i ≤ 2, u (1) n (x, t, k, ∞) = u (1) n (x, t, k, T 0 ), t ≤ T 0 , 1 k N t > T 0 , u (2) n (x, t, k, ∞) = u (1) n (x, t, k, T 0 ), t ≤ T 0 , (2µ+λ) k N −1 t > T 0 .\nObviously, for N ≥ 2,\nf (x, t, k, ∞) −→ k→∞ 0 in [C b (Ω T )] n , u (0) , (x, k, ∞) −→ k→∞ 0 in [C ∞ (Ω)] n , u (1) (x, t, k, ∞) −→ k→∞ 0 in [C 1,0 (Γ T ) ∩ C b (Γ T )] n , u (2) (x, t, k, ∞) −→ k→∞ 0 in [C b (Γ T )] n .\nThe Uniqueness Theorem 3 for Problem 3 implies that\nu n (x, t, k, ∞) = u n (x, t, k, T 0 ) for 0 < t ≤ T 0 .\nThen, for all x n > 0 and all 2 ≤ N ∈ N, we have lim k→+∞ u n (x, T 0 , k, ∞) = +∞. Thus, if the data f , u (0) , u (2) , u (2) admits the solution to (14) in Ω T with boundary conditions (15), (16) for T = +∞ and the initial condition (17) then there is no continuity with respect to the data in the chosen space. Otherwise there is no solutions to the problem for some data in the data's spaces. In any case, the problem is ill-posed for T = +∞, too.\n\nAs both Problems 1 and 2 are ill-posed, we will not study Problem 2 because in addition to (14)- (16) to investigate it one needs to know also the data related to initial condition (17). Besides, we will consider the case 0 < T < +∞ only.\n\n\nSolvability Conditions\n\nFrom now on we will study Problem 1 under the assumption that its data belong to Hölder spaces (cf., [3,Ch. 1,§1] for other boundary problems for parabolic equations). We recall that a function u(x), defined on a set M ∈ R m , is called Hölder continuous with a power 0 < λ < 1 on M , if there is such a constant C > 0 that\n|u(x) − u(y)| ≤ C|x − y| λ for all x, y ∈ M (20) (|x − y| = m j=1 (x j − y j ) 2\nbeing Euclidean distance between points x and y in R m ). Let C λ (Ω T ) stand for the set of Hölder continuous functions with a power λ over Ω T . Besides, let C 1+λ,λ (Ω T ) be the set of Hölder continuous functions with a power λ over Ω T , having Hölder continuous derivatives u xi , 1 ≤ i ≤ n, with the same power in Ω T .\n\nWe choose a set Ω + in such a way that the set D = Ω ∪ Γ ∪ Ω + would be a bounded domain with piece-wise smooth boundary. It is possible since Γ is an open connected set. It is convenient to set Ω − = Ω. For a function v on D T we denote by v + its restriction to Ω + and, similarly, we denote by v − its restriction to Ω. It is natural to denote limit values of v ± on Γ T , when they are defined, by v ± |ΓT .\n\n\nTheorem 4 (Solvability criterion). Let\nΓ ∈ C 1+λ , f ∈ [C λ (Ω T )] n , u 1 ∈ [C 1+λ,λ (Γ T )] n , u 2 ∈ [C λ (Γ T )] n .\nProblem 1 is solvable if and only if there is a vector-function F ∈ [C 2,1 (D T )] n satisfying the following conditions:\n1) L n+1 F = 0 in D T , 2) F = G Ω,0 (f ) + V Γ,0 (u 2 ) + W Γ,0 (u 1 ) in Ω + T .\nProof. Necessity. Let a function u(x, t) ∈ [C 2,1 (Ω T ) ∩ C 1,0 (Ω T ∪ Γ T )] n satisfies (14), (15), (15). Consider the function\nF = G Ω,0 (f ) + V Γ,0 (u 2 ) + W Γ,0 (u 1 ) − χ ΩT u.\nin the domain D T , where χ M is a characteristic function of the set M ⊂ R n+1 . By the very construction condition 2) is fulfilled for it.\n\nClearly, the function u(x, t) belongs to the space [C 1,2 (Ω ′ T )] n for each cylindrical domain Ω ′ T with such a base Ω ′ that Ω ′ ⊂ Ω and Ω ′ ∩ ∂Ω ⊂ Γ. Besides, L n+1 u = f ∈ [C λ (Ω ′ T )] n . Without loss of the generality we may assume that the interior part Γ ′ of the set Ω ′ ∩ ∂Ω is non-empty.\n\nWe note that\nχ ΩT u = χ Ω ′ T u in D ′ T , where D ′ = Ω ′ ∪ Γ ′ ∪ Ω + .\nThen using Lemma 2 we obtain:\nF = G Ω\\Ω ′ ,0 (f ) + V Γ\\Γ ′ ,0 (u 2 ) + W Γ\\Γ ′ ,0 (u 1 ) − I Ω ′ ,0 (u) in D ′ T .(21)\nArguing as in the proof of Theorem 3 we conclude that each of the integrals in the right hand side of (21) satisfies homogeneous Lamé type equation outside the corresponding integration set. In particular, we see that L n+1 F = 0 in D ′ T . Obviously, for any point (x, t) ∈ D T there is a domain D ′ T containing (x, t). That is why L n+1 F = 0 in D T , and hence F belongs to the space [C 2,1 (D T )] n . Thus this function satisfies condition 1), too.\n\nSufficiency. Let there be a function F ∈ [C 2,1 (D T )] n , satisfying conditions 1) and 2) of the theorem. Consider on the set D T the function\nU = G Ω,0 (f ) + V Γ,0 (u 2 ) + W Γ,0 (u 1 ) − F.(22)\nAs f ∈ [C λ (Ω T )] n then the results of [3, Ch. 1, §3] imply\nG Ω,0 (f ) ∈ [C 2,1 (Ω ± T ) ∩ C 1,0 (D T ) ∩ C(D T )] n(23)\nand, moreover,\nL n+1 G − Ω,0 (f ) = f in Ω T , L n+1 G + Ω,0 (f ) = 0 in Ω + T .(24)\nSince u 2 ∈ [C λ (Γ T )] n then the results of [3, Ch. 5, §2]yield\nV Γ,0 (u 2 ) ∈ [C 2,1 (Ω ± T ) ∩ C 1,0 ((Ω ± ∪ Γ) T ) ∩ C(D T \\ (∂Γ) T )] n ,(25)L n+1 V Γ,0 (u 2 ) = 0 in Ω T ∪ Ω + T .(26)\nOn the other hand, the behavior of the Double Layer Potential W Γ,0 (u 1 ) is similar to the behavior of the normal derivative of Single Layer Potential V Γ,0 (u 1 ). Hence\nW Γ,0 (u 1 ) ∈ [C 2,1 (Ω ± T ) ∩ C(Ω ± T \\ (∂Ω ± \\ Γ) T )] n ,(27)L n+1 W Γ,0 (u 1 ) = 0 in Ω T ∪ Ω + T .(28)\nLemma 2. Let S ⊂ Γ ∈ C 1+λ . If u 1 ∈ [C 1+λ,λ (Γ T )] n , then the potential W − Γ,0 (u 1 ) belongs to the space [C 1,0 (Ω T ∪ S T )] n if and only if W + Γ,0 (u 2 ) ∈ [C 1,0 (Ω + T ∪ S T )] n .\n\nProof. It is similar to the proof of the analogous lemma for Newton Double Layer Potential (see, for instance, [14, lemma 1.1]). Actually, one needs to use Lemma 2 instead of the standard Green formula for the Laplace operator.\n\nSince F ∈ [C 1,0 (D T )] n then it follows from the discussion above that W + Γ,0 (u 2 ) ∈ [C 1,0 ((Ω + ∪ Γ) T )] n . Thus, formulas (22)-(28) and Lemma 2 imply that\nU ∈ [C 2,1 (Ω ± T ) ∩ C 1,0 ((Ω ± ∪ Γ) T ) ∩ C(Ω ± T \\ (∂Ω \\ Γ) T )] n , L n+1 U = χ DT f in Ω T ∪ Ω + T . In particular, (14) is fulfilled for U − .\nLet us show that the function U − satisfies (15) and (16).\nSince F ∈ [C 1,0 (D T )] n we see that ∂ α F − = ∂ α F + on Γ T for α ∈ Z + with |α| ≤ 1 and ∂ α F + |ΓT = ∂ α G + Ω,0 (f ) + ∂ α V + Γ,0 (u 2 ) + ∂ α W + Γ,0 (u 1 ) |ΓT .\nIt follows from formulas (23) and (25) that the Parabolic Volume Potential and the Single Layer Parabolic Potential are continuous if the point (x, t) passes over the surface Γ T . Then Formula (23) means that that the surface stress of the Parabolic Volume Potential is continuous if the point (x, t) passes over the surface Γ T . Therefore\nU − |ΓT = W − Γ,0 (u 1 ) |ΓT − W + Γ,0 (u 1 ) |ΓT = u 1 .(σU ) − |ΓT = σV − Γ,0 u 2 |ΓT − σV + Γ,0 u 2 |ΓT + σW − Γ,0 u 1 |ΓT − σW + Γ,0 u 1 |ΓT .(29)\nBy theorem on jump behavior of the stress of the Parabolic Single Layer Potential (see, for instance, [15,Ch. 3, §10, theorem 10.1])\nσV − Γ,0 u 2 |ΓT − σV + Γ,0 u 2 |ΓT = u 2 .(30)\nFinally, we need the following lemma which is an analogue of the famous Theorem on jump behavior of the normal derivative of the Newton's Double Layer Potential.\nLemma 3. Let Γ ∈ C 1+λ and u 2 ∈ [C λ (Γ T )] n . If W − Γ,0 (u 1 ) ∈ [C 1,0 ((Ω ∪ Γ) T )] n or W + Γ,0 (u 1 ) ∈ [C 1,0 ((Ω + ∪ Γ) T )] n then σW − Γ,0 u 1 − σW + Γ,0 u 1 |ΓT = 0.(31)\nProof. Really, let, for instance, W − Γ,0 (u 1 ) ∈ [C 1,0 ((Ω ∪ Γ) T )] n . Then using Lemma 2 we obtain W + Γ,0 u 1 ∈ [C 1,0 ((Ω + ∪ Γ) T )] n and σW ± Γ,0 (u 1 )\n|ΓT ∈ [C(Γ T )] n .\nLet φ ∈ [C ∞ 0 (D T )] n be a function with compact support in D T . Then formulas (9)-(11) yield:\nΓT φ * σW − Γ,0 u 1 − σW + Γ,0 u 1 ds(x)dt = (32) ΩT ∪Ω + T φ * (L n + a)W Γ,0 u 1 dxdt + T2 T1 D Ω∪Ω + (W Γ,0 u 1 , φ)dt = ΩT ∪Ω + T φ * ∂ ∂t − A + a W Γ,0 u 1 dxdt + T2 T1 D Ω∪Ω + (W Γ,0 u 1 , φ)dt\nbecause L n+1 W ± Γ,0 u 1 = 0 in Ω ± according to (28). Again, integrating by parts and using formulas (9)- (11) and Theorem on jump behavior of the Parabolic Double Layer Potential, we see that\nΩT ∪Ω + T φ * ∂ ∂t − A + a W Γ,0 u 1 dxdt + T2 T1 D Ω∪Ω + (W Γ,0 u 1 , φ)dt = (33) − ΩT ∪Ω + T ∂φ ∂t * W Γ,0 u 1 dxdt − ΩT ∪Ω + T ((L n + A * )φ) * W Γ,0 u 1 dxdt+ ΓT (σφ) * (W − Γ,0 u 1 − W + Γ,0 u 1 )ds(x)dt = ΓT (σφ) * u 1 ds(x)dt − ΩT ∪Ω + T (L * n+1 φ) * W Γ,0 u 1 dxdt.\nBut the kernel Φ(x, y, t, τ ) is a fundamental solution of the backward parabolic operator L * n+1 with respect to variables (y, τ ). Hence DT (L * n+1 φ(x, t)) * Φ(x, y, t, τ )dxdt = φ * (y, τ ), (y, τ ) ∈ D T .\n\nThen the type of the singularity of the fundamental solution allows us to apply Fubini Theorem and to conclude that\nΩT ∪Ω + T (L * n+1 φ) * W Γ,0 u 1 dxdt = (34) ΓTσ DT (L * n+1 φ(x, t)) * Φ(x, y, t, τ )dxdt u 1 ds(y)dτ = ΓT (σφ) * u 1 ds(y)dτ.\nFinally, formulas (32)-(34) imply that In particular, if S = ∅ then corollary 2 gives criterion for the existence of solution to Problem 1 in the space [C(Ω T )] n .\nΓT φ * σW − Γ,0 u 1 − σW + Γ,0 u 1 ds = 0 for all φ ∈ [C ∞ 0 (D T )] n .\nWe note that Theorem 4 is an analogue of Theorem by Aizenberg and Kytmanov [8]) describing solvability conditions of the Cauchy problem for the Cauchy-Riemann system (cf. also [14] in the Cauchy Problem for Laplace Equation or [12] in the Cauchy problem for general elliptic systems). Formula (22), obtained in the proof of Theorem 4, gives the unique solution to Problem 1. Clearly, if we will be able to write the extension F of the sum of potentials G Ω,0 (f )+ V Γ,0 (u 2 )+ W Γ,0 (u 1 ) from Ω + onto D T as a series with respect to special functions or a limit of parameter depending integrals then we will get a Carleman type formula for solutions to Problem 1 (cf. [8]). However this is a topic for another paper. In the sequel we will discuss polynomial and formal solutions for operators with constant coefficients only. Proof. First let us prove that if in this case the data of Problem 1 are polynomials then the problem is solvable and its solution is a polynomial.\n\n\nPolynomial solutions and dense solvability\n\nIndeed, Problem 1 is easily can be reduced to the following one (see Example 3):\nL n+1 v = g in Ω T (35) v(x 1 , . . . , x n−1 , 0, t) = 0 on Γ T ,(36)µ ∂v j ∂x n (x 1 , . . . , x n−1 , 0, t) = 0 on Γ T , 1 ≤ j ≤ n − 1,(37)(2µ + λ) ∂v n ∂x n (x 1 , . . . , x n−1 , 0, t) = 0 on Γ T , 1 ≤ j ≤ n − 1,(38)\nwith\ng(x, t) = f (x, t) − (L n u 1 )(x 1 , . . . , x n−1 , t) − x n J(µ, λ)(L n u 2 )(x 1 , . . . , x n−1 , t).\nwhere J(µ, λ) is the diagonal matrix with the components\nJ j,j (µ, λ) = µ −1 , 1 ≤ j ≤ n − 1, J n,n (µ, λ) = (2µ + λ) −1 . Besides, u(x, t) = v(x, t) + u 1 (x 1 , . . . , x n−1 , t) + J(µ, λ)x n u 2 (x 1 , . . . , x n−1 , t). Now consider data g (j,α) (x, t) = t j x α with a multi-index α ∈ Z n + . If 0 ≤ α 1 + . . . α n−1 ≤ 1, we easily obtain (unique) polynomial solutions v (j,α) (x, t) = x α1 1 · · · x αn−1 n−1 w (j,αn) (x n , t), α n , j ∈ Z + ,(39)\nto problem (35)-(37) where\nw (0,k) (y, t) = − y k+2 k! (k + 2)! , w (1,k) (y, t) = − ty k+2 k! (k + 2)! − y k+4 k! (k + 4)! , k ∈ Z + , y ∈ R\nand, by the induction with respect to j ∈ Z + ,\nw (j,k) (y, t) = − j µ=0 t j−µ y k+2µ+2 k!j! (k + 2µ + 2)!(j − µ)! , k ∈ Z + , y ∈ R.(40)\nTo finish the arguments we use the induction with respect to |α ′ | ∈ Z + where α ′ = (α 1 , . . . , α n−1 ) ∈ Z n−1 + . Namely, let for s ≥ 2 and all α ′ with |α ′ | = s the solutions to the problem are polynomial. If |α ′ | = s + 1 then L n+1 x α1 1 · · · x αn−1 n−1 w (j,αn) (x n , t) = t j x α − w (j,αn) (x n , t)∆ n−1 x α1 1 · · · x αn−1 n−1 .\n\nClearly, the degree of the polynomial p j,α (x, t) = w (j,αn) (x n , t)∆ n−1 x α1 1 · · · x αn−1 n−1 with respect to x ′ ∈ R n−1 equals to s − 1. Then, by the induction, problem (14)- (16) with data p j,α (x, t) admits a polynomial solution, say, r j,α (x, t). Therefore the solution v (j,α) (x, t) to problem (14)-(16) with data g (j,α) (x, t) = t j x α , |α ′ | = s + 1, is given as follows:\n\nv (j,α) (x, t) = x α1 1 · · · x αn−1 n−1 w (j,αn) (x n , t) + r j,α (x, t), i.e. it is a polynomial, too. Now Problem 1 with zero boundary data in the case Γ ⊂ {x n = 0} is densely solvable because any continuous function g on the compact set Ω T can be approximated by polynomials. But the reducing to zero boundary data was organized in such a way that one easily sees, in this case Problem 1 is densely solvable for non-zero boundary data, too.\n\nThe dense solvability of Problem 1 in general setting is natural to expect if the set ∂Ω \\ Γ has at least one interior point in ∂Ω (cf. [10] in the Cauchy Problem for elliptic equations).\n\nFinally, we note that polynomial solutions indicated in the proof of Lemma 4 can be used in order to construct formal solutions to Problem 1.\n\n\nBasis with double orthogonality\n\nDenote by {h (s) ν } the set of harmonic homogeneous polynomials (spherical harmonics) forming an orthonormal basis in L 2 (∂B(0, 1)); here ν is the degree of the homogeneity and 1 ≤ s ≤ J(s) where J(s) = (2n + 2s − 2)(2n + s − 3)! s!(2n − 2)! (see [18]). On the other hand, ∂|x| 2k ∂x j = 2kx j |x| 2k−2 , ∂ 2 |x| 2k ∂x 2 j = 2k|x| 2k−2 + 2k(2k − 2)x 2 j |x| 2k−4 ∆ n |x| 2k = 2k(n + 2k − 2)|x| 2k−2 , k ≥ 1.\n\nHence, for ν ≥ 1, k ≥ 1, ∆ n (|x| 2k h ν (x)) = 2k(n + 2k + 2ν − 2)|x| 2k−2 h (s) ν (x) because of Euler's formula \n\nProblem 1 .\n1Find a vector-function u(x, t) ∈ [C 2,1 (Ω T )∩C 1,0 (Ω T ∪Γ T )] n satisfying the Lamé type equationL n+1 u = f in Ω T(14) \n\n\nbecause of the theorem on jump behavior of the Parabolic Double Layer Potential (see, for instance, [3, Ch. 5, §2, theorem 1]), i.e. equality (15) is valid for U − .\n\n\nAs such functions are dense in the Lebesgue space [L 1 (K)] n for any compact K ⊂ Γ T then formula (31) holds true. Now using lemma 3 and formulas (29), (30), we conclude that (σU ) − |ΓT = u 2 , i.e. (16) is fulfilled for U − .Thus, function u(x, t) = U − (x, t) satisfies conditions(14)-(16). The proof is complete.It follows from formula (22) that properties of a solution to Problem 1 depend on properties of the extension F of the sum of the parabolic potentials, described in Theorem 4.\n\nCorollary 2 .\n2Let S ⊂ ∂Ω \\ Γ. Under the hypotheses of Theorem 4, Problem 1 is solvable in the space[C 2,1 (Ω T ) ∩ C 1,0 (Ω T ∪ Γ T ) ∩ C(Ω T \\ S T )] nif and only if there exists a functionF ∈ [C 2,1 (D T ) ∩ C 1,0 (Ω T ∪ Γ T ) ∩ C(Ω T \\ S T )] n ,satisfying conditions 1) and 2) of Theorem 4.\n\n\nIt is not difficult to prove dense solvability of Problem 1 in the case where Γ is an open connected set of the hyperplane {x n = 0}.\n\nLemma 4 .\n4If Γ is an open connected set if the hyperplane {x n = 0} the Problem 1 is densely solvable.\n\nLemma 5 . 1 .ν\n51Let µ = 1, λ = −Then (x)(2N )!!(n + 2N − 2 + 2ν)!! j!(2N − 2j)!!(n + 2N − 2j + 2ν)!! , ν ≥ 1, N ≥ 1 are solutions to the heat equations in R n+1 . Besides, H µ are L 2 (B(0, R) T )-orthogonal for all R > 0 and T > 0 if (ν, s) = (µ, p).Proof. Indeed, for ν ≥ 0, N ≥ 1, and 1 ≤ i ≤ n we have:\n\n\nj t j |x| 2N −2j h (s) ν (x) with constants c j . Then L n+1 H = N j=1 c j jt j−1 |x| 2N −2j h (s) ν (x)−\n\nT2 )\nT2(see, for instance, [2, Ch. 4, §1], [15, Ch. 3, §10] or [3, Ch. 1, §3 and Ch. 5, §2]). Moreover, as Φ is a fundamental solution to Lamé type operator then using\nThus, we get a recurrent formulafor the coefficients in the case L n+1 H = 0. Choosing c 0 = 1 we easily obtainFinally the statement on L 2 (B(0, R) T )-orthogonality follows from Fubini Theorem and the homogeneity of the polynomials hThe work was supported by RFBR grant 11-01-00852a.\n. L D Landau, E M Lifshitz, A Course of Theoretical Physics. 6Pergamon PressFluid MechanicsLandau, L.D., Lifshitz, E.M. Fluid Mechanics (Volume 6 of A Course of Theoretical Physics), Pergamon Press, 1959.\n\nLinear and Quasilinear Equations of Parabolic Type. O A Ladyzhenskaya, V A Solonnikov, N N Ural'tseva, Moscow, NaukaLadyzhenskaya, O.A., Solonnikov V.A., Ural'tseva, N.N. Linear and Quasilinear Equations of Parabolic Type, Moscow, Nauka, 1967.\n\nPartial Differential Equations of Parabolic Type. A Friedman, Prentice-Hall, IncEnglewood Cliffs, NJFriedman, A. Partial Differential Equations of Parabolic Type, Englewood Cliffs, NJ, Prentice- Hall, Inc., 1964.\n\nParabolic equations, Partial differential equations 6. S D Eidel'man, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. 63VINITIEidel'man S.D., Parabolic equations, Partial differential equations 6, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 63, VINITI, Moscow, 1990, 201-313.\n\nOn boundary value problems for linear parabolic systems of differential equations of general form. Boundary value problems of mathematical physics. Part 3. On boundary value problems for linear parabolic systems of differential equations of general form. V A Solonnikov, Trudy Mat. Inst. Steklov. 83Solonnikov V.A., On boundary value problems for linear parabolic systems of differential equa- tions of general form. Boundary value problems of mathematical physics. Part 3. On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Inst. Steklov, 83, 1965, 3-163\n\nA strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain Ω × O of R N+M. A Lorenzi, L Lorenzi, 10.1088/0266-5611/29/2/025007Inverse Problems. 292Lorenzi A., Lorenzi, L. A strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain Ω × O of R N+M , Inverse Problems, 29:2 (2013), doi:10.1088/0266-5611/29/2/025007.\n\nOn an ill-posed problem for the heat equation. R Puzyrev, A Shlapunov, J. Siberian Fed. Univ. 42Puzyrev, R., Shlapunov, A., On an ill-posed problem for the heat equation. J. Siberian Fed. Univ., 4:2 (2011), 218-229.\n\nOn the possibility of holomorphic continuation to a domain of functions given on a part of its boundary. L A Aizenberg, A M Kytmanov, Matem. sbornik. 182Aizenberg, L.A., Kytmanov, A.M., On the possibility of holomorphic continuation to a domain of functions given on a part of its boundary, Matem. sbornik. 182:5 (1991), 490-597.\n\nOn the Cauchy Problem for the Lamé system. A Shlapunov, Zeitschrift für Angewandte Mathematik und Mechanik. 76Shlapunov, A., On the Cauchy Problem for the Lamé system. Zeitschrift für Angewandte Mathematik und Mechanik, 76:4 (1996), 215-221.\n\nBases with double orthogonality in the Cauchy problem for systems with injective symbols. A A Shlapunov, N Tarkhanov, Proc. London. Math. Soc. 71Shlapunov, A.A., Tarkhanov, N. Bases with double orthogonality in the Cauchy problem for systems with injective symbols, Proc. London. Math. Soc., 71:1 (1995), 1-54.\n\nOn Cauchy problem for operators with injective symbols in the spaces of distributions. I V Shestakov, A A Shlapunov, Journal Inverse and Ill-posed Problems. 19Shestakov, I.V., Shlapunov A.A. On Cauchy problem for operators with injective symbols in the spaces of distributions, Journal Inverse and Ill-posed Problems, 19 (2011), 127-150.\n\nThe Cauchy Problem for Solutions of Elliptic Equations. N N Tarkhanov, Akademie-VerlagBerlinTarkhanov, N.N. The Cauchy Problem for Solutions of Elliptic Equations. Berlin: Akademie- Verlag, 1995.\n\nLectures on Cauchy's Problem in Linear Partial Differential Equations. J Hadamard, Yale Univ. PressNew Haven-LondonHadamard, J. Lectures on Cauchy's Problem in Linear Partial Differential Equations. / J. Hadamard. -Yale Univ. Press, New Haven-London, 1923.\n\nOn the Cauchy Problem for the Laplace Equation. A A Shlapunov, Siberian Math. J. 33Shlapunov, A.A. On the Cauchy Problem for the Laplace Equation, Siberian Math. J., 33:3 (1992), 205-215.\n\nSecond Order Equations of Elliptic and Parabolic Types. E M Landis, Moscow, NaukaLandis, E.M. Second Order Equations of Elliptic and Parabolic Types, Moscow, Nauka, 1971.\n\nV P Mihailov, Partial Differential Equations. Moscow, NaukaMihailov, V.P. Partial Differential Equations, Moscow, Nauka, 1976.\n\n. A G Sveshnikov, A N Bogolyubov, V V Kravtsov, Lectures on Mathematical Physics. Sveshnikov, A.G., Bogolyubov, A.N., Kravtsov, V.V. Lectures on Mathematical Physics, Moscow, Nauka, 2004.\n\nSvobodnyi 79, 660041 Krasnoyarsk, Russia E-mail address: effervesce@mail.ru. S L Sobolev, pr. Svobodnyi. 79Roman Puzyrev) Siberian Federal University, Institute of Mathematics and Computer Science, pr ; Alexander Shlapunov) Siberian Federal University, Institute of Mathematics and Computer ScienceRussia E-mail address: ashlapunov@sfu-kras.ruSobolev, S.L., (Roman Puzyrev) Siberian Federal University, Institute of Mathematics and Com- puter Science, pr. Svobodnyi 79, 660041 Krasnoyarsk, Russia E-mail address: effervesce@mail.ru (Alexander Shlapunov) Siberian Federal University, Institute of Mathematics and Computer Science, pr. Svobodnyi 79, 660041 Krasnoyarsk, Russia E-mail address: ashlapunov@sfu-kras.ru\n"},"corpus_id":{"kind":"number","value":119112683,"string":"119,112,683"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1904.06797\",\n \"mag\": 2333818229,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1515/jiip-2014-0043\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1904.06797v1.pdf\"\n ],\n \"pdf_sha\": \"51a86184b95e0f61821791dd8f1c3134c77ec07f\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1904.06797\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"ON A MIXED PROBLEM FOR THE PARABOLIC LAMÉ TYPE OPERATOR"},"authors":{"kind":"list like","value":["R Puzyrev","A Shlapunov"],"string":"[\n \"R Puzyrev\",\n \"A Shlapunov\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We consider a boundary value problem for the parabolic Lamé type operator being a linearization of the Navier-Stokes' equations for compressible flow of Newtonian fluids. It consists of recovering a vector-function, satisfying the parabolic Lamé type system in a cylindrical domain, via its values and the values of the boundary stress tensor on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural spaces of smooth functions and in the corresponding Hölder spaces; besides, additional initial data do not turn the problem to a well-posed one. Using the Integral Representation's Method we obtain the Uniqueness Theorem and solvability conditions for the problem.with ν = (ν 1 , ν 2 , ν 3 , ν 4 ) being the unit normal vector to the surface ∂D and δ i,j being the Kronecker symbol (see [1, §15, formula (15.12)])."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Let, as usual, ∆ n be the Laplace operator, ∇ n be the gradient operator and div n be the divergence operator in R n , n ≥ 2. The Navier-Stokes' equations for compressible flow of Newtonian fluids over the four-dimensional domain D ⊂ R 3\nx ×R t under the action of a body force F (x, t) = (F 1 (x, t), F 2 (x, t), F 3 (x, t)) can be written in the following form (see [1, §15, formulas (15.5), (15.6)]):\nwhere v(x, t) = (v 1 (x, t), v 2 (x, t), v 3 (x, t)) is the flow velocity, ρ(x, t) is the fluid density, p(x, t) is the pressure, µ j (x, t) are (positive) viscosity coefficients and av = [(∇ 3 µ 1 ) * ⊗ ∇ 3 − (∇ 3 µ 1 )div 3 ]v is the linear first order summand with M * 1 being the adjoint matrix for a matrix M 1 and M * 1 ⊗M 2 being the Kronecker product of matrices M * 1 and M 2 . If the boundary ∂D of D is piece-wise smooth then the boundary conditions for this system often involve the force νp − σ ′ v acting on the unit surface area where the force friction (or the boundary viscosity tensor) σ ′ has the components\nSince the density ρ is positive, a proper linearization of the substantial derivative term v · ∇ 3 v turns (1) into a parabolic Lamé type system related to an unknown vector u:\nwhere a j (x, t), 0 ≤ j ≤ 3, are (3 × 3) matrices with functional entries and L n = div n µ∇ n + ∇ n (µ + λ)div n , n ≥ 2 is the strongly elliptic (with respect to the space variables) formally self-adjoint Lamé type operator with the Lamé coefficients satisfying µ(x, t) > 0, (µ(x, t) + λ(x, t)) ≥ 0.\nThe regularity (smoothness) of the Lamé coefficients and the matrices a j (x, t) depends upon the regularity of the density ρ and the viscosity coefficients µ j . Note that if µ is constant, λ + µ = 0 and a j = 0, 0 ≤ j ≤ 3, then L 4 reduces to the heat operator, though, of course, it is known that the heat equation is not ideal to model the process of the heat conduction.\nLet Ω be a bounded domain (i.e. bounded open connected set) in n-dimensional real space R n with the coordinates x = (x 1 , . . . , x n ). As usual we denote by Ω the closure of Ω, and we denote by ∂Ω its boundary. In the sequel we assume that ∂Ω is piece-wise smooth. As ∂Ω is piece-wise smooth, the normal vector ν = (ν 1 , ..., ν n ) is defined almost everywhere on ∂Ω and satisfies n j=1 ν 2 j = 0. Let Ω T = {x ∈ Ω, 0 < t < T } an open cylinder, having the altitude 0 < T ≤ +∞ and the base Ω, in (n + 1)-dimensional real space R n+1 = R n × {−∞ < t < +∞}. Let also Γ ⊂ ∂Ω be a non empty connected open (in the topology of ∂Ω) subset of ∂Ω and Γ T = Γ × (0, T ).\nIn the present paper we consider a mixed boundary problem for the parabolic system in the cylindrical domain Ω T\nthe Lamé coefficients and the entries of the (n × n)-matrices a j (x, t), 0 ≤ j ≤ n, are C ∞ -smooth in a neighborhood of Ω T and real analytic with respect to the space variables in a neighborhood of Ω. Instead of classical boundary value problems for parabolic equations (see, for instance, [2], [3], [4], [5]) we consider the ill-posed problem, consisting in finding a vector-function satisfying the corresponding parabolic equation in the cylinder via its values and the values of the boundary stress tensor with the components\non the given part Γ T of the lateral surface of the cylinder Ω T (cf. [6]). Using parabolic potentials we prove Uniqueness Theorem and obtain solvability conditions for the problem (cf. [7] related to similar results for the heat equation). Actually, the approach was invented for the investigation of the famous ill-posed Cauchy problem for elliptic equations (see, for instance, [8] for the Cauchy-Riemann operator, [9] for the elliptic Lamé operator and [10], [11], [12], for general systems with injective principal symbols)."},{"title":"Preliminaries","content":"As usual, for s ∈ Z + (here Z + = N ∪ {0}) and an open subset D ⊂ R m we denote C s (D) the set of all s times continuously differentiable functions in D. The standard topology of this metrisable space induces uniform convergence on compact subsets in D together with all the partial derivatives up to order s.\nFor S ⊂ ∂D we denote C s (D ∪S) the set of such functions from the space C s (D) that all their derivatives up to order s can be extended continuously onto D ∪ S. The standard topology of this metrisable space induces uniform convergence on compact subsets in D ∪ S together with all the partial derivatives up to order s. In particular, for bounded domains, C s (D ∪ ∂D) = C s (D) is a Banach space. If D is an unbounded then the Banach space C s b (D ∪ S) consists of s-times differentiable functions in D ∪ S with bounded derivatives up to order s and it is endowed with the standard sup-norm. Then C s b (D) = C s (D) for a bounded domain D. Apart from the standard functional spaces, we need also spaces taking into account the specific properties of parabolic equations in R n+1 = R n × {−∞ < t < +∞}. Namely, let C 1,0 (Ω T ) be the set of continuous functions u in Ω T , having in Ω T continuous partial derivatives u xi , and let C 2,1 (Ω T ) denote the set of continuous functions in Ω T , having in Ω T continuous partial derivatives u xi , u xixj , u t . The standard topology of this metrisable space induces uniform convergence on compact subsets in D together with all the partial derivatives used in its definition.\nAs before, for S ⊂ ∂Ω T we denote by C 1,0 (Ω T ∪S) the set of such functions u from the space C 1,0 (Ω T ) that their derivatives u xi can be extended continuously onto Ω T ∪S. The standard topology of this metrisable space induces uniform convergence on compact subsets of Ω T ∪S of both the functional sequences and the corresponding sequences of first partial derivatives x i . Clearly, C 1,0 (Ω T ∪ ∂Ω T ) = C 1,0 (Ω T ) is a Banach space. Similarly to the standard spaces, if D ⊂ R n+1 the Banach space C 1,0 b (D ∪ S) consisting of bounded C 0,1 (D ∪ S)-functions with bounded derivatives u xi in D ∪ S and it is endowed with the norm\nThen C 0,1 b (D) = C 0,1 (D) for a bounded domain D. The space of n-vector-functions u = (u 1 , . . . , u n ) of a class C will be denoted by\nLet now θ be such positive constant that for all (x, t) ∈ Ω T we have\nThen a direct calculation shows that for all ζ ∈ R n we have det µ(x, t)|ζ| 2 ( √ −1) 2 I n + (λ(x, t) + µ(x, t))ζζ T ( √ −1) 2 − κI n = (−1) n (µ(x, t)|ζ| 2 + κ) n−1 ((2µ(x, t) + λ(x, t))|ζ| 2 + κ)\nwhere I n is the unit (n × n) matrix and ζ T is the transposed vector for ζ. Hence the roots of this polynomial (with respect to κ) are\nand, for all (x, t) ∈ Ω T , we have\ni.e. the operator L n is uniformly parabolic (according to Petrovskii) on Ω T . Now we assume that there is a n-dimensional domain U ⊃ Ω such that the Lamé coefficients µ(x, t), λ(x, t) and the entries of the (n×n)-matrices a j (x, t), 0 ≤ j ≤ n, are C ∞ -smooth in U T and real analytic with respect to the space variables in U .\nUnder the assumptions, the following properties hold true for parabolic operator L n+1 , which will be crucial for the approach below. Theorem 1. Each weak solution u to L n+1 u = 0 in the domain Ω T ⊂ U T belongs to C ∞ (Ω T ) and it is actually real analytic with respect to variables x in Ω.\nTheorem 2. The operator L n+1 has a fundamental solution in U T , i.e. a (n × n)matrix Φ(x, t, y, τ ) satisfying\njj with the formal adjoint operators\nProof See, for instance, [4,Ch. 2]. We need a sort of an integral representation, similar to the famous Green Formula for the Laplace Operator, constructed with the use the fundamental solutions. More precisely, consider the cylinder type domain Ω T1,T2 = Ω T2 \\ Ω T1 and a closed measurable set S ⊂ ∂Ω.\nLet σ be the tensor with the components given by (2) and\n.\nwhere ds is the volume form on S induced from R n . All these functions are called Parabolic Potentials with densities f , v, w and h respectively. In our situation these are convergent improper integrals depending on vector parameter (x, t) in the neighborhood U of the cylinder Ω T1,T2 in R n+1 (see, for instance, [\n) the following formula holds:\nProof. Indeed, it follows from Gauß-Ostrogradskii Formula that\nOn the other hand, by Gauß-Ostrogradskii Formula,\nHence, again by Gauß-Ostrogradskii Formula, we obtain the (first) Green formula for the Lamé type operator:\nIt follows from the definition of the fundamental solution, that\nThen, using the standard argument (see, for instance, [17,Ch. 6, §12] for the heat equation), we see that Green's Formula (8) follows from (12) and Fubini Theorem.\nTheorem 3 (Uniqueness Theorem). If Γ has at least one interior point (on ∂Ω),\nProof. Under the hypothesis of the theorem there is an interior point x 0 on Γ.\nThen there is such a number\nunder the hypothesis of the theorem. Hence formula (8) implies:\nTaking into account the character of the singularity of the kernel (see [4, Theorem 2.2]) Φ(x, y, t, τ ) we conclude that the following properties are fulfilled for the integrals, depending on parameter, from the right hand side of identity (13): (3) and Leibniz rule for differentiation of integrals depending on parameter we obtain:\nHence the function\nsatisfies the Lamé type equation\nThis implies that the function P (x, t) is real analytic with respect to the space variable x ∈ U \\ (∂Ω ′ \\ Γ) for any T 1 < t < T 2 (see, for instance, [16, Ch. VI, §1, Theorem 1]). In particular, by the construction the function P (x, t) is real analytic with respect to x in the ball B(x 0 , r) and it equals to zero for x ∈ B(x 0 , R) \\ Ω for all T 1 < t < T 2 . Therefore, the Uniqueness Theorem for real analytic functions yields P (x, t) ≡ 0 in (U \\ (∂Ω ′ \\ Γ)) T1,T2 , and in the cylinder Ω ′ T1,T2 , containing the point (x ′ , t ′ ). Now it follows from (13) that u(x ′ , t ′ ) = P (x ′ , t ′ ) = 0 and then, since the point (x ′ , t ′ ) ∈ Ω T is arbitrary we conclude that u ≡ 0 in Ω T . The proof is complete. Example 1. Let µ = 1, λ = −1 and a j = 0, 0 ≤ j ≤ n. Then L n+1 reduces to the heat operator:\nand corresponding fundamental solution is given by\nIn this caseσ = σ = ∂ ∂ν . Example 2. Let µ, λ be constant and a j = 0, 0 ≤ j ≤ n. Then L n+1 reduces to the parabolic Lamé operator\n(see, for instance, [4]). In this caseσ = σ = µ ∂ ∂ν + µ ν * ⊗ ∇ n + λ ν div n ."},{"title":"The boundary problem","content":"Green formula (8) and the Uniqueness Theorem 3 suggest us to consider two kind of problems for the parabolic Lamé type equation.\nLet vector-functions\nbe given. and boundary conditions\nNote that, if the surface Γ and the data of the problem are real analytic then the Cauchy-Kovalevsky Theorem implies that Problem 1 can not have more than one solution in the class of (even formal) power series. However the theorem does not imply the existence of solutions to Problem 1 because it grants the solution in a small neighborhood of the surface Γ T only (but not in a given domain Ω T !). In any case, we do not assume the real analyticity of Γ and the data u (1) , u (2) and f . Corollary 1. If Γ has at least one interior point (on ∂Ω) then Problem 1 has no more that one solution.\nProof. Let v(x, t) and w(x, t) be two solutions to Problem 1. Then function\nis a solution to the corresponding problem with f = 0, u 1 = 0, u 2 = 0. Using 3 we conclude that u is identically zero in Ω T .\nThus, the Uniqueness Theorem 3 implies that the data of Problem 1 are suitable in order to uniquely define its solution.\nEasily, Problem 1 is ill-posed because this is the property of the Cauchy problem for elliptic systems in R n (see, for instance [13] or [16, Ch. 1, §2]). Of course, in this case the boundary data should be taken independent on t. The Uniqueness Theorem clarify why the problem is ill-posed. The reason is the redundant data. Indeed, if Γ has at least one interior point (on ∂Ω), then taking a smaller set Γ ′ ⊂ Γ we again obtain a problem with no more than one solution.\nAnother problem involves the initial data.\nin Ω T Lamé type equation (14), boundary conditions (15), (16) and initial condition\nOf course one should also take care on the compatibility of the data u (0) , u (1) , u (2) : at least\nand, if u (0) ∈ C 1 (Ω), even\nThe motivation of Problems 1 and 2 is transparent. The space C b (Ω T ) is chosen because u represents the \"velocity\". The first problem describes the situation where for some reasons at each time t ≥ 0 only part Γ of the solid surface ∂Ω bounding the fluid is available for measurements. The second one describes the situation where the continuity up to ∂Ω T is postulated, the \"velocity\" u is known at every point x ∈ Ω at the initial time t = 0 but the data on ∂Ω \\ Γ were lost for t > 0.\nClearly, Problem 2 has no more than one solution, too, if Γ has at least one interior point (on ∂Ω).\nWe note that in classical theory of (initial and) boundary problems for the parabolic equation (14), initial condition (17) and boundary condition αu + βσ = u (3) on the whole lateral surface ∂Ω T of the cylinder Ω T are usually considered. As a rule, such a problem is well-posed in proper spaces (Hölder spaces, Sobolev spaces etc.), see, for instance, [2].\nLet us show that Problem 2 is ill-posed, too.\nExample 3. Let the Lamé coefficients µ, λ be constant and a j = 0, 0 ≤ j ≤ n. Take a cube Q n = {0 < x j < 1, 1 ≤ j ≤ n} as base Ω of the cylinder Ω T . Let Γ be the face {x n = 0} of the cube Q n . Then Γ T = Q n−1 × (0, T ) and the stress tensor σ is given by the diagonal matrix with the non-zero entries\nFix N ∈ N and consider the sequence of functions u(x, t, k, r) ∈ [C ∞ (R n+1 )] n with the components: u 1 (x, t, k, r) = 0, . . . , u n−1 (x, t, k, r) = 0, u n (x, t, k, r) = e k 2 (2µ+λ)(t−r)+kxn k N , depending on a parameter 0 < r < +∞. Consider also the data f (x, t, k, r), u (0) (x, t, k, r), u (1) (x, t, k, r), u (2) (x, t, k, r) having the following components:\nj (x 1 , . . . , x n−1 , t, k, r) = 0, 1 ≤ j ≤ n − 1,\nj (x 1 , . . . , x n−1 , t, k, r) = 0, 1 ≤ j ≤ n − 1,\nThen, for 0 < T < +∞, each function u(x, t, k, T ) is a solution to problem (14), (15), (16), (17) with the data f (x, t, k, T ),"},{"title":"It is clear, that compatibility conditions (18), (19) hold and","content":"if N > 2s + 1. On the other hand, for all x n > 0 and all N ∈ N we have:\nThus, there is no continuity with respect to the data and hence Problem 2 is ill-posed for 0 < T < +∞. Let now T = +∞. Then we may consider the data with a fixed 0 < T 0 < +∞:\nObviously, for N ≥ 2,\nThe Uniqueness Theorem 3 for Problem 3 implies that\nThen, for all x n > 0 and all 2 ≤ N ∈ N, we have lim k→+∞ u n (x, T 0 , k, ∞) = +∞. Thus, if the data f , u (0) , u (2) , u (2) admits the solution to (14) in Ω T with boundary conditions (15), (16) for T = +∞ and the initial condition (17) then there is no continuity with respect to the data in the chosen space. Otherwise there is no solutions to the problem for some data in the data's spaces. In any case, the problem is ill-posed for T = +∞, too.\nAs both Problems 1 and 2 are ill-posed, we will not study Problem 2 because in addition to (14)- (16) to investigate it one needs to know also the data related to initial condition (17). Besides, we will consider the case 0 < T < +∞ only."},{"title":"Solvability Conditions","content":"From now on we will study Problem 1 under the assumption that its data belong to Hölder spaces (cf., [3,Ch. 1,§1] for other boundary problems for parabolic equations). We recall that a function u(x), defined on a set M ∈ R m , is called Hölder continuous with a power 0 < λ < 1 on M , if there is such a constant C > 0 that\nbeing Euclidean distance between points x and y in R m ). Let C λ (Ω T ) stand for the set of Hölder continuous functions with a power λ over Ω T . Besides, let C 1+λ,λ (Ω T ) be the set of Hölder continuous functions with a power λ over Ω T , having Hölder continuous derivatives u xi , 1 ≤ i ≤ n, with the same power in Ω T .\nWe choose a set Ω + in such a way that the set D = Ω ∪ Γ ∪ Ω + would be a bounded domain with piece-wise smooth boundary. It is possible since Γ is an open connected set. It is convenient to set Ω − = Ω. For a function v on D T we denote by v + its restriction to Ω + and, similarly, we denote by v − its restriction to Ω. It is natural to denote limit values of v ± on Γ T , when they are defined, by v ± |ΓT ."},{"title":"Theorem 4 (Solvability criterion). Let","content":"Problem 1 is solvable if and only if there is a vector-function F ∈ [C 2,1 (D T )] n satisfying the following conditions:\nProof. Necessity. Let a function u(x, t) ∈ [C 2,1 (Ω T ) ∩ C 1,0 (Ω T ∪ Γ T )] n satisfies (14), (15), (15). Consider the function\nin the domain D T , where χ M is a characteristic function of the set M ⊂ R n+1 . By the very construction condition 2) is fulfilled for it.\nClearly, the function u(x, t) belongs to the space [C 1,2 (Ω ′ T )] n for each cylindrical domain Ω ′ T with such a base Ω ′ that Ω ′ ⊂ Ω and Ω ′ ∩ ∂Ω ⊂ Γ. Besides, L n+1 u = f ∈ [C λ (Ω ′ T )] n . Without loss of the generality we may assume that the interior part Γ ′ of the set Ω ′ ∩ ∂Ω is non-empty.\nWe note that\nThen using Lemma 2 we obtain:\nArguing as in the proof of Theorem 3 we conclude that each of the integrals in the right hand side of (21) satisfies homogeneous Lamé type equation outside the corresponding integration set. In particular, we see that L n+1 F = 0 in D ′ T . Obviously, for any point (x, t) ∈ D T there is a domain D ′ T containing (x, t). That is why L n+1 F = 0 in D T , and hence F belongs to the space [C 2,1 (D T )] n . Thus this function satisfies condition 1), too.\nSufficiency. Let there be a function F ∈ [C 2,1 (D T )] n , satisfying conditions 1) and 2) of the theorem. Consider on the set D T the function\nAs f ∈ [C λ (Ω T )] n then the results of [3, Ch. 1, §3] imply\nand, moreover,\nSince u 2 ∈ [C λ (Γ T )] n then the results of [3, Ch. 5, §2]yield\nOn the other hand, the behavior of the Double Layer Potential W Γ,0 (u 1 ) is similar to the behavior of the normal derivative of Single Layer Potential V Γ,0 (u 1 ). Hence\nLemma 2. Let S ⊂ Γ ∈ C 1+λ . If u 1 ∈ [C 1+λ,λ (Γ T )] n , then the potential W − Γ,0 (u 1 ) belongs to the space [C 1,0 (Ω T ∪ S T )] n if and only if W + Γ,0 (u 2 ) ∈ [C 1,0 (Ω + T ∪ S T )] n .\nProof. It is similar to the proof of the analogous lemma for Newton Double Layer Potential (see, for instance, [14, lemma 1.1]). Actually, one needs to use Lemma 2 instead of the standard Green formula for the Laplace operator.\nSince F ∈ [C 1,0 (D T )] n then it follows from the discussion above that W + Γ,0 (u 2 ) ∈ [C 1,0 ((Ω + ∪ Γ) T )] n . Thus, formulas (22)-(28) and Lemma 2 imply that\nLet us show that the function U − satisfies (15) and (16).\nIt follows from formulas (23) and (25) that the Parabolic Volume Potential and the Single Layer Parabolic Potential are continuous if the point (x, t) passes over the surface Γ T . Then Formula (23) means that that the surface stress of the Parabolic Volume Potential is continuous if the point (x, t) passes over the surface Γ T . Therefore\nBy theorem on jump behavior of the stress of the Parabolic Single Layer Potential (see, for instance, [15,Ch. 3, §10, theorem 10.1])\nFinally, we need the following lemma which is an analogue of the famous Theorem on jump behavior of the normal derivative of the Newton's Double Layer Potential.\nProof. Really, let, for instance, W − Γ,0 (u 1 ) ∈ [C 1,0 ((Ω ∪ Γ) T )] n . Then using Lemma 2 we obtain W + Γ,0 u 1 ∈ [C 1,0 ((Ω + ∪ Γ) T )] n and σW ± Γ,0 (u 1 )\nLet φ ∈ [C ∞ 0 (D T )] n be a function with compact support in D T . Then formulas (9)-(11) yield:\nbecause L n+1 W ± Γ,0 u 1 = 0 in Ω ± according to (28). Again, integrating by parts and using formulas (9)- (11) and Theorem on jump behavior of the Parabolic Double Layer Potential, we see that\nBut the kernel Φ(x, y, t, τ ) is a fundamental solution of the backward parabolic operator L * n+1 with respect to variables (y, τ ). Hence DT (L * n+1 φ(x, t)) * Φ(x, y, t, τ )dxdt = φ * (y, τ ), (y, τ ) ∈ D T .\nThen the type of the singularity of the fundamental solution allows us to apply Fubini Theorem and to conclude that\nFinally, formulas (32)-(34) imply that In particular, if S = ∅ then corollary 2 gives criterion for the existence of solution to Problem 1 in the space [C(Ω T )] n .\nWe note that Theorem 4 is an analogue of Theorem by Aizenberg and Kytmanov [8]) describing solvability conditions of the Cauchy problem for the Cauchy-Riemann system (cf. also [14] in the Cauchy Problem for Laplace Equation or [12] in the Cauchy problem for general elliptic systems). Formula (22), obtained in the proof of Theorem 4, gives the unique solution to Problem 1. Clearly, if we will be able to write the extension F of the sum of potentials G Ω,0 (f )+ V Γ,0 (u 2 )+ W Γ,0 (u 1 ) from Ω + onto D T as a series with respect to special functions or a limit of parameter depending integrals then we will get a Carleman type formula for solutions to Problem 1 (cf. [8]). However this is a topic for another paper. In the sequel we will discuss polynomial and formal solutions for operators with constant coefficients only. Proof. First let us prove that if in this case the data of Problem 1 are polynomials then the problem is solvable and its solution is a polynomial."},{"title":"Polynomial solutions and dense solvability","content":"Indeed, Problem 1 is easily can be reduced to the following one (see Example 3):\nwith\nwhere J(µ, λ) is the diagonal matrix with the components\nto problem (35)-(37) where\nand, by the induction with respect to j ∈ Z + ,\nTo finish the arguments we use the induction with respect to |α ′ | ∈ Z + where α ′ = (α 1 , . . . , α n−1 ) ∈ Z n−1 + . Namely, let for s ≥ 2 and all α ′ with |α ′ | = s the solutions to the problem are polynomial. If |α ′ | = s + 1 then L n+1 x α1 1 · · · x αn−1 n−1 w (j,αn) (x n , t) = t j x α − w (j,αn) (x n , t)∆ n−1 x α1 1 · · · x αn−1 n−1 .\nClearly, the degree of the polynomial p j,α (x, t) = w (j,αn) (x n , t)∆ n−1 x α1 1 · · · x αn−1 n−1 with respect to x ′ ∈ R n−1 equals to s − 1. Then, by the induction, problem (14)- (16) with data p j,α (x, t) admits a polynomial solution, say, r j,α (x, t). Therefore the solution v (j,α) (x, t) to problem (14)-(16) with data g (j,α) (x, t) = t j x α , |α ′ | = s + 1, is given as follows:\nv (j,α) (x, t) = x α1 1 · · · x αn−1 n−1 w (j,αn) (x n , t) + r j,α (x, t), i.e. it is a polynomial, too. Now Problem 1 with zero boundary data in the case Γ ⊂ {x n = 0} is densely solvable because any continuous function g on the compact set Ω T can be approximated by polynomials. But the reducing to zero boundary data was organized in such a way that one easily sees, in this case Problem 1 is densely solvable for non-zero boundary data, too.\nThe dense solvability of Problem 1 in general setting is natural to expect if the set ∂Ω \\ Γ has at least one interior point in ∂Ω (cf. [10] in the Cauchy Problem for elliptic equations).\nFinally, we note that polynomial solutions indicated in the proof of Lemma 4 can be used in order to construct formal solutions to Problem 1."},{"title":"Basis with double orthogonality","content":"Denote by {h (s) ν } the set of harmonic homogeneous polynomials (spherical harmonics) forming an orthonormal basis in L 2 (∂B(0, 1)); here ν is the degree of the homogeneity and 1 ≤ s ≤ J(s) where J(s) = (2n + 2s − 2)(2n + s − 3)! s!(2n − 2)! (see [18]). On the other hand, ∂|x| 2k ∂x j = 2kx j |x| 2k−2 , ∂ 2 |x| 2k ∂x 2 j = 2k|x| 2k−2 + 2k(2k − 2)x 2 j |x| 2k−4 ∆ n |x| 2k = 2k(n + 2k − 2)|x| 2k−2 , k ≥ 1.\nHence, for ν ≥ 1, k ≥ 1, ∆ n (|x| 2k h ν (x)) = 2k(n + 2k + 2ν − 2)|x| 2k−2 h (s) ν (x) because of Euler's formula"},{"title":"Problem 1 .","content":""},{"title":"Corollary 2 .","content":""},{"title":"Lemma 4 .","content":""},{"title":"Lemma 5 . 1 .ν","content":""},{"title":"T2 )","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Let, as usual, ∆ n be the Laplace operator, ∇ n be the gradient operator and div n be the divergence operator in R n , n ≥ 2. The Navier-Stokes' equations for compressible flow of Newtonian fluids over the four-dimensional domain D ⊂ R 3\\nx ×R t under the action of a body force F (x, t) = (F 1 (x, t), F 2 (x, t), F 3 (x, t)) can be written in the following form (see [1, §15, formulas (15.5), (15.6)]):\\nwhere v(x, t) = (v 1 (x, t), v 2 (x, t), v 3 (x, t)) is the flow velocity, ρ(x, t) is the fluid density, p(x, t) is the pressure, µ j (x, t) are (positive) viscosity coefficients and av = [(∇ 3 µ 1 ) * ⊗ ∇ 3 − (∇ 3 µ 1 )div 3 ]v is the linear first order summand with M * 1 being the adjoint matrix for a matrix M 1 and M * 1 ⊗M 2 being the Kronecker product of matrices M * 1 and M 2 . If the boundary ∂D of D is piece-wise smooth then the boundary conditions for this system often involve the force νp − σ ′ v acting on the unit surface area where the force friction (or the boundary viscosity tensor) σ ′ has the components\\nSince the density ρ is positive, a proper linearization of the substantial derivative term v · ∇ 3 v turns (1) into a parabolic Lamé type system related to an unknown vector u:\\nwhere a j (x, t), 0 ≤ j ≤ 3, are (3 × 3) matrices with functional entries and L n = div n µ∇ n + ∇ n (µ + λ)div n , n ≥ 2 is the strongly elliptic (with respect to the space variables) formally self-adjoint Lamé type operator with the Lamé coefficients satisfying µ(x, t) > 0, (µ(x, t) + λ(x, t)) ≥ 0.\\nThe regularity (smoothness) of the Lamé coefficients and the matrices a j (x, t) depends upon the regularity of the density ρ and the viscosity coefficients µ j . Note that if µ is constant, λ + µ = 0 and a j = 0, 0 ≤ j ≤ 3, then L 4 reduces to the heat operator, though, of course, it is known that the heat equation is not ideal to model the process of the heat conduction.\\nLet Ω be a bounded domain (i.e. bounded open connected set) in n-dimensional real space R n with the coordinates x = (x 1 , . . . , x n ). As usual we denote by Ω the closure of Ω, and we denote by ∂Ω its boundary. In the sequel we assume that ∂Ω is piece-wise smooth. As ∂Ω is piece-wise smooth, the normal vector ν = (ν 1 , ..., ν n ) is defined almost everywhere on ∂Ω and satisfies n j=1 ν 2 j = 0. Let Ω T = {x ∈ Ω, 0 < t < T } an open cylinder, having the altitude 0 < T ≤ +∞ and the base Ω, in (n + 1)-dimensional real space R n+1 = R n × {−∞ < t < +∞}. Let also Γ ⊂ ∂Ω be a non empty connected open (in the topology of ∂Ω) subset of ∂Ω and Γ T = Γ × (0, T ).\\nIn the present paper we consider a mixed boundary problem for the parabolic system in the cylindrical domain Ω T\\nthe Lamé coefficients and the entries of the (n × n)-matrices a j (x, t), 0 ≤ j ≤ n, are C ∞ -smooth in a neighborhood of Ω T and real analytic with respect to the space variables in a neighborhood of Ω. Instead of classical boundary value problems for parabolic equations (see, for instance, [2], [3], [4], [5]) we consider the ill-posed problem, consisting in finding a vector-function satisfying the corresponding parabolic equation in the cylinder via its values and the values of the boundary stress tensor with the components\\non the given part Γ T of the lateral surface of the cylinder Ω T (cf. [6]). Using parabolic potentials we prove Uniqueness Theorem and obtain solvability conditions for the problem (cf. [7] related to similar results for the heat equation). Actually, the approach was invented for the investigation of the famous ill-posed Cauchy problem for elliptic equations (see, for instance, [8] for the Cauchy-Riemann operator, [9] for the elliptic Lamé operator and [10], [11], [12], for general systems with injective principal symbols).\"\n },\n {\n \"title\": \"Preliminaries\",\n \"content\": \"As usual, for s ∈ Z + (here Z + = N ∪ {0}) and an open subset D ⊂ R m we denote C s (D) the set of all s times continuously differentiable functions in D. The standard topology of this metrisable space induces uniform convergence on compact subsets in D together with all the partial derivatives up to order s.\\nFor S ⊂ ∂D we denote C s (D ∪S) the set of such functions from the space C s (D) that all their derivatives up to order s can be extended continuously onto D ∪ S. The standard topology of this metrisable space induces uniform convergence on compact subsets in D ∪ S together with all the partial derivatives up to order s. In particular, for bounded domains, C s (D ∪ ∂D) = C s (D) is a Banach space. If D is an unbounded then the Banach space C s b (D ∪ S) consists of s-times differentiable functions in D ∪ S with bounded derivatives up to order s and it is endowed with the standard sup-norm. Then C s b (D) = C s (D) for a bounded domain D. Apart from the standard functional spaces, we need also spaces taking into account the specific properties of parabolic equations in R n+1 = R n × {−∞ < t < +∞}. Namely, let C 1,0 (Ω T ) be the set of continuous functions u in Ω T , having in Ω T continuous partial derivatives u xi , and let C 2,1 (Ω T ) denote the set of continuous functions in Ω T , having in Ω T continuous partial derivatives u xi , u xixj , u t . The standard topology of this metrisable space induces uniform convergence on compact subsets in D together with all the partial derivatives used in its definition.\\nAs before, for S ⊂ ∂Ω T we denote by C 1,0 (Ω T ∪S) the set of such functions u from the space C 1,0 (Ω T ) that their derivatives u xi can be extended continuously onto Ω T ∪S. The standard topology of this metrisable space induces uniform convergence on compact subsets of Ω T ∪S of both the functional sequences and the corresponding sequences of first partial derivatives x i . Clearly, C 1,0 (Ω T ∪ ∂Ω T ) = C 1,0 (Ω T ) is a Banach space. Similarly to the standard spaces, if D ⊂ R n+1 the Banach space C 1,0 b (D ∪ S) consisting of bounded C 0,1 (D ∪ S)-functions with bounded derivatives u xi in D ∪ S and it is endowed with the norm\\nThen C 0,1 b (D) = C 0,1 (D) for a bounded domain D. The space of n-vector-functions u = (u 1 , . . . , u n ) of a class C will be denoted by\\nLet now θ be such positive constant that for all (x, t) ∈ Ω T we have\\nThen a direct calculation shows that for all ζ ∈ R n we have det µ(x, t)|ζ| 2 ( √ −1) 2 I n + (λ(x, t) + µ(x, t))ζζ T ( √ −1) 2 − κI n = (−1) n (µ(x, t)|ζ| 2 + κ) n−1 ((2µ(x, t) + λ(x, t))|ζ| 2 + κ)\\nwhere I n is the unit (n × n) matrix and ζ T is the transposed vector for ζ. Hence the roots of this polynomial (with respect to κ) are\\nand, for all (x, t) ∈ Ω T , we have\\ni.e. the operator L n is uniformly parabolic (according to Petrovskii) on Ω T . Now we assume that there is a n-dimensional domain U ⊃ Ω such that the Lamé coefficients µ(x, t), λ(x, t) and the entries of the (n×n)-matrices a j (x, t), 0 ≤ j ≤ n, are C ∞ -smooth in U T and real analytic with respect to the space variables in U .\\nUnder the assumptions, the following properties hold true for parabolic operator L n+1 , which will be crucial for the approach below. Theorem 1. Each weak solution u to L n+1 u = 0 in the domain Ω T ⊂ U T belongs to C ∞ (Ω T ) and it is actually real analytic with respect to variables x in Ω.\\nTheorem 2. The operator L n+1 has a fundamental solution in U T , i.e. a (n × n)matrix Φ(x, t, y, τ ) satisfying\\njj with the formal adjoint operators\\nProof See, for instance, [4,Ch. 2]. We need a sort of an integral representation, similar to the famous Green Formula for the Laplace Operator, constructed with the use the fundamental solutions. More precisely, consider the cylinder type domain Ω T1,T2 = Ω T2 \\\\ Ω T1 and a closed measurable set S ⊂ ∂Ω.\\nLet σ be the tensor with the components given by (2) and\\n.\\nwhere ds is the volume form on S induced from R n . All these functions are called Parabolic Potentials with densities f , v, w and h respectively. In our situation these are convergent improper integrals depending on vector parameter (x, t) in the neighborhood U of the cylinder Ω T1,T2 in R n+1 (see, for instance, [\\n) the following formula holds:\\nProof. Indeed, it follows from Gauß-Ostrogradskii Formula that\\nOn the other hand, by Gauß-Ostrogradskii Formula,\\nHence, again by Gauß-Ostrogradskii Formula, we obtain the (first) Green formula for the Lamé type operator:\\nIt follows from the definition of the fundamental solution, that\\nThen, using the standard argument (see, for instance, [17,Ch. 6, §12] for the heat equation), we see that Green's Formula (8) follows from (12) and Fubini Theorem.\\nTheorem 3 (Uniqueness Theorem). If Γ has at least one interior point (on ∂Ω),\\nProof. Under the hypothesis of the theorem there is an interior point x 0 on Γ.\\nThen there is such a number\\nunder the hypothesis of the theorem. Hence formula (8) implies:\\nTaking into account the character of the singularity of the kernel (see [4, Theorem 2.2]) Φ(x, y, t, τ ) we conclude that the following properties are fulfilled for the integrals, depending on parameter, from the right hand side of identity (13): (3) and Leibniz rule for differentiation of integrals depending on parameter we obtain:\\nHence the function\\nsatisfies the Lamé type equation\\nThis implies that the function P (x, t) is real analytic with respect to the space variable x ∈ U \\\\ (∂Ω ′ \\\\ Γ) for any T 1 < t < T 2 (see, for instance, [16, Ch. VI, §1, Theorem 1]). In particular, by the construction the function P (x, t) is real analytic with respect to x in the ball B(x 0 , r) and it equals to zero for x ∈ B(x 0 , R) \\\\ Ω for all T 1 < t < T 2 . Therefore, the Uniqueness Theorem for real analytic functions yields P (x, t) ≡ 0 in (U \\\\ (∂Ω ′ \\\\ Γ)) T1,T2 , and in the cylinder Ω ′ T1,T2 , containing the point (x ′ , t ′ ). Now it follows from (13) that u(x ′ , t ′ ) = P (x ′ , t ′ ) = 0 and then, since the point (x ′ , t ′ ) ∈ Ω T is arbitrary we conclude that u ≡ 0 in Ω T . The proof is complete. Example 1. Let µ = 1, λ = −1 and a j = 0, 0 ≤ j ≤ n. Then L n+1 reduces to the heat operator:\\nand corresponding fundamental solution is given by\\nIn this caseσ = σ = ∂ ∂ν . Example 2. Let µ, λ be constant and a j = 0, 0 ≤ j ≤ n. Then L n+1 reduces to the parabolic Lamé operator\\n(see, for instance, [4]). In this caseσ = σ = µ ∂ ∂ν + µ ν * ⊗ ∇ n + λ ν div n .\"\n },\n {\n \"title\": \"The boundary problem\",\n \"content\": \"Green formula (8) and the Uniqueness Theorem 3 suggest us to consider two kind of problems for the parabolic Lamé type equation.\\nLet vector-functions\\nbe given. and boundary conditions\\nNote that, if the surface Γ and the data of the problem are real analytic then the Cauchy-Kovalevsky Theorem implies that Problem 1 can not have more than one solution in the class of (even formal) power series. However the theorem does not imply the existence of solutions to Problem 1 because it grants the solution in a small neighborhood of the surface Γ T only (but not in a given domain Ω T !). In any case, we do not assume the real analyticity of Γ and the data u (1) , u (2) and f . Corollary 1. If Γ has at least one interior point (on ∂Ω) then Problem 1 has no more that one solution.\\nProof. Let v(x, t) and w(x, t) be two solutions to Problem 1. Then function\\nis a solution to the corresponding problem with f = 0, u 1 = 0, u 2 = 0. Using 3 we conclude that u is identically zero in Ω T .\\nThus, the Uniqueness Theorem 3 implies that the data of Problem 1 are suitable in order to uniquely define its solution.\\nEasily, Problem 1 is ill-posed because this is the property of the Cauchy problem for elliptic systems in R n (see, for instance [13] or [16, Ch. 1, §2]). Of course, in this case the boundary data should be taken independent on t. The Uniqueness Theorem clarify why the problem is ill-posed. The reason is the redundant data. Indeed, if Γ has at least one interior point (on ∂Ω), then taking a smaller set Γ ′ ⊂ Γ we again obtain a problem with no more than one solution.\\nAnother problem involves the initial data.\\nin Ω T Lamé type equation (14), boundary conditions (15), (16) and initial condition\\nOf course one should also take care on the compatibility of the data u (0) , u (1) , u (2) : at least\\nand, if u (0) ∈ C 1 (Ω), even\\nThe motivation of Problems 1 and 2 is transparent. The space C b (Ω T ) is chosen because u represents the \\\"velocity\\\". The first problem describes the situation where for some reasons at each time t ≥ 0 only part Γ of the solid surface ∂Ω bounding the fluid is available for measurements. The second one describes the situation where the continuity up to ∂Ω T is postulated, the \\\"velocity\\\" u is known at every point x ∈ Ω at the initial time t = 0 but the data on ∂Ω \\\\ Γ were lost for t > 0.\\nClearly, Problem 2 has no more than one solution, too, if Γ has at least one interior point (on ∂Ω).\\nWe note that in classical theory of (initial and) boundary problems for the parabolic equation (14), initial condition (17) and boundary condition αu + βσ = u (3) on the whole lateral surface ∂Ω T of the cylinder Ω T are usually considered. As a rule, such a problem is well-posed in proper spaces (Hölder spaces, Sobolev spaces etc.), see, for instance, [2].\\nLet us show that Problem 2 is ill-posed, too.\\nExample 3. Let the Lamé coefficients µ, λ be constant and a j = 0, 0 ≤ j ≤ n. Take a cube Q n = {0 < x j < 1, 1 ≤ j ≤ n} as base Ω of the cylinder Ω T . Let Γ be the face {x n = 0} of the cube Q n . Then Γ T = Q n−1 × (0, T ) and the stress tensor σ is given by the diagonal matrix with the non-zero entries\\nFix N ∈ N and consider the sequence of functions u(x, t, k, r) ∈ [C ∞ (R n+1 )] n with the components: u 1 (x, t, k, r) = 0, . . . , u n−1 (x, t, k, r) = 0, u n (x, t, k, r) = e k 2 (2µ+λ)(t−r)+kxn k N , depending on a parameter 0 < r < +∞. Consider also the data f (x, t, k, r), u (0) (x, t, k, r), u (1) (x, t, k, r), u (2) (x, t, k, r) having the following components:\\nj (x 1 , . . . , x n−1 , t, k, r) = 0, 1 ≤ j ≤ n − 1,\\nj (x 1 , . . . , x n−1 , t, k, r) = 0, 1 ≤ j ≤ n − 1,\\nThen, for 0 < T < +∞, each function u(x, t, k, T ) is a solution to problem (14), (15), (16), (17) with the data f (x, t, k, T ),\"\n },\n {\n \"title\": \"It is clear, that compatibility conditions (18), (19) hold and\",\n \"content\": \"if N > 2s + 1. On the other hand, for all x n > 0 and all N ∈ N we have:\\nThus, there is no continuity with respect to the data and hence Problem 2 is ill-posed for 0 < T < +∞. Let now T = +∞. Then we may consider the data with a fixed 0 < T 0 < +∞:\\nObviously, for N ≥ 2,\\nThe Uniqueness Theorem 3 for Problem 3 implies that\\nThen, for all x n > 0 and all 2 ≤ N ∈ N, we have lim k→+∞ u n (x, T 0 , k, ∞) = +∞. Thus, if the data f , u (0) , u (2) , u (2) admits the solution to (14) in Ω T with boundary conditions (15), (16) for T = +∞ and the initial condition (17) then there is no continuity with respect to the data in the chosen space. Otherwise there is no solutions to the problem for some data in the data's spaces. In any case, the problem is ill-posed for T = +∞, too.\\nAs both Problems 1 and 2 are ill-posed, we will not study Problem 2 because in addition to (14)- (16) to investigate it one needs to know also the data related to initial condition (17). Besides, we will consider the case 0 < T < +∞ only.\"\n },\n {\n \"title\": \"Solvability Conditions\",\n \"content\": \"From now on we will study Problem 1 under the assumption that its data belong to Hölder spaces (cf., [3,Ch. 1,§1] for other boundary problems for parabolic equations). We recall that a function u(x), defined on a set M ∈ R m , is called Hölder continuous with a power 0 < λ < 1 on M , if there is such a constant C > 0 that\\nbeing Euclidean distance between points x and y in R m ). Let C λ (Ω T ) stand for the set of Hölder continuous functions with a power λ over Ω T . Besides, let C 1+λ,λ (Ω T ) be the set of Hölder continuous functions with a power λ over Ω T , having Hölder continuous derivatives u xi , 1 ≤ i ≤ n, with the same power in Ω T .\\nWe choose a set Ω + in such a way that the set D = Ω ∪ Γ ∪ Ω + would be a bounded domain with piece-wise smooth boundary. It is possible since Γ is an open connected set. It is convenient to set Ω − = Ω. For a function v on D T we denote by v + its restriction to Ω + and, similarly, we denote by v − its restriction to Ω. It is natural to denote limit values of v ± on Γ T , when they are defined, by v ± |ΓT .\"\n },\n {\n \"title\": \"Theorem 4 (Solvability criterion). Let\",\n \"content\": \"Problem 1 is solvable if and only if there is a vector-function F ∈ [C 2,1 (D T )] n satisfying the following conditions:\\nProof. Necessity. Let a function u(x, t) ∈ [C 2,1 (Ω T ) ∩ C 1,0 (Ω T ∪ Γ T )] n satisfies (14), (15), (15). Consider the function\\nin the domain D T , where χ M is a characteristic function of the set M ⊂ R n+1 . By the very construction condition 2) is fulfilled for it.\\nClearly, the function u(x, t) belongs to the space [C 1,2 (Ω ′ T )] n for each cylindrical domain Ω ′ T with such a base Ω ′ that Ω ′ ⊂ Ω and Ω ′ ∩ ∂Ω ⊂ Γ. Besides, L n+1 u = f ∈ [C λ (Ω ′ T )] n . Without loss of the generality we may assume that the interior part Γ ′ of the set Ω ′ ∩ ∂Ω is non-empty.\\nWe note that\\nThen using Lemma 2 we obtain:\\nArguing as in the proof of Theorem 3 we conclude that each of the integrals in the right hand side of (21) satisfies homogeneous Lamé type equation outside the corresponding integration set. In particular, we see that L n+1 F = 0 in D ′ T . Obviously, for any point (x, t) ∈ D T there is a domain D ′ T containing (x, t). That is why L n+1 F = 0 in D T , and hence F belongs to the space [C 2,1 (D T )] n . Thus this function satisfies condition 1), too.\\nSufficiency. Let there be a function F ∈ [C 2,1 (D T )] n , satisfying conditions 1) and 2) of the theorem. Consider on the set D T the function\\nAs f ∈ [C λ (Ω T )] n then the results of [3, Ch. 1, §3] imply\\nand, moreover,\\nSince u 2 ∈ [C λ (Γ T )] n then the results of [3, Ch. 5, §2]yield\\nOn the other hand, the behavior of the Double Layer Potential W Γ,0 (u 1 ) is similar to the behavior of the normal derivative of Single Layer Potential V Γ,0 (u 1 ). Hence\\nLemma 2. Let S ⊂ Γ ∈ C 1+λ . If u 1 ∈ [C 1+λ,λ (Γ T )] n , then the potential W − Γ,0 (u 1 ) belongs to the space [C 1,0 (Ω T ∪ S T )] n if and only if W + Γ,0 (u 2 ) ∈ [C 1,0 (Ω + T ∪ S T )] n .\\nProof. It is similar to the proof of the analogous lemma for Newton Double Layer Potential (see, for instance, [14, lemma 1.1]). Actually, one needs to use Lemma 2 instead of the standard Green formula for the Laplace operator.\\nSince F ∈ [C 1,0 (D T )] n then it follows from the discussion above that W + Γ,0 (u 2 ) ∈ [C 1,0 ((Ω + ∪ Γ) T )] n . Thus, formulas (22)-(28) and Lemma 2 imply that\\nLet us show that the function U − satisfies (15) and (16).\\nIt follows from formulas (23) and (25) that the Parabolic Volume Potential and the Single Layer Parabolic Potential are continuous if the point (x, t) passes over the surface Γ T . Then Formula (23) means that that the surface stress of the Parabolic Volume Potential is continuous if the point (x, t) passes over the surface Γ T . Therefore\\nBy theorem on jump behavior of the stress of the Parabolic Single Layer Potential (see, for instance, [15,Ch. 3, §10, theorem 10.1])\\nFinally, we need the following lemma which is an analogue of the famous Theorem on jump behavior of the normal derivative of the Newton's Double Layer Potential.\\nProof. Really, let, for instance, W − Γ,0 (u 1 ) ∈ [C 1,0 ((Ω ∪ Γ) T )] n . Then using Lemma 2 we obtain W + Γ,0 u 1 ∈ [C 1,0 ((Ω + ∪ Γ) T )] n and σW ± Γ,0 (u 1 )\\nLet φ ∈ [C ∞ 0 (D T )] n be a function with compact support in D T . Then formulas (9)-(11) yield:\\nbecause L n+1 W ± Γ,0 u 1 = 0 in Ω ± according to (28). Again, integrating by parts and using formulas (9)- (11) and Theorem on jump behavior of the Parabolic Double Layer Potential, we see that\\nBut the kernel Φ(x, y, t, τ ) is a fundamental solution of the backward parabolic operator L * n+1 with respect to variables (y, τ ). Hence DT (L * n+1 φ(x, t)) * Φ(x, y, t, τ )dxdt = φ * (y, τ ), (y, τ ) ∈ D T .\\nThen the type of the singularity of the fundamental solution allows us to apply Fubini Theorem and to conclude that\\nFinally, formulas (32)-(34) imply that In particular, if S = ∅ then corollary 2 gives criterion for the existence of solution to Problem 1 in the space [C(Ω T )] n .\\nWe note that Theorem 4 is an analogue of Theorem by Aizenberg and Kytmanov [8]) describing solvability conditions of the Cauchy problem for the Cauchy-Riemann system (cf. also [14] in the Cauchy Problem for Laplace Equation or [12] in the Cauchy problem for general elliptic systems). Formula (22), obtained in the proof of Theorem 4, gives the unique solution to Problem 1. Clearly, if we will be able to write the extension F of the sum of potentials G Ω,0 (f )+ V Γ,0 (u 2 )+ W Γ,0 (u 1 ) from Ω + onto D T as a series with respect to special functions or a limit of parameter depending integrals then we will get a Carleman type formula for solutions to Problem 1 (cf. [8]). However this is a topic for another paper. In the sequel we will discuss polynomial and formal solutions for operators with constant coefficients only. Proof. First let us prove that if in this case the data of Problem 1 are polynomials then the problem is solvable and its solution is a polynomial.\"\n },\n {\n \"title\": \"Polynomial solutions and dense solvability\",\n \"content\": \"Indeed, Problem 1 is easily can be reduced to the following one (see Example 3):\\nwith\\nwhere J(µ, λ) is the diagonal matrix with the components\\nto problem (35)-(37) where\\nand, by the induction with respect to j ∈ Z + ,\\nTo finish the arguments we use the induction with respect to |α ′ | ∈ Z + where α ′ = (α 1 , . . . , α n−1 ) ∈ Z n−1 + . Namely, let for s ≥ 2 and all α ′ with |α ′ | = s the solutions to the problem are polynomial. If |α ′ | = s + 1 then L n+1 x α1 1 · · · x αn−1 n−1 w (j,αn) (x n , t) = t j x α − w (j,αn) (x n , t)∆ n−1 x α1 1 · · · x αn−1 n−1 .\\nClearly, the degree of the polynomial p j,α (x, t) = w (j,αn) (x n , t)∆ n−1 x α1 1 · · · x αn−1 n−1 with respect to x ′ ∈ R n−1 equals to s − 1. Then, by the induction, problem (14)- (16) with data p j,α (x, t) admits a polynomial solution, say, r j,α (x, t). Therefore the solution v (j,α) (x, t) to problem (14)-(16) with data g (j,α) (x, t) = t j x α , |α ′ | = s + 1, is given as follows:\\nv (j,α) (x, t) = x α1 1 · · · x αn−1 n−1 w (j,αn) (x n , t) + r j,α (x, t), i.e. it is a polynomial, too. Now Problem 1 with zero boundary data in the case Γ ⊂ {x n = 0} is densely solvable because any continuous function g on the compact set Ω T can be approximated by polynomials. But the reducing to zero boundary data was organized in such a way that one easily sees, in this case Problem 1 is densely solvable for non-zero boundary data, too.\\nThe dense solvability of Problem 1 in general setting is natural to expect if the set ∂Ω \\\\ Γ has at least one interior point in ∂Ω (cf. [10] in the Cauchy Problem for elliptic equations).\\nFinally, we note that polynomial solutions indicated in the proof of Lemma 4 can be used in order to construct formal solutions to Problem 1.\"\n },\n {\n \"title\": \"Basis with double orthogonality\",\n \"content\": \"Denote by {h (s) ν } the set of harmonic homogeneous polynomials (spherical harmonics) forming an orthonormal basis in L 2 (∂B(0, 1)); here ν is the degree of the homogeneity and 1 ≤ s ≤ J(s) where J(s) = (2n + 2s − 2)(2n + s − 3)! s!(2n − 2)! (see [18]). On the other hand, ∂|x| 2k ∂x j = 2kx j |x| 2k−2 , ∂ 2 |x| 2k ∂x 2 j = 2k|x| 2k−2 + 2k(2k − 2)x 2 j |x| 2k−4 ∆ n |x| 2k = 2k(n + 2k − 2)|x| 2k−2 , k ≥ 1.\\nHence, for ν ≥ 1, k ≥ 1, ∆ n (|x| 2k h ν (x)) = 2k(n + 2k + 2ν − 2)|x| 2k−2 h (s) ν (x) because of Euler's formula\"\n },\n {\n \"title\": \"Problem 1 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Corollary 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 4 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"Lemma 5 . 1 .ν\",\n \"content\": \"\"\n },\n {\n \"title\": \"T2 )\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Find a vector-function u(x, t) ∈ [C 2,1 (Ω T )∩C 1,0 (Ω T ∪Γ T )] n satisfying the Lamé type equationL n+1 u = f in Ω T(14)","content":"Problem 1 .\n1Find a vector-function u(x, t) ∈ [C 2,1 (Ω T )∩C 1,0 (Ω T ∪Γ T )] n satisfying the Lamé type equationL n+1 u = f in Ω T(14)"},{"id":"","caption":"because of the theorem on jump behavior of the Parabolic Double Layer Potential (see, for instance, [3, Ch. 5, §2, theorem 1]), i.e. equality (15) is valid for U − .","content":"because of the theorem on jump behavior of the Parabolic Double Layer Potential (see, for instance, [3, Ch. 5, §2, theorem 1]), i.e. equality (15) is valid for U − ."},{"id":"","caption":"As such functions are dense in the Lebesgue space [L 1 (K)] n for any compact K ⊂ Γ T then formula (31) holds true. Now using lemma 3 and formulas (29), (30), we conclude that (σU ) − |ΓT = u 2 , i.e. (16) is fulfilled for U − .Thus, function u(x, t) = U − (x, t) satisfies conditions(14)-(16). The proof is complete.It follows from formula (22) that properties of a solution to Problem 1 depend on properties of the extension F of the sum of the parabolic potentials, described in Theorem 4.","content":"As such functions are dense in the Lebesgue space [L 1 (K)] n for any compact K ⊂ Γ T then formula (31) holds true. Now using lemma 3 and formulas (29), (30), we conclude that (σU ) − |ΓT = u 2 , i.e. (16) is fulfilled for U − .Thus, function u(x, t) = U − (x, t) satisfies conditions(14)-(16). The proof is complete.It follows from formula (22) that properties of a solution to Problem 1 depend on properties of the extension F of the sum of the parabolic potentials, described in Theorem 4."},{"id":"","caption":"Let S ⊂ ∂Ω \\ Γ. Under the hypotheses of Theorem 4, Problem 1 is solvable in the space[C 2,1 (Ω T ) ∩ C 1,0 (Ω T ∪ Γ T ) ∩ C(Ω T \\ S T )] nif and only if there exists a functionF ∈ [C 2,1 (D T ) ∩ C 1,0 (Ω T ∪ Γ T ) ∩ C(Ω T \\ S T )] n ,satisfying conditions 1) and 2) of Theorem 4.","content":"Corollary 2 .\n2Let S ⊂ ∂Ω \\ Γ. Under the hypotheses of Theorem 4, Problem 1 is solvable in the space[C 2,1 (Ω T ) ∩ C 1,0 (Ω T ∪ Γ T ) ∩ C(Ω T \\ S T )] nif and only if there exists a functionF ∈ [C 2,1 (D T ) ∩ C 1,0 (Ω T ∪ Γ T ) ∩ C(Ω T \\ S T )] n ,satisfying conditions 1) and 2) of Theorem 4."},{"id":"","caption":"It is not difficult to prove dense solvability of Problem 1 in the case where Γ is an open connected set of the hyperplane {x n = 0}.","content":"It is not difficult to prove dense solvability of Problem 1 in the case where Γ is an open connected set of the hyperplane {x n = 0}."},{"id":"","caption":"If Γ is an open connected set if the hyperplane {x n = 0} the Problem 1 is densely solvable.","content":"Lemma 4 .\n4If Γ is an open connected set if the hyperplane {x n = 0} the Problem 1 is densely solvable."},{"id":"","caption":"Let µ = 1, λ = −Then (x)(2N )!!(n + 2N − 2 + 2ν)!! j!(2N − 2j)!!(n + 2N − 2j + 2ν)!! , ν ≥ 1, N ≥ 1 are solutions to the heat equations in R n+1 . Besides, H µ are L 2 (B(0, R) T )-orthogonal for all R > 0 and T > 0 if (ν, s) = (µ, p).Proof. Indeed, for ν ≥ 0, N ≥ 1, and 1 ≤ i ≤ n we have:","content":"Lemma 5 . 1 .ν\n51Let µ = 1, λ = −Then (x)(2N )!!(n + 2N − 2 + 2ν)!! j!(2N − 2j)!!(n + 2N − 2j + 2ν)!! , ν ≥ 1, N ≥ 1 are solutions to the heat equations in R n+1 . Besides, H µ are L 2 (B(0, R) T )-orthogonal for all R > 0 and T > 0 if (ν, s) = (µ, p).Proof. Indeed, for ν ≥ 0, N ≥ 1, and 1 ≤ i ≤ n we have:"},{"id":"","caption":"j t j |x| 2N −2j h (s) ν (x) with constants c j . Then L n+1 H = N j=1 c j jt j−1 |x| 2N −2j h (s) ν (x)−","content":"j t j |x| 2N −2j h (s) ν (x) with constants c j . Then L n+1 H = N j=1 c j jt j−1 |x| 2N −2j h (s) ν (x)−"},{"id":"","caption":"(see, for instance, [2, Ch. 4, §1], [15, Ch. 3, §10] or [3, Ch. 1, §3 and Ch. 5, §2]). Moreover, as Φ is a fundamental solution to Lamé type operator then using","content":"T2 )\nT2(see, for instance, [2, Ch. 4, §1], [15, Ch. 3, §10] or [3, Ch. 1, §3 and Ch. 5, §2]). Moreover, as Φ is a fundamental solution to Lamé type operator then using"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Find a vector-function u(x, t) ∈ [C 2,1 (Ω T )∩C 1,0 (Ω T ∪Γ T )] n satisfying the Lamé type equationL n+1 u = f in Ω T(14)\",\n \"content\": \"Problem 1 .\\n1Find a vector-function u(x, t) ∈ [C 2,1 (Ω T )∩C 1,0 (Ω T ∪Γ T )] n satisfying the Lamé type equationL n+1 u = f in Ω T(14)\"\n },\n {\n \"id\": \"\",\n \"caption\": \"because of the theorem on jump behavior of the Parabolic Double Layer Potential (see, for instance, [3, Ch. 5, §2, theorem 1]), i.e. equality (15) is valid for U − .\",\n \"content\": \"because of the theorem on jump behavior of the Parabolic Double Layer Potential (see, for instance, [3, Ch. 5, §2, theorem 1]), i.e. equality (15) is valid for U − .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"As such functions are dense in the Lebesgue space [L 1 (K)] n for any compact K ⊂ Γ T then formula (31) holds true. Now using lemma 3 and formulas (29), (30), we conclude that (σU ) − |ΓT = u 2 , i.e. (16) is fulfilled for U − .Thus, function u(x, t) = U − (x, t) satisfies conditions(14)-(16). The proof is complete.It follows from formula (22) that properties of a solution to Problem 1 depend on properties of the extension F of the sum of the parabolic potentials, described in Theorem 4.\",\n \"content\": \"As such functions are dense in the Lebesgue space [L 1 (K)] n for any compact K ⊂ Γ T then formula (31) holds true. Now using lemma 3 and formulas (29), (30), we conclude that (σU ) − |ΓT = u 2 , i.e. (16) is fulfilled for U − .Thus, function u(x, t) = U − (x, t) satisfies conditions(14)-(16). The proof is complete.It follows from formula (22) that properties of a solution to Problem 1 depend on properties of the extension F of the sum of the parabolic potentials, described in Theorem 4.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let S ⊂ ∂Ω \\\\ Γ. Under the hypotheses of Theorem 4, Problem 1 is solvable in the space[C 2,1 (Ω T ) ∩ C 1,0 (Ω T ∪ Γ T ) ∩ C(Ω T \\\\ S T )] nif and only if there exists a functionF ∈ [C 2,1 (D T ) ∩ C 1,0 (Ω T ∪ Γ T ) ∩ C(Ω T \\\\ S T )] n ,satisfying conditions 1) and 2) of Theorem 4.\",\n \"content\": \"Corollary 2 .\\n2Let S ⊂ ∂Ω \\\\ Γ. Under the hypotheses of Theorem 4, Problem 1 is solvable in the space[C 2,1 (Ω T ) ∩ C 1,0 (Ω T ∪ Γ T ) ∩ C(Ω T \\\\ S T )] nif and only if there exists a functionF ∈ [C 2,1 (D T ) ∩ C 1,0 (Ω T ∪ Γ T ) ∩ C(Ω T \\\\ S T )] n ,satisfying conditions 1) and 2) of Theorem 4.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"It is not difficult to prove dense solvability of Problem 1 in the case where Γ is an open connected set of the hyperplane {x n = 0}.\",\n \"content\": \"It is not difficult to prove dense solvability of Problem 1 in the case where Γ is an open connected set of the hyperplane {x n = 0}.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"If Γ is an open connected set if the hyperplane {x n = 0} the Problem 1 is densely solvable.\",\n \"content\": \"Lemma 4 .\\n4If Γ is an open connected set if the hyperplane {x n = 0} the Problem 1 is densely solvable.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let µ = 1, λ = −Then (x)(2N )!!(n + 2N − 2 + 2ν)!! j!(2N − 2j)!!(n + 2N − 2j + 2ν)!! , ν ≥ 1, N ≥ 1 are solutions to the heat equations in R n+1 . Besides, H µ are L 2 (B(0, R) T )-orthogonal for all R > 0 and T > 0 if (ν, s) = (µ, p).Proof. Indeed, for ν ≥ 0, N ≥ 1, and 1 ≤ i ≤ n we have:\",\n \"content\": \"Lemma 5 . 1 .ν\\n51Let µ = 1, λ = −Then (x)(2N )!!(n + 2N − 2 + 2ν)!! j!(2N − 2j)!!(n + 2N − 2j + 2ν)!! , ν ≥ 1, N ≥ 1 are solutions to the heat equations in R n+1 . Besides, H µ are L 2 (B(0, R) T )-orthogonal for all R > 0 and T > 0 if (ν, s) = (µ, p).Proof. Indeed, for ν ≥ 0, N ≥ 1, and 1 ≤ i ≤ n we have:\"\n },\n {\n \"id\": \"\",\n \"caption\": \"j t j |x| 2N −2j h (s) ν (x) with constants c j . Then L n+1 H = N j=1 c j jt j−1 |x| 2N −2j h (s) ν (x)−\",\n \"content\": \"j t j |x| 2N −2j h (s) ν (x) with constants c j . Then L n+1 H = N j=1 c j jt j−1 |x| 2N −2j h (s) ν (x)−\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(see, for instance, [2, Ch. 4, §1], [15, Ch. 3, §10] or [3, Ch. 1, §3 and Ch. 5, §2]). Moreover, as Φ is a fundamental solution to Lamé type operator then using\",\n \"content\": \"T2 )\\nT2(see, for instance, [2, Ch. 4, §1], [15, Ch. 3, §10] or [3, Ch. 1, §3 and Ch. 5, §2]). Moreover, as Φ is a fundamental solution to Lamé type operator then using\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". L D Landau, E M Lifshitz, A Course of Theoretical Physics. 6Pergamon PressFluid MechanicsLandau, L.D., Lifshitz, E.M. Fluid Mechanics (Volume 6 of A Course of Theoretical Physics), Pergamon Press, 1959.","doi":""},{"text":"Linear and Quasilinear Equations of Parabolic Type. O A Ladyzhenskaya, V A Solonnikov, N N Ural'tseva, Moscow, NaukaLadyzhenskaya, O.A., Solonnikov V.A., Ural'tseva, N.N. Linear and Quasilinear Equations of Parabolic Type, Moscow, Nauka, 1967.","doi":""},{"text":"Partial Differential Equations of Parabolic Type. A Friedman, Prentice-Hall, IncEnglewood Cliffs, NJFriedman, A. Partial Differential Equations of Parabolic Type, Englewood Cliffs, NJ, Prentice- Hall, Inc., 1964.","doi":""},{"text":"Parabolic equations, Partial differential equations 6. S D Eidel'man, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. 63VINITIEidel'man S.D., Parabolic equations, Partial differential equations 6, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 63, VINITI, Moscow, 1990, 201-313.","doi":""},{"text":"On boundary value problems for linear parabolic systems of differential equations of general form. Boundary value problems of mathematical physics. Part 3. On boundary value problems for linear parabolic systems of differential equations of general form. V A Solonnikov, Trudy Mat. Inst. Steklov. 83Solonnikov V.A., On boundary value problems for linear parabolic systems of differential equa- tions of general form. Boundary value problems of mathematical physics. Part 3. On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Inst. Steklov, 83, 1965, 3-163","doi":""},{"text":"A strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain Ω × O of R N+M. A Lorenzi, L Lorenzi, 10.1088/0266-5611/29/2/025007Inverse Problems. 292Lorenzi A., Lorenzi, L. A strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain Ω × O of R N+M , Inverse Problems, 29:2 (2013), doi:10.1088/0266-5611/29/2/025007.","doi":"10.1088/0266-5611/29/2/025007"},{"text":"On an ill-posed problem for the heat equation. R Puzyrev, A Shlapunov, J. Siberian Fed. Univ. 42Puzyrev, R., Shlapunov, A., On an ill-posed problem for the heat equation. J. Siberian Fed. Univ., 4:2 (2011), 218-229.","doi":""},{"text":"On the possibility of holomorphic continuation to a domain of functions given on a part of its boundary. L A Aizenberg, A M Kytmanov, Matem. sbornik. 182Aizenberg, L.A., Kytmanov, A.M., On the possibility of holomorphic continuation to a domain of functions given on a part of its boundary, Matem. sbornik. 182:5 (1991), 490-597.","doi":""},{"text":"On the Cauchy Problem for the Lamé system. A Shlapunov, Zeitschrift für Angewandte Mathematik und Mechanik. 76Shlapunov, A., On the Cauchy Problem for the Lamé system. Zeitschrift für Angewandte Mathematik und Mechanik, 76:4 (1996), 215-221.","doi":""},{"text":"Bases with double orthogonality in the Cauchy problem for systems with injective symbols. A A Shlapunov, N Tarkhanov, Proc. London. Math. Soc. 71Shlapunov, A.A., Tarkhanov, N. Bases with double orthogonality in the Cauchy problem for systems with injective symbols, Proc. London. Math. Soc., 71:1 (1995), 1-54.","doi":""},{"text":"On Cauchy problem for operators with injective symbols in the spaces of distributions. I V Shestakov, A A Shlapunov, Journal Inverse and Ill-posed Problems. 19Shestakov, I.V., Shlapunov A.A. On Cauchy problem for operators with injective symbols in the spaces of distributions, Journal Inverse and Ill-posed Problems, 19 (2011), 127-150.","doi":""},{"text":"The Cauchy Problem for Solutions of Elliptic Equations. N N Tarkhanov, Akademie-VerlagBerlinTarkhanov, N.N. The Cauchy Problem for Solutions of Elliptic Equations. Berlin: Akademie- Verlag, 1995.","doi":""},{"text":"Lectures on Cauchy's Problem in Linear Partial Differential Equations. J Hadamard, Yale Univ. PressNew Haven-LondonHadamard, J. Lectures on Cauchy's Problem in Linear Partial Differential Equations. / J. Hadamard. -Yale Univ. Press, New Haven-London, 1923.","doi":""},{"text":"On the Cauchy Problem for the Laplace Equation. A A Shlapunov, Siberian Math. J. 33Shlapunov, A.A. On the Cauchy Problem for the Laplace Equation, Siberian Math. J., 33:3 (1992), 205-215.","doi":""},{"text":"Second Order Equations of Elliptic and Parabolic Types. E M Landis, Moscow, NaukaLandis, E.M. Second Order Equations of Elliptic and Parabolic Types, Moscow, Nauka, 1971.","doi":""},{"text":"V P Mihailov, Partial Differential Equations. Moscow, NaukaMihailov, V.P. Partial Differential Equations, Moscow, Nauka, 1976.","doi":""},{"text":". A G Sveshnikov, A N Bogolyubov, V V Kravtsov, Lectures on Mathematical Physics. Sveshnikov, A.G., Bogolyubov, A.N., Kravtsov, V.V. Lectures on Mathematical Physics, Moscow, Nauka, 2004.","doi":""},{"text":"Svobodnyi 79, 660041 Krasnoyarsk, Russia E-mail address: effervesce@mail.ru. S L Sobolev, pr. Svobodnyi. 79Roman Puzyrev) Siberian Federal University, Institute of Mathematics and Computer Science, pr ; Alexander Shlapunov) Siberian Federal University, Institute of Mathematics and Computer ScienceRussia E-mail address: ashlapunov@sfu-kras.ruSobolev, S.L., (Roman Puzyrev) Siberian Federal University, Institute of Mathematics and Com- puter Science, pr. Svobodnyi 79, 660041 Krasnoyarsk, Russia E-mail address: effervesce@mail.ru (Alexander Shlapunov) Siberian Federal University, Institute of Mathematics and Computer Science, pr. Svobodnyi 79, 660041 Krasnoyarsk, Russia E-mail address: ashlapunov@sfu-kras.ru","doi":""}],"string":"[\n {\n \"text\": \". L D Landau, E M Lifshitz, A Course of Theoretical Physics. 6Pergamon PressFluid MechanicsLandau, L.D., Lifshitz, E.M. Fluid Mechanics (Volume 6 of A Course of Theoretical Physics), Pergamon Press, 1959.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Linear and Quasilinear Equations of Parabolic Type. O A Ladyzhenskaya, V A Solonnikov, N N Ural'tseva, Moscow, NaukaLadyzhenskaya, O.A., Solonnikov V.A., Ural'tseva, N.N. Linear and Quasilinear Equations of Parabolic Type, Moscow, Nauka, 1967.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Partial Differential Equations of Parabolic Type. A Friedman, Prentice-Hall, IncEnglewood Cliffs, NJFriedman, A. Partial Differential Equations of Parabolic Type, Englewood Cliffs, NJ, Prentice- Hall, Inc., 1964.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Parabolic equations, Partial differential equations 6. S D Eidel'man, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. 63VINITIEidel'man S.D., Parabolic equations, Partial differential equations 6, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 63, VINITI, Moscow, 1990, 201-313.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On boundary value problems for linear parabolic systems of differential equations of general form. Boundary value problems of mathematical physics. Part 3. On boundary value problems for linear parabolic systems of differential equations of general form. V A Solonnikov, Trudy Mat. Inst. Steklov. 83Solonnikov V.A., On boundary value problems for linear parabolic systems of differential equa- tions of general form. Boundary value problems of mathematical physics. Part 3. On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Inst. Steklov, 83, 1965, 3-163\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain Ω × O of R N+M. A Lorenzi, L Lorenzi, 10.1088/0266-5611/29/2/025007Inverse Problems. 292Lorenzi A., Lorenzi, L. A strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain Ω × O of R N+M , Inverse Problems, 29:2 (2013), doi:10.1088/0266-5611/29/2/025007.\",\n \"doi\": \"10.1088/0266-5611/29/2/025007\"\n },\n {\n \"text\": \"On an ill-posed problem for the heat equation. R Puzyrev, A Shlapunov, J. Siberian Fed. Univ. 42Puzyrev, R., Shlapunov, A., On an ill-posed problem for the heat equation. J. Siberian Fed. Univ., 4:2 (2011), 218-229.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the possibility of holomorphic continuation to a domain of functions given on a part of its boundary. L A Aizenberg, A M Kytmanov, Matem. sbornik. 182Aizenberg, L.A., Kytmanov, A.M., On the possibility of holomorphic continuation to a domain of functions given on a part of its boundary, Matem. sbornik. 182:5 (1991), 490-597.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the Cauchy Problem for the Lamé system. A Shlapunov, Zeitschrift für Angewandte Mathematik und Mechanik. 76Shlapunov, A., On the Cauchy Problem for the Lamé system. Zeitschrift für Angewandte Mathematik und Mechanik, 76:4 (1996), 215-221.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Bases with double orthogonality in the Cauchy problem for systems with injective symbols. A A Shlapunov, N Tarkhanov, Proc. London. Math. Soc. 71Shlapunov, A.A., Tarkhanov, N. Bases with double orthogonality in the Cauchy problem for systems with injective symbols, Proc. London. Math. Soc., 71:1 (1995), 1-54.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On Cauchy problem for operators with injective symbols in the spaces of distributions. I V Shestakov, A A Shlapunov, Journal Inverse and Ill-posed Problems. 19Shestakov, I.V., Shlapunov A.A. On Cauchy problem for operators with injective symbols in the spaces of distributions, Journal Inverse and Ill-posed Problems, 19 (2011), 127-150.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Cauchy Problem for Solutions of Elliptic Equations. N N Tarkhanov, Akademie-VerlagBerlinTarkhanov, N.N. The Cauchy Problem for Solutions of Elliptic Equations. Berlin: Akademie- Verlag, 1995.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Lectures on Cauchy's Problem in Linear Partial Differential Equations. J Hadamard, Yale Univ. PressNew Haven-LondonHadamard, J. Lectures on Cauchy's Problem in Linear Partial Differential Equations. / J. Hadamard. -Yale Univ. Press, New Haven-London, 1923.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the Cauchy Problem for the Laplace Equation. A A Shlapunov, Siberian Math. J. 33Shlapunov, A.A. On the Cauchy Problem for the Laplace Equation, Siberian Math. J., 33:3 (1992), 205-215.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Second Order Equations of Elliptic and Parabolic Types. E M Landis, Moscow, NaukaLandis, E.M. Second Order Equations of Elliptic and Parabolic Types, Moscow, Nauka, 1971.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"V P Mihailov, Partial Differential Equations. Moscow, NaukaMihailov, V.P. Partial Differential Equations, Moscow, Nauka, 1976.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A G Sveshnikov, A N Bogolyubov, V V Kravtsov, Lectures on Mathematical Physics. Sveshnikov, A.G., Bogolyubov, A.N., Kravtsov, V.V. Lectures on Mathematical Physics, Moscow, Nauka, 2004.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Svobodnyi 79, 660041 Krasnoyarsk, Russia E-mail address: effervesce@mail.ru. S L Sobolev, pr. Svobodnyi. 79Roman Puzyrev) Siberian Federal University, Institute of Mathematics and Computer Science, pr ; Alexander Shlapunov) Siberian Federal University, Institute of Mathematics and Computer ScienceRussia E-mail address: ashlapunov@sfu-kras.ruSobolev, S.L., (Roman Puzyrev) Siberian Federal University, Institute of Mathematics and Com- puter Science, pr. Svobodnyi 79, 660041 Krasnoyarsk, Russia E-mail address: effervesce@mail.ru (Alexander Shlapunov) Siberian Federal University, Institute of Mathematics and Computer Science, pr. Svobodnyi 79, 660041 Krasnoyarsk, Russia E-mail address: ashlapunov@sfu-kras.ru\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":595,"cells":{"text":{"kind":"string","value":"\nAnisotropic magnetic properties of CeAg 2 Ge 2 single crystals\n10 Jan 2007\n\nA Thamizhavel \nDepartment of Condensed Matter Physics and Materials Science\nTata Institute of Fundamental Research\nHomi Bhabha Road400 005ColabaMumbaiIndia\n\nR Kulkarni \nDepartment of Condensed Matter Physics and Materials Science\nTata Institute of Fundamental Research\nHomi Bhabha Road400 005ColabaMumbaiIndia\n\nS K Dhar \nDepartment of Condensed Matter Physics and Materials Science\nTata Institute of Fundamental Research\nHomi Bhabha Road400 005ColabaMumbaiIndia\n\nAnisotropic magnetic properties of CeAg 2 Ge 2 single crystals\n10 Jan 2007(Dated: March 23, 2022)numbers: 8110-h7127+a7170Ch7510Dg7550Ee\nIn order to investigate the anisotropic magnetic properties of CeAg2Ge2, we have successfully grown the single crystals, for the first time, by high temperature solution growth (flux) method. We have performed a detailed study of the grown single crystals by measuring their electrical resistivity, magnetic susceptibility, magnetization, specific heat and magnetoresistance. A clear anisotropy and an antiferromagnetic transition at TN = 4.6 K have been observed in the magnetic properties. The magnetic entropy reaches R ln 4 at 20 K indicating that the ground state and the first excited state are very closely spaced (a quasi-quartet state). From the specific heat measurements and crystalline electric field (CEF) analysis of the magnetic susceptibility, we have found the level splitting energies as 5 K and 130 K. The magnetization measurements reveal that the a-axis is the easy axis of magnetization and the saturation moment is µs = 1.6 µB/Ce, corroborating the previous neutron scattering measurements on a polycrystalline sample.\n\nI. INTRODUCTION\n\nIn the Ce-based intermetallic compounds, the competition between the RKKY interaction and the Kondo effect leads to diverse ground states. This competition can be readily studied in the multifarious CeT 2 X 2 compounds, where T is a transition metal and X is a group IV element namely Si or Ge. CeT 2 X 2 compounds crystallize in the well known ThCr 2 Si 2 type tetragonal crystal structure and exhibit a wide range of interesting magnetic properties like heavy fermion superconductivity in CeCu 2 Si 2 1 , pressure induced superconductivity in CePd 2 Si 2 2 , CeRh 2 Si 2 3 , unconventional metamagnetic transition in CeRu 2 Si 2 4 etc. Similarly the isostructural germanides also show interesting magnetic properties 5,6,7,8 . While most of the above series of silicides and germanides have been grown in single crystalline form and the anisotropic magnetic properties have been investigated, there are no reports available on single crystalline CeAg 2 Ge 2 owing to the difficulty in growing the single crystal from a stoichiometric melt. Moreover, the polycrystalline data are also limited. The first report on a polycrystalline CeAg 2 Ge 2 was made by Rauchschwalbe et al. 9 in which they have mentioned that this compound undergoes an antiferromagnetic ordering below 8 K. From neutron scattering experiments an antiferromagnetic ordering temperature of T N = 7 K was reported by Knopp et al. 10 and Loidl et al. 11 . Furthermore, an ordered moment of 1.85 µ B /Ce at 1.5 K oriented along the [100] direction was estimated from the neutron scattering experiments. From the specific heat measurements Böhm et al., 12 have reported that CeAg 2 Ge 2 orders antiferromagnetically below 5 K and they observed a peak in the specific heat data at 350 mK when plotted as δC/T vs. T (δC = C − C nuclear − C magnon ) which they attributed to coherent electronic quasi-particles of medium heavy mass, coexisting with long range magnetic order. However, a recent report by Cordruwisch et al. 13 have re-ported a Néel temperature of 4.5 K on a polycrystalline sample. In view of these conflicting reports on the magnetic ordering temperature and to study the magnetic properties more precisely, we have succeeded in growing a single crystal of CeAg 2 Ge 2 for the first time and investigated the anisotropic physical properties by means of electrical resistivity, magnetic susceptibility, magnetization, specific heat and magnetoresistance.\n\n\nII. EXPERIMENT\n\nCeAg 2 Ge 2 single crystals were grown by self flux method. Since the use of fourth element as flux normally introduces some inclusions in the grown single crystals, we have grown the single crystals of CeAg 2 Ge 2 from an off-stoichiometric melt, with excess of Ag and Ge. The binary phase diagram of Ag and Ge shows an eutectic at 650 • C. We have taken advantage of this eutectic composition and used it as a flux for the growth of CeAg 2 Ge 2 single crystal. Similar kind of binary eutectic compositions have been successfully used as flux for the growth of several intermetallic compounds like Au-Si binary eutectic for the crystal growth of CeAu 4 Si 2 14 , Ag-Ge eutectic flux for the growth of YbAgGe, 15 Ni-Ge eutectic composition for the growth of several RNi 2 Ge 2 16 and Ni-B eutectic composition for the crystal growth of borocarbides 17 . The starting materials with 3N-Ce, 5N-Ag and 5N-Ge were taken in the ratio 1 : 16.25 : 6.75 which includes the eutectic composition of the excess flux Ag-Ge. The contents were placed in an alumina crucible, and subsequently sealed in an evacuated quartz ampoule. The temperature of the furnace was raised to 1050 • C and after homogenizing the mixture for two days, the furnace was cooled down to the eutectic temperature of the binary flux Ag-Ge over a period of 3 weeks time and then rapidly to room temperature. The crystals were separated from the flux by means of centrifuging. The typical size of the crystal was 6 × 3 × 0.3 mm 3 , with (001) plane as the flat plane.\n\nThe dc magnetic susceptibility and the magnetization measurements were performed in the temperature range 1.8-300 K and in magnetic fields up to 7 T along the principal directions using a Quantum Design SQUID magnetometer. The temperature dependence of electrical resistivity in the range 1.8-300 K was measured using a home made DC electrical resistivity set up. The heat capacity and magnetoresistance measurements were performed using a Quantum Design PPMS instrument in the temperature range from 0.5 K to room temperature and for fields up to 12 T.\n\n\nIII. EXPERIMENTAL RESULTS\n\n\nA. X-ray studies\n\nSince the growth of the single crystals of CeAg 2 Ge 2 was performed from an off-stoichiometric starting composition we performed powder X-ray diffraction by crushing a few small pieces of the single crystals to confirm the phase purity of CeAg 2 Ge 2 . The powder X-ray pattern to- gether with the Rietveld refinement are shown in Fig. 1. The X-ray pattern clearly reveals that the grown single crystals are single phase and no detectable traces of impurity phases are seen. From the Rietveld refinement the ThCr 2 Si 2 -type crystal structure of CeAg 2 Ge 2 is confirmed and the lattice constants were estimated to be a = 4.301(8)Å and c = 10.973(7)Å. We have also performed the energy dispersive X-ray analysis (EDAX) and confirmed the stoichiometry of CeAg 2 Ge 2 single crystals. The crystals were then oriented along the principal directions, namely [100] and [001] directions, by means of the Laue back reflection method. Well defined Laue diffrac-tion spots, together with the tetragonal symmetry pattern, indicated the good quality of the single crystals. The crystals were cut along the principal direction using a spark erosion cutting machine for the anisotropic physical property measurements.\n\n\nB. Electrical resistivity\n\nThe dc electrical resistivity of CeAg 2 Ge 2 in the temperature range from 1.8 to 300 K is shown in Fig. 2. The . The electrical resistivity is anisotropic reflecting the tetragonal symmetry of the crystal structure. As it can be seen from the Fig. 2 the absolute value of electrical resistivity at 295 K is 52 µΩ·cm and 106 µΩ·cm, respectively for J\n\n[100] and [001] and at 1.8 K is 12 µΩ·cm and 18 µΩ·cm for J\n\n[100] and [001], respectively. At high temperatures the scattering is phonon dominated and the resistivity decreases linearly with decreasing temperature typical of a metallic sample. The electrical resistivity shows a shallow minimum around 20 K and then increases with decrease in temperature up to 4.6 K. This increase in the electrical resistivity at low temperature can be attributed to short range antiferromagnetic order and/or the presence of weak Kondo-type interaction. It may be mentioned here that the corresponding silicide, CeAg 2 Si 2 has been reported to be a dense Kondo lattice antiferromagnet 18 . With further decrease in temperature below 4.6 K, the resistivity changes its slope and drops due to the reduction in spin-disorder scattering caused by the antiferromagnetic ordering of the magnetic moments. In the limited temperature range from 1.8 -4.0 K the resisitivity follows a power law relation ρ = ρ 0 + AT n with ρ 0 = 10.15 µΩ·cm, A = 0.99 µΩ·cm/K 1.75 and n = 1.75 and ρ 0 = 15.97 µΩ·cm, A = 0.53 µΩ·cm/K 2.21 and n = 2.21 for the currents along J\n\n[100] and [001] directions, respectively. Here the exponent n is close to 2 which can in principle be explained on the basis of electron-electron scattering. Since our data do not extend for T << T N we have not attempted to fit our data using sophisticated model of spin-waves.\n\n\nC. Magnetic susceptibility and magnetization\n\nThe temperature dependence of magnetic susceptibility in the temperature range from 1.8 to 300 K measured in a field of 1 kOe along the two principal directions viz., H parallel to [100] and [001] is shown in Fig. 3. The antiferromagnetic ordering at T N = 4.6 K is clearly seen as indicated by the arrow. The susceptibility below T N remains almost T -independent and at the lowest temperature measured there is a small rise in the susceptibility, indicating that the antiferromagnetism observed in CeAg 2 Ge 2 is not a simple two sublattice antiferromagnetism. For example, canted antiferromagnetism may show a weak residual ferromagnetic magnetization in the Néel state. Also, this type of temperature independent susceptibility at low temperature may be attributed to the crystalline electric field (CEF) effect. The inverse magnetic susceptibility of CeAg 2 Ge 2 does not obey the simple Curie-Weiss law (not shown here), on the other hand, it can be very well fitted to a modified Curie-Weiss law which is given by χ = χ 0 + C T −θp , where χ 0 is the temperature-independent part of the magnetic susceptibility and C is the Curie constant. The main contributions to χ 0 includes the core-electron diamagnetism, and the susceptibility of the conduction electrons. The details of the inverse magnetic susceptibility is discussed later in the discussion part. For an effective magnetic moment of 2.54 µ B /Ce we have estimated the θ p values as -7.2 K and -42 K for H\n\n[100] and [001], respectively.\n\nThe field dependence of isothermal magnetization at T = 2 K measured in a SQUID magnetometer up to a field of 70 kOe is shown in Fig. 4 curves show large uniaxial magnetocrystalline anisotropy.\n\nThe magnetization for H [100] is linear for low fields and shows metamagnetic transitions at critical fields H m1 = 31 kOe and at H m2 = 44.7 kOe and nearly saturates at 70 kOe with a saturation moment µ s = 1.6 µ B /Ce, this indicates that [100]-axis is the easy axis of magnetization. Here, the saturation moment is smaller than the free ion value of 2.1 µ B /Ce which could be attributed to the crystal field effects. However, one can achieve the saturation value at high applied magnetic fields. On the other hand, the magnetization for H\n\n[001] is very small and varies linearly with field reaching a value of 0.32 µ B /Ce at 50 kOe, indicating a hard axis of magnetization. We have also performed the isothermal magnetization at 3 K, 4 K, 5 K and 10 K for H\n\n[100]. From the differential plots of the isothermal magnetization measurements, we have constructed the magnetic phase diagram as shown in Fig. 5. The two metamagnetic transitions are clearly seen for 2 K and 3 K magnetization curves; however at 3 K only one metamagnetic transition is seen. For temperatures above the magnetic ordering temperature the magnetization curves did not show any metamagnetic behaviour and the magnetization curves were linear indicating a paramagnetic state. and β is the phonon contribution to the heat capacity. The γ and β values thus obtained are estimated to be 2.8 mJ/K 2 ·mol and 0.59 mJ/K 4 ·mol, respectively. The inset of Fig. 6(a) shows the low temperature part of the specific heat and the antiferromagnetic ordering is manifested by the clear jump in the specific heat at T N = 4.6 K as indicated by the arrow. The inset of Fig. 6(b) shows the specific heat in the form of C/T versus T . Just below the magnetic ordering the specific heat shows a broad peak in the C/T versus T curve which presumably indicates the presence of low lying crystal field levels.\n\nAssuming the lattice heat capacity of CeAg 2 Ge 2 is the same as that of LaAg 2 Ge 2 , the 4f -derived contribution to the heat capacity C mag was obtained by subtracting the specific heat of LaAg 2 Ge 2 from the total specific heat of CeAg 2 Ge 2 . Figure 6(c) shows C mag /T versus T together with the entropy S mag which is obtained by integrating C mag /T . As it can be seen from the figure, the entropy of CeAg 2 Ge 2 is very high at the magnetic ordering temperature and reaches R ln 4 near 20 K. In tetragonal symmetry, the degenerate six fold levels of the groundstate multiplet of Ce 3+ split into three doublets and ∆ 1 and ∆ 2 are the excitation energies of the first and second excited states, respectively. Since the entropy change reaches R ln 4, not too far above T N , one can come to a conclusion that the ground state and the first excited state are very closely spaced or nearly degenerate. This finding clearly corroborates the earlier neutron scattering results by Loidl et al. 11 in which they could observe only one crystal field transition at 11 meV and concluded that the ground state is almost degenerate with the first excited state. From the crystalline electric field analysis of the magnetic susceptibility data, to be discussed later, we found that the energies of the exicted states ∆ 1 and ∆ 2 as 5 K and 130 K, respectively. Due to the very small splitting energy between the ground state and the first excited state the estimation of the Sommerfeld coefficient γ, by the usual method, from the low temperature data will lead to ambiguity. Hence we estimated the γ value from the high temperature data in the paramagnetic region above the magnetic ordering after subtracting the Schottky contribution and linearly extrapolating the C/T versus T 2 behaviour to T = 0 K and is shown in Fig. 6(b). The γ value thus estimated is 45 mJ/K 2 ·mol.\n\n\nE. Magnetoresistance\n\nWe have also studied the effect of magnetic field on the resistivity of CeAg 2 Ge 2 . The magnetic field did not have any appreciable effect on the resistivity for the field perpendicular to the easy axis direction. On the other hand, when the field was applied parallel to the easy axis direction, we found that the resistivity gradually decreased with increasing field. In Fig. 7 \n\n\nmagnetic field for H\n\n[100], ∆ρ/ρ 0 at 2 K initially increases in the positive direction and then turns negative at higher magnetic fields, giving rise to a maximum at 3.1 T. This field value coincides with the metamagnetic transition observed in the magnetization measurement at T = 2 K. Such a behaviour of the magnetoresistance is qualitatively consistent with the theoretical calculation given by Yamada and Takada 19 . In the antiferromagnetic state (H < H m ), the magnetic moment fluctuation in one magnetic sublattice is enhanced by the field while, in the field induced ferromagnetic state (H > H m ), the fluctuation is suppressed by the field. The change in the fluctuation is reflected in the magnetoresistance. With the increase in the temperature, the peak in the magnetoresistance moves toward lower fields and decreases, finally disappears for temperatures above T N . In the paramagnetic region the negative magnetoresistance is due to the freezing out of the spin-flip scattering by the magnetic field. The normalized magnetoresistance for T > T N can be mapped onto a single curve using the scaling relation ∆ρ/ρ(0) = f [B/(T + T * )] derived by Schlottmann 20 within the Bethe-ansatz approach, as shown in Fig. 7(b). Here T * is the characteristic temperature which is an ap-proximate measure of the Kondo temperature T K and is estimated to be 0.36 K. This indicates that the Kondo effect is very weak in CeAg 2 Ge 2 , which substantiates our earlier prediction from the zero field resistivity data.\n\n\nIV. DISCUSSION\n\nFrom the results of the electrical resistivity, susceptibility and specific heat measurements it can be clearly seen that CeAg 2 Ge 2 undergoes an antiferromagnetic ordering at 4.6 K with the easy axis of magnetization as [100]. The magnetization at 70 kOe reaches 1.6 µ B /Ce thus corroborating the earlier neutron scattering experiment 11 on a polycrystalline sample of CeAg 2 Ge 2 . The electrical resistivity at high temperature shows a typical metallic behaviour and at sufficiently low temperature it shows a weak minimum before ordering magnetically. This behaviour is quite different from what one has observed in the CeCu 2 Ge 2 which is similar to CeAg 2 Ge 2 both structurally and magnetically, although the antiferromagnetic ordering temperature is nearly equal (T N = 4.1 K for CeCu 2 Ge 2 and T N = 4.6 K for CeAg 2 Ge 2 ). The logarithmic temperature dependence of electrical resistivity in CeCu 2 Ge 2 exhibit a double peak structure which is presumably attributed to the combined influence of the Kondo and crystalline electric field (CEF) effects. The Kondo temperature of CeCu 2 Ge 2 was estimated to be about 6 K 21 , whereas for CeAg 2 Ge 2 the Kondo temperature is very small. Since the unit cell volume of CeCu 2 Ge 2 is smaller (V ≈ 178Å 3 ), a larger value of the Kondo coupling constant J sf is expected and hence the Kondo interaction dominates in CeCu 2 Ge 2 compared to that in CeAg 2 Ge 2 (V ≈ 203Å 3 ). In CeCu 2 Ge 2 superconductivity occurs when the unit cell volume attains a favourable value of 168 ± 3Å 3 . This is achieved with an external pressure of 7 GPa. Considering this fact the unit cell volume of CeAg 2 Ge 2 is quite large and one would require a very high pressure to reduce the unit cell volume to nearly 168 ± 3Å 3 for probable observation of superconductivity. Based on this it can be said that CeAg 2 Ge 2 lies on the left hand side of the Doniach phase diagram in which the RKKY energy scale is dominant and Kondo interaction is weak.\n\nThe heat capacity measurement of CeAg 2 Ge 2 single crystal clearly reveals the presence of low lying crystal field levels with a very small separation between the ground state and the first excited state indicating that the ground state is a quasi-quartet state instead of a doublet which is usually observed for a tetragonal site symmetry. In order to further analyze the crystal field levels and to understand the present anisotropy in the magnetic susceptibility we have performed the CEF analysis on the susceptibility data. For the purpose of CEF analysis in Fig. 8 \nH CEF = B 0 2 O 0 2 + B 0 4 O 0 4 + B 4 4 O 4 4 ,(2)\nThe magnetic susceptibility including the molecular field contribution λ is given by\nχ −1 = χ −1 CEF − λ,(3)\nwhere χ CEF is the CEF susceptibility. Diagonalization of the CEF Hamiltonian gives us the eignvalues and eigenfunctions. For Ce 3+ J = 5/2 wave function splits into three doublets, Γ\n\n(1) 7 = a |±5/2 + b |∓3/2 , Γ\n\n(2) 7 = a |±3/2 − b |∓5/2 and Γ 6 = |±1/2 , where a and b are mixing parameters with the condition a 2 + b 2 = 1. The CEF parameters were estimated from the fits to the magnetic susceptibility. Solid lines in Fig. 8 show the least square fitting to Eqn. 3, the CEF parameters thus obtained are listed in Table I. The corresponding crystal field level scheme together with that of CeCu 2 Ge 2 is shown in the bottom part of Fig. 8. The crystal field level scheme for CeCu 2 Ge 2 is taken from Ref. 21. The ground state of CeAg 2 Ge 2 shows a mixing of |∓3/2 and |±5/2 wave functions. From Fig. 8 it is obvious that the present set of CEF parameters gives a good fit to the experimental data there by explaining the anisotropy in the magnetic susceptibility. The CEF parameters have resulted in the first and second excited states at ∆ 1 = 5 K and ∆ 2 = 130 K. It is interesting to note here that the CEF level scheme of CeCu 2 Ge 2 is qualitatively opposite to the present case where the ground state is a doublet and the first and second excited states are nearly degenerate. This can be explained on the basis of the sign of the B 0 2 parameter. For CeCu 2 Ge 2 it is negative while it is positive for CeAg 2 Ge 2 . This change in sign of B 0 2 suggests that the CEF potential in CeT 2 Ge 2 is largely dependent on the hybridization between localized f -electron states and the conduction-electron bands.\n\n\nV. CONCLUSION\n\nSingle crystals of CeAg 2 Ge 2 have been grown for the first time, by flux method by using a Ag-Ge binary eutectic composition as flux. The antiferromagnetic ordering temperature T N = 4.6 K is clearly manifested by the resistivity, heat capacity and susceptibility measurements. Thus the ambiguity about the magnetic ordering temperature of this compound, reflected in the conflicting reports earlier in the literature, has been removed. A large anisotropy in the electrical resistivity, magnetic susceptibility and magnetization is observed. The susceptibility and magnetization clearly reveals that [100]-axis as the easy axis of magnetization with a moment of 1.6 µ B /Ce at 70 kOe. Metamagnetic transitions have been observed at the critical fields, H m1 = 31 kOe and at H m2 = 44.7 kOe. The heat capacity and the susceptibility data clearly support the closely spaced ground and first excited states, which have been analyzed by the CEF calculations. \n\nFIG. 1 :\n1(Color online) Powder X-ray diffraction pattern recorded for crushed single crystals of CeAg2Ge2 at room temperature. The solid line through the experimental data points is the Rietveld refinement profile calculated for the tetragonal CeAg2Ge2.\n\n\nonline) Logarithmic temperature dependence of the dc electrical resistivity of CeAg2Ge2 for J [100] and [001]. The inset shows the linear temperature dependence of the electrical resistivity. The solid lines are least square fitting toa power law relation. resistivity was measured for the current direction parallel to [100] and [001]\n\n\nonline)Temperature dependence of magnetic susceptibility χ from 1.8 -300 K. The inset shows the low temperature magnetic susceptibility.\n\n\nonline) Isothermal magnetization curves of CeAg2Ge2 measured at T = 2 K along the two principal directions.\n\nFigure 6 (\n6a) shows the temperature dependence of the specific heat of single crystalline CeAg 2 Ge 2 together with the specific heat of a polycrytalline reference sample LaAg 2 Ge 2 . The low temperature data (∼ 1.5 to ∼ 10 K) of LaAg 2 Ge 2 have been fitted to the expression C = γT + βT 3 where γ is the electronic contribution\n\n\nonline)(a) Temperature dependence of the specific heat of CeAg2Ge2 and LaAg2Ge2, the inset shows the low temperature part, (b) specific heat of CeAg2Ge2 in the form of C/T vs. T 2 after subtracting the schottky and the 4f contribution, the solid line shows the extrapolation of the high temperature specific heat to 0 K, inset shows C/T versus T and (c) magnetic specific heat Cmag in the form of Cmag/T vs. T together with the magnetic entropy Smag.\n\nFIG. 7 :\n7we have plotted the normalized magnetoresistance ∆ρ/ρ 0 = [ρ(B) -ρ(B = 0)]/ρ(B = 0) as a function of applied magnetic field at various fixed temperatures. With increasing (Color online) (a)Isothermal normalized magnetoresistance for CeAg2Ge2 as function of applied magnetic field for J [100] and H [100]. (b) Normalized magnetoresistivity, in the paramagnetic state, plotted as a function of B/(T + T * ).\n\n\nwe have plotted the experimental results on susceptibility in the form of 1/(χ − χ 0 ), where χ 0 was determined as 1.33 × 10 −3 and 1.41 × 10 −3 emu/mol for H [001] and [100], respectively, so an effective magneticFIG. 8: (Color online) Temperature dependence of the inverse magnetic susceptibility in CeAg2Ge2. Solid lines are fitting to the CEF level scheme with a molecular field term. The bottom figure shows the crystal-field level scheme of Ce 3+ in the CeCu2Ge2 (taken from Ref.21) and CeAg2Ge2. moment of 2.54 µ B /Ce is obtained above 100 K. Similar kind of treatment has been made for CePt 3 Si 22 while performing the CEF analysis. The Ce atoms in CeAg 2 Ge 2 occupy the 2a Wyckoff position with the point symmetry 4/mmm (D 4h ) and hence possess the tetragonal site symmetry. The CEF hamiltonian for a tetragonal site symmetry is given by, are the CEF parameters and the Stevens operators, respectively 23,24 . For Ce atom, the 6th order terms O 0 6 and O 4 6 vanishes and hence CEF Hamiltonian reduces to,0.90|±5/2〉 \n\n|±1/2〉 \n\n|±1/2〉 \n\n0.44|±5/2〉+0.90| 3/2〉 \n0.66|±5/2〉 +0.75| 3/2〉 \n\n-0.44| 3/2〉 \n0.75|±5/2〉 -0.66| 3/2〉 \n\n5 Κ \n\n130 Κ \n191 Κ \n\nCF level scheme for Ce 3+ in \n\nCeCu 2 Ge 2 \nCeAg 2 Ge 2 \n\n400 \n\n200 \n\n0 \n\n1/(χ− χ \n0 ) (mol/emu) \n\n300 \n200 \n100 \n0 \n\nTemperature (K) \n\nCeAg 2 Ge 2 \nH // [001] \n\n[100] \n\nH CEF = B 0 \n2 O 0 \n2 + B 0 \n4 O 0 \n4 + B 4 \n4 O 4 \n4 + B 0 \n6 O 0 \n6 + B 4 \n6 O 4 \n6 , (1) \n\nwhere B m \nℓ and O m \nℓ \n\nTABLE I :\nICEF parameters, energy level schemes and the corresponding wave functions for CeAg2Ge2.CEF parameters \nB 0 \n2 (K) \nB 0 \n4 (K) \nB 4 \n4 (K) \nλi (emu/mol) −1 \n2.24 \n−0.19 \n2.40 \nλ [100] = −7.2 \nλ [001] = −26 \nenergy levels and wave functions \nE(K) \n| +5/2 \n| +3/2 \n| +1/2 \n| −1/2 \n| −3/2 \n| −5/2 \n130 \n−0.\nF. Steglich, J. Aarts, C. D. Bredl, W. Lieke, D. Meschede, W. Franz and H. 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Mu- rani, J. Magn. Magn. Mater. 63 & 64, 88 (1987).\n\n. A Loidl, K Knorr, G Knopp, A Krimmel, R Caspary, A Böhm, G Sparn, C Geibel, F Steglich, A P Murani, Phys. Rev. B. 469341A. Loidl, K. Knorr, G. Knopp, A. Krimmel, R. Caspary, A. Böhm, G. Sparn, C. Geibel, F. Steglich and A. P. Murani, Phys. Rev. B 46, 9341 (1992).\n\n. A Böhm, R Caspary, U Habel, L Pawlak, A Zuber, F Steglich, A Loidl, J. Magn. Magn. Mater. 76 &. 77150A. Böhm, R. Caspary, U. Habel, L. Pawlak, A. Zuber F. Steglich and A. Loidl, J. Magn. Magn. Mater. 76 & 77, 150 (1988).\n\n. E Cordruwish, D Kaczorowski, A Saccone, P Rogl, R Ferro, J. Phase Equilibria. 20407E. Cordruwish, D. Kaczorowski, A. Saccone, P. Rogl and R. Ferro, J. Phase Equilibria 20, 407 (1999).\n\n. H Nakashima, A Thamizhavel, T D Matsuda, Y Haga, T Takeuchi, K Sugiyama, R Settai, Y Ōnuki, J Compd, 424H. Nakashima, A. Thamizhavel, T. D. Matsuda, Y. Haga, T. Takeuchi, K. Sugiyama, R. Settai and Y.Ōnuki, J. Al- loys Compd. 424, 7 (2006).\n\n. 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G Knopp, A Loidl, K Knorr, L Pawlak, M Duczmal, R Caspary, U Gottwick, H Spille, F Steglich, A P Murani, Z. Phys. B -Condensed Matter. 7795G. Knopp, A. Loidl, K. Knorr, L. Pawlak, M. Duczmal, R. Caspary, U. Gottwick, H. Spille, F. Steglich and A. P. Murani, Z. Phys. B -Condensed Matter 77, 95 (1989).\n\n. T Takeuchi, S Hashimoto, T Yasuda, H Shishido, T Ueda, M Yamada, Y Obiraki, M Shiimoto, H Kohara, T Yamamoto, K Sugiyama, K Kindo, T D Matsuda, Y Haga, Y Aoki, H Sato, R Settai, Y Ōnuki, J. Phys.: Condens. Matter. 16333T. Takeuchi, S. Hashimoto, T. Yasuda, H. Shishido, T. Ueda, M. Yamada, Y. Obiraki, M. Shiimoto, H. Kohara, T. Yamamoto, K. Sugiyama, K. Kindo, T. D. Matsuda, Y. Haga, Y. Aoki, H. Sato, R. Settai and Y.Ōnuki, J. Phys.: Condens. Matter 16, L333 (2004).\n\n. K W H Stevens, Proc. Phys. Soc. 65209K. W. H. Stevens, Proc. Phys. Soc., London, Sect. A65, 209 (1952).\n\nM T Hutchings, Solid State Physics: Advances in Research and Applications. F. Seitz and B. TurnbullNew YorkAcademic16227M. T. Hutchings, in Solid State Physics: Advances in Re- search and Applications, edited by F. Seitz and B. Turnbull (Academic, New York, 1965), Vol.16, p.227.\n"},"corpus_id":{"kind":"number","value":1244314,"string":"1,244,314"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"cond-mat/0611197\",\n \"mag\": 3021840504,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1016/j.physb.2006.01.483\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/cond-mat/0611197v2.pdf\"\n ],\n \"pdf_sha\": \"a3cc9dc83e47764b5784372c5d6d438d9e8b2fc1\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/cond-mat/0611197\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Anisotropic magnetic properties of CeAg 2 Ge 2 single crystals"},"authors":{"kind":"list like","value":["A Thamizhavel \nDepartment of Condensed Matter Physics and Materials Science\nTata Institute of Fundamental Research\nHomi Bhabha Road400 005ColabaMumbaiIndia","R Kulkarni \nDepartment of Condensed Matter Physics and Materials Science\nTata Institute of Fundamental Research\nHomi Bhabha Road400 005ColabaMumbaiIndia","S K Dhar \nDepartment of Condensed Matter Physics and Materials Science\nTata Institute of Fundamental Research\nHomi Bhabha Road400 005ColabaMumbaiIndia"],"string":"[\n \"A Thamizhavel \\nDepartment of Condensed Matter Physics and Materials Science\\nTata Institute of Fundamental Research\\nHomi Bhabha Road400 005ColabaMumbaiIndia\",\n \"R Kulkarni \\nDepartment of Condensed Matter Physics and Materials Science\\nTata Institute of Fundamental Research\\nHomi Bhabha Road400 005ColabaMumbaiIndia\",\n \"S K Dhar \\nDepartment of Condensed Matter Physics and Materials Science\\nTata Institute of Fundamental Research\\nHomi Bhabha Road400 005ColabaMumbaiIndia\"\n]","isConversational":false},"abstract":{"kind":"string","value":"In order to investigate the anisotropic magnetic properties of CeAg2Ge2, we have successfully grown the single crystals, for the first time, by high temperature solution growth (flux) method. We have performed a detailed study of the grown single crystals by measuring their electrical resistivity, magnetic susceptibility, magnetization, specific heat and magnetoresistance. A clear anisotropy and an antiferromagnetic transition at TN = 4.6 K have been observed in the magnetic properties. The magnetic entropy reaches R ln 4 at 20 K indicating that the ground state and the first excited state are very closely spaced (a quasi-quartet state). From the specific heat measurements and crystalline electric field (CEF) analysis of the magnetic susceptibility, we have found the level splitting energies as 5 K and 130 K. The magnetization measurements reveal that the a-axis is the easy axis of magnetization and the saturation moment is µs = 1.6 µB/Ce, corroborating the previous neutron scattering measurements on a polycrystalline sample."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"In the Ce-based intermetallic compounds, the competition between the RKKY interaction and the Kondo effect leads to diverse ground states. This competition can be readily studied in the multifarious CeT 2 X 2 compounds, where T is a transition metal and X is a group IV element namely Si or Ge. CeT 2 X 2 compounds crystallize in the well known ThCr 2 Si 2 type tetragonal crystal structure and exhibit a wide range of interesting magnetic properties like heavy fermion superconductivity in CeCu 2 Si 2 1 , pressure induced superconductivity in CePd 2 Si 2 2 , CeRh 2 Si 2 3 , unconventional metamagnetic transition in CeRu 2 Si 2 4 etc. Similarly the isostructural germanides also show interesting magnetic properties 5,6,7,8 . While most of the above series of silicides and germanides have been grown in single crystalline form and the anisotropic magnetic properties have been investigated, there are no reports available on single crystalline CeAg 2 Ge 2 owing to the difficulty in growing the single crystal from a stoichiometric melt. Moreover, the polycrystalline data are also limited. The first report on a polycrystalline CeAg 2 Ge 2 was made by Rauchschwalbe et al. 9 in which they have mentioned that this compound undergoes an antiferromagnetic ordering below 8 K. From neutron scattering experiments an antiferromagnetic ordering temperature of T N = 7 K was reported by Knopp et al. 10 and Loidl et al. 11 . Furthermore, an ordered moment of 1.85 µ B /Ce at 1.5 K oriented along the [100] direction was estimated from the neutron scattering experiments. From the specific heat measurements Böhm et al., 12 have reported that CeAg 2 Ge 2 orders antiferromagnetically below 5 K and they observed a peak in the specific heat data at 350 mK when plotted as δC/T vs. T (δC = C − C nuclear − C magnon ) which they attributed to coherent electronic quasi-particles of medium heavy mass, coexisting with long range magnetic order. However, a recent report by Cordruwisch et al. 13 have re-ported a Néel temperature of 4.5 K on a polycrystalline sample. In view of these conflicting reports on the magnetic ordering temperature and to study the magnetic properties more precisely, we have succeeded in growing a single crystal of CeAg 2 Ge 2 for the first time and investigated the anisotropic physical properties by means of electrical resistivity, magnetic susceptibility, magnetization, specific heat and magnetoresistance."},{"title":"II. EXPERIMENT","content":"CeAg 2 Ge 2 single crystals were grown by self flux method. Since the use of fourth element as flux normally introduces some inclusions in the grown single crystals, we have grown the single crystals of CeAg 2 Ge 2 from an off-stoichiometric melt, with excess of Ag and Ge. The binary phase diagram of Ag and Ge shows an eutectic at 650 • C. We have taken advantage of this eutectic composition and used it as a flux for the growth of CeAg 2 Ge 2 single crystal. Similar kind of binary eutectic compositions have been successfully used as flux for the growth of several intermetallic compounds like Au-Si binary eutectic for the crystal growth of CeAu 4 Si 2 14 , Ag-Ge eutectic flux for the growth of YbAgGe, 15 Ni-Ge eutectic composition for the growth of several RNi 2 Ge 2 16 and Ni-B eutectic composition for the crystal growth of borocarbides 17 . The starting materials with 3N-Ce, 5N-Ag and 5N-Ge were taken in the ratio 1 : 16.25 : 6.75 which includes the eutectic composition of the excess flux Ag-Ge. The contents were placed in an alumina crucible, and subsequently sealed in an evacuated quartz ampoule. The temperature of the furnace was raised to 1050 • C and after homogenizing the mixture for two days, the furnace was cooled down to the eutectic temperature of the binary flux Ag-Ge over a period of 3 weeks time and then rapidly to room temperature. The crystals were separated from the flux by means of centrifuging. The typical size of the crystal was 6 × 3 × 0.3 mm 3 , with (001) plane as the flat plane.\nThe dc magnetic susceptibility and the magnetization measurements were performed in the temperature range 1.8-300 K and in magnetic fields up to 7 T along the principal directions using a Quantum Design SQUID magnetometer. The temperature dependence of electrical resistivity in the range 1.8-300 K was measured using a home made DC electrical resistivity set up. The heat capacity and magnetoresistance measurements were performed using a Quantum Design PPMS instrument in the temperature range from 0.5 K to room temperature and for fields up to 12 T."},{"title":"III. EXPERIMENTAL RESULTS","content":""},{"title":"A. X-ray studies","content":"Since the growth of the single crystals of CeAg 2 Ge 2 was performed from an off-stoichiometric starting composition we performed powder X-ray diffraction by crushing a few small pieces of the single crystals to confirm the phase purity of CeAg 2 Ge 2 . The powder X-ray pattern to- gether with the Rietveld refinement are shown in Fig. 1. The X-ray pattern clearly reveals that the grown single crystals are single phase and no detectable traces of impurity phases are seen. From the Rietveld refinement the ThCr 2 Si 2 -type crystal structure of CeAg 2 Ge 2 is confirmed and the lattice constants were estimated to be a = 4.301(8)Å and c = 10.973(7)Å. We have also performed the energy dispersive X-ray analysis (EDAX) and confirmed the stoichiometry of CeAg 2 Ge 2 single crystals. The crystals were then oriented along the principal directions, namely [100] and [001] directions, by means of the Laue back reflection method. Well defined Laue diffrac-tion spots, together with the tetragonal symmetry pattern, indicated the good quality of the single crystals. The crystals were cut along the principal direction using a spark erosion cutting machine for the anisotropic physical property measurements."},{"title":"B. Electrical resistivity","content":"The dc electrical resistivity of CeAg 2 Ge 2 in the temperature range from 1.8 to 300 K is shown in Fig. 2. The . The electrical resistivity is anisotropic reflecting the tetragonal symmetry of the crystal structure. As it can be seen from the Fig. 2 the absolute value of electrical resistivity at 295 K is 52 µΩ·cm and 106 µΩ·cm, respectively for J\n[100] and [001] and at 1.8 K is 12 µΩ·cm and 18 µΩ·cm for J\n[100] and [001], respectively. At high temperatures the scattering is phonon dominated and the resistivity decreases linearly with decreasing temperature typical of a metallic sample. The electrical resistivity shows a shallow minimum around 20 K and then increases with decrease in temperature up to 4.6 K. This increase in the electrical resistivity at low temperature can be attributed to short range antiferromagnetic order and/or the presence of weak Kondo-type interaction. It may be mentioned here that the corresponding silicide, CeAg 2 Si 2 has been reported to be a dense Kondo lattice antiferromagnet 18 . With further decrease in temperature below 4.6 K, the resistivity changes its slope and drops due to the reduction in spin-disorder scattering caused by the antiferromagnetic ordering of the magnetic moments. In the limited temperature range from 1.8 -4.0 K the resisitivity follows a power law relation ρ = ρ 0 + AT n with ρ 0 = 10.15 µΩ·cm, A = 0.99 µΩ·cm/K 1.75 and n = 1.75 and ρ 0 = 15.97 µΩ·cm, A = 0.53 µΩ·cm/K 2.21 and n = 2.21 for the currents along J\n[100] and [001] directions, respectively. Here the exponent n is close to 2 which can in principle be explained on the basis of electron-electron scattering. Since our data do not extend for T << T N we have not attempted to fit our data using sophisticated model of spin-waves."},{"title":"C. Magnetic susceptibility and magnetization","content":"The temperature dependence of magnetic susceptibility in the temperature range from 1.8 to 300 K measured in a field of 1 kOe along the two principal directions viz., H parallel to [100] and [001] is shown in Fig. 3. The antiferromagnetic ordering at T N = 4.6 K is clearly seen as indicated by the arrow. The susceptibility below T N remains almost T -independent and at the lowest temperature measured there is a small rise in the susceptibility, indicating that the antiferromagnetism observed in CeAg 2 Ge 2 is not a simple two sublattice antiferromagnetism. For example, canted antiferromagnetism may show a weak residual ferromagnetic magnetization in the Néel state. Also, this type of temperature independent susceptibility at low temperature may be attributed to the crystalline electric field (CEF) effect. The inverse magnetic susceptibility of CeAg 2 Ge 2 does not obey the simple Curie-Weiss law (not shown here), on the other hand, it can be very well fitted to a modified Curie-Weiss law which is given by χ = χ 0 + C T −θp , where χ 0 is the temperature-independent part of the magnetic susceptibility and C is the Curie constant. The main contributions to χ 0 includes the core-electron diamagnetism, and the susceptibility of the conduction electrons. The details of the inverse magnetic susceptibility is discussed later in the discussion part. For an effective magnetic moment of 2.54 µ B /Ce we have estimated the θ p values as -7.2 K and -42 K for H\n[100] and [001], respectively.\nThe field dependence of isothermal magnetization at T = 2 K measured in a SQUID magnetometer up to a field of 70 kOe is shown in Fig. 4 curves show large uniaxial magnetocrystalline anisotropy.\nThe magnetization for H [100] is linear for low fields and shows metamagnetic transitions at critical fields H m1 = 31 kOe and at H m2 = 44.7 kOe and nearly saturates at 70 kOe with a saturation moment µ s = 1.6 µ B /Ce, this indicates that [100]-axis is the easy axis of magnetization. Here, the saturation moment is smaller than the free ion value of 2.1 µ B /Ce which could be attributed to the crystal field effects. However, one can achieve the saturation value at high applied magnetic fields. On the other hand, the magnetization for H\n[001] is very small and varies linearly with field reaching a value of 0.32 µ B /Ce at 50 kOe, indicating a hard axis of magnetization. We have also performed the isothermal magnetization at 3 K, 4 K, 5 K and 10 K for H\n[100]. From the differential plots of the isothermal magnetization measurements, we have constructed the magnetic phase diagram as shown in Fig. 5. The two metamagnetic transitions are clearly seen for 2 K and 3 K magnetization curves; however at 3 K only one metamagnetic transition is seen. For temperatures above the magnetic ordering temperature the magnetization curves did not show any metamagnetic behaviour and the magnetization curves were linear indicating a paramagnetic state. and β is the phonon contribution to the heat capacity. The γ and β values thus obtained are estimated to be 2.8 mJ/K 2 ·mol and 0.59 mJ/K 4 ·mol, respectively. The inset of Fig. 6(a) shows the low temperature part of the specific heat and the antiferromagnetic ordering is manifested by the clear jump in the specific heat at T N = 4.6 K as indicated by the arrow. The inset of Fig. 6(b) shows the specific heat in the form of C/T versus T . Just below the magnetic ordering the specific heat shows a broad peak in the C/T versus T curve which presumably indicates the presence of low lying crystal field levels.\nAssuming the lattice heat capacity of CeAg 2 Ge 2 is the same as that of LaAg 2 Ge 2 , the 4f -derived contribution to the heat capacity C mag was obtained by subtracting the specific heat of LaAg 2 Ge 2 from the total specific heat of CeAg 2 Ge 2 . Figure 6(c) shows C mag /T versus T together with the entropy S mag which is obtained by integrating C mag /T . As it can be seen from the figure, the entropy of CeAg 2 Ge 2 is very high at the magnetic ordering temperature and reaches R ln 4 near 20 K. In tetragonal symmetry, the degenerate six fold levels of the groundstate multiplet of Ce 3+ split into three doublets and ∆ 1 and ∆ 2 are the excitation energies of the first and second excited states, respectively. Since the entropy change reaches R ln 4, not too far above T N , one can come to a conclusion that the ground state and the first excited state are very closely spaced or nearly degenerate. This finding clearly corroborates the earlier neutron scattering results by Loidl et al. 11 in which they could observe only one crystal field transition at 11 meV and concluded that the ground state is almost degenerate with the first excited state. From the crystalline electric field analysis of the magnetic susceptibility data, to be discussed later, we found that the energies of the exicted states ∆ 1 and ∆ 2 as 5 K and 130 K, respectively. Due to the very small splitting energy between the ground state and the first excited state the estimation of the Sommerfeld coefficient γ, by the usual method, from the low temperature data will lead to ambiguity. Hence we estimated the γ value from the high temperature data in the paramagnetic region above the magnetic ordering after subtracting the Schottky contribution and linearly extrapolating the C/T versus T 2 behaviour to T = 0 K and is shown in Fig. 6(b). The γ value thus estimated is 45 mJ/K 2 ·mol."},{"title":"E. Magnetoresistance","content":"We have also studied the effect of magnetic field on the resistivity of CeAg 2 Ge 2 . The magnetic field did not have any appreciable effect on the resistivity for the field perpendicular to the easy axis direction. On the other hand, when the field was applied parallel to the easy axis direction, we found that the resistivity gradually decreased with increasing field. In Fig. 7"},{"title":"magnetic field for H","content":"[100], ∆ρ/ρ 0 at 2 K initially increases in the positive direction and then turns negative at higher magnetic fields, giving rise to a maximum at 3.1 T. This field value coincides with the metamagnetic transition observed in the magnetization measurement at T = 2 K. Such a behaviour of the magnetoresistance is qualitatively consistent with the theoretical calculation given by Yamada and Takada 19 . In the antiferromagnetic state (H < H m ), the magnetic moment fluctuation in one magnetic sublattice is enhanced by the field while, in the field induced ferromagnetic state (H > H m ), the fluctuation is suppressed by the field. The change in the fluctuation is reflected in the magnetoresistance. With the increase in the temperature, the peak in the magnetoresistance moves toward lower fields and decreases, finally disappears for temperatures above T N . In the paramagnetic region the negative magnetoresistance is due to the freezing out of the spin-flip scattering by the magnetic field. The normalized magnetoresistance for T > T N can be mapped onto a single curve using the scaling relation ∆ρ/ρ(0) = f [B/(T + T * )] derived by Schlottmann 20 within the Bethe-ansatz approach, as shown in Fig. 7(b). Here T * is the characteristic temperature which is an ap-proximate measure of the Kondo temperature T K and is estimated to be 0.36 K. This indicates that the Kondo effect is very weak in CeAg 2 Ge 2 , which substantiates our earlier prediction from the zero field resistivity data."},{"title":"IV. DISCUSSION","content":"From the results of the electrical resistivity, susceptibility and specific heat measurements it can be clearly seen that CeAg 2 Ge 2 undergoes an antiferromagnetic ordering at 4.6 K with the easy axis of magnetization as [100]. The magnetization at 70 kOe reaches 1.6 µ B /Ce thus corroborating the earlier neutron scattering experiment 11 on a polycrystalline sample of CeAg 2 Ge 2 . The electrical resistivity at high temperature shows a typical metallic behaviour and at sufficiently low temperature it shows a weak minimum before ordering magnetically. This behaviour is quite different from what one has observed in the CeCu 2 Ge 2 which is similar to CeAg 2 Ge 2 both structurally and magnetically, although the antiferromagnetic ordering temperature is nearly equal (T N = 4.1 K for CeCu 2 Ge 2 and T N = 4.6 K for CeAg 2 Ge 2 ). The logarithmic temperature dependence of electrical resistivity in CeCu 2 Ge 2 exhibit a double peak structure which is presumably attributed to the combined influence of the Kondo and crystalline electric field (CEF) effects. The Kondo temperature of CeCu 2 Ge 2 was estimated to be about 6 K 21 , whereas for CeAg 2 Ge 2 the Kondo temperature is very small. Since the unit cell volume of CeCu 2 Ge 2 is smaller (V ≈ 178Å 3 ), a larger value of the Kondo coupling constant J sf is expected and hence the Kondo interaction dominates in CeCu 2 Ge 2 compared to that in CeAg 2 Ge 2 (V ≈ 203Å 3 ). In CeCu 2 Ge 2 superconductivity occurs when the unit cell volume attains a favourable value of 168 ± 3Å 3 . This is achieved with an external pressure of 7 GPa. Considering this fact the unit cell volume of CeAg 2 Ge 2 is quite large and one would require a very high pressure to reduce the unit cell volume to nearly 168 ± 3Å 3 for probable observation of superconductivity. Based on this it can be said that CeAg 2 Ge 2 lies on the left hand side of the Doniach phase diagram in which the RKKY energy scale is dominant and Kondo interaction is weak.\nThe heat capacity measurement of CeAg 2 Ge 2 single crystal clearly reveals the presence of low lying crystal field levels with a very small separation between the ground state and the first excited state indicating that the ground state is a quasi-quartet state instead of a doublet which is usually observed for a tetragonal site symmetry. In order to further analyze the crystal field levels and to understand the present anisotropy in the magnetic susceptibility we have performed the CEF analysis on the susceptibility data. For the purpose of CEF analysis in Fig. 8\nThe magnetic susceptibility including the molecular field contribution λ is given by\nwhere χ CEF is the CEF susceptibility. Diagonalization of the CEF Hamiltonian gives us the eignvalues and eigenfunctions. For Ce 3+ J = 5/2 wave function splits into three doublets, Γ\n(1) 7 = a |±5/2 + b |∓3/2 , Γ\n(2) 7 = a |±3/2 − b |∓5/2 and Γ 6 = |±1/2 , where a and b are mixing parameters with the condition a 2 + b 2 = 1. The CEF parameters were estimated from the fits to the magnetic susceptibility. Solid lines in Fig. 8 show the least square fitting to Eqn. 3, the CEF parameters thus obtained are listed in Table I. The corresponding crystal field level scheme together with that of CeCu 2 Ge 2 is shown in the bottom part of Fig. 8. The crystal field level scheme for CeCu 2 Ge 2 is taken from Ref. 21. The ground state of CeAg 2 Ge 2 shows a mixing of |∓3/2 and |±5/2 wave functions. From Fig. 8 it is obvious that the present set of CEF parameters gives a good fit to the experimental data there by explaining the anisotropy in the magnetic susceptibility. The CEF parameters have resulted in the first and second excited states at ∆ 1 = 5 K and ∆ 2 = 130 K. It is interesting to note here that the CEF level scheme of CeCu 2 Ge 2 is qualitatively opposite to the present case where the ground state is a doublet and the first and second excited states are nearly degenerate. This can be explained on the basis of the sign of the B 0 2 parameter. For CeCu 2 Ge 2 it is negative while it is positive for CeAg 2 Ge 2 . This change in sign of B 0 2 suggests that the CEF potential in CeT 2 Ge 2 is largely dependent on the hybridization between localized f -electron states and the conduction-electron bands."},{"title":"V. CONCLUSION","content":"Single crystals of CeAg 2 Ge 2 have been grown for the first time, by flux method by using a Ag-Ge binary eutectic composition as flux. The antiferromagnetic ordering temperature T N = 4.6 K is clearly manifested by the resistivity, heat capacity and susceptibility measurements. Thus the ambiguity about the magnetic ordering temperature of this compound, reflected in the conflicting reports earlier in the literature, has been removed. A large anisotropy in the electrical resistivity, magnetic susceptibility and magnetization is observed. The susceptibility and magnetization clearly reveals that [100]-axis as the easy axis of magnetization with a moment of 1.6 µ B /Ce at 70 kOe. Metamagnetic transitions have been observed at the critical fields, H m1 = 31 kOe and at H m2 = 44.7 kOe. The heat capacity and the susceptibility data clearly support the closely spaced ground and first excited states, which have been analyzed by the CEF calculations."},{"title":"FIG. 1 :","content":""},{"title":"Figure 6 (","content":""},{"title":"FIG. 7 :","content":""},{"title":"TABLE I :","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"In the Ce-based intermetallic compounds, the competition between the RKKY interaction and the Kondo effect leads to diverse ground states. This competition can be readily studied in the multifarious CeT 2 X 2 compounds, where T is a transition metal and X is a group IV element namely Si or Ge. CeT 2 X 2 compounds crystallize in the well known ThCr 2 Si 2 type tetragonal crystal structure and exhibit a wide range of interesting magnetic properties like heavy fermion superconductivity in CeCu 2 Si 2 1 , pressure induced superconductivity in CePd 2 Si 2 2 , CeRh 2 Si 2 3 , unconventional metamagnetic transition in CeRu 2 Si 2 4 etc. Similarly the isostructural germanides also show interesting magnetic properties 5,6,7,8 . While most of the above series of silicides and germanides have been grown in single crystalline form and the anisotropic magnetic properties have been investigated, there are no reports available on single crystalline CeAg 2 Ge 2 owing to the difficulty in growing the single crystal from a stoichiometric melt. Moreover, the polycrystalline data are also limited. The first report on a polycrystalline CeAg 2 Ge 2 was made by Rauchschwalbe et al. 9 in which they have mentioned that this compound undergoes an antiferromagnetic ordering below 8 K. From neutron scattering experiments an antiferromagnetic ordering temperature of T N = 7 K was reported by Knopp et al. 10 and Loidl et al. 11 . Furthermore, an ordered moment of 1.85 µ B /Ce at 1.5 K oriented along the [100] direction was estimated from the neutron scattering experiments. From the specific heat measurements Böhm et al., 12 have reported that CeAg 2 Ge 2 orders antiferromagnetically below 5 K and they observed a peak in the specific heat data at 350 mK when plotted as δC/T vs. T (δC = C − C nuclear − C magnon ) which they attributed to coherent electronic quasi-particles of medium heavy mass, coexisting with long range magnetic order. However, a recent report by Cordruwisch et al. 13 have re-ported a Néel temperature of 4.5 K on a polycrystalline sample. In view of these conflicting reports on the magnetic ordering temperature and to study the magnetic properties more precisely, we have succeeded in growing a single crystal of CeAg 2 Ge 2 for the first time and investigated the anisotropic physical properties by means of electrical resistivity, magnetic susceptibility, magnetization, specific heat and magnetoresistance.\"\n },\n {\n \"title\": \"II. EXPERIMENT\",\n \"content\": \"CeAg 2 Ge 2 single crystals were grown by self flux method. Since the use of fourth element as flux normally introduces some inclusions in the grown single crystals, we have grown the single crystals of CeAg 2 Ge 2 from an off-stoichiometric melt, with excess of Ag and Ge. The binary phase diagram of Ag and Ge shows an eutectic at 650 • C. We have taken advantage of this eutectic composition and used it as a flux for the growth of CeAg 2 Ge 2 single crystal. Similar kind of binary eutectic compositions have been successfully used as flux for the growth of several intermetallic compounds like Au-Si binary eutectic for the crystal growth of CeAu 4 Si 2 14 , Ag-Ge eutectic flux for the growth of YbAgGe, 15 Ni-Ge eutectic composition for the growth of several RNi 2 Ge 2 16 and Ni-B eutectic composition for the crystal growth of borocarbides 17 . The starting materials with 3N-Ce, 5N-Ag and 5N-Ge were taken in the ratio 1 : 16.25 : 6.75 which includes the eutectic composition of the excess flux Ag-Ge. The contents were placed in an alumina crucible, and subsequently sealed in an evacuated quartz ampoule. The temperature of the furnace was raised to 1050 • C and after homogenizing the mixture for two days, the furnace was cooled down to the eutectic temperature of the binary flux Ag-Ge over a period of 3 weeks time and then rapidly to room temperature. The crystals were separated from the flux by means of centrifuging. The typical size of the crystal was 6 × 3 × 0.3 mm 3 , with (001) plane as the flat plane.\\nThe dc magnetic susceptibility and the magnetization measurements were performed in the temperature range 1.8-300 K and in magnetic fields up to 7 T along the principal directions using a Quantum Design SQUID magnetometer. The temperature dependence of electrical resistivity in the range 1.8-300 K was measured using a home made DC electrical resistivity set up. The heat capacity and magnetoresistance measurements were performed using a Quantum Design PPMS instrument in the temperature range from 0.5 K to room temperature and for fields up to 12 T.\"\n },\n {\n \"title\": \"III. EXPERIMENTAL RESULTS\",\n \"content\": \"\"\n },\n {\n \"title\": \"A. X-ray studies\",\n \"content\": \"Since the growth of the single crystals of CeAg 2 Ge 2 was performed from an off-stoichiometric starting composition we performed powder X-ray diffraction by crushing a few small pieces of the single crystals to confirm the phase purity of CeAg 2 Ge 2 . The powder X-ray pattern to- gether with the Rietveld refinement are shown in Fig. 1. The X-ray pattern clearly reveals that the grown single crystals are single phase and no detectable traces of impurity phases are seen. From the Rietveld refinement the ThCr 2 Si 2 -type crystal structure of CeAg 2 Ge 2 is confirmed and the lattice constants were estimated to be a = 4.301(8)Å and c = 10.973(7)Å. We have also performed the energy dispersive X-ray analysis (EDAX) and confirmed the stoichiometry of CeAg 2 Ge 2 single crystals. The crystals were then oriented along the principal directions, namely [100] and [001] directions, by means of the Laue back reflection method. Well defined Laue diffrac-tion spots, together with the tetragonal symmetry pattern, indicated the good quality of the single crystals. The crystals were cut along the principal direction using a spark erosion cutting machine for the anisotropic physical property measurements.\"\n },\n {\n \"title\": \"B. Electrical resistivity\",\n \"content\": \"The dc electrical resistivity of CeAg 2 Ge 2 in the temperature range from 1.8 to 300 K is shown in Fig. 2. The . The electrical resistivity is anisotropic reflecting the tetragonal symmetry of the crystal structure. As it can be seen from the Fig. 2 the absolute value of electrical resistivity at 295 K is 52 µΩ·cm and 106 µΩ·cm, respectively for J\\n[100] and [001] and at 1.8 K is 12 µΩ·cm and 18 µΩ·cm for J\\n[100] and [001], respectively. At high temperatures the scattering is phonon dominated and the resistivity decreases linearly with decreasing temperature typical of a metallic sample. The electrical resistivity shows a shallow minimum around 20 K and then increases with decrease in temperature up to 4.6 K. This increase in the electrical resistivity at low temperature can be attributed to short range antiferromagnetic order and/or the presence of weak Kondo-type interaction. It may be mentioned here that the corresponding silicide, CeAg 2 Si 2 has been reported to be a dense Kondo lattice antiferromagnet 18 . With further decrease in temperature below 4.6 K, the resistivity changes its slope and drops due to the reduction in spin-disorder scattering caused by the antiferromagnetic ordering of the magnetic moments. In the limited temperature range from 1.8 -4.0 K the resisitivity follows a power law relation ρ = ρ 0 + AT n with ρ 0 = 10.15 µΩ·cm, A = 0.99 µΩ·cm/K 1.75 and n = 1.75 and ρ 0 = 15.97 µΩ·cm, A = 0.53 µΩ·cm/K 2.21 and n = 2.21 for the currents along J\\n[100] and [001] directions, respectively. Here the exponent n is close to 2 which can in principle be explained on the basis of electron-electron scattering. Since our data do not extend for T << T N we have not attempted to fit our data using sophisticated model of spin-waves.\"\n },\n {\n \"title\": \"C. Magnetic susceptibility and magnetization\",\n \"content\": \"The temperature dependence of magnetic susceptibility in the temperature range from 1.8 to 300 K measured in a field of 1 kOe along the two principal directions viz., H parallel to [100] and [001] is shown in Fig. 3. The antiferromagnetic ordering at T N = 4.6 K is clearly seen as indicated by the arrow. The susceptibility below T N remains almost T -independent and at the lowest temperature measured there is a small rise in the susceptibility, indicating that the antiferromagnetism observed in CeAg 2 Ge 2 is not a simple two sublattice antiferromagnetism. For example, canted antiferromagnetism may show a weak residual ferromagnetic magnetization in the Néel state. Also, this type of temperature independent susceptibility at low temperature may be attributed to the crystalline electric field (CEF) effect. The inverse magnetic susceptibility of CeAg 2 Ge 2 does not obey the simple Curie-Weiss law (not shown here), on the other hand, it can be very well fitted to a modified Curie-Weiss law which is given by χ = χ 0 + C T −θp , where χ 0 is the temperature-independent part of the magnetic susceptibility and C is the Curie constant. The main contributions to χ 0 includes the core-electron diamagnetism, and the susceptibility of the conduction electrons. The details of the inverse magnetic susceptibility is discussed later in the discussion part. For an effective magnetic moment of 2.54 µ B /Ce we have estimated the θ p values as -7.2 K and -42 K for H\\n[100] and [001], respectively.\\nThe field dependence of isothermal magnetization at T = 2 K measured in a SQUID magnetometer up to a field of 70 kOe is shown in Fig. 4 curves show large uniaxial magnetocrystalline anisotropy.\\nThe magnetization for H [100] is linear for low fields and shows metamagnetic transitions at critical fields H m1 = 31 kOe and at H m2 = 44.7 kOe and nearly saturates at 70 kOe with a saturation moment µ s = 1.6 µ B /Ce, this indicates that [100]-axis is the easy axis of magnetization. Here, the saturation moment is smaller than the free ion value of 2.1 µ B /Ce which could be attributed to the crystal field effects. However, one can achieve the saturation value at high applied magnetic fields. On the other hand, the magnetization for H\\n[001] is very small and varies linearly with field reaching a value of 0.32 µ B /Ce at 50 kOe, indicating a hard axis of magnetization. We have also performed the isothermal magnetization at 3 K, 4 K, 5 K and 10 K for H\\n[100]. From the differential plots of the isothermal magnetization measurements, we have constructed the magnetic phase diagram as shown in Fig. 5. The two metamagnetic transitions are clearly seen for 2 K and 3 K magnetization curves; however at 3 K only one metamagnetic transition is seen. For temperatures above the magnetic ordering temperature the magnetization curves did not show any metamagnetic behaviour and the magnetization curves were linear indicating a paramagnetic state. and β is the phonon contribution to the heat capacity. The γ and β values thus obtained are estimated to be 2.8 mJ/K 2 ·mol and 0.59 mJ/K 4 ·mol, respectively. The inset of Fig. 6(a) shows the low temperature part of the specific heat and the antiferromagnetic ordering is manifested by the clear jump in the specific heat at T N = 4.6 K as indicated by the arrow. The inset of Fig. 6(b) shows the specific heat in the form of C/T versus T . Just below the magnetic ordering the specific heat shows a broad peak in the C/T versus T curve which presumably indicates the presence of low lying crystal field levels.\\nAssuming the lattice heat capacity of CeAg 2 Ge 2 is the same as that of LaAg 2 Ge 2 , the 4f -derived contribution to the heat capacity C mag was obtained by subtracting the specific heat of LaAg 2 Ge 2 from the total specific heat of CeAg 2 Ge 2 . Figure 6(c) shows C mag /T versus T together with the entropy S mag which is obtained by integrating C mag /T . As it can be seen from the figure, the entropy of CeAg 2 Ge 2 is very high at the magnetic ordering temperature and reaches R ln 4 near 20 K. In tetragonal symmetry, the degenerate six fold levels of the groundstate multiplet of Ce 3+ split into three doublets and ∆ 1 and ∆ 2 are the excitation energies of the first and second excited states, respectively. Since the entropy change reaches R ln 4, not too far above T N , one can come to a conclusion that the ground state and the first excited state are very closely spaced or nearly degenerate. This finding clearly corroborates the earlier neutron scattering results by Loidl et al. 11 in which they could observe only one crystal field transition at 11 meV and concluded that the ground state is almost degenerate with the first excited state. From the crystalline electric field analysis of the magnetic susceptibility data, to be discussed later, we found that the energies of the exicted states ∆ 1 and ∆ 2 as 5 K and 130 K, respectively. Due to the very small splitting energy between the ground state and the first excited state the estimation of the Sommerfeld coefficient γ, by the usual method, from the low temperature data will lead to ambiguity. Hence we estimated the γ value from the high temperature data in the paramagnetic region above the magnetic ordering after subtracting the Schottky contribution and linearly extrapolating the C/T versus T 2 behaviour to T = 0 K and is shown in Fig. 6(b). The γ value thus estimated is 45 mJ/K 2 ·mol.\"\n },\n {\n \"title\": \"E. Magnetoresistance\",\n \"content\": \"We have also studied the effect of magnetic field on the resistivity of CeAg 2 Ge 2 . The magnetic field did not have any appreciable effect on the resistivity for the field perpendicular to the easy axis direction. On the other hand, when the field was applied parallel to the easy axis direction, we found that the resistivity gradually decreased with increasing field. In Fig. 7\"\n },\n {\n \"title\": \"magnetic field for H\",\n \"content\": \"[100], ∆ρ/ρ 0 at 2 K initially increases in the positive direction and then turns negative at higher magnetic fields, giving rise to a maximum at 3.1 T. This field value coincides with the metamagnetic transition observed in the magnetization measurement at T = 2 K. Such a behaviour of the magnetoresistance is qualitatively consistent with the theoretical calculation given by Yamada and Takada 19 . In the antiferromagnetic state (H < H m ), the magnetic moment fluctuation in one magnetic sublattice is enhanced by the field while, in the field induced ferromagnetic state (H > H m ), the fluctuation is suppressed by the field. The change in the fluctuation is reflected in the magnetoresistance. With the increase in the temperature, the peak in the magnetoresistance moves toward lower fields and decreases, finally disappears for temperatures above T N . In the paramagnetic region the negative magnetoresistance is due to the freezing out of the spin-flip scattering by the magnetic field. The normalized magnetoresistance for T > T N can be mapped onto a single curve using the scaling relation ∆ρ/ρ(0) = f [B/(T + T * )] derived by Schlottmann 20 within the Bethe-ansatz approach, as shown in Fig. 7(b). Here T * is the characteristic temperature which is an ap-proximate measure of the Kondo temperature T K and is estimated to be 0.36 K. This indicates that the Kondo effect is very weak in CeAg 2 Ge 2 , which substantiates our earlier prediction from the zero field resistivity data.\"\n },\n {\n \"title\": \"IV. DISCUSSION\",\n \"content\": \"From the results of the electrical resistivity, susceptibility and specific heat measurements it can be clearly seen that CeAg 2 Ge 2 undergoes an antiferromagnetic ordering at 4.6 K with the easy axis of magnetization as [100]. The magnetization at 70 kOe reaches 1.6 µ B /Ce thus corroborating the earlier neutron scattering experiment 11 on a polycrystalline sample of CeAg 2 Ge 2 . The electrical resistivity at high temperature shows a typical metallic behaviour and at sufficiently low temperature it shows a weak minimum before ordering magnetically. This behaviour is quite different from what one has observed in the CeCu 2 Ge 2 which is similar to CeAg 2 Ge 2 both structurally and magnetically, although the antiferromagnetic ordering temperature is nearly equal (T N = 4.1 K for CeCu 2 Ge 2 and T N = 4.6 K for CeAg 2 Ge 2 ). The logarithmic temperature dependence of electrical resistivity in CeCu 2 Ge 2 exhibit a double peak structure which is presumably attributed to the combined influence of the Kondo and crystalline electric field (CEF) effects. The Kondo temperature of CeCu 2 Ge 2 was estimated to be about 6 K 21 , whereas for CeAg 2 Ge 2 the Kondo temperature is very small. Since the unit cell volume of CeCu 2 Ge 2 is smaller (V ≈ 178Å 3 ), a larger value of the Kondo coupling constant J sf is expected and hence the Kondo interaction dominates in CeCu 2 Ge 2 compared to that in CeAg 2 Ge 2 (V ≈ 203Å 3 ). In CeCu 2 Ge 2 superconductivity occurs when the unit cell volume attains a favourable value of 168 ± 3Å 3 . This is achieved with an external pressure of 7 GPa. Considering this fact the unit cell volume of CeAg 2 Ge 2 is quite large and one would require a very high pressure to reduce the unit cell volume to nearly 168 ± 3Å 3 for probable observation of superconductivity. Based on this it can be said that CeAg 2 Ge 2 lies on the left hand side of the Doniach phase diagram in which the RKKY energy scale is dominant and Kondo interaction is weak.\\nThe heat capacity measurement of CeAg 2 Ge 2 single crystal clearly reveals the presence of low lying crystal field levels with a very small separation between the ground state and the first excited state indicating that the ground state is a quasi-quartet state instead of a doublet which is usually observed for a tetragonal site symmetry. In order to further analyze the crystal field levels and to understand the present anisotropy in the magnetic susceptibility we have performed the CEF analysis on the susceptibility data. For the purpose of CEF analysis in Fig. 8\\nThe magnetic susceptibility including the molecular field contribution λ is given by\\nwhere χ CEF is the CEF susceptibility. Diagonalization of the CEF Hamiltonian gives us the eignvalues and eigenfunctions. For Ce 3+ J = 5/2 wave function splits into three doublets, Γ\\n(1) 7 = a |±5/2 + b |∓3/2 , Γ\\n(2) 7 = a |±3/2 − b |∓5/2 and Γ 6 = |±1/2 , where a and b are mixing parameters with the condition a 2 + b 2 = 1. The CEF parameters were estimated from the fits to the magnetic susceptibility. Solid lines in Fig. 8 show the least square fitting to Eqn. 3, the CEF parameters thus obtained are listed in Table I. The corresponding crystal field level scheme together with that of CeCu 2 Ge 2 is shown in the bottom part of Fig. 8. The crystal field level scheme for CeCu 2 Ge 2 is taken from Ref. 21. The ground state of CeAg 2 Ge 2 shows a mixing of |∓3/2 and |±5/2 wave functions. From Fig. 8 it is obvious that the present set of CEF parameters gives a good fit to the experimental data there by explaining the anisotropy in the magnetic susceptibility. The CEF parameters have resulted in the first and second excited states at ∆ 1 = 5 K and ∆ 2 = 130 K. It is interesting to note here that the CEF level scheme of CeCu 2 Ge 2 is qualitatively opposite to the present case where the ground state is a doublet and the first and second excited states are nearly degenerate. This can be explained on the basis of the sign of the B 0 2 parameter. For CeCu 2 Ge 2 it is negative while it is positive for CeAg 2 Ge 2 . This change in sign of B 0 2 suggests that the CEF potential in CeT 2 Ge 2 is largely dependent on the hybridization between localized f -electron states and the conduction-electron bands.\"\n },\n {\n \"title\": \"V. CONCLUSION\",\n \"content\": \"Single crystals of CeAg 2 Ge 2 have been grown for the first time, by flux method by using a Ag-Ge binary eutectic composition as flux. The antiferromagnetic ordering temperature T N = 4.6 K is clearly manifested by the resistivity, heat capacity and susceptibility measurements. Thus the ambiguity about the magnetic ordering temperature of this compound, reflected in the conflicting reports earlier in the literature, has been removed. A large anisotropy in the electrical resistivity, magnetic susceptibility and magnetization is observed. The susceptibility and magnetization clearly reveals that [100]-axis as the easy axis of magnetization with a moment of 1.6 µ B /Ce at 70 kOe. Metamagnetic transitions have been observed at the critical fields, H m1 = 31 kOe and at H m2 = 44.7 kOe. The heat capacity and the susceptibility data clearly support the closely spaced ground and first excited states, which have been analyzed by the CEF calculations.\"\n },\n {\n \"title\": \"FIG. 1 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 6 (\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 7 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"TABLE I :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"(Color online) Powder X-ray diffraction pattern recorded for crushed single crystals of CeAg2Ge2 at room temperature. The solid line through the experimental data points is the Rietveld refinement profile calculated for the tetragonal CeAg2Ge2.","content":"FIG. 1 :\n1(Color online) Powder X-ray diffraction pattern recorded for crushed single crystals of CeAg2Ge2 at room temperature. The solid line through the experimental data points is the Rietveld refinement profile calculated for the tetragonal CeAg2Ge2."},{"id":"","caption":"online) Logarithmic temperature dependence of the dc electrical resistivity of CeAg2Ge2 for J [100] and [001]. The inset shows the linear temperature dependence of the electrical resistivity. The solid lines are least square fitting toa power law relation. resistivity was measured for the current direction parallel to [100] and [001]","content":"online) Logarithmic temperature dependence of the dc electrical resistivity of CeAg2Ge2 for J [100] and [001]. The inset shows the linear temperature dependence of the electrical resistivity. The solid lines are least square fitting toa power law relation. resistivity was measured for the current direction parallel to [100] and [001]"},{"id":"","caption":"online)Temperature dependence of magnetic susceptibility χ from 1.8 -300 K. The inset shows the low temperature magnetic susceptibility.","content":"online)Temperature dependence of magnetic susceptibility χ from 1.8 -300 K. The inset shows the low temperature magnetic susceptibility."},{"id":"","caption":"online) Isothermal magnetization curves of CeAg2Ge2 measured at T = 2 K along the two principal directions.","content":"online) Isothermal magnetization curves of CeAg2Ge2 measured at T = 2 K along the two principal directions."},{"id":"","caption":"a) shows the temperature dependence of the specific heat of single crystalline CeAg 2 Ge 2 together with the specific heat of a polycrytalline reference sample LaAg 2 Ge 2 . The low temperature data (∼ 1.5 to ∼ 10 K) of LaAg 2 Ge 2 have been fitted to the expression C = γT + βT 3 where γ is the electronic contribution","content":"Figure 6 (\n6a) shows the temperature dependence of the specific heat of single crystalline CeAg 2 Ge 2 together with the specific heat of a polycrytalline reference sample LaAg 2 Ge 2 . The low temperature data (∼ 1.5 to ∼ 10 K) of LaAg 2 Ge 2 have been fitted to the expression C = γT + βT 3 where γ is the electronic contribution"},{"id":"","caption":"online)(a) Temperature dependence of the specific heat of CeAg2Ge2 and LaAg2Ge2, the inset shows the low temperature part, (b) specific heat of CeAg2Ge2 in the form of C/T vs. T 2 after subtracting the schottky and the 4f contribution, the solid line shows the extrapolation of the high temperature specific heat to 0 K, inset shows C/T versus T and (c) magnetic specific heat Cmag in the form of Cmag/T vs. T together with the magnetic entropy Smag.","content":"online)(a) Temperature dependence of the specific heat of CeAg2Ge2 and LaAg2Ge2, the inset shows the low temperature part, (b) specific heat of CeAg2Ge2 in the form of C/T vs. T 2 after subtracting the schottky and the 4f contribution, the solid line shows the extrapolation of the high temperature specific heat to 0 K, inset shows C/T versus T and (c) magnetic specific heat Cmag in the form of Cmag/T vs. T together with the magnetic entropy Smag."},{"id":"","caption":"we have plotted the normalized magnetoresistance ∆ρ/ρ 0 = [ρ(B) -ρ(B = 0)]/ρ(B = 0) as a function of applied magnetic field at various fixed temperatures. With increasing (Color online) (a)Isothermal normalized magnetoresistance for CeAg2Ge2 as function of applied magnetic field for J [100] and H [100]. (b) Normalized magnetoresistivity, in the paramagnetic state, plotted as a function of B/(T + T * ).","content":"FIG. 7 :\n7we have plotted the normalized magnetoresistance ∆ρ/ρ 0 = [ρ(B) -ρ(B = 0)]/ρ(B = 0) as a function of applied magnetic field at various fixed temperatures. With increasing (Color online) (a)Isothermal normalized magnetoresistance for CeAg2Ge2 as function of applied magnetic field for J [100] and H [100]. (b) Normalized magnetoresistivity, in the paramagnetic state, plotted as a function of B/(T + T * )."},{"id":"","caption":"we have plotted the experimental results on susceptibility in the form of 1/(χ − χ 0 ), where χ 0 was determined as 1.33 × 10 −3 and 1.41 × 10 −3 emu/mol for H [001] and [100], respectively, so an effective magneticFIG. 8: (Color online) Temperature dependence of the inverse magnetic susceptibility in CeAg2Ge2. Solid lines are fitting to the CEF level scheme with a molecular field term. The bottom figure shows the crystal-field level scheme of Ce 3+ in the CeCu2Ge2 (taken from Ref.21) and CeAg2Ge2. moment of 2.54 µ B /Ce is obtained above 100 K. Similar kind of treatment has been made for CePt 3 Si 22 while performing the CEF analysis. The Ce atoms in CeAg 2 Ge 2 occupy the 2a Wyckoff position with the point symmetry 4/mmm (D 4h ) and hence possess the tetragonal site symmetry. The CEF hamiltonian for a tetragonal site symmetry is given by, are the CEF parameters and the Stevens operators, respectively 23,24 . For Ce atom, the 6th order terms O 0 6 and O 4 6 vanishes and hence CEF Hamiltonian reduces to,","content":"we have plotted the experimental results on susceptibility in the form of 1/(χ − χ 0 ), where χ 0 was determined as 1.33 × 10 −3 and 1.41 × 10 −3 emu/mol for H [001] and [100], respectively, so an effective magneticFIG. 8: (Color online) Temperature dependence of the inverse magnetic susceptibility in CeAg2Ge2. Solid lines are fitting to the CEF level scheme with a molecular field term. The bottom figure shows the crystal-field level scheme of Ce 3+ in the CeCu2Ge2 (taken from Ref.21) and CeAg2Ge2. moment of 2.54 µ B /Ce is obtained above 100 K. Similar kind of treatment has been made for CePt 3 Si 22 while performing the CEF analysis. The Ce atoms in CeAg 2 Ge 2 occupy the 2a Wyckoff position with the point symmetry 4/mmm (D 4h ) and hence possess the tetragonal site symmetry. The CEF hamiltonian for a tetragonal site symmetry is given by, are the CEF parameters and the Stevens operators, respectively 23,24 . For Ce atom, the 6th order terms O 0 6 and O 4 6 vanishes and hence CEF Hamiltonian reduces to,0.90|±5/2〉 \n\n|±1/2〉 \n\n|±1/2〉 \n\n0.44|±5/2〉+0.90| 3/2〉 \n0.66|±5/2〉 +0.75| 3/2〉 \n\n-0.44| 3/2〉 \n0.75|±5/2〉 -0.66| 3/2〉 \n\n5 Κ \n\n130 Κ \n191 Κ \n\nCF level scheme for Ce 3+ in \n\nCeCu 2 Ge 2 \nCeAg 2 Ge 2 \n\n400 \n\n200 \n\n0 \n\n1/(χ− χ \n0 ) (mol/emu) \n\n300 \n200 \n100 \n0 \n\nTemperature (K) \n\nCeAg 2 Ge 2 \nH // [001] \n\n[100] \n\nH CEF = B 0 \n2 O 0 \n2 + B 0 \n4 O 0 \n4 + B 4 \n4 O 4 \n4 + B 0 \n6 O 0 \n6 + B 4 \n6 O 4 \n6 , (1) \n\nwhere B m \nℓ and O m \nℓ"},{"id":"","caption":"CEF parameters, energy level schemes and the corresponding wave functions for CeAg2Ge2.","content":"TABLE I :\nICEF parameters, energy level schemes and the corresponding wave functions for CeAg2Ge2.CEF parameters \nB 0 \n2 (K) \nB 0 \n4 (K) \nB 4 \n4 (K) \nλi (emu/mol) −1 \n2.24 \n−0.19 \n2.40 \nλ [100] = −7.2 \nλ [001] = −26 \nenergy levels and wave functions \nE(K) \n| +5/2 \n| +3/2 \n| +1/2 \n| −1/2 \n| −3/2 \n| −5/2 \n130 \n−0."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"(Color online) Powder X-ray diffraction pattern recorded for crushed single crystals of CeAg2Ge2 at room temperature. The solid line through the experimental data points is the Rietveld refinement profile calculated for the tetragonal CeAg2Ge2.\",\n \"content\": \"FIG. 1 :\\n1(Color online) Powder X-ray diffraction pattern recorded for crushed single crystals of CeAg2Ge2 at room temperature. The solid line through the experimental data points is the Rietveld refinement profile calculated for the tetragonal CeAg2Ge2.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"online) Logarithmic temperature dependence of the dc electrical resistivity of CeAg2Ge2 for J [100] and [001]. The inset shows the linear temperature dependence of the electrical resistivity. The solid lines are least square fitting toa power law relation. resistivity was measured for the current direction parallel to [100] and [001]\",\n \"content\": \"online) Logarithmic temperature dependence of the dc electrical resistivity of CeAg2Ge2 for J [100] and [001]. The inset shows the linear temperature dependence of the electrical resistivity. The solid lines are least square fitting toa power law relation. resistivity was measured for the current direction parallel to [100] and [001]\"\n },\n {\n \"id\": \"\",\n \"caption\": \"online)Temperature dependence of magnetic susceptibility χ from 1.8 -300 K. The inset shows the low temperature magnetic susceptibility.\",\n \"content\": \"online)Temperature dependence of magnetic susceptibility χ from 1.8 -300 K. The inset shows the low temperature magnetic susceptibility.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"online) Isothermal magnetization curves of CeAg2Ge2 measured at T = 2 K along the two principal directions.\",\n \"content\": \"online) Isothermal magnetization curves of CeAg2Ge2 measured at T = 2 K along the two principal directions.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"a) shows the temperature dependence of the specific heat of single crystalline CeAg 2 Ge 2 together with the specific heat of a polycrytalline reference sample LaAg 2 Ge 2 . The low temperature data (∼ 1.5 to ∼ 10 K) of LaAg 2 Ge 2 have been fitted to the expression C = γT + βT 3 where γ is the electronic contribution\",\n \"content\": \"Figure 6 (\\n6a) shows the temperature dependence of the specific heat of single crystalline CeAg 2 Ge 2 together with the specific heat of a polycrytalline reference sample LaAg 2 Ge 2 . The low temperature data (∼ 1.5 to ∼ 10 K) of LaAg 2 Ge 2 have been fitted to the expression C = γT + βT 3 where γ is the electronic contribution\"\n },\n {\n \"id\": \"\",\n \"caption\": \"online)(a) Temperature dependence of the specific heat of CeAg2Ge2 and LaAg2Ge2, the inset shows the low temperature part, (b) specific heat of CeAg2Ge2 in the form of C/T vs. T 2 after subtracting the schottky and the 4f contribution, the solid line shows the extrapolation of the high temperature specific heat to 0 K, inset shows C/T versus T and (c) magnetic specific heat Cmag in the form of Cmag/T vs. T together with the magnetic entropy Smag.\",\n \"content\": \"online)(a) Temperature dependence of the specific heat of CeAg2Ge2 and LaAg2Ge2, the inset shows the low temperature part, (b) specific heat of CeAg2Ge2 in the form of C/T vs. T 2 after subtracting the schottky and the 4f contribution, the solid line shows the extrapolation of the high temperature specific heat to 0 K, inset shows C/T versus T and (c) magnetic specific heat Cmag in the form of Cmag/T vs. T together with the magnetic entropy Smag.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"we have plotted the normalized magnetoresistance ∆ρ/ρ 0 = [ρ(B) -ρ(B = 0)]/ρ(B = 0) as a function of applied magnetic field at various fixed temperatures. With increasing (Color online) (a)Isothermal normalized magnetoresistance for CeAg2Ge2 as function of applied magnetic field for J [100] and H [100]. (b) Normalized magnetoresistivity, in the paramagnetic state, plotted as a function of B/(T + T * ).\",\n \"content\": \"FIG. 7 :\\n7we have plotted the normalized magnetoresistance ∆ρ/ρ 0 = [ρ(B) -ρ(B = 0)]/ρ(B = 0) as a function of applied magnetic field at various fixed temperatures. With increasing (Color online) (a)Isothermal normalized magnetoresistance for CeAg2Ge2 as function of applied magnetic field for J [100] and H [100]. (b) Normalized magnetoresistivity, in the paramagnetic state, plotted as a function of B/(T + T * ).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"we have plotted the experimental results on susceptibility in the form of 1/(χ − χ 0 ), where χ 0 was determined as 1.33 × 10 −3 and 1.41 × 10 −3 emu/mol for H [001] and [100], respectively, so an effective magneticFIG. 8: (Color online) Temperature dependence of the inverse magnetic susceptibility in CeAg2Ge2. Solid lines are fitting to the CEF level scheme with a molecular field term. The bottom figure shows the crystal-field level scheme of Ce 3+ in the CeCu2Ge2 (taken from Ref.21) and CeAg2Ge2. moment of 2.54 µ B /Ce is obtained above 100 K. Similar kind of treatment has been made for CePt 3 Si 22 while performing the CEF analysis. The Ce atoms in CeAg 2 Ge 2 occupy the 2a Wyckoff position with the point symmetry 4/mmm (D 4h ) and hence possess the tetragonal site symmetry. The CEF hamiltonian for a tetragonal site symmetry is given by, are the CEF parameters and the Stevens operators, respectively 23,24 . For Ce atom, the 6th order terms O 0 6 and O 4 6 vanishes and hence CEF Hamiltonian reduces to,\",\n \"content\": \"we have plotted the experimental results on susceptibility in the form of 1/(χ − χ 0 ), where χ 0 was determined as 1.33 × 10 −3 and 1.41 × 10 −3 emu/mol for H [001] and [100], respectively, so an effective magneticFIG. 8: (Color online) Temperature dependence of the inverse magnetic susceptibility in CeAg2Ge2. Solid lines are fitting to the CEF level scheme with a molecular field term. The bottom figure shows the crystal-field level scheme of Ce 3+ in the CeCu2Ge2 (taken from Ref.21) and CeAg2Ge2. moment of 2.54 µ B /Ce is obtained above 100 K. Similar kind of treatment has been made for CePt 3 Si 22 while performing the CEF analysis. The Ce atoms in CeAg 2 Ge 2 occupy the 2a Wyckoff position with the point symmetry 4/mmm (D 4h ) and hence possess the tetragonal site symmetry. The CEF hamiltonian for a tetragonal site symmetry is given by, are the CEF parameters and the Stevens operators, respectively 23,24 . For Ce atom, the 6th order terms O 0 6 and O 4 6 vanishes and hence CEF Hamiltonian reduces to,0.90|±5/2〉 \\n\\n|±1/2〉 \\n\\n|±1/2〉 \\n\\n0.44|±5/2〉+0.90| 3/2〉 \\n0.66|±5/2〉 +0.75| 3/2〉 \\n\\n-0.44| 3/2〉 \\n0.75|±5/2〉 -0.66| 3/2〉 \\n\\n5 Κ \\n\\n130 Κ \\n191 Κ \\n\\nCF level scheme for Ce 3+ in \\n\\nCeCu 2 Ge 2 \\nCeAg 2 Ge 2 \\n\\n400 \\n\\n200 \\n\\n0 \\n\\n1/(χ− χ \\n0 ) (mol/emu) \\n\\n300 \\n200 \\n100 \\n0 \\n\\nTemperature (K) \\n\\nCeAg 2 Ge 2 \\nH // [001] \\n\\n[100] \\n\\nH CEF = B 0 \\n2 O 0 \\n2 + B 0 \\n4 O 0 \\n4 + B 4 \\n4 O 4 \\n4 + B 0 \\n6 O 0 \\n6 + B 4 \\n6 O 4 \\n6 , (1) \\n\\nwhere B m \\nℓ and O m \\nℓ\"\n },\n {\n \"id\": \"\",\n \"caption\": \"CEF parameters, energy level schemes and the corresponding wave functions for CeAg2Ge2.\",\n \"content\": \"TABLE I :\\nICEF parameters, energy level schemes and the corresponding wave functions for CeAg2Ge2.CEF parameters \\nB 0 \\n2 (K) \\nB 0 \\n4 (K) \\nB 4 \\n4 (K) \\nλi (emu/mol) −1 \\n2.24 \\n−0.19 \\n2.40 \\nλ [100] = −7.2 \\nλ [001] = −26 \\nenergy levels and wave functions \\nE(K) \\n| +5/2 \\n| +3/2 \\n| +1/2 \\n| −1/2 \\n| −3/2 \\n| −5/2 \\n130 \\n−0.\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". D Mond, J Jaccard, Flouquet, J. Low Temp. Phys. 122591mond, D. Jaccard and J. Flouquet, J. Low Temp. Phys. 122, 591 (2001).","doi":""},{"text":". R Movshovich, T Graf, D Mandrus, J D Thompson, J L Smith, Z Fisk, Phys. Rev. B. 538241R. Movshovich, T. Graf, D. Mandrus, J. D. Thompson, J. L. Smith and Z. Fisk, Phys. Rev. B 53, 8241 (1996).","doi":""},{"text":". P Haen, J Flouquet, F Labpierre, P Lejay, G Remenyi, J. Low Temp. Phys. 67391P. Haen, J. Flouquet, F. Labpierre, P. Lejay and G. Re- menyi, J. Low Temp. Phys. 67, (1987) 391.","doi":""},{"text":". H Abe, K Yoshii, H Kitazawa, 253Physica B 312-313H. Abe, K. Yoshii and H. Kitazawa, Physica B 312-313, 253 (2002).","doi":""},{"text":". S Raymond, P Haen, R Calemczuk, S Kame, B Fak, P Lejay, T Fukuhara, J Flouquet, J. Phys.: Condens. Matter. 115547S. Raymond, P. Haen, R. Calemczuk, S. Kame, B. Fak, P. Lejay, T. Fukuhara and J. Flouquet, J. Phys.: Condens. Matter 11, 5547 (1999).","doi":""},{"text":". T Fukuhara, K Maezawa, H Ohkuni, J Sakurai, H Sato, J. Magn. Magn. Mater. 889T. Fukuhara, K. Maezawa, H. Ohkuni, J. Sakurai and H. Sato, J. Magn. Magn. Mater 140-144, 889 (1995).","doi":""},{"text":". F M Grosche, P Agarwal, S R Julian, N J Wilson, R K W Haselwimmer, S J S Lister, N D Mathur, F V Carter, S S Saxena, G G Lonzarich, J. Phys.: Condens. Matter. 12533F. M. Grosche, P. Agarwal, S. R. Julian, N. J. Wilson, R. K. W. Haselwimmer, S. J. S. Lister, N. D. Mathur, F. V. Carter, S. S. Saxena and G. G. Lonzarich, J. Phys.: Condens. Matter 12, L533 (2000).","doi":""},{"text":". R Rauchschwalbe, U Gottwick, U Ahlheim, H M Mayer, F Steglich, J. Less Common. Metals. 111265R. Rauchschwalbe, U. Gottwick, U. Ahlheim, H. M. Mayer and F. Steglich, J. Less Common. Metals 111, 265 (1985).","doi":""},{"text":". G Knopp, H Spille, A Loidl, K Knorr, U Rauchschwalbe, R Felten, G Weber, F Steglich, A P Murani, J. Magn. Magn. Mater. 63 &. 6488G. Knopp, H. Spille, A. Loidl, K. Knorr, U. Rauch- schwalbe, R. Felten, G. Weber, F. Steglich and A. P. Mu- rani, J. Magn. Magn. Mater. 63 & 64, 88 (1987).","doi":""},{"text":". A Loidl, K Knorr, G Knopp, A Krimmel, R Caspary, A Böhm, G Sparn, C Geibel, F Steglich, A P Murani, Phys. Rev. B. 469341A. Loidl, K. Knorr, G. Knopp, A. Krimmel, R. Caspary, A. Böhm, G. Sparn, C. Geibel, F. Steglich and A. P. Murani, Phys. Rev. B 46, 9341 (1992).","doi":""},{"text":". A Böhm, R Caspary, U Habel, L Pawlak, A Zuber, F Steglich, A Loidl, J. Magn. Magn. Mater. 76 &. 77150A. Böhm, R. Caspary, U. Habel, L. Pawlak, A. Zuber F. Steglich and A. Loidl, J. Magn. Magn. Mater. 76 & 77, 150 (1988).","doi":""},{"text":". E Cordruwish, D Kaczorowski, A Saccone, P Rogl, R Ferro, J. Phase Equilibria. 20407E. Cordruwish, D. Kaczorowski, A. Saccone, P. Rogl and R. Ferro, J. Phase Equilibria 20, 407 (1999).","doi":""},{"text":". H Nakashima, A Thamizhavel, T D Matsuda, Y Haga, T Takeuchi, K Sugiyama, R Settai, Y Ōnuki, J Compd, 424H. Nakashima, A. Thamizhavel, T. D. Matsuda, Y. Haga, T. Takeuchi, K. Sugiyama, R. Settai and Y.Ōnuki, J. Al- loys Compd. 424, 7 (2006).","doi":""},{"text":". E Morosan, S L Bud'ko, P C Canfield, M S Torikachvili, A H Lacerda, J. Magn. Magn. Mater. 227298E. Morosan, S. L. Bud'ko, P. C. Canfield, M. S. Torikachvili and A. H. Lacerda, J. Magn. Magn. Mater. 227, 298 (2004).","doi":""},{"text":". S L Bud'ko, Z Islam, T A Wiener, I R Fisher, A H Lacerda, P C Canfield, J. Magn. Magn. Mater. 20553S. L. Bud'ko, Z. Islam, T. A. Wiener, I. R. Fisher, A. H. Lacerda, P. C. Canfield, J. Magn. Magn. Mater. 205, 53 (1999).","doi":""},{"text":". P C Canfield, I R Fisher, J. Crystal Growth. 225155P. C. Canfield, I. R. Fisher, J. Crystal Growth 225, 155 (2001).","doi":""},{"text":". C S Garde, J Ray, J. Phys. Condens. Matter. 68585C. S. Garde and J. Ray, J. Phys. Condens. Matter 6, 8585 (1994).","doi":""},{"text":". H Yamada, S Takada, Prog. Theor. Phys. 491401H. Yamada and S. Takada, Prog. Theor. Phys. 49, 1401 (1973).","doi":""},{"text":". P Schlottman, Z. Phys. B: Condens. Matter. 51223P. Schlottman, Z. Phys. B: Condens. Matter 51, 223 (1983).","doi":""},{"text":". G Knopp, A Loidl, K Knorr, L Pawlak, M Duczmal, R Caspary, U Gottwick, H Spille, F Steglich, A P Murani, Z. Phys. B -Condensed Matter. 7795G. Knopp, A. Loidl, K. Knorr, L. Pawlak, M. Duczmal, R. Caspary, U. Gottwick, H. Spille, F. Steglich and A. P. Murani, Z. Phys. B -Condensed Matter 77, 95 (1989).","doi":""},{"text":". T Takeuchi, S Hashimoto, T Yasuda, H Shishido, T Ueda, M Yamada, Y Obiraki, M Shiimoto, H Kohara, T Yamamoto, K Sugiyama, K Kindo, T D Matsuda, Y Haga, Y Aoki, H Sato, R Settai, Y Ōnuki, J. Phys.: Condens. Matter. 16333T. Takeuchi, S. Hashimoto, T. Yasuda, H. Shishido, T. Ueda, M. Yamada, Y. Obiraki, M. Shiimoto, H. Kohara, T. Yamamoto, K. Sugiyama, K. Kindo, T. D. Matsuda, Y. Haga, Y. Aoki, H. Sato, R. Settai and Y.Ōnuki, J. Phys.: Condens. Matter 16, L333 (2004).","doi":""},{"text":". K W H Stevens, Proc. Phys. Soc. 65209K. W. H. Stevens, Proc. Phys. Soc., London, Sect. A65, 209 (1952).","doi":""},{"text":"M T Hutchings, Solid State Physics: Advances in Research and Applications. F. Seitz and B. TurnbullNew YorkAcademic16227M. T. Hutchings, in Solid State Physics: Advances in Re- search and Applications, edited by F. Seitz and B. Turnbull (Academic, New York, 1965), Vol.16, p.227.","doi":""}],"string":"[\n {\n \"text\": \". D Mond, J Jaccard, Flouquet, J. Low Temp. Phys. 122591mond, D. Jaccard and J. Flouquet, J. Low Temp. Phys. 122, 591 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Movshovich, T Graf, D Mandrus, J D Thompson, J L Smith, Z Fisk, Phys. Rev. B. 538241R. Movshovich, T. Graf, D. Mandrus, J. D. Thompson, J. L. Smith and Z. Fisk, Phys. Rev. B 53, 8241 (1996).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Haen, J Flouquet, F Labpierre, P Lejay, G Remenyi, J. Low Temp. Phys. 67391P. Haen, J. Flouquet, F. Labpierre, P. Lejay and G. Re- menyi, J. Low Temp. Phys. 67, (1987) 391.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Abe, K Yoshii, H Kitazawa, 253Physica B 312-313H. Abe, K. Yoshii and H. Kitazawa, Physica B 312-313, 253 (2002).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Raymond, P Haen, R Calemczuk, S Kame, B Fak, P Lejay, T Fukuhara, J Flouquet, J. Phys.: Condens. Matter. 115547S. Raymond, P. Haen, R. Calemczuk, S. Kame, B. Fak, P. Lejay, T. Fukuhara and J. Flouquet, J. Phys.: Condens. Matter 11, 5547 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Fukuhara, K Maezawa, H Ohkuni, J Sakurai, H Sato, J. Magn. Magn. Mater. 889T. Fukuhara, K. Maezawa, H. Ohkuni, J. Sakurai and H. Sato, J. Magn. Magn. Mater 140-144, 889 (1995).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". F M Grosche, P Agarwal, S R Julian, N J Wilson, R K W Haselwimmer, S J S Lister, N D Mathur, F V Carter, S S Saxena, G G Lonzarich, J. Phys.: Condens. Matter. 12533F. M. Grosche, P. Agarwal, S. R. Julian, N. J. Wilson, R. K. W. Haselwimmer, S. J. S. Lister, N. D. Mathur, F. V. Carter, S. S. Saxena and G. G. Lonzarich, J. Phys.: Condens. Matter 12, L533 (2000).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Rauchschwalbe, U Gottwick, U Ahlheim, H M Mayer, F Steglich, J. Less Common. Metals. 111265R. Rauchschwalbe, U. Gottwick, U. Ahlheim, H. M. Mayer and F. Steglich, J. Less Common. Metals 111, 265 (1985).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Knopp, H Spille, A Loidl, K Knorr, U Rauchschwalbe, R Felten, G Weber, F Steglich, A P Murani, J. Magn. Magn. Mater. 63 &. 6488G. Knopp, H. Spille, A. Loidl, K. Knorr, U. Rauch- schwalbe, R. Felten, G. Weber, F. Steglich and A. P. Mu- rani, J. Magn. Magn. Mater. 63 & 64, 88 (1987).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Loidl, K Knorr, G Knopp, A Krimmel, R Caspary, A Böhm, G Sparn, C Geibel, F Steglich, A P Murani, Phys. Rev. B. 469341A. Loidl, K. Knorr, G. Knopp, A. Krimmel, R. Caspary, A. Böhm, G. Sparn, C. Geibel, F. Steglich and A. P. Murani, Phys. Rev. B 46, 9341 (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Böhm, R Caspary, U Habel, L Pawlak, A Zuber, F Steglich, A Loidl, J. Magn. Magn. Mater. 76 &. 77150A. Böhm, R. Caspary, U. Habel, L. Pawlak, A. Zuber F. Steglich and A. Loidl, J. Magn. Magn. Mater. 76 & 77, 150 (1988).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Cordruwish, D Kaczorowski, A Saccone, P Rogl, R Ferro, J. Phase Equilibria. 20407E. Cordruwish, D. Kaczorowski, A. Saccone, P. Rogl and R. Ferro, J. Phase Equilibria 20, 407 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Nakashima, A Thamizhavel, T D Matsuda, Y Haga, T Takeuchi, K Sugiyama, R Settai, Y Ōnuki, J Compd, 424H. Nakashima, A. Thamizhavel, T. D. Matsuda, Y. Haga, T. Takeuchi, K. Sugiyama, R. Settai and Y.Ōnuki, J. Al- loys Compd. 424, 7 (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Morosan, S L Bud'ko, P C Canfield, M S Torikachvili, A H Lacerda, J. Magn. Magn. Mater. 227298E. Morosan, S. L. Bud'ko, P. C. Canfield, M. S. Torikachvili and A. H. Lacerda, J. Magn. Magn. Mater. 227, 298 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S L Bud'ko, Z Islam, T A Wiener, I R Fisher, A H Lacerda, P C Canfield, J. Magn. Magn. Mater. 20553S. L. Bud'ko, Z. Islam, T. A. Wiener, I. R. Fisher, A. H. Lacerda, P. C. Canfield, J. Magn. Magn. Mater. 205, 53 (1999).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P C Canfield, I R Fisher, J. Crystal Growth. 225155P. C. Canfield, I. R. Fisher, J. Crystal Growth 225, 155 (2001).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C S Garde, J Ray, J. Phys. Condens. Matter. 68585C. S. Garde and J. Ray, J. Phys. Condens. Matter 6, 8585 (1994).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Yamada, S Takada, Prog. Theor. Phys. 491401H. Yamada and S. Takada, Prog. Theor. Phys. 49, 1401 (1973).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Schlottman, Z. Phys. B: Condens. Matter. 51223P. Schlottman, Z. Phys. B: Condens. Matter 51, 223 (1983).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Knopp, A Loidl, K Knorr, L Pawlak, M Duczmal, R Caspary, U Gottwick, H Spille, F Steglich, A P Murani, Z. Phys. B -Condensed Matter. 7795G. Knopp, A. Loidl, K. Knorr, L. Pawlak, M. Duczmal, R. Caspary, U. Gottwick, H. Spille, F. Steglich and A. P. Murani, Z. Phys. B -Condensed Matter 77, 95 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Takeuchi, S Hashimoto, T Yasuda, H Shishido, T Ueda, M Yamada, Y Obiraki, M Shiimoto, H Kohara, T Yamamoto, K Sugiyama, K Kindo, T D Matsuda, Y Haga, Y Aoki, H Sato, R Settai, Y Ōnuki, J. Phys.: Condens. Matter. 16333T. Takeuchi, S. Hashimoto, T. Yasuda, H. Shishido, T. Ueda, M. Yamada, Y. Obiraki, M. Shiimoto, H. Kohara, T. Yamamoto, K. Sugiyama, K. Kindo, T. D. Matsuda, Y. Haga, Y. Aoki, H. Sato, R. Settai and Y.Ōnuki, J. Phys.: Condens. Matter 16, L333 (2004).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K W H Stevens, Proc. Phys. Soc. 65209K. W. H. Stevens, Proc. Phys. Soc., London, Sect. A65, 209 (1952).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M T Hutchings, Solid State Physics: Advances in Research and Applications. F. Seitz and B. TurnbullNew YorkAcademic16227M. T. Hutchings, in Solid State Physics: Advances in Re- search and Applications, edited by F. Seitz and B. Turnbull (Academic, New York, 1965), Vol.16, p.227.\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":596,"cells":{"text":{"kind":"string","value":"\narXiv:cond-mat/9409090v1 21 Sep 1994 Semi-classical description of the frustrated antiferromagnetic chain Typeset using REVT E X 1\n\n\nD Allen \nCentre de Recherche en Physique du Solide et Département de Physique\nUniversité de Sherbrooke\nJ1K 2R1SherbrookeQuébecCanada\n\nD Sénéchal \nCentre de Recherche en Physique du Solide et Département de Physique\nUniversité de Sherbrooke\nJ1K 2R1SherbrookeQuébecCanada\n\narXiv:cond-mat/9409090v1 21 Sep 1994 Semi-classical description of the frustrated antiferromagnetic chain Typeset using REVT E X 1\n(September 1994)\nThe antiferromagnetic Heisenberg model on a chain with nearest and next nearest neighbor couplings is mapped onto the SO(3) nonlinear sigma model in the continuum limit. In one spatial dimension this model is always in its disordered phase and a gap opens to excited states. The latter form a doubly degenerate spin-1 branch at all orders in 1/N . We argue that this feature should be present in the spin-1 Heisenberg model itself. Exact diagonalizations are used to support this claim. The inapplicability of this model to half-integer spin chains is discussed.\n\nI. INTRODUCTION\n\nInterest in low-dimensional magnetic systems has been great in recent years, partly because of the widespread belief that magnetism plays a key role in high-temperature superconductivity, but also because of the successful application of field-theoretic methods to these systems, in particular to spin chains. Indeed, a mapping from the spin-s antiferromagnetic \nL σ = 1 2g 1 v (∂ t e) 2 − v(∂ x e) 2 .(1)\nThe coupling g and the velocity v are related to the spin s, the AF coupling J and the lattice spacing: v = 2Jas and g = 2/s. To this action one must add a topological term S top for half-integer spin (cf. Sec. IV). An extension of this mapping to dimensions higher than one has been obtained by many authors, with the difference that no topological term arises, hence no distinction between half-integral and integral spin.\n\nDombre and Read 4 have conducted a similar analysis for the antiferromagnetic Heisenberg model on a triangular lattice. The essential difference here is the presence of frustration, leading to a classical ground state characterized by a 120 • order. The local order must then be specified by a rotation matrix instead of a unit vector and the order parameter is thus an element of SO (3). The long-wavelength action found in this case is the SO(3) nonlinear sigma model, with Lagrangian density\nL = 1 2g 1 v tr(∂ t R −1 ∂ t R) − vtr(P ∇R −1 · ∇R) .(2)\nHere R(x, t) is a position-and time-dependent rotation matrix and P is the constant diagonal matrix diag [1,1,0]. Again, the constants g and v depend on the spin s, the lattice spacing a and the AF couplings J. At zero temperature and in dimension two, this model has an ordered phase for g < g c and a disordered phase otherwise. Since g ∼ a/s, the spin-1 2 case is the closest to the disordered phase. However, it is now widely believed 5 that the ground state of the spin-1 2 antiferromagnet on a triangular lattice has long range order and consequently the disordered phase of the SO(3) model is not physically realized to twodimensional antiferromagnets, at least at half-filling (one spin per site).\n\nIn this work we will argue that this disordered phase of the SO(3) sigma model could be realized in a frustrated antiferromagnetic chain, with Hamiltonian\nH = J n S n · S n+1 + J ′ n S n · S n+2 .(3)\nIntroducing a next-nearest-neighbor (NNN) coupling J ′ modifies the classical ground state if J ′ /J > 1 4 : the spins still lie on a common plane, but instead of being collinear (as in the Néel state), they are arranged in a canted configuration (see Fig. 1) in which each spin makes an angle α with its predecessor, given by\ncos α = − J 4J ′ .(4)\nIn the special case J ′ /J = 1 2 , the classical order is quite similar to that of the triangular lattice, with its 120 • angle from site to site and its periodicity of three sites. For spin-1 2 this particular case constitutes the Majumdar-Gosh model, whose ground state is exactly known. 6 For a generic value of J ′ /J the classical order is incommensurable, with infinite periodicity. The spin-1 case has been studied previously using various methods: exact diagonalizations 7 , approximate mappings to field theories involving fermions 8 or bosons 9,10 , among others. This paper is organized as follows. In Sec. II we explain how to obtain the SO (3) nonlinear sigma model as the continuum limit of the model defined in Eq. (3). In Sec. III we argue that the disordered phase of this model is characterized by a singlet ground state with a gap to two degenerate triplets of excited states. Hence this model cannot adequately represent the half-integer NNN chain, but may be correct for integer spin. In Sec. IV we discuss our results with the help of exact diagonalization results and discuss the apparent lack of distinction between integer and half-integer spin in this model.\n\n\nII. SEMI-CLASSICAL MAPPING TO THE SO(3) NONLINEAR SIGMA MODEL\n\nIn this section we argue that the long wavelength, continuum theory describing the frustrated antiferromagnetic chain is specified by the Lagrangian density of Eq. (17), with the notation explained thereafter.\n\nThe Lagrangian description of spin dynamics, as used in path integrals, requires the introduction of spin coherent states. The unfamiliar reader is referred to Fradkin's text 11 or to Manousakis' review 12 . Each quantum spin S i is described by a fluctuating unit vector n i and the action associated to a spin Hamiltonian H is\nS = dt s i A(n i ) · ∂ t n i − H(5)\nwherein S i is replaced by sn i in the Hamiltonian. A is the vector potential of a magnetic monopole of flux 4π. Thus, given a closed curve n(t) on the unit sphere, the integral A · dn equals the area of the sphere enclosed by the curve, modulo 4π.\n\nWe shall use a description of each spin S i in terms of a slowly varying SO(3) order parameter R(x, t) and a local magnetization vector l(x, t):\nS k = s R (n c k + al) |n c k + al| (6a) n c k =î cos(kα) + sin(kα) (6b)\nHere n c k is the orientation of the spin S k in a classical ground state taken as reference, in which all spins lie on the plane defined by two mutually orthogonal unit vectorsî and; α is the pitch of this ground state, as given in Eq. (4); finally, a is the lattice spacing: It is assumed that the spin configurations that contribute significantly to the path integral are locally close to the classical ground state, and the approximation that the deviation al is small will be controlled at the same time as the continuum approximation.\n\nSome comments are in order concerning this representation. If J ′ = J/2 (or α = 2π/3), the periodicity is 3 and the relation (6) may be considered as a bona fide change of variables if we group the spins in sets of three and assume that the fields R and l do not vary within such sets. We check that the number of degrees of freedom match: 6 per set of three spins.\n\nHowever, it is more convenient to assume that R and l vary slowly from site to site, in which case the representation (6) may not be regarded as a change of variables, but simply as long wavelength description of the fluctuations around the classical ground state; then we need not restrict ourselves to the case J ′ = J/2. The next step is to substitute the representation (6) into the action (5) and to Taylor expand when needed in order to get a continuum action in terms of R and l only. We will treat the kinetic term first, and then the Hamiltonian. For the sake of convenience, we will assume a periodicity of N spins in the classical ground state. The case N = 3 (or J ′ = J/2) is the simplest, leading to a calculation almost identical to that of Ref. 4. The next simplest is N = 5, with J ′ ≈ 0.809J, and so on.\n\nAn explicit calculation of the kinetic term is not needed here, since it would be almost identical to that of Ref. 4, to which the reader is referred. The difference here lies in the number of reference vectors n c i , which is N instead of 3. The kinetic part of the Lagrangian density is then\nL K = as(T l) · V(7)\nwith the definitions\nT ab = δ ab − n c i,a n c i,b = 1 2 0 0 0 1 2 0 0 0 0 (8a) V a = − 1 2 ǫ abc (R −1 ∂ t R) bc (8b)\nwherein the indices in spin space correspond to the axes defined byî, andî ×, and · · · means an average over N contiguous sites.\n\nIn order to write down the continuum limit of the Hamiltonian H, we need to calculate the interaction H (k) of a spin S k with its four neighbors and then average the result over the N spins S k of the period. Specifically, the Hamiltonian density is\nH = 1 2aN k H (k) = 1 2a H (k) .(9)\nThe contribution H (k) may be written as follows:\nH (k) = J S k · S k+1 + S k · S k−1 + J ′ J S k · S k+2 + J ′ J S k · S k−2 + aJ S k · ∂ x S k+1 − S k · ∂ x S k−1 + 2J ′ J S k · ∂ x S k+2 − 2J ′ J S k · ∂ x S k−2 + a 2 J 2 S k · ∂ 2 x S k+1 + S k · ∂ 2 x S k−1 + 4J ′ J S k · ∂ 2 x S k+2 + 4J ′ J S k · ∂ 2 x S k−2 + · · ·(10)\nwhere the spin S n stands in fact for its expression in terms of R and l (Eq. (6)) and where the derivatives act on these same fields. We will carry the expansion to second order only, enough to yield a non trivial Lagrangian. We shall also use a Taylor expanded version of the representation (6):\nS k = sR(n c k ) + saR(l − [l · n c k ]n c k ) + sa 2 R(n c k ) 3 2 (n c k · l) 2 − 1 2 l 2 − sa 2 R(l)[n c k · l] + O(a 3 ) .(11)\nWe may then write the Hamiltonian density as an expansion in powers of a:\nH = H (0) + aH (1) + a 2 H (2) + · · ·(12)\nH (0) is a constant, independent of R and l. H (1) vanishes for the following reasons: Part of it depends on l, but it is proportional to n c k , which is zero. Another part depends on R, and is given by\nH (1) = s 2 2a (R −1 ∂ x R) ab J( n c k,a n c k+1,b − n c k,a n c k−1,b ) + 2J ′ ( n c k,a n c k+2,b − n c k,a n c k−2,b ) .(13)\nBut one easily calculates that n c k,a n c\nk+n,b = 1 2 cos nα sin nα 0 − sin nα cos nα 0 0 0 0 .(14)\nAfter substituting and using the relation (4), we find out that the expression (13) vanishes.\n\nFinally, we are left with the second order part H (2) . It is a straightforward exercise to\n\nshow that\nH = Jas 2 2 l a M ab l b − Jas 2 4 ξ − 1 ξ tr[P ∂ x R −1 ∂ x R](15)\nwhere ξ ≡ − cos α = J/4J ′ and where the diagonal matrix M has elements\nM 33 = (1 + ξ) 2 ξ M 11 = M 22 = M 33 (ξ 2 − ξ + 1 2 ) .(16)\nAgain, the matrix P is the projector onto the plane defined byî and, as in Eq. (2). The full Lagrangian density is then L = L K − H.\n\nThe last step consists in integrating out the field l.\n\nSince the latter appears quadratically in L, this operation amounts to substituting into L the solution of the classical equations of motion for l. The final result for the Lagrangian density is\nL = 1 2g 1 c tr[Q∂ t R −1 ∂ t R] −c tr[P ∂ x R −1 ∂ x R](17)\nwhere the matrix Q is diagonal, with elements\nQ 11 = Q 22 = 1 Q 33 = 2ξ(1 − ξ) ξ 2 + (ξ − 1) 2(18)\nand where the constantsg andc are defined as\ng = 2 s 1 + ξ 1 − ξ (19a) c = Jas 1 + ξ ξ 1 − ξ 2 .(19b)\nIn the special case ξ = 1 2 , which corresponds to J ′ = J/2 and α = 2π/3, one recovers precisely the form of the Lagrangian (2) since Q 33 = 1. The parameter ξ ranges from 0 (J ′ → ∞) to 1 (J ′ = J/4). The characteristic speedc diverges as ξ → 0, whereas the coupling constantg diverges as ξ → 1, the collinear phase boundary. In all cases, the Lagrangian (17) Its symmetries comprise global left rotations R → U L R with U L ∈ SO(3) and global right rotations R → RU R with U R ∈ SO(2) (i.e. U R commuting with P and Q). The first of these reflects invariance under rotations in spin space, whereas the other will have consequences on the spectrum of the theory: a degeneracy of the excitation branches.\n\nIt is important to stress here that in going from the discrete Heisenberg Hamiltonian (3) to the continuum Lagrangian (17) we have assumed that smooth configurations dominate the path integral. There seems to be no distinction between integer and half-integer spins within this mapping. In particular, there is no topological term of the type arising in the collinear antiferromagnetic chain (J ′ < J/4). In fact, none could exist, since the relevant homotopy group is trivial: π 2 (SO(3)) = 0. However, there are qualitative differences between integer and half-integer spin in this system. The question as to why they do not appear explicitly in the field theory will be discussed in Sec. IV.\n\n\nIII. SPECTRUM OF THE NONLINEAR SIGMA MODEL\n\nIn this section we discuss the spectrum of the model defined in Eq. (17), in particular regarding the gap to excited states and the degeneracy of the excitation branches. To this end it is preferable to rewrite the Lagrangian (17) in terms of two orthonormal vectors e 1 and e 2 specifying the rotation matrix R. Defining e 3 = e 1 × e 2 , the elements of the matrix R may be expressed as R ab = (e b ) a and tr[Q∂ t R −1 ∂ t R] = a Q aa (∂ t e a ) 2 . Using the fact that\n(∂ t e 3 ) 2 = (∂ t e 1 ) 2 + (∂ t e 2 ) 2 − 2(e 1 · ∂ t e 2 ) 2 ,(20)\nwe may finally express the Lagrangian in the following form:\nL = 1 2g 1 c (∂ t e 1 ) 2 + (∂ t e 2 ) 2 − γ c (e 1 · ∂ t e 2 ) 2 − c (∂ x e 1 ) 2 + (∂ x e 2 ) 2(21)\nwherein the coupling constants g, γ and the velocity c are now\ng = 2 s (1 + ξ)(2ξ 2 − 2ξ + 1) 1 − ξ (22a) c = Jas 1 + ξ ξ (1 − ξ 2 )(2ξ 2 − 2ξ + 1) (22b) γ = 4ξ(1 − ξ)(22c)\nThe functional integration measure associated with the fields e 1,2 must incorporate the constraints e 2 1 = e 2 2 = 1, e 1 · e 2 = 0. This may be done by introducing three Lagrange parameters σ 11 , σ 22 and σ 12 = σ 21 , and by adding the following constraint term to the Lagrangian:\nL constr = 1 2 σ ab (e a · e b − δ ab ) .(23)\nSince this model is defined in one spatial dimension, the Mermin-Wagner-Coleman theorem applies and the global SO(3) symmetry is not broken. If we extended the model to dimensions greater than one, a broken symmetry phase could be realized, for g below some critical value. In such a phase, three Goldstone modes would appear, corresponding to the three parameters of the broken SO(3) symmetry. These three Goldstone modes would have a linear dispersion relation, two of them with speed c and a third with speed c/ 1 − γ/2.\n\nThe most elegant way to see this is to go back to the form (17) of the Lagrangian and to substitute, in the small oscillation approximation, R ab ≈ δ ab + ε abc Ω c , keeping the terms of order Ω 2 c only. Since the field Ω c is unconstrained, the dispersion relations are easily read off. In the case of an antiferromagnet on a triangular or hexagonal lattice (ξ = 1 2 ), the two speeds are c and c √ 2. This may also be observed within spin-wave theory, since it is a feature of the ordered phase.\n\nHowever, we are interested here in the disordered phase of the model, which we shall study within the large-N approach (here N is the number of components of the vectors e a , normally 3). In the imaginary-time formalism, the partition function of the model is\nZ = [de 1 ][de 2 ][dσ ab ] exp − dxdτ L E(24)\nwith the Euclidian Lagrangian density\nL E = 1 2 (∂ τ e 1 ) 2 + (∂ τ e 2 ) 2 − gγ(e 1 · ∂ τ e 2 ) 2 + (∂ x e 1 ) 2 + (∂ x e 2 ) 2 + σ ab (e a · e b − 1 g δ ab )(25)\nwhere we have rescaled e a by a factor √ g in order to recover the standard normalization for the kinetic term. The characteristic speed c has been set to unity in order to lighten the notation; it may be restored by dimensional analysis. We will also use a Hubbard-Stratonovich decomposition of the quartic term:\nexp − dxdτ gγ 2 (e 1 · ∂ τ e 2 ) 2 = [dφ] exp − dxdτ 1 2 φ 2 − √ gγφ(e 1 · ∂ τ e 2 )(26)\nWe then proceed to find the large-N saddle point. In other words, we assume that the auxiliary fields φ and σ ab take a constant value, which is determined by extremizing the effective potential obtained by integrating the fields e a . This exercise is better done in Fourier space, in which the Euclidian action with constant auxiliary fields may be written as\nS E = 1 2 dω 2π dk 2π e * a (ω, k)K ab e b (ω, k) + 1 2 L 2 φ 2 − 1 g σ ab δ ab (27a) K(ω, k) = ω 2 + k 2 + σ 11 σ 12 − iωφ √ gγ σ 12 − iωφ √ gγ ω 2 + k 2 + σ 22 (27b)\nThe effective potential is then\nV ef f = 1 2 φ 2 − 1 2g (σ 11 + σ 22 ) + N 2 dω 2π dk 2π log det K(ω, k)(28)\nwhere we assume the system to be limited by a box of side L.\n\nIn terms of the variables σ ≡ 1 2 (σ 11 + σ 22 ) and\nη 2 ≡ 1 4 (σ 11 − σ 22 ) 2 + σ 2 12(29)\nthe saddle-point equations are the following:\n∂V ef f ∂σ = 0 = − 1 g + N 2 dω 2π dk 2π 2(ω 2 + k 2 + σ) (ω 2 + k 2 + σ) 2 − η 2 − gγφ 2 ω 2 (30a) ∂V ef f ∂η = 0 = Nη dω 2π dk 2π 1 (ω 2 + k 2 + σ) 2 − η 2 − gγφ 2 ω 2 (30b) ∂V ef f ∂φ = 0 = φ − Nφgγ dω 2π dk 2π ω 2 (ω 2 + k 2 + σ) 2 − η 2 − gγφ 2 ω 2 (30c)\nThe immediate solution to these equations is φ = η = 0 and σ = 0, determined by the simpler equation \n1 g = N 2 dω 2π dk 2π 1 ω 2 + k 2 + σ(31)σ = σ 0 ≡ Λ 2 exp − 8π 3g (32)\nwherein Λ is a momentum cutoff. The fields σ 11 and σ 22 must therefore be shifted by the constant σ 0 in order to fluctuate about zero. Then σ 0 multiplies 1 2 e 2 1 and 1 2 e 2 2 in the large-N effective action; thus, it has the interpretation of a quantum fluctuation-induced mass squared for the fields e a . The excitations of these fields are then triplets with energy gap ∆ = Λ exp −(4π/3g). After restoring the characteristic speed c, and the dependence on ξ, this becomes \n∆ = (Λa)Js 1 + ξ ξ (1 − ξ 2 )(2ξ 2 − 2ξ + 1) exp − 2πs 3 1 − ξ (1 + ξ)(2ξ 2 − 2ξ + 1)(33)\n\nIV. DISCUSSION\n\nIn Sec II we argued that the frustrated AF spin chain may be described, in the continuum limit, by the SO(3) nonlinear sigma model defined by the Lagrangian (17). In Sec III, we argued that the main feature of the spectrum of this continuum model is a doubly degenerate triplet of spin-1 excitations, separated from the singlet ground-state by a gap. However, this conclusion is incompatible with what is known of the spectrum of the frustrated spin-1 2 chain. The Lieb-Schultz-Mattis (LSM) theorem 17 states that a spin-1 2 chain has either no gap or degenerate ground states (corresponding to spontaneously broken parity). More recently, it has been shown 18 that this theorem applies to half-integer spin chains in general. In the special case J ′ = J/2 the exact ground state of the spin-1 2 chain obtained by Majumdar and Gosh 6 is two-fold degenerate. This is compatible with the LSM theorem, but in contradiction with the field theory. Thus, the latter does not correctly describe frustrated half-integer spin chains. Consequently, this minimum also occurs at −k 0 , which is distinct from k 0 except at the ferromagnetic (k = 0) and antiferromagnetic (k = π) points. The existence of these minima away from the parity-invariant points should not surprise us. If it were not for the quantum fluctuation disordering the system, there would be massless Goldstone branches around the ordering wave-vectors k 0 = α and −k 0 (α is given by Eq. (4)). The effect of quantum fluctuations is to produce a singlet ground state and to raise the minima of the Goldstone branches. We expect that the positions of these minima be approximately the same as the classical ordering wave-vectors k 0 and −k 0 , although we have no way to prove that they coincide exactly with these values. The SO(3) field theory effectively describes low-energy excitations about the minima of these excitation branches, and the degeneracy is essentially linked to the fact that the classical ordering wavevectors k 0 and −k 0 are inequivalent. This feature of the excitation spectrum supports the view that the SO(3) field theory describes the long wavelength behavior of the frustrated spin-1 chain.\n\nIt should be pointed out that, in a chain of length N, there is critical value ξ (N ) c of the ratio J/4J ′ , above which the minimum in the triplet dispersion relation occurs at k = π.\n\nThis critical value depends on the length of the chain. If one trusts the heuristic mapping of section II, it should be equal to ξ (∞) c = 1 when N → ∞. For finite chains it is closer to 5 8 , but it is conceivable that it reaches the predicted value as N grows. Figure 6 shows the evolution of ξ c for small sizes, as well as the value of the mass gap ∆. \nL top = − s 2 e · (∂ t e × ∂ x e) .(34)\nSince the effective lattice spacing for each chain is nowã = 2a, the characteristic velocity would be c = 4J ′ as, while the coupling constant is still g = 2/s. One checks that this agrees perfectly with the ξ → 0 limit of Eq. The interaction L int between the two chains may be easily expressed in terms of e 1 and e 2 . Indeed, recall that in taking the continuum limit of a single AF chain, one uses the decomposition S i = s[(−1) i e +ãl], where l is the local magnetization andã = 2a is the effective lattice spacing for each chain. Neglecting derivatives, the interaction Lagrangian may be written as\nL int = − 2Js 2ã l 1 · l 2 .(35)\nThe elimination of l 1 and l 2 is done by substituting the equation of motion 11\nl a = − 1 4J ′ sã (e a × ∂ t e a ) .(36)\nThe interaction Lagrangian then becomes\nL int = − 1 16aJ ′ J J ′ (e 1 × ∂ t e 1 ) · (e 2 × ∂ t e 2 ) (37) = − 1 aJ ξ 2 {e 1 · e 2 ∂ t e 1 · ∂ t e 2 − (e 1 · ∂ t e 2 )(e 2 · ∂ t e 1 )} .(38)\nIf we assume that the two vectors e 1 and e 2 are identically orthogonal, as the classical ground state suggests if ξ is small, then the above interaction exactly agrees with Eq. (21) and the ξ → 0 limit of γ as given in Eq. (22). Moreover, the topological terms S top [e 1 ] and S top [e 2 ] cancel each other if e 1 is identically orthogonal to e 2 . This is particularly easy to understand if the configuration e 1 belongs to the homotopy class of the identity, since the orthogonality constraint then effectively makes e 2 a mapping from the sphere S 2 to the circle S 1 and π 2 (S 1 ) = 0. In general, the two topological terms cancel simply because of the fact that two orthogonal unit vectors specify a rotation and π 2 (SO(3)) = 0. Thus, if we assume that e 1 ⊥ e 2 in the small ξ limit, the Lagrangian (21) is recovered with exactly the same parameters, since the topological terms are absent.\n\nHowever, the orthogonality e 1 ⊥ e 2 is not strict, but only favored energetically. A local deviation from this orthogonality gives back their full importance to the topological terms and causes a distinction between integer and half-integer spins. Such a distinction does not occur in the SO(3) formulation of the problem since the orthogonality is then 'built-in'. In that formulation, a local deviation from orthogonality corresponds to a discontinuity in the order parameter, which was not allowed from the start.\n\nFinally, let us point out that the above remarks concerning parity-breaking ordering wavevectors in frustrated antiferromagnets and their consequences on the excitation spectrum may apply to other systems, such as the nearest-neighbor Heisenberg antiferromagnet on a triangular lattice. In this case it is believed that the ground state has long-range order.\n\nHowever, it is conceivable that the introduction of vacancies destroys this order without affecting the applicability of the SO (3) \n\n(\nAF) Heisenberg chain with nearest-neighbor (NN) coupling to the O(3)/O(2) sigma model has led Haldane to conjecture 1 that integer spin chains should exhibit a gap to a triplet of excited states, whereas half-integer spin chains should not. This conjecture has later been confirmed by numerical calculations. 2,3 Such a mapping may be termed semi-classical, since it is constructed by introducing a local field characterizing the order expected in the classical ground state and by taking into account fluctuations of the spin variables around this local order. In the case of the simple AF chain, a collinear order exists in the classical ground state and the local field introduced is a unit vector e(x, t) representing the staggered magnetization. The long wavelength effective action obtained for this field is that of the O(3)/O(2) sigma model, with Lagrangian density\n\n\ninvariance, which makes it a qualitatively distinct theory from the versions of the SO(3) nonlinear sigma model studied in the context of classical critical phenomena 13 .\n\n\nIt can be shown explicitly 14 that the solution φ = η = 0 is the only acceptable one in this case. Eq. (31) coincides with the saddle-point equation for the O(3)/O(2) nonlinear sigma model, whose solution is\n\n\nOf course, this result neglects 1/N corrections. As ξ → 1 (the collinear point) the gap goes to zero. This signals a continuous phase transition, beyond which a different description ofthe system is needed, for instance in terms of the O(3)/O(2) sigma model. From ξ = 1 up to ξ = 0, which corresponds to J ′ → ∞, the gap increases and becomes eventually proportional to J ′ . The main feature of the excitations of this model is the degeneracy of the triplets. This ultimately comes from the SO(2) symmetry existing between the vectors e 1 and e 2 , which may be mixed with each other without affecting the action. It is then not surprising if this degeneracy survives the large-N approximation. Indeed, 1/N corrections to the dispersion of the excitations can in principle be calculated, like they are in the O(3)/O(2) sigma model (see, for instance, Polyakov's text 15 ). The quantity of interest is then the self-energy of the fields e a . In general, this is a matrix Σ ab (ω, k), which may be calculated diagrammatically with the help of the vertices shown in Fig. 2 and of the propagators for the auxiliary fields σ ab and φ. The latter scale as 1/N since they are inverse polarization operators, and the order in 1/N is simply determined by the number of auxiliary field internal lines. However, the propagators for σ 11 and σ 22 are identical, since the large-N propagators of e 1 and e 2 are the same. Moreover, the structure of the vertices of Fig. 2 makes it impossible for a non-diagonal self-energy Σ 12 to arise: even though the auxiliary fields σ 12 and φ may change a e 1 quanta into a e 2 quanta, the number of such vertices must be even, without tadpoles. The net result is that no self-energy diagram exists in which an entering e 1 line is turned into an exiting e 2 line, and consequently Σ 12 = 0. The 1/N contributions to the Σ 11 are show diagrammatically on Fig. 3. At all orders in 1/N the two self-energies Σ 11 and Σ 22 are identical and therefore the degeneracy of the excitations branches seems a exact feature of the model.\n\n\nBefore throwing the SO(3) model away, one should see if it describes the long wavelength behavior of integer spin chains. Exact results for the spin-1 chain are lacking and thus we are reduced to numerical study of finite systems. Quantum Monte-Carlo studies of frustrated spin systems are quite difficult because of a sign problem, therefore we limited ourselves to numerical diagonalizations of small chains (up to 14 sites). We used various methods, of which the most practical turned out to be a variant of the Lanczos method 19 applied to subspaces of fixed momentum. Diagonalizations of spin-1 2 chains with up to 22 sites have also been performed, for the sake of comparison. It turns out that the spectra of spin-1 and spin-1 2 chains of small lengths are qualitatively different. The ground state of the spin-1 chains is non-degenerate, with a gap to a doubly degenerate triplet of spin-1 states, exactly as in the field theory. Excitation spectra for J ′ = J/2 and various chain lengths are shown in Fig. 4, whereas spectra for 14 spins and various values of J ′ /J are shown inFig. 5. The dispersion relation is symmetric with respect to parity (k → −k) and the minimum of the spin-1 branch occurs at a wavenumber k 0 smaller than π (in units of 1/a).\n\n\nThe question remains as to why this semi-classical treatment fails in the half-integer spin case. In other words, where in the argument of Sec. II should the distinction between integer and half-integer spin come into play? The answer may lie in our neglect of configurations with discontinuities. This has already been pointed out in Ref.4 in relation with the kinetic term.An easier way to see this is to consider the limit J ′ > J. In that limit one may reconsider the problem and view the system as a pair of interwoven, antiferromagnetic chains with a small interchain interaction J. Each chain could be semi-classically described by the usualO(3)/O(2) nonlinear sigma model, with the Lagrangian density (1), plus a topological term L top expressed as\n\n\n(22). Thus, as ξ = J/4J ′ decreases, the two unit vectors e 1 and e 2 of the SO(3) model become the fields of two weakly coupled O(3)/O(2) sigma models, with the correct values of c and g.\n\nFIG. 1 .FIG. 2 .\n12theory: the effect of the vacancies would then be to increase the value of the effective coupling constant g beyond the critical value g c . A priori, it is not clear if the SO(3) field theory describes the long wavelength behavior of antiferromagnets on the Kagomé lattice or on the see-saw chain 20 . A semi-classical study of these systems may be interesting. Classical ground state of the frustrated AF chain with J ′ /J = 1 2 . Each spin makes a 120 • angle with its neighbors and the periodicity is the smallest possible, i.e., three sites. Below is different look at the same spin chain, using a 'railroad trestle' geometry. Vertices used in the 1/N expansion of the model (27).\n\nFIGFIG. 4 .FIG. 6 .\n46. 3. 1/N contributions to the self-energy Σ 11 of the field e 1 . Excitation spectrum of the frustrated spin-1 chain with J ′ = J/2 for chains lengths L = 8, 10, 12 and 14. The white circles represent spin-triplet states, while the black squares represent spin-singlets, when lower in energy. Dependence on chain length N of the critical coupling ratio ξ c = (J/4J ′ ) c at which the triplet degeneracy appears (error bars). Also illustrated is the energy gap ∆/J for ξ = 1 2 (circles).\n\n. F D M Haldane, Phys. Lett. 93464F. D. M. Haldane, Phys. Lett. 93A, 464 (1983);\n\n. Phys. Rev. Lett. 501153Phys. Rev. Lett. 50, 1153 (1983).\n\n. M P Nightingale, H W J Blöte, Phys. Rev. B. 33659M. P. Nightingale, H. W. J. Blöte, Phys. Rev. B 33, 659 (1986).\n\n. S R White, Phys. Rev. Lett. 692863S. R. White, Phys. Rev. Lett. 69, 2863 (1992);\n\n. S R White, D A Huse, Phys. Rev. B. 483844S. R. White and D. A. Huse, Phys. Rev. B 48, 3844 (1993).\n\n. T Dombre, N Read, Phys. Rev. B. 396797T. Dombre and N. Read, Phys. Rev. B 39, 6797 (1989).\n\n. S Miyashita, ; B Bernu, C Lhuillier, L Pierre, J. Phys. Soc. Jpn. P. Azaria, B. Delamotte and D.Mouhanna532483Phys. Rev. Lett.S. Miyashita, J. Phys. Soc. Jpn 53, 44 (1984). B. Bernu, C. Lhuillier and L. Pierre, Phys. Rev. Lett. 69, 2590 (1992). P. Azaria, B. Delamotte and D.Mouhanna, Phys. Rev. Lett. 70, 2483 (1993).\n\n. C K Majumdar, D K Gosh, J. Math. Phys. 101399C. K. Majumdar and D. K. Gosh, J. Math. Phys. 10, 1388, 1399 (1969).\n\n. T Tonegawa, M Kaburagi, N Ichikawa, I Harada, J. Phys. Soc. Jap. 612890T. Tonegawa, M. Kaburagi, N. Ichikawa and I. Harada, J. Phys. Soc. Jap. 61, 2890 (1992).\n\n. I Harada, M Fujikawa, I Mannari, J. Phys. Soc. Jap. 623694I. Harada, M. Fujikawa, I. Mannari, J. Phys. Soc. Jap. 62, 3694 (1993).\n\n. H Shimaoka, K Kuboki, J. Phys. Soc. Jap. 623257H. Shimaoka and K. Kuboki, J. Phys. Soc. Jap. 62, 3257 (1993).\n\n. S Rao, D Sen, Phys. Rev. B. 4812763S. Rao and D. Sen, Phys. Rev. B 48, 12763 (1993).\n\nE Fradkin, Field Theories of Condensed Matter Systems. Reading, MAAddison-WesleyE. Fradkin, Field Theories of Condensed Matter Systems, Addison-Wesley, Reading, MA, 1991.\n\n. E Manousakis, Rev. Mod. Phys. 631E. Manousakis, Rev. Mod. Phys. 63, 1 (1991).\n\n. P Azaria, D Delamotte, F Delduc, T Jolicoeur, Nucl. Phys. 408485P. Azaria, D. Delamotte, F. Delduc and T. Jolicoeur, Nucl. Phys. B408, 485 (1993).\n\n. D Allen, M Sc, Université de SherbrookethesisD. Allen, M.Sc. thesis, Université de Sherbrooke (1994).\n\nA M Polyakov, Gauge Fields and Strings. Harwood, New YorkA. M. Polyakov, Gauge Fields and Strings, Harwood, New York (1987).\n\n. H Kawamura, Phys. Rev. B. 384916H. Kawamura, Phys. Rev. B 38, 4916 (1988).\n\n. E H Lieb, T Schultz, D J Mattis, Ann. Phys. (N.Y.). 16407E. H. Lieb, T. Schultz and D. J. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961).\n\n. I Affleck, E H Lieb, Lett. Math. Phys. 1257I. Affleck and E. H. Lieb, Lett. Math. Phys. 12, 57 (1986).\n\n. E R Gagliano, E Dagotto, A Moreo, F C Alcaraz, Phys. Rev. B. 341677E. R. Gagliano, E. Dagotto, A. Moreo and F. C. Alcaraz, Phys. Rev. B 34, 1677 (1986).\n\n. M W Long, S Siak, J. Phys.: Condens. Matter. 34901M. W. Long and S. Siak, J. Phys.: Condens. Matter 3, 4901 (1991).\n"},"corpus_id":{"kind":"number","value":9529653,"string":"9,529,653"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"cond-mat/9409090\",\n \"mag\": 2035979179,\n \"acl\": null,\n \"pubmed\": \"9977179\",\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1103/physrevb.51.6394\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/cond-mat/9409090v1.pdf\"\n ],\n \"pdf_sha\": \"3703467003d6d3e16a956152091d9c147a480ea7\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/cond-mat/9409090\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"arXiv:cond-mat/9409090v1 21 Sep 1994 Semi-classical description of the frustrated antiferromagnetic chain Typeset using REVT E X 1"},"authors":{"kind":"list like","value":["D Allen \nCentre de Recherche en Physique du Solide et Département de Physique\nUniversité de Sherbrooke\nJ1K 2R1SherbrookeQuébecCanada","D Sénéchal \nCentre de Recherche en Physique du Solide et Département de Physique\nUniversité de Sherbrooke\nJ1K 2R1SherbrookeQuébecCanada"],"string":"[\n \"D Allen \\nCentre de Recherche en Physique du Solide et Département de Physique\\nUniversité de Sherbrooke\\nJ1K 2R1SherbrookeQuébecCanada\",\n \"D Sénéchal \\nCentre de Recherche en Physique du Solide et Département de Physique\\nUniversité de Sherbrooke\\nJ1K 2R1SherbrookeQuébecCanada\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The antiferromagnetic Heisenberg model on a chain with nearest and next nearest neighbor couplings is mapped onto the SO(3) nonlinear sigma model in the continuum limit. In one spatial dimension this model is always in its disordered phase and a gap opens to excited states. The latter form a doubly degenerate spin-1 branch at all orders in 1/N . We argue that this feature should be present in the spin-1 Heisenberg model itself. Exact diagonalizations are used to support this claim. The inapplicability of this model to half-integer spin chains is discussed."},"sections":{"kind":"list like","value":[{"title":"I. INTRODUCTION","content":"Interest in low-dimensional magnetic systems has been great in recent years, partly because of the widespread belief that magnetism plays a key role in high-temperature superconductivity, but also because of the successful application of field-theoretic methods to these systems, in particular to spin chains. Indeed, a mapping from the spin-s antiferromagnetic\nThe coupling g and the velocity v are related to the spin s, the AF coupling J and the lattice spacing: v = 2Jas and g = 2/s. To this action one must add a topological term S top for half-integer spin (cf. Sec. IV). An extension of this mapping to dimensions higher than one has been obtained by many authors, with the difference that no topological term arises, hence no distinction between half-integral and integral spin.\nDombre and Read 4 have conducted a similar analysis for the antiferromagnetic Heisenberg model on a triangular lattice. The essential difference here is the presence of frustration, leading to a classical ground state characterized by a 120 • order. The local order must then be specified by a rotation matrix instead of a unit vector and the order parameter is thus an element of SO (3). The long-wavelength action found in this case is the SO(3) nonlinear sigma model, with Lagrangian density\nHere R(x, t) is a position-and time-dependent rotation matrix and P is the constant diagonal matrix diag [1,1,0]. Again, the constants g and v depend on the spin s, the lattice spacing a and the AF couplings J. At zero temperature and in dimension two, this model has an ordered phase for g < g c and a disordered phase otherwise. Since g ∼ a/s, the spin-1 2 case is the closest to the disordered phase. However, it is now widely believed 5 that the ground state of the spin-1 2 antiferromagnet on a triangular lattice has long range order and consequently the disordered phase of the SO(3) model is not physically realized to twodimensional antiferromagnets, at least at half-filling (one spin per site).\nIn this work we will argue that this disordered phase of the SO(3) sigma model could be realized in a frustrated antiferromagnetic chain, with Hamiltonian\nIntroducing a next-nearest-neighbor (NNN) coupling J ′ modifies the classical ground state if J ′ /J > 1 4 : the spins still lie on a common plane, but instead of being collinear (as in the Néel state), they are arranged in a canted configuration (see Fig. 1) in which each spin makes an angle α with its predecessor, given by\nIn the special case J ′ /J = 1 2 , the classical order is quite similar to that of the triangular lattice, with its 120 • angle from site to site and its periodicity of three sites. For spin-1 2 this particular case constitutes the Majumdar-Gosh model, whose ground state is exactly known. 6 For a generic value of J ′ /J the classical order is incommensurable, with infinite periodicity. The spin-1 case has been studied previously using various methods: exact diagonalizations 7 , approximate mappings to field theories involving fermions 8 or bosons 9,10 , among others. This paper is organized as follows. In Sec. II we explain how to obtain the SO (3) nonlinear sigma model as the continuum limit of the model defined in Eq. (3). In Sec. III we argue that the disordered phase of this model is characterized by a singlet ground state with a gap to two degenerate triplets of excited states. Hence this model cannot adequately represent the half-integer NNN chain, but may be correct for integer spin. In Sec. IV we discuss our results with the help of exact diagonalization results and discuss the apparent lack of distinction between integer and half-integer spin in this model."},{"title":"II. SEMI-CLASSICAL MAPPING TO THE SO(3) NONLINEAR SIGMA MODEL","content":"In this section we argue that the long wavelength, continuum theory describing the frustrated antiferromagnetic chain is specified by the Lagrangian density of Eq. (17), with the notation explained thereafter.\nThe Lagrangian description of spin dynamics, as used in path integrals, requires the introduction of spin coherent states. The unfamiliar reader is referred to Fradkin's text 11 or to Manousakis' review 12 . Each quantum spin S i is described by a fluctuating unit vector n i and the action associated to a spin Hamiltonian H is\nwherein S i is replaced by sn i in the Hamiltonian. A is the vector potential of a magnetic monopole of flux 4π. Thus, given a closed curve n(t) on the unit sphere, the integral A · dn equals the area of the sphere enclosed by the curve, modulo 4π.\nWe shall use a description of each spin S i in terms of a slowly varying SO(3) order parameter R(x, t) and a local magnetization vector l(x, t):\nHere n c k is the orientation of the spin S k in a classical ground state taken as reference, in which all spins lie on the plane defined by two mutually orthogonal unit vectorsî and; α is the pitch of this ground state, as given in Eq. (4); finally, a is the lattice spacing: It is assumed that the spin configurations that contribute significantly to the path integral are locally close to the classical ground state, and the approximation that the deviation al is small will be controlled at the same time as the continuum approximation.\nSome comments are in order concerning this representation. If J ′ = J/2 (or α = 2π/3), the periodicity is 3 and the relation (6) may be considered as a bona fide change of variables if we group the spins in sets of three and assume that the fields R and l do not vary within such sets. We check that the number of degrees of freedom match: 6 per set of three spins.\nHowever, it is more convenient to assume that R and l vary slowly from site to site, in which case the representation (6) may not be regarded as a change of variables, but simply as long wavelength description of the fluctuations around the classical ground state; then we need not restrict ourselves to the case J ′ = J/2. The next step is to substitute the representation (6) into the action (5) and to Taylor expand when needed in order to get a continuum action in terms of R and l only. We will treat the kinetic term first, and then the Hamiltonian. For the sake of convenience, we will assume a periodicity of N spins in the classical ground state. The case N = 3 (or J ′ = J/2) is the simplest, leading to a calculation almost identical to that of Ref. 4. The next simplest is N = 5, with J ′ ≈ 0.809J, and so on.\nAn explicit calculation of the kinetic term is not needed here, since it would be almost identical to that of Ref. 4, to which the reader is referred. The difference here lies in the number of reference vectors n c i , which is N instead of 3. The kinetic part of the Lagrangian density is then\nwith the definitions\nwherein the indices in spin space correspond to the axes defined byî, andî ×, and · · · means an average over N contiguous sites.\nIn order to write down the continuum limit of the Hamiltonian H, we need to calculate the interaction H (k) of a spin S k with its four neighbors and then average the result over the N spins S k of the period. Specifically, the Hamiltonian density is\nThe contribution H (k) may be written as follows:\nwhere the spin S n stands in fact for its expression in terms of R and l (Eq. (6)) and where the derivatives act on these same fields. We will carry the expansion to second order only, enough to yield a non trivial Lagrangian. We shall also use a Taylor expanded version of the representation (6):\nWe may then write the Hamiltonian density as an expansion in powers of a:\nH (0) is a constant, independent of R and l. H (1) vanishes for the following reasons: Part of it depends on l, but it is proportional to n c k , which is zero. Another part depends on R, and is given by\nBut one easily calculates that n c k,a n c\nAfter substituting and using the relation (4), we find out that the expression (13) vanishes.\nFinally, we are left with the second order part H (2) . It is a straightforward exercise to\nshow that\nwhere ξ ≡ − cos α = J/4J ′ and where the diagonal matrix M has elements\nAgain, the matrix P is the projector onto the plane defined byî and, as in Eq. (2). The full Lagrangian density is then L = L K − H.\nThe last step consists in integrating out the field l.\nSince the latter appears quadratically in L, this operation amounts to substituting into L the solution of the classical equations of motion for l. The final result for the Lagrangian density is\nwhere the matrix Q is diagonal, with elements\nand where the constantsg andc are defined as\nIn the special case ξ = 1 2 , which corresponds to J ′ = J/2 and α = 2π/3, one recovers precisely the form of the Lagrangian (2) since Q 33 = 1. The parameter ξ ranges from 0 (J ′ → ∞) to 1 (J ′ = J/4). The characteristic speedc diverges as ξ → 0, whereas the coupling constantg diverges as ξ → 1, the collinear phase boundary. In all cases, the Lagrangian (17) Its symmetries comprise global left rotations R → U L R with U L ∈ SO(3) and global right rotations R → RU R with U R ∈ SO(2) (i.e. U R commuting with P and Q). The first of these reflects invariance under rotations in spin space, whereas the other will have consequences on the spectrum of the theory: a degeneracy of the excitation branches.\nIt is important to stress here that in going from the discrete Heisenberg Hamiltonian (3) to the continuum Lagrangian (17) we have assumed that smooth configurations dominate the path integral. There seems to be no distinction between integer and half-integer spins within this mapping. In particular, there is no topological term of the type arising in the collinear antiferromagnetic chain (J ′ < J/4). In fact, none could exist, since the relevant homotopy group is trivial: π 2 (SO(3)) = 0. However, there are qualitative differences between integer and half-integer spin in this system. The question as to why they do not appear explicitly in the field theory will be discussed in Sec. IV."},{"title":"III. SPECTRUM OF THE NONLINEAR SIGMA MODEL","content":"In this section we discuss the spectrum of the model defined in Eq. (17), in particular regarding the gap to excited states and the degeneracy of the excitation branches. To this end it is preferable to rewrite the Lagrangian (17) in terms of two orthonormal vectors e 1 and e 2 specifying the rotation matrix R. Defining e 3 = e 1 × e 2 , the elements of the matrix R may be expressed as R ab = (e b ) a and tr[Q∂ t R −1 ∂ t R] = a Q aa (∂ t e a ) 2 . Using the fact that\nwe may finally express the Lagrangian in the following form:\nwherein the coupling constants g, γ and the velocity c are now\nThe functional integration measure associated with the fields e 1,2 must incorporate the constraints e 2 1 = e 2 2 = 1, e 1 · e 2 = 0. This may be done by introducing three Lagrange parameters σ 11 , σ 22 and σ 12 = σ 21 , and by adding the following constraint term to the Lagrangian:\nSince this model is defined in one spatial dimension, the Mermin-Wagner-Coleman theorem applies and the global SO(3) symmetry is not broken. If we extended the model to dimensions greater than one, a broken symmetry phase could be realized, for g below some critical value. In such a phase, three Goldstone modes would appear, corresponding to the three parameters of the broken SO(3) symmetry. These three Goldstone modes would have a linear dispersion relation, two of them with speed c and a third with speed c/ 1 − γ/2.\nThe most elegant way to see this is to go back to the form (17) of the Lagrangian and to substitute, in the small oscillation approximation, R ab ≈ δ ab + ε abc Ω c , keeping the terms of order Ω 2 c only. Since the field Ω c is unconstrained, the dispersion relations are easily read off. In the case of an antiferromagnet on a triangular or hexagonal lattice (ξ = 1 2 ), the two speeds are c and c √ 2. This may also be observed within spin-wave theory, since it is a feature of the ordered phase.\nHowever, we are interested here in the disordered phase of the model, which we shall study within the large-N approach (here N is the number of components of the vectors e a , normally 3). In the imaginary-time formalism, the partition function of the model is\nwith the Euclidian Lagrangian density\nwhere we have rescaled e a by a factor √ g in order to recover the standard normalization for the kinetic term. The characteristic speed c has been set to unity in order to lighten the notation; it may be restored by dimensional analysis. We will also use a Hubbard-Stratonovich decomposition of the quartic term:\nWe then proceed to find the large-N saddle point. In other words, we assume that the auxiliary fields φ and σ ab take a constant value, which is determined by extremizing the effective potential obtained by integrating the fields e a . This exercise is better done in Fourier space, in which the Euclidian action with constant auxiliary fields may be written as\nThe effective potential is then\nwhere we assume the system to be limited by a box of side L.\nIn terms of the variables σ ≡ 1 2 (σ 11 + σ 22 ) and\nthe saddle-point equations are the following:\nThe immediate solution to these equations is φ = η = 0 and σ = 0, determined by the simpler equation\nwherein Λ is a momentum cutoff. The fields σ 11 and σ 22 must therefore be shifted by the constant σ 0 in order to fluctuate about zero. Then σ 0 multiplies 1 2 e 2 1 and 1 2 e 2 2 in the large-N effective action; thus, it has the interpretation of a quantum fluctuation-induced mass squared for the fields e a . The excitations of these fields are then triplets with energy gap ∆ = Λ exp −(4π/3g). After restoring the characteristic speed c, and the dependence on ξ, this becomes"},{"title":"IV. DISCUSSION","content":"In Sec II we argued that the frustrated AF spin chain may be described, in the continuum limit, by the SO(3) nonlinear sigma model defined by the Lagrangian (17). In Sec III, we argued that the main feature of the spectrum of this continuum model is a doubly degenerate triplet of spin-1 excitations, separated from the singlet ground-state by a gap. However, this conclusion is incompatible with what is known of the spectrum of the frustrated spin-1 2 chain. The Lieb-Schultz-Mattis (LSM) theorem 17 states that a spin-1 2 chain has either no gap or degenerate ground states (corresponding to spontaneously broken parity). More recently, it has been shown 18 that this theorem applies to half-integer spin chains in general. In the special case J ′ = J/2 the exact ground state of the spin-1 2 chain obtained by Majumdar and Gosh 6 is two-fold degenerate. This is compatible with the LSM theorem, but in contradiction with the field theory. Thus, the latter does not correctly describe frustrated half-integer spin chains. Consequently, this minimum also occurs at −k 0 , which is distinct from k 0 except at the ferromagnetic (k = 0) and antiferromagnetic (k = π) points. The existence of these minima away from the parity-invariant points should not surprise us. If it were not for the quantum fluctuation disordering the system, there would be massless Goldstone branches around the ordering wave-vectors k 0 = α and −k 0 (α is given by Eq. (4)). The effect of quantum fluctuations is to produce a singlet ground state and to raise the minima of the Goldstone branches. We expect that the positions of these minima be approximately the same as the classical ordering wave-vectors k 0 and −k 0 , although we have no way to prove that they coincide exactly with these values. The SO(3) field theory effectively describes low-energy excitations about the minima of these excitation branches, and the degeneracy is essentially linked to the fact that the classical ordering wavevectors k 0 and −k 0 are inequivalent. This feature of the excitation spectrum supports the view that the SO(3) field theory describes the long wavelength behavior of the frustrated spin-1 chain.\nIt should be pointed out that, in a chain of length N, there is critical value ξ (N ) c of the ratio J/4J ′ , above which the minimum in the triplet dispersion relation occurs at k = π.\nThis critical value depends on the length of the chain. If one trusts the heuristic mapping of section II, it should be equal to ξ (∞) c = 1 when N → ∞. For finite chains it is closer to 5 8 , but it is conceivable that it reaches the predicted value as N grows. Figure 6 shows the evolution of ξ c for small sizes, as well as the value of the mass gap ∆.\nSince the effective lattice spacing for each chain is nowã = 2a, the characteristic velocity would be c = 4J ′ as, while the coupling constant is still g = 2/s. One checks that this agrees perfectly with the ξ → 0 limit of Eq. The interaction L int between the two chains may be easily expressed in terms of e 1 and e 2 . Indeed, recall that in taking the continuum limit of a single AF chain, one uses the decomposition S i = s[(−1) i e +ãl], where l is the local magnetization andã = 2a is the effective lattice spacing for each chain. Neglecting derivatives, the interaction Lagrangian may be written as\nThe elimination of l 1 and l 2 is done by substituting the equation of motion 11\nThe interaction Lagrangian then becomes\nIf we assume that the two vectors e 1 and e 2 are identically orthogonal, as the classical ground state suggests if ξ is small, then the above interaction exactly agrees with Eq. (21) and the ξ → 0 limit of γ as given in Eq. (22). Moreover, the topological terms S top [e 1 ] and S top [e 2 ] cancel each other if e 1 is identically orthogonal to e 2 . This is particularly easy to understand if the configuration e 1 belongs to the homotopy class of the identity, since the orthogonality constraint then effectively makes e 2 a mapping from the sphere S 2 to the circle S 1 and π 2 (S 1 ) = 0. In general, the two topological terms cancel simply because of the fact that two orthogonal unit vectors specify a rotation and π 2 (SO(3)) = 0. Thus, if we assume that e 1 ⊥ e 2 in the small ξ limit, the Lagrangian (21) is recovered with exactly the same parameters, since the topological terms are absent.\nHowever, the orthogonality e 1 ⊥ e 2 is not strict, but only favored energetically. A local deviation from this orthogonality gives back their full importance to the topological terms and causes a distinction between integer and half-integer spins. Such a distinction does not occur in the SO(3) formulation of the problem since the orthogonality is then 'built-in'. In that formulation, a local deviation from orthogonality corresponds to a discontinuity in the order parameter, which was not allowed from the start.\nFinally, let us point out that the above remarks concerning parity-breaking ordering wavevectors in frustrated antiferromagnets and their consequences on the excitation spectrum may apply to other systems, such as the nearest-neighbor Heisenberg antiferromagnet on a triangular lattice. In this case it is believed that the ground state has long-range order.\nHowever, it is conceivable that the introduction of vacancies destroys this order without affecting the applicability of the SO (3)"},{"title":"(","content":""},{"title":"FIG. 1 .FIG. 2 .","content":""},{"title":"FIGFIG. 4 .FIG. 6 .","content":""}],"string":"[\n {\n \"title\": \"I. INTRODUCTION\",\n \"content\": \"Interest in low-dimensional magnetic systems has been great in recent years, partly because of the widespread belief that magnetism plays a key role in high-temperature superconductivity, but also because of the successful application of field-theoretic methods to these systems, in particular to spin chains. Indeed, a mapping from the spin-s antiferromagnetic\\nThe coupling g and the velocity v are related to the spin s, the AF coupling J and the lattice spacing: v = 2Jas and g = 2/s. To this action one must add a topological term S top for half-integer spin (cf. Sec. IV). An extension of this mapping to dimensions higher than one has been obtained by many authors, with the difference that no topological term arises, hence no distinction between half-integral and integral spin.\\nDombre and Read 4 have conducted a similar analysis for the antiferromagnetic Heisenberg model on a triangular lattice. The essential difference here is the presence of frustration, leading to a classical ground state characterized by a 120 • order. The local order must then be specified by a rotation matrix instead of a unit vector and the order parameter is thus an element of SO (3). The long-wavelength action found in this case is the SO(3) nonlinear sigma model, with Lagrangian density\\nHere R(x, t) is a position-and time-dependent rotation matrix and P is the constant diagonal matrix diag [1,1,0]. Again, the constants g and v depend on the spin s, the lattice spacing a and the AF couplings J. At zero temperature and in dimension two, this model has an ordered phase for g < g c and a disordered phase otherwise. Since g ∼ a/s, the spin-1 2 case is the closest to the disordered phase. However, it is now widely believed 5 that the ground state of the spin-1 2 antiferromagnet on a triangular lattice has long range order and consequently the disordered phase of the SO(3) model is not physically realized to twodimensional antiferromagnets, at least at half-filling (one spin per site).\\nIn this work we will argue that this disordered phase of the SO(3) sigma model could be realized in a frustrated antiferromagnetic chain, with Hamiltonian\\nIntroducing a next-nearest-neighbor (NNN) coupling J ′ modifies the classical ground state if J ′ /J > 1 4 : the spins still lie on a common plane, but instead of being collinear (as in the Néel state), they are arranged in a canted configuration (see Fig. 1) in which each spin makes an angle α with its predecessor, given by\\nIn the special case J ′ /J = 1 2 , the classical order is quite similar to that of the triangular lattice, with its 120 • angle from site to site and its periodicity of three sites. For spin-1 2 this particular case constitutes the Majumdar-Gosh model, whose ground state is exactly known. 6 For a generic value of J ′ /J the classical order is incommensurable, with infinite periodicity. The spin-1 case has been studied previously using various methods: exact diagonalizations 7 , approximate mappings to field theories involving fermions 8 or bosons 9,10 , among others. This paper is organized as follows. In Sec. II we explain how to obtain the SO (3) nonlinear sigma model as the continuum limit of the model defined in Eq. (3). In Sec. III we argue that the disordered phase of this model is characterized by a singlet ground state with a gap to two degenerate triplets of excited states. Hence this model cannot adequately represent the half-integer NNN chain, but may be correct for integer spin. In Sec. IV we discuss our results with the help of exact diagonalization results and discuss the apparent lack of distinction between integer and half-integer spin in this model.\"\n },\n {\n \"title\": \"II. SEMI-CLASSICAL MAPPING TO THE SO(3) NONLINEAR SIGMA MODEL\",\n \"content\": \"In this section we argue that the long wavelength, continuum theory describing the frustrated antiferromagnetic chain is specified by the Lagrangian density of Eq. (17), with the notation explained thereafter.\\nThe Lagrangian description of spin dynamics, as used in path integrals, requires the introduction of spin coherent states. The unfamiliar reader is referred to Fradkin's text 11 or to Manousakis' review 12 . Each quantum spin S i is described by a fluctuating unit vector n i and the action associated to a spin Hamiltonian H is\\nwherein S i is replaced by sn i in the Hamiltonian. A is the vector potential of a magnetic monopole of flux 4π. Thus, given a closed curve n(t) on the unit sphere, the integral A · dn equals the area of the sphere enclosed by the curve, modulo 4π.\\nWe shall use a description of each spin S i in terms of a slowly varying SO(3) order parameter R(x, t) and a local magnetization vector l(x, t):\\nHere n c k is the orientation of the spin S k in a classical ground state taken as reference, in which all spins lie on the plane defined by two mutually orthogonal unit vectorsî and; α is the pitch of this ground state, as given in Eq. (4); finally, a is the lattice spacing: It is assumed that the spin configurations that contribute significantly to the path integral are locally close to the classical ground state, and the approximation that the deviation al is small will be controlled at the same time as the continuum approximation.\\nSome comments are in order concerning this representation. If J ′ = J/2 (or α = 2π/3), the periodicity is 3 and the relation (6) may be considered as a bona fide change of variables if we group the spins in sets of three and assume that the fields R and l do not vary within such sets. We check that the number of degrees of freedom match: 6 per set of three spins.\\nHowever, it is more convenient to assume that R and l vary slowly from site to site, in which case the representation (6) may not be regarded as a change of variables, but simply as long wavelength description of the fluctuations around the classical ground state; then we need not restrict ourselves to the case J ′ = J/2. The next step is to substitute the representation (6) into the action (5) and to Taylor expand when needed in order to get a continuum action in terms of R and l only. We will treat the kinetic term first, and then the Hamiltonian. For the sake of convenience, we will assume a periodicity of N spins in the classical ground state. The case N = 3 (or J ′ = J/2) is the simplest, leading to a calculation almost identical to that of Ref. 4. The next simplest is N = 5, with J ′ ≈ 0.809J, and so on.\\nAn explicit calculation of the kinetic term is not needed here, since it would be almost identical to that of Ref. 4, to which the reader is referred. The difference here lies in the number of reference vectors n c i , which is N instead of 3. The kinetic part of the Lagrangian density is then\\nwith the definitions\\nwherein the indices in spin space correspond to the axes defined byî, andî ×, and · · · means an average over N contiguous sites.\\nIn order to write down the continuum limit of the Hamiltonian H, we need to calculate the interaction H (k) of a spin S k with its four neighbors and then average the result over the N spins S k of the period. Specifically, the Hamiltonian density is\\nThe contribution H (k) may be written as follows:\\nwhere the spin S n stands in fact for its expression in terms of R and l (Eq. (6)) and where the derivatives act on these same fields. We will carry the expansion to second order only, enough to yield a non trivial Lagrangian. We shall also use a Taylor expanded version of the representation (6):\\nWe may then write the Hamiltonian density as an expansion in powers of a:\\nH (0) is a constant, independent of R and l. H (1) vanishes for the following reasons: Part of it depends on l, but it is proportional to n c k , which is zero. Another part depends on R, and is given by\\nBut one easily calculates that n c k,a n c\\nAfter substituting and using the relation (4), we find out that the expression (13) vanishes.\\nFinally, we are left with the second order part H (2) . It is a straightforward exercise to\\nshow that\\nwhere ξ ≡ − cos α = J/4J ′ and where the diagonal matrix M has elements\\nAgain, the matrix P is the projector onto the plane defined byî and, as in Eq. (2). The full Lagrangian density is then L = L K − H.\\nThe last step consists in integrating out the field l.\\nSince the latter appears quadratically in L, this operation amounts to substituting into L the solution of the classical equations of motion for l. The final result for the Lagrangian density is\\nwhere the matrix Q is diagonal, with elements\\nand where the constantsg andc are defined as\\nIn the special case ξ = 1 2 , which corresponds to J ′ = J/2 and α = 2π/3, one recovers precisely the form of the Lagrangian (2) since Q 33 = 1. The parameter ξ ranges from 0 (J ′ → ∞) to 1 (J ′ = J/4). The characteristic speedc diverges as ξ → 0, whereas the coupling constantg diverges as ξ → 1, the collinear phase boundary. In all cases, the Lagrangian (17) Its symmetries comprise global left rotations R → U L R with U L ∈ SO(3) and global right rotations R → RU R with U R ∈ SO(2) (i.e. U R commuting with P and Q). The first of these reflects invariance under rotations in spin space, whereas the other will have consequences on the spectrum of the theory: a degeneracy of the excitation branches.\\nIt is important to stress here that in going from the discrete Heisenberg Hamiltonian (3) to the continuum Lagrangian (17) we have assumed that smooth configurations dominate the path integral. There seems to be no distinction between integer and half-integer spins within this mapping. In particular, there is no topological term of the type arising in the collinear antiferromagnetic chain (J ′ < J/4). In fact, none could exist, since the relevant homotopy group is trivial: π 2 (SO(3)) = 0. However, there are qualitative differences between integer and half-integer spin in this system. The question as to why they do not appear explicitly in the field theory will be discussed in Sec. IV.\"\n },\n {\n \"title\": \"III. SPECTRUM OF THE NONLINEAR SIGMA MODEL\",\n \"content\": \"In this section we discuss the spectrum of the model defined in Eq. (17), in particular regarding the gap to excited states and the degeneracy of the excitation branches. To this end it is preferable to rewrite the Lagrangian (17) in terms of two orthonormal vectors e 1 and e 2 specifying the rotation matrix R. Defining e 3 = e 1 × e 2 , the elements of the matrix R may be expressed as R ab = (e b ) a and tr[Q∂ t R −1 ∂ t R] = a Q aa (∂ t e a ) 2 . Using the fact that\\nwe may finally express the Lagrangian in the following form:\\nwherein the coupling constants g, γ and the velocity c are now\\nThe functional integration measure associated with the fields e 1,2 must incorporate the constraints e 2 1 = e 2 2 = 1, e 1 · e 2 = 0. This may be done by introducing three Lagrange parameters σ 11 , σ 22 and σ 12 = σ 21 , and by adding the following constraint term to the Lagrangian:\\nSince this model is defined in one spatial dimension, the Mermin-Wagner-Coleman theorem applies and the global SO(3) symmetry is not broken. If we extended the model to dimensions greater than one, a broken symmetry phase could be realized, for g below some critical value. In such a phase, three Goldstone modes would appear, corresponding to the three parameters of the broken SO(3) symmetry. These three Goldstone modes would have a linear dispersion relation, two of them with speed c and a third with speed c/ 1 − γ/2.\\nThe most elegant way to see this is to go back to the form (17) of the Lagrangian and to substitute, in the small oscillation approximation, R ab ≈ δ ab + ε abc Ω c , keeping the terms of order Ω 2 c only. Since the field Ω c is unconstrained, the dispersion relations are easily read off. In the case of an antiferromagnet on a triangular or hexagonal lattice (ξ = 1 2 ), the two speeds are c and c √ 2. This may also be observed within spin-wave theory, since it is a feature of the ordered phase.\\nHowever, we are interested here in the disordered phase of the model, which we shall study within the large-N approach (here N is the number of components of the vectors e a , normally 3). In the imaginary-time formalism, the partition function of the model is\\nwith the Euclidian Lagrangian density\\nwhere we have rescaled e a by a factor √ g in order to recover the standard normalization for the kinetic term. The characteristic speed c has been set to unity in order to lighten the notation; it may be restored by dimensional analysis. We will also use a Hubbard-Stratonovich decomposition of the quartic term:\\nWe then proceed to find the large-N saddle point. In other words, we assume that the auxiliary fields φ and σ ab take a constant value, which is determined by extremizing the effective potential obtained by integrating the fields e a . This exercise is better done in Fourier space, in which the Euclidian action with constant auxiliary fields may be written as\\nThe effective potential is then\\nwhere we assume the system to be limited by a box of side L.\\nIn terms of the variables σ ≡ 1 2 (σ 11 + σ 22 ) and\\nthe saddle-point equations are the following:\\nThe immediate solution to these equations is φ = η = 0 and σ = 0, determined by the simpler equation\\nwherein Λ is a momentum cutoff. The fields σ 11 and σ 22 must therefore be shifted by the constant σ 0 in order to fluctuate about zero. Then σ 0 multiplies 1 2 e 2 1 and 1 2 e 2 2 in the large-N effective action; thus, it has the interpretation of a quantum fluctuation-induced mass squared for the fields e a . The excitations of these fields are then triplets with energy gap ∆ = Λ exp −(4π/3g). After restoring the characteristic speed c, and the dependence on ξ, this becomes\"\n },\n {\n \"title\": \"IV. DISCUSSION\",\n \"content\": \"In Sec II we argued that the frustrated AF spin chain may be described, in the continuum limit, by the SO(3) nonlinear sigma model defined by the Lagrangian (17). In Sec III, we argued that the main feature of the spectrum of this continuum model is a doubly degenerate triplet of spin-1 excitations, separated from the singlet ground-state by a gap. However, this conclusion is incompatible with what is known of the spectrum of the frustrated spin-1 2 chain. The Lieb-Schultz-Mattis (LSM) theorem 17 states that a spin-1 2 chain has either no gap or degenerate ground states (corresponding to spontaneously broken parity). More recently, it has been shown 18 that this theorem applies to half-integer spin chains in general. In the special case J ′ = J/2 the exact ground state of the spin-1 2 chain obtained by Majumdar and Gosh 6 is two-fold degenerate. This is compatible with the LSM theorem, but in contradiction with the field theory. Thus, the latter does not correctly describe frustrated half-integer spin chains. Consequently, this minimum also occurs at −k 0 , which is distinct from k 0 except at the ferromagnetic (k = 0) and antiferromagnetic (k = π) points. The existence of these minima away from the parity-invariant points should not surprise us. If it were not for the quantum fluctuation disordering the system, there would be massless Goldstone branches around the ordering wave-vectors k 0 = α and −k 0 (α is given by Eq. (4)). The effect of quantum fluctuations is to produce a singlet ground state and to raise the minima of the Goldstone branches. We expect that the positions of these minima be approximately the same as the classical ordering wave-vectors k 0 and −k 0 , although we have no way to prove that they coincide exactly with these values. The SO(3) field theory effectively describes low-energy excitations about the minima of these excitation branches, and the degeneracy is essentially linked to the fact that the classical ordering wavevectors k 0 and −k 0 are inequivalent. This feature of the excitation spectrum supports the view that the SO(3) field theory describes the long wavelength behavior of the frustrated spin-1 chain.\\nIt should be pointed out that, in a chain of length N, there is critical value ξ (N ) c of the ratio J/4J ′ , above which the minimum in the triplet dispersion relation occurs at k = π.\\nThis critical value depends on the length of the chain. If one trusts the heuristic mapping of section II, it should be equal to ξ (∞) c = 1 when N → ∞. For finite chains it is closer to 5 8 , but it is conceivable that it reaches the predicted value as N grows. Figure 6 shows the evolution of ξ c for small sizes, as well as the value of the mass gap ∆.\\nSince the effective lattice spacing for each chain is nowã = 2a, the characteristic velocity would be c = 4J ′ as, while the coupling constant is still g = 2/s. One checks that this agrees perfectly with the ξ → 0 limit of Eq. The interaction L int between the two chains may be easily expressed in terms of e 1 and e 2 . Indeed, recall that in taking the continuum limit of a single AF chain, one uses the decomposition S i = s[(−1) i e +ãl], where l is the local magnetization andã = 2a is the effective lattice spacing for each chain. Neglecting derivatives, the interaction Lagrangian may be written as\\nThe elimination of l 1 and l 2 is done by substituting the equation of motion 11\\nThe interaction Lagrangian then becomes\\nIf we assume that the two vectors e 1 and e 2 are identically orthogonal, as the classical ground state suggests if ξ is small, then the above interaction exactly agrees with Eq. (21) and the ξ → 0 limit of γ as given in Eq. (22). Moreover, the topological terms S top [e 1 ] and S top [e 2 ] cancel each other if e 1 is identically orthogonal to e 2 . This is particularly easy to understand if the configuration e 1 belongs to the homotopy class of the identity, since the orthogonality constraint then effectively makes e 2 a mapping from the sphere S 2 to the circle S 1 and π 2 (S 1 ) = 0. In general, the two topological terms cancel simply because of the fact that two orthogonal unit vectors specify a rotation and π 2 (SO(3)) = 0. Thus, if we assume that e 1 ⊥ e 2 in the small ξ limit, the Lagrangian (21) is recovered with exactly the same parameters, since the topological terms are absent.\\nHowever, the orthogonality e 1 ⊥ e 2 is not strict, but only favored energetically. A local deviation from this orthogonality gives back their full importance to the topological terms and causes a distinction between integer and half-integer spins. Such a distinction does not occur in the SO(3) formulation of the problem since the orthogonality is then 'built-in'. In that formulation, a local deviation from orthogonality corresponds to a discontinuity in the order parameter, which was not allowed from the start.\\nFinally, let us point out that the above remarks concerning parity-breaking ordering wavevectors in frustrated antiferromagnets and their consequences on the excitation spectrum may apply to other systems, such as the nearest-neighbor Heisenberg antiferromagnet on a triangular lattice. In this case it is believed that the ground state has long-range order.\\nHowever, it is conceivable that the introduction of vacancies destroys this order without affecting the applicability of the SO (3)\"\n },\n {\n \"title\": \"(\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIG. 1 .FIG. 2 .\",\n \"content\": \"\"\n },\n {\n \"title\": \"FIGFIG. 4 .FIG. 6 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"AF) Heisenberg chain with nearest-neighbor (NN) coupling to the O(3)/O(2) sigma model has led Haldane to conjecture 1 that integer spin chains should exhibit a gap to a triplet of excited states, whereas half-integer spin chains should not. This conjecture has later been confirmed by numerical calculations. 2,3 Such a mapping may be termed semi-classical, since it is constructed by introducing a local field characterizing the order expected in the classical ground state and by taking into account fluctuations of the spin variables around this local order. In the case of the simple AF chain, a collinear order exists in the classical ground state and the local field introduced is a unit vector e(x, t) representing the staggered magnetization. The long wavelength effective action obtained for this field is that of the O(3)/O(2) sigma model, with Lagrangian density","content":"(\nAF) Heisenberg chain with nearest-neighbor (NN) coupling to the O(3)/O(2) sigma model has led Haldane to conjecture 1 that integer spin chains should exhibit a gap to a triplet of excited states, whereas half-integer spin chains should not. This conjecture has later been confirmed by numerical calculations. 2,3 Such a mapping may be termed semi-classical, since it is constructed by introducing a local field characterizing the order expected in the classical ground state and by taking into account fluctuations of the spin variables around this local order. In the case of the simple AF chain, a collinear order exists in the classical ground state and the local field introduced is a unit vector e(x, t) representing the staggered magnetization. The long wavelength effective action obtained for this field is that of the O(3)/O(2) sigma model, with Lagrangian density"},{"id":"","caption":"invariance, which makes it a qualitatively distinct theory from the versions of the SO(3) nonlinear sigma model studied in the context of classical critical phenomena 13 .","content":"invariance, which makes it a qualitatively distinct theory from the versions of the SO(3) nonlinear sigma model studied in the context of classical critical phenomena 13 ."},{"id":"","caption":"It can be shown explicitly 14 that the solution φ = η = 0 is the only acceptable one in this case. Eq. (31) coincides with the saddle-point equation for the O(3)/O(2) nonlinear sigma model, whose solution is","content":"It can be shown explicitly 14 that the solution φ = η = 0 is the only acceptable one in this case. Eq. (31) coincides with the saddle-point equation for the O(3)/O(2) nonlinear sigma model, whose solution is"},{"id":"","caption":"Of course, this result neglects 1/N corrections. As ξ → 1 (the collinear point) the gap goes to zero. This signals a continuous phase transition, beyond which a different description ofthe system is needed, for instance in terms of the O(3)/O(2) sigma model. From ξ = 1 up to ξ = 0, which corresponds to J ′ → ∞, the gap increases and becomes eventually proportional to J ′ . The main feature of the excitations of this model is the degeneracy of the triplets. This ultimately comes from the SO(2) symmetry existing between the vectors e 1 and e 2 , which may be mixed with each other without affecting the action. It is then not surprising if this degeneracy survives the large-N approximation. Indeed, 1/N corrections to the dispersion of the excitations can in principle be calculated, like they are in the O(3)/O(2) sigma model (see, for instance, Polyakov's text 15 ). The quantity of interest is then the self-energy of the fields e a . In general, this is a matrix Σ ab (ω, k), which may be calculated diagrammatically with the help of the vertices shown in Fig. 2 and of the propagators for the auxiliary fields σ ab and φ. The latter scale as 1/N since they are inverse polarization operators, and the order in 1/N is simply determined by the number of auxiliary field internal lines. However, the propagators for σ 11 and σ 22 are identical, since the large-N propagators of e 1 and e 2 are the same. Moreover, the structure of the vertices of Fig. 2 makes it impossible for a non-diagonal self-energy Σ 12 to arise: even though the auxiliary fields σ 12 and φ may change a e 1 quanta into a e 2 quanta, the number of such vertices must be even, without tadpoles. The net result is that no self-energy diagram exists in which an entering e 1 line is turned into an exiting e 2 line, and consequently Σ 12 = 0. The 1/N contributions to the Σ 11 are show diagrammatically on Fig. 3. At all orders in 1/N the two self-energies Σ 11 and Σ 22 are identical and therefore the degeneracy of the excitations branches seems a exact feature of the model.","content":"Of course, this result neglects 1/N corrections. As ξ → 1 (the collinear point) the gap goes to zero. This signals a continuous phase transition, beyond which a different description ofthe system is needed, for instance in terms of the O(3)/O(2) sigma model. From ξ = 1 up to ξ = 0, which corresponds to J ′ → ∞, the gap increases and becomes eventually proportional to J ′ . The main feature of the excitations of this model is the degeneracy of the triplets. This ultimately comes from the SO(2) symmetry existing between the vectors e 1 and e 2 , which may be mixed with each other without affecting the action. It is then not surprising if this degeneracy survives the large-N approximation. Indeed, 1/N corrections to the dispersion of the excitations can in principle be calculated, like they are in the O(3)/O(2) sigma model (see, for instance, Polyakov's text 15 ). The quantity of interest is then the self-energy of the fields e a . In general, this is a matrix Σ ab (ω, k), which may be calculated diagrammatically with the help of the vertices shown in Fig. 2 and of the propagators for the auxiliary fields σ ab and φ. The latter scale as 1/N since they are inverse polarization operators, and the order in 1/N is simply determined by the number of auxiliary field internal lines. However, the propagators for σ 11 and σ 22 are identical, since the large-N propagators of e 1 and e 2 are the same. Moreover, the structure of the vertices of Fig. 2 makes it impossible for a non-diagonal self-energy Σ 12 to arise: even though the auxiliary fields σ 12 and φ may change a e 1 quanta into a e 2 quanta, the number of such vertices must be even, without tadpoles. The net result is that no self-energy diagram exists in which an entering e 1 line is turned into an exiting e 2 line, and consequently Σ 12 = 0. The 1/N contributions to the Σ 11 are show diagrammatically on Fig. 3. At all orders in 1/N the two self-energies Σ 11 and Σ 22 are identical and therefore the degeneracy of the excitations branches seems a exact feature of the model."},{"id":"","caption":"Before throwing the SO(3) model away, one should see if it describes the long wavelength behavior of integer spin chains. Exact results for the spin-1 chain are lacking and thus we are reduced to numerical study of finite systems. Quantum Monte-Carlo studies of frustrated spin systems are quite difficult because of a sign problem, therefore we limited ourselves to numerical diagonalizations of small chains (up to 14 sites). We used various methods, of which the most practical turned out to be a variant of the Lanczos method 19 applied to subspaces of fixed momentum. Diagonalizations of spin-1 2 chains with up to 22 sites have also been performed, for the sake of comparison. It turns out that the spectra of spin-1 and spin-1 2 chains of small lengths are qualitatively different. The ground state of the spin-1 chains is non-degenerate, with a gap to a doubly degenerate triplet of spin-1 states, exactly as in the field theory. Excitation spectra for J ′ = J/2 and various chain lengths are shown in Fig. 4, whereas spectra for 14 spins and various values of J ′ /J are shown inFig. 5. The dispersion relation is symmetric with respect to parity (k → −k) and the minimum of the spin-1 branch occurs at a wavenumber k 0 smaller than π (in units of 1/a).","content":"Before throwing the SO(3) model away, one should see if it describes the long wavelength behavior of integer spin chains. Exact results for the spin-1 chain are lacking and thus we are reduced to numerical study of finite systems. Quantum Monte-Carlo studies of frustrated spin systems are quite difficult because of a sign problem, therefore we limited ourselves to numerical diagonalizations of small chains (up to 14 sites). We used various methods, of which the most practical turned out to be a variant of the Lanczos method 19 applied to subspaces of fixed momentum. Diagonalizations of spin-1 2 chains with up to 22 sites have also been performed, for the sake of comparison. It turns out that the spectra of spin-1 and spin-1 2 chains of small lengths are qualitatively different. The ground state of the spin-1 chains is non-degenerate, with a gap to a doubly degenerate triplet of spin-1 states, exactly as in the field theory. Excitation spectra for J ′ = J/2 and various chain lengths are shown in Fig. 4, whereas spectra for 14 spins and various values of J ′ /J are shown inFig. 5. The dispersion relation is symmetric with respect to parity (k → −k) and the minimum of the spin-1 branch occurs at a wavenumber k 0 smaller than π (in units of 1/a)."},{"id":"","caption":"The question remains as to why this semi-classical treatment fails in the half-integer spin case. In other words, where in the argument of Sec. II should the distinction between integer and half-integer spin come into play? The answer may lie in our neglect of configurations with discontinuities. This has already been pointed out in Ref.4 in relation with the kinetic term.An easier way to see this is to consider the limit J ′ > J. In that limit one may reconsider the problem and view the system as a pair of interwoven, antiferromagnetic chains with a small interchain interaction J. Each chain could be semi-classically described by the usualO(3)/O(2) nonlinear sigma model, with the Lagrangian density (1), plus a topological term L top expressed as","content":"The question remains as to why this semi-classical treatment fails in the half-integer spin case. In other words, where in the argument of Sec. II should the distinction between integer and half-integer spin come into play? The answer may lie in our neglect of configurations with discontinuities. This has already been pointed out in Ref.4 in relation with the kinetic term.An easier way to see this is to consider the limit J ′ > J. In that limit one may reconsider the problem and view the system as a pair of interwoven, antiferromagnetic chains with a small interchain interaction J. Each chain could be semi-classically described by the usualO(3)/O(2) nonlinear sigma model, with the Lagrangian density (1), plus a topological term L top expressed as"},{"id":"","caption":"(22). Thus, as ξ = J/4J ′ decreases, the two unit vectors e 1 and e 2 of the SO(3) model become the fields of two weakly coupled O(3)/O(2) sigma models, with the correct values of c and g.","content":"(22). Thus, as ξ = J/4J ′ decreases, the two unit vectors e 1 and e 2 of the SO(3) model become the fields of two weakly coupled O(3)/O(2) sigma models, with the correct values of c and g."},{"id":"","caption":"theory: the effect of the vacancies would then be to increase the value of the effective coupling constant g beyond the critical value g c . A priori, it is not clear if the SO(3) field theory describes the long wavelength behavior of antiferromagnets on the Kagomé lattice or on the see-saw chain 20 . A semi-classical study of these systems may be interesting. Classical ground state of the frustrated AF chain with J ′ /J = 1 2 . Each spin makes a 120 • angle with its neighbors and the periodicity is the smallest possible, i.e., three sites. Below is different look at the same spin chain, using a 'railroad trestle' geometry. Vertices used in the 1/N expansion of the model (27).","content":"FIG. 1 .FIG. 2 .\n12theory: the effect of the vacancies would then be to increase the value of the effective coupling constant g beyond the critical value g c . A priori, it is not clear if the SO(3) field theory describes the long wavelength behavior of antiferromagnets on the Kagomé lattice or on the see-saw chain 20 . A semi-classical study of these systems may be interesting. Classical ground state of the frustrated AF chain with J ′ /J = 1 2 . Each spin makes a 120 • angle with its neighbors and the periodicity is the smallest possible, i.e., three sites. Below is different look at the same spin chain, using a 'railroad trestle' geometry. Vertices used in the 1/N expansion of the model (27)."},{"id":"","caption":". 3. 1/N contributions to the self-energy Σ 11 of the field e 1 . Excitation spectrum of the frustrated spin-1 chain with J ′ = J/2 for chains lengths L = 8, 10, 12 and 14. The white circles represent spin-triplet states, while the black squares represent spin-singlets, when lower in energy. Dependence on chain length N of the critical coupling ratio ξ c = (J/4J ′ ) c at which the triplet degeneracy appears (error bars). Also illustrated is the energy gap ∆/J for ξ = 1 2 (circles).","content":"FIGFIG. 4 .FIG. 6 .\n46. 3. 1/N contributions to the self-energy Σ 11 of the field e 1 . Excitation spectrum of the frustrated spin-1 chain with J ′ = J/2 for chains lengths L = 8, 10, 12 and 14. The white circles represent spin-triplet states, while the black squares represent spin-singlets, when lower in energy. Dependence on chain length N of the critical coupling ratio ξ c = (J/4J ′ ) c at which the triplet degeneracy appears (error bars). Also illustrated is the energy gap ∆/J for ξ = 1 2 (circles)."}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"AF) Heisenberg chain with nearest-neighbor (NN) coupling to the O(3)/O(2) sigma model has led Haldane to conjecture 1 that integer spin chains should exhibit a gap to a triplet of excited states, whereas half-integer spin chains should not. This conjecture has later been confirmed by numerical calculations. 2,3 Such a mapping may be termed semi-classical, since it is constructed by introducing a local field characterizing the order expected in the classical ground state and by taking into account fluctuations of the spin variables around this local order. In the case of the simple AF chain, a collinear order exists in the classical ground state and the local field introduced is a unit vector e(x, t) representing the staggered magnetization. The long wavelength effective action obtained for this field is that of the O(3)/O(2) sigma model, with Lagrangian density\",\n \"content\": \"(\\nAF) Heisenberg chain with nearest-neighbor (NN) coupling to the O(3)/O(2) sigma model has led Haldane to conjecture 1 that integer spin chains should exhibit a gap to a triplet of excited states, whereas half-integer spin chains should not. This conjecture has later been confirmed by numerical calculations. 2,3 Such a mapping may be termed semi-classical, since it is constructed by introducing a local field characterizing the order expected in the classical ground state and by taking into account fluctuations of the spin variables around this local order. In the case of the simple AF chain, a collinear order exists in the classical ground state and the local field introduced is a unit vector e(x, t) representing the staggered magnetization. The long wavelength effective action obtained for this field is that of the O(3)/O(2) sigma model, with Lagrangian density\"\n },\n {\n \"id\": \"\",\n \"caption\": \"invariance, which makes it a qualitatively distinct theory from the versions of the SO(3) nonlinear sigma model studied in the context of classical critical phenomena 13 .\",\n \"content\": \"invariance, which makes it a qualitatively distinct theory from the versions of the SO(3) nonlinear sigma model studied in the context of classical critical phenomena 13 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"It can be shown explicitly 14 that the solution φ = η = 0 is the only acceptable one in this case. Eq. (31) coincides with the saddle-point equation for the O(3)/O(2) nonlinear sigma model, whose solution is\",\n \"content\": \"It can be shown explicitly 14 that the solution φ = η = 0 is the only acceptable one in this case. Eq. (31) coincides with the saddle-point equation for the O(3)/O(2) nonlinear sigma model, whose solution is\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Of course, this result neglects 1/N corrections. As ξ → 1 (the collinear point) the gap goes to zero. This signals a continuous phase transition, beyond which a different description ofthe system is needed, for instance in terms of the O(3)/O(2) sigma model. From ξ = 1 up to ξ = 0, which corresponds to J ′ → ∞, the gap increases and becomes eventually proportional to J ′ . The main feature of the excitations of this model is the degeneracy of the triplets. This ultimately comes from the SO(2) symmetry existing between the vectors e 1 and e 2 , which may be mixed with each other without affecting the action. It is then not surprising if this degeneracy survives the large-N approximation. Indeed, 1/N corrections to the dispersion of the excitations can in principle be calculated, like they are in the O(3)/O(2) sigma model (see, for instance, Polyakov's text 15 ). The quantity of interest is then the self-energy of the fields e a . In general, this is a matrix Σ ab (ω, k), which may be calculated diagrammatically with the help of the vertices shown in Fig. 2 and of the propagators for the auxiliary fields σ ab and φ. The latter scale as 1/N since they are inverse polarization operators, and the order in 1/N is simply determined by the number of auxiliary field internal lines. However, the propagators for σ 11 and σ 22 are identical, since the large-N propagators of e 1 and e 2 are the same. Moreover, the structure of the vertices of Fig. 2 makes it impossible for a non-diagonal self-energy Σ 12 to arise: even though the auxiliary fields σ 12 and φ may change a e 1 quanta into a e 2 quanta, the number of such vertices must be even, without tadpoles. The net result is that no self-energy diagram exists in which an entering e 1 line is turned into an exiting e 2 line, and consequently Σ 12 = 0. The 1/N contributions to the Σ 11 are show diagrammatically on Fig. 3. At all orders in 1/N the two self-energies Σ 11 and Σ 22 are identical and therefore the degeneracy of the excitations branches seems a exact feature of the model.\",\n \"content\": \"Of course, this result neglects 1/N corrections. As ξ → 1 (the collinear point) the gap goes to zero. This signals a continuous phase transition, beyond which a different description ofthe system is needed, for instance in terms of the O(3)/O(2) sigma model. From ξ = 1 up to ξ = 0, which corresponds to J ′ → ∞, the gap increases and becomes eventually proportional to J ′ . The main feature of the excitations of this model is the degeneracy of the triplets. This ultimately comes from the SO(2) symmetry existing between the vectors e 1 and e 2 , which may be mixed with each other without affecting the action. It is then not surprising if this degeneracy survives the large-N approximation. Indeed, 1/N corrections to the dispersion of the excitations can in principle be calculated, like they are in the O(3)/O(2) sigma model (see, for instance, Polyakov's text 15 ). The quantity of interest is then the self-energy of the fields e a . In general, this is a matrix Σ ab (ω, k), which may be calculated diagrammatically with the help of the vertices shown in Fig. 2 and of the propagators for the auxiliary fields σ ab and φ. The latter scale as 1/N since they are inverse polarization operators, and the order in 1/N is simply determined by the number of auxiliary field internal lines. However, the propagators for σ 11 and σ 22 are identical, since the large-N propagators of e 1 and e 2 are the same. Moreover, the structure of the vertices of Fig. 2 makes it impossible for a non-diagonal self-energy Σ 12 to arise: even though the auxiliary fields σ 12 and φ may change a e 1 quanta into a e 2 quanta, the number of such vertices must be even, without tadpoles. The net result is that no self-energy diagram exists in which an entering e 1 line is turned into an exiting e 2 line, and consequently Σ 12 = 0. The 1/N contributions to the Σ 11 are show diagrammatically on Fig. 3. At all orders in 1/N the two self-energies Σ 11 and Σ 22 are identical and therefore the degeneracy of the excitations branches seems a exact feature of the model.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Before throwing the SO(3) model away, one should see if it describes the long wavelength behavior of integer spin chains. Exact results for the spin-1 chain are lacking and thus we are reduced to numerical study of finite systems. Quantum Monte-Carlo studies of frustrated spin systems are quite difficult because of a sign problem, therefore we limited ourselves to numerical diagonalizations of small chains (up to 14 sites). We used various methods, of which the most practical turned out to be a variant of the Lanczos method 19 applied to subspaces of fixed momentum. Diagonalizations of spin-1 2 chains with up to 22 sites have also been performed, for the sake of comparison. It turns out that the spectra of spin-1 and spin-1 2 chains of small lengths are qualitatively different. The ground state of the spin-1 chains is non-degenerate, with a gap to a doubly degenerate triplet of spin-1 states, exactly as in the field theory. Excitation spectra for J ′ = J/2 and various chain lengths are shown in Fig. 4, whereas spectra for 14 spins and various values of J ′ /J are shown inFig. 5. The dispersion relation is symmetric with respect to parity (k → −k) and the minimum of the spin-1 branch occurs at a wavenumber k 0 smaller than π (in units of 1/a).\",\n \"content\": \"Before throwing the SO(3) model away, one should see if it describes the long wavelength behavior of integer spin chains. Exact results for the spin-1 chain are lacking and thus we are reduced to numerical study of finite systems. Quantum Monte-Carlo studies of frustrated spin systems are quite difficult because of a sign problem, therefore we limited ourselves to numerical diagonalizations of small chains (up to 14 sites). We used various methods, of which the most practical turned out to be a variant of the Lanczos method 19 applied to subspaces of fixed momentum. Diagonalizations of spin-1 2 chains with up to 22 sites have also been performed, for the sake of comparison. It turns out that the spectra of spin-1 and spin-1 2 chains of small lengths are qualitatively different. The ground state of the spin-1 chains is non-degenerate, with a gap to a doubly degenerate triplet of spin-1 states, exactly as in the field theory. Excitation spectra for J ′ = J/2 and various chain lengths are shown in Fig. 4, whereas spectra for 14 spins and various values of J ′ /J are shown inFig. 5. The dispersion relation is symmetric with respect to parity (k → −k) and the minimum of the spin-1 branch occurs at a wavenumber k 0 smaller than π (in units of 1/a).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The question remains as to why this semi-classical treatment fails in the half-integer spin case. In other words, where in the argument of Sec. II should the distinction between integer and half-integer spin come into play? The answer may lie in our neglect of configurations with discontinuities. This has already been pointed out in Ref.4 in relation with the kinetic term.An easier way to see this is to consider the limit J ′ > J. In that limit one may reconsider the problem and view the system as a pair of interwoven, antiferromagnetic chains with a small interchain interaction J. Each chain could be semi-classically described by the usualO(3)/O(2) nonlinear sigma model, with the Lagrangian density (1), plus a topological term L top expressed as\",\n \"content\": \"The question remains as to why this semi-classical treatment fails in the half-integer spin case. In other words, where in the argument of Sec. II should the distinction between integer and half-integer spin come into play? The answer may lie in our neglect of configurations with discontinuities. This has already been pointed out in Ref.4 in relation with the kinetic term.An easier way to see this is to consider the limit J ′ > J. In that limit one may reconsider the problem and view the system as a pair of interwoven, antiferromagnetic chains with a small interchain interaction J. Each chain could be semi-classically described by the usualO(3)/O(2) nonlinear sigma model, with the Lagrangian density (1), plus a topological term L top expressed as\"\n },\n {\n \"id\": \"\",\n \"caption\": \"(22). Thus, as ξ = J/4J ′ decreases, the two unit vectors e 1 and e 2 of the SO(3) model become the fields of two weakly coupled O(3)/O(2) sigma models, with the correct values of c and g.\",\n \"content\": \"(22). Thus, as ξ = J/4J ′ decreases, the two unit vectors e 1 and e 2 of the SO(3) model become the fields of two weakly coupled O(3)/O(2) sigma models, with the correct values of c and g.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"theory: the effect of the vacancies would then be to increase the value of the effective coupling constant g beyond the critical value g c . A priori, it is not clear if the SO(3) field theory describes the long wavelength behavior of antiferromagnets on the Kagomé lattice or on the see-saw chain 20 . A semi-classical study of these systems may be interesting. Classical ground state of the frustrated AF chain with J ′ /J = 1 2 . Each spin makes a 120 • angle with its neighbors and the periodicity is the smallest possible, i.e., three sites. Below is different look at the same spin chain, using a 'railroad trestle' geometry. Vertices used in the 1/N expansion of the model (27).\",\n \"content\": \"FIG. 1 .FIG. 2 .\\n12theory: the effect of the vacancies would then be to increase the value of the effective coupling constant g beyond the critical value g c . A priori, it is not clear if the SO(3) field theory describes the long wavelength behavior of antiferromagnets on the Kagomé lattice or on the see-saw chain 20 . A semi-classical study of these systems may be interesting. Classical ground state of the frustrated AF chain with J ′ /J = 1 2 . Each spin makes a 120 • angle with its neighbors and the periodicity is the smallest possible, i.e., three sites. Below is different look at the same spin chain, using a 'railroad trestle' geometry. Vertices used in the 1/N expansion of the model (27).\"\n },\n {\n \"id\": \"\",\n \"caption\": \". 3. 1/N contributions to the self-energy Σ 11 of the field e 1 . Excitation spectrum of the frustrated spin-1 chain with J ′ = J/2 for chains lengths L = 8, 10, 12 and 14. The white circles represent spin-triplet states, while the black squares represent spin-singlets, when lower in energy. Dependence on chain length N of the critical coupling ratio ξ c = (J/4J ′ ) c at which the triplet degeneracy appears (error bars). Also illustrated is the energy gap ∆/J for ξ = 1 2 (circles).\",\n \"content\": \"FIGFIG. 4 .FIG. 6 .\\n46. 3. 1/N contributions to the self-energy Σ 11 of the field e 1 . Excitation spectrum of the frustrated spin-1 chain with J ′ = J/2 for chains lengths L = 8, 10, 12 and 14. The white circles represent spin-triplet states, while the black squares represent spin-singlets, when lower in energy. Dependence on chain length N of the critical coupling ratio ξ c = (J/4J ′ ) c at which the triplet degeneracy appears (error bars). Also illustrated is the energy gap ∆/J for ξ = 1 2 (circles).\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":". F D M Haldane, Phys. Lett. 93464F. D. M. Haldane, Phys. Lett. 93A, 464 (1983);","doi":""},{"text":". Phys. Rev. Lett. 501153Phys. Rev. Lett. 50, 1153 (1983).","doi":""},{"text":". M P Nightingale, H W J Blöte, Phys. Rev. B. 33659M. P. Nightingale, H. W. J. Blöte, Phys. Rev. B 33, 659 (1986).","doi":""},{"text":". S R White, Phys. Rev. Lett. 692863S. R. White, Phys. Rev. Lett. 69, 2863 (1992);","doi":""},{"text":". S R White, D A Huse, Phys. Rev. B. 483844S. R. White and D. A. Huse, Phys. Rev. B 48, 3844 (1993).","doi":""},{"text":". T Dombre, N Read, Phys. Rev. B. 396797T. Dombre and N. Read, Phys. Rev. B 39, 6797 (1989).","doi":""},{"text":". S Miyashita, ; B Bernu, C Lhuillier, L Pierre, J. Phys. Soc. Jpn. P. Azaria, B. Delamotte and D.Mouhanna532483Phys. Rev. Lett.S. Miyashita, J. Phys. Soc. Jpn 53, 44 (1984). B. Bernu, C. Lhuillier and L. Pierre, Phys. Rev. Lett. 69, 2590 (1992). P. Azaria, B. Delamotte and D.Mouhanna, Phys. Rev. Lett. 70, 2483 (1993).","doi":""},{"text":". C K Majumdar, D K Gosh, J. Math. Phys. 101399C. K. Majumdar and D. K. Gosh, J. Math. Phys. 10, 1388, 1399 (1969).","doi":""},{"text":". T Tonegawa, M Kaburagi, N Ichikawa, I Harada, J. Phys. Soc. Jap. 612890T. Tonegawa, M. Kaburagi, N. Ichikawa and I. Harada, J. Phys. Soc. Jap. 61, 2890 (1992).","doi":""},{"text":". I Harada, M Fujikawa, I Mannari, J. Phys. Soc. Jap. 623694I. Harada, M. Fujikawa, I. Mannari, J. Phys. Soc. Jap. 62, 3694 (1993).","doi":""},{"text":". H Shimaoka, K Kuboki, J. Phys. Soc. Jap. 623257H. Shimaoka and K. Kuboki, J. Phys. Soc. Jap. 62, 3257 (1993).","doi":""},{"text":". S Rao, D Sen, Phys. Rev. B. 4812763S. Rao and D. Sen, Phys. Rev. B 48, 12763 (1993).","doi":""},{"text":"E Fradkin, Field Theories of Condensed Matter Systems. Reading, MAAddison-WesleyE. Fradkin, Field Theories of Condensed Matter Systems, Addison-Wesley, Reading, MA, 1991.","doi":""},{"text":". E Manousakis, Rev. Mod. Phys. 631E. Manousakis, Rev. Mod. Phys. 63, 1 (1991).","doi":""},{"text":". P Azaria, D Delamotte, F Delduc, T Jolicoeur, Nucl. Phys. 408485P. Azaria, D. Delamotte, F. Delduc and T. Jolicoeur, Nucl. Phys. B408, 485 (1993).","doi":""},{"text":". D Allen, M Sc, Université de SherbrookethesisD. Allen, M.Sc. thesis, Université de Sherbrooke (1994).","doi":""},{"text":"A M Polyakov, Gauge Fields and Strings. Harwood, New YorkA. M. Polyakov, Gauge Fields and Strings, Harwood, New York (1987).","doi":""},{"text":". H Kawamura, Phys. Rev. B. 384916H. Kawamura, Phys. Rev. B 38, 4916 (1988).","doi":""},{"text":". E H Lieb, T Schultz, D J Mattis, Ann. Phys. (N.Y.). 16407E. H. Lieb, T. Schultz and D. J. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961).","doi":""},{"text":". I Affleck, E H Lieb, Lett. Math. Phys. 1257I. Affleck and E. H. Lieb, Lett. Math. Phys. 12, 57 (1986).","doi":""},{"text":". E R Gagliano, E Dagotto, A Moreo, F C Alcaraz, Phys. Rev. B. 341677E. R. Gagliano, E. Dagotto, A. Moreo and F. C. Alcaraz, Phys. Rev. B 34, 1677 (1986).","doi":""},{"text":". M W Long, S Siak, J. Phys.: Condens. Matter. 34901M. W. Long and S. Siak, J. Phys.: Condens. Matter 3, 4901 (1991).","doi":""}],"string":"[\n {\n \"text\": \". F D M Haldane, Phys. Lett. 93464F. D. M. Haldane, Phys. Lett. 93A, 464 (1983);\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Phys. Rev. Lett. 501153Phys. Rev. Lett. 50, 1153 (1983).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M P Nightingale, H W J Blöte, Phys. Rev. B. 33659M. P. Nightingale, H. W. J. Blöte, Phys. Rev. B 33, 659 (1986).\",\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\": \". S R White, D A Huse, Phys. Rev. B. 483844S. R. White and D. A. Huse, Phys. Rev. B 48, 3844 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Dombre, N Read, Phys. Rev. B. 396797T. Dombre and N. Read, Phys. Rev. B 39, 6797 (1989).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Miyashita, ; B Bernu, C Lhuillier, L Pierre, J. Phys. Soc. Jpn. P. Azaria, B. Delamotte and D.Mouhanna532483Phys. Rev. Lett.S. Miyashita, J. Phys. Soc. Jpn 53, 44 (1984). B. Bernu, C. Lhuillier and L. Pierre, Phys. Rev. Lett. 69, 2590 (1992). P. Azaria, B. Delamotte and D.Mouhanna, Phys. Rev. Lett. 70, 2483 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". C K Majumdar, D K Gosh, J. Math. Phys. 101399C. K. Majumdar and D. K. Gosh, J. Math. Phys. 10, 1388, 1399 (1969).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Tonegawa, M Kaburagi, N Ichikawa, I Harada, J. Phys. Soc. Jap. 612890T. Tonegawa, M. Kaburagi, N. Ichikawa and I. Harada, J. Phys. Soc. Jap. 61, 2890 (1992).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Harada, M Fujikawa, I Mannari, J. Phys. Soc. Jap. 623694I. Harada, M. Fujikawa, I. Mannari, J. Phys. Soc. Jap. 62, 3694 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Shimaoka, K Kuboki, J. Phys. Soc. Jap. 623257H. Shimaoka and K. Kuboki, J. Phys. Soc. Jap. 62, 3257 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Rao, D Sen, Phys. Rev. B. 4812763S. Rao and D. Sen, Phys. Rev. B 48, 12763 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"E Fradkin, Field Theories of Condensed Matter Systems. Reading, MAAddison-WesleyE. Fradkin, Field Theories of Condensed Matter Systems, Addison-Wesley, Reading, MA, 1991.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Manousakis, Rev. Mod. Phys. 631E. Manousakis, Rev. Mod. Phys. 63, 1 (1991).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". P Azaria, D Delamotte, F Delduc, T Jolicoeur, Nucl. Phys. 408485P. Azaria, D. Delamotte, F. Delduc and T. Jolicoeur, Nucl. Phys. B408, 485 (1993).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Allen, M Sc, Université de SherbrookethesisD. Allen, M.Sc. thesis, Université de Sherbrooke (1994).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A M Polyakov, Gauge Fields and Strings. Harwood, New YorkA. M. Polyakov, Gauge Fields and Strings, Harwood, New York (1987).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". H Kawamura, Phys. Rev. B. 384916H. Kawamura, Phys. Rev. B 38, 4916 (1988).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E H Lieb, T Schultz, D J Mattis, Ann. Phys. (N.Y.). 16407E. H. Lieb, T. Schultz and D. J. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". I Affleck, E H Lieb, Lett. Math. Phys. 1257I. Affleck and E. H. Lieb, Lett. Math. Phys. 12, 57 (1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E R Gagliano, E Dagotto, A Moreo, F C Alcaraz, Phys. Rev. B. 341677E. R. Gagliano, E. Dagotto, A. Moreo and F. C. Alcaraz, Phys. Rev. B 34, 1677 (1986).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M W Long, S Siak, J. Phys.: Condens. Matter. 34901M. W. Long and S. Siak, J. Phys.: Condens. Matter 3, 4901 (1991).\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":597,"cells":{"text":{"kind":"string","value":"\nSub-systems in nearby solar-type wide binaries 1\n7 Jun 2010\n\nAndrei Tokovinin atokovinin@ctio.noao.edu \nGemini Observatory\nCerro Tololo Inter-American Observatory\nCasilla 603, Southern Operations Center, c/o AURA, Casilla 603, La SerenaLa SerenaChile, Chile\n\nMarkus Hartung mhartung@gemini.edu \nGemini Observatory\nCerro Tololo Inter-American Observatory\nCasilla 603, Southern Operations Center, c/o AURA, Casilla 603, La SerenaLa SerenaChile, Chile\n\nThomas L Hayward thayward@gemini.edu \nGemini Observatory\nCerro Tololo Inter-American Observatory\nCasilla 603, Southern Operations Center, c/o AURA, Casilla 603, La SerenaLa SerenaChile, Chile\n\nSub-systems in nearby solar-type wide binaries 1\n7 Jun 2010Subject headings: stars: binaries\nWe conducted a deep survey of resolved sub-systems among wide binaries with solar-type components within 67 pc from the Sun. Images of 61 stars in the K and H bands were obtained with the NICI adaptive-optics instrument on the 8-m Gemini-South telescope. Our maximum detectable magnitude difference is about 5 m and 7.8 m at 0.15 ′′ and 0.9 ′′ separations, respectively. This enables a complete census of sub-systems with stellar companions in the projected separation range from 5 to 100 AU. Out of 7 such companions found in our sample, only one was known previously. We determine that the fraction of sub-systems with projected separations above 5 AU is 0.12 ± 0.04 and that the distribution of their mass ratio is flat, with a power-law index 0.2 ± 0.5. Comparing this with the properties of closer spectroscopic sub-systems (separations below 1 AU), it appears that the mass-ratio distribution does not depend on the separation. The frequency of sub-systems in the separation ranges below 1 AU and between 5 and 100 AU is similar, about 0.15. Unbiased statistics of multiplicity higher than two, advanced by this work, provide constraints on star-formation theory.\n\nIntroduction\n\nThe quest for exo-planets has attracted attention to stars in the solar neighbourhood. A complete understanding of their formation requires a quantitative and predictive description of stellar multiplicity. Empirical data on stellar multiplicity serve as a benchmark for star formation theories, for example to compare with recent large-scale hydrodynamical simulations (Bate 2009). Although multiplicity statistics of solar-type dwarfs were assumed to be known since the survey of Duquennoy & Mayor (1991), in fact that survey did not constrain multiplicities higher than two because (i) the sample of 164 stars was too small and (ii) the discovery of wide binaries relied on traditional visual techniques. Raghavan (2009) shows that Duquennoy & Mayor (1991) missed half of higher-order multiples within 25 pc from the Sun.\n\nSeveral studies of large samples of nearby stars are now in progress. Radial-velocity (RV) techniques ensure a secure detection of all short-period stellar companions (see summary in Grether & Lineweaver 2006). However, for wider separations (> 10 AU) RV variations are too slow or too small, and such companions are better discovered by direct imaging. Several recent deep imaging surveys, summarized by Metchev & Hillenbrandt (2009), aimed to detect substellar or planetary-mass companions to nearby stars, obtaining statistics of stellar companions as a by-product. The typical sample size of imaging surveys is still small, on the order of 100 targets.\n\nMost imaging and RV surveys for low-mass companions explicitly avoid known visual binaries (e.g. Chauvin et al. 2010), so important starformation effects may not be represented in their samples. For example, catalogued triples show that the mass-ratio distributions in close inner sub-systems discovered spectroscopically and in wider \"visual\" sub-systems are radically different (see Section 5). One of our goals is to investigate if this is a selection effect or a genuine feature of the star-formation process.\n\nIn order to improve the multiplicity statistics of a well-defined and sufficiently large distancelimited sample, we focus on ∼5000 dwarfs with 0.5 < V − I < 0.8 within 67 pc from the Sun. According to preliminary estimates, there should be some 400 triples and 100 quadruples in this sample. Surveying such a large sample to detect those multiples with a reasonably deep and known completeness limit is a large task; here we address only a small part of it. Instead of avoiding known binaries, we purposefuly selected wide visual binaries from this sample. Some of these targets are also monitored in RV, giving complementary constraints on sub-systems with short periods (Desidera et al. 2006).\n\nWe imaged the chosen wide binaries with Adaptive Optics (AO) to study the properties of inner sub-systems. AO imaging on 8-m class telescopes provides nearly diffraction-limited resolution of 0.06 ′′ in the near-infrared, which corresponds to a projected orbital radius of 4 AU for the farthest target stars in our sample, filling an important gap between RV and seeing-limited imaging surveys.\n\nOur sample is detailed in Section 2. Observations, data reduction, and detection limits are described in Section 3, the results are presented in Section 4. Sections 5 and 6 contain the discussion and conclusions, respectively.\n\n\nThe sample of wide binaries\n\nThe wide binaries chosen for our survey belong to a large sample of nearby solar-type dwarfs (Nsample) selected from the latest version of Hipparcos catalog (van Leeuwen 2007) by the following criteria:\n\n1. trigonometric parallax π HIP < 15 mas (within 67 pc from the Sun, distance modulus < 4.12 m ); 2. color 0.5 < V − I < 0.8 (this corresponds approximately to spectral types from F5V to K0V);\n\n3. unevolved, satisfying the condition M Hp > 9(V − I) − 3.5, where M Hp is the absolute magnitude in the Hipparcos band calculated with π HIP .\n\nThere are 5040 stars (i.e. Hipparcos catalog entries) satisfying these conditions. Some of these stars with erroneous parallaxes or colors will be eventually removed from the sample, but this contamination is expected to be minor. The Nsample defined above is reasonably complete because the cumulative object count versus distance d follows the d 3 law, deviating from it by only 10% at d = 67 pc. Visual binaries with separations 5 ′′ < ρ < 20 ′′ and one or both components belonging to the Nsample were selected for AO observations from the Washington Double-Star Catalog, WDS (Mason et al. 2001). Additionally, we require each pair to have at least 3 observations and magnitudes of both components listed in the WDS, and that apparent motion of the wide pair be compatible with π HIP . As most targets have large proper motions, relative stability of the binary position over time permits rejection of optical pairs. An additional check was made by placing the components on the (M V , V − K) color-magnitude diagram (CMD). There are ∼200 wide binaries in this list, 101 with negative declinations.\n\nIn Table 1 we list basic data on the components of the 33 wide pairs observed in this run. Column 1 gives the Hipparcos numbers of the primary and, in some cases, secondary components. The next three columns contain π HIP , binary separation ρ, and projected separation r = ρ/π HIP . Then follow the magnitudes of the primary (A) and sec- ondary (B) components in the V , H, and K s bands as listed in the WDS and 2MASS (Cutri et al. 2003). The last two columns contain the date of observations and remarks where we indicate 5 cases when only one component was observed and 3 cases when the coronagraphic mask was used.\n\n3. Observations and data reduction 3.1. Observing procedure\n\nThe Near-Infrared Coronagraphic Imager, NICI, on the Gemini South telescope is an 85-element curvature AO instrument based on natural guide stars (Toomey & Ftaclas 2003;Chun et al. 2008). We used NICI in normal (non-coronagraphic) mode, as a classical AO system. However, simultaneous imaging in the two infrared bands H and K offered by NICI helps to distinguish faint companions from static speckles (the radial distance of a static speckle scales with wavelength while the position of a real detection does not change). The two detectors have 1024 2 pixels of 18 mas (milliarcseconds) size, covering a square field of 18 ′′ .\n\nTo avoid saturation, we used narrow-band filters with central wavelengths 2.272 µm and 1.587 µm in the red and blue channels, respectively. The minimum possible exposure time is 0.38 s, still causing saturation for bright targets. We observed three of the brightest stars with the coronagraphic mask of 0.32 ′′ radius, using in this instance broadband K and H filters. For the remaining bright targets we allowed moderate saturation. As the tradeoff between slower observations with more complex data reduction in coronagraphic mode and slightly saturated PSFs is not obvious, we tried both.\n\nTypically, each target was observed at 5-6 dithered positions for a total accumulation time of about 5 min. The dither offsets were designed to take images of both wide-binary components in one frame whenever possible (ρ < 10 ′′ ) while closing the AO loop on the primary. Components of wider pairs were observed separately, except secondaries with V > 12 m , too faint to serve as AO guide stars.\n\nOf the two nights allocated to this program, one was lost due to a technical problem. Three targets were observed on 2009 December 10 in service mode. The remaining objects were observed on the night of 2010 January 31 to February 1 in classical mode. The seeing during that night was good. As measured from the NICI AO loop data, it ranged between 0.34 ′′ and 0.94 ′′ , with a median of 0.50 ′′ (at 500 nm, not corrected to the zenith). High AO performance was achieved (see next Section). The transparency was variable (light cirrus). Out of 202 binary components accessible to NICI, we were able to observe 61, or 30%. The efficiency was high, with an average of 12 min per target including telescope slew, tuning of the primary mirror, acquisition of the object on the science camera, closing the AO loop, and data-taking.\n\n\nData processing\n\nThe data were processed in a standard way using custom IDL programs. Individual images written to separate FITS files were combined into data cubes, separately in red and blue channels. The sky image was obtained for each target as a median of the cube, then subtracted from each plane of the cube. After correcting for bad pixels and dividing by the flat field, the images were re-centered by cross-correlation and median-combined.\n\nMost frames contain just one or two bright point sources corresponding to the components of the wide binary. The position of each point source was recorded by \"clicking\" on its image, then refined by fitting a 2-dimensional Gaussian to the central part of the Point Spread Function (PSF) within 5-pixel radius. The ratio of the central intensity of this Gaussian to the total flux in a 0.9 ′′ diameter circle gives an estimate of the Strehl ratio. The typical Strehls 2 are 43% and 22% in the red and blue channel, respectively, with maximum values reaching 54% and 38%. The median Full Width at Half Maximum (FWHM) of the PSF core was 65 mas and 57 mas in the red and blue channels, respectively. The diffraction limit λ/D is 63 mas and 44 mas for the effective aperture diameter D = 7.5 m set by the coronagraphic mask. Figure 1 shows the mosaic of 7 close subsystems found around some primary components of wide binaries. Most companions are rather obvious, well above the detection threshold. The PSFs contain also a faint secondary reflex at 13 pixels (0.24 ′′ ), with a magnitude difference in the red channel of about 4.3 m . This ghost (illustrated in Fig. 1 in the case of HIP 37332) is detected in all targets, and can be taken advantage of as a fiducial (see Section 3.4). Its position is the same in both channels, whereas the static speckle pattern scales in proportion to the wavelength.\n\n\nMeasurement of binaries\n\nTo measure the relative position and flux ratio of wide binaries more precisely, we fit the scaled and shifted image of the primary component (A) to the secondary (B). Such fits work very well, leaving only small residuals, and produce mutually consistent measurements in the two channels. The rms scatter between the wide-binary positions measured in two NICI channels is 5.5 mas in both radial and tangential directions.\n\nWe checked the pixel scale and detector orientation by comparing the measured positions of 16 wide pairs (the widest in Table 2 after exclusion of HIP 21963, 43947 and 50883, which were not used) with the positions listed in the Hipparcos Double and Multiple Star Catalogue (ESA 1997). We did not find any systematic effect above noise and therefore used the nominal pixel scale of 18.0 mas for both channels. We found that the detector orientation of the channels differs by 1.01 • ± 0.01 • , so 1.0 • were added to measured angles in the blue channel.\n\nFor close pairs where the PSFs overlap, we use a variant of iterative blind deconvolution. Good preliminary estimates of the relative component's position and intensity are already available from the Gaussian fits of the PSF cores. The image Fourier Transform (FT) is divided by the FT of the binary with preliminary parameters to get an estimate of the PSF. Negative parts of this PSF are set to zero and it is replaced by its azimuthal average at radial distances over 5 pixels from the center. The binary parameters are refined by fitting again the image with this synthetic PSF. Then the process is repeated, obtaining a second approximation to the PSF and using it again (without azimuthal averaging) to obtain the final binary parameters. The success of this procedure is verified by inspecting the resulting PSF, which should be \"clean,\" with the least possible trace of the companion. To determine detection limits, we use mostly the data from the red NICI channel where the detection of low-mass components is deeper because of the smaller magnitude difference and larger Strehl ratios. The maximum magnitude difference of detectable companions ∆m lim is estimated by calculating the rms flux variations σ(ρ) in circular zones of increasing radii ρ centered on the main component. It is assumed that a detectable companion will have maximum intensity exceeding 5σ, therefore\n\n\nDetection limits\n∆m lim (ρ) < −2.5 log 10 [5σ(ρ)/I max ].\n(1)\n\nThis assumption is, of course, simplistic. We do not account for the existence of two simultaneous images in red and blue channels and for the good knowledge of the PSF from other companions of the same target or from other targets observed shortly before or after. Nevertheless, this simple strategy of estimating detection limits is sufficient for our purpose and has been tried before with good results (e.g. Tokovinin et al. 2006). The limits computed for HIP 37322 are shown in Fig. 2. A bump is produced in the curves at ρ ≈ 13 pixels by the ghost and by fixed speckle pattern (see the PSF in Fig. 1). To confirm the validity of (1), we performed Monte-Carlo simulations, adding artificial companions within ±1.5 m from the estimated ∆m lim . Secure detection of the ghost (∆m = 4.3, ρ = 0.24 ′′ ) in all images gives confidence in our estimation of ∆m lim .\n\nThe 5σ detection limits ∆m lim (ρ) were fitted by two straight lines intersecting at ρ = 8 pixels (0.144 ′′ ). The second segment extends from 8 to 50 pixels (0.9 ′′ ), but the region between 8 and 16 pixel radius affected by the ghost and strong speckle is avoided in the fit. At ρ > 50 pixels ∆m lim is assumed to be constant. Only two numbers ∆m(8) and ∆m(50) adequately describe the actual ∆m lim (ρ) curves, with typical rms error of only 0.3 m (excluding the zone between 8 and 16 pixel radii where the 5σ method does not properly account for the static speckle).\n\nDetection limits were determined by the above procedure for all targets, although the Monte Carlo simulation checks were only performed for a subset. The two detection parameters ∆m(8) and ∆m(50) vary little between targets. Their median values are 5.02 and 7.79, respectively; the quartiles are (4.29, 5.22) and (7.03, 8.06). In the following we apply median detection parameters to all targets, including the three coronagraphic images where the actual limits should be deeper.\n\nThe detection limits found here match other similar studies done with \"standard\" AO. For example, companion detection was complete down to ∆K < 3.9 at 0.14 ′′ and ∆K < 7.9 at 0.9 ′′ in (Tokovinin et al. 2006). Similar results were obtained by Eggenberger et al. (2007). The deep companion survey at Palomar and Keck conducted by Metchev & Hillenbrandt (2009) also reached ∆K < 8 at 0.9 ′′ . Note that typical detection limits with NICI are much deeper for dedicated planet search programs. In these cases long exposure times and techniques such as angular and spectral differential imaging are applied Artigau et al. 2008). \n\n\nResults\n\n\nCompanion data\n\nWe have found 6 new close companions, one previously known companion, and several faint optical companions (Fig. 1). Table 2 lists the relative positions and magnitude differences in known and new pairs. Each pair is identified by the HIP number and component designations. The measured position angles in degrees, separations in arcseconds, and magnitude differences are listed separately for the two NICI channels.\n\nThe large rms difference between Hipparcos and our measurements of the wide pairs (130 mas in ρ and 1.05 • in θ) is mostly caused by their motion between the two epochs. The comparison with the relative positions of the same binaries deduced from the 2MASS Point Source Catalog (Cutri et al. 2003) has confirmed that no systematic corrections are needed and has shown a scatter of similar magnitude.\n\nRemarks in the last column of Table 2 indicate cases when the image was saturated (the relative photometry may be compromised), newly detected close companions, and likely optical companions. These faint companions with separations of a few arcseconds would be dynamically unstable within wide binaries, should they be physical 3 .\n\n\nNotes on individual systems\n\nHIP 28790 has a common-proper-motion companion C at 196 ′′ .\n\nHIP 36165 A is suspected to be a spectroscopic binary (SB).\n\nHIP 43947 is unexpectedly discovered to be a resolved quadruple system. Both components are located ∼ 2 m below the Main Sequence in the (M V , V − K) CMD, possibly because the Hipparcos parallax measurement was biased by the motions in the sub-systems. The position and ∆m of the wide pair Aa,Ba is derived from the Gaussian fits, therefore it may be less accurate.\n\nHIP 45734: This is a young pre-Mainsequence system and X-ray source RX J0919.4-7738. It projects on the Chamaeleon star-forming region, but is located much closer than the corresponding association. Covino et al. (1997) found from a single spectrum that component B (southern) is a double-lined SB, with a large RV difference from A (see also Desidera et al. 2006). A itself is a close visual binary KOH83 resolved in March 1996 by Köhler (2001). He measured the position (173.9 • , 0.109 ′′ ) and ∆K = 0.45. This is the only previously known resolved sub-system among the observed stars. Follow-up observations will soon lead to the visual orbit and mass measurement, adding another empirical point to the evolutionary tracks of young stars.\n\nHIP 49520: The faint companion Bb is close to the detection limit (see Fig. 1). Yet, it is securely detected in both channels and stands up clearly in residuals when fitting the image of B with the image of A.\n\nHIP 50638/36: The wide pair has been observed since 1835; it is definitely physical. Component B is variable. For this reason, probably, it is located on the CMD some 6 m below the MS, while A is on the MS with the parallax π HIP = 83 mas listed in Table 1. According to van Leeuwen (2007), the parallaxes and proper motions of the two components are discordant, but have large errors. Future RV measurements can settle the controversy.\n\nHIP 59021: Component A is a suspected SB and also an astrometric binary (acceleration solution in Hipparcos). Non-detection of the subsystem with NICI means that its period is likely less than 10 y, or it is a white dwarf.\n\n\nEstimation of the mass ratios\n\nK-band absolute magnitudes M K of all companions were computed based on the photometry from 2MASS (see Table 1) and trigonometric parallaxes π HIP from Hipparcos. For resolved pairs, we assume that the ∆m in the red NICI channel correspond to magnitude differences in the K-band. The masses were estimated from M K using the empirical relations of Henry & McCarthy (1993). We also tried a quadratic approximation of relation between mass and M K from Girardi et al. (2000). The relative difference with Henry & McCarthy (1993) is less than 10% for masses above 0.1 M ⊙ . We checked that the influence of using either of these relations on the final result is not critical. Table 3 lists the estimated masses of the companions in the resolved sub-systems, their projected separations r = ρ/π HIP and order-of-magnitude period estimates P * = r 3/2 (M 1 + M 1 ) −1/2 . Figure 3 shows the mass ratios q = M 2 /M 1 in the resolved inner sub-systems as a function of projected separation. The median parallax of observed targets is 19 mas, so the upper limit of the separation range, 100 AU, corresponds to ρ ≈ 1.9 ′′ . Maximum separations of dynamically stable sub-systems should not exceed 100 AU in most cases because the projected separations of the wide binaries range from 150 to 1000 AU (Ta- ). Indeed, all detected sub-systems have r < 44 AU. The lower limit of 5 AU corresponds to ρ ≈ 0.1 ′′ , well above the NICI resolution limit.\n\nThe detection limits in the red channel were converted into mass ratios and averaged over the 61 observed components, producing a smooth detection probability p det (r, q). Contours of this function at 0.1, 0.5, and 0.9 levels are over-plotted in Fig. 3. We see that the chosen region of the (r, q) parameter space is well covered with NICI.\n\n\nStatistical analysis\n\nThe purpose of this study is to reach statistical conclusions about the frequency and the properties of sub-systems in solar-type wide binaries. Considering the small number of components, we keep this analysis at a basic level, and assume that the mass ratios in secondary sub-systems have a power-law distribution independent of separation:\nf (q) = ǫ(β + 1)q β .\n(2)\n\nThis model has only two free parameters, the total companion frequency ǫ and the power-law index β. Neglecting incomplete detection, the companion count gives ǫ = 7/61 = 0.12 ± 0.04. The distribution of the companion projected separations in the considered range from 5 to 100 AU matters only because the detection limits depend on r. We assume that f (log r) = const.\n\nThe input data are the (r, q) values for K = 7 sub-systems, the total number of surveyed targets N = 61, and the array of detection probabilities p det (r, q). We find the estimates of parameters (ǫ, β) by the Maximum Likelihood (ML) method (see e.g. Tokovinin et al. 2006, Apendix B). The likelihood function L is a product of the probabilities to obtain observations for each component (non-detections for N − K systems or detections of K subsystems with mass ratios q k ).\n\nThe minimum of S = −2 ln L is sought. This function depends on the input data and the model parameters as\nS = 2N f 0 − 2 K k=1 ln[f (q k )p det (r k , q k )].(3)\nThe summation over k is done for the 7 detected companions. The companion probability f 0 equals the product f (r, q)p det (r, q) averaged over the considered part of the parameter space (r, q). If the detection probability p det equals one everywhere, the result is f 0 = ǫ, otherwise f 0 < ǫ. The contours of S in the parameter space define confidence limits: ∆S = 1 corresponds to the 68% interval (\"1σ\"), ∆S = 2.71 to 90% and ∆S = 4 to 95%, in direct analogy with the Gaussian probability distribution. We verified the ML method on a simulated data set. The input parameters were recovered and the confidence intervals were as expected.\n\nMinimization of (3) leads to (ǫ, β) = (0.12, 0.18). If we ignore the detection limits and set p det = 1, the result changes little: (ǫ, β) = (0.11, 0.40). The companion fraction 0.12 is not different from its naive estimate. As to the power index β, the ML points to a uniform distribution in q as being most likely, although the uncertainty is large. The confidence areas of the parameters are shown in Fig. 4. Considering that the contours are not very elongated, we can determine the 68% and 90% confidence intervals for each parameter by a simple cross-section through the minimum (Table 4). The data are marginally compatible with β ∼ −0.5, in which case the fraction of sub-systems may exceed 0.2 (see the lower-right extension of the 90% contour in Fig. 4).\n\n\nDiscussion\n\n\nCompleteness of the existing catalog\n\nThe fact that, in a sample of only 33 systems, six of the seven observed companions are new, shows that current knowledge of multiplicity among nearby solar-type systems is very incomplete, and that near-IR AO imaging is a powerful technique for detecting new companions. We plot in Fig. 5 the mass-ratio distribution in the inner sub-systems of known triple stars. The data are extracted from the updated version of the Multiple-Star Catalog (MSC, Tokovinin 1997), where component masses are estimated by a variety of methods. We selected 220 systems within 200 pc from the Sun with masses of primary components from 0.5 to 1.5 M ⊙ . They were sub-divided by the separation in the inner subsystems in two groups, 123 close (semi-major axis < 1 AU) and 61 wide (between 5 and 100 AU). The remaining 36 pairs have even wider separations. The mass-ratio distributions in these two groups are markedly different, but our un-biased survey results in a flat mass-ratio distribution, within errors. Therefore, the difference seen in Fig. 5 is caused by observational selection. Given an approximately equal number of close and wide sub-systems with large mass ratios (two last bins in Fig. 5) and assuming that the mass-ratio distributions in these two groups are indeed similar, we expect that the number of close and wide sub-systems (with separations below 1 AU and between 5 and 100 AU, respectively) are similar, too. The completeness of the MSC for wide subsystems is thus about 1/2.\n\n\nCompanion fraction\n\nIs the presence of a wide binary companion influencing the frequency of inner sub-systems? To answer this question, we compare our results on triples with surveys of solar-type binaries. Unfortunately, binary-star statistics are usually derived regardless of higher-order multiplicity, so a cleaner comparison must await new study of a large unbiased sample.\n\nThe separation range from 5 to 100 AU explored here corresponds to orbital periods between 10 y and 1000 y. According to Duquennoy & Mayor (1991), about 20% of solar-type dwarfs have companions with such periods. In the actual, narrower range of projected separations from 6.8 AU to 44 AU, 12% of solar-type dwarfs have companions. Therefore, the frequency of sub-systems around components of wide binaries is similar to the frequency of binary companions to single stars in the same separation range.\n\nThe frequency of sub-systems with periods below 3 y in visual multiples has been determined by Tokovinin & Smekhov (2002) to be between 11% and 18%. Again, it turned out to equal within observational errors the companion frequency among single dwarfs of similar masses in the solar neighborhood and in some open clusters. Desidera et al. (2006) surveyed RVs of 56 wide visual pairs and found that 0.135 ± 0.05 fraction of companions have spectroscopic sub-systems.\n\nTherefore, one might surmise that the presence of a wide companion does not affect the formation of inner, closer sub-systems in a wide range of separations. This question is however far from having been settled for both stellar and planetary subsystems (see the discussion in Eggenberger et al. 2007). We know that almost all close binaries with periods below 3 d do have additional outer companions, although the mass ratios of close binaries with and without tertiary companions have identical distributions (Tokovinin et al. 2006).\n\n\nIs the mass-ratio distribution universal?\n\nMetchev & Hillenbrandt (2009) advocate the idea of a companion mass function with a universal power law over a wide range of separations and masses. They fit the mass-ratio distribution in solar-mass visual binaries to a power law with β = −0.39±0.36. Visual companions to stars more massive than the Sun follow a power law f (q) ∝ q β with β from −0.3 to −0.5, as established in AO imaging surveys by Shatsky & Tokovinin (2002) and Kouwenhoven et al. (2005). Certainly, these distributions do not correspond to the most simplistic case of random companion pairing, as described in detail by Kouwenhoven et al. (2009).\n\nOur survey of triples points to a flat massratio distribution with β ∼ 0. The errors are large, so the difference with visual binaries is not yet significant. The mass-ratio distribution in closer, spectroscopic binaries is approximately flat (Mazeh et al. 2003;Halbwachs et al. 2003), similar to the mass-ratio distribution in close subsystems of triple stars (Fig. 5). If we confirm on a larger sample that sub-systems in triples do differ from pure binaries in their mass-ratio distribution, its universality will be seriously challenged.\n\n\nBrown-dwarf desert\n\nThe mass function of companions to binary and multiple stars is radically different from the Initial Mass Function. Therefore, the fraction of lowmass companions is much smaller than the fraction of low-mass stars in the field. This fact, known as brown dwarf (BD) desert, is well established (Grether & Lineweaver 2006). Adopting a flat mass-ratio distribution for our sample, also typical for spectroscopic binaries, we estimate that only 7% of companions to a solar-mass star will be in the BD regime. Considering the companion frequency of 0.12 in the studied separation range, one BD companion per ∼100 primaries is expected. Our result thus suggests that the BD desert in multiple systems extends to separations of 100 AU.\n\nThe BD desert and the tendency to equal companion masses are explained in the current starformation scenario by continuing accretion onto a binary. Whenever a low-mass companion forms by fragmentation of a massive disk, continuing accretion usually brings its mass into stellar regime (e.g. Kratter et al. 2010). Companions of substellar masses should be intrinsically rare. At the same time the orbital drag (inward migration) reduces the separation, producing close sub-systems. It appears that about half of the companions settle in close orbits with separation < 1 AU, the rest of them remain in wider orbits.\n\n\nConclusions\n\nWe surveyed 33 nearby wide binaries with solartype primaries and found 7 resolved sub-systems, most of them previously unknown. We derive the fraction of sub-systems with projected separations from 5 to 100 AU to be 0.12 ± 0.04. The massratio distribution in these sub-systems appears to be flat, to within a large statistical error.\n\nThe sample can be increased 6 times if we continue observations with NICI and extend them to the Northern sky using some other AO facility. Such investment of 5 nights on 8-m telescopes will enable to reduce the size of the confidence intervals in Fig. 4 by 2.5 times. Then, it will become possible to distinguish between the negative power index β ∼ −0.3 typical for resolved binaries and flat mass-ratio distribution β ∼ 0 characteristic of close binaries.\n\nThe joint fraction of both visual and spectroscopic sub-systems in wide solar-type binaries is about 0.25. In consequence, the majority of the components are single, and provide an accomodating environment to host planetary systems.\n\nWe thank M. Chun and F. Rigaut for helping us diagnosing a NICI AO misbehaviour, and F. Rantakyro for his assistance in the preparation and execution of the observations. We are thankful for discussions on the manuscript to W. Brandner and N. Huelamo. This work used the NASA's Astrophysics Data System, data products from the 2MASS funded by the NASA and the NSF, and the SIMBAD database maintained by the University of Strasbourg, France. The comments of the anonymous referee helped us to improve the presentation.\n\nFacilities: Gemini:South (NICI)\n\n1\nBased on observations obtained at the Gemini Observatory (Program ID GS-2009B-Q-49), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência e Tecnologia (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina).\n\nFig. 1 .\n1-Mosaic of resolved-component images in the r-channel, labeled by the Hipparcos numbers. Each frame is 81 pixels (1.46 ′′ ) across, oriented North down and East to the right. The intensity scaling is negative logarithmic. The first 7 frames show all binary components where we detected a close companion. The last frame shows the PSF of the unresolved target HIP 37332, with the ghost marked. The faint companions to HIP 24711A and HIP 49520B are marked by arrows. Note that at good AO correction the typical features of the NICI PSF become visible. These are particularly the double diffraction spikes and the four spots on the first Airy ring, giving it a \"boxy\" look. These features are caused by the oversized vanes of a spider mask placed in the pupil plane to control scatter.\n\nFig. 2 .\n2-Estimated 5σ companion detection limit (solid line) in the red channel for HIP 37332. The simulated companions are over-plotted as asterisks (detected) or empty squares (missed). The dotted line shows the 2-segment model.\n\nFig. 3 .\n3-Mass ratios q = M 2 /M 1 of 7 the resolved sub-systems versus their projected separations. The shaded region indicates incomplete detections (light grey corresponds to detection probabilities from 0.5 to 0.9, dark grey from 0.1 to 0.5).\n\nFig. 4 .\n4-Contours of the S-function in the ǫ, β space corresponding to the 68% and 90% confidence areas. The minimum is marked with a cross.\n\nFig. 5 .\n5-Histograms of the mass-ratios in inner systems of triple stars. Solid lines correspond to the 123 close pairs and dashed lines to the 61 wide pairs. The vertical bars show ± √ N statistical errors for each bin of width 0.2. All points are slightly displaced horizontally for clarity.\n\nTable 1\n1Data on observed targetsHIP (A/B) \nπHIP \nρ \nr \nComponent A \nComponent B \nDate \nRem \nmas \n\n′′ \n\nAU \nV \nH \nKs \nV \nH \nKs \n2000+ \n\n\nTable 2\n2Measured parameters of known and new pairsHIP \nComp \nRed channel (2.272 µm) \nBlue channel (1.587 µm) \nRem \nθ \nρ \n∆m \nθ \nρ \n∆m \n\n• \n′′ \n\nmag \n\n• \n′′ \n\nmag \n\n10579 \nA,B \n288.58 \n6.710 \n0.47 \n288.50 \n6.703 \n0.71 \n20552 \nA,B \n247.35 \n5.405 \n0.16 \n247.55 \n5.403 \n0.26 \nsat? \n21963 \nA,B \n92.33 \n8.184 \n2.35 \n92.26 \n8.169 \n2.64 \n23926 \nA,B \n259.93 \n10.255 \n2.35 \n259.96 \n10.246 \n2.66 \nsat \n24711 \nAa,Ab \n17.70 \n0.670 \n5.28 \n17.71 \n0.663 \n5.66 \nnew \n24711 \nB,C \n324.85 \n11.135 \n6.29 \n324.87 \n11.132 \n6.45 \nopt \n27922 \nA,B \n19.35 \n10.742 \n1.55 \n19.37 \n10.727 \n1.88 \n28790 \nA,B \n215.28 \n5.945 \n1.29 \n215.18 \n5.941 \n1.48 \nsat \n30158 \nA,B \n7.48 \n7.096 \n1.18 \n7.38 \n7.093 \n1.33 \n32644 \nA,B \n160.49 \n5.029 \n0.72 \n160.43 \n5.039 \n0.73 \nsat? \n37735 \nA,B \n128.43 \n6.210 \n1.57 \n128.49 \n6.200 \n1.76 \n39409 \nA,B \n329.64 \n5.187 \n0.10 \n329.67 \n5.193 \n0.14 \n43652 \nA,Ca \n128.47 \n9.206 \n6.16 \n128.48 \n9.188 \n6.41 \nopt \n43652 \nCa,Cb \n328.86 \n0.124 \n0.67 \n330.90 \n0.126 \n0.64 \nopt \n43947 \nAa,Ab \n260.79 \n0.424 \n0.31 \n260.78 \n0.423 \n0.30 \nnew \n43947 \nBa,Bb \n152.16 \n0.289 \n1.65 \n152.16 \n0.289 \n1.73 \nnew \n43947 \nAa,Ba \n202.89 \n10.377 \n0.82 \n202.78 \n10.365 \n0.90 \n44804 \nAa,B \n136.01 \n7.175 \n1.16 \n136.08 \n7.167 \n1.21 \n44804 \nAa,Ab \n313.09 \n0.453 \n1.84 \n313.09 \n0.452 \n2.02 \nnew \n45734 \nAa,Ab \n132.69 \n0.119 \n0.38 \n133.33 \n0.117 \n0.36 \n45734 \nAa,B \n193.63 \n9.077 \n0.71 \n193.65 \n9.076 \n1.23 \n45940 \nA,B \n69.28 \n6.713 \n1.72 \n69.29 \n6.705 \n1.98 \n47836 \nA,C \n328.29 \n9.809 \n5.31 \n328.36 \n9.797 \n5.77 \nopt \n49520 \nA,B \n326.95 \n9.552 \n0.29 \n326.95 \n9.543 \n0.44 \n49520 \nBa,Bb \n44.21 \n0.213 \n3.61 \n46.69 \n0.210 \n3.40 \nnew \n50883 \nA,B \n192.65 \n6.839 \n2.33 \n192.57 \n6.830 \n2.57 \n50883 \nA,C \n299.87 \n7.379 \n6.23 \n299.82 \n7.363 \n6.55 \nopt \n55288 \nA,B \n253.94 \n9.462 \n0.54 \n253.96 \n9.456 \n0.84 \n58241 \nA,C \n118.32 \n4.015 \n6.38 \n118.09 \n4.008 \n6.52 \n64498 \nAa,B \n302.20 \n9.464 \n1.59 \n302.24 \n9.458 \n1.91 \n64498 \nAa,Ab \n296.55 \n0.372 \n1.76 \n296.31 \n0.370 \n2.01 \nnew \n65176 \nA,B \n96.89 \n5.168 \n0.61 \n96.95 \n5.164 \n0.62 \n\n\nTable 3 :\n3Estimated parameters of the 7 observed \nsub-systems in wide binaries \nHIP \nComp M 1 \nM 2 \nr \nP * \nM ⊙ M ⊙ AU \nyr \n24711 Aa,Ab 1.04 0.15 43.6 260 \n43947 Aa,Ab 0.64 0.59 15.9 60 \n43947 Ba,Bb 0.55 0.24 10.9 40 \n44804 Aa,Ab 0.95 0.61 29.3 130 \n45734 Aa,Ab 0.88 0.81 6.8 \n14 \n49520 Ba,Bb 0.92 0.26 12.6 40 \n64498 Aa,Ab 1.09 0.71 20.6 70 \n\n\n\nTable 4 :\n4Confidence intervalsInterval \nǫ \nβ \nMin. 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A. 2009, ApJS, 181, 62\n"},"corpus_id":{"kind":"number","value":73629363,"string":"73,629,363"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"1006.1253\",\n \"mag\": 2085464080,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1088/0004-6256/140/2/510\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/1006.1253v1.pdf\"\n ],\n \"pdf_sha\": \"f7144aa52d1c979febd9b6c98f3be65b48ddc156\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/1006.1253\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Sub-systems in nearby solar-type wide binaries 1"},"authors":{"kind":"list like","value":["Andrei Tokovinin atokovinin@ctio.noao.edu \nGemini Observatory\nCerro Tololo Inter-American Observatory\nCasilla 603, Southern Operations Center, c/o AURA, Casilla 603, La SerenaLa SerenaChile, Chile","Markus Hartung mhartung@gemini.edu \nGemini Observatory\nCerro Tololo Inter-American Observatory\nCasilla 603, Southern Operations Center, c/o AURA, Casilla 603, La SerenaLa SerenaChile, Chile","Thomas L Hayward thayward@gemini.edu \nGemini Observatory\nCerro Tololo Inter-American Observatory\nCasilla 603, Southern Operations Center, c/o AURA, Casilla 603, La SerenaLa SerenaChile, Chile"],"string":"[\n \"Andrei Tokovinin atokovinin@ctio.noao.edu \\nGemini Observatory\\nCerro Tololo Inter-American Observatory\\nCasilla 603, Southern Operations Center, c/o AURA, Casilla 603, La SerenaLa SerenaChile, Chile\",\n \"Markus Hartung mhartung@gemini.edu \\nGemini Observatory\\nCerro Tololo Inter-American Observatory\\nCasilla 603, Southern Operations Center, c/o AURA, Casilla 603, La SerenaLa SerenaChile, Chile\",\n \"Thomas L Hayward thayward@gemini.edu \\nGemini Observatory\\nCerro Tololo Inter-American Observatory\\nCasilla 603, Southern Operations Center, c/o AURA, Casilla 603, La SerenaLa SerenaChile, Chile\"\n]","isConversational":false},"abstract":{"kind":"string","value":"We conducted a deep survey of resolved sub-systems among wide binaries with solar-type components within 67 pc from the Sun. Images of 61 stars in the K and H bands were obtained with the NICI adaptive-optics instrument on the 8-m Gemini-South telescope. Our maximum detectable magnitude difference is about 5 m and 7.8 m at 0.15 ′′ and 0.9 ′′ separations, respectively. This enables a complete census of sub-systems with stellar companions in the projected separation range from 5 to 100 AU. Out of 7 such companions found in our sample, only one was known previously. We determine that the fraction of sub-systems with projected separations above 5 AU is 0.12 ± 0.04 and that the distribution of their mass ratio is flat, with a power-law index 0.2 ± 0.5. Comparing this with the properties of closer spectroscopic sub-systems (separations below 1 AU), it appears that the mass-ratio distribution does not depend on the separation. The frequency of sub-systems in the separation ranges below 1 AU and between 5 and 100 AU is similar, about 0.15. Unbiased statistics of multiplicity higher than two, advanced by this work, provide constraints on star-formation theory."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"The quest for exo-planets has attracted attention to stars in the solar neighbourhood. A complete understanding of their formation requires a quantitative and predictive description of stellar multiplicity. Empirical data on stellar multiplicity serve as a benchmark for star formation theories, for example to compare with recent large-scale hydrodynamical simulations (Bate 2009). Although multiplicity statistics of solar-type dwarfs were assumed to be known since the survey of Duquennoy & Mayor (1991), in fact that survey did not constrain multiplicities higher than two because (i) the sample of 164 stars was too small and (ii) the discovery of wide binaries relied on traditional visual techniques. Raghavan (2009) shows that Duquennoy & Mayor (1991) missed half of higher-order multiples within 25 pc from the Sun.\nSeveral studies of large samples of nearby stars are now in progress. Radial-velocity (RV) techniques ensure a secure detection of all short-period stellar companions (see summary in Grether & Lineweaver 2006). However, for wider separations (> 10 AU) RV variations are too slow or too small, and such companions are better discovered by direct imaging. Several recent deep imaging surveys, summarized by Metchev & Hillenbrandt (2009), aimed to detect substellar or planetary-mass companions to nearby stars, obtaining statistics of stellar companions as a by-product. The typical sample size of imaging surveys is still small, on the order of 100 targets.\nMost imaging and RV surveys for low-mass companions explicitly avoid known visual binaries (e.g. Chauvin et al. 2010), so important starformation effects may not be represented in their samples. For example, catalogued triples show that the mass-ratio distributions in close inner sub-systems discovered spectroscopically and in wider \"visual\" sub-systems are radically different (see Section 5). One of our goals is to investigate if this is a selection effect or a genuine feature of the star-formation process.\nIn order to improve the multiplicity statistics of a well-defined and sufficiently large distancelimited sample, we focus on ∼5000 dwarfs with 0.5 < V − I < 0.8 within 67 pc from the Sun. According to preliminary estimates, there should be some 400 triples and 100 quadruples in this sample. Surveying such a large sample to detect those multiples with a reasonably deep and known completeness limit is a large task; here we address only a small part of it. Instead of avoiding known binaries, we purposefuly selected wide visual binaries from this sample. Some of these targets are also monitored in RV, giving complementary constraints on sub-systems with short periods (Desidera et al. 2006).\nWe imaged the chosen wide binaries with Adaptive Optics (AO) to study the properties of inner sub-systems. AO imaging on 8-m class telescopes provides nearly diffraction-limited resolution of 0.06 ′′ in the near-infrared, which corresponds to a projected orbital radius of 4 AU for the farthest target stars in our sample, filling an important gap between RV and seeing-limited imaging surveys.\nOur sample is detailed in Section 2. Observations, data reduction, and detection limits are described in Section 3, the results are presented in Section 4. Sections 5 and 6 contain the discussion and conclusions, respectively."},{"title":"The sample of wide binaries","content":"The wide binaries chosen for our survey belong to a large sample of nearby solar-type dwarfs (Nsample) selected from the latest version of Hipparcos catalog (van Leeuwen 2007) by the following criteria:\n1. trigonometric parallax π HIP < 15 mas (within 67 pc from the Sun, distance modulus < 4.12 m ); 2. color 0.5 < V − I < 0.8 (this corresponds approximately to spectral types from F5V to K0V);\n3. unevolved, satisfying the condition M Hp > 9(V − I) − 3.5, where M Hp is the absolute magnitude in the Hipparcos band calculated with π HIP .\nThere are 5040 stars (i.e. Hipparcos catalog entries) satisfying these conditions. Some of these stars with erroneous parallaxes or colors will be eventually removed from the sample, but this contamination is expected to be minor. The Nsample defined above is reasonably complete because the cumulative object count versus distance d follows the d 3 law, deviating from it by only 10% at d = 67 pc. Visual binaries with separations 5 ′′ < ρ < 20 ′′ and one or both components belonging to the Nsample were selected for AO observations from the Washington Double-Star Catalog, WDS (Mason et al. 2001). Additionally, we require each pair to have at least 3 observations and magnitudes of both components listed in the WDS, and that apparent motion of the wide pair be compatible with π HIP . As most targets have large proper motions, relative stability of the binary position over time permits rejection of optical pairs. An additional check was made by placing the components on the (M V , V − K) color-magnitude diagram (CMD). There are ∼200 wide binaries in this list, 101 with negative declinations.\nIn Table 1 we list basic data on the components of the 33 wide pairs observed in this run. Column 1 gives the Hipparcos numbers of the primary and, in some cases, secondary components. The next three columns contain π HIP , binary separation ρ, and projected separation r = ρ/π HIP . Then follow the magnitudes of the primary (A) and sec- ondary (B) components in the V , H, and K s bands as listed in the WDS and 2MASS (Cutri et al. 2003). The last two columns contain the date of observations and remarks where we indicate 5 cases when only one component was observed and 3 cases when the coronagraphic mask was used.\n3. Observations and data reduction 3.1. Observing procedure\nThe Near-Infrared Coronagraphic Imager, NICI, on the Gemini South telescope is an 85-element curvature AO instrument based on natural guide stars (Toomey & Ftaclas 2003;Chun et al. 2008). We used NICI in normal (non-coronagraphic) mode, as a classical AO system. However, simultaneous imaging in the two infrared bands H and K offered by NICI helps to distinguish faint companions from static speckles (the radial distance of a static speckle scales with wavelength while the position of a real detection does not change). The two detectors have 1024 2 pixels of 18 mas (milliarcseconds) size, covering a square field of 18 ′′ .\nTo avoid saturation, we used narrow-band filters with central wavelengths 2.272 µm and 1.587 µm in the red and blue channels, respectively. The minimum possible exposure time is 0.38 s, still causing saturation for bright targets. We observed three of the brightest stars with the coronagraphic mask of 0.32 ′′ radius, using in this instance broadband K and H filters. For the remaining bright targets we allowed moderate saturation. As the tradeoff between slower observations with more complex data reduction in coronagraphic mode and slightly saturated PSFs is not obvious, we tried both.\nTypically, each target was observed at 5-6 dithered positions for a total accumulation time of about 5 min. The dither offsets were designed to take images of both wide-binary components in one frame whenever possible (ρ < 10 ′′ ) while closing the AO loop on the primary. Components of wider pairs were observed separately, except secondaries with V > 12 m , too faint to serve as AO guide stars.\nOf the two nights allocated to this program, one was lost due to a technical problem. Three targets were observed on 2009 December 10 in service mode. The remaining objects were observed on the night of 2010 January 31 to February 1 in classical mode. The seeing during that night was good. As measured from the NICI AO loop data, it ranged between 0.34 ′′ and 0.94 ′′ , with a median of 0.50 ′′ (at 500 nm, not corrected to the zenith). High AO performance was achieved (see next Section). The transparency was variable (light cirrus). Out of 202 binary components accessible to NICI, we were able to observe 61, or 30%. The efficiency was high, with an average of 12 min per target including telescope slew, tuning of the primary mirror, acquisition of the object on the science camera, closing the AO loop, and data-taking."},{"title":"Data processing","content":"The data were processed in a standard way using custom IDL programs. Individual images written to separate FITS files were combined into data cubes, separately in red and blue channels. The sky image was obtained for each target as a median of the cube, then subtracted from each plane of the cube. After correcting for bad pixels and dividing by the flat field, the images were re-centered by cross-correlation and median-combined.\nMost frames contain just one or two bright point sources corresponding to the components of the wide binary. The position of each point source was recorded by \"clicking\" on its image, then refined by fitting a 2-dimensional Gaussian to the central part of the Point Spread Function (PSF) within 5-pixel radius. The ratio of the central intensity of this Gaussian to the total flux in a 0.9 ′′ diameter circle gives an estimate of the Strehl ratio. The typical Strehls 2 are 43% and 22% in the red and blue channel, respectively, with maximum values reaching 54% and 38%. The median Full Width at Half Maximum (FWHM) of the PSF core was 65 mas and 57 mas in the red and blue channels, respectively. The diffraction limit λ/D is 63 mas and 44 mas for the effective aperture diameter D = 7.5 m set by the coronagraphic mask. Figure 1 shows the mosaic of 7 close subsystems found around some primary components of wide binaries. Most companions are rather obvious, well above the detection threshold. The PSFs contain also a faint secondary reflex at 13 pixels (0.24 ′′ ), with a magnitude difference in the red channel of about 4.3 m . This ghost (illustrated in Fig. 1 in the case of HIP 37332) is detected in all targets, and can be taken advantage of as a fiducial (see Section 3.4). Its position is the same in both channels, whereas the static speckle pattern scales in proportion to the wavelength."},{"title":"Measurement of binaries","content":"To measure the relative position and flux ratio of wide binaries more precisely, we fit the scaled and shifted image of the primary component (A) to the secondary (B). Such fits work very well, leaving only small residuals, and produce mutually consistent measurements in the two channels. The rms scatter between the wide-binary positions measured in two NICI channels is 5.5 mas in both radial and tangential directions.\nWe checked the pixel scale and detector orientation by comparing the measured positions of 16 wide pairs (the widest in Table 2 after exclusion of HIP 21963, 43947 and 50883, which were not used) with the positions listed in the Hipparcos Double and Multiple Star Catalogue (ESA 1997). We did not find any systematic effect above noise and therefore used the nominal pixel scale of 18.0 mas for both channels. We found that the detector orientation of the channels differs by 1.01 • ± 0.01 • , so 1.0 • were added to measured angles in the blue channel.\nFor close pairs where the PSFs overlap, we use a variant of iterative blind deconvolution. Good preliminary estimates of the relative component's position and intensity are already available from the Gaussian fits of the PSF cores. The image Fourier Transform (FT) is divided by the FT of the binary with preliminary parameters to get an estimate of the PSF. Negative parts of this PSF are set to zero and it is replaced by its azimuthal average at radial distances over 5 pixels from the center. The binary parameters are refined by fitting again the image with this synthetic PSF. Then the process is repeated, obtaining a second approximation to the PSF and using it again (without azimuthal averaging) to obtain the final binary parameters. The success of this procedure is verified by inspecting the resulting PSF, which should be \"clean,\" with the least possible trace of the companion. To determine detection limits, we use mostly the data from the red NICI channel where the detection of low-mass components is deeper because of the smaller magnitude difference and larger Strehl ratios. The maximum magnitude difference of detectable companions ∆m lim is estimated by calculating the rms flux variations σ(ρ) in circular zones of increasing radii ρ centered on the main component. It is assumed that a detectable companion will have maximum intensity exceeding 5σ, therefore"},{"title":"Detection limits","content":"(1)\nThis assumption is, of course, simplistic. We do not account for the existence of two simultaneous images in red and blue channels and for the good knowledge of the PSF from other companions of the same target or from other targets observed shortly before or after. Nevertheless, this simple strategy of estimating detection limits is sufficient for our purpose and has been tried before with good results (e.g. Tokovinin et al. 2006). The limits computed for HIP 37322 are shown in Fig. 2. A bump is produced in the curves at ρ ≈ 13 pixels by the ghost and by fixed speckle pattern (see the PSF in Fig. 1). To confirm the validity of (1), we performed Monte-Carlo simulations, adding artificial companions within ±1.5 m from the estimated ∆m lim . Secure detection of the ghost (∆m = 4.3, ρ = 0.24 ′′ ) in all images gives confidence in our estimation of ∆m lim .\nThe 5σ detection limits ∆m lim (ρ) were fitted by two straight lines intersecting at ρ = 8 pixels (0.144 ′′ ). The second segment extends from 8 to 50 pixels (0.9 ′′ ), but the region between 8 and 16 pixel radius affected by the ghost and strong speckle is avoided in the fit. At ρ > 50 pixels ∆m lim is assumed to be constant. Only two numbers ∆m(8) and ∆m(50) adequately describe the actual ∆m lim (ρ) curves, with typical rms error of only 0.3 m (excluding the zone between 8 and 16 pixel radii where the 5σ method does not properly account for the static speckle).\nDetection limits were determined by the above procedure for all targets, although the Monte Carlo simulation checks were only performed for a subset. The two detection parameters ∆m(8) and ∆m(50) vary little between targets. Their median values are 5.02 and 7.79, respectively; the quartiles are (4.29, 5.22) and (7.03, 8.06). In the following we apply median detection parameters to all targets, including the three coronagraphic images where the actual limits should be deeper.\nThe detection limits found here match other similar studies done with \"standard\" AO. For example, companion detection was complete down to ∆K < 3.9 at 0.14 ′′ and ∆K < 7.9 at 0.9 ′′ in (Tokovinin et al. 2006). Similar results were obtained by Eggenberger et al. (2007). The deep companion survey at Palomar and Keck conducted by Metchev & Hillenbrandt (2009) also reached ∆K < 8 at 0.9 ′′ . Note that typical detection limits with NICI are much deeper for dedicated planet search programs. In these cases long exposure times and techniques such as angular and spectral differential imaging are applied Artigau et al. 2008)."},{"title":"Results","content":""},{"title":"Companion data","content":"We have found 6 new close companions, one previously known companion, and several faint optical companions (Fig. 1). Table 2 lists the relative positions and magnitude differences in known and new pairs. Each pair is identified by the HIP number and component designations. The measured position angles in degrees, separations in arcseconds, and magnitude differences are listed separately for the two NICI channels.\nThe large rms difference between Hipparcos and our measurements of the wide pairs (130 mas in ρ and 1.05 • in θ) is mostly caused by their motion between the two epochs. The comparison with the relative positions of the same binaries deduced from the 2MASS Point Source Catalog (Cutri et al. 2003) has confirmed that no systematic corrections are needed and has shown a scatter of similar magnitude.\nRemarks in the last column of Table 2 indicate cases when the image was saturated (the relative photometry may be compromised), newly detected close companions, and likely optical companions. These faint companions with separations of a few arcseconds would be dynamically unstable within wide binaries, should they be physical 3 ."},{"title":"Notes on individual systems","content":"HIP 28790 has a common-proper-motion companion C at 196 ′′ .\nHIP 36165 A is suspected to be a spectroscopic binary (SB).\nHIP 43947 is unexpectedly discovered to be a resolved quadruple system. Both components are located ∼ 2 m below the Main Sequence in the (M V , V − K) CMD, possibly because the Hipparcos parallax measurement was biased by the motions in the sub-systems. The position and ∆m of the wide pair Aa,Ba is derived from the Gaussian fits, therefore it may be less accurate.\nHIP 45734: This is a young pre-Mainsequence system and X-ray source RX J0919.4-7738. It projects on the Chamaeleon star-forming region, but is located much closer than the corresponding association. Covino et al. (1997) found from a single spectrum that component B (southern) is a double-lined SB, with a large RV difference from A (see also Desidera et al. 2006). A itself is a close visual binary KOH83 resolved in March 1996 by Köhler (2001). He measured the position (173.9 • , 0.109 ′′ ) and ∆K = 0.45. This is the only previously known resolved sub-system among the observed stars. Follow-up observations will soon lead to the visual orbit and mass measurement, adding another empirical point to the evolutionary tracks of young stars.\nHIP 49520: The faint companion Bb is close to the detection limit (see Fig. 1). Yet, it is securely detected in both channels and stands up clearly in residuals when fitting the image of B with the image of A.\nHIP 50638/36: The wide pair has been observed since 1835; it is definitely physical. Component B is variable. For this reason, probably, it is located on the CMD some 6 m below the MS, while A is on the MS with the parallax π HIP = 83 mas listed in Table 1. According to van Leeuwen (2007), the parallaxes and proper motions of the two components are discordant, but have large errors. Future RV measurements can settle the controversy.\nHIP 59021: Component A is a suspected SB and also an astrometric binary (acceleration solution in Hipparcos). Non-detection of the subsystem with NICI means that its period is likely less than 10 y, or it is a white dwarf."},{"title":"Estimation of the mass ratios","content":"K-band absolute magnitudes M K of all companions were computed based on the photometry from 2MASS (see Table 1) and trigonometric parallaxes π HIP from Hipparcos. For resolved pairs, we assume that the ∆m in the red NICI channel correspond to magnitude differences in the K-band. The masses were estimated from M K using the empirical relations of Henry & McCarthy (1993). We also tried a quadratic approximation of relation between mass and M K from Girardi et al. (2000). The relative difference with Henry & McCarthy (1993) is less than 10% for masses above 0.1 M ⊙ . We checked that the influence of using either of these relations on the final result is not critical. Table 3 lists the estimated masses of the companions in the resolved sub-systems, their projected separations r = ρ/π HIP and order-of-magnitude period estimates P * = r 3/2 (M 1 + M 1 ) −1/2 . Figure 3 shows the mass ratios q = M 2 /M 1 in the resolved inner sub-systems as a function of projected separation. The median parallax of observed targets is 19 mas, so the upper limit of the separation range, 100 AU, corresponds to ρ ≈ 1.9 ′′ . Maximum separations of dynamically stable sub-systems should not exceed 100 AU in most cases because the projected separations of the wide binaries range from 150 to 1000 AU (Ta- ). Indeed, all detected sub-systems have r < 44 AU. The lower limit of 5 AU corresponds to ρ ≈ 0.1 ′′ , well above the NICI resolution limit.\nThe detection limits in the red channel were converted into mass ratios and averaged over the 61 observed components, producing a smooth detection probability p det (r, q). Contours of this function at 0.1, 0.5, and 0.9 levels are over-plotted in Fig. 3. We see that the chosen region of the (r, q) parameter space is well covered with NICI."},{"title":"Statistical analysis","content":"The purpose of this study is to reach statistical conclusions about the frequency and the properties of sub-systems in solar-type wide binaries. Considering the small number of components, we keep this analysis at a basic level, and assume that the mass ratios in secondary sub-systems have a power-law distribution independent of separation:\n(2)\nThis model has only two free parameters, the total companion frequency ǫ and the power-law index β. Neglecting incomplete detection, the companion count gives ǫ = 7/61 = 0.12 ± 0.04. The distribution of the companion projected separations in the considered range from 5 to 100 AU matters only because the detection limits depend on r. We assume that f (log r) = const.\nThe input data are the (r, q) values for K = 7 sub-systems, the total number of surveyed targets N = 61, and the array of detection probabilities p det (r, q). We find the estimates of parameters (ǫ, β) by the Maximum Likelihood (ML) method (see e.g. Tokovinin et al. 2006, Apendix B). The likelihood function L is a product of the probabilities to obtain observations for each component (non-detections for N − K systems or detections of K subsystems with mass ratios q k ).\nThe minimum of S = −2 ln L is sought. This function depends on the input data and the model parameters as\nThe summation over k is done for the 7 detected companions. The companion probability f 0 equals the product f (r, q)p det (r, q) averaged over the considered part of the parameter space (r, q). If the detection probability p det equals one everywhere, the result is f 0 = ǫ, otherwise f 0 < ǫ. The contours of S in the parameter space define confidence limits: ∆S = 1 corresponds to the 68% interval (\"1σ\"), ∆S = 2.71 to 90% and ∆S = 4 to 95%, in direct analogy with the Gaussian probability distribution. We verified the ML method on a simulated data set. The input parameters were recovered and the confidence intervals were as expected.\nMinimization of (3) leads to (ǫ, β) = (0.12, 0.18). If we ignore the detection limits and set p det = 1, the result changes little: (ǫ, β) = (0.11, 0.40). The companion fraction 0.12 is not different from its naive estimate. As to the power index β, the ML points to a uniform distribution in q as being most likely, although the uncertainty is large. The confidence areas of the parameters are shown in Fig. 4. Considering that the contours are not very elongated, we can determine the 68% and 90% confidence intervals for each parameter by a simple cross-section through the minimum (Table 4). The data are marginally compatible with β ∼ −0.5, in which case the fraction of sub-systems may exceed 0.2 (see the lower-right extension of the 90% contour in Fig. 4)."},{"title":"Discussion","content":""},{"title":"Completeness of the existing catalog","content":"The fact that, in a sample of only 33 systems, six of the seven observed companions are new, shows that current knowledge of multiplicity among nearby solar-type systems is very incomplete, and that near-IR AO imaging is a powerful technique for detecting new companions. We plot in Fig. 5 the mass-ratio distribution in the inner sub-systems of known triple stars. The data are extracted from the updated version of the Multiple-Star Catalog (MSC, Tokovinin 1997), where component masses are estimated by a variety of methods. We selected 220 systems within 200 pc from the Sun with masses of primary components from 0.5 to 1.5 M ⊙ . They were sub-divided by the separation in the inner subsystems in two groups, 123 close (semi-major axis < 1 AU) and 61 wide (between 5 and 100 AU). The remaining 36 pairs have even wider separations. The mass-ratio distributions in these two groups are markedly different, but our un-biased survey results in a flat mass-ratio distribution, within errors. Therefore, the difference seen in Fig. 5 is caused by observational selection. Given an approximately equal number of close and wide sub-systems with large mass ratios (two last bins in Fig. 5) and assuming that the mass-ratio distributions in these two groups are indeed similar, we expect that the number of close and wide sub-systems (with separations below 1 AU and between 5 and 100 AU, respectively) are similar, too. The completeness of the MSC for wide subsystems is thus about 1/2."},{"title":"Companion fraction","content":"Is the presence of a wide binary companion influencing the frequency of inner sub-systems? To answer this question, we compare our results on triples with surveys of solar-type binaries. Unfortunately, binary-star statistics are usually derived regardless of higher-order multiplicity, so a cleaner comparison must await new study of a large unbiased sample.\nThe separation range from 5 to 100 AU explored here corresponds to orbital periods between 10 y and 1000 y. According to Duquennoy & Mayor (1991), about 20% of solar-type dwarfs have companions with such periods. In the actual, narrower range of projected separations from 6.8 AU to 44 AU, 12% of solar-type dwarfs have companions. Therefore, the frequency of sub-systems around components of wide binaries is similar to the frequency of binary companions to single stars in the same separation range.\nThe frequency of sub-systems with periods below 3 y in visual multiples has been determined by Tokovinin & Smekhov (2002) to be between 11% and 18%. Again, it turned out to equal within observational errors the companion frequency among single dwarfs of similar masses in the solar neighborhood and in some open clusters. Desidera et al. (2006) surveyed RVs of 56 wide visual pairs and found that 0.135 ± 0.05 fraction of companions have spectroscopic sub-systems.\nTherefore, one might surmise that the presence of a wide companion does not affect the formation of inner, closer sub-systems in a wide range of separations. This question is however far from having been settled for both stellar and planetary subsystems (see the discussion in Eggenberger et al. 2007). We know that almost all close binaries with periods below 3 d do have additional outer companions, although the mass ratios of close binaries with and without tertiary companions have identical distributions (Tokovinin et al. 2006)."},{"title":"Is the mass-ratio distribution universal?","content":"Metchev & Hillenbrandt (2009) advocate the idea of a companion mass function with a universal power law over a wide range of separations and masses. They fit the mass-ratio distribution in solar-mass visual binaries to a power law with β = −0.39±0.36. Visual companions to stars more massive than the Sun follow a power law f (q) ∝ q β with β from −0.3 to −0.5, as established in AO imaging surveys by Shatsky & Tokovinin (2002) and Kouwenhoven et al. (2005). Certainly, these distributions do not correspond to the most simplistic case of random companion pairing, as described in detail by Kouwenhoven et al. (2009).\nOur survey of triples points to a flat massratio distribution with β ∼ 0. The errors are large, so the difference with visual binaries is not yet significant. The mass-ratio distribution in closer, spectroscopic binaries is approximately flat (Mazeh et al. 2003;Halbwachs et al. 2003), similar to the mass-ratio distribution in close subsystems of triple stars (Fig. 5). If we confirm on a larger sample that sub-systems in triples do differ from pure binaries in their mass-ratio distribution, its universality will be seriously challenged."},{"title":"Brown-dwarf desert","content":"The mass function of companions to binary and multiple stars is radically different from the Initial Mass Function. Therefore, the fraction of lowmass companions is much smaller than the fraction of low-mass stars in the field. This fact, known as brown dwarf (BD) desert, is well established (Grether & Lineweaver 2006). Adopting a flat mass-ratio distribution for our sample, also typical for spectroscopic binaries, we estimate that only 7% of companions to a solar-mass star will be in the BD regime. Considering the companion frequency of 0.12 in the studied separation range, one BD companion per ∼100 primaries is expected. Our result thus suggests that the BD desert in multiple systems extends to separations of 100 AU.\nThe BD desert and the tendency to equal companion masses are explained in the current starformation scenario by continuing accretion onto a binary. Whenever a low-mass companion forms by fragmentation of a massive disk, continuing accretion usually brings its mass into stellar regime (e.g. Kratter et al. 2010). Companions of substellar masses should be intrinsically rare. At the same time the orbital drag (inward migration) reduces the separation, producing close sub-systems. It appears that about half of the companions settle in close orbits with separation < 1 AU, the rest of them remain in wider orbits."},{"title":"Conclusions","content":"We surveyed 33 nearby wide binaries with solartype primaries and found 7 resolved sub-systems, most of them previously unknown. We derive the fraction of sub-systems with projected separations from 5 to 100 AU to be 0.12 ± 0.04. The massratio distribution in these sub-systems appears to be flat, to within a large statistical error.\nThe sample can be increased 6 times if we continue observations with NICI and extend them to the Northern sky using some other AO facility. Such investment of 5 nights on 8-m telescopes will enable to reduce the size of the confidence intervals in Fig. 4 by 2.5 times. Then, it will become possible to distinguish between the negative power index β ∼ −0.3 typical for resolved binaries and flat mass-ratio distribution β ∼ 0 characteristic of close binaries.\nThe joint fraction of both visual and spectroscopic sub-systems in wide solar-type binaries is about 0.25. In consequence, the majority of the components are single, and provide an accomodating environment to host planetary systems.\nWe thank M. Chun and F. Rigaut for helping us diagnosing a NICI AO misbehaviour, and F. Rantakyro for his assistance in the preparation and execution of the observations. We are thankful for discussions on the manuscript to W. Brandner and N. Huelamo. This work used the NASA's Astrophysics Data System, data products from the 2MASS funded by the NASA and the NSF, and the SIMBAD database maintained by the University of Strasbourg, France. The comments of the anonymous referee helped us to improve the presentation.\nFacilities: Gemini:South (NICI)"},{"title":"1","content":""},{"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":""},{"title":"Table 3 :","content":""},{"title":"Table 4 :","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"The quest for exo-planets has attracted attention to stars in the solar neighbourhood. A complete understanding of their formation requires a quantitative and predictive description of stellar multiplicity. Empirical data on stellar multiplicity serve as a benchmark for star formation theories, for example to compare with recent large-scale hydrodynamical simulations (Bate 2009). Although multiplicity statistics of solar-type dwarfs were assumed to be known since the survey of Duquennoy & Mayor (1991), in fact that survey did not constrain multiplicities higher than two because (i) the sample of 164 stars was too small and (ii) the discovery of wide binaries relied on traditional visual techniques. Raghavan (2009) shows that Duquennoy & Mayor (1991) missed half of higher-order multiples within 25 pc from the Sun.\\nSeveral studies of large samples of nearby stars are now in progress. Radial-velocity (RV) techniques ensure a secure detection of all short-period stellar companions (see summary in Grether & Lineweaver 2006). However, for wider separations (> 10 AU) RV variations are too slow or too small, and such companions are better discovered by direct imaging. Several recent deep imaging surveys, summarized by Metchev & Hillenbrandt (2009), aimed to detect substellar or planetary-mass companions to nearby stars, obtaining statistics of stellar companions as a by-product. The typical sample size of imaging surveys is still small, on the order of 100 targets.\\nMost imaging and RV surveys for low-mass companions explicitly avoid known visual binaries (e.g. Chauvin et al. 2010), so important starformation effects may not be represented in their samples. For example, catalogued triples show that the mass-ratio distributions in close inner sub-systems discovered spectroscopically and in wider \\\"visual\\\" sub-systems are radically different (see Section 5). One of our goals is to investigate if this is a selection effect or a genuine feature of the star-formation process.\\nIn order to improve the multiplicity statistics of a well-defined and sufficiently large distancelimited sample, we focus on ∼5000 dwarfs with 0.5 < V − I < 0.8 within 67 pc from the Sun. According to preliminary estimates, there should be some 400 triples and 100 quadruples in this sample. Surveying such a large sample to detect those multiples with a reasonably deep and known completeness limit is a large task; here we address only a small part of it. Instead of avoiding known binaries, we purposefuly selected wide visual binaries from this sample. Some of these targets are also monitored in RV, giving complementary constraints on sub-systems with short periods (Desidera et al. 2006).\\nWe imaged the chosen wide binaries with Adaptive Optics (AO) to study the properties of inner sub-systems. AO imaging on 8-m class telescopes provides nearly diffraction-limited resolution of 0.06 ′′ in the near-infrared, which corresponds to a projected orbital radius of 4 AU for the farthest target stars in our sample, filling an important gap between RV and seeing-limited imaging surveys.\\nOur sample is detailed in Section 2. Observations, data reduction, and detection limits are described in Section 3, the results are presented in Section 4. Sections 5 and 6 contain the discussion and conclusions, respectively.\"\n },\n {\n \"title\": \"The sample of wide binaries\",\n \"content\": \"The wide binaries chosen for our survey belong to a large sample of nearby solar-type dwarfs (Nsample) selected from the latest version of Hipparcos catalog (van Leeuwen 2007) by the following criteria:\\n1. trigonometric parallax π HIP < 15 mas (within 67 pc from the Sun, distance modulus < 4.12 m ); 2. color 0.5 < V − I < 0.8 (this corresponds approximately to spectral types from F5V to K0V);\\n3. unevolved, satisfying the condition M Hp > 9(V − I) − 3.5, where M Hp is the absolute magnitude in the Hipparcos band calculated with π HIP .\\nThere are 5040 stars (i.e. Hipparcos catalog entries) satisfying these conditions. Some of these stars with erroneous parallaxes or colors will be eventually removed from the sample, but this contamination is expected to be minor. The Nsample defined above is reasonably complete because the cumulative object count versus distance d follows the d 3 law, deviating from it by only 10% at d = 67 pc. Visual binaries with separations 5 ′′ < ρ < 20 ′′ and one or both components belonging to the Nsample were selected for AO observations from the Washington Double-Star Catalog, WDS (Mason et al. 2001). Additionally, we require each pair to have at least 3 observations and magnitudes of both components listed in the WDS, and that apparent motion of the wide pair be compatible with π HIP . As most targets have large proper motions, relative stability of the binary position over time permits rejection of optical pairs. An additional check was made by placing the components on the (M V , V − K) color-magnitude diagram (CMD). There are ∼200 wide binaries in this list, 101 with negative declinations.\\nIn Table 1 we list basic data on the components of the 33 wide pairs observed in this run. Column 1 gives the Hipparcos numbers of the primary and, in some cases, secondary components. The next three columns contain π HIP , binary separation ρ, and projected separation r = ρ/π HIP . Then follow the magnitudes of the primary (A) and sec- ondary (B) components in the V , H, and K s bands as listed in the WDS and 2MASS (Cutri et al. 2003). The last two columns contain the date of observations and remarks where we indicate 5 cases when only one component was observed and 3 cases when the coronagraphic mask was used.\\n3. Observations and data reduction 3.1. Observing procedure\\nThe Near-Infrared Coronagraphic Imager, NICI, on the Gemini South telescope is an 85-element curvature AO instrument based on natural guide stars (Toomey & Ftaclas 2003;Chun et al. 2008). We used NICI in normal (non-coronagraphic) mode, as a classical AO system. However, simultaneous imaging in the two infrared bands H and K offered by NICI helps to distinguish faint companions from static speckles (the radial distance of a static speckle scales with wavelength while the position of a real detection does not change). The two detectors have 1024 2 pixels of 18 mas (milliarcseconds) size, covering a square field of 18 ′′ .\\nTo avoid saturation, we used narrow-band filters with central wavelengths 2.272 µm and 1.587 µm in the red and blue channels, respectively. The minimum possible exposure time is 0.38 s, still causing saturation for bright targets. We observed three of the brightest stars with the coronagraphic mask of 0.32 ′′ radius, using in this instance broadband K and H filters. For the remaining bright targets we allowed moderate saturation. As the tradeoff between slower observations with more complex data reduction in coronagraphic mode and slightly saturated PSFs is not obvious, we tried both.\\nTypically, each target was observed at 5-6 dithered positions for a total accumulation time of about 5 min. The dither offsets were designed to take images of both wide-binary components in one frame whenever possible (ρ < 10 ′′ ) while closing the AO loop on the primary. Components of wider pairs were observed separately, except secondaries with V > 12 m , too faint to serve as AO guide stars.\\nOf the two nights allocated to this program, one was lost due to a technical problem. Three targets were observed on 2009 December 10 in service mode. The remaining objects were observed on the night of 2010 January 31 to February 1 in classical mode. The seeing during that night was good. As measured from the NICI AO loop data, it ranged between 0.34 ′′ and 0.94 ′′ , with a median of 0.50 ′′ (at 500 nm, not corrected to the zenith). High AO performance was achieved (see next Section). The transparency was variable (light cirrus). Out of 202 binary components accessible to NICI, we were able to observe 61, or 30%. The efficiency was high, with an average of 12 min per target including telescope slew, tuning of the primary mirror, acquisition of the object on the science camera, closing the AO loop, and data-taking.\"\n },\n {\n \"title\": \"Data processing\",\n \"content\": \"The data were processed in a standard way using custom IDL programs. Individual images written to separate FITS files were combined into data cubes, separately in red and blue channels. The sky image was obtained for each target as a median of the cube, then subtracted from each plane of the cube. After correcting for bad pixels and dividing by the flat field, the images were re-centered by cross-correlation and median-combined.\\nMost frames contain just one or two bright point sources corresponding to the components of the wide binary. The position of each point source was recorded by \\\"clicking\\\" on its image, then refined by fitting a 2-dimensional Gaussian to the central part of the Point Spread Function (PSF) within 5-pixel radius. The ratio of the central intensity of this Gaussian to the total flux in a 0.9 ′′ diameter circle gives an estimate of the Strehl ratio. The typical Strehls 2 are 43% and 22% in the red and blue channel, respectively, with maximum values reaching 54% and 38%. The median Full Width at Half Maximum (FWHM) of the PSF core was 65 mas and 57 mas in the red and blue channels, respectively. The diffraction limit λ/D is 63 mas and 44 mas for the effective aperture diameter D = 7.5 m set by the coronagraphic mask. Figure 1 shows the mosaic of 7 close subsystems found around some primary components of wide binaries. Most companions are rather obvious, well above the detection threshold. The PSFs contain also a faint secondary reflex at 13 pixels (0.24 ′′ ), with a magnitude difference in the red channel of about 4.3 m . This ghost (illustrated in Fig. 1 in the case of HIP 37332) is detected in all targets, and can be taken advantage of as a fiducial (see Section 3.4). Its position is the same in both channels, whereas the static speckle pattern scales in proportion to the wavelength.\"\n },\n {\n \"title\": \"Measurement of binaries\",\n \"content\": \"To measure the relative position and flux ratio of wide binaries more precisely, we fit the scaled and shifted image of the primary component (A) to the secondary (B). Such fits work very well, leaving only small residuals, and produce mutually consistent measurements in the two channels. The rms scatter between the wide-binary positions measured in two NICI channels is 5.5 mas in both radial and tangential directions.\\nWe checked the pixel scale and detector orientation by comparing the measured positions of 16 wide pairs (the widest in Table 2 after exclusion of HIP 21963, 43947 and 50883, which were not used) with the positions listed in the Hipparcos Double and Multiple Star Catalogue (ESA 1997). We did not find any systematic effect above noise and therefore used the nominal pixel scale of 18.0 mas for both channels. We found that the detector orientation of the channels differs by 1.01 • ± 0.01 • , so 1.0 • were added to measured angles in the blue channel.\\nFor close pairs where the PSFs overlap, we use a variant of iterative blind deconvolution. Good preliminary estimates of the relative component's position and intensity are already available from the Gaussian fits of the PSF cores. The image Fourier Transform (FT) is divided by the FT of the binary with preliminary parameters to get an estimate of the PSF. Negative parts of this PSF are set to zero and it is replaced by its azimuthal average at radial distances over 5 pixels from the center. The binary parameters are refined by fitting again the image with this synthetic PSF. Then the process is repeated, obtaining a second approximation to the PSF and using it again (without azimuthal averaging) to obtain the final binary parameters. The success of this procedure is verified by inspecting the resulting PSF, which should be \\\"clean,\\\" with the least possible trace of the companion. To determine detection limits, we use mostly the data from the red NICI channel where the detection of low-mass components is deeper because of the smaller magnitude difference and larger Strehl ratios. The maximum magnitude difference of detectable companions ∆m lim is estimated by calculating the rms flux variations σ(ρ) in circular zones of increasing radii ρ centered on the main component. It is assumed that a detectable companion will have maximum intensity exceeding 5σ, therefore\"\n },\n {\n \"title\": \"Detection limits\",\n \"content\": \"(1)\\nThis assumption is, of course, simplistic. We do not account for the existence of two simultaneous images in red and blue channels and for the good knowledge of the PSF from other companions of the same target or from other targets observed shortly before or after. Nevertheless, this simple strategy of estimating detection limits is sufficient for our purpose and has been tried before with good results (e.g. Tokovinin et al. 2006). The limits computed for HIP 37322 are shown in Fig. 2. A bump is produced in the curves at ρ ≈ 13 pixels by the ghost and by fixed speckle pattern (see the PSF in Fig. 1). To confirm the validity of (1), we performed Monte-Carlo simulations, adding artificial companions within ±1.5 m from the estimated ∆m lim . Secure detection of the ghost (∆m = 4.3, ρ = 0.24 ′′ ) in all images gives confidence in our estimation of ∆m lim .\\nThe 5σ detection limits ∆m lim (ρ) were fitted by two straight lines intersecting at ρ = 8 pixels (0.144 ′′ ). The second segment extends from 8 to 50 pixels (0.9 ′′ ), but the region between 8 and 16 pixel radius affected by the ghost and strong speckle is avoided in the fit. At ρ > 50 pixels ∆m lim is assumed to be constant. Only two numbers ∆m(8) and ∆m(50) adequately describe the actual ∆m lim (ρ) curves, with typical rms error of only 0.3 m (excluding the zone between 8 and 16 pixel radii where the 5σ method does not properly account for the static speckle).\\nDetection limits were determined by the above procedure for all targets, although the Monte Carlo simulation checks were only performed for a subset. The two detection parameters ∆m(8) and ∆m(50) vary little between targets. Their median values are 5.02 and 7.79, respectively; the quartiles are (4.29, 5.22) and (7.03, 8.06). In the following we apply median detection parameters to all targets, including the three coronagraphic images where the actual limits should be deeper.\\nThe detection limits found here match other similar studies done with \\\"standard\\\" AO. For example, companion detection was complete down to ∆K < 3.9 at 0.14 ′′ and ∆K < 7.9 at 0.9 ′′ in (Tokovinin et al. 2006). Similar results were obtained by Eggenberger et al. (2007). The deep companion survey at Palomar and Keck conducted by Metchev & Hillenbrandt (2009) also reached ∆K < 8 at 0.9 ′′ . Note that typical detection limits with NICI are much deeper for dedicated planet search programs. In these cases long exposure times and techniques such as angular and spectral differential imaging are applied Artigau et al. 2008).\"\n },\n {\n \"title\": \"Results\",\n \"content\": \"\"\n },\n {\n \"title\": \"Companion data\",\n \"content\": \"We have found 6 new close companions, one previously known companion, and several faint optical companions (Fig. 1). Table 2 lists the relative positions and magnitude differences in known and new pairs. Each pair is identified by the HIP number and component designations. The measured position angles in degrees, separations in arcseconds, and magnitude differences are listed separately for the two NICI channels.\\nThe large rms difference between Hipparcos and our measurements of the wide pairs (130 mas in ρ and 1.05 • in θ) is mostly caused by their motion between the two epochs. The comparison with the relative positions of the same binaries deduced from the 2MASS Point Source Catalog (Cutri et al. 2003) has confirmed that no systematic corrections are needed and has shown a scatter of similar magnitude.\\nRemarks in the last column of Table 2 indicate cases when the image was saturated (the relative photometry may be compromised), newly detected close companions, and likely optical companions. These faint companions with separations of a few arcseconds would be dynamically unstable within wide binaries, should they be physical 3 .\"\n },\n {\n \"title\": \"Notes on individual systems\",\n \"content\": \"HIP 28790 has a common-proper-motion companion C at 196 ′′ .\\nHIP 36165 A is suspected to be a spectroscopic binary (SB).\\nHIP 43947 is unexpectedly discovered to be a resolved quadruple system. Both components are located ∼ 2 m below the Main Sequence in the (M V , V − K) CMD, possibly because the Hipparcos parallax measurement was biased by the motions in the sub-systems. The position and ∆m of the wide pair Aa,Ba is derived from the Gaussian fits, therefore it may be less accurate.\\nHIP 45734: This is a young pre-Mainsequence system and X-ray source RX J0919.4-7738. It projects on the Chamaeleon star-forming region, but is located much closer than the corresponding association. Covino et al. (1997) found from a single spectrum that component B (southern) is a double-lined SB, with a large RV difference from A (see also Desidera et al. 2006). A itself is a close visual binary KOH83 resolved in March 1996 by Köhler (2001). He measured the position (173.9 • , 0.109 ′′ ) and ∆K = 0.45. This is the only previously known resolved sub-system among the observed stars. Follow-up observations will soon lead to the visual orbit and mass measurement, adding another empirical point to the evolutionary tracks of young stars.\\nHIP 49520: The faint companion Bb is close to the detection limit (see Fig. 1). Yet, it is securely detected in both channels and stands up clearly in residuals when fitting the image of B with the image of A.\\nHIP 50638/36: The wide pair has been observed since 1835; it is definitely physical. Component B is variable. For this reason, probably, it is located on the CMD some 6 m below the MS, while A is on the MS with the parallax π HIP = 83 mas listed in Table 1. According to van Leeuwen (2007), the parallaxes and proper motions of the two components are discordant, but have large errors. Future RV measurements can settle the controversy.\\nHIP 59021: Component A is a suspected SB and also an astrometric binary (acceleration solution in Hipparcos). Non-detection of the subsystem with NICI means that its period is likely less than 10 y, or it is a white dwarf.\"\n },\n {\n \"title\": \"Estimation of the mass ratios\",\n \"content\": \"K-band absolute magnitudes M K of all companions were computed based on the photometry from 2MASS (see Table 1) and trigonometric parallaxes π HIP from Hipparcos. For resolved pairs, we assume that the ∆m in the red NICI channel correspond to magnitude differences in the K-band. The masses were estimated from M K using the empirical relations of Henry & McCarthy (1993). We also tried a quadratic approximation of relation between mass and M K from Girardi et al. (2000). The relative difference with Henry & McCarthy (1993) is less than 10% for masses above 0.1 M ⊙ . We checked that the influence of using either of these relations on the final result is not critical. Table 3 lists the estimated masses of the companions in the resolved sub-systems, their projected separations r = ρ/π HIP and order-of-magnitude period estimates P * = r 3/2 (M 1 + M 1 ) −1/2 . Figure 3 shows the mass ratios q = M 2 /M 1 in the resolved inner sub-systems as a function of projected separation. The median parallax of observed targets is 19 mas, so the upper limit of the separation range, 100 AU, corresponds to ρ ≈ 1.9 ′′ . Maximum separations of dynamically stable sub-systems should not exceed 100 AU in most cases because the projected separations of the wide binaries range from 150 to 1000 AU (Ta- ). Indeed, all detected sub-systems have r < 44 AU. The lower limit of 5 AU corresponds to ρ ≈ 0.1 ′′ , well above the NICI resolution limit.\\nThe detection limits in the red channel were converted into mass ratios and averaged over the 61 observed components, producing a smooth detection probability p det (r, q). Contours of this function at 0.1, 0.5, and 0.9 levels are over-plotted in Fig. 3. We see that the chosen region of the (r, q) parameter space is well covered with NICI.\"\n },\n {\n \"title\": \"Statistical analysis\",\n \"content\": \"The purpose of this study is to reach statistical conclusions about the frequency and the properties of sub-systems in solar-type wide binaries. Considering the small number of components, we keep this analysis at a basic level, and assume that the mass ratios in secondary sub-systems have a power-law distribution independent of separation:\\n(2)\\nThis model has only two free parameters, the total companion frequency ǫ and the power-law index β. Neglecting incomplete detection, the companion count gives ǫ = 7/61 = 0.12 ± 0.04. The distribution of the companion projected separations in the considered range from 5 to 100 AU matters only because the detection limits depend on r. We assume that f (log r) = const.\\nThe input data are the (r, q) values for K = 7 sub-systems, the total number of surveyed targets N = 61, and the array of detection probabilities p det (r, q). We find the estimates of parameters (ǫ, β) by the Maximum Likelihood (ML) method (see e.g. Tokovinin et al. 2006, Apendix B). The likelihood function L is a product of the probabilities to obtain observations for each component (non-detections for N − K systems or detections of K subsystems with mass ratios q k ).\\nThe minimum of S = −2 ln L is sought. This function depends on the input data and the model parameters as\\nThe summation over k is done for the 7 detected companions. The companion probability f 0 equals the product f (r, q)p det (r, q) averaged over the considered part of the parameter space (r, q). If the detection probability p det equals one everywhere, the result is f 0 = ǫ, otherwise f 0 < ǫ. The contours of S in the parameter space define confidence limits: ∆S = 1 corresponds to the 68% interval (\\\"1σ\\\"), ∆S = 2.71 to 90% and ∆S = 4 to 95%, in direct analogy with the Gaussian probability distribution. We verified the ML method on a simulated data set. The input parameters were recovered and the confidence intervals were as expected.\\nMinimization of (3) leads to (ǫ, β) = (0.12, 0.18). If we ignore the detection limits and set p det = 1, the result changes little: (ǫ, β) = (0.11, 0.40). The companion fraction 0.12 is not different from its naive estimate. As to the power index β, the ML points to a uniform distribution in q as being most likely, although the uncertainty is large. The confidence areas of the parameters are shown in Fig. 4. Considering that the contours are not very elongated, we can determine the 68% and 90% confidence intervals for each parameter by a simple cross-section through the minimum (Table 4). The data are marginally compatible with β ∼ −0.5, in which case the fraction of sub-systems may exceed 0.2 (see the lower-right extension of the 90% contour in Fig. 4).\"\n },\n {\n \"title\": \"Discussion\",\n \"content\": \"\"\n },\n {\n \"title\": \"Completeness of the existing catalog\",\n \"content\": \"The fact that, in a sample of only 33 systems, six of the seven observed companions are new, shows that current knowledge of multiplicity among nearby solar-type systems is very incomplete, and that near-IR AO imaging is a powerful technique for detecting new companions. We plot in Fig. 5 the mass-ratio distribution in the inner sub-systems of known triple stars. The data are extracted from the updated version of the Multiple-Star Catalog (MSC, Tokovinin 1997), where component masses are estimated by a variety of methods. We selected 220 systems within 200 pc from the Sun with masses of primary components from 0.5 to 1.5 M ⊙ . They were sub-divided by the separation in the inner subsystems in two groups, 123 close (semi-major axis < 1 AU) and 61 wide (between 5 and 100 AU). The remaining 36 pairs have even wider separations. The mass-ratio distributions in these two groups are markedly different, but our un-biased survey results in a flat mass-ratio distribution, within errors. Therefore, the difference seen in Fig. 5 is caused by observational selection. Given an approximately equal number of close and wide sub-systems with large mass ratios (two last bins in Fig. 5) and assuming that the mass-ratio distributions in these two groups are indeed similar, we expect that the number of close and wide sub-systems (with separations below 1 AU and between 5 and 100 AU, respectively) are similar, too. The completeness of the MSC for wide subsystems is thus about 1/2.\"\n },\n {\n \"title\": \"Companion fraction\",\n \"content\": \"Is the presence of a wide binary companion influencing the frequency of inner sub-systems? To answer this question, we compare our results on triples with surveys of solar-type binaries. Unfortunately, binary-star statistics are usually derived regardless of higher-order multiplicity, so a cleaner comparison must await new study of a large unbiased sample.\\nThe separation range from 5 to 100 AU explored here corresponds to orbital periods between 10 y and 1000 y. According to Duquennoy & Mayor (1991), about 20% of solar-type dwarfs have companions with such periods. In the actual, narrower range of projected separations from 6.8 AU to 44 AU, 12% of solar-type dwarfs have companions. Therefore, the frequency of sub-systems around components of wide binaries is similar to the frequency of binary companions to single stars in the same separation range.\\nThe frequency of sub-systems with periods below 3 y in visual multiples has been determined by Tokovinin & Smekhov (2002) to be between 11% and 18%. Again, it turned out to equal within observational errors the companion frequency among single dwarfs of similar masses in the solar neighborhood and in some open clusters. Desidera et al. (2006) surveyed RVs of 56 wide visual pairs and found that 0.135 ± 0.05 fraction of companions have spectroscopic sub-systems.\\nTherefore, one might surmise that the presence of a wide companion does not affect the formation of inner, closer sub-systems in a wide range of separations. This question is however far from having been settled for both stellar and planetary subsystems (see the discussion in Eggenberger et al. 2007). We know that almost all close binaries with periods below 3 d do have additional outer companions, although the mass ratios of close binaries with and without tertiary companions have identical distributions (Tokovinin et al. 2006).\"\n },\n {\n \"title\": \"Is the mass-ratio distribution universal?\",\n \"content\": \"Metchev & Hillenbrandt (2009) advocate the idea of a companion mass function with a universal power law over a wide range of separations and masses. They fit the mass-ratio distribution in solar-mass visual binaries to a power law with β = −0.39±0.36. Visual companions to stars more massive than the Sun follow a power law f (q) ∝ q β with β from −0.3 to −0.5, as established in AO imaging surveys by Shatsky & Tokovinin (2002) and Kouwenhoven et al. (2005). Certainly, these distributions do not correspond to the most simplistic case of random companion pairing, as described in detail by Kouwenhoven et al. (2009).\\nOur survey of triples points to a flat massratio distribution with β ∼ 0. The errors are large, so the difference with visual binaries is not yet significant. The mass-ratio distribution in closer, spectroscopic binaries is approximately flat (Mazeh et al. 2003;Halbwachs et al. 2003), similar to the mass-ratio distribution in close subsystems of triple stars (Fig. 5). If we confirm on a larger sample that sub-systems in triples do differ from pure binaries in their mass-ratio distribution, its universality will be seriously challenged.\"\n },\n {\n \"title\": \"Brown-dwarf desert\",\n \"content\": \"The mass function of companions to binary and multiple stars is radically different from the Initial Mass Function. Therefore, the fraction of lowmass companions is much smaller than the fraction of low-mass stars in the field. This fact, known as brown dwarf (BD) desert, is well established (Grether & Lineweaver 2006). Adopting a flat mass-ratio distribution for our sample, also typical for spectroscopic binaries, we estimate that only 7% of companions to a solar-mass star will be in the BD regime. Considering the companion frequency of 0.12 in the studied separation range, one BD companion per ∼100 primaries is expected. Our result thus suggests that the BD desert in multiple systems extends to separations of 100 AU.\\nThe BD desert and the tendency to equal companion masses are explained in the current starformation scenario by continuing accretion onto a binary. Whenever a low-mass companion forms by fragmentation of a massive disk, continuing accretion usually brings its mass into stellar regime (e.g. Kratter et al. 2010). Companions of substellar masses should be intrinsically rare. At the same time the orbital drag (inward migration) reduces the separation, producing close sub-systems. It appears that about half of the companions settle in close orbits with separation < 1 AU, the rest of them remain in wider orbits.\"\n },\n {\n \"title\": \"Conclusions\",\n \"content\": \"We surveyed 33 nearby wide binaries with solartype primaries and found 7 resolved sub-systems, most of them previously unknown. We derive the fraction of sub-systems with projected separations from 5 to 100 AU to be 0.12 ± 0.04. The massratio distribution in these sub-systems appears to be flat, to within a large statistical error.\\nThe sample can be increased 6 times if we continue observations with NICI and extend them to the Northern sky using some other AO facility. Such investment of 5 nights on 8-m telescopes will enable to reduce the size of the confidence intervals in Fig. 4 by 2.5 times. Then, it will become possible to distinguish between the negative power index β ∼ −0.3 typical for resolved binaries and flat mass-ratio distribution β ∼ 0 characteristic of close binaries.\\nThe joint fraction of both visual and spectroscopic sub-systems in wide solar-type binaries is about 0.25. In consequence, the majority of the components are single, and provide an accomodating environment to host planetary systems.\\nWe thank M. Chun and F. Rigaut for helping us diagnosing a NICI AO misbehaviour, and F. Rantakyro for his assistance in the preparation and execution of the observations. We are thankful for discussions on the manuscript to W. Brandner and N. Huelamo. This work used the NASA's Astrophysics Data System, data products from the 2MASS funded by the NASA and the NSF, and the SIMBAD database maintained by the University of Strasbourg, France. The comments of the anonymous referee helped us to improve the presentation.\\nFacilities: Gemini:South (NICI)\"\n },\n {\n \"title\": \"1\",\n \"content\": \"\"\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 {\n \"title\": \"Table 3 :\",\n \"content\": \"\"\n },\n {\n \"title\": \"Table 4 :\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Based on observations obtained at the Gemini Observatory (Program ID GS-2009B-Q-49), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência e Tecnologia (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina).","content":"1\nBased on observations obtained at the Gemini Observatory (Program ID GS-2009B-Q-49), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência e Tecnologia (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina)."},{"id":"","caption":"-Mosaic of resolved-component images in the r-channel, labeled by the Hipparcos numbers. Each frame is 81 pixels (1.46 ′′ ) across, oriented North down and East to the right. The intensity scaling is negative logarithmic. The first 7 frames show all binary components where we detected a close companion. The last frame shows the PSF of the unresolved target HIP 37332, with the ghost marked. The faint companions to HIP 24711A and HIP 49520B are marked by arrows. Note that at good AO correction the typical features of the NICI PSF become visible. These are particularly the double diffraction spikes and the four spots on the first Airy ring, giving it a \"boxy\" look. These features are caused by the oversized vanes of a spider mask placed in the pupil plane to control scatter.","content":"Fig. 1 .\n1-Mosaic of resolved-component images in the r-channel, labeled by the Hipparcos numbers. Each frame is 81 pixels (1.46 ′′ ) across, oriented North down and East to the right. The intensity scaling is negative logarithmic. The first 7 frames show all binary components where we detected a close companion. The last frame shows the PSF of the unresolved target HIP 37332, with the ghost marked. The faint companions to HIP 24711A and HIP 49520B are marked by arrows. Note that at good AO correction the typical features of the NICI PSF become visible. These are particularly the double diffraction spikes and the four spots on the first Airy ring, giving it a \"boxy\" look. These features are caused by the oversized vanes of a spider mask placed in the pupil plane to control scatter."},{"id":"","caption":"-Estimated 5σ companion detection limit (solid line) in the red channel for HIP 37332. The simulated companions are over-plotted as asterisks (detected) or empty squares (missed). The dotted line shows the 2-segment model.","content":"Fig. 2 .\n2-Estimated 5σ companion detection limit (solid line) in the red channel for HIP 37332. The simulated companions are over-plotted as asterisks (detected) or empty squares (missed). The dotted line shows the 2-segment model."},{"id":"","caption":"-Mass ratios q = M 2 /M 1 of 7 the resolved sub-systems versus their projected separations. The shaded region indicates incomplete detections (light grey corresponds to detection probabilities from 0.5 to 0.9, dark grey from 0.1 to 0.5).","content":"Fig. 3 .\n3-Mass ratios q = M 2 /M 1 of 7 the resolved sub-systems versus their projected separations. The shaded region indicates incomplete detections (light grey corresponds to detection probabilities from 0.5 to 0.9, dark grey from 0.1 to 0.5)."},{"id":"","caption":"-Contours of the S-function in the ǫ, β space corresponding to the 68% and 90% confidence areas. The minimum is marked with a cross.","content":"Fig. 4 .\n4-Contours of the S-function in the ǫ, β space corresponding to the 68% and 90% confidence areas. The minimum is marked with a cross."},{"id":"","caption":"-Histograms of the mass-ratios in inner systems of triple stars. Solid lines correspond to the 123 close pairs and dashed lines to the 61 wide pairs. The vertical bars show ± √ N statistical errors for each bin of width 0.2. All points are slightly displaced horizontally for clarity.","content":"Fig. 5 .\n5-Histograms of the mass-ratios in inner systems of triple stars. Solid lines correspond to the 123 close pairs and dashed lines to the 61 wide pairs. The vertical bars show ± √ N statistical errors for each bin of width 0.2. All points are slightly displaced horizontally for clarity."},{"id":"","caption":"Data on observed targets","content":"Table 1\n1Data on observed targetsHIP (A/B) \nπHIP \nρ \nr \nComponent A \nComponent B \nDate \nRem \nmas \n\n′′ \n\nAU \nV \nH \nKs \nV \nH \nKs \n2000+"},{"id":"","caption":"Measured parameters of known and new pairs","content":"Table 2\n2Measured parameters of known and new pairsHIP \nComp \nRed channel (2.272 µm) \nBlue channel (1.587 µm) \nRem \nθ \nρ \n∆m \nθ \nρ \n∆m \n\n• \n′′ \n\nmag \n\n• \n′′ \n\nmag \n\n10579 \nA,B \n288.58 \n6.710 \n0.47 \n288.50 \n6.703 \n0.71 \n20552 \nA,B \n247.35 \n5.405 \n0.16 \n247.55 \n5.403 \n0.26 \nsat? \n21963 \nA,B \n92.33 \n8.184 \n2.35 \n92.26 \n8.169 \n2.64 \n23926 \nA,B \n259.93 \n10.255 \n2.35 \n259.96 \n10.246 \n2.66 \nsat \n24711 \nAa,Ab \n17.70 \n0.670 \n5.28 \n17.71 \n0.663 \n5.66 \nnew \n24711 \nB,C \n324.85 \n11.135 \n6.29 \n324.87 \n11.132 \n6.45 \nopt \n27922 \nA,B \n19.35 \n10.742 \n1.55 \n19.37 \n10.727 \n1.88 \n28790 \nA,B \n215.28 \n5.945 \n1.29 \n215.18 \n5.941 \n1.48 \nsat \n30158 \nA,B \n7.48 \n7.096 \n1.18 \n7.38 \n7.093 \n1.33 \n32644 \nA,B \n160.49 \n5.029 \n0.72 \n160.43 \n5.039 \n0.73 \nsat? \n37735 \nA,B \n128.43 \n6.210 \n1.57 \n128.49 \n6.200 \n1.76 \n39409 \nA,B \n329.64 \n5.187 \n0.10 \n329.67 \n5.193 \n0.14 \n43652 \nA,Ca \n128.47 \n9.206 \n6.16 \n128.48 \n9.188 \n6.41 \nopt \n43652 \nCa,Cb \n328.86 \n0.124 \n0.67 \n330.90 \n0.126 \n0.64 \nopt \n43947 \nAa,Ab \n260.79 \n0.424 \n0.31 \n260.78 \n0.423 \n0.30 \nnew \n43947 \nBa,Bb \n152.16 \n0.289 \n1.65 \n152.16 \n0.289 \n1.73 \nnew \n43947 \nAa,Ba \n202.89 \n10.377 \n0.82 \n202.78 \n10.365 \n0.90 \n44804 \nAa,B \n136.01 \n7.175 \n1.16 \n136.08 \n7.167 \n1.21 \n44804 \nAa,Ab \n313.09 \n0.453 \n1.84 \n313.09 \n0.452 \n2.02 \nnew \n45734 \nAa,Ab \n132.69 \n0.119 \n0.38 \n133.33 \n0.117 \n0.36 \n45734 \nAa,B \n193.63 \n9.077 \n0.71 \n193.65 \n9.076 \n1.23 \n45940 \nA,B \n69.28 \n6.713 \n1.72 \n69.29 \n6.705 \n1.98 \n47836 \nA,C \n328.29 \n9.809 \n5.31 \n328.36 \n9.797 \n5.77 \nopt \n49520 \nA,B \n326.95 \n9.552 \n0.29 \n326.95 \n9.543 \n0.44 \n49520 \nBa,Bb \n44.21 \n0.213 \n3.61 \n46.69 \n0.210 \n3.40 \nnew \n50883 \nA,B \n192.65 \n6.839 \n2.33 \n192.57 \n6.830 \n2.57 \n50883 \nA,C \n299.87 \n7.379 \n6.23 \n299.82 \n7.363 \n6.55 \nopt \n55288 \nA,B \n253.94 \n9.462 \n0.54 \n253.96 \n9.456 \n0.84 \n58241 \nA,C \n118.32 \n4.015 \n6.38 \n118.09 \n4.008 \n6.52 \n64498 \nAa,B \n302.20 \n9.464 \n1.59 \n302.24 \n9.458 \n1.91 \n64498 \nAa,Ab \n296.55 \n0.372 \n1.76 \n296.31 \n0.370 \n2.01 \nnew \n65176 \nA,B \n96.89 \n5.168 \n0.61 \n96.95 \n5.164 \n0.62"},{"id":"","caption":"Confidence intervals","content":"Table 3 :\n3Estimated parameters of the 7 observed \nsub-systems in wide binaries \nHIP \nComp M 1 \nM 2 \nr \nP * \nM ⊙ M ⊙ AU \nyr \n24711 Aa,Ab 1.04 0.15 43.6 260 \n43947 Aa,Ab 0.64 0.59 15.9 60 \n43947 Ba,Bb 0.55 0.24 10.9 40 \n44804 Aa,Ab 0.95 0.61 29.3 130 \n45734 Aa,Ab 0.88 0.81 6.8 \n14 \n49520 Ba,Bb 0.92 0.26 12.6 40 \n64498 Aa,Ab 1.09 0.71 20.6 70"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Based on observations obtained at the Gemini Observatory (Program ID GS-2009B-Q-49), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência e Tecnologia (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina).\",\n \"content\": \"1\\nBased on observations obtained at the Gemini Observatory (Program ID GS-2009B-Q-49), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência e Tecnologia (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Mosaic of resolved-component images in the r-channel, labeled by the Hipparcos numbers. Each frame is 81 pixels (1.46 ′′ ) across, oriented North down and East to the right. The intensity scaling is negative logarithmic. The first 7 frames show all binary components where we detected a close companion. The last frame shows the PSF of the unresolved target HIP 37332, with the ghost marked. The faint companions to HIP 24711A and HIP 49520B are marked by arrows. Note that at good AO correction the typical features of the NICI PSF become visible. These are particularly the double diffraction spikes and the four spots on the first Airy ring, giving it a \\\"boxy\\\" look. These features are caused by the oversized vanes of a spider mask placed in the pupil plane to control scatter.\",\n \"content\": \"Fig. 1 .\\n1-Mosaic of resolved-component images in the r-channel, labeled by the Hipparcos numbers. Each frame is 81 pixels (1.46 ′′ ) across, oriented North down and East to the right. The intensity scaling is negative logarithmic. The first 7 frames show all binary components where we detected a close companion. The last frame shows the PSF of the unresolved target HIP 37332, with the ghost marked. The faint companions to HIP 24711A and HIP 49520B are marked by arrows. Note that at good AO correction the typical features of the NICI PSF become visible. These are particularly the double diffraction spikes and the four spots on the first Airy ring, giving it a \\\"boxy\\\" look. These features are caused by the oversized vanes of a spider mask placed in the pupil plane to control scatter.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Estimated 5σ companion detection limit (solid line) in the red channel for HIP 37332. The simulated companions are over-plotted as asterisks (detected) or empty squares (missed). The dotted line shows the 2-segment model.\",\n \"content\": \"Fig. 2 .\\n2-Estimated 5σ companion detection limit (solid line) in the red channel for HIP 37332. The simulated companions are over-plotted as asterisks (detected) or empty squares (missed). The dotted line shows the 2-segment model.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Mass ratios q = M 2 /M 1 of 7 the resolved sub-systems versus their projected separations. The shaded region indicates incomplete detections (light grey corresponds to detection probabilities from 0.5 to 0.9, dark grey from 0.1 to 0.5).\",\n \"content\": \"Fig. 3 .\\n3-Mass ratios q = M 2 /M 1 of 7 the resolved sub-systems versus their projected separations. The shaded region indicates incomplete detections (light grey corresponds to detection probabilities from 0.5 to 0.9, dark grey from 0.1 to 0.5).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Contours of the S-function in the ǫ, β space corresponding to the 68% and 90% confidence areas. The minimum is marked with a cross.\",\n \"content\": \"Fig. 4 .\\n4-Contours of the S-function in the ǫ, β space corresponding to the 68% and 90% confidence areas. The minimum is marked with a cross.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"-Histograms of the mass-ratios in inner systems of triple stars. Solid lines correspond to the 123 close pairs and dashed lines to the 61 wide pairs. The vertical bars show ± √ N statistical errors for each bin of width 0.2. All points are slightly displaced horizontally for clarity.\",\n \"content\": \"Fig. 5 .\\n5-Histograms of the mass-ratios in inner systems of triple stars. Solid lines correspond to the 123 close pairs and dashed lines to the 61 wide pairs. The vertical bars show ± √ N statistical errors for each bin of width 0.2. All points are slightly displaced horizontally for clarity.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Data on observed targets\",\n \"content\": \"Table 1\\n1Data on observed targetsHIP (A/B) \\nπHIP \\nρ \\nr \\nComponent A \\nComponent B \\nDate \\nRem \\nmas \\n\\n′′ \\n\\nAU \\nV \\nH \\nKs \\nV \\nH \\nKs \\n2000+\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Measured parameters of known and new pairs\",\n \"content\": \"Table 2\\n2Measured parameters of known and new pairsHIP \\nComp \\nRed channel (2.272 µm) \\nBlue channel (1.587 µm) \\nRem \\nθ \\nρ \\n∆m \\nθ \\nρ \\n∆m \\n\\n• \\n′′ \\n\\nmag \\n\\n• \\n′′ \\n\\nmag \\n\\n10579 \\nA,B \\n288.58 \\n6.710 \\n0.47 \\n288.50 \\n6.703 \\n0.71 \\n20552 \\nA,B \\n247.35 \\n5.405 \\n0.16 \\n247.55 \\n5.403 \\n0.26 \\nsat? \\n21963 \\nA,B \\n92.33 \\n8.184 \\n2.35 \\n92.26 \\n8.169 \\n2.64 \\n23926 \\nA,B \\n259.93 \\n10.255 \\n2.35 \\n259.96 \\n10.246 \\n2.66 \\nsat \\n24711 \\nAa,Ab \\n17.70 \\n0.670 \\n5.28 \\n17.71 \\n0.663 \\n5.66 \\nnew \\n24711 \\nB,C \\n324.85 \\n11.135 \\n6.29 \\n324.87 \\n11.132 \\n6.45 \\nopt \\n27922 \\nA,B \\n19.35 \\n10.742 \\n1.55 \\n19.37 \\n10.727 \\n1.88 \\n28790 \\nA,B \\n215.28 \\n5.945 \\n1.29 \\n215.18 \\n5.941 \\n1.48 \\nsat \\n30158 \\nA,B \\n7.48 \\n7.096 \\n1.18 \\n7.38 \\n7.093 \\n1.33 \\n32644 \\nA,B \\n160.49 \\n5.029 \\n0.72 \\n160.43 \\n5.039 \\n0.73 \\nsat? \\n37735 \\nA,B \\n128.43 \\n6.210 \\n1.57 \\n128.49 \\n6.200 \\n1.76 \\n39409 \\nA,B \\n329.64 \\n5.187 \\n0.10 \\n329.67 \\n5.193 \\n0.14 \\n43652 \\nA,Ca \\n128.47 \\n9.206 \\n6.16 \\n128.48 \\n9.188 \\n6.41 \\nopt \\n43652 \\nCa,Cb \\n328.86 \\n0.124 \\n0.67 \\n330.90 \\n0.126 \\n0.64 \\nopt \\n43947 \\nAa,Ab \\n260.79 \\n0.424 \\n0.31 \\n260.78 \\n0.423 \\n0.30 \\nnew \\n43947 \\nBa,Bb \\n152.16 \\n0.289 \\n1.65 \\n152.16 \\n0.289 \\n1.73 \\nnew \\n43947 \\nAa,Ba \\n202.89 \\n10.377 \\n0.82 \\n202.78 \\n10.365 \\n0.90 \\n44804 \\nAa,B \\n136.01 \\n7.175 \\n1.16 \\n136.08 \\n7.167 \\n1.21 \\n44804 \\nAa,Ab \\n313.09 \\n0.453 \\n1.84 \\n313.09 \\n0.452 \\n2.02 \\nnew \\n45734 \\nAa,Ab \\n132.69 \\n0.119 \\n0.38 \\n133.33 \\n0.117 \\n0.36 \\n45734 \\nAa,B \\n193.63 \\n9.077 \\n0.71 \\n193.65 \\n9.076 \\n1.23 \\n45940 \\nA,B \\n69.28 \\n6.713 \\n1.72 \\n69.29 \\n6.705 \\n1.98 \\n47836 \\nA,C \\n328.29 \\n9.809 \\n5.31 \\n328.36 \\n9.797 \\n5.77 \\nopt \\n49520 \\nA,B \\n326.95 \\n9.552 \\n0.29 \\n326.95 \\n9.543 \\n0.44 \\n49520 \\nBa,Bb \\n44.21 \\n0.213 \\n3.61 \\n46.69 \\n0.210 \\n3.40 \\nnew \\n50883 \\nA,B \\n192.65 \\n6.839 \\n2.33 \\n192.57 \\n6.830 \\n2.57 \\n50883 \\nA,C \\n299.87 \\n7.379 \\n6.23 \\n299.82 \\n7.363 \\n6.55 \\nopt \\n55288 \\nA,B \\n253.94 \\n9.462 \\n0.54 \\n253.96 \\n9.456 \\n0.84 \\n58241 \\nA,C \\n118.32 \\n4.015 \\n6.38 \\n118.09 \\n4.008 \\n6.52 \\n64498 \\nAa,B \\n302.20 \\n9.464 \\n1.59 \\n302.24 \\n9.458 \\n1.91 \\n64498 \\nAa,Ab \\n296.55 \\n0.372 \\n1.76 \\n296.31 \\n0.370 \\n2.01 \\nnew \\n65176 \\nA,B \\n96.89 \\n5.168 \\n0.61 \\n96.95 \\n5.164 \\n0.62\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Confidence intervals\",\n \"content\": \"Table 3 :\\n3Estimated parameters of the 7 observed \\nsub-systems in wide binaries \\nHIP \\nComp M 1 \\nM 2 \\nr \\nP * \\nM ⊙ M ⊙ AU \\nyr \\n24711 Aa,Ab 1.04 0.15 43.6 260 \\n43947 Aa,Ab 0.64 0.59 15.9 60 \\n43947 Ba,Bb 0.55 0.24 10.9 40 \\n44804 Aa,Ab 0.95 0.61 29.3 130 \\n45734 Aa,Ab 0.88 0.81 6.8 \\n14 \\n49520 Ba,Bb 0.92 0.26 12.6 40 \\n64498 Aa,Ab 1.09 0.71 20.6 70\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"E Artigau, B A Biller, Z Wahhaj, M Hartung, T L Hayward, L M Close, M R Chun, M C Liu, G Trancho, F Rigaut, Ground-based and Airborne Instrumentation for Astronomy II. Ed. McLean, I. S. & Casali, M. Proc. SPIE. 701466Artigau, E., Biller, B. A., Wahhaj, Z., Hartung, M., Hayward, T. L., Close, L. M., Chun, M. R., Liu, M. C., Trancho, G., Rigaut, F. et al. 2008, in: Ground-based and Airborne Instru- mentation for Astronomy II. Ed. McLean, I. S. & Casali, M. Proc. SPIE, 7014, 66","doi":""},{"text":". M R Bate, MNRAS. 392590Bate, M. R. 2009, MNRAS, 392, 590","doi":""},{"text":"B Biller, É Artigaut, Z Wahhaj, M Hartung, M.-C Liu, L M Close, M R Chun, C Ftaclas, D W Toomey, T Hayward, Proc. SPIE. SPIE7015184Biller, B., Artigaut,É., Wahhaj, Z., Hartung, M., Liu, M.-C., Close, L. M., Chun, M. R., Ftaclas, C., Toomey, D. W., & Hayward, T. 2008, In: Proc. SPIE, 7015, 184","doi":""},{"text":". G Chauvin, A.-M Lagrange, M Bonavita, B Zuckerman, C Dumas, M S Bessell, J.-L Beuzit, M Bonnefoy, S Desidera, J Farihi, P Lowrance, D Mouillet, I Song, A&A. 50952Chauvin, G., Lagrange, A.-M., Bonavita, M., Zuckerman, B., Dumas, C., Bessell, M. 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M B N Kouwenhoven, A G A Brown, S P Goodwin, S F Portegies Zwart, L Kaper, A&A. 493979Kouwenhoven, M. B. N., Brown, A. G. A., Good- win, S. P., Portegies Zwart, S. F., & Kaper, L. 2009, A&A, 493, 979","doi":""},{"text":". K M Kratter, R A Murray-Clay, A N Youdin, ApJ. 7101375Kratter, K. M., Murray-Clay, R. A., & Youdin, A. N. 2010, ApJ, 710, 1375","doi":""},{"text":". B D Mason, G L Wycoff, W I Hartkopf, G G Douglass, C E Worley, AJ. 122Mason, B. D., Wycoff, G. L., Hartkopf, W. I., Douglass, G. G. & Worley, C. E. 2001, AJ 122, 3466 (see the current version at http://www.usno.navy.mil/USNO/astrometry/optical-IR-prod/wds/wds.html)","doi":""},{"text":". T Mazeh, M Simon, L Prato, B Markus, S Zucker, ApJ. 5991344Mazeh, T., Simon, M., Prato, L., Markus, B., & Zucker, S. 2003, ApJ, 599, 1344","doi":""},{"text":". S A Metchev, L A Hillenbrandt, ApJS. 18162Metchev, S. A. & Hillenbrandt, L. A. 2009, ApJS, 181, 62","doi":""}],"string":"[\n {\n \"text\": \"E Artigau, B A Biller, Z Wahhaj, M Hartung, T L Hayward, L M Close, M R Chun, M C Liu, G Trancho, F Rigaut, Ground-based and Airborne Instrumentation for Astronomy II. Ed. McLean, I. S. & Casali, M. Proc. SPIE. 701466Artigau, E., Biller, B. A., Wahhaj, Z., Hartung, M., Hayward, T. L., Close, L. M., Chun, M. R., Liu, M. C., Trancho, G., Rigaut, F. et al. 2008, in: Ground-based and Airborne Instru- mentation for Astronomy II. Ed. McLean, I. S. & Casali, M. Proc. SPIE, 7014, 66\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M R Bate, MNRAS. 392590Bate, M. R. 2009, MNRAS, 392, 590\",\n \"doi\": \"\"\n },\n {\n \"text\": \"B Biller, É Artigaut, Z Wahhaj, M Hartung, M.-C Liu, L M Close, M R Chun, C Ftaclas, D W Toomey, T Hayward, Proc. SPIE. SPIE7015184Biller, B., Artigaut,É., Wahhaj, Z., Hartung, M., Liu, M.-C., Close, L. M., Chun, M. R., Ftaclas, C., Toomey, D. W., & Hayward, T. 2008, In: Proc. SPIE, 7015, 184\",\n \"doi\": \"\"\n },\n {\n \"text\": \". G Chauvin, A.-M Lagrange, M Bonavita, B Zuckerman, C Dumas, M S Bessell, J.-L Beuzit, M Bonnefoy, S Desidera, J Farihi, P Lowrance, D Mouillet, I Song, A&A. 50952Chauvin, G., Lagrange, A.-M., Bonavita, M., Zuckerman, B., Dumas, C., Bessell, M. S., Beuzit, J.-L., Bonnefoy, M., Desidera, S., Far- ihi, J., Lowrance, P., Mouillet, D., & Song, I. 2010, A&A, 509, A52\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Chun, D Toomey, Z Wahhaj, B Biller, E Artigau, T Hayward, M Liu, L Close, M Hartung, F Rigaut, Ch Ftaclas, Proc. SPIE. Adaptive Optics Systems. Ed. Hubin, N., Max, C. E., Wizinowich, P. LSPIE701570151Chun, M. Toomey, D., Wahhaj, Z., Biller, B., Ar- tigau, E., Hayward, T., Liu, M., Close, L., Har- tung, M., Rigaut, F., & Ftaclas, Ch. 2008, in: Adaptive Optics Systems. Ed. Hubin, N., Max, C. E., Wizinowich, P. L. Proc. SPIE, 7015, 70151V\",\n \"doi\": \"\"\n },\n {\n \"text\": \". E Covino, J M Alcala, S Allain, A&A. 328187Covino, E., Alcala, J. M., Allain, S., et al. 1997, A&A, 328, 187\",\n \"doi\": \"\"\n },\n {\n \"text\": \"R M Cutri, M F Skrutskie, S Van Dyk, C A Beichman, J M Carpenter, T Chester, L Cambresy, T Evans, J Fowler, J Gizis, The IRSA 2MASS All-Sky Point Source Catalog. NASA/IPAC Infrared Science Archive. Cutri, R. M., Skrutskie, M. F., van Dyk, S., Be- ichman, C. A., Carpenter, J. M., Chester, T., Cambresy, L., Evans, T., Fowler, J., Gizis, J. et al. 2003, The IRSA 2MASS All-Sky Point Source Catalog. NASA/IPAC Infrared Science Archive.\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S Desidera, R G Gratton, S Lucatello, R U Claudi, T H Dall, A&A. 454553Desidera, S., Gratton, R. G., Lucatello, S., Claudi, R. U., & Dall, T.H. 2006, A&A, 454, 553\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Duquennoy, M Mayor, A&A. 248485Duquennoy, A. & Mayor, M. 1991, A&A 248, 485\",\n \"doi\": \"\"\n },\n {\n \"text\": \". A Eggenberger, S Udry, G Chauvin, J.-L Beuzit, A.-M Lagrange, D Ségransan, M Mayor, A&A. 474273Eggenberger, A., Udry, S., Chauvin, G., Beuzit, J.-L., Lagrange, A.-M., Ségransan, D., & Mayor, M. 2007, A&A, 474, 273\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The Hipparcos and Tycho Catalogues. ESA SP-1200ESA 1997. ESA 1997, The Hipparcos and Tycho Catalogues, ESA SP-1200\",\n \"doi\": \"ESA SP-1200\"\n },\n {\n \"text\": \". L Girardi, A Bressan, G Bertelli, C Chiosi, A&AA. 141371Girardi, L., Bressan, A., Bertelli, G., & Chiosi, C., 2000, A&AA, 141, 371\",\n \"doi\": \"\"\n },\n {\n \"text\": \". D Grether, C H Lineweaver, ApJ. 6401051Grether, D. & Lineweaver, C.H. 2006, ApJ, 640, 1051\",\n \"doi\": \"\"\n },\n {\n \"text\": \". J L Halbwachs, M Mayor, S Udry, F Arenou, A&A. 397159Halbwachs, J. L., Mayor, M., Udry, S., & Arenou, F. 2003, A&A, 397, 159\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R S Harrington, Celest. Mech. 6322Harrington, R. S. 1972, Celest. Mech., 6, 322\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T J Henry, D W Mccarthy, AJ. 106773Henry, T. J. & McCarthy, D. W. 1993, AJ, 106, 773\",\n \"doi\": \"\"\n },\n {\n \"text\": \". R Köhler, AJ. 1223325Köhler, R. 2001, AJ, 122, 3325\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M B N Kouwenhoven, A G A Brown, H Zinnecker, L Kaper, S F Zwart, A&A. 430137Kouwenhoven, M. B. N., Brown, A. G. A., Zin- necker, H., Kaper, L., & Portegies Zwart, S. F. 2005, A&A, 430, 137\",\n \"doi\": \"\"\n },\n {\n \"text\": \". M B N Kouwenhoven, A G A Brown, S P Goodwin, S F Portegies Zwart, L Kaper, A&A. 493979Kouwenhoven, M. B. N., Brown, A. G. A., Good- win, S. P., Portegies Zwart, S. F., & Kaper, L. 2009, A&A, 493, 979\",\n \"doi\": \"\"\n },\n {\n \"text\": \". K M Kratter, R A Murray-Clay, A N Youdin, ApJ. 7101375Kratter, K. M., Murray-Clay, R. A., & Youdin, A. N. 2010, ApJ, 710, 1375\",\n \"doi\": \"\"\n },\n {\n \"text\": \". B D Mason, G L Wycoff, W I Hartkopf, G G Douglass, C E Worley, AJ. 122Mason, B. D., Wycoff, G. L., Hartkopf, W. I., Douglass, G. G. & Worley, C. E. 2001, AJ 122, 3466 (see the current version at http://www.usno.navy.mil/USNO/astrometry/optical-IR-prod/wds/wds.html)\",\n \"doi\": \"\"\n },\n {\n \"text\": \". T Mazeh, M Simon, L Prato, B Markus, S Zucker, ApJ. 5991344Mazeh, T., Simon, M., Prato, L., Markus, B., & Zucker, S. 2003, ApJ, 599, 1344\",\n \"doi\": \"\"\n },\n {\n \"text\": \". S A Metchev, L A Hillenbrandt, ApJS. 18162Metchev, S. A. & Hillenbrandt, L. A. 2009, ApJS, 181, 62\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":598,"cells":{"text":{"kind":"string","value":"\n\n1943-2007\n\nOlaf Post 1943-2007FIRST ORDER OPERATORS AND BOUNDARY TRIPLES 1\nThe aim of the present paper is to introduce a first order approach to the abstract concept of boundary triples for Laplace operators. Our main application is the Laplace operator on a manifold with boundary; a case in which the ordinary concept of boundary triples does not apply directly. In our first order approach, we show that we can use the usual boundary operators also in the abstract Green's formula. Another motivation for the first order approach is to give an intrinsic definition of the Dirichlet-to-Neumann map and intrinsic norms on the corresponding boundary spaces. We also show how the first order boundary triples can be used to define a usual boundary triple leading to a Dirac operator.\n\nIntroduction\n\nThe concept of boundary triples, originally introduced in [V63], has successfully be applied to the theory of self-adjoint extensions of symmetric operators, for example on quantum graphs, singular perturbations or point interactions on manifolds (see e.g. [BGP06]). For a general treatment of boundary triples we refer to [BGP06,DHMdS06] and the references therein.\n\nOur main purpose here is not to characterise all self-adjoint extensions of a given symmetric operator, but to show that the concept of boundary triples can also be used in the PDE case, namely to Laplacians on a manifold with boundary. The standard theory of boundary triples does not directly apply in this case, since Green's formula\nX ∆f g dx − X f ∆g dx = ∂X (∂ n fg − f ∂ n g)↾ ∂X\ndoes not extend to f, g in the maximal operator domain dom ∆ max = { f ∈ L 2 (X) | ∆ max f ∈ L 2 (X) (distributional sense) } (cf. Remark 4.2 for details). A solution to overcome this problem is either to modify the boundary operators (restriction of the function and the normal derivative onto ∂X) as e.g. in [BMNW07,Pc07], or to introduce the concept of quasi boundary triples as in [BL07] (cf. also the references therein for further treatments of boundary triples in the PDE case).\n\nHere, we use a different approach: we start with first order operators, namely the exterior derivative d taking functions (0-forms) to 1-forms and its adjoint, the divergence operator δ, mapping 1-forms into functions, since the first order operator domains are simpler. The Laplacian (on functions) is then defined as ∆ 0 := δd. Certainly, in our approach we do not cover all selfadjoint extensions of the minimal Laplacian.\n\nThe abstract approach also allows to define the Dirichlet-to-Neumann map in an intrinsic manner, and also the norm of G 1/2 = H 1/2 (∂X) is defined intrinsicly. This might be a great advantage when dealing with parameter-depending manifolds, as it is the case for graph-like manifolds (see e.g. [EP07,P06]). We will treat this question in a forthcoming publication. Our approach is related to the recent works of Arlinskii [A00], Posilicano [Pc07] and Brown et al. [BMNW07], where also a PDE example is treated in the context of boundary triples.\n\nTo precise our idea of the first order approach we sketch the construction here. The given data are 1 H 0 , H 1 , d : H 0 H 1 , H 1 0 := dom d, where H p are Hilbert spaces (\"p-forms\"), and H 1 0 carries the graph norm. Guided by our main application (a manifold with boundary), we call d an exterior derivative.\n\nA boundary map (of order 0) is a bounded operator γ 0 : H 1 0 −→ G , G 1/2 := ran γ 0 with dense range G 1/2 ⊂ G , where G is another Hilbert space (usually over the boundary). For these data, we define d 0 := d restricted toH 1 0 := ker γ 0 and the divergence operator δ := d * 0 with domain H 1 1 := dom δ. Furthermore, we can define a natural norm on G 1/2 using γ 0 . In addition, we have a boundary operator of order 1, namely, γ 1 : H 1 1 −→ G , with the same range ran γ 1 = ran γ 0 = G 1/2 . Moreover, an abstract Green's formula is valid, i.e.,\ndf 0 , g 1 − f 0 , δg 1 = γ 0 f 0 , γ 1 g 1 G 1/2 .\nFinally, h p = β z p ϕ is the solution of the Dirichlet and Neumann problem ∆ p h p = zh p , γ p h p = ϕ, respectively; we call β z p also a Krein Γ-field of order p. The Krein Q-function is defined as Q z 0 ϕ := γ 1 dβ z 0 ; a bounded operator (on the boundary space G 1/2 ), closely related to the usual Dirichlet-to-Neumann map Λ(z) on a manifold with boundary defined in Eq. (4.1).\n\nThe main idea here is to consider the Laplacian ∆ 0 f 0 := δdf 0 on the space\nH 2 0 := dom δd := f 0 ∈ dom d df 0 ∈ dom δ instead of the maximal domain dom ∆ max 0 = { f 0 ∈ H 0 | ∆ 0 f 0 ∈ H 0 }. Although ∆ 0 is not closed on H 2\n0 , we can develop a suitable theory of boundary spaces. In particular, for a bounded and self-adjoint operator B in G 1/2 we can show that the Laplacian ∆ 0 restricted to dom ∆ B 0 := { f 0 ∈ H 2 0 | γ 1 df 0 = Bγ 0 f 0 } (Robin-type boundary conditions) is self-adjoint under a suitable condition on the domain of the adjoint (fulfilled in our example of the Laplacian on a manifold with boundary). Our main result is Krein's resolvent formula for the resolvents of ∆ B 0 and the Dirichlet Laplacian ∆ D 0 ; and a spectral relation between the operators\n∆ B 0 and Q z 0 − B, namely σ(∆ B 0 ) \\ σ(∆ D 0 ) = { z / ∈ σ(∆ D 0 ) | 0 ∈ σ(Q z 0 − B) }.\n(see Theorem 2.30). The main advantage of our approach is that it can almost immediately be applied to the case of the Laplacian on a manifold with boundary, using the standard boundary operator (restriction of a function to the boundary and restriction of the normal component of a 1-form to the boundary).\n\nThe paper is organised as follows: In the next section, we develop the concept of first order boundary triples. In Section 3 we show how this concept fits into the usual theory of boundary triples. Section 4 contains our motivating example, namely, the Laplacian on a manifold with boundary.\n\nAcknowledgements. The author would like to dedicate this contribution to his dear colleague at the Humboldt University, Vladimir A. Geyler, who passed away very suddenly. The author acknowledges the financial support of the Collaborative Research Center SFB 647 \"Space -Time -Matter. Analytic and Geometric Structures\" in which Vladimir Geyler also was a member. Finally, the author would like to thank Y. Arlinskii for drawing his attention to the article [A00], where a very similar approach was used.\n\n\nFirst order approach\n\nIn this section, we develop the concept of boundary triples for operators acting in different Hilbert spaces; guided by our main example of the exterior derivative on a manifold with boundary. \n:= f 0 2 + df 0 2 .\n(iii) We call γ 0 : H 1 0 −→ G a boundary map (of degree 0) associated to d iff γ 0 is bounded with dense image, and ifH 1 0 := ker γ 0 ⊂ H 1 0 = dom d is dense in H 0 . The auxiliary Hilbert space is also referred to as a boundary space. We say that \nγ 0 is proper, if γ 0 is not surjective, i.e., if G 1/2 := γ 0 (H 1 0 ) G . (iv) The data (H , G , γ 0 ) define:= f 0 2 + df 0 2 + δdf 0 2 , f 1 2 H 2 0 := f 1 2 + δf 1 2 + dδf 1 2 .\nWe denote the eigenspaces by N z\np := ker(∆ p − z) ⊂ H 2 p . For z = −1, we set N p := N −1 p . (ii) We call ∆ D 0 := d * 0 d 0 , ∆ N 0 := d * d, ∆ D 1 := d 0 d * 0 ,\n∆ N 1 := dd * with the appropriate domains the Dirichlet Laplacian of degree p = 0, 1 and the Neumann Laplacian of degree p = 0, 1, respectively. Clearly, all these operators are self-adjoint and non-negative. We denote the corresponding resolvents by R D p := (∆ D p + 1) −1 and R N p := (∆ N p + 1) −1 .\n\nThe following diagram tries to illustrate the two scales of Hilbert spaces associated to d, d * and d 0 , d * 0 = δ (dotted arrows). Note that only at order 1, 0 and −1, we have relations between the two scales: \nH 1 0 . . . -H −1 0 (R N 0 ) 1/2 -H 0 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( R D 0 ) 1 / 2 - H 1 0 ? ∩ -. . .= f p ∈ H p ∆ p f p ∈ H p (2.2)\nin the distributional sense). In general, H 2 p dom ∆ max p . This observation is one of the motivations for our first order approach (see Section 4).\nLemma 2.5. We have H 1 p =H 1 p ⊕ N p (orthogonal sum). Proof. Let p = 0 and f 0 ∈ H 1 0 . In this case, f 0 ∈ (H 1 0 ) ⊥ is equivalent to 0 = f 0 , g 0 H 1 0 = f 0 , g 0 H 0 + df 0 , dg 0 H 1 , ∀ g 0 ∈H 1 0 . (2.3)\nHowever, by definition of the adjoint operator\nδ = d * 0 , we have h 1 ∈ dom d * 0 iff there exists h 0 ∈ H 0 such that h 1 , d 0 g 0 H 0 = h 0 , g 0 H ∀ g 0 ∈H 1 0 . (2.4) Choosing h 0 = −f 0 , the orthogonality relation (2.3) reads h 1 = df 0 ∈ dom d * 0 and d * 0 df 0 = −f 0 , i.e., f 0 ∈ N z 0 .\nThe argument for p = 1 is similar. Lemma 2.6. The maps d : N 0 −→ N 1 and δ : N 1 −→ N 0 are unitary.\n\nProof.\nIf f 0 ∈ N 0 then dδdf 0 = −df 0 , i.e, df 0 ∈ N 1 . Similarly, f 1 ∈ N 1 implies δf 1 ∈ N 0 . Furthermore, −δdf 0 = f 0 and d(−δf 1 ) = f 1 implies that −δ is the inverse of d. Finally, d is an isometry because df 0 2 H 1 1 = df 0 2 H 1 + δdf 0 2 H 0 = df 0 2 H 1 + f 0 2 H 0 = f 0 2 H 1 0 .\nSince d is surjective, it is therefore unitary with unitary inverse −δ.\n\nLemma 2.7. Assume that the boundary map γ 0 is proper (i.e., G 1/2 = ran γ 0 G ). Definê\nγ 0 := γ 0 ↾ N 0 , thenγ 0 is invertible and (γ 0 ) −1 : G N 0 is an unbounded operator with domain dom(γ 0 ) −1 = G 1/2 . Furthermore, (γ 0 ) −1 ϕ = h 0 is the (unique) solution of the Dirichlet problem (∆ 0 + 1)h 0 = 0, γ 0 h 0 = ϕ.\nProof. The operatorγ 0 is invertible since (ker γ 0 ) ⊥ = (H 1 0 ) ⊥ = N 0 by Lemma 2.5. If (γ 0 ) −1 were be bounded, thenγ 0 would be a topological isomorphism of N 0 and ran γ 0 = G 1/2 , in particular, G 1/2 would be closed in G , and by the density, we would have G 1/2 = G -a contradiction. The last assertion is an immediate consequence of Lemma 2.5 and the definition of the inverse map (γ 0 ) −1 .\n\nDefinition 2.8. We endow G 1/2 with the norm\nϕ G 1/2 := (γ 0 ) −1 ϕ H 1 0 . Lemma 2.9.\nAssume that the boundary map γ 0 is proper (i.e., G 1/2 = ran γ 0 G ), then the following assertions hold:\n(i) We have ϕ G ≤ γ 0 ϕ G 1/2 for ϕ ∈ G 1/2 . (ii) The operator γ 0 γ * 0 ≥ 0 is invertible in G , and Λ := (γ 0 γ * 0 ) −1 = ((γ 0 ) −1 ) * (γ 0 ) −1 ≥ 1 γ 0 2 .\nWe define the associated scale of Hilbert spaces by\nG s := dom Λ s , ϕ G s := Λ s ϕ G\nfor s ≥ 0 (and the dual with respect to (·, ·) G for s < 0).\n(iii) The operator ((γ 0 ) −1 ) * : N 0 G is unbounded with domain dom((γ 0 ) −1 ) * = { f 0 ∈ N 0 | γ 0 f 0 ∈ dom Λ = G 1 }. (iv) The operator γ * 0 : G −→ H 1 0 is bounded, and γ * 0 ϕ = h 0 is the unique Neumann solution, i.e., (∆ 0 + 1)h 0 = 0, γ 0 h 0 ∈ G 1 , Λγ 0 h 0 = ϕ.\nRemark 2.10. If γ 0 is not proper (i.e., if γ 0 is surjective, i.e., G 1/2 = G ), then all the above assertions remain valid except for the fact that (γ 0 ) −1 , ((γ 0 ) −1 ) * and Λ are bounded operators.\n\nProof. The first assertion follows from\nϕ G = γ 0 (γ 0 ) −1 ϕ G ≤ γ 0 (γ 0 ) −1 ϕ H 1 = γ 0 ϕ G 1/2 .\nTo prove the second, note that γ 0 γ * 0 =γ 0γ * 0 is bijective and\nϕ, ϕ G 1/2 = (γ 0 ) −1 ϕ, (γ 0 ) −1 ϕ H 1 = ϕ, ((γ 0 ) −1 ) * (γ 0 ) −1 ϕ G = ϕ, Λϕ G if (γ 0 ) −1 ϕ ∈ dom((γ 0 ) −1 ) * , i.e, ϕ ∈ dom Λ. Furthermore, Λ −1 ≤ γ 0 2 .\nThe third assertion is a consequence of Lemma 2.7, and the domain characterisation can be seen readily. To prove the fourth assertion, take\nh 0 = γ * 0 ϕ ∈ ran γ * 0 ⊂ (ker γ 0 ) ⊥ = N 0 ; in this case h 0 , f 0 H 1 0 = ϕ, γ 0 f 0 G for all f 0 ∈ H 1 0 . If f 0 ∈ N 0 , then h 0 , f 0 H 1 0 = γ 0 h 0 , γ 0 f 0 G 1/2 by definition of the norm on G 1/2 . But the latter term equals Λγ 0 h 0 , γ 0 f 0 G if γ 0 h 0 ∈ dom Λ, and thus ϕ = Λγ 0 h 0 .\nRemark 2.11. Note that G −1/2 is the completion of G with respect to the norm ϕ G −1/2 = γ 0 ϕ H 1 0 . Definition 2.12. We define the boundary map of order 1 as\nγ 1 : H 1 1 −→ G , γ 1 := −γ 0 δP 1\nwhere P p is the orthogonal projection in H 1 p onto the subspace N p . Lemma 2.13. We have ker γ 1 =H 1 1 , and γ 1 : H 1 1 −→ G is bounded with norm γ 1 = γ 0 . Furthermore, ran γ 1 = G 1/2 andγ 1 := γ 1 ↾ N 1 is a unitary map from N 1 onto G 1/2 .\nProof. If f 1 ∈H 1 1 = (N 1 ) ⊥ , then γ 1 f 1 = 0 since P 1 f 1 = 0. If f 1 ∈ N 1 , then γ 1 f 1 = −γ 0 δf 1 = 0 iff f 1 = 0 since δ is unitary from N 1 onto N 0 = (ker γ 0 ) ⊥ .\nThe boundedness follows from\nγ 1 f 1 G ≤ γ 0 δP 1 f 1 G ≤ γ 0 δP 1 f 1 H 1 0 = γ 0 P 1 f 1 H 1 1 ≤ γ 0 f 1 H 1 1\nby Lemma 2.6. Furthermore, for f 0 ∈ N 0 set f 1 := df 0 , then γ 1 f 1 = −γ 0 δdf 0 = γ 0 f 0 . In particular, γ 1 = γ 0 . Finally,\nγ 1 f 1 G 1/2 = γ 0 δf 1 G 1/2 = δf 1 H 1 0 = f 1 H 1 1\nfor f 1 ∈ N 1 , since δf 1 ∈ N 0 and by Lemma 2.6.\n\nLemma 2.14. The (abstract) Green's formula holds, namely,\ndf 0 , g 1 − f 0 , δg 1 = γ 0 f 0 , γ 1 g 1 G 1/2 = (γ 0 f 0 , γ 1 g 1 ) G\nwhere γ 1 := Λγ 1 : H 1 1 −→ G −1/2 . Proof. If f 0 ∈H 1 0 , then the LHS vanishes since δ = d * 0 , and so is the RHS, since γ 0 f 0 = 0. Similarly, if g 1 ∈H 1 1 = dom d * , then the LHS vanishes since δg 1 = d * g 1 and so is the RHS, because γ 1 g 1 = 0 by Lemma 2.13. For f 0 ∈ N 0 and g 1 ∈ N 1 , we have \ndf 0 , g 1 − f 0 , δg 1 = − df 0 , dδg 1 − f 0 , δg 1 = − f 0 , δg 1 H 1 0 = γ 0 f 0 , −γ 0 δg 1 G 1∆ 0 f 0 , g 0 − f 0 , ∆ 0 g 0 = γ 0 f 0 , γ 1 dg 0 G 1/2 − γ 1 df 0 , γ 0 g 0 G 1/2 = γ 0 f 0 , γ 1 dg 0 G − γ 1 df 0 , γ 0 g 0 G for f 0 , g 0 ∈ H 2\n0 . The following lemma shows that Λ = Λ(−1) is the Dirichlet-to-Neumann map for the operator ∆ 0 + 1: Lemma 2.16. For ϕ ∈ G 1/2 and h 0 := (γ 0 ) −1 ϕ we have\nΛϕ = γ 1 dh 0 .\nProof. By Lemma 2.14, we have\ndf 0 , dh 0 − f 0 , ∆ 0 h 0 = (γ 0 f 0 , γ 1 dh 0 ) G .\nOn the other hand, we have\ndf 0 , dh 0 − f 0 , ∆ 0 h 0 = f 0 , h 0 H 1 0 = γ 0 f 0 , γ 0 h 0 G 1/2 = γ 0 f 0 , ϕ G 1/2 = (γ 0 f 0 , Λϕ) G . for f 0 , h 0 ∈ N 0 .\nRemark 2.17. The map γ 1 is indeed the boundary map occuring in the applications (see Section 4). Namely, the Green's formula is usually formulated with a boundary integral given as an inner product of G rather than G 1/2 . In particular, γ 1 dh 0 is the \"normal derivative at the boundary\" (in the case of a manifold with boundary).\n\nThe boundary maps are also bounded as maps with target space G 1/2 : Lemma 2.18. The operators γ p : H 1 p −→ G 1/2 are bounded with norm bounded by 1. Proof. For p = 0, we have\nγ 0 f 0 G 1/2 = (γ 0 ) −1 γ 0 f 0 H 1 0 = (γ 0 ) −1 γ 0 P 0 f 0 H 1 0 = P 0 f 0 H 1 0 ≤ f 0 H 1 0 , since γ 0 f 0 = γ 0 P 0 f 0 . For p = 1, we obtain γ 1 f 1 G 1/2 = (γ 0 ) −1 γ 0 δP 1 f 1 H 1 0 = δP 1 f 1 H 1 0 = P 1 f 1 H 1 1 ≤ f 1 H 1 1\nusing Lemmas 2.6-2.7.\n\nIn order to define the Dirichlet-to-Neumann map also for other resolvent values z, we need to provide results similar to those in Lemmas 2.5-2.7 for general z. Write\nΣ 0 := σ(∆ D 0 ), Σ 1 := σ(∆ N 1 ). (2.5) Lemma 2.19. For z / ∈ Σ p , we have H 1 p =H 1 p+ N z p (topological direct sum). In particular, γ z p := γ p ↾ N z p is a topological isomorphism from N z p onto G 1/2 . Proof. For z / ∈ σ(∆ D 0 ), we define P z 0 := 1 − ι 0 (∆ D 0 − z) −1 (∆ 0 − z) : H 1 0 −→ H 1 0 where ∆ 0 = δd : H 1 0 −→H −1 0 , (∆ D 0 − z) −1 = (δd 0 − z) −1 :H −1 0 −→H 1 0\nand ι 0 :H 1 0 ֒→ H 1 0 . A simple calculation shows that (1 − P z 0 ) 2 = (1 − P z 0 ), i.e., 1 − P z 0 and therefore P z 0 are projections. Furthermore, f 0 = P z 0 f 0 is equivalent to ∆ 0 f 0 = zf 0 . In order to show that f 0 = P z 0 f 0 ∈ N z 0 let us first show that f 0 ∈ H 2 0 , i.e., that h 1 := df 0 ∈ H 1 1 = dom δ. To this end, recall the definition of the domain dom δ = dom d * 0 in (2.4). We have here df 0 , d 0 g 0 = δdf 0 , g 0 = zf 0 , g 0 by Lemma 2.14 (note that γ 0 g 0 = 0) and the fact that δdf 0 = zf 0 ; we can choose h 0 = zf 0 and therefore f 0 ∈ H 2 0 . A straightforward calculation shows now that (∆ 0 − z)f 0 = 0, and finally, f 0 ∈ N z 0 . By the definition of P z 0 , it is also clear that ran(1 − P z 0 ) ⊂H 1 0 , and therefore H 1 0 splits into the direct sum. The direct sum is also a topological sum, since 1 − P z 0 and P z 0 are bounded maps. Therefore f 0 → ((1 − P z 0 )f 0 , P z 0 f 0 ) is a bounded bijection, and also a topological isomorphism. The argument for 1-forms is similar, using\nP z 1 := 1 − ι 1 (∆ N 1 − z) −1 (∆ 1 − z) : H 1 1 −→ H 1 1 where ∆ 1 = dδ : H 1 1 −→H −1 1 , (∆ N 1 − z) −1 = (dd * − z) −1 :H −1 1 −→H 1 1\nand ι 1 :H 1 1 ֒→ H 1 1 . For the last assertion, note that ker γ p =H 1 p and that ran γ p = G 1/2 (see Lemma 2.13); in particular,γ z p is bijective. Furthermore,γ z p is bounded as restriction of the bounded map γ p : H 1 p −→ G 1/2 (cf. Lemma 2.18), and therefore,γ z p is a topological isomorphism.\n\nLemma 2.20. For z = 0, the maps d : N z 0 −→ N z 1 and δ : N z 1 −→ N z 0 are topological isomorphisms.\n\nProof. If f 0 ∈ N z 0 then dδdf 0 = zdf 0 , i.e, df 0 ∈ N z 1 . Similarly, f 1 ∈ N z 1 implies δf 1 ∈ N z 0 . Furthermore, 1 z δdf 0 = f 0 and d( 1 z δf 1 ) = f 1 implies that 1 z δ is the inverse of d. Finally, d is bounded on N z 0 , since df 0\n2 H 1 1 = df 0 2 H 1 + δdf 0 2 H 0 = df 0 2 H 1 + |z| 2 f 0 2 H 0 ≤ (1 + |z| 2 ) f 0 2 H 1 0\nand therefore a topological isomorphism. The assertion for δ follows similarly.\n\nDefinition 2.21. We call z → β z 0 := (γ z 0 ) −1 , z / ∈ Σ 0 the Dirichlet solution map or the Krein Γ-field of order 0 associated to the first order boundary triple (H , G , γ 0 ). Similarly, we call z → β z 1 := (γ z 1 ) −1 , z / ∈ Σ 1 the Neumann solution map or the Krein Γ-field of order 1.\n\nRemark 2.22.\n\n(i) We prefer to use the symbol β instead of γ for the Krein Γ-field in order to avoid confusion with our boundary maps γ p . (ii) The maps β z p : G 1/2 −→ N z p ⊂ H 1 p are topological isomorphisms, since the inversesγ z p are. (iii) The names \"Dirichlet/Neumann solution map\" are due to the following fact: The p-form h p := β z p ϕ is the solution of (∆ p − z)h p = 0, and γ p h p = ϕ. For p = 0, this is the solution of the \"Dirichlet problem\" (γ 0 h 0 prescribed), and for p = 1, the solution of the \"Neumann problem\" (γ 1 h 1 prescribed). We will see in Lemma 3.7 that the Krein Γ-fields are related to a Krein Γ-field in the sense of an ordinary boundary triple. (iv) The map β z 0 : G 1/2 −→ H 1 0 regarded as an operator β z 0 : G 1/2 −→ H 0 into H 0 is bounded, as well as its adjoint, denoted by (β z 0 ) * : \nH 0 −→ G 1/2 . Lemma 2.23. We have γ 1 df 0 = (β z 0 ) * (∆ 0 − z)f 0 for f 0 ∈ dom ∆ D 0 = H 2 0 ∩H 1 0 where (β z 0 ) * isϕ, (β z 0 ) * (∆ 0 − z)f 0 G 1/2 = β z 0 ϕ, (∆ 0 − z)f 0 H = (∆ 0 − z)β z 0 ϕ, f 0 H + γ 0 β z 0 ϕ, γ 1 df 0 G 1/2 − γ 1 dβ z 0 ϕ, γ 0 f 0 G 1/2 = ϕ, γ 1 df 0 G 1/2\nby Corollary 2.15 for the second equality. As far as the third equality is concerned, note that the first term vanishes since β z 0 ϕ solves the eigenvalue equation; the same holds for the third term since γ 0 f 0 = 0 for f 0 ∈H 1 0 . For the second term, we have γ 0 β z 0 ϕ = ϕ by the definition of β z 0 . The last assertion follows from ran(β z 0 ) * = (ker β z 0 ) ⊥ and from the fact that β z 0 :\nG 1/2 −→ H 0 is injective.\nWe can now define the Dirichlet-to-Neumann map and a closely related map for arbitrary resolvent values z:\n\nDefinition 2.24. The Krein Q-function associated to the first order boundary triple (H , G , γ 0 ) is the map z → Q z 0 := γ 1 d(γ z 0 ) −1 = γ 1 dβ z 0 , z / ∈ Σ 0 = σ(∆ D 0 ). For z / ∈ Σ 0 , the abstract Dirichlet-to-Neumann map at z is defined by\nΛ(z) := ΛQ z 0 = Λγ 1 dβ z 0 = γ 1 dβ z 0 : G 1/2 −→ G −1/2 .\nRemark 2.25. (i) We shall see in Section 3 that Q z 0 is indeed a Krein Q-function for an ordinary boundary triple. Note that Q z 0 : G 1/2 −→ G 1/2 is a bounded map (cf. Lemmas 2.18-2.20). In addition, we have\nQ −1 0 = γ 1 d(γ 0 ) −1 = −γ 0 δP 1 d(γ 0 ) −1 = γ 0 (γ 0 ) −1 = id G 1/2 at z = −1.\n(ii) Note that Λ(z) is indeed the Dirichlet-to-Neumann map: We solve the Dirichlet problem h 0 = β z 0 ϕ, i.e, ∆ 0 h 0 = zh 0 , γ 0 h 0 = ϕ; and the Dirichlet-to-Neumann map is the \"normal derivative at the boundary\" of h 0 (cf. Remark 2.17), i.e., Λ(z)ϕ = γ 1 dh 0 .\n\nLet us now define self-adjoint restrictions of ∆ 0 .\n\nDefinition 2.26. Let B be a bounded operator in G 1/2 . We set\ndom ∆ B 0 := { f 0 ∈ H 2 0 | γ 1 df 0 = Bγ 0 f 0 } dom ∆ B 1 := { f 1 ∈ H 2 1 | γ 1 f 1 = Bγ 0 δf 1 } and denote by ∆ B p the restriction of ∆ p onto dom ∆ B p . Lemma 2.27. Assume that dom(∆ B 0 ) * ⊂ H 1 0 , then the operator ∆ B 0 is self-adjoint iff B is self- adjoint in G 1/2 .\nRemark 2.28. The domain condition does not seem to follow from abstract (\"soft\") arguments; in our manifold example, it follows from elliptic regularity (\"hard\" arguments). Note that in general, dom ∆ max 0 defined in (2.2) is even not a subset of H 1 0 (see Remark 2.4 (ii) and Remark 4.2). Proof. The graph of the operator (∆ B 0 ) * is given as\ngraph(∆ B 0 ) * = (f 0 , ∆ 0 f 0 ) f 0 ∈ dom ∆ max 0 , ∀ g 0 ∈ dom ∆ B 0 : ∆ max 0 f 0 , g 0 = f 0 , ∆ max 0 g 0 ⊂ H 1 0 × H 0\n, and the latter inclusion holds by our assumption on the domain of the adjoint. In particular, f 0 , g 0 ∈ H 2 0 and we can apply Corollary 2.15, namely, ∆ max\n0 f 0 , g 0 − f 0 , ∆ max 0 g 0 = γ 0 f 0 , γ 1 dg 0 G 1/2 − γ 1 df 0 , γ 0 g 0 G 1/2 = γ 0 f 0 , Bγ 0 g 0 G 1/2 − Bγ 0 f 0 , γ 0 g 0 G 1/2 ,\nand the latter equality follows from f 0 , g 0 ∈ dom ∆ B 0 . The assertion is now obvious. The self-adjointness of B in G 1/2 can be shown as follows:\n\nLemma 2.29. Let B be a bounded and self-adjoint operator on G . In this case, B := Λ −1 B is bounded and self-adjoint as operator on G 1/2 . Proof. We have\nB B(G 1/2 ) = Λ 1/2 BΛ −1/2 B(G ) = Λ −1/2 BΛ −1/2 B(G ) ≤ Λ −1 B(G ) B B(G )\n, so that B is bounded on G 1/2 , and Bϕ, ψ G 1/2 = Λ 1/2 Bϕ, Λ 1/2 ψ G = Λ −1/2 Bϕ, Λ 1/2 ψ G = Bϕ, ψ G and the similar symmetric expression shows the self-adjointness.\n\nWe can now formulate our main result. For brevity, we restrict ourselves here to 0-forms. Similar results hold also for 1-forms. \n∆ B 0 ) * ⊂ H 1 0 . (i) For z / ∈ σ(∆ D 0 ) we have ker(∆ B 0 − z) = β z 0 ker(Q z 0 − B). (ii) For z / ∈ σ(∆ B 0 ) ∪ σ(∆ D 0 ) we have 0 / ∈ σ(Q z 0 − B) and Krein's formula (∆ D 0 − z) −1 − (∆ B 0 − z) −1 = β z 0 (Q z 0 − B) −1 (β z 0 ) * is valid, where (β z 0 ) * is the adjoint of β z 0 as operator β z 0 : G 1/2 −→ H 0 . (iii) We have σ(∆ B 0 ) \\ σ(∆ D 0 ) = { z / ∈ σ(∆ D 0 ) | 0 ∈ σ(Q z 0 − B) }. Proof.\nThe proof is again closely related to the proof for ordinary boundary triples (cf. [BGP06,Thm. 1.29]). For the first assertion, take ϕ ∈ ker(Q z 0 − B) and set f 0 = β z 0 ϕ. By the definition of the solution map β z 0 , we have (∆ 0 − z)f 0 = 0 and γ 0 f 0 = ϕ. Furthermore, Q z 0 ϕ = Bϕ is equivalent to γ 1 df 0 = Bγ 0 f 0 by the definition of Q z 0 . However, the last equation shows that f 0 ∈ dom ∆ B 0 , i.e., f 0 ∈ ker(∆ B 0 − z). The opposite inclusion follows similarly. To prove the second assertion, take h 0 ∈ H 0 and f 0 :\n= (∆ B 0 −z) −1 h 0 ∈ dom ∆ B 0 . By Lemma 2.19 we can decompose f 0 = f z 0+ g z 0 ∈H 1 0+ N z 0 . Since f 0 , g z 0 ∈ H 2 0 we also have f z 0 ∈ H 2 0 and h 0 = (∆ B 0 − z)f 0 = (∆ 0 − z)f 0 = (∆ 0 − z)f z 0 = (∆ D 0 − z)f z 0 , i.e., f z 0 = (∆ D 0 − z) −1 h 0 . Furthermore, γ 0 f z 0 = 0, therefore γ 0 f 0 = γ 0 g z 0 , i.e., g z 0 = β z 0 γ 0 f 0 and we have (∆ B 0 − z) −1 h 0 = f 0 = f z 0 + g z 0 = (∆ D 0 − z) −1 h 0 + β z 0 γ 0 f 0 .\n(2.6) Now we apply γ 1 d to the decomposition of f 0 ∈ dom ∆ B 0 and obtain\nBγ 0 f 0 = γ 1 df 0 = γ 1 df z 0 + γ 1 dβ z 0 γ 0 f 0 = (β z 0 ) * (∆ 0 − z)f z 0 + Q z 0 γ 0 f 0 = (β z 0 ) * h 0 + Q z 0 γ 0 f 0 .\nusing the definition of Q z 0 (cf. Definition 2.24) and Lemma 2.23 for the third equality. In particular,\n(Q z 0 − B)γ 0 f 0 = (β z 0 ) * h 0 , (2.7)\nand the RHS covers the entire space G 1/2 since h 0 covers H 0 (see again Lemma 2.23). In particular, (Q z 0 − B) is surjective. By (i), this operator is also injective, i.e., 0 / ∈ σ(Q z 0 − B). Krein's formula now follows from (2.6)-(2.7). The last assertion is a consequence of (ii).\n\nReturning to the original boundary space G and the Dirichlet-to-Neumann map Λ(z) = ΛQ z 0regarded as an unbounded operator in G -, we obtain the following result: The next lemma is a well known fact; we give a proof for completeness.\ndom ∆ B 0 := { f 0 ∈ H 2 0 | γ 1 df 0 = Bγ 0 f 0 } (Robin type boundary conditions). Assume that dom(∆ B 0 ) * ⊂ H 1 0 (i) For z / ∈ σ(∆ D 0 ) we have ker(∆ B 0 − z) = β z 0 Λ −1 ker(Λ(z) − B). (ii) For z / ∈ σ(∆ B 0 ) ∪ σ(∆ D 0 ) we have 0 / ∈ σ(Λ(z) − B) and Krein's formula (∆ D 0 − z) −1 − (∆ B 0 − z) −1 = β z 0 (Λ(z) − B) −1 ( β z 0 ) * is valid, where ( β z 0 ) * is the adjoint of β z 0 : G 1/2 −→ H 0 considered as an unbounded operator β z 0 : G H 0 with domain G 1/2 . (iii) We have σ(∆ B 0 ) \\ σ(∆ D 0 ) = { z / ∈ σ(∆ D 0 ) | 0 ∈ σ(Λ(z) − B) }. Proof. The proof follows from Theorem 2.30 because Λ(z)− B = Λ(Q z 0 −B) and ( β z 0 ) * = Λ(β z 0 ) * .\nLemma 3.5. If w = 0 then H 1 =H 1+ N w+ N −w (topological direct sum), and the projection P w onto N w is given by\nP w = 1 2 P w 2 0 1 w δP w 2 1 1 w dP w 2 0 P w 2 1 .\nIf w = ±i, then we have H 1 =H 1 ⊕ N i ⊕ N −i (orthogonal direct sum), and P ±i are orthogonal projections (in H 1 ).\n\nProof. Recall that P z p is the projection onto N z p = ker(∆ p − z). Denote byP p := 1 − P z p the projection ontoH 1 p and setP :=P 0 ⊕P 1 . Then we can decompose f ∈ H 1 as f =P f + P w f + P −w f, since P w + P −w = P w 2 0 ⊕ P w 2 1 andP + (P w 2 0 ⊕ P w 2 1 ) = 1. A simple calculation shows that DP w = wP w , i.e., that P w f ∈ N w ; in addition, (P w ) 2 = P w , i.e., P w is a projection; and P f ∈H 1 .\n\nThe sum of eigenspaces associated to different eigenvalues is direct, and N w+ N −w = N w 2 0 ⊕ N w 2 1 (Lemma 3.3). Since in addition, H 1 p =H 1 p+ N w 2 p , it follows that the sum H 1 =H 1+ N w+ N −w is direct. The direct sum is also topological since the projections are bounded operators. The orthogonality for w = ±i can be checked easily.\n\nLemma 3.6. Let D min be the restriction of D onto dom D min =H 1 :=H 1 0 ⊕H 1 1 = ker(Γ 0 ⊕ Γ 1 ). Then (D min ) * = D.\n\nProof. We refer to [BGP06, Thm. 1.13 (1)⇒(4)] for a proof. Note that D has self-adjoint restrictions since the defect index is 0 by Corollary 3.4.\n\nWe write D D := D↾ ker Γ 0 , the Dirichlet Dirac operator, and D N := D↾ ker Γ 1 , the Neumann Dirichlet operator. Note that (D D ) 2 = ∆ D 0 ⊕ ∆ D 1 and (D N ) 2 = ∆ N 0 ⊕ ∆ N 1 . Lemma 3.7. Let w / ∈ σ(D D ). The operator Γ 0 ↾ N w : N w −→ G 1/2 has a bounded inverse β w , and w → β w is a Γ-Krein field, i.e., β w : G 1/2 −→ N w is a topological isomorphism and (3.2a)\nβ w 1 = U w 1 ,w 2 β w 2 , w 1 , w 2 / ∈ σ(D D ), (3.2b) where U w 1 ,w 2 := (D D − w 2 )(D D − w 1 ) −1 = 1 + (w 1 − w 2 )(D D − w 1 ) −1 .\nFurthermore, β w = √ 2ψ w 0 β w 2 0 , where β z 0 is the Krein γ-function of order 0 associated with the first order boundary triple (H , G , γ 0 ) (cf. Definition 2.21) and ψ w 0 is defined in Lemma 3.3. Proof. For the proof of the first assertion, we refer again to [BGP06, Thm. 1.23 (2a-b)]. The relation with β w 2 0 follows from the fact that Γ 0 = γ 0 π 0 , where π 0 : H 1 −→ H 1 0 , f → f 0 ; and the inverse of √ 2ψ w 0 is π 0 (restricted to the appropriate subspaces). Lemma 3.8. The operator Q w := Γ 1 β w : G 1/2 −→ G 1/2 defines the Krein Q-function w → Q w , i.e., Q w 1 − (Q w 2 ) * = (w 1 − w 2 )(β w 2 ) * β w 1 w 1 , w 2 / ∈ σ(D D ).\n\nFurthermore, Q w = 1 w Q w 2 0 , where Q z 0 is the Krein Q-function associated to the first order boundary triple (H , G , γ 0 ) (cf. Definition 2.24).\n\nProof. For the proof of the first assertion, we refer again to [BGP06, Thm. 1.23 (2c)]. The other follows straightforward.\n\nFurther results like Krein's resolvent formula or the spectral relation for self-adjoint restrictions D B of D can be found e.g. in [BGP06]. In particular, if B is bounded and self-adjoint in G 1/2 then the restriction of D to Remark 4.2. The first order approach to boundary triples enables us to use the natural boundary maps γ 0 f = f ↾ ∂X and γ 1 η = η n ↾ ∂X , in contrast to the second order approach using the Laplacian as e.g. in [BMNW07,Pc07]. In the second order approach, the maximal domain of the Laplacian dom ∆ max = { f ∈ L 2 (X) | ∆f ∈ L 2 (X) } is not a subset of the Sobolev space H 1 (X). In particular, f ↾ ∂X is not in L 2 (∂X), but only in H −1/2 (∂X); and ∂ n f ↾ ∂X ∈ H −3/2 (∂X) (see e.g. [G68,G06,LM72]). In particular, Green's formula (cf. (3.1a)) fails to hold with the natural boundary maps.\ndom D B := { f ∈ H 1 | Γ 1 f = BΓ 0 f } = { f ∈ H 1 | γ 1 f 1 = Bγ 0 f 0 }\n....\n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . δ . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n.\n. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. d 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The spaces H 2 p are complete, i.e., Hilbert spaces with their natural norms. (ii) Note that ∆ p is a bounded operator on H 2 p . However, ∆ p with dom ∆ p = H 2 p is not closed. Although we call ∆ p the maximal Laplacian, it is not the maximal operator ∆ max p in the usual sens (which is the operator closure of ∆ p with domain dom ∆ max p :\n\n\nthe adjoint of β z 0 as operator β z 0 : G 1/2 −→ H 0 . Furthermore, ran(β z 0 ) * = G 1/2 . Proof. The assertion follows from (see also [BGP06, Thm. 1.23 (2d)])\n\nTheorem 2. 31 .\n31Let B be a self-adjoint and bounded operator in G and ∆ e B 0 the corresponding self-adjoint restriction of the Laplacian with domain\n\n\ntopological isomorphisms. In particular, for w = ±i, they are unitary.Corollary 3.4. The operator D has zero defect index, i.e., N i = ker(D −i) and N −i = ker(D +i) are isomorphic.\n\n\ndefines a self-adjoint operator D B . The Laplacian (D B ) 2 acts on each component as the Laplacian∆ p f p , but with domain dom(D B ) 2 = { f ∈ dom D B | Df ∈ dom D B } = { f ∈ H 2 | γ 1 f 1 = Bγ 0 f 0 , γ 1 df 0 = Bγ 0 δf 1 }. Note that this domain is different from dom ∆ B 0 ⊕ dom ∆ B 1 (cf.Definition 2.26) since the two components in dom(D B ) 2 are coupled.\n\n\nDefinition 2.1. Let H = H 0 ⊕ H 1 and G be Hilbert spaces.(i) Elements of H p are referred to as p-forms. \n(ii) A partial map d : H 0 \nH 1 is called an exterior derivative if d is a closed map with \ndense domain H 1 \n0 := dom d ⊂ H 0 . We endow H 1 \n0 with the natural norm defined by \nf 0 \n\n2 \nH 1 \n\n0 \n\n\n\n\na first order boundary triple for the exterior derivative d : H 0 H 1 if γ 0 a boundary map associated to d.Definition 2.2. We set d 0 := d↾H 1 \n\n0 \n\n, and call δ := d * \n0 : H 1 \nH 0 the divergence operator with \n\ndomain H 1 \n1 := dom δ andH 1 \n1 := dom d * (clearly,H 1 \n1 ⊂ H 1 \n1 , andH 1 \n1 is dense in H 1 since d is \ndensely defined). We endow H 1 \n1 with the natural norm \nf 1 \n\n2 \nH 1 \n\n1 \n\n:= f 1 \n2 + δf 1 \n2 . \n\nDefinition 2.3. \n(i) We call ∆ 0 := δd the Laplacian of degree 0 with domain \n\nH 2 \n0 := dom δd := f 0 ∈ dom d df 0 ∈ dom δ \nSimilarly, ∆ 1 := dδ is called the (maximal) Laplacian of degree 1 with domain \n\nH 2 \n1 := dom dδ := f 1 ∈ dom δ δf 1 ∈ dom d . \nWe endow H 2 \np with the norms \nf 0 \n\n2 \nH 2 \n\n0 \n\n\n\n\n/2 by Definition 2.8. The last assertion is obvious. Corollary 2.15. We have\nBoundary triplesIn this section we show how the first order approach of the last section fits into the setting of boundary triples in the usual sense. We only sketch the ideas here; for more details on boundary triples, we refer to[BGP06,DHMdS06]and the references therein.Definition 3.1. Let H be a Hilbert space with a closed operator D in H . Assume furthermore that G is another Hilbert space, and Γ 0 , Γ 1 : dom D −→ G are two linear maps. We say that ( G , Γ 0 , Γ 1 ) is an (ordinary) boundary triple for D iffandis an ordinary boundary triple for D.Proof. The Green's formula (3.1a) follows fromby Lemma 2.14. The second condition (3.1b) follows from Γ 0 ⊕ Γ 1 = γ 0 ⊕ γ 1 and the surjectivity of γ p : H 1 p −→ G 1/2 . The last condition (3.1c), i.e., the density ofH 1 :=H 1 0 ⊕H 1 1 in H , is a consequence of Definition 2.1 (iii).The next lemma can be proved readily:Manifolds with boundaryIn this section we present our main example and show how it fits into the abstract setting of first order boundary triples of Section 2 (see also[A00]).Let X be a Riemannian manifold with boundary ∂X equipped with their natural volume measures. Denote the cotangential bundle (or bundle of 1-forms) by T * X. The data we need to fix are the following:where L 2 (X) and L 2 (T * X) are the spaces of square-integrable functions and sections over the cotangent (1-form) bundle, and where d stands for the usual exterior derivative with domain H 1 0 := H 1 (X), the Sobolev space of functions f ∈ L 2 (X) such that |df | ∈ L 2 (X) (or df ∈ L 2 (T * X), what is the same).For the boundary map, we need to fix the boundary space G := L 2 (∂X), and we defineNote that the norm of γ 0 depends on the local geometry of X near ∂X. The range of γ 0 is G 1/2 = H 1/2 (∂X) together with the intrinsic norm defined in Section 2, namely, where f 0 is the solution of the Dirichlet problem (∆ 0 + 1)f 0 = 0 and γ 0 f 0 = ϕ. Since H 1/2 (∂X) = L 2 (∂X), the boundary map γ 0 is proper.After defining these data, we obtainH 1 0 =H 1 (X) = ker γ 0 and d 0 := d↾H 1 (X) . Furthermore, δ = d * 0 is the divergence operator. Comparing the abstract Green's formula in Lemma 2.14 with Green's formulawhere η n stands for the normal component of the 1-form η near ∂X, we see that γ 1 η = η n ↾ ∂X Remark 4.1. Note that H 1 1 := dom δ ⊂ L 2 (T * X) is not the Sobolev space of order 1 on 1forms, defined locally via charts. Therefore, γ 1 : dom δ −→ H −1/2 (∂X), and γ 1 does not map into H 1/2 (∂X), as one could naively guess.The Dirichlet-to-Neumann map in this case isfor ϕ ∈ H 1/2 (∂X) and z / ∈ σ(∆ D 0 ) (cf. Definition 2.24). Self-adjoint boundary conditions of the Laplacian on 0-forms like Robin boundary conditions are now given as follows: Let B be a bounded, real-valued function on ∂X and set B := Λ −1 B. Then B is bounded and self-adjoint on G 1/2 (Lemma 2.29) and In particular, the domain condition dom(∆ B 0 ) * ⊂ H 1 0 = H 1 (X) is fulfilled, and the above domain defines a self-adjoint Laplace operator (cf. Lemma 2.27).Note that in general, the Robin boundary conditions cannot be expressed as (D B ) 2 where D B is a self-adjoint restriction of the Dirac operator (cf. the end of Section 3). This is another justification of our first order approach (instead of directly starting from an ordinary boundary triple as in Section 3).\nAbstract boundary conditions for maximal sectorial extensions of sectorial operators. Y Arlinskii, Math. Nachr. 209Y. Arlinskii, Abstract boundary conditions for maximal sectorial extensions of sectorial operators, Math. Nachr. 209 (2000), 5-36.\n\nBoundary value problems for elliptic partial differential operators on bounded domains. J Behrndt, M Langer, J. Funct. Anal. 2432J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal. 243 (2007), no. 2, 536-565.\n\nBoundary value spaces and m-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices. M Brown, M Marletta, S Naboko, I Wood, arXiv:0704.2562PreprintM. Brown, M. Marletta, S. Naboko, and I. Wood, Boundary value spaces and m-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices, Preprint (arXiv:0704.2562) (2007).\n\nSpectra of self-adjoint extensions and applications to solvable Schrödinger operators. J Brüning, V Geyler, K Pankrashkin, PreprintJ. Brüning, V. Geyler, and K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schrödinger operators, Preprint (2006).\n\nBoundary relations and their Weyl families. V Derkach, S Hassi, M Malamud, H De Snoo, Trans. Amer. Math. Soc. 35812V. Derkach, S. Hassi, M. Malamud, and H. de Snoo, Boundary relations and their Weyl families, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5351-5400\n\nP Exner, O Post, arXiv:0706.0481Quantum networks modelled by graphs, Preprint. P. Exner and O. Post, Quantum networks modelled by graphs, Preprint (arXiv:0706.0481) (2007).\n\nA characterization of the non-local boundary value problems associated with an elliptic operator. G Grubb, Ann. Scuola Norm. Sup. Pisa. 3G. Grubb, A characterization of the non-local boundary value problems associated with an elliptic operator, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 425-513.\n\nKnown and unknown results on elliptic boundary problems. Bull. Amer. Math. Soc. (N.S.). 432electronic, Known and unknown results on elliptic boundary problems, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 2, 227-230 (electronic).\n\nNon-homogeneous boundary value problems and applications. J.-L Lions, E Magenes, Die Grundlehren der mathematischen Wissenschaften. P. KennethNew YorkSpringer-VerlagI181J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.\n\nSelf-adjoint extensions of restrictions. A Posilicano, arXiv:math-ph/0703078PreprintA. Posilicano, Self-adjoint extensions of restrictions, Preprint (arXiv:math-ph/0703078) (2007).\n\nSpectral convergence of quasi-one-dimensional spaces. O Post, Ann. Henri Poincaré. 75O. Post, Spectral convergence of quasi-one-dimensional spaces, Ann. Henri Poincaré 7 (2006), no. 5, 933-973.\n\nOn general boundary problems for elliptic differential equations. M I Vishik, Am. Math. Soc., Transl., II. Ser. 24EnglishM. I. Vishik, On general boundary problems for elliptic differential equations., Am. Math. Soc., Transl., II. Ser. 24 (1963), 107-172 (English. Russian original from 1952).\n\n. Rudower Chaussee. 25deInstitut für Mathematik, Humboldt-Universität zu Berlinpost@math.hu-berlinInstitut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany E-mail address: post@math.hu-berlin.de\n"},"corpus_id":{"kind":"number","value":17085199,"string":"17,085,199"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"0711.2218\",\n \"mag\": 2162102317,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": null,\n \"doi\": \"10.1134/s1061920807040139\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://arxiv.org/pdf/0711.2218v3.pdf\"\n ],\n \"pdf_sha\": \"3eda14ebb2a7c1ce7cfd4d8819a9ff7175aa4357\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/0711.2218\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":""},"authors":{"kind":"list like","value":["Olaf Post"],"string":"[\n \"Olaf Post\"\n]","isConversational":false},"abstract":{"kind":"string","value":"The aim of the present paper is to introduce a first order approach to the abstract concept of boundary triples for Laplace operators. Our main application is the Laplace operator on a manifold with boundary; a case in which the ordinary concept of boundary triples does not apply directly. In our first order approach, we show that we can use the usual boundary operators also in the abstract Green's formula. Another motivation for the first order approach is to give an intrinsic definition of the Dirichlet-to-Neumann map and intrinsic norms on the corresponding boundary spaces. We also show how the first order boundary triples can be used to define a usual boundary triple leading to a Dirac operator."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"The concept of boundary triples, originally introduced in [V63], has successfully be applied to the theory of self-adjoint extensions of symmetric operators, for example on quantum graphs, singular perturbations or point interactions on manifolds (see e.g. [BGP06]). For a general treatment of boundary triples we refer to [BGP06,DHMdS06] and the references therein.\nOur main purpose here is not to characterise all self-adjoint extensions of a given symmetric operator, but to show that the concept of boundary triples can also be used in the PDE case, namely to Laplacians on a manifold with boundary. The standard theory of boundary triples does not directly apply in this case, since Green's formula\ndoes not extend to f, g in the maximal operator domain dom ∆ max = { f ∈ L 2 (X) | ∆ max f ∈ L 2 (X) (distributional sense) } (cf. Remark 4.2 for details). A solution to overcome this problem is either to modify the boundary operators (restriction of the function and the normal derivative onto ∂X) as e.g. in [BMNW07,Pc07], or to introduce the concept of quasi boundary triples as in [BL07] (cf. also the references therein for further treatments of boundary triples in the PDE case).\nHere, we use a different approach: we start with first order operators, namely the exterior derivative d taking functions (0-forms) to 1-forms and its adjoint, the divergence operator δ, mapping 1-forms into functions, since the first order operator domains are simpler. The Laplacian (on functions) is then defined as ∆ 0 := δd. Certainly, in our approach we do not cover all selfadjoint extensions of the minimal Laplacian.\nThe abstract approach also allows to define the Dirichlet-to-Neumann map in an intrinsic manner, and also the norm of G 1/2 = H 1/2 (∂X) is defined intrinsicly. This might be a great advantage when dealing with parameter-depending manifolds, as it is the case for graph-like manifolds (see e.g. [EP07,P06]). We will treat this question in a forthcoming publication. Our approach is related to the recent works of Arlinskii [A00], Posilicano [Pc07] and Brown et al. [BMNW07], where also a PDE example is treated in the context of boundary triples.\nTo precise our idea of the first order approach we sketch the construction here. The given data are 1 H 0 , H 1 , d : H 0 H 1 , H 1 0 := dom d, where H p are Hilbert spaces (\"p-forms\"), and H 1 0 carries the graph norm. Guided by our main application (a manifold with boundary), we call d an exterior derivative.\nA boundary map (of order 0) is a bounded operator γ 0 : H 1 0 −→ G , G 1/2 := ran γ 0 with dense range G 1/2 ⊂ G , where G is another Hilbert space (usually over the boundary). For these data, we define d 0 := d restricted toH 1 0 := ker γ 0 and the divergence operator δ := d * 0 with domain H 1 1 := dom δ. Furthermore, we can define a natural norm on G 1/2 using γ 0 . In addition, we have a boundary operator of order 1, namely, γ 1 : H 1 1 −→ G , with the same range ran γ 1 = ran γ 0 = G 1/2 . Moreover, an abstract Green's formula is valid, i.e.,\nFinally, h p = β z p ϕ is the solution of the Dirichlet and Neumann problem ∆ p h p = zh p , γ p h p = ϕ, respectively; we call β z p also a Krein Γ-field of order p. The Krein Q-function is defined as Q z 0 ϕ := γ 1 dβ z 0 ; a bounded operator (on the boundary space G 1/2 ), closely related to the usual Dirichlet-to-Neumann map Λ(z) on a manifold with boundary defined in Eq. (4.1).\nThe main idea here is to consider the Laplacian ∆ 0 f 0 := δdf 0 on the space\n0 , we can develop a suitable theory of boundary spaces. In particular, for a bounded and self-adjoint operator B in G 1/2 we can show that the Laplacian ∆ 0 restricted to dom ∆ B 0 := { f 0 ∈ H 2 0 | γ 1 df 0 = Bγ 0 f 0 } (Robin-type boundary conditions) is self-adjoint under a suitable condition on the domain of the adjoint (fulfilled in our example of the Laplacian on a manifold with boundary). Our main result is Krein's resolvent formula for the resolvents of ∆ B 0 and the Dirichlet Laplacian ∆ D 0 ; and a spectral relation between the operators\n(see Theorem 2.30). The main advantage of our approach is that it can almost immediately be applied to the case of the Laplacian on a manifold with boundary, using the standard boundary operator (restriction of a function to the boundary and restriction of the normal component of a 1-form to the boundary).\nThe paper is organised as follows: In the next section, we develop the concept of first order boundary triples. In Section 3 we show how this concept fits into the usual theory of boundary triples. Section 4 contains our motivating example, namely, the Laplacian on a manifold with boundary.\nAcknowledgements. The author would like to dedicate this contribution to his dear colleague at the Humboldt University, Vladimir A. Geyler, who passed away very suddenly. The author acknowledges the financial support of the Collaborative Research Center SFB 647 \"Space -Time -Matter. Analytic and Geometric Structures\" in which Vladimir Geyler also was a member. Finally, the author would like to thank Y. Arlinskii for drawing his attention to the article [A00], where a very similar approach was used."},{"title":"First order approach","content":"In this section, we develop the concept of boundary triples for operators acting in different Hilbert spaces; guided by our main example of the exterior derivative on a manifold with boundary.\n(iii) We call γ 0 : H 1 0 −→ G a boundary map (of degree 0) associated to d iff γ 0 is bounded with dense image, and ifH 1 0 := ker γ 0 ⊂ H 1 0 = dom d is dense in H 0 . The auxiliary Hilbert space is also referred to as a boundary space. We say that\nWe denote the eigenspaces by N z\n∆ N 1 := dd * with the appropriate domains the Dirichlet Laplacian of degree p = 0, 1 and the Neumann Laplacian of degree p = 0, 1, respectively. Clearly, all these operators are self-adjoint and non-negative. We denote the corresponding resolvents by R D p := (∆ D p + 1) −1 and R N p := (∆ N p + 1) −1 .\nThe following diagram tries to illustrate the two scales of Hilbert spaces associated to d, d * and d 0 , d * 0 = δ (dotted arrows). Note that only at order 1, 0 and −1, we have relations between the two scales:\nin the distributional sense). In general, H 2 p dom ∆ max p . This observation is one of the motivations for our first order approach (see Section 4).\nHowever, by definition of the adjoint operator\nThe argument for p = 1 is similar. Lemma 2.6. The maps d : N 0 −→ N 1 and δ : N 1 −→ N 0 are unitary.\nProof.\nSince d is surjective, it is therefore unitary with unitary inverse −δ.\nLemma 2.7. Assume that the boundary map γ 0 is proper (i.e., G 1/2 = ran γ 0 G ). Definê\nProof. The operatorγ 0 is invertible since (ker γ 0 ) ⊥ = (H 1 0 ) ⊥ = N 0 by Lemma 2.5. If (γ 0 ) −1 were be bounded, thenγ 0 would be a topological isomorphism of N 0 and ran γ 0 = G 1/2 , in particular, G 1/2 would be closed in G , and by the density, we would have G 1/2 = G -a contradiction. The last assertion is an immediate consequence of Lemma 2.5 and the definition of the inverse map (γ 0 ) −1 .\nDefinition 2.8. We endow G 1/2 with the norm\nAssume that the boundary map γ 0 is proper (i.e., G 1/2 = ran γ 0 G ), then the following assertions hold:\nWe define the associated scale of Hilbert spaces by\nfor s ≥ 0 (and the dual with respect to (·, ·) G for s < 0).\nRemark 2.10. If γ 0 is not proper (i.e., if γ 0 is surjective, i.e., G 1/2 = G ), then all the above assertions remain valid except for the fact that (γ 0 ) −1 , ((γ 0 ) −1 ) * and Λ are bounded operators.\nProof. The first assertion follows from\nTo prove the second, note that γ 0 γ * 0 =γ 0γ * 0 is bijective and\nThe third assertion is a consequence of Lemma 2.7, and the domain characterisation can be seen readily. To prove the fourth assertion, take\nRemark 2.11. Note that G −1/2 is the completion of G with respect to the norm ϕ G −1/2 = γ 0 ϕ H 1 0 . Definition 2.12. We define the boundary map of order 1 as\nwhere P p is the orthogonal projection in H 1 p onto the subspace N p . Lemma 2.13. We have ker γ 1 =H 1 1 , and γ 1 : H 1 1 −→ G is bounded with norm γ 1 = γ 0 . Furthermore, ran γ 1 = G 1/2 andγ 1 := γ 1 ↾ N 1 is a unitary map from N 1 onto G 1/2 .\nThe boundedness follows from\nby Lemma 2.6. Furthermore, for f 0 ∈ N 0 set f 1 := df 0 , then γ 1 f 1 = −γ 0 δdf 0 = γ 0 f 0 . In particular, γ 1 = γ 0 . Finally,\nfor f 1 ∈ N 1 , since δf 1 ∈ N 0 and by Lemma 2.6.\nLemma 2.14. The (abstract) Green's formula holds, namely,\nwhere γ 1 := Λγ 1 : H 1 1 −→ G −1/2 . Proof. If f 0 ∈H 1 0 , then the LHS vanishes since δ = d * 0 , and so is the RHS, since γ 0 f 0 = 0. Similarly, if g 1 ∈H 1 1 = dom d * , then the LHS vanishes since δg 1 = d * g 1 and so is the RHS, because γ 1 g 1 = 0 by Lemma 2.13. For f 0 ∈ N 0 and g 1 ∈ N 1 , we have\n0 . The following lemma shows that Λ = Λ(−1) is the Dirichlet-to-Neumann map for the operator ∆ 0 + 1: Lemma 2.16. For ϕ ∈ G 1/2 and h 0 := (γ 0 ) −1 ϕ we have\nProof. By Lemma 2.14, we have\nOn the other hand, we have\nRemark 2.17. The map γ 1 is indeed the boundary map occuring in the applications (see Section 4). Namely, the Green's formula is usually formulated with a boundary integral given as an inner product of G rather than G 1/2 . In particular, γ 1 dh 0 is the \"normal derivative at the boundary\" (in the case of a manifold with boundary).\nThe boundary maps are also bounded as maps with target space G 1/2 : Lemma 2.18. The operators γ p : H 1 p −→ G 1/2 are bounded with norm bounded by 1. Proof. For p = 0, we have\nusing Lemmas 2.6-2.7.\nIn order to define the Dirichlet-to-Neumann map also for other resolvent values z, we need to provide results similar to those in Lemmas 2.5-2.7 for general z. Write\nand ι 0 :H 1 0 ֒→ H 1 0 . A simple calculation shows that (1 − P z 0 ) 2 = (1 − P z 0 ), i.e., 1 − P z 0 and therefore P z 0 are projections. Furthermore, f 0 = P z 0 f 0 is equivalent to ∆ 0 f 0 = zf 0 . In order to show that f 0 = P z 0 f 0 ∈ N z 0 let us first show that f 0 ∈ H 2 0 , i.e., that h 1 := df 0 ∈ H 1 1 = dom δ. To this end, recall the definition of the domain dom δ = dom d * 0 in (2.4). We have here df 0 , d 0 g 0 = δdf 0 , g 0 = zf 0 , g 0 by Lemma 2.14 (note that γ 0 g 0 = 0) and the fact that δdf 0 = zf 0 ; we can choose h 0 = zf 0 and therefore f 0 ∈ H 2 0 . A straightforward calculation shows now that (∆ 0 − z)f 0 = 0, and finally, f 0 ∈ N z 0 . By the definition of P z 0 , it is also clear that ran(1 − P z 0 ) ⊂H 1 0 , and therefore H 1 0 splits into the direct sum. The direct sum is also a topological sum, since 1 − P z 0 and P z 0 are bounded maps. Therefore f 0 → ((1 − P z 0 )f 0 , P z 0 f 0 ) is a bounded bijection, and also a topological isomorphism. The argument for 1-forms is similar, using\nand ι 1 :H 1 1 ֒→ H 1 1 . For the last assertion, note that ker γ p =H 1 p and that ran γ p = G 1/2 (see Lemma 2.13); in particular,γ z p is bijective. Furthermore,γ z p is bounded as restriction of the bounded map γ p : H 1 p −→ G 1/2 (cf. Lemma 2.18), and therefore,γ z p is a topological isomorphism.\nLemma 2.20. For z = 0, the maps d : N z 0 −→ N z 1 and δ : N z 1 −→ N z 0 are topological isomorphisms.\nProof. If f 0 ∈ N z 0 then dδdf 0 = zdf 0 , i.e, df 0 ∈ N z 1 . Similarly, f 1 ∈ N z 1 implies δf 1 ∈ N z 0 . Furthermore, 1 z δdf 0 = f 0 and d( 1 z δf 1 ) = f 1 implies that 1 z δ is the inverse of d. Finally, d is bounded on N z 0 , since df 0\nand therefore a topological isomorphism. The assertion for δ follows similarly.\nDefinition 2.21. We call z → β z 0 := (γ z 0 ) −1 , z / ∈ Σ 0 the Dirichlet solution map or the Krein Γ-field of order 0 associated to the first order boundary triple (H , G , γ 0 ). Similarly, we call z → β z 1 := (γ z 1 ) −1 , z / ∈ Σ 1 the Neumann solution map or the Krein Γ-field of order 1.\nRemark 2.22.\n(i) We prefer to use the symbol β instead of γ for the Krein Γ-field in order to avoid confusion with our boundary maps γ p . (ii) The maps β z p : G 1/2 −→ N z p ⊂ H 1 p are topological isomorphisms, since the inversesγ z p are. (iii) The names \"Dirichlet/Neumann solution map\" are due to the following fact: The p-form h p := β z p ϕ is the solution of (∆ p − z)h p = 0, and γ p h p = ϕ. For p = 0, this is the solution of the \"Dirichlet problem\" (γ 0 h 0 prescribed), and for p = 1, the solution of the \"Neumann problem\" (γ 1 h 1 prescribed). We will see in Lemma 3.7 that the Krein Γ-fields are related to a Krein Γ-field in the sense of an ordinary boundary triple. (iv) The map β z 0 : G 1/2 −→ H 1 0 regarded as an operator β z 0 : G 1/2 −→ H 0 into H 0 is bounded, as well as its adjoint, denoted by (β z 0 ) * :\nby Corollary 2.15 for the second equality. As far as the third equality is concerned, note that the first term vanishes since β z 0 ϕ solves the eigenvalue equation; the same holds for the third term since γ 0 f 0 = 0 for f 0 ∈H 1 0 . For the second term, we have γ 0 β z 0 ϕ = ϕ by the definition of β z 0 . The last assertion follows from ran(β z 0 ) * = (ker β z 0 ) ⊥ and from the fact that β z 0 :\nWe can now define the Dirichlet-to-Neumann map and a closely related map for arbitrary resolvent values z:\nDefinition 2.24. The Krein Q-function associated to the first order boundary triple (H , G , γ 0 ) is the map z → Q z 0 := γ 1 d(γ z 0 ) −1 = γ 1 dβ z 0 , z / ∈ Σ 0 = σ(∆ D 0 ). For z / ∈ Σ 0 , the abstract Dirichlet-to-Neumann map at z is defined by\nRemark 2.25. (i) We shall see in Section 3 that Q z 0 is indeed a Krein Q-function for an ordinary boundary triple. Note that Q z 0 : G 1/2 −→ G 1/2 is a bounded map (cf. Lemmas 2.18-2.20). In addition, we have\n(ii) Note that Λ(z) is indeed the Dirichlet-to-Neumann map: We solve the Dirichlet problem h 0 = β z 0 ϕ, i.e, ∆ 0 h 0 = zh 0 , γ 0 h 0 = ϕ; and the Dirichlet-to-Neumann map is the \"normal derivative at the boundary\" of h 0 (cf. Remark 2.17), i.e., Λ(z)ϕ = γ 1 dh 0 .\nLet us now define self-adjoint restrictions of ∆ 0 .\nDefinition 2.26. Let B be a bounded operator in G 1/2 . We set\nRemark 2.28. The domain condition does not seem to follow from abstract (\"soft\") arguments; in our manifold example, it follows from elliptic regularity (\"hard\" arguments). Note that in general, dom ∆ max 0 defined in (2.2) is even not a subset of H 1 0 (see Remark 2.4 (ii) and Remark 4.2). Proof. The graph of the operator (∆ B 0 ) * is given as\n, and the latter inclusion holds by our assumption on the domain of the adjoint. In particular, f 0 , g 0 ∈ H 2 0 and we can apply Corollary 2.15, namely, ∆ max\nand the latter equality follows from f 0 , g 0 ∈ dom ∆ B 0 . The assertion is now obvious. The self-adjointness of B in G 1/2 can be shown as follows:\nLemma 2.29. Let B be a bounded and self-adjoint operator on G . In this case, B := Λ −1 B is bounded and self-adjoint as operator on G 1/2 . Proof. We have\n, so that B is bounded on G 1/2 , and Bϕ, ψ G 1/2 = Λ 1/2 Bϕ, Λ 1/2 ψ G = Λ −1/2 Bϕ, Λ 1/2 ψ G = Bϕ, ψ G and the similar symmetric expression shows the self-adjointness.\nWe can now formulate our main result. For brevity, we restrict ourselves here to 0-forms. Similar results hold also for 1-forms.\nThe proof is again closely related to the proof for ordinary boundary triples (cf. [BGP06,Thm. 1.29]). For the first assertion, take ϕ ∈ ker(Q z 0 − B) and set f 0 = β z 0 ϕ. By the definition of the solution map β z 0 , we have (∆ 0 − z)f 0 = 0 and γ 0 f 0 = ϕ. Furthermore, Q z 0 ϕ = Bϕ is equivalent to γ 1 df 0 = Bγ 0 f 0 by the definition of Q z 0 . However, the last equation shows that f 0 ∈ dom ∆ B 0 , i.e., f 0 ∈ ker(∆ B 0 − z). The opposite inclusion follows similarly. To prove the second assertion, take h 0 ∈ H 0 and f 0 :\n(2.6) Now we apply γ 1 d to the decomposition of f 0 ∈ dom ∆ B 0 and obtain\nusing the definition of Q z 0 (cf. Definition 2.24) and Lemma 2.23 for the third equality. In particular,\nand the RHS covers the entire space G 1/2 since h 0 covers H 0 (see again Lemma 2.23). In particular, (Q z 0 − B) is surjective. By (i), this operator is also injective, i.e., 0 / ∈ σ(Q z 0 − B). Krein's formula now follows from (2.6)-(2.7). The last assertion is a consequence of (ii).\nReturning to the original boundary space G and the Dirichlet-to-Neumann map Λ(z) = ΛQ z 0regarded as an unbounded operator in G -, we obtain the following result: The next lemma is a well known fact; we give a proof for completeness.\nLemma 3.5. If w = 0 then H 1 =H 1+ N w+ N −w (topological direct sum), and the projection P w onto N w is given by\nIf w = ±i, then we have H 1 =H 1 ⊕ N i ⊕ N −i (orthogonal direct sum), and P ±i are orthogonal projections (in H 1 ).\nProof. Recall that P z p is the projection onto N z p = ker(∆ p − z). Denote byP p := 1 − P z p the projection ontoH 1 p and setP :=P 0 ⊕P 1 . Then we can decompose f ∈ H 1 as f =P f + P w f + P −w f, since P w + P −w = P w 2 0 ⊕ P w 2 1 andP + (P w 2 0 ⊕ P w 2 1 ) = 1. A simple calculation shows that DP w = wP w , i.e., that P w f ∈ N w ; in addition, (P w ) 2 = P w , i.e., P w is a projection; and P f ∈H 1 .\nThe sum of eigenspaces associated to different eigenvalues is direct, and N w+ N −w = N w 2 0 ⊕ N w 2 1 (Lemma 3.3). Since in addition, H 1 p =H 1 p+ N w 2 p , it follows that the sum H 1 =H 1+ N w+ N −w is direct. The direct sum is also topological since the projections are bounded operators. The orthogonality for w = ±i can be checked easily.\nLemma 3.6. Let D min be the restriction of D onto dom D min =H 1 :=H 1 0 ⊕H 1 1 = ker(Γ 0 ⊕ Γ 1 ). Then (D min ) * = D.\nProof. We refer to [BGP06, Thm. 1.13 (1)⇒(4)] for a proof. Note that D has self-adjoint restrictions since the defect index is 0 by Corollary 3.4.\nWe write D D := D↾ ker Γ 0 , the Dirichlet Dirac operator, and D N := D↾ ker Γ 1 , the Neumann Dirichlet operator. Note that (D D ) 2 = ∆ D 0 ⊕ ∆ D 1 and (D N ) 2 = ∆ N 0 ⊕ ∆ N 1 . Lemma 3.7. Let w / ∈ σ(D D ). The operator Γ 0 ↾ N w : N w −→ G 1/2 has a bounded inverse β w , and w → β w is a Γ-Krein field, i.e., β w : G 1/2 −→ N w is a topological isomorphism and (3.2a)\nFurthermore, β w = √ 2ψ w 0 β w 2 0 , where β z 0 is the Krein γ-function of order 0 associated with the first order boundary triple (H , G , γ 0 ) (cf. Definition 2.21) and ψ w 0 is defined in Lemma 3.3. Proof. For the proof of the first assertion, we refer again to [BGP06, Thm. 1.23 (2a-b)]. The relation with β w 2 0 follows from the fact that Γ 0 = γ 0 π 0 , where π 0 : H 1 −→ H 1 0 , f → f 0 ; and the inverse of √ 2ψ w 0 is π 0 (restricted to the appropriate subspaces). Lemma 3.8. The operator Q w := Γ 1 β w : G 1/2 −→ G 1/2 defines the Krein Q-function w → Q w , i.e., Q w 1 − (Q w 2 ) * = (w 1 − w 2 )(β w 2 ) * β w 1 w 1 , w 2 / ∈ σ(D D ).\nFurthermore, Q w = 1 w Q w 2 0 , where Q z 0 is the Krein Q-function associated to the first order boundary triple (H , G , γ 0 ) (cf. Definition 2.24).\nProof. For the proof of the first assertion, we refer again to [BGP06, Thm. 1.23 (2c)]. The other follows straightforward.\nFurther results like Krein's resolvent formula or the spectral relation for self-adjoint restrictions D B of D can be found e.g. in [BGP06]. In particular, if B is bounded and self-adjoint in G 1/2 then the restriction of D to Remark 4.2. The first order approach to boundary triples enables us to use the natural boundary maps γ 0 f = f ↾ ∂X and γ 1 η = η n ↾ ∂X , in contrast to the second order approach using the Laplacian as e.g. in [BMNW07,Pc07]. In the second order approach, the maximal domain of the Laplacian dom ∆ max = { f ∈ L 2 (X) | ∆f ∈ L 2 (X) } is not a subset of the Sobolev space H 1 (X). In particular, f ↾ ∂X is not in L 2 (∂X), but only in H −1/2 (∂X); and ∂ n f ↾ ∂X ∈ H −3/2 (∂X) (see e.g. [G68,G06,LM72]). In particular, Green's formula (cf. (3.1a)) fails to hold with the natural boundary maps."},{"title":"....","content":""},{"title":".","content":""},{"title":"Theorem 2. 31 .","content":""}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"The concept of boundary triples, originally introduced in [V63], has successfully be applied to the theory of self-adjoint extensions of symmetric operators, for example on quantum graphs, singular perturbations or point interactions on manifolds (see e.g. [BGP06]). For a general treatment of boundary triples we refer to [BGP06,DHMdS06] and the references therein.\\nOur main purpose here is not to characterise all self-adjoint extensions of a given symmetric operator, but to show that the concept of boundary triples can also be used in the PDE case, namely to Laplacians on a manifold with boundary. The standard theory of boundary triples does not directly apply in this case, since Green's formula\\ndoes not extend to f, g in the maximal operator domain dom ∆ max = { f ∈ L 2 (X) | ∆ max f ∈ L 2 (X) (distributional sense) } (cf. Remark 4.2 for details). A solution to overcome this problem is either to modify the boundary operators (restriction of the function and the normal derivative onto ∂X) as e.g. in [BMNW07,Pc07], or to introduce the concept of quasi boundary triples as in [BL07] (cf. also the references therein for further treatments of boundary triples in the PDE case).\\nHere, we use a different approach: we start with first order operators, namely the exterior derivative d taking functions (0-forms) to 1-forms and its adjoint, the divergence operator δ, mapping 1-forms into functions, since the first order operator domains are simpler. The Laplacian (on functions) is then defined as ∆ 0 := δd. Certainly, in our approach we do not cover all selfadjoint extensions of the minimal Laplacian.\\nThe abstract approach also allows to define the Dirichlet-to-Neumann map in an intrinsic manner, and also the norm of G 1/2 = H 1/2 (∂X) is defined intrinsicly. This might be a great advantage when dealing with parameter-depending manifolds, as it is the case for graph-like manifolds (see e.g. [EP07,P06]). We will treat this question in a forthcoming publication. Our approach is related to the recent works of Arlinskii [A00], Posilicano [Pc07] and Brown et al. [BMNW07], where also a PDE example is treated in the context of boundary triples.\\nTo precise our idea of the first order approach we sketch the construction here. The given data are 1 H 0 , H 1 , d : H 0 H 1 , H 1 0 := dom d, where H p are Hilbert spaces (\\\"p-forms\\\"), and H 1 0 carries the graph norm. Guided by our main application (a manifold with boundary), we call d an exterior derivative.\\nA boundary map (of order 0) is a bounded operator γ 0 : H 1 0 −→ G , G 1/2 := ran γ 0 with dense range G 1/2 ⊂ G , where G is another Hilbert space (usually over the boundary). For these data, we define d 0 := d restricted toH 1 0 := ker γ 0 and the divergence operator δ := d * 0 with domain H 1 1 := dom δ. Furthermore, we can define a natural norm on G 1/2 using γ 0 . In addition, we have a boundary operator of order 1, namely, γ 1 : H 1 1 −→ G , with the same range ran γ 1 = ran γ 0 = G 1/2 . Moreover, an abstract Green's formula is valid, i.e.,\\nFinally, h p = β z p ϕ is the solution of the Dirichlet and Neumann problem ∆ p h p = zh p , γ p h p = ϕ, respectively; we call β z p also a Krein Γ-field of order p. The Krein Q-function is defined as Q z 0 ϕ := γ 1 dβ z 0 ; a bounded operator (on the boundary space G 1/2 ), closely related to the usual Dirichlet-to-Neumann map Λ(z) on a manifold with boundary defined in Eq. (4.1).\\nThe main idea here is to consider the Laplacian ∆ 0 f 0 := δdf 0 on the space\\n0 , we can develop a suitable theory of boundary spaces. In particular, for a bounded and self-adjoint operator B in G 1/2 we can show that the Laplacian ∆ 0 restricted to dom ∆ B 0 := { f 0 ∈ H 2 0 | γ 1 df 0 = Bγ 0 f 0 } (Robin-type boundary conditions) is self-adjoint under a suitable condition on the domain of the adjoint (fulfilled in our example of the Laplacian on a manifold with boundary). Our main result is Krein's resolvent formula for the resolvents of ∆ B 0 and the Dirichlet Laplacian ∆ D 0 ; and a spectral relation between the operators\\n(see Theorem 2.30). The main advantage of our approach is that it can almost immediately be applied to the case of the Laplacian on a manifold with boundary, using the standard boundary operator (restriction of a function to the boundary and restriction of the normal component of a 1-form to the boundary).\\nThe paper is organised as follows: In the next section, we develop the concept of first order boundary triples. In Section 3 we show how this concept fits into the usual theory of boundary triples. Section 4 contains our motivating example, namely, the Laplacian on a manifold with boundary.\\nAcknowledgements. The author would like to dedicate this contribution to his dear colleague at the Humboldt University, Vladimir A. Geyler, who passed away very suddenly. The author acknowledges the financial support of the Collaborative Research Center SFB 647 \\\"Space -Time -Matter. Analytic and Geometric Structures\\\" in which Vladimir Geyler also was a member. Finally, the author would like to thank Y. Arlinskii for drawing his attention to the article [A00], where a very similar approach was used.\"\n },\n {\n \"title\": \"First order approach\",\n \"content\": \"In this section, we develop the concept of boundary triples for operators acting in different Hilbert spaces; guided by our main example of the exterior derivative on a manifold with boundary.\\n(iii) We call γ 0 : H 1 0 −→ G a boundary map (of degree 0) associated to d iff γ 0 is bounded with dense image, and ifH 1 0 := ker γ 0 ⊂ H 1 0 = dom d is dense in H 0 . The auxiliary Hilbert space is also referred to as a boundary space. We say that\\nWe denote the eigenspaces by N z\\n∆ N 1 := dd * with the appropriate domains the Dirichlet Laplacian of degree p = 0, 1 and the Neumann Laplacian of degree p = 0, 1, respectively. Clearly, all these operators are self-adjoint and non-negative. We denote the corresponding resolvents by R D p := (∆ D p + 1) −1 and R N p := (∆ N p + 1) −1 .\\nThe following diagram tries to illustrate the two scales of Hilbert spaces associated to d, d * and d 0 , d * 0 = δ (dotted arrows). Note that only at order 1, 0 and −1, we have relations between the two scales:\\nin the distributional sense). In general, H 2 p dom ∆ max p . This observation is one of the motivations for our first order approach (see Section 4).\\nHowever, by definition of the adjoint operator\\nThe argument for p = 1 is similar. Lemma 2.6. The maps d : N 0 −→ N 1 and δ : N 1 −→ N 0 are unitary.\\nProof.\\nSince d is surjective, it is therefore unitary with unitary inverse −δ.\\nLemma 2.7. Assume that the boundary map γ 0 is proper (i.e., G 1/2 = ran γ 0 G ). Definê\\nProof. The operatorγ 0 is invertible since (ker γ 0 ) ⊥ = (H 1 0 ) ⊥ = N 0 by Lemma 2.5. If (γ 0 ) −1 were be bounded, thenγ 0 would be a topological isomorphism of N 0 and ran γ 0 = G 1/2 , in particular, G 1/2 would be closed in G , and by the density, we would have G 1/2 = G -a contradiction. The last assertion is an immediate consequence of Lemma 2.5 and the definition of the inverse map (γ 0 ) −1 .\\nDefinition 2.8. We endow G 1/2 with the norm\\nAssume that the boundary map γ 0 is proper (i.e., G 1/2 = ran γ 0 G ), then the following assertions hold:\\nWe define the associated scale of Hilbert spaces by\\nfor s ≥ 0 (and the dual with respect to (·, ·) G for s < 0).\\nRemark 2.10. If γ 0 is not proper (i.e., if γ 0 is surjective, i.e., G 1/2 = G ), then all the above assertions remain valid except for the fact that (γ 0 ) −1 , ((γ 0 ) −1 ) * and Λ are bounded operators.\\nProof. The first assertion follows from\\nTo prove the second, note that γ 0 γ * 0 =γ 0γ * 0 is bijective and\\nThe third assertion is a consequence of Lemma 2.7, and the domain characterisation can be seen readily. To prove the fourth assertion, take\\nRemark 2.11. Note that G −1/2 is the completion of G with respect to the norm ϕ G −1/2 = γ 0 ϕ H 1 0 . Definition 2.12. We define the boundary map of order 1 as\\nwhere P p is the orthogonal projection in H 1 p onto the subspace N p . Lemma 2.13. We have ker γ 1 =H 1 1 , and γ 1 : H 1 1 −→ G is bounded with norm γ 1 = γ 0 . Furthermore, ran γ 1 = G 1/2 andγ 1 := γ 1 ↾ N 1 is a unitary map from N 1 onto G 1/2 .\\nThe boundedness follows from\\nby Lemma 2.6. Furthermore, for f 0 ∈ N 0 set f 1 := df 0 , then γ 1 f 1 = −γ 0 δdf 0 = γ 0 f 0 . In particular, γ 1 = γ 0 . Finally,\\nfor f 1 ∈ N 1 , since δf 1 ∈ N 0 and by Lemma 2.6.\\nLemma 2.14. The (abstract) Green's formula holds, namely,\\nwhere γ 1 := Λγ 1 : H 1 1 −→ G −1/2 . Proof. If f 0 ∈H 1 0 , then the LHS vanishes since δ = d * 0 , and so is the RHS, since γ 0 f 0 = 0. Similarly, if g 1 ∈H 1 1 = dom d * , then the LHS vanishes since δg 1 = d * g 1 and so is the RHS, because γ 1 g 1 = 0 by Lemma 2.13. For f 0 ∈ N 0 and g 1 ∈ N 1 , we have\\n0 . The following lemma shows that Λ = Λ(−1) is the Dirichlet-to-Neumann map for the operator ∆ 0 + 1: Lemma 2.16. For ϕ ∈ G 1/2 and h 0 := (γ 0 ) −1 ϕ we have\\nProof. By Lemma 2.14, we have\\nOn the other hand, we have\\nRemark 2.17. The map γ 1 is indeed the boundary map occuring in the applications (see Section 4). Namely, the Green's formula is usually formulated with a boundary integral given as an inner product of G rather than G 1/2 . In particular, γ 1 dh 0 is the \\\"normal derivative at the boundary\\\" (in the case of a manifold with boundary).\\nThe boundary maps are also bounded as maps with target space G 1/2 : Lemma 2.18. The operators γ p : H 1 p −→ G 1/2 are bounded with norm bounded by 1. Proof. For p = 0, we have\\nusing Lemmas 2.6-2.7.\\nIn order to define the Dirichlet-to-Neumann map also for other resolvent values z, we need to provide results similar to those in Lemmas 2.5-2.7 for general z. Write\\nand ι 0 :H 1 0 ֒→ H 1 0 . A simple calculation shows that (1 − P z 0 ) 2 = (1 − P z 0 ), i.e., 1 − P z 0 and therefore P z 0 are projections. Furthermore, f 0 = P z 0 f 0 is equivalent to ∆ 0 f 0 = zf 0 . In order to show that f 0 = P z 0 f 0 ∈ N z 0 let us first show that f 0 ∈ H 2 0 , i.e., that h 1 := df 0 ∈ H 1 1 = dom δ. To this end, recall the definition of the domain dom δ = dom d * 0 in (2.4). We have here df 0 , d 0 g 0 = δdf 0 , g 0 = zf 0 , g 0 by Lemma 2.14 (note that γ 0 g 0 = 0) and the fact that δdf 0 = zf 0 ; we can choose h 0 = zf 0 and therefore f 0 ∈ H 2 0 . A straightforward calculation shows now that (∆ 0 − z)f 0 = 0, and finally, f 0 ∈ N z 0 . By the definition of P z 0 , it is also clear that ran(1 − P z 0 ) ⊂H 1 0 , and therefore H 1 0 splits into the direct sum. The direct sum is also a topological sum, since 1 − P z 0 and P z 0 are bounded maps. Therefore f 0 → ((1 − P z 0 )f 0 , P z 0 f 0 ) is a bounded bijection, and also a topological isomorphism. The argument for 1-forms is similar, using\\nand ι 1 :H 1 1 ֒→ H 1 1 . For the last assertion, note that ker γ p =H 1 p and that ran γ p = G 1/2 (see Lemma 2.13); in particular,γ z p is bijective. Furthermore,γ z p is bounded as restriction of the bounded map γ p : H 1 p −→ G 1/2 (cf. Lemma 2.18), and therefore,γ z p is a topological isomorphism.\\nLemma 2.20. For z = 0, the maps d : N z 0 −→ N z 1 and δ : N z 1 −→ N z 0 are topological isomorphisms.\\nProof. If f 0 ∈ N z 0 then dδdf 0 = zdf 0 , i.e, df 0 ∈ N z 1 . Similarly, f 1 ∈ N z 1 implies δf 1 ∈ N z 0 . Furthermore, 1 z δdf 0 = f 0 and d( 1 z δf 1 ) = f 1 implies that 1 z δ is the inverse of d. Finally, d is bounded on N z 0 , since df 0\\nand therefore a topological isomorphism. The assertion for δ follows similarly.\\nDefinition 2.21. We call z → β z 0 := (γ z 0 ) −1 , z / ∈ Σ 0 the Dirichlet solution map or the Krein Γ-field of order 0 associated to the first order boundary triple (H , G , γ 0 ). Similarly, we call z → β z 1 := (γ z 1 ) −1 , z / ∈ Σ 1 the Neumann solution map or the Krein Γ-field of order 1.\\nRemark 2.22.\\n(i) We prefer to use the symbol β instead of γ for the Krein Γ-field in order to avoid confusion with our boundary maps γ p . (ii) The maps β z p : G 1/2 −→ N z p ⊂ H 1 p are topological isomorphisms, since the inversesγ z p are. (iii) The names \\\"Dirichlet/Neumann solution map\\\" are due to the following fact: The p-form h p := β z p ϕ is the solution of (∆ p − z)h p = 0, and γ p h p = ϕ. For p = 0, this is the solution of the \\\"Dirichlet problem\\\" (γ 0 h 0 prescribed), and for p = 1, the solution of the \\\"Neumann problem\\\" (γ 1 h 1 prescribed). We will see in Lemma 3.7 that the Krein Γ-fields are related to a Krein Γ-field in the sense of an ordinary boundary triple. (iv) The map β z 0 : G 1/2 −→ H 1 0 regarded as an operator β z 0 : G 1/2 −→ H 0 into H 0 is bounded, as well as its adjoint, denoted by (β z 0 ) * :\\nby Corollary 2.15 for the second equality. As far as the third equality is concerned, note that the first term vanishes since β z 0 ϕ solves the eigenvalue equation; the same holds for the third term since γ 0 f 0 = 0 for f 0 ∈H 1 0 . For the second term, we have γ 0 β z 0 ϕ = ϕ by the definition of β z 0 . The last assertion follows from ran(β z 0 ) * = (ker β z 0 ) ⊥ and from the fact that β z 0 :\\nWe can now define the Dirichlet-to-Neumann map and a closely related map for arbitrary resolvent values z:\\nDefinition 2.24. The Krein Q-function associated to the first order boundary triple (H , G , γ 0 ) is the map z → Q z 0 := γ 1 d(γ z 0 ) −1 = γ 1 dβ z 0 , z / ∈ Σ 0 = σ(∆ D 0 ). For z / ∈ Σ 0 , the abstract Dirichlet-to-Neumann map at z is defined by\\nRemark 2.25. (i) We shall see in Section 3 that Q z 0 is indeed a Krein Q-function for an ordinary boundary triple. Note that Q z 0 : G 1/2 −→ G 1/2 is a bounded map (cf. Lemmas 2.18-2.20). In addition, we have\\n(ii) Note that Λ(z) is indeed the Dirichlet-to-Neumann map: We solve the Dirichlet problem h 0 = β z 0 ϕ, i.e, ∆ 0 h 0 = zh 0 , γ 0 h 0 = ϕ; and the Dirichlet-to-Neumann map is the \\\"normal derivative at the boundary\\\" of h 0 (cf. Remark 2.17), i.e., Λ(z)ϕ = γ 1 dh 0 .\\nLet us now define self-adjoint restrictions of ∆ 0 .\\nDefinition 2.26. Let B be a bounded operator in G 1/2 . We set\\nRemark 2.28. The domain condition does not seem to follow from abstract (\\\"soft\\\") arguments; in our manifold example, it follows from elliptic regularity (\\\"hard\\\" arguments). Note that in general, dom ∆ max 0 defined in (2.2) is even not a subset of H 1 0 (see Remark 2.4 (ii) and Remark 4.2). Proof. The graph of the operator (∆ B 0 ) * is given as\\n, and the latter inclusion holds by our assumption on the domain of the adjoint. In particular, f 0 , g 0 ∈ H 2 0 and we can apply Corollary 2.15, namely, ∆ max\\nand the latter equality follows from f 0 , g 0 ∈ dom ∆ B 0 . The assertion is now obvious. The self-adjointness of B in G 1/2 can be shown as follows:\\nLemma 2.29. Let B be a bounded and self-adjoint operator on G . In this case, B := Λ −1 B is bounded and self-adjoint as operator on G 1/2 . Proof. We have\\n, so that B is bounded on G 1/2 , and Bϕ, ψ G 1/2 = Λ 1/2 Bϕ, Λ 1/2 ψ G = Λ −1/2 Bϕ, Λ 1/2 ψ G = Bϕ, ψ G and the similar symmetric expression shows the self-adjointness.\\nWe can now formulate our main result. For brevity, we restrict ourselves here to 0-forms. Similar results hold also for 1-forms.\\nThe proof is again closely related to the proof for ordinary boundary triples (cf. [BGP06,Thm. 1.29]). For the first assertion, take ϕ ∈ ker(Q z 0 − B) and set f 0 = β z 0 ϕ. By the definition of the solution map β z 0 , we have (∆ 0 − z)f 0 = 0 and γ 0 f 0 = ϕ. Furthermore, Q z 0 ϕ = Bϕ is equivalent to γ 1 df 0 = Bγ 0 f 0 by the definition of Q z 0 . However, the last equation shows that f 0 ∈ dom ∆ B 0 , i.e., f 0 ∈ ker(∆ B 0 − z). The opposite inclusion follows similarly. To prove the second assertion, take h 0 ∈ H 0 and f 0 :\\n(2.6) Now we apply γ 1 d to the decomposition of f 0 ∈ dom ∆ B 0 and obtain\\nusing the definition of Q z 0 (cf. Definition 2.24) and Lemma 2.23 for the third equality. In particular,\\nand the RHS covers the entire space G 1/2 since h 0 covers H 0 (see again Lemma 2.23). In particular, (Q z 0 − B) is surjective. By (i), this operator is also injective, i.e., 0 / ∈ σ(Q z 0 − B). Krein's formula now follows from (2.6)-(2.7). The last assertion is a consequence of (ii).\\nReturning to the original boundary space G and the Dirichlet-to-Neumann map Λ(z) = ΛQ z 0regarded as an unbounded operator in G -, we obtain the following result: The next lemma is a well known fact; we give a proof for completeness.\\nLemma 3.5. If w = 0 then H 1 =H 1+ N w+ N −w (topological direct sum), and the projection P w onto N w is given by\\nIf w = ±i, then we have H 1 =H 1 ⊕ N i ⊕ N −i (orthogonal direct sum), and P ±i are orthogonal projections (in H 1 ).\\nProof. Recall that P z p is the projection onto N z p = ker(∆ p − z). Denote byP p := 1 − P z p the projection ontoH 1 p and setP :=P 0 ⊕P 1 . Then we can decompose f ∈ H 1 as f =P f + P w f + P −w f, since P w + P −w = P w 2 0 ⊕ P w 2 1 andP + (P w 2 0 ⊕ P w 2 1 ) = 1. A simple calculation shows that DP w = wP w , i.e., that P w f ∈ N w ; in addition, (P w ) 2 = P w , i.e., P w is a projection; and P f ∈H 1 .\\nThe sum of eigenspaces associated to different eigenvalues is direct, and N w+ N −w = N w 2 0 ⊕ N w 2 1 (Lemma 3.3). Since in addition, H 1 p =H 1 p+ N w 2 p , it follows that the sum H 1 =H 1+ N w+ N −w is direct. The direct sum is also topological since the projections are bounded operators. The orthogonality for w = ±i can be checked easily.\\nLemma 3.6. Let D min be the restriction of D onto dom D min =H 1 :=H 1 0 ⊕H 1 1 = ker(Γ 0 ⊕ Γ 1 ). Then (D min ) * = D.\\nProof. We refer to [BGP06, Thm. 1.13 (1)⇒(4)] for a proof. Note that D has self-adjoint restrictions since the defect index is 0 by Corollary 3.4.\\nWe write D D := D↾ ker Γ 0 , the Dirichlet Dirac operator, and D N := D↾ ker Γ 1 , the Neumann Dirichlet operator. Note that (D D ) 2 = ∆ D 0 ⊕ ∆ D 1 and (D N ) 2 = ∆ N 0 ⊕ ∆ N 1 . Lemma 3.7. Let w / ∈ σ(D D ). The operator Γ 0 ↾ N w : N w −→ G 1/2 has a bounded inverse β w , and w → β w is a Γ-Krein field, i.e., β w : G 1/2 −→ N w is a topological isomorphism and (3.2a)\\nFurthermore, β w = √ 2ψ w 0 β w 2 0 , where β z 0 is the Krein γ-function of order 0 associated with the first order boundary triple (H , G , γ 0 ) (cf. Definition 2.21) and ψ w 0 is defined in Lemma 3.3. Proof. For the proof of the first assertion, we refer again to [BGP06, Thm. 1.23 (2a-b)]. The relation with β w 2 0 follows from the fact that Γ 0 = γ 0 π 0 , where π 0 : H 1 −→ H 1 0 , f → f 0 ; and the inverse of √ 2ψ w 0 is π 0 (restricted to the appropriate subspaces). Lemma 3.8. The operator Q w := Γ 1 β w : G 1/2 −→ G 1/2 defines the Krein Q-function w → Q w , i.e., Q w 1 − (Q w 2 ) * = (w 1 − w 2 )(β w 2 ) * β w 1 w 1 , w 2 / ∈ σ(D D ).\\nFurthermore, Q w = 1 w Q w 2 0 , where Q z 0 is the Krein Q-function associated to the first order boundary triple (H , G , γ 0 ) (cf. Definition 2.24).\\nProof. For the proof of the first assertion, we refer again to [BGP06, Thm. 1.23 (2c)]. The other follows straightforward.\\nFurther results like Krein's resolvent formula or the spectral relation for self-adjoint restrictions D B of D can be found e.g. in [BGP06]. In particular, if B is bounded and self-adjoint in G 1/2 then the restriction of D to Remark 4.2. The first order approach to boundary triples enables us to use the natural boundary maps γ 0 f = f ↾ ∂X and γ 1 η = η n ↾ ∂X , in contrast to the second order approach using the Laplacian as e.g. in [BMNW07,Pc07]. In the second order approach, the maximal domain of the Laplacian dom ∆ max = { f ∈ L 2 (X) | ∆f ∈ L 2 (X) } is not a subset of the Sobolev space H 1 (X). In particular, f ↾ ∂X is not in L 2 (∂X), but only in H −1/2 (∂X); and ∂ n f ↾ ∂X ∈ H −3/2 (∂X) (see e.g. [G68,G06,LM72]). In particular, Green's formula (cf. (3.1a)) fails to hold with the natural boundary maps.\"\n },\n {\n \"title\": \"....\",\n \"content\": \"\"\n },\n {\n \"title\": \".\",\n \"content\": \"\"\n },\n {\n \"title\": \"Theorem 2. 31 .\",\n \"content\": \"\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . δ . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .","content":"....\n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . δ . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ."},{"id":"","caption":". .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. d 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The spaces H 2 p are complete, i.e., Hilbert spaces with their natural norms. (ii) Note that ∆ p is a bounded operator on H 2 p . However, ∆ p with dom ∆ p = H 2 p is not closed. Although we call ∆ p the maximal Laplacian, it is not the maximal operator ∆ max p in the usual sens (which is the operator closure of ∆ p with domain dom ∆ max p :","content":".\n. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. d 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The spaces H 2 p are complete, i.e., Hilbert spaces with their natural norms. (ii) Note that ∆ p is a bounded operator on H 2 p . However, ∆ p with dom ∆ p = H 2 p is not closed. Although we call ∆ p the maximal Laplacian, it is not the maximal operator ∆ max p in the usual sens (which is the operator closure of ∆ p with domain dom ∆ max p :"},{"id":"","caption":"the adjoint of β z 0 as operator β z 0 : G 1/2 −→ H 0 . Furthermore, ran(β z 0 ) * = G 1/2 . Proof. The assertion follows from (see also [BGP06, Thm. 1.23 (2d)])","content":"the adjoint of β z 0 as operator β z 0 : G 1/2 −→ H 0 . Furthermore, ran(β z 0 ) * = G 1/2 . Proof. The assertion follows from (see also [BGP06, Thm. 1.23 (2d)])"},{"id":"","caption":"Let B be a self-adjoint and bounded operator in G and ∆ e B 0 the corresponding self-adjoint restriction of the Laplacian with domain","content":"Theorem 2. 31 .\n31Let B be a self-adjoint and bounded operator in G and ∆ e B 0 the corresponding self-adjoint restriction of the Laplacian with domain"},{"id":"","caption":"topological isomorphisms. In particular, for w = ±i, they are unitary.Corollary 3.4. The operator D has zero defect index, i.e., N i = ker(D −i) and N −i = ker(D +i) are isomorphic.","content":"topological isomorphisms. In particular, for w = ±i, they are unitary.Corollary 3.4. The operator D has zero defect index, i.e., N i = ker(D −i) and N −i = ker(D +i) are isomorphic."},{"id":"","caption":"defines a self-adjoint operator D B . The Laplacian (D B ) 2 acts on each component as the Laplacian∆ p f p , but with domain dom(D B ) 2 = { f ∈ dom D B | Df ∈ dom D B } = { f ∈ H 2 | γ 1 f 1 = Bγ 0 f 0 , γ 1 df 0 = Bγ 0 δf 1 }. Note that this domain is different from dom ∆ B 0 ⊕ dom ∆ B 1 (cf.Definition 2.26) since the two components in dom(D B ) 2 are coupled.","content":"defines a self-adjoint operator D B . The Laplacian (D B ) 2 acts on each component as the Laplacian∆ p f p , but with domain dom(D B ) 2 = { f ∈ dom D B | Df ∈ dom D B } = { f ∈ H 2 | γ 1 f 1 = Bγ 0 f 0 , γ 1 df 0 = Bγ 0 δf 1 }. Note that this domain is different from dom ∆ B 0 ⊕ dom ∆ B 1 (cf.Definition 2.26) since the two components in dom(D B ) 2 are coupled."},{"id":"","caption":"Definition 2.1. Let H = H 0 ⊕ H 1 and G be Hilbert spaces.","content":"Definition 2.1. Let H = H 0 ⊕ H 1 and G be Hilbert spaces.(i) Elements of H p are referred to as p-forms. \n(ii) A partial map d : H 0 \nH 1 is called an exterior derivative if d is a closed map with \ndense domain H 1 \n0 := dom d ⊂ H 0 . We endow H 1 \n0 with the natural norm defined by \nf 0 \n\n2 \nH 1 \n\n0"},{"id":"","caption":"a first order boundary triple for the exterior derivative d : H 0 H 1 if γ 0 a boundary map associated to d.","content":"a first order boundary triple for the exterior derivative d : H 0 H 1 if γ 0 a boundary map associated to d.Definition 2.2. We set d 0 := d↾H 1 \n\n0 \n\n, and call δ := d * \n0 : H 1 \nH 0 the divergence operator with \n\ndomain H 1 \n1 := dom δ andH 1 \n1 := dom d * (clearly,H 1 \n1 ⊂ H 1 \n1 , andH 1 \n1 is dense in H 1 since d is \ndensely defined). We endow H 1 \n1 with the natural norm \nf 1 \n\n2 \nH 1 \n\n1 \n\n:= f 1 \n2 + δf 1 \n2 . \n\nDefinition 2.3. \n(i) We call ∆ 0 := δd the Laplacian of degree 0 with domain \n\nH 2 \n0 := dom δd := f 0 ∈ dom d df 0 ∈ dom δ \nSimilarly, ∆ 1 := dδ is called the (maximal) Laplacian of degree 1 with domain \n\nH 2 \n1 := dom dδ := f 1 ∈ dom δ δf 1 ∈ dom d . \nWe endow H 2 \np with the norms \nf 0 \n\n2 \nH 2 \n\n0"},{"id":"","caption":"/2 by Definition 2.8. The last assertion is obvious. Corollary 2.15. We have","content":"/2 by Definition 2.8. The last assertion is obvious. Corollary 2.15. We have"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . δ . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\",\n \"content\": \"....\\n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . δ . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\"\n },\n {\n \"id\": \"\",\n \"caption\": \". .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. d 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The spaces H 2 p are complete, i.e., Hilbert spaces with their natural norms. (ii) Note that ∆ p is a bounded operator on H 2 p . However, ∆ p with dom ∆ p = H 2 p is not closed. Although we call ∆ p the maximal Laplacian, it is not the maximal operator ∆ max p in the usual sens (which is the operator closure of ∆ p with domain dom ∆ max p :\",\n \"content\": \".\\n. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. d 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The spaces H 2 p are complete, i.e., Hilbert spaces with their natural norms. (ii) Note that ∆ p is a bounded operator on H 2 p . However, ∆ p with dom ∆ p = H 2 p is not closed. Although we call ∆ p the maximal Laplacian, it is not the maximal operator ∆ max p in the usual sens (which is the operator closure of ∆ p with domain dom ∆ max p :\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the adjoint of β z 0 as operator β z 0 : G 1/2 −→ H 0 . Furthermore, ran(β z 0 ) * = G 1/2 . Proof. The assertion follows from (see also [BGP06, Thm. 1.23 (2d)])\",\n \"content\": \"the adjoint of β z 0 as operator β z 0 : G 1/2 −→ H 0 . Furthermore, ran(β z 0 ) * = G 1/2 . Proof. The assertion follows from (see also [BGP06, Thm. 1.23 (2d)])\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Let B be a self-adjoint and bounded operator in G and ∆ e B 0 the corresponding self-adjoint restriction of the Laplacian with domain\",\n \"content\": \"Theorem 2. 31 .\\n31Let B be a self-adjoint and bounded operator in G and ∆ e B 0 the corresponding self-adjoint restriction of the Laplacian with domain\"\n },\n {\n \"id\": \"\",\n \"caption\": \"topological isomorphisms. In particular, for w = ±i, they are unitary.Corollary 3.4. The operator D has zero defect index, i.e., N i = ker(D −i) and N −i = ker(D +i) are isomorphic.\",\n \"content\": \"topological isomorphisms. In particular, for w = ±i, they are unitary.Corollary 3.4. The operator D has zero defect index, i.e., N i = ker(D −i) and N −i = ker(D +i) are isomorphic.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"defines a self-adjoint operator D B . The Laplacian (D B ) 2 acts on each component as the Laplacian∆ p f p , but with domain dom(D B ) 2 = { f ∈ dom D B | Df ∈ dom D B } = { f ∈ H 2 | γ 1 f 1 = Bγ 0 f 0 , γ 1 df 0 = Bγ 0 δf 1 }. Note that this domain is different from dom ∆ B 0 ⊕ dom ∆ B 1 (cf.Definition 2.26) since the two components in dom(D B ) 2 are coupled.\",\n \"content\": \"defines a self-adjoint operator D B . The Laplacian (D B ) 2 acts on each component as the Laplacian∆ p f p , but with domain dom(D B ) 2 = { f ∈ dom D B | Df ∈ dom D B } = { f ∈ H 2 | γ 1 f 1 = Bγ 0 f 0 , γ 1 df 0 = Bγ 0 δf 1 }. Note that this domain is different from dom ∆ B 0 ⊕ dom ∆ B 1 (cf.Definition 2.26) since the two components in dom(D B ) 2 are coupled.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Definition 2.1. Let H = H 0 ⊕ H 1 and G be Hilbert spaces.\",\n \"content\": \"Definition 2.1. Let H = H 0 ⊕ H 1 and G be Hilbert spaces.(i) Elements of H p are referred to as p-forms. \\n(ii) A partial map d : H 0 \\nH 1 is called an exterior derivative if d is a closed map with \\ndense domain H 1 \\n0 := dom d ⊂ H 0 . We endow H 1 \\n0 with the natural norm defined by \\nf 0 \\n\\n2 \\nH 1 \\n\\n0\"\n },\n {\n \"id\": \"\",\n \"caption\": \"a first order boundary triple for the exterior derivative d : H 0 H 1 if γ 0 a boundary map associated to d.\",\n \"content\": \"a first order boundary triple for the exterior derivative d : H 0 H 1 if γ 0 a boundary map associated to d.Definition 2.2. We set d 0 := d↾H 1 \\n\\n0 \\n\\n, and call δ := d * \\n0 : H 1 \\nH 0 the divergence operator with \\n\\ndomain H 1 \\n1 := dom δ andH 1 \\n1 := dom d * (clearly,H 1 \\n1 ⊂ H 1 \\n1 , andH 1 \\n1 is dense in H 1 since d is \\ndensely defined). We endow H 1 \\n1 with the natural norm \\nf 1 \\n\\n2 \\nH 1 \\n\\n1 \\n\\n:= f 1 \\n2 + δf 1 \\n2 . \\n\\nDefinition 2.3. \\n(i) We call ∆ 0 := δd the Laplacian of degree 0 with domain \\n\\nH 2 \\n0 := dom δd := f 0 ∈ dom d df 0 ∈ dom δ \\nSimilarly, ∆ 1 := dδ is called the (maximal) Laplacian of degree 1 with domain \\n\\nH 2 \\n1 := dom dδ := f 1 ∈ dom δ δf 1 ∈ dom d . \\nWe endow H 2 \\np with the norms \\nf 0 \\n\\n2 \\nH 2 \\n\\n0\"\n },\n {\n \"id\": \"\",\n \"caption\": \"/2 by Definition 2.8. The last assertion is obvious. Corollary 2.15. We have\",\n \"content\": \"/2 by Definition 2.8. The last assertion is obvious. Corollary 2.15. We have\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"Abstract boundary conditions for maximal sectorial extensions of sectorial operators. Y Arlinskii, Math. Nachr. 209Y. Arlinskii, Abstract boundary conditions for maximal sectorial extensions of sectorial operators, Math. Nachr. 209 (2000), 5-36.","doi":""},{"text":"Boundary value problems for elliptic partial differential operators on bounded domains. J Behrndt, M Langer, J. Funct. Anal. 2432J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal. 243 (2007), no. 2, 536-565.","doi":""},{"text":"Boundary value spaces and m-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices. M Brown, M Marletta, S Naboko, I Wood, arXiv:0704.2562PreprintM. Brown, M. Marletta, S. Naboko, and I. Wood, Boundary value spaces and m-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices, Preprint (arXiv:0704.2562) (2007).","doi":"arXiv:0704.2562"},{"text":"Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. J Brüning, V Geyler, K Pankrashkin, PreprintJ. Brüning, V. Geyler, and K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schrödinger operators, Preprint (2006).","doi":""},{"text":"Boundary relations and their Weyl families. V Derkach, S Hassi, M Malamud, H De Snoo, Trans. Amer. Math. Soc. 35812V. Derkach, S. Hassi, M. Malamud, and H. de Snoo, Boundary relations and their Weyl families, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5351-5400","doi":""},{"text":"P Exner, O Post, arXiv:0706.0481Quantum networks modelled by graphs, Preprint. P. Exner and O. Post, Quantum networks modelled by graphs, Preprint (arXiv:0706.0481) (2007).","doi":"arXiv:0706.0481"},{"text":"A characterization of the non-local boundary value problems associated with an elliptic operator. G Grubb, Ann. Scuola Norm. Sup. Pisa. 3G. Grubb, A characterization of the non-local boundary value problems associated with an elliptic operator, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 425-513.","doi":""},{"text":"Known and unknown results on elliptic boundary problems. Bull. Amer. Math. Soc. (N.S.). 432electronic, Known and unknown results on elliptic boundary problems, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 2, 227-230 (electronic).","doi":""},{"text":"Non-homogeneous boundary value problems and applications. J.-L Lions, E Magenes, Die Grundlehren der mathematischen Wissenschaften. P. KennethNew YorkSpringer-VerlagI181J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.","doi":""},{"text":"Self-adjoint extensions of restrictions. A Posilicano, arXiv:math-ph/0703078PreprintA. Posilicano, Self-adjoint extensions of restrictions, Preprint (arXiv:math-ph/0703078) (2007).","doi":"arXiv:math-ph/0703078"},{"text":"Spectral convergence of quasi-one-dimensional spaces. O Post, Ann. Henri Poincaré. 75O. Post, Spectral convergence of quasi-one-dimensional spaces, Ann. Henri Poincaré 7 (2006), no. 5, 933-973.","doi":""},{"text":"On general boundary problems for elliptic differential equations. M I Vishik, Am. Math. Soc., Transl., II. Ser. 24EnglishM. I. Vishik, On general boundary problems for elliptic differential equations., Am. Math. Soc., Transl., II. Ser. 24 (1963), 107-172 (English. Russian original from 1952).","doi":""},{"text":". Rudower Chaussee. 25deInstitut für Mathematik, Humboldt-Universität zu Berlinpost@math.hu-berlinInstitut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany E-mail address: post@math.hu-berlin.de","doi":""}],"string":"[\n {\n \"text\": \"Abstract boundary conditions for maximal sectorial extensions of sectorial operators. Y Arlinskii, Math. Nachr. 209Y. Arlinskii, Abstract boundary conditions for maximal sectorial extensions of sectorial operators, Math. Nachr. 209 (2000), 5-36.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Boundary value problems for elliptic partial differential operators on bounded domains. J Behrndt, M Langer, J. Funct. Anal. 2432J. Behrndt and M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal. 243 (2007), no. 2, 536-565.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Boundary value spaces and m-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices. M Brown, M Marletta, S Naboko, I Wood, arXiv:0704.2562PreprintM. Brown, M. Marletta, S. Naboko, and I. Wood, Boundary value spaces and m-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices, Preprint (arXiv:0704.2562) (2007).\",\n \"doi\": \"arXiv:0704.2562\"\n },\n {\n \"text\": \"Spectra of self-adjoint extensions and applications to solvable Schrödinger operators. J Brüning, V Geyler, K Pankrashkin, PreprintJ. Brüning, V. Geyler, and K. Pankrashkin, Spectra of self-adjoint extensions and applications to solvable Schrödinger operators, Preprint (2006).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Boundary relations and their Weyl families. V Derkach, S Hassi, M Malamud, H De Snoo, Trans. Amer. Math. Soc. 35812V. Derkach, S. Hassi, M. Malamud, and H. de Snoo, Boundary relations and their Weyl families, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5351-5400\",\n \"doi\": \"\"\n },\n {\n \"text\": \"P Exner, O Post, arXiv:0706.0481Quantum networks modelled by graphs, Preprint. P. Exner and O. Post, Quantum networks modelled by graphs, Preprint (arXiv:0706.0481) (2007).\",\n \"doi\": \"arXiv:0706.0481\"\n },\n {\n \"text\": \"A characterization of the non-local boundary value problems associated with an elliptic operator. G Grubb, Ann. Scuola Norm. Sup. Pisa. 3G. Grubb, A characterization of the non-local boundary value problems associated with an elliptic operator, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 425-513.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Known and unknown results on elliptic boundary problems. Bull. Amer. Math. Soc. (N.S.). 432electronic, Known and unknown results on elliptic boundary problems, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 2, 227-230 (electronic).\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Non-homogeneous boundary value problems and applications. J.-L Lions, E Magenes, Die Grundlehren der mathematischen Wissenschaften. P. KennethNew YorkSpringer-VerlagI181J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Springer-Verlag, New York, 1972, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Self-adjoint extensions of restrictions. A Posilicano, arXiv:math-ph/0703078PreprintA. Posilicano, Self-adjoint extensions of restrictions, Preprint (arXiv:math-ph/0703078) (2007).\",\n \"doi\": \"arXiv:math-ph/0703078\"\n },\n {\n \"text\": \"Spectral convergence of quasi-one-dimensional spaces. O Post, Ann. Henri Poincaré. 75O. Post, Spectral convergence of quasi-one-dimensional spaces, Ann. Henri Poincaré 7 (2006), no. 5, 933-973.\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On general boundary problems for elliptic differential equations. M I Vishik, Am. Math. Soc., Transl., II. Ser. 24EnglishM. I. Vishik, On general boundary problems for elliptic differential equations., Am. Math. Soc., Transl., II. Ser. 24 (1963), 107-172 (English. Russian original from 1952).\",\n \"doi\": \"\"\n },\n {\n \"text\": \". Rudower Chaussee. 25deInstitut für Mathematik, Humboldt-Universität zu Berlinpost@math.hu-berlinInstitut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany E-mail address: post@math.hu-berlin.de\",\n \"doi\": \"\"\n }\n]","isConversational":false}}},{"rowIdx":599,"cells":{"text":{"kind":"string","value":"\nGraph Neural Tangent Kernel: Convergence on Large Graphs\n31 May 2023\n\nSanjukta Krishnagopal \nLuana Ruiz luanaruiz<ruizl@mit.edu>. \nGraph Neural Tangent Kernel: Convergence on Large Graphs\n31 May 2023D705755388A517EBAE3EC3909971A543arXiv:2301.10808v2[cs.LG]\nGraph neural networks (GNNs) achieve remarkable performance in graph machine learning tasks but can be hard to train on large-graph data, where their learning dynamics are not well understood.We investigate the training dynamics of largegraph GNNs using graph neural tangent kernels (GNTKs) and graphons.In the limit of large width, optimization of an overparametrized NN is equivalent to kernel regression on the NTK.Here, we investigate how the GNTK evolves as another independent dimension is varied: the graph size.We use graphons to define limit objects-graphon NNs for GNNs, and graphon NTKs for GNTKs-, and prove that, on a sequence of graphs, the GN-TKs converge to the graphon NTK.We further prove that the spectrum of the GNTK, which is related to the directions of fastest learning which becomes relevant during early stopping, converges to the spectrum of the graphon NTK.This implies that in the large-graph limit, the GNTK fitted on a graph of moderate size can be used to solve the same task on the large graph, and to infer the learning dynamics of the large-graph GNN.These results are verified empirically on node regression and classification tasks.* Equal contribution 1 Dept. of Electrical Engineering and Computer Science 2 Dept. of Mathematics, UCLA 3 Work done in part while visiting the Simons Institute for the Theory of Computing.\n\nIntroduction\n\nSeveral real-world systems such as social-interactions, brainconnectome, epidemic spread, recommender systems, and traffic patterns are best represented by structured data in the form of large graphs.Graph neural networks (GNNs) are deep neural network architectures that leverage these graph structures to learn meaningful representations of node and edge data (Kipf & Welling, 2017;Hamilton et al., 2017;Defferrard et al., 2016;Gama et al., 2018).GNNs have shown remarkable empirical performance in a number of graph machine learning tasks, but can be hard to train on large-graph data.Recent research efforts have attempted to understand the large-graph behavior of GNNs, and in particular why GNNs trained on small graphs scale well to large networks (Ruiz et al., 2020;Levie et al., 2021;Keriven et al., 2020).However, the specific learning dynamics of a GNN trained directly on the large network, which are known to be challenging, are not as well understood.\n\nModern deep neural networks (DNNs) are typically overparametrized.The benefits of overparametrization include faster convergence (Allen-Zhu et al., 2019;Arora et al., 2018) and better generalization (Cao & Gu, 2019), but on the other hand the parameters can be more difficult to interpret and the learning dynamics harder to understand.A remarkable contribution of Jacot et al. (2018) was the observation that, in the infinite width limit, learning the weights of a DNN via gradient descent reduces to kernel regression with a deterministic and fixed kernel called the neural tangent kernel (NTK), which captures the first-order approximation of the neural network's evolution during gradient descent (Lee et al., 2019).Since its introduction, the NTK has been an important and widely-studied tool in the machine learning toolbox, and kernel regression using the NTK has shown strong performance on small datasets (Arora et al., 2019b).\n\nIn the graph case, it is straightforward to define the NTK associated with a GNN, or the graph neural tangent kernel (GNTK) (Du et al., 2019).The GNTK allows studying the training dynamics of the GNN when the number of features (the analog of width in the DNN) is large.Nonetheless, the effect of graph size on the GNN learning dynamics, which can itself be thought of as another 'width dimension', is not well understood, and there is limited research that investigates this rigorously.Herein, we propose to to understand the training dynamics of GNNs that are wide in both of these senses by combining the NTK formalism with the theory of graphons (Lovász, 2012;Borgs et al., 2008).\n\nA graphon is a symmetric bounded measurable function W : [0, 1] 2 → [0, 1] representing the limit of a sequence of dense graphs.It can also be interpreted as a random graph model, in which case we can use the graphon to sample stochastic graphs.The interpretation of W as both a graph limit and a random graph model makes it so that each graphon defines a family of similar graphs.Hence, one can expect properties of a graphon to generalize, in a probabilistic sense, the properties of graphs belonging to its family.Graphons have been used to study the limit behavior of GNNs, which converge to so-called graphon neural networks (WNNs) (Ruiz et al., 2020).The fact that GNNs have a limit on the graphon implies that they are transferable across graphs in the same family, thus allowing a GNN to be trained on a graph of moderate size and transferred to a larger graph.\n\n\nContributions\n\nIn this paper, our first contribution is to define the graphon NTK (WNTK) associated with the WNN (Sec.4).We then prove, using mathematical induction, that the GNTK converges to the WNTK (Thm.5.1).In practice, this implies that GNTKs, like GNNs, are transferable across graphs of different sizes associated with the same graphon.That is to say, one can subsample a small graph and the corresponding data from a large graph, then fit the subsampled data to the small-graph GNTK via kernel regression, and then transfer the fitted model to the large-graph GNTK, which is particularly important for large graphs.\n\nA more important implication of the convergence of the GNTK is that it is possible to understand the training dynamics of GNNs on large graphs by analyzing the behavior of the corresponding GNTK in graphs of moderate size.For instance, the eigenvalues of the NTK are associated with the speed of convergence along the corresponding eigendirections (Jacot et al., 2018).In Thm.5.2, we show that the eigenvalues of the graph GNTK, which indicate the directions of fastest convergence of the GNN, converge to the eigenvalues of the WNTK.This allows these eigenvalues to be estimated from GNTKs associated with smaller graphs.Lastly, we verify our theoretical results in three numerical applications: prediction of opinion dynamics on random graphs, movie recommendation using the Movie-Lens dataset, and node classification on the Cora, CiteSeer and PubMed networks.We observe the convergence of the GNTK (Sec.6.1), the effect of width in kernel regression and GNN training (Sec.6.2), and the convergence of the GNTK eigenvalues on sequences of graphs (Sec.6.3).\n\n\nRelated Work\n\nNeural tangent kernels.The connection between infinitely wide DNNs and kernel methods (Gaussian processes) has been known since the 1990s (Neal & Neal, 1996;Williams, 1996), but a more theoretical formulation was presented by Jacot et al. (2018), which introduced the NTK and proved its constancy property in the infinite width limit, with Liu et al. (2020) later showing that constancy only holds for architectures with linear output layer.Several works have derived NTKs for a generalized classes of neural networks, including convolutional neural networks (Arora et al., 2019a;Li et al., 2019), ResNets (Huang et al., 2020) and, most closely related to our work, GNNs (Du et al., 2019).\n\nGraphons and size generalization in deep learning.Graphons have been used to understand GNN convergence and transferability (Ruiz et al., 2020;2021a;Maskey et al., 2023), to analyze the generalization properties of GNNs in large graphs (Maskey et al., 2022), and to propose more computationally efficient training algorithms for large-scale GNNs (Cerviño et al., 2023).More recently, Xia et al. (2022) proposed implicit graphon neural representations, which use neural networks to estimate graphons.Yehudai et al. (2021) study graphs in which the local structure depends on the graph size-which is analogous to the dense graphs associated with graphons-, and find that GNNs are not guaranteed to scale to large graph sizes.However, their result foregoes the normalization of the adjacency matrix by the graph size, while we leverage this normalization to show operator norm convergence of the kernels associated with the graph convolution.It is also worth noting that our work is fundamentally different than GNN convergence or transferability results, because it relates not to the GNN architecture but to the GNN learning dynamics, and uses a mathematical induction proof as opposed to the spectral convergence argument typically used in GNN convergence proofs.\n\n\nGraph and Graphon Neural Networks\n\nLet G n = (V, E, w) be a graph where V, |V| = n, is the set of nodes or vertices, E ⊆ V × V is the set of edges and w : E → R is a map assigning weights to the edges in E. The size-normalized adjacency matrix of G n , denoted A n ∈ R n×n , is given by [A] ij = w(i, j)/n.In this paper we focus on undirected graphs G n , with symmetric A n .\n\n\nGraph Neural Networks\n\nSupported on the graph G n with n nodes, we define node data x n ∈ R n -also called graph signals (Shuman et al., 2013)-where [x n ] i is the value of the signal at node i. GNNs iteratively update each node's data by aggregating the data from its neighbors using graph convolutions.An order-K graph convolution is defined as\ny n = H(x n ) = K−1 k=0 h k A k n x n (1)\nwhere h 0 , . . ., h K−1 are the convolution coefficients.When K = 2 and A n is binary, (1) can be seen as an aggregation operation akin to the AGGREGATE operation in, e.g., (Xu et al., 2019;Hamilton et al., 2017), or as the message-passing operation in message-passing neural net-\nY n = H(X n ) = K−1 k=0 A k n X n H k (2)\nwith weights H 0 , H 1 , . . ., H K−1 ∈ R F ×G .Note that the number of parameters of the graph convolutions in (1) and\n\n(2), K and KF G respectively, are independent of n.\n\nA GNN consists of L layers, each of which composes a graph convolution and a nonlinear activation function.Explicitly, the lth layer of a GNN can be written as\nX l,n = σ (U l,n ) U l,n = H(X l−1,n ) = K−1 k=0 A k n X l−1,n H l,k(3)\nwhere X l−1,n ∈ R n×F l−1 is the layer input, X l,n ∈ R n×F l is the layer output, H l,k ∈ R F l−1 ×F l are the layer weights and σ is a pointwise nonlinear activation function, i.e., [σ(X)] ij = σ([X] ij ).Typical choices for σ include ReLU, tanh, and sigmoid.At the first layer of the GNN, the input X 0,n is the input data X.Similarly, the GNN output Y n is given by the last layer output X n,L .\n\nIn the following, we represent the entire GNN consisting of the concatenation of L layers like (3) as the parametric map Y n = f (X n ; A n , H), where H = {H l,k } l,k groups the learnable weights H l,k at all layers.This more concise representation highlights the independence between the adjacency matrix A n (i.e., the graph) and the parameters H, which the GNN inherits from the graph convolution.We note that while we describe a general class of GNNs here, the methods in our paper can be directly adapted to other types of GNNs.\n\n\nGraphon Neural Networks\n\nGraphons are bounded, symmetric, measurable functions W : [0, 1] 2 → [0, 1] representing limits of sequences of dense graphs (Lovász, 2012;Borgs et al., 2008).A graph sequence {G n } converges to a graphon in the sense that the densities of homomorphisms of any finite, unweighted and undirected graph F = (V ′ , E ′ ) into G n converge to the densities of homomorphisms of F into W.\n\nThe graphs F can be thought of as motifs, such as triangles, k-cycles, k-cliques, etc. Explicitly, let hom(F, G n ) denote the total number of homomorphisms between V ′ and V n .The density of such homomorphisms is given by\nt(F, G n ) = hom(F, G n )/n |V ′ |\nand we can similarly define t(F, W) (see (Borgs et al., 2008)).Then, we say that G n → W if and only if\nlim n→∞ t(F, G n ) = t(F, W)(4)\nfor all simple F (Lovász & Szegedy, 2006).\n\nAlternatively, the graphon can also be seen as a generative model for stochastic (also called W-random) graphs.Nodes are picked by sampling points u i , 1 ≤ i ≤ n, from the unit interval and connecting edges between nodes i and j with probability W(u i , u j ).Importantly, sequences of stochastic graphs generated in this way converge almost surely to the graphon (Lovász, 2012)[Cor. 10.4].\n\nThe notion of graph signals is extended to graphons by defining graphon signals, which are functions X : [0, 1] → R (Ruiz et al., 2021a).We restrict attention to graphon signals with finite energy, i.e., X ∈ L 2 ([0, 1]).\n\nAnalogously to (2), given F -and G-dimensional graphon signals X : [0, 1] → R F and Y : [0, 1] → R G , the graphon convolution is defined as (Ruiz et al., 2021a)\nY = T H X = K−1 k=0 T (k) W XH k T (k) W X = 1 0 W(u, v)T (k−1) W X(u)du(5)\nwhere T (0) W = I is the identity and the convolution weights are collected in the matrices\nH 0 , H 1 , . . . , H K−1 ∈ R F ×G .\nThe extension of the GNN to graphon data is the graphon neural network (WNN).Akin to the GNN, the WNN is formed by L layers each of which composes a graphon convolution and a nonlinear activation function.Explicitly, the lth layer of the WNN is given by\nX l = σ (U l ) U l = T H l X l−1 = K−1 k=0 T (k) W X l−1 H l,k (6) where X l−1 : [0, 1] → R F l−1 is the layer input, X l : [0, 1] → R F l is its output, H l,k ∈ R F l−1 ×F l are\nits weights and σ is a pointwise nonlinearity (e.g., the ReLU).The input of the first layer of the WNN is X 0 = X, and the output of the WNN is the last layer ouput, i.e., Y = X L .\n\nWe can describe the WNN more compactly as the map Y = f (X; W, H), with H = {H l,k } l,k the set of learnable parameters at all layers.Note that, if the weights H are the same, the WNN map f (X; W, H) is the same as the GNN map f (X; A n , H) with A n swapped with W. This is important because it implies that, similarly to how graphons are generative models for graphs, WNNs are generative models for GNNs.Indeed, we can use the WNN f (X; W, H) to sample the GNN\ny n = f (x n ; A n , H) with [A n ] ij ∼ Ber(W(u i , u j )) [x n ] i = X(u i ) (7)\nwhere the u i are sampled uniformly and independently at random from [0, 1] and Ber() is the Bernoulli distribution.\n\nAs n → ∞, sequences of GNNs sampled from the WNN as described above converge to the WNN.In fact, as long as H is the same, any GNN f et al., 2021a).A more important result in practice is that this convergence implies that GNNs are transferable across graphs associated with the same graphon, i.e., they can be trained on graphs G n and executed on graphs G m with an error that decreases asymptotically with n and m (Ruiz et al., 2021b;2020).\n(x n ; A n , H) applied to a sequence {(G n , x n )} converging to (W, X) converges to f (X; W, H) (Ruiz\nIn Sec. 5, we prove a similar convergence result for the neural tangent kernels (NTKs) associated with f (x n ; A n , H) and f (X; W, H), but before doing so, we need to introduce induced WNNs.The WNN induced by the GNN\nf (x n ; A n , H) is defined as Y n = f (X n ; W n , H), with W n (u, v) = n i=1 n j=1 [A n ] ij I(u ∈ I i )I(v ∈ I j ), X n (u) = n i=1 [x n ] i I(u ∈ I i ) (8)\nand where I is the indicator function,\nI i = [(i − 1)/n, i/n) for 1 ≤ i ≤ n − 1, and I n = [(n − 1)/n, 1].\n\nGraph and Graphon Neural Tangent Kernel\n\nConsider a general, fully-connected neural network f (x; H), with layers given by\nx l = σ(H l x l−1 ), input x 0 = x ∈ R d0 and learnable parameters H = {H l } l ∈ R d l−1 ×d l , [H l ] pq = h l,pq . For a training set {x i , ỹi } M i=1\n, assume that the loss to be minimized is the mean squared error (MSE) or quadratic loss\nmin H ℓ(H) = min H M i=1 (f (x i ; H) − ỹi ) 2 . (9)\nAs the training progresses, the output of the neural network for input x i is updated as y i (t) = f (x i ; H(t)).As such, the weight update rule is given by ∂H ∂t = −∇ℓ(H(t)) and so the output evolves as\n∂y i (t) ∂t = j Θ(x i , x j ; H(t))(f (x i ; H(t)) − ŷi ), Θ(x i , x j ; H) = l,p,q ∂f (x i ; H) ∂h l,pq ∂f (x j ; H) ∂h l,pq .(10)\nIn the infinite-width limit as d l → ∞, Jacot et al. ( 2018); Liu et al. (2020) showed that, provided that the last layer has linear output (i.e., it does not have an activation σ), Θ(x i , x j ; H) converges to a limiting kernel, the NTK.This kernel stays constant during training-a property called constancy-, which is equivalent to replacing the outputs of the neural network by their first-order Taylor expansion in the parameter space (Lee et al., 2019).Hence, in the infinite-width limit the training dynamics of (10) reduce to kernel ridge regression1 on the NTK, which has a closedform solution.This facilitates understanding the learning dynamics of overparametrized neural networks, which are notoriously difficult to study directly, by analysis of the corresponding NTK.\n\n\nGraph Neural Tangent Kernel\n\nFor simplicity, we will consider a GNN with only one feature per layer; the generalization to multiple features is more involved but straightforward.Recall that the L-layer GNN supported on A n ∈ R n×n is written as\nx n,0 = x n u l,n = H l (x l−1,n ) x l,n = σ(u l,n ) f (x n ; A n , H) = x L,n .\nThe graph NTK (GNTK) associated with this GNN is\nΘ(x n , x ′ n ; A n , H) =∇ H f (x n ; A n , H) T ∇ H f (x ′ n ; A n , H) = k σ ′ (u L,n )A k n x L−1,n ⊗ σ ′ (u ′ L,n )A k n x ′ L−1,n + σ ′ (u L,n )(σ ′ (u L−1,n )A k n x L−2,n ) ⊗ σ ′ (u ′ L,n )H L (σ ′ (u ′ L−1,n )A k n x ′ L−2,n ) + . . .\nwhere there are L terms; the first term corresponds to the last layer, the second term to the second-last layer, and so on.We have used σ ′ (u) to denote the diagonal matrix with jjth entry equal to σ ′ ([u] j ), where σ ′ is the derivative of σ.\nNote that ∇ H f is a |H| × n matrix, hence the GNTK is a matrix Θ n (x n , x ′ n ; A n , H) ∈ R n×n .\n\nGraphon Neural Tangent Kernel\n\nTo derive the WNTK, we will also consider a WNN with only one feature per layer for simplicity.Recall that the L-layer WNN associated with the graphon W is given by\nX 0 = X U l = T H l X l−1 X l = σ(U l ) f (X; W, H) = X L .\nTherefore, the WNTK is given by\nΘ(X, X ′ ; W, H) = = l,k ∂ h l,k f (X; W, H) ⊗ ∂ h l,k f (X ′ ; W, H).(11)\nCalculating the derivative with respect to the kth weight in layer l j = L − j, we get\n∂ h l j ,k u L = T (k) W X L−1\nfor j = 0, and similarly for 0 < j < L,\n∂ h l j ,k u L = T H L σ ′ (U L−1 ) . . . T H lj−1 σ ′ (U L−j )T (k) W X lj−1 .\nHence, (11) has L terms in total, explicitly\nΘ(X, X ′ ; W, H) = = k σ ′ (U L )T (k) W X L−1 ⊗ σ ′ (U ′ L )T (k) W X ′ L−1 + σ ′ (U L )T H L (σ ′ (U L−1 )T (k) W X L−2 ) ⊗ σ ′ (U ′ L )T H L (σ ′ (U ′ L−1 )T (k) W X ′ L−2 ) + . . .\nwhere σ ′ (U L ) is now evaluated and multiplied pointwise.Note that Θ is a linear operator on graphon signals, i.e., given a signal Y ∈ L 2 ([0, 1]), the WNTK applied to this signal yields\nΘ(X, X ′ ; W, H)Y = 1 0 k σ ′ (U L (u))T (k) W X L−1 (u) × σ ′ (U ′ L (v))T (k) W X ′ L−1 (v)Y (v) + σ ′ (U L (u))T H L (σ ′ (U L−1 )T (k) W X L−2 )(u) × σ ′ (U L (v))T H L (σ ′ (U L−1 )T (k) W X L−2 )(v)Y (v) + . . . dv\nwhere the first term corresponds to the last layer, the second corrresponds to the second-last layer and so on.\n\nFor a graph A n sampled from W with associated induced graphon W n , and for input features x n sampled from a graphon signal X and associated induced graphon signal X n [cf.( 8)], the induced WNTK is given by the same formula as (11) but with W, X replaced by W n , X n .\n\n\nGraph NTK Converges to Graphon NTK\n\nNext, we consider a sequence of graph signals {(G n , x n )} converging to a graphon signal (W, X) to evaluate the convergence of the GNTK Θ(x n , x ′ n ; A n , H) to the WNTK Θ(X, X ′ ; W, H).First, for each pair of signals X, X ′ , the WNTK Θ(X, X ′ ; W, H)-which we denote Θ(X, X ′ ) to simplify notation-is a linear operator on functions in L 2 .Hence, to prove convergence to this operator we introduce the operator norm\n||Θ(X, X ′ )|| = sup Y ̸ =0 ||Θ(X, X ′ )Y || ||Y || . (12)\nLet W be a graphon and X, X ′ be fixed graphon signals.\n\nConsider a sequence of graph signals (Ruiz et al., 2021a)[Def. 2].The notion of convergence of graph signals on a sequence of dense graphs is simple; G n converges to W in the homomorphism density sense [cf.(4)], and X n , the graphon signal induced by x n , converges to X in the L 2 norm, i.e.\n{(G n , x n )} (respec- tively {(G n , x ′ n )}) converging to (W, X) (respectively (W, X ′ )) in the sense of\n∥X n − X∥ → 0 (13) up to node relabelings (see (Ruiz et al., 2021a) for further details, and in particular Lemma 2 for the relationship between (G n , x n ) and the induced signal (W n , X n )).\n\nLet (W n , X n ) be the graphon signal induced by the graph signal\n(G n , x n ) [cf. (8)].\nThe main result of this paper is the following theorem that shows that the WNTK induced by the GNTK converges to the limiting WNTK.The proof is deferred to Appendix A. Theorem 5.1.Let W be a graphon and X, X ′ ∈ L 2 ([0, 1]) be arbitrary graphon signals.Suppose that {G n } is a sequence of graphs converging to W [cf. (4)] and {x n } , {x ′ n } are sequences of graph signals converging to X [cf.(13)].Then, for any L-layer GNN with finite K, fixed weights H and 1-Lipschitz nonlinearity σ, the associated GNTKs Θ(x n , x ′ n ; A n , H) converge in the operator norm:\nlim n→∞ ||Θ(X n , X ′ n ; W n , H) − Θ(X, X ′ ; W, H)|| = 0 where Θ(X n , X ′ n ; W n , H) is the WNTK induced by the GNTK Θ(x n , x ′ n ; A n , H).\nThe convergence of the GNTK to a limit object-the WNTK-has two important implications for machine learning on large-scale graphs.First, consider that we have a large graph G N on which we want to predict signals y N using the GNTK, but that we do not have enough computational resources to compute the full-sized GNTK Θ(x N , x ′ N ; A N , H).This is expected, since calculating the kernel regression weights on this graph requires inverting a matrix with dimension proportional to M , the number of training samples, and the graph size N , see Sec. 4. Thm.5.1 implies that we can subsample the GNTK and the labels Y N on a smaller graph G n , n ≪ N -as Θ(x n , x ′ n ; A n , H) and y n respectively-and fit the data to the GNTK on this smaller graph.Once the kernel regression weights are obtained, they can then be transferred to make predictions on G N .Naturally, there will be an error associated with this transference; however, due to convergence, this error vanishes asymptotically in n.\n\nThe precise convergence bounds for different graphs can be computed by plugging into (26).For instance, for graphs sampled from the graphon as in (Ruiz et al., 2021b)\n[Def. 1], the bound is given by ||Θ(X n , X ′ n ) − Θ(X, X ′ )|| ≤ C(K 4+L 2A w n + K 2+L A x n ).\nwhere C is a constant, and A w and A x are the Lipschitz constants of the graphon and the graphon signal respectively.\n\nIn practice, transferability is useful in a variety of real-world applications, especially on large graphs.For instance, one may be interested in the learning dynamics of a GNN trained to provide ad recommendations on a growing, dense social media network.Our theoretical results show that one can calculate the GNTK on a smaller subgraph, e.g., the same social media network at an earlier time, and then transfer it to the large target graph with theoretical performance guarantees.This works because dense networks growing under the same underlying 'rule' or 'process' should have similar homomorphism densities, and converge to similar graphons; or, inversely, they can be thought of as being sampled from similar graphons.The more similar these limiting graphons are (under a metric such as the cut metric or the L 2 norm), the lower the error when transferring the GNTK across graphs.\n\nA second implication of Thm.5.1, which is perhaps more important, is that GNTK convergence implies that a GNTK fitted on a smaller graph can be used to infer details about the training dynamics of wide/overparametrized GNNs on larger graphs.This follows from Thm. 5.1 in conjunction with theoretical results by Liu et al. (2020), which proves that in the infinite-width limit the training dynamics of general convolutional models (including GNNs) reduces to kernel ridge regression on the corresponding NTK; and empirical results by Sabanayagam et al. (2021) showing that even when the output layer is nonlinear, constancy of the GNTK can still be observed.\n\nAn example of property of a large-graph GNTK Θ(x N , x ′ N ; A N , H) that can be estimated from a smallgraph GNTK Θ(x n , x ′ n ; A n , H) is its eigenvalue spectrum.As we show in Thm.5.2, the spectrum of Θ(X n , X ′ n ; W n , H) converges to the the spectrum of Θ(X, X ′ ; W, H).The proof is deferred to Appendix B.\n\nTheorem 5.2.Let {x i,n } M i=1 be a set of M signals on the graph G n , and {X i,n } M i=1 the corresponding induced signals on the induced graphon W n .Assume that G n → W [cf. (4)] and X i,n → X [cf.( 13)] for all i.Define the operators Θ n and Θ where\n[Θ n ] ij = Θ(X i,n , X j,n ; W n , H) and [Θ] ij = Θ(X i , X j ; W, H). Let λ p (T ), p ∈ Z \\ {0}, denote the pth eigenvalue of a compact, self-adjoint opera- tor T , with λ −1 < λ −2 < . . . ≤ 0 ≤ . . . < λ 2 < λ 1 . If thelim n→∞ |λ p (Θ n ) − λ p (Θ)| → 0.\nMoreover, the corresponding eigenspaces converge in the sense that the spectral projectors converge.\n\nThis is an important result because, as proved by Jacot et al. (2018), the convergence of kernel gradient descent follows the kernel principal components.Thus, if the GNN f (H) is sufficiently wide, and if f (H 0 ) − f (H ⋆ ) is aligned with the ith GNTK principal component, we can expect gradient descent to converge with rate proportional to λ i , the ith eigenvalue of the GNTK.Thm.5.2 shows that the GNTK eigenvalues converge to the eigenvalues of the WNTK.As such, we can estimate the speed of convergence of a largegraph GNN f (x N ; A N , H) from the eigenvalues of a smallgraph GNTK Θ(x n , x ′ n ; A n , H).A numerical example of this application of Thms.5.1 and 5.2 is given in Sec.6.3.Spectral convergence of the GNTK is particularly relevant for practical applications on large graphs.Convergence of the spectrum implies convergence of the directions of learning.Intuitively, one can calculate the GNTK on the smaller graph, then transfer it to the large graph with the guarantee that the dominant directions of fastest convergence are being captured.We can learn GNN architectures that harness this property by pretraining on a smaller graph, and making adjustments on the large graph; or, by iterating between different resolutions of graph size.\n\n\nResults\n\nIn the following, we illustrate the convergence of the GNTK to the WNTK on a simulated opinion dynamics model on random graphs; on a movie recommendation graph; and on citation networks (Sec.6.1).In the opinion dynamics problem, we further vary the width of the network to analyze the large width behavior of the GNTK and how it relates to the behavior of the GNN (Sec.6.2).We conclude with an example of how the convergence of the GNTK can be used in practice to estimate the eigenvalues of the GNTK, and thus the speed of GNN training along its principal components, on large-scale graphs (Sec.6.3).\n\nOpinion dynamics.This is a node-level task modeling the outcomes of a mathematical model for studying the evolution of ideologies, affiliations, and opinions in society (Lorenz, 2007), including topics of important practical interest such as political ideologies and misinformation spread.On an undirected n-node graph G = (V, E), we consider an opinion dynamics process x t ∈ R n .The node data [x t ] i is the opinion of individual i under a standard boundedconfidence model described by Hegselmann et al. (2002):\n[x t ] i = j∈Si,t−1 c[x t−1 ] j . (14)\nwhere c > 0 is the so-called influence parameter and After drawing the initial opinions x 0 from a Gaussian distribution with µ = 0 and Σ = 2I, we run the process (14) until convergence (for at most T = 1000 iterations) to obtain the final opinions x T .The goal of the learning problem is then to predict y = x T from x = x 0 .We fix the training set size to 300 samples, and use 30 samples for both validation and testing.\nS i,t−1 = N i ∩ X i,t−1 is the intersection of N i = {j ∈ V | (i, j) ∈ E}, the neighborhood of i, and X i,t = {j ∈ V | |[x t ] j − [x t ] i | ≤ ϵ},\nMovie recommendation.In this problem, the graph is a movie similarity network, the data are existing user ratings (ranging from 1 to 5), and the task is to use the existing ratings to fill in the missing ratings for movie recommendation.The movie similarity graph is a dense correlation graph built by computing pairwise correlations between the rating vectors of the movies rated in the MovieLens-100k dataset (Harper & Konstan, 2016).We only consider movies with at least 10 ratings, and the large graph has N = 1000 nodes sampled at random from the set of admissible movies.\n\nThe learning problem is to predict the ratings of the movie \"Contact\".We use a 90-10 train-test split of the data, and build the graph using only the training samples.Due to memory limitations, we only use 25% of the users (samples) to compute the GNTK.\n\nCitation networks.The third problem we consider is a node classification problem on the Cora, CiteSeer and PubMed networks, where nodes represent documents and undirected edges represent citations between papers in either direction.\n\nThe nodes a bag-of-words representations are grouped in X ∈ {0, 1} n×F and each node is associated with one of C classes.We consider the networks and train-test-splits from the full distribution (Bojchevski & Günnemann, 2017) available in PyTorch Geometric, but only sample F = 1000 features for CiteSeer and N = 10000 nodes for PubMed due to memory limitations.Through studying citation networks, we illustrate the empirical manifestation of our results on real-world graphs.Although citation networks are not best modeled by graphon limits, they certainly have limits, and our results suggest that convergence is a global property of GNTKs and graph limits.\n\nArchitectures and experiment details.In all experiments, the GNNs have L = 1 layer (3) with ReLU nonlinearity followed by a perceptron layer.For opinion dynamics, K = 2, and for movie recommendation, K = 5.In both cases, we consider the MSE and fit the GNTK using linear regression.In the citation network experiment, the GNN architecture has K = 2 and includes a softmax layer followed by argmax; and we consider the cross-entropy (CE) and fit the GNTK using logistic regression.All reported results are averaged over 5 and 10 realizations for opinion dynamics and movie recommendation/citation networks respectively.Additional architecture and experiment details are listed in each subsection.The code can be found in this repository.\n\nAll experiments were run on a NVIDIA RTX A6000 GPU.\n\n\nConvergence\n\nTo visualize the convergence of the GNTK, we fit a GNTK with F 1 = 10 in the opinon dynamics and citation network experiments, and F 1 = 16 in the recommendation experiment, on the training set of a small n-node graph; and transfer this GNTK, without retraining, to predict the outputs on the test set of both the same n-node graph and a larger N -node graph.To visualize convergence, we plot in Fig. 1 the absolute difference between the test errors attained on the n-node and the N -node graph (normalized by the test error on the N -node graph) as a function of the size of the smaller graph n, for both the opinion dynamics (left) and node classification (right) experiments.The results of the recommendation experiment are reported in Table 2.\n\nIn opinion dynamics, N = 300.We consider two types of graphs, corresponding to two graphon families: (a) a symmetrized geometric k-nearest neighbor graph where nodes are drawn at random from a 50 × 50 square and k = n/10; and (b) a stochastic block model (SBM) graph with intra-community probability p = 0.1 and inter-community probability q = 0.05.As n increases from 20 to 100, we observe that the difference between the MSEs for the n-node and the N -node graphs decrease steadily from ∼ 70% to ∼ 20% on the geometric graphs and from ∼ 90% to ∼ 40% on the SBM graphs.\n\nIn the recommendation experiment, N = 1000.As n increases from 100 to 500, we see from Table 2 that the average test MSE difference on the large graph decreases from 6.8% to 0.3%, and so does the standard deviation, thus confirming our theoretical findings.In particular, n = 300 is enough to achieve an MSE difference of less than 1%.\n\nIn the node classification experiment, N = 2708 for Cora, N = 3327 for CiteSeer, and N = 10000 for PubMed (sampled from the 19717 nodes in the original due to memory limitations).For all datasets, the difference between the CE loss for the n-node and the N -node graph also decreases as n increases.This can be interpreted to mean that node classi-fication on these citation networks depends largely on local information, so beyond a critical size n, the transference error saturates.An interesting point to make is that although these citation networks are not dense (i.e., are not best modeled by a graphon), we still see the empirical manifestation of our theoretical results.\n\n\nWide Network Behavior\n\nNext, we analyze the effect of width on both the GNN and the GNTK when they are trained/fitted on a small graph and transferred to a large graph.From the NTK analysis by Jacot et al. (2018), as the GNN width increases we expect (i) the GNTK and the GNN to exhibit less variance over multiple weight initializations and (ii) the GNN outputs to approach those of kernel regression with the GNTK.\n\nFor the opinion dynamics experiment on SBMs, we consider three GNNs with widths (number of features) F\n\n(1) 1 = 10, F\n\n(2) 1 = 50, and F\n\n(3) 1 = 250 supported on the same 80-node graph.Using their initial weights, we construct the corresponding GNTKs.We train the three GNNs by minimizing the MSE loss over 20 epochs and with batch size 32, using ADAM (Kingma & Ba, 2015) with learning rate 1e −3 and weight decay 5e −3 .Simultaneously, we fit the GNTKs to the training set, and then transfer both the GNNs and the corresponding GNTKs to the N -node graph and compute their outputs on the test set.\n\nIn Fig. 2, we plot the projection of the outputs y onto the second graph eigenvector v 2 , [ŷ] 2 = v T 2 y against the projections of the inputs onto the same vector [x] 2 = v T 2 x (sorted in ascending order) for both the GNNs and the GNTKs for F\n\n(1) 1 = 10 (left), F\n\n(2) 1 = 50 (center), and F\n\n(3) 1 = 250 (right).The second graph eigenvector was chosen as it captures the community structure, however, note that the same behavior was observed upon projecting onto other eigenvectors, as shown in Appendix C. The results are averaged over 5 GNN initializations, with the solid lines representing the mean and the shaded areas representing the standard deviation.The behavior is as expected: as the width increases, the GNN and the GNTK have smaller variance in behavior for different weight initializations, and the GNN and GNTK curves align.In Appendix C, we include additional plots displaying the mean and variance over different initializations as we vary the graph size.\n\n\nApplication: Eigenvalue Convergence\n\nIn this section, we elucidate an important application of the convergence of the GNTK.Jacot et al. (2018) shows that the convergence of kernel gradient descent follows the kernel principal components.Therefore, if the GNN is sufficiently wide, and f (X n ; A n , H 0 ) − f (X n ; S n , H ⋆ ) is aligned with the ith GNTK principal component, we can expect gradient descent to converge with rate proportional to λ i , the ith eigenvalue of the GNTK2 .\n\nRecall that we prove, in Thm. 2, that the spectrum of the GNTKs converge in the graphon limit.In this context, a simple application of the convergence of the GNTK spectrum is as follows: say that we know that f (X N ; A N , H 0 ) − f (X N ; A N , H ⋆ ) is aligned with the pth GNTK eigenvector, and we want to estimate its speed of convergence, but the kernel Θ(X N , X ′ N ; A N , H) is too expensive to compute.We could in this case sample a graph G n , n ≪ N from the induced graphon W N and calculate Θ(X n , X ′ n ; A n , H).From Thm. 5.2, λ n,p converges to λ N,p .Hence, for large enough n, λ n,p should provide a good approximation of λ N,p .\n\nTo illustrate this empirically, in Fig. 3 we plot the dominant eigenvalues of the GNTK associated with a GNN with F 1 = 10 features supported on graphs of increasing size n sampled from the Cora, CiteSeer, and PubMed networks (averaged over 10 initializations).As n grows, we see that the difference between the dominant eigenvalues of consecutive GNTKs reduces.I.e., for each citation dataset, the value of the dominant eigenvalue converges as n increases, indicating that the speed of convergence of the GNN along the dominant GNTK eigenvector converges as the graph grows.For all but the PubMed dataset, the 1600-node graph gives an approximation of λ N,1 that is over 70% good.\n\n\nConclusions\n\nIn this paper, we define WNTKs as the limiting objects of GNTKs, and study how the GNTK evolves as the underlying graph of the corresponding GNN grows.We show that GNTKs converge to the WNTK and that their spectra also converges, thus providing theoretical insight into how the learning dynamics of a GNN evolves as an important dimension-the graph size-grows.In practice, these convergences imply that one can transfer the GNTK of a smaller graph to solve the same task on a larger graph without any further optimization, with theoretical guarantees of performance and insight on the rates of learning along eigendirections that converge as the graph grows.These results were demonstrated through simulations on synthetic and real-world graphs.To conclude, a limitation of this work is that graphons are only good models for dense graphs.In future work, we plan to extend our results to more general graph models, and to better understand the relationship between the spectra of the graph and of the GNTK.\n\n\nA. Proof of GNTK Convergence\n\nProof of Thm.5.1.To simplify notation, we will write Θ(X, X ′ ; W, H) = Θ(X, X ′ ) and Θ(X n , X ′ n ; W n , H) = Θ(X n , X ′ n ).Let Y be an arbitrary L 2 signal.Using the triangle inequality,\n||Θ(X, X ′ )Y − Θ(X n , X ′ n )Y || ≤ k ||σ ′ (U L )T (k) W X L−1 ⟨σ ′ (U ′ L )T (k) W X ′ L−1 , Y ⟩ − σ ′ (U L,n )T (k) Wn X n,L−1 ⟨σ ′ (U ′ L,n )T (k) Wn X ′ L−1,n Y ⟩|| + . . .(15)\nwhere there are L terms for each k.\n\nSince σ is Lipschitz, from the formula we see that it suffices to prove the following convergence statements for all k ∈ 1, ..., K − 1 as n → ∞:\n∥U l,n − U l ∥ → 0 (16) ∥U ′ l,n − U ′ l ∥ → 0 (17)\nas well as\nT (k) Wn − T (k) W → 0 (18) T h l,n − T h l → 0.(19)\nHere, we prove these statements for each finite K using mathematical induction.For fixed, finite K, (18) implies ( 19).The convergence in ( 16) and ( 17) are essentially the same, so let us focus on the first without loss of generality.Expanding the first layer, we have\nU 1,n − U 1 = k h 1,k (T (k) Wn X n − T (k) W X) = k h 1,k (T (k) Wn X n − T (k) W X + T (k) W X n − T (k) W X n ) = k h 1,k ((T (k) Wn − T k W )X n + T (k) W (X n − X)).\nTherefore, by the triangle inequality and the definition of the operator norm, we have From 21, using mathematical induction, we have, for arbitrary l,\n∥U l,n − U l ∥ ≤ (∥X n ∥∥X n − X∥ + ∥T (k) Wn − T k W ∥) • (max(|h|)k) l (22) + l−1 j=1 (max(|h|)k) l−1−j max(|h|)∥T (k) Wn − T k W ∥ (23)\nwhere h is the max of h l,k over all l and k, and so convergence of T (k)\nWn → T (k)\nW as in (18) and X n → X implies convergence of U 1,n → U 1 and U l,n → U l .Since the nonlinearity σ is Lipschitz, convergence of (17) follows from convergence of (16).\n\nWe now focus on proving (18).By Cauchy-Schwarz,\n\n\n||(T\n\nFigure 1 .\n1\nFigure 1.Difference between the test error achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph.(left) Opinion dynamics on geometric and SBM graphs, where the test error is the MSE N = 300.(right) Node classification on Cora, CiteSeer and Pubmed, where the test error is the CE and N = 2708, 3327, and 10000 respectively.\n\n\n\n\nthe set of nodes whose opinion at time t diverges by at most 0 ≤ ϵ ≤ 1 of the opinion of node i.This model reduces to the classicDe-Groot (DeGroot, 1974) opinion model for ϵ = 1.We fix ϵ = 0.3 and c = 0.1.\n\n\nFigure 2 .\n2\nFigure 2. Projections of the inputs and outputs of the GNTK and GNN, and of the target or true labels onto the second eigenvector of the adjacency matrix of a 80-node SBM graph for widths F = 10 (left), F = 50 (center) and F = 250 (right).\n\n\nFigure 3 .\n3\nFigure 3.The leading eigenvalue of the GNTK with width F = 10 for Cora, CiteSeer and PubMed as a function of the graph size n.\n\n\nW\n\n∥∥X n − X∥. (21)Looking at layer l, we haveU l,n − U l = k h l,k (T (k) Wn σ(U l−1,n ) − T (k) W σ(U l−1 )).\n\n\n\n\nWn − T W )Y || 2 = n (u, v) − W(u, v))Y (v)dv n (u, v) − W(u, v)) 2 dvdu ||Y || 2 .\n\n\nTable 1 .\n1\nDescription of variables.\nVariableDescriptionA nAdjacency matrix of graph G nx nGraph signal on G nW nGraphon induced by G nX nGraphon signal induced by x nWLimiting graphonXLimiting graphon signalworks (MPNNs) (Gilmer et al., 2017).More generally, let X n ∈ R n×F and Y n ∈ R n×G be F -and G-dimensional signals respectively, where the f th(gth) column is a feature x f n (y g n ). In this case, the graphconvolution generalizes to\n\n\n\nThe graphon W n is induced by the graph G n , and the graphon signals X n and Y n are induced by the graph signals x n and y n .A succinct description of all graph and graphon variables is provided in Table 1.\n\n\nTable 2 .\n2\nMovie recommendation results.Relative difference between the test MSE achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph, where N = 1000.\nNumber of nodes 100 200 300 400 500Mean (%)6.80 0.78 0.09 0.03 0.03Std. dev. (p.p.)3.51 1.16 0.03 0.01 0.02weights in H are bounded, then, for all p,\nIn practice, it is not necessary to consider the MSE for this derivation. It suffices for the loss to be such that the norm of the training direction f (H0) − f (H ⋆ ) is strictly decreasing during training(Jacot et al., \n). E.g., if we consider the cross-entropy loss, the training dynamics reduce to kernel logistic regression.\nTheoretically, constancy of the NTK in the infinite-width limit and equivalence between GNN training and kernel regression only holds for architectures with linear output layer(Liu et al., 2020). Although this is not the case of the GNNs used here, GNTKs associated with GNNs with nonlinear output layers seem to exhibit constancy empirically(Sabanayagam et al., 2021) \nTherefore, taking the square root, dividing by ||Y ||, and using (12), we getIt follows that convergence of the induced graphons W n to the limiting graphon W implies (18) for k = 1.For higher iterates we noteTherefore, by the same Cauchy-Schwarz argument above, Tis upper bounded byand so convergence W n → W implies the convergence of all the iterates T (k)W by induction.In fact, one can writeFrom the proof one can, with some simple algebra, compute a quantitative estimate of ||Θ(X n , XPrecisely, by adding and subtracting cross-terms for each term in (15) and using the triangle inequality, we getwhere C is a constant in terms of ∥X∥, ∥X ′ ∥, max(|h|), l.For different choices of graphs, one can then compute these quantities for an explicit estimate of the convergence bound.(Ruiz et al., 2021b)[Appendix B] provides quantitative estimates on ||W n − W|| and ||X n − X|| for random graph models such as so-called template graphs, weighted graphs and stochastic graphs.B. Proof of Spectrum ConvergenceProof of Theorem 5.2.The proof relies on(Anselone & Palmer, 1968)[Prop. 7.1]which proves that if a sequence of compact operators T n converges to a compact operator T in the appropriate operator norm, the spectrum of T n converges to the spectrum of T .Additionally, from(Anselone & Palmer, 1968)[Thm.6.3 and Prop 7.1], for every p, the eigenspace associated with the p th eigenvalue of T n also converge to the corresponding eigenspace of T .From Thm. 5.1, we know that Θ n → Θ.It remains to show that Θ n and Θ are compact.For any graphon W, the operators T (k)W and T H [cf. (5)] are compact as they are linear compositions of compact operators with bounded linear weights.From (11), we thus conclude ∥Θ(X,C. Wide Network Behavior: Additional ExperimentsUnder the same expeirmental setting of Sec.6.2, in Fig.4, we plot the projection of the outputs y onto the first graph eigenvector v 1 , [ŷ] 1 = v T 1 y against the projections of the inputs onto the same vector [x] 1 = v T 1 x (sorted in ascending order) for both the GNNs and the GNTKs for F = 250 (right), and n = 20, 40, 60, 80, 100 (top to bottom).The results are averaged over 5 GNN initializations, with the solid lines representing the mean and the shaded areas representing the standard deviation.The behavior is as expected: as the width increases, the GNN and the GNTK have smaller variance in behavior for different weight initializations, and the GNN and GNTK curves align.As n increases, we also observe a slight improvement in the variance of the GNN curves.This is expected as, when GNNs are trained on larger graphs, they have better transferability(Ruiz et al., 2020).No significant trends are observed for the GNTK curves for increasing n.Graph Neural Tangent Kernel: Convergence on Large Graphs\nA convergence theory for deep learning via over-parameterization. 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PMLR2021\n"},"corpus_id":{"kind":"number","value":256274591,"string":"256,274,591"},"external_ids":{"kind":"string","value":"{\n \"arxiv\": \"2301.10808\",\n \"mag\": null,\n \"acl\": null,\n \"pubmed\": null,\n \"pubmedcentral\": null,\n \"dblp\": \"conf/icml/KrishnagopalR23\",\n \"doi\": \"10.48550/arxiv.2301.10808\"\n}"},"source":{"kind":"string","value":"{\n \"pdf_urls\": [\n \"https://export.arxiv.org/pdf/2301.10808v2.pdf\"\n ],\n \"pdf_sha\": \"4e0cef1f767e9a8a643eb529a6ba8584cbdd68c7\",\n \"open_access\": {\n \"license\": null,\n \"url\": \"http://arxiv.org/pdf/2301.10808\",\n \"status\": \"GREEN\"\n }\n}"},"title":{"kind":"string","value":"Graph Neural Tangent Kernel: Convergence on Large Graphs"},"authors":{"kind":"list like","value":["Sanjukta Krishnagopal","Luana Ruiz luanaruiz<ruizl@mit.edu>."],"string":"[\n \"Sanjukta Krishnagopal\",\n \"Luana Ruiz luanaruiz<ruizl@mit.edu>.\"\n]","isConversational":false},"abstract":{"kind":"string","value":"Graph neural networks (GNNs) achieve remarkable performance in graph machine learning tasks but can be hard to train on large-graph data, where their learning dynamics are not well understood.We investigate the training dynamics of largegraph GNNs using graph neural tangent kernels (GNTKs) and graphons.In the limit of large width, optimization of an overparametrized NN is equivalent to kernel regression on the NTK.Here, we investigate how the GNTK evolves as another independent dimension is varied: the graph size.We use graphons to define limit objects-graphon NNs for GNNs, and graphon NTKs for GNTKs-, and prove that, on a sequence of graphs, the GN-TKs converge to the graphon NTK.We further prove that the spectrum of the GNTK, which is related to the directions of fastest learning which becomes relevant during early stopping, converges to the spectrum of the graphon NTK.This implies that in the large-graph limit, the GNTK fitted on a graph of moderate size can be used to solve the same task on the large graph, and to infer the learning dynamics of the large-graph GNN.These results are verified empirically on node regression and classification tasks.* Equal contribution 1 Dept. of Electrical Engineering and Computer Science 2 Dept. of Mathematics, UCLA 3 Work done in part while visiting the Simons Institute for the Theory of Computing."},"sections":{"kind":"list like","value":[{"title":"Introduction","content":"Several real-world systems such as social-interactions, brainconnectome, epidemic spread, recommender systems, and traffic patterns are best represented by structured data in the form of large graphs.Graph neural networks (GNNs) are deep neural network architectures that leverage these graph structures to learn meaningful representations of node and edge data (Kipf & Welling, 2017;Hamilton et al., 2017;Defferrard et al., 2016;Gama et al., 2018).GNNs have shown remarkable empirical performance in a number of graph machine learning tasks, but can be hard to train on large-graph data.Recent research efforts have attempted to understand the large-graph behavior of GNNs, and in particular why GNNs trained on small graphs scale well to large networks (Ruiz et al., 2020;Levie et al., 2021;Keriven et al., 2020).However, the specific learning dynamics of a GNN trained directly on the large network, which are known to be challenging, are not as well understood.\nModern deep neural networks (DNNs) are typically overparametrized.The benefits of overparametrization include faster convergence (Allen-Zhu et al., 2019;Arora et al., 2018) and better generalization (Cao & Gu, 2019), but on the other hand the parameters can be more difficult to interpret and the learning dynamics harder to understand.A remarkable contribution of Jacot et al. (2018) was the observation that, in the infinite width limit, learning the weights of a DNN via gradient descent reduces to kernel regression with a deterministic and fixed kernel called the neural tangent kernel (NTK), which captures the first-order approximation of the neural network's evolution during gradient descent (Lee et al., 2019).Since its introduction, the NTK has been an important and widely-studied tool in the machine learning toolbox, and kernel regression using the NTK has shown strong performance on small datasets (Arora et al., 2019b).\nIn the graph case, it is straightforward to define the NTK associated with a GNN, or the graph neural tangent kernel (GNTK) (Du et al., 2019).The GNTK allows studying the training dynamics of the GNN when the number of features (the analog of width in the DNN) is large.Nonetheless, the effect of graph size on the GNN learning dynamics, which can itself be thought of as another 'width dimension', is not well understood, and there is limited research that investigates this rigorously.Herein, we propose to to understand the training dynamics of GNNs that are wide in both of these senses by combining the NTK formalism with the theory of graphons (Lovász, 2012;Borgs et al., 2008).\nA graphon is a symmetric bounded measurable function W : [0, 1] 2 → [0, 1] representing the limit of a sequence of dense graphs.It can also be interpreted as a random graph model, in which case we can use the graphon to sample stochastic graphs.The interpretation of W as both a graph limit and a random graph model makes it so that each graphon defines a family of similar graphs.Hence, one can expect properties of a graphon to generalize, in a probabilistic sense, the properties of graphs belonging to its family.Graphons have been used to study the limit behavior of GNNs, which converge to so-called graphon neural networks (WNNs) (Ruiz et al., 2020).The fact that GNNs have a limit on the graphon implies that they are transferable across graphs in the same family, thus allowing a GNN to be trained on a graph of moderate size and transferred to a larger graph."},{"title":"Contributions","content":"In this paper, our first contribution is to define the graphon NTK (WNTK) associated with the WNN (Sec.4).We then prove, using mathematical induction, that the GNTK converges to the WNTK (Thm.5.1).In practice, this implies that GNTKs, like GNNs, are transferable across graphs of different sizes associated with the same graphon.That is to say, one can subsample a small graph and the corresponding data from a large graph, then fit the subsampled data to the small-graph GNTK via kernel regression, and then transfer the fitted model to the large-graph GNTK, which is particularly important for large graphs.\nA more important implication of the convergence of the GNTK is that it is possible to understand the training dynamics of GNNs on large graphs by analyzing the behavior of the corresponding GNTK in graphs of moderate size.For instance, the eigenvalues of the NTK are associated with the speed of convergence along the corresponding eigendirections (Jacot et al., 2018).In Thm.5.2, we show that the eigenvalues of the graph GNTK, which indicate the directions of fastest convergence of the GNN, converge to the eigenvalues of the WNTK.This allows these eigenvalues to be estimated from GNTKs associated with smaller graphs.Lastly, we verify our theoretical results in three numerical applications: prediction of opinion dynamics on random graphs, movie recommendation using the Movie-Lens dataset, and node classification on the Cora, CiteSeer and PubMed networks.We observe the convergence of the GNTK (Sec.6.1), the effect of width in kernel regression and GNN training (Sec.6.2), and the convergence of the GNTK eigenvalues on sequences of graphs (Sec.6.3)."},{"title":"Related Work","content":"Neural tangent kernels.The connection between infinitely wide DNNs and kernel methods (Gaussian processes) has been known since the 1990s (Neal & Neal, 1996;Williams, 1996), but a more theoretical formulation was presented by Jacot et al. (2018), which introduced the NTK and proved its constancy property in the infinite width limit, with Liu et al. (2020) later showing that constancy only holds for architectures with linear output layer.Several works have derived NTKs for a generalized classes of neural networks, including convolutional neural networks (Arora et al., 2019a;Li et al., 2019), ResNets (Huang et al., 2020) and, most closely related to our work, GNNs (Du et al., 2019).\nGraphons and size generalization in deep learning.Graphons have been used to understand GNN convergence and transferability (Ruiz et al., 2020;2021a;Maskey et al., 2023), to analyze the generalization properties of GNNs in large graphs (Maskey et al., 2022), and to propose more computationally efficient training algorithms for large-scale GNNs (Cerviño et al., 2023).More recently, Xia et al. (2022) proposed implicit graphon neural representations, which use neural networks to estimate graphons.Yehudai et al. (2021) study graphs in which the local structure depends on the graph size-which is analogous to the dense graphs associated with graphons-, and find that GNNs are not guaranteed to scale to large graph sizes.However, their result foregoes the normalization of the adjacency matrix by the graph size, while we leverage this normalization to show operator norm convergence of the kernels associated with the graph convolution.It is also worth noting that our work is fundamentally different than GNN convergence or transferability results, because it relates not to the GNN architecture but to the GNN learning dynamics, and uses a mathematical induction proof as opposed to the spectral convergence argument typically used in GNN convergence proofs."},{"title":"Graph and Graphon Neural Networks","content":"Let G n = (V, E, w) be a graph where V, |V| = n, is the set of nodes or vertices, E ⊆ V × V is the set of edges and w : E → R is a map assigning weights to the edges in E. The size-normalized adjacency matrix of G n , denoted A n ∈ R n×n , is given by [A] ij = w(i, j)/n.In this paper we focus on undirected graphs G n , with symmetric A n ."},{"title":"Graph Neural Networks","content":"Supported on the graph G n with n nodes, we define node data x n ∈ R n -also called graph signals (Shuman et al., 2013)-where [x n ] i is the value of the signal at node i. GNNs iteratively update each node's data by aggregating the data from its neighbors using graph convolutions.An order-K graph convolution is defined as\nwhere h 0 , . . ., h K−1 are the convolution coefficients.When K = 2 and A n is binary, (1) can be seen as an aggregation operation akin to the AGGREGATE operation in, e.g., (Xu et al., 2019;Hamilton et al., 2017), or as the message-passing operation in message-passing neural net-\nwith weights H 0 , H 1 , . . ., H K−1 ∈ R F ×G .Note that the number of parameters of the graph convolutions in (1) and\n(2), K and KF G respectively, are independent of n.\nA GNN consists of L layers, each of which composes a graph convolution and a nonlinear activation function.Explicitly, the lth layer of a GNN can be written as\nwhere X l−1,n ∈ R n×F l−1 is the layer input, X l,n ∈ R n×F l is the layer output, H l,k ∈ R F l−1 ×F l are the layer weights and σ is a pointwise nonlinear activation function, i.e., [σ(X)] ij = σ([X] ij ).Typical choices for σ include ReLU, tanh, and sigmoid.At the first layer of the GNN, the input X 0,n is the input data X.Similarly, the GNN output Y n is given by the last layer output X n,L .\nIn the following, we represent the entire GNN consisting of the concatenation of L layers like (3) as the parametric map Y n = f (X n ; A n , H), where H = {H l,k } l,k groups the learnable weights H l,k at all layers.This more concise representation highlights the independence between the adjacency matrix A n (i.e., the graph) and the parameters H, which the GNN inherits from the graph convolution.We note that while we describe a general class of GNNs here, the methods in our paper can be directly adapted to other types of GNNs."},{"title":"Graphon Neural Networks","content":"Graphons are bounded, symmetric, measurable functions W : [0, 1] 2 → [0, 1] representing limits of sequences of dense graphs (Lovász, 2012;Borgs et al., 2008).A graph sequence {G n } converges to a graphon in the sense that the densities of homomorphisms of any finite, unweighted and undirected graph F = (V ′ , E ′ ) into G n converge to the densities of homomorphisms of F into W.\nThe graphs F can be thought of as motifs, such as triangles, k-cycles, k-cliques, etc. Explicitly, let hom(F, G n ) denote the total number of homomorphisms between V ′ and V n .The density of such homomorphisms is given by\nand we can similarly define t(F, W) (see (Borgs et al., 2008)).Then, we say that G n → W if and only if\nfor all simple F (Lovász & Szegedy, 2006).\nAlternatively, the graphon can also be seen as a generative model for stochastic (also called W-random) graphs.Nodes are picked by sampling points u i , 1 ≤ i ≤ n, from the unit interval and connecting edges between nodes i and j with probability W(u i , u j ).Importantly, sequences of stochastic graphs generated in this way converge almost surely to the graphon (Lovász, 2012)[Cor. 10.4].\nThe notion of graph signals is extended to graphons by defining graphon signals, which are functions X : [0, 1] → R (Ruiz et al., 2021a).We restrict attention to graphon signals with finite energy, i.e., X ∈ L 2 ([0, 1]).\nAnalogously to (2), given F -and G-dimensional graphon signals X : [0, 1] → R F and Y : [0, 1] → R G , the graphon convolution is defined as (Ruiz et al., 2021a)\nwhere T (0) W = I is the identity and the convolution weights are collected in the matrices\nThe extension of the GNN to graphon data is the graphon neural network (WNN).Akin to the GNN, the WNN is formed by L layers each of which composes a graphon convolution and a nonlinear activation function.Explicitly, the lth layer of the WNN is given by\nits weights and σ is a pointwise nonlinearity (e.g., the ReLU).The input of the first layer of the WNN is X 0 = X, and the output of the WNN is the last layer ouput, i.e., Y = X L .\nWe can describe the WNN more compactly as the map Y = f (X; W, H), with H = {H l,k } l,k the set of learnable parameters at all layers.Note that, if the weights H are the same, the WNN map f (X; W, H) is the same as the GNN map f (X; A n , H) with A n swapped with W. This is important because it implies that, similarly to how graphons are generative models for graphs, WNNs are generative models for GNNs.Indeed, we can use the WNN f (X; W, H) to sample the GNN\nwhere the u i are sampled uniformly and independently at random from [0, 1] and Ber() is the Bernoulli distribution.\nAs n → ∞, sequences of GNNs sampled from the WNN as described above converge to the WNN.In fact, as long as H is the same, any GNN f et al., 2021a).A more important result in practice is that this convergence implies that GNNs are transferable across graphs associated with the same graphon, i.e., they can be trained on graphs G n and executed on graphs G m with an error that decreases asymptotically with n and m (Ruiz et al., 2021b;2020).\nIn Sec. 5, we prove a similar convergence result for the neural tangent kernels (NTKs) associated with f (x n ; A n , H) and f (X; W, H), but before doing so, we need to introduce induced WNNs.The WNN induced by the GNN\nand where I is the indicator function,"},{"title":"Graph and Graphon Neural Tangent Kernel","content":"Consider a general, fully-connected neural network f (x; H), with layers given by\n, assume that the loss to be minimized is the mean squared error (MSE) or quadratic loss\nAs the training progresses, the output of the neural network for input x i is updated as y i (t) = f (x i ; H(t)).As such, the weight update rule is given by ∂H ∂t = −∇ℓ(H(t)) and so the output evolves as\nIn the infinite-width limit as d l → ∞, Jacot et al. ( 2018); Liu et al. (2020) showed that, provided that the last layer has linear output (i.e., it does not have an activation σ), Θ(x i , x j ; H) converges to a limiting kernel, the NTK.This kernel stays constant during training-a property called constancy-, which is equivalent to replacing the outputs of the neural network by their first-order Taylor expansion in the parameter space (Lee et al., 2019).Hence, in the infinite-width limit the training dynamics of (10) reduce to kernel ridge regression1 on the NTK, which has a closedform solution.This facilitates understanding the learning dynamics of overparametrized neural networks, which are notoriously difficult to study directly, by analysis of the corresponding NTK."},{"title":"Graph Neural Tangent Kernel","content":"For simplicity, we will consider a GNN with only one feature per layer; the generalization to multiple features is more involved but straightforward.Recall that the L-layer GNN supported on A n ∈ R n×n is written as\nThe graph NTK (GNTK) associated with this GNN is\nwhere there are L terms; the first term corresponds to the last layer, the second term to the second-last layer, and so on.We have used σ ′ (u) to denote the diagonal matrix with jjth entry equal to σ ′ ([u] j ), where σ ′ is the derivative of σ."},{"title":"Graphon Neural Tangent Kernel","content":"To derive the WNTK, we will also consider a WNN with only one feature per layer for simplicity.Recall that the L-layer WNN associated with the graphon W is given by\nTherefore, the WNTK is given by\nCalculating the derivative with respect to the kth weight in layer l j = L − j, we get\nfor j = 0, and similarly for 0 < j < L,\nHence, (11) has L terms in total, explicitly\nwhere σ ′ (U L ) is now evaluated and multiplied pointwise.Note that Θ is a linear operator on graphon signals, i.e., given a signal Y ∈ L 2 ([0, 1]), the WNTK applied to this signal yields\nwhere the first term corresponds to the last layer, the second corrresponds to the second-last layer and so on.\nFor a graph A n sampled from W with associated induced graphon W n , and for input features x n sampled from a graphon signal X and associated induced graphon signal X n [cf.( 8)], the induced WNTK is given by the same formula as (11) but with W, X replaced by W n , X n ."},{"title":"Graph NTK Converges to Graphon NTK","content":"Next, we consider a sequence of graph signals {(G n , x n )} converging to a graphon signal (W, X) to evaluate the convergence of the GNTK Θ(x n , x ′ n ; A n , H) to the WNTK Θ(X, X ′ ; W, H).First, for each pair of signals X, X ′ , the WNTK Θ(X, X ′ ; W, H)-which we denote Θ(X, X ′ ) to simplify notation-is a linear operator on functions in L 2 .Hence, to prove convergence to this operator we introduce the operator norm\nLet W be a graphon and X, X ′ be fixed graphon signals.\nConsider a sequence of graph signals (Ruiz et al., 2021a)[Def. 2].The notion of convergence of graph signals on a sequence of dense graphs is simple; G n converges to W in the homomorphism density sense [cf.(4)], and X n , the graphon signal induced by x n , converges to X in the L 2 norm, i.e.\n∥X n − X∥ → 0 (13) up to node relabelings (see (Ruiz et al., 2021a) for further details, and in particular Lemma 2 for the relationship between (G n , x n ) and the induced signal (W n , X n )).\nLet (W n , X n ) be the graphon signal induced by the graph signal\nThe main result of this paper is the following theorem that shows that the WNTK induced by the GNTK converges to the limiting WNTK.The proof is deferred to Appendix A. Theorem 5.1.Let W be a graphon and X, X ′ ∈ L 2 ([0, 1]) be arbitrary graphon signals.Suppose that {G n } is a sequence of graphs converging to W [cf. (4)] and {x n } , {x ′ n } are sequences of graph signals converging to X [cf.(13)].Then, for any L-layer GNN with finite K, fixed weights H and 1-Lipschitz nonlinearity σ, the associated GNTKs Θ(x n , x ′ n ; A n , H) converge in the operator norm:\nThe convergence of the GNTK to a limit object-the WNTK-has two important implications for machine learning on large-scale graphs.First, consider that we have a large graph G N on which we want to predict signals y N using the GNTK, but that we do not have enough computational resources to compute the full-sized GNTK Θ(x N , x ′ N ; A N , H).This is expected, since calculating the kernel regression weights on this graph requires inverting a matrix with dimension proportional to M , the number of training samples, and the graph size N , see Sec. 4. Thm.5.1 implies that we can subsample the GNTK and the labels Y N on a smaller graph G n , n ≪ N -as Θ(x n , x ′ n ; A n , H) and y n respectively-and fit the data to the GNTK on this smaller graph.Once the kernel regression weights are obtained, they can then be transferred to make predictions on G N .Naturally, there will be an error associated with this transference; however, due to convergence, this error vanishes asymptotically in n.\nThe precise convergence bounds for different graphs can be computed by plugging into (26).For instance, for graphs sampled from the graphon as in (Ruiz et al., 2021b)\nwhere C is a constant, and A w and A x are the Lipschitz constants of the graphon and the graphon signal respectively.\nIn practice, transferability is useful in a variety of real-world applications, especially on large graphs.For instance, one may be interested in the learning dynamics of a GNN trained to provide ad recommendations on a growing, dense social media network.Our theoretical results show that one can calculate the GNTK on a smaller subgraph, e.g., the same social media network at an earlier time, and then transfer it to the large target graph with theoretical performance guarantees.This works because dense networks growing under the same underlying 'rule' or 'process' should have similar homomorphism densities, and converge to similar graphons; or, inversely, they can be thought of as being sampled from similar graphons.The more similar these limiting graphons are (under a metric such as the cut metric or the L 2 norm), the lower the error when transferring the GNTK across graphs.\nA second implication of Thm.5.1, which is perhaps more important, is that GNTK convergence implies that a GNTK fitted on a smaller graph can be used to infer details about the training dynamics of wide/overparametrized GNNs on larger graphs.This follows from Thm. 5.1 in conjunction with theoretical results by Liu et al. (2020), which proves that in the infinite-width limit the training dynamics of general convolutional models (including GNNs) reduces to kernel ridge regression on the corresponding NTK; and empirical results by Sabanayagam et al. (2021) showing that even when the output layer is nonlinear, constancy of the GNTK can still be observed.\nAn example of property of a large-graph GNTK Θ(x N , x ′ N ; A N , H) that can be estimated from a smallgraph GNTK Θ(x n , x ′ n ; A n , H) is its eigenvalue spectrum.As we show in Thm.5.2, the spectrum of Θ(X n , X ′ n ; W n , H) converges to the the spectrum of Θ(X, X ′ ; W, H).The proof is deferred to Appendix B.\nTheorem 5.2.Let {x i,n } M i=1 be a set of M signals on the graph G n , and {X i,n } M i=1 the corresponding induced signals on the induced graphon W n .Assume that G n → W [cf. (4)] and X i,n → X [cf.( 13)] for all i.Define the operators Θ n and Θ where\nMoreover, the corresponding eigenspaces converge in the sense that the spectral projectors converge.\nThis is an important result because, as proved by Jacot et al. (2018), the convergence of kernel gradient descent follows the kernel principal components.Thus, if the GNN f (H) is sufficiently wide, and if f (H 0 ) − f (H ⋆ ) is aligned with the ith GNTK principal component, we can expect gradient descent to converge with rate proportional to λ i , the ith eigenvalue of the GNTK.Thm.5.2 shows that the GNTK eigenvalues converge to the eigenvalues of the WNTK.As such, we can estimate the speed of convergence of a largegraph GNN f (x N ; A N , H) from the eigenvalues of a smallgraph GNTK Θ(x n , x ′ n ; A n , H).A numerical example of this application of Thms.5.1 and 5.2 is given in Sec.6.3.Spectral convergence of the GNTK is particularly relevant for practical applications on large graphs.Convergence of the spectrum implies convergence of the directions of learning.Intuitively, one can calculate the GNTK on the smaller graph, then transfer it to the large graph with the guarantee that the dominant directions of fastest convergence are being captured.We can learn GNN architectures that harness this property by pretraining on a smaller graph, and making adjustments on the large graph; or, by iterating between different resolutions of graph size."},{"title":"Results","content":"In the following, we illustrate the convergence of the GNTK to the WNTK on a simulated opinion dynamics model on random graphs; on a movie recommendation graph; and on citation networks (Sec.6.1).In the opinion dynamics problem, we further vary the width of the network to analyze the large width behavior of the GNTK and how it relates to the behavior of the GNN (Sec.6.2).We conclude with an example of how the convergence of the GNTK can be used in practice to estimate the eigenvalues of the GNTK, and thus the speed of GNN training along its principal components, on large-scale graphs (Sec.6.3).\nOpinion dynamics.This is a node-level task modeling the outcomes of a mathematical model for studying the evolution of ideologies, affiliations, and opinions in society (Lorenz, 2007), including topics of important practical interest such as political ideologies and misinformation spread.On an undirected n-node graph G = (V, E), we consider an opinion dynamics process x t ∈ R n .The node data [x t ] i is the opinion of individual i under a standard boundedconfidence model described by Hegselmann et al. (2002):\nwhere c > 0 is the so-called influence parameter and After drawing the initial opinions x 0 from a Gaussian distribution with µ = 0 and Σ = 2I, we run the process (14) until convergence (for at most T = 1000 iterations) to obtain the final opinions x T .The goal of the learning problem is then to predict y = x T from x = x 0 .We fix the training set size to 300 samples, and use 30 samples for both validation and testing.\nMovie recommendation.In this problem, the graph is a movie similarity network, the data are existing user ratings (ranging from 1 to 5), and the task is to use the existing ratings to fill in the missing ratings for movie recommendation.The movie similarity graph is a dense correlation graph built by computing pairwise correlations between the rating vectors of the movies rated in the MovieLens-100k dataset (Harper & Konstan, 2016).We only consider movies with at least 10 ratings, and the large graph has N = 1000 nodes sampled at random from the set of admissible movies.\nThe learning problem is to predict the ratings of the movie \"Contact\".We use a 90-10 train-test split of the data, and build the graph using only the training samples.Due to memory limitations, we only use 25% of the users (samples) to compute the GNTK.\nCitation networks.The third problem we consider is a node classification problem on the Cora, CiteSeer and PubMed networks, where nodes represent documents and undirected edges represent citations between papers in either direction.\nThe nodes a bag-of-words representations are grouped in X ∈ {0, 1} n×F and each node is associated with one of C classes.We consider the networks and train-test-splits from the full distribution (Bojchevski & Günnemann, 2017) available in PyTorch Geometric, but only sample F = 1000 features for CiteSeer and N = 10000 nodes for PubMed due to memory limitations.Through studying citation networks, we illustrate the empirical manifestation of our results on real-world graphs.Although citation networks are not best modeled by graphon limits, they certainly have limits, and our results suggest that convergence is a global property of GNTKs and graph limits.\nArchitectures and experiment details.In all experiments, the GNNs have L = 1 layer (3) with ReLU nonlinearity followed by a perceptron layer.For opinion dynamics, K = 2, and for movie recommendation, K = 5.In both cases, we consider the MSE and fit the GNTK using linear regression.In the citation network experiment, the GNN architecture has K = 2 and includes a softmax layer followed by argmax; and we consider the cross-entropy (CE) and fit the GNTK using logistic regression.All reported results are averaged over 5 and 10 realizations for opinion dynamics and movie recommendation/citation networks respectively.Additional architecture and experiment details are listed in each subsection.The code can be found in this repository.\nAll experiments were run on a NVIDIA RTX A6000 GPU."},{"title":"Convergence","content":"To visualize the convergence of the GNTK, we fit a GNTK with F 1 = 10 in the opinon dynamics and citation network experiments, and F 1 = 16 in the recommendation experiment, on the training set of a small n-node graph; and transfer this GNTK, without retraining, to predict the outputs on the test set of both the same n-node graph and a larger N -node graph.To visualize convergence, we plot in Fig. 1 the absolute difference between the test errors attained on the n-node and the N -node graph (normalized by the test error on the N -node graph) as a function of the size of the smaller graph n, for both the opinion dynamics (left) and node classification (right) experiments.The results of the recommendation experiment are reported in Table 2.\nIn opinion dynamics, N = 300.We consider two types of graphs, corresponding to two graphon families: (a) a symmetrized geometric k-nearest neighbor graph where nodes are drawn at random from a 50 × 50 square and k = n/10; and (b) a stochastic block model (SBM) graph with intra-community probability p = 0.1 and inter-community probability q = 0.05.As n increases from 20 to 100, we observe that the difference between the MSEs for the n-node and the N -node graphs decrease steadily from ∼ 70% to ∼ 20% on the geometric graphs and from ∼ 90% to ∼ 40% on the SBM graphs.\nIn the recommendation experiment, N = 1000.As n increases from 100 to 500, we see from Table 2 that the average test MSE difference on the large graph decreases from 6.8% to 0.3%, and so does the standard deviation, thus confirming our theoretical findings.In particular, n = 300 is enough to achieve an MSE difference of less than 1%.\nIn the node classification experiment, N = 2708 for Cora, N = 3327 for CiteSeer, and N = 10000 for PubMed (sampled from the 19717 nodes in the original due to memory limitations).For all datasets, the difference between the CE loss for the n-node and the N -node graph also decreases as n increases.This can be interpreted to mean that node classi-fication on these citation networks depends largely on local information, so beyond a critical size n, the transference error saturates.An interesting point to make is that although these citation networks are not dense (i.e., are not best modeled by a graphon), we still see the empirical manifestation of our theoretical results."},{"title":"Wide Network Behavior","content":"Next, we analyze the effect of width on both the GNN and the GNTK when they are trained/fitted on a small graph and transferred to a large graph.From the NTK analysis by Jacot et al. (2018), as the GNN width increases we expect (i) the GNTK and the GNN to exhibit less variance over multiple weight initializations and (ii) the GNN outputs to approach those of kernel regression with the GNTK.\nFor the opinion dynamics experiment on SBMs, we consider three GNNs with widths (number of features) F\n(1) 1 = 10, F\n(2) 1 = 50, and F\n(3) 1 = 250 supported on the same 80-node graph.Using their initial weights, we construct the corresponding GNTKs.We train the three GNNs by minimizing the MSE loss over 20 epochs and with batch size 32, using ADAM (Kingma & Ba, 2015) with learning rate 1e −3 and weight decay 5e −3 .Simultaneously, we fit the GNTKs to the training set, and then transfer both the GNNs and the corresponding GNTKs to the N -node graph and compute their outputs on the test set.\nIn Fig. 2, we plot the projection of the outputs y onto the second graph eigenvector v 2 , [ŷ] 2 = v T 2 y against the projections of the inputs onto the same vector [x] 2 = v T 2 x (sorted in ascending order) for both the GNNs and the GNTKs for F\n(1) 1 = 10 (left), F\n(2) 1 = 50 (center), and F\n(3) 1 = 250 (right).The second graph eigenvector was chosen as it captures the community structure, however, note that the same behavior was observed upon projecting onto other eigenvectors, as shown in Appendix C. The results are averaged over 5 GNN initializations, with the solid lines representing the mean and the shaded areas representing the standard deviation.The behavior is as expected: as the width increases, the GNN and the GNTK have smaller variance in behavior for different weight initializations, and the GNN and GNTK curves align.In Appendix C, we include additional plots displaying the mean and variance over different initializations as we vary the graph size."},{"title":"Application: Eigenvalue Convergence","content":"In this section, we elucidate an important application of the convergence of the GNTK.Jacot et al. (2018) shows that the convergence of kernel gradient descent follows the kernel principal components.Therefore, if the GNN is sufficiently wide, and f (X n ; A n , H 0 ) − f (X n ; S n , H ⋆ ) is aligned with the ith GNTK principal component, we can expect gradient descent to converge with rate proportional to λ i , the ith eigenvalue of the GNTK2 .\nRecall that we prove, in Thm. 2, that the spectrum of the GNTKs converge in the graphon limit.In this context, a simple application of the convergence of the GNTK spectrum is as follows: say that we know that f (X N ; A N , H 0 ) − f (X N ; A N , H ⋆ ) is aligned with the pth GNTK eigenvector, and we want to estimate its speed of convergence, but the kernel Θ(X N , X ′ N ; A N , H) is too expensive to compute.We could in this case sample a graph G n , n ≪ N from the induced graphon W N and calculate Θ(X n , X ′ n ; A n , H).From Thm. 5.2, λ n,p converges to λ N,p .Hence, for large enough n, λ n,p should provide a good approximation of λ N,p .\nTo illustrate this empirically, in Fig. 3 we plot the dominant eigenvalues of the GNTK associated with a GNN with F 1 = 10 features supported on graphs of increasing size n sampled from the Cora, CiteSeer, and PubMed networks (averaged over 10 initializations).As n grows, we see that the difference between the dominant eigenvalues of consecutive GNTKs reduces.I.e., for each citation dataset, the value of the dominant eigenvalue converges as n increases, indicating that the speed of convergence of the GNN along the dominant GNTK eigenvector converges as the graph grows.For all but the PubMed dataset, the 1600-node graph gives an approximation of λ N,1 that is over 70% good."},{"title":"Conclusions","content":"In this paper, we define WNTKs as the limiting objects of GNTKs, and study how the GNTK evolves as the underlying graph of the corresponding GNN grows.We show that GNTKs converge to the WNTK and that their spectra also converges, thus providing theoretical insight into how the learning dynamics of a GNN evolves as an important dimension-the graph size-grows.In practice, these convergences imply that one can transfer the GNTK of a smaller graph to solve the same task on a larger graph without any further optimization, with theoretical guarantees of performance and insight on the rates of learning along eigendirections that converge as the graph grows.These results were demonstrated through simulations on synthetic and real-world graphs.To conclude, a limitation of this work is that graphons are only good models for dense graphs.In future work, we plan to extend our results to more general graph models, and to better understand the relationship between the spectra of the graph and of the GNTK."},{"title":"A. Proof of GNTK Convergence","content":"Proof of Thm.5.1.To simplify notation, we will write Θ(X, X ′ ; W, H) = Θ(X, X ′ ) and Θ(X n , X ′ n ; W n , H) = Θ(X n , X ′ n ).Let Y be an arbitrary L 2 signal.Using the triangle inequality,\nwhere there are L terms for each k.\nSince σ is Lipschitz, from the formula we see that it suffices to prove the following convergence statements for all k ∈ 1, ..., K − 1 as n → ∞:\nas well as\nHere, we prove these statements for each finite K using mathematical induction.For fixed, finite K, (18) implies ( 19).The convergence in ( 16) and ( 17) are essentially the same, so let us focus on the first without loss of generality.Expanding the first layer, we have\nTherefore, by the triangle inequality and the definition of the operator norm, we have From 21, using mathematical induction, we have, for arbitrary l,\nwhere h is the max of h l,k over all l and k, and so convergence of T (k)\nW as in (18) and X n → X implies convergence of U 1,n → U 1 and U l,n → U l .Since the nonlinearity σ is Lipschitz, convergence of (17) follows from convergence of (16).\nWe now focus on proving (18).By Cauchy-Schwarz,"},{"title":"||(T","content":""},{"title":"Figure 1 .","content":"Figure 1.Difference between the test error achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph.(left) Opinion dynamics on geometric and SBM graphs, where the test error is the MSE N = 300.(right) Node classification on Cora, CiteSeer and Pubmed, where the test error is the CE and N = 2708, 3327, and 10000 respectively.\nthe set of nodes whose opinion at time t diverges by at most 0 ≤ ϵ ≤ 1 of the opinion of node i.This model reduces to the classicDe-Groot (DeGroot, 1974) opinion model for ϵ = 1.We fix ϵ = 0.3 and c = 0.1."},{"title":"Figure 2 .","content":"Figure 2. Projections of the inputs and outputs of the GNTK and GNN, and of the target or true labels onto the second eigenvector of the adjacency matrix of a 80-node SBM graph for widths F = 10 (left), F = 50 (center) and F = 250 (right)."},{"title":"Figure 3 .","content":"Figure 3.The leading eigenvalue of the GNTK with width F = 10 for Cora, CiteSeer and PubMed as a function of the graph size n."},{"title":"W","content":"∥∥X n − X∥. (21)Looking at layer l, we haveU l,n − U l = k h l,k (T (k) Wn σ(U l−1,n ) − T (k) W σ(U l−1 )).\nWn − T W )Y || 2 = n (u, v) − W(u, v))Y (v)dv n (u, v) − W(u, v)) 2 dvdu ||Y || 2 ."},{"title":"Table 1 .","content":"Description of variables.\nThe graphon W n is induced by the graph G n , and the graphon signals X n and Y n are induced by the graph signals x n and y n .A succinct description of all graph and graphon variables is provided in Table 1."},{"title":"Table 2 .","content":"Movie recommendation results.Relative difference between the test MSE achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph, where N = 1000."}],"string":"[\n {\n \"title\": \"Introduction\",\n \"content\": \"Several real-world systems such as social-interactions, brainconnectome, epidemic spread, recommender systems, and traffic patterns are best represented by structured data in the form of large graphs.Graph neural networks (GNNs) are deep neural network architectures that leverage these graph structures to learn meaningful representations of node and edge data (Kipf & Welling, 2017;Hamilton et al., 2017;Defferrard et al., 2016;Gama et al., 2018).GNNs have shown remarkable empirical performance in a number of graph machine learning tasks, but can be hard to train on large-graph data.Recent research efforts have attempted to understand the large-graph behavior of GNNs, and in particular why GNNs trained on small graphs scale well to large networks (Ruiz et al., 2020;Levie et al., 2021;Keriven et al., 2020).However, the specific learning dynamics of a GNN trained directly on the large network, which are known to be challenging, are not as well understood.\\nModern deep neural networks (DNNs) are typically overparametrized.The benefits of overparametrization include faster convergence (Allen-Zhu et al., 2019;Arora et al., 2018) and better generalization (Cao & Gu, 2019), but on the other hand the parameters can be more difficult to interpret and the learning dynamics harder to understand.A remarkable contribution of Jacot et al. (2018) was the observation that, in the infinite width limit, learning the weights of a DNN via gradient descent reduces to kernel regression with a deterministic and fixed kernel called the neural tangent kernel (NTK), which captures the first-order approximation of the neural network's evolution during gradient descent (Lee et al., 2019).Since its introduction, the NTK has been an important and widely-studied tool in the machine learning toolbox, and kernel regression using the NTK has shown strong performance on small datasets (Arora et al., 2019b).\\nIn the graph case, it is straightforward to define the NTK associated with a GNN, or the graph neural tangent kernel (GNTK) (Du et al., 2019).The GNTK allows studying the training dynamics of the GNN when the number of features (the analog of width in the DNN) is large.Nonetheless, the effect of graph size on the GNN learning dynamics, which can itself be thought of as another 'width dimension', is not well understood, and there is limited research that investigates this rigorously.Herein, we propose to to understand the training dynamics of GNNs that are wide in both of these senses by combining the NTK formalism with the theory of graphons (Lovász, 2012;Borgs et al., 2008).\\nA graphon is a symmetric bounded measurable function W : [0, 1] 2 → [0, 1] representing the limit of a sequence of dense graphs.It can also be interpreted as a random graph model, in which case we can use the graphon to sample stochastic graphs.The interpretation of W as both a graph limit and a random graph model makes it so that each graphon defines a family of similar graphs.Hence, one can expect properties of a graphon to generalize, in a probabilistic sense, the properties of graphs belonging to its family.Graphons have been used to study the limit behavior of GNNs, which converge to so-called graphon neural networks (WNNs) (Ruiz et al., 2020).The fact that GNNs have a limit on the graphon implies that they are transferable across graphs in the same family, thus allowing a GNN to be trained on a graph of moderate size and transferred to a larger graph.\"\n },\n {\n \"title\": \"Contributions\",\n \"content\": \"In this paper, our first contribution is to define the graphon NTK (WNTK) associated with the WNN (Sec.4).We then prove, using mathematical induction, that the GNTK converges to the WNTK (Thm.5.1).In practice, this implies that GNTKs, like GNNs, are transferable across graphs of different sizes associated with the same graphon.That is to say, one can subsample a small graph and the corresponding data from a large graph, then fit the subsampled data to the small-graph GNTK via kernel regression, and then transfer the fitted model to the large-graph GNTK, which is particularly important for large graphs.\\nA more important implication of the convergence of the GNTK is that it is possible to understand the training dynamics of GNNs on large graphs by analyzing the behavior of the corresponding GNTK in graphs of moderate size.For instance, the eigenvalues of the NTK are associated with the speed of convergence along the corresponding eigendirections (Jacot et al., 2018).In Thm.5.2, we show that the eigenvalues of the graph GNTK, which indicate the directions of fastest convergence of the GNN, converge to the eigenvalues of the WNTK.This allows these eigenvalues to be estimated from GNTKs associated with smaller graphs.Lastly, we verify our theoretical results in three numerical applications: prediction of opinion dynamics on random graphs, movie recommendation using the Movie-Lens dataset, and node classification on the Cora, CiteSeer and PubMed networks.We observe the convergence of the GNTK (Sec.6.1), the effect of width in kernel regression and GNN training (Sec.6.2), and the convergence of the GNTK eigenvalues on sequences of graphs (Sec.6.3).\"\n },\n {\n \"title\": \"Related Work\",\n \"content\": \"Neural tangent kernels.The connection between infinitely wide DNNs and kernel methods (Gaussian processes) has been known since the 1990s (Neal & Neal, 1996;Williams, 1996), but a more theoretical formulation was presented by Jacot et al. (2018), which introduced the NTK and proved its constancy property in the infinite width limit, with Liu et al. (2020) later showing that constancy only holds for architectures with linear output layer.Several works have derived NTKs for a generalized classes of neural networks, including convolutional neural networks (Arora et al., 2019a;Li et al., 2019), ResNets (Huang et al., 2020) and, most closely related to our work, GNNs (Du et al., 2019).\\nGraphons and size generalization in deep learning.Graphons have been used to understand GNN convergence and transferability (Ruiz et al., 2020;2021a;Maskey et al., 2023), to analyze the generalization properties of GNNs in large graphs (Maskey et al., 2022), and to propose more computationally efficient training algorithms for large-scale GNNs (Cerviño et al., 2023).More recently, Xia et al. (2022) proposed implicit graphon neural representations, which use neural networks to estimate graphons.Yehudai et al. (2021) study graphs in which the local structure depends on the graph size-which is analogous to the dense graphs associated with graphons-, and find that GNNs are not guaranteed to scale to large graph sizes.However, their result foregoes the normalization of the adjacency matrix by the graph size, while we leverage this normalization to show operator norm convergence of the kernels associated with the graph convolution.It is also worth noting that our work is fundamentally different than GNN convergence or transferability results, because it relates not to the GNN architecture but to the GNN learning dynamics, and uses a mathematical induction proof as opposed to the spectral convergence argument typically used in GNN convergence proofs.\"\n },\n {\n \"title\": \"Graph and Graphon Neural Networks\",\n \"content\": \"Let G n = (V, E, w) be a graph where V, |V| = n, is the set of nodes or vertices, E ⊆ V × V is the set of edges and w : E → R is a map assigning weights to the edges in E. The size-normalized adjacency matrix of G n , denoted A n ∈ R n×n , is given by [A] ij = w(i, j)/n.In this paper we focus on undirected graphs G n , with symmetric A n .\"\n },\n {\n \"title\": \"Graph Neural Networks\",\n \"content\": \"Supported on the graph G n with n nodes, we define node data x n ∈ R n -also called graph signals (Shuman et al., 2013)-where [x n ] i is the value of the signal at node i. GNNs iteratively update each node's data by aggregating the data from its neighbors using graph convolutions.An order-K graph convolution is defined as\\nwhere h 0 , . . ., h K−1 are the convolution coefficients.When K = 2 and A n is binary, (1) can be seen as an aggregation operation akin to the AGGREGATE operation in, e.g., (Xu et al., 2019;Hamilton et al., 2017), or as the message-passing operation in message-passing neural net-\\nwith weights H 0 , H 1 , . . ., H K−1 ∈ R F ×G .Note that the number of parameters of the graph convolutions in (1) and\\n(2), K and KF G respectively, are independent of n.\\nA GNN consists of L layers, each of which composes a graph convolution and a nonlinear activation function.Explicitly, the lth layer of a GNN can be written as\\nwhere X l−1,n ∈ R n×F l−1 is the layer input, X l,n ∈ R n×F l is the layer output, H l,k ∈ R F l−1 ×F l are the layer weights and σ is a pointwise nonlinear activation function, i.e., [σ(X)] ij = σ([X] ij ).Typical choices for σ include ReLU, tanh, and sigmoid.At the first layer of the GNN, the input X 0,n is the input data X.Similarly, the GNN output Y n is given by the last layer output X n,L .\\nIn the following, we represent the entire GNN consisting of the concatenation of L layers like (3) as the parametric map Y n = f (X n ; A n , H), where H = {H l,k } l,k groups the learnable weights H l,k at all layers.This more concise representation highlights the independence between the adjacency matrix A n (i.e., the graph) and the parameters H, which the GNN inherits from the graph convolution.We note that while we describe a general class of GNNs here, the methods in our paper can be directly adapted to other types of GNNs.\"\n },\n {\n \"title\": \"Graphon Neural Networks\",\n \"content\": \"Graphons are bounded, symmetric, measurable functions W : [0, 1] 2 → [0, 1] representing limits of sequences of dense graphs (Lovász, 2012;Borgs et al., 2008).A graph sequence {G n } converges to a graphon in the sense that the densities of homomorphisms of any finite, unweighted and undirected graph F = (V ′ , E ′ ) into G n converge to the densities of homomorphisms of F into W.\\nThe graphs F can be thought of as motifs, such as triangles, k-cycles, k-cliques, etc. Explicitly, let hom(F, G n ) denote the total number of homomorphisms between V ′ and V n .The density of such homomorphisms is given by\\nand we can similarly define t(F, W) (see (Borgs et al., 2008)).Then, we say that G n → W if and only if\\nfor all simple F (Lovász & Szegedy, 2006).\\nAlternatively, the graphon can also be seen as a generative model for stochastic (also called W-random) graphs.Nodes are picked by sampling points u i , 1 ≤ i ≤ n, from the unit interval and connecting edges between nodes i and j with probability W(u i , u j ).Importantly, sequences of stochastic graphs generated in this way converge almost surely to the graphon (Lovász, 2012)[Cor. 10.4].\\nThe notion of graph signals is extended to graphons by defining graphon signals, which are functions X : [0, 1] → R (Ruiz et al., 2021a).We restrict attention to graphon signals with finite energy, i.e., X ∈ L 2 ([0, 1]).\\nAnalogously to (2), given F -and G-dimensional graphon signals X : [0, 1] → R F and Y : [0, 1] → R G , the graphon convolution is defined as (Ruiz et al., 2021a)\\nwhere T (0) W = I is the identity and the convolution weights are collected in the matrices\\nThe extension of the GNN to graphon data is the graphon neural network (WNN).Akin to the GNN, the WNN is formed by L layers each of which composes a graphon convolution and a nonlinear activation function.Explicitly, the lth layer of the WNN is given by\\nits weights and σ is a pointwise nonlinearity (e.g., the ReLU).The input of the first layer of the WNN is X 0 = X, and the output of the WNN is the last layer ouput, i.e., Y = X L .\\nWe can describe the WNN more compactly as the map Y = f (X; W, H), with H = {H l,k } l,k the set of learnable parameters at all layers.Note that, if the weights H are the same, the WNN map f (X; W, H) is the same as the GNN map f (X; A n , H) with A n swapped with W. This is important because it implies that, similarly to how graphons are generative models for graphs, WNNs are generative models for GNNs.Indeed, we can use the WNN f (X; W, H) to sample the GNN\\nwhere the u i are sampled uniformly and independently at random from [0, 1] and Ber() is the Bernoulli distribution.\\nAs n → ∞, sequences of GNNs sampled from the WNN as described above converge to the WNN.In fact, as long as H is the same, any GNN f et al., 2021a).A more important result in practice is that this convergence implies that GNNs are transferable across graphs associated with the same graphon, i.e., they can be trained on graphs G n and executed on graphs G m with an error that decreases asymptotically with n and m (Ruiz et al., 2021b;2020).\\nIn Sec. 5, we prove a similar convergence result for the neural tangent kernels (NTKs) associated with f (x n ; A n , H) and f (X; W, H), but before doing so, we need to introduce induced WNNs.The WNN induced by the GNN\\nand where I is the indicator function,\"\n },\n {\n \"title\": \"Graph and Graphon Neural Tangent Kernel\",\n \"content\": \"Consider a general, fully-connected neural network f (x; H), with layers given by\\n, assume that the loss to be minimized is the mean squared error (MSE) or quadratic loss\\nAs the training progresses, the output of the neural network for input x i is updated as y i (t) = f (x i ; H(t)).As such, the weight update rule is given by ∂H ∂t = −∇ℓ(H(t)) and so the output evolves as\\nIn the infinite-width limit as d l → ∞, Jacot et al. ( 2018); Liu et al. (2020) showed that, provided that the last layer has linear output (i.e., it does not have an activation σ), Θ(x i , x j ; H) converges to a limiting kernel, the NTK.This kernel stays constant during training-a property called constancy-, which is equivalent to replacing the outputs of the neural network by their first-order Taylor expansion in the parameter space (Lee et al., 2019).Hence, in the infinite-width limit the training dynamics of (10) reduce to kernel ridge regression1 on the NTK, which has a closedform solution.This facilitates understanding the learning dynamics of overparametrized neural networks, which are notoriously difficult to study directly, by analysis of the corresponding NTK.\"\n },\n {\n \"title\": \"Graph Neural Tangent Kernel\",\n \"content\": \"For simplicity, we will consider a GNN with only one feature per layer; the generalization to multiple features is more involved but straightforward.Recall that the L-layer GNN supported on A n ∈ R n×n is written as\\nThe graph NTK (GNTK) associated with this GNN is\\nwhere there are L terms; the first term corresponds to the last layer, the second term to the second-last layer, and so on.We have used σ ′ (u) to denote the diagonal matrix with jjth entry equal to σ ′ ([u] j ), where σ ′ is the derivative of σ.\"\n },\n {\n \"title\": \"Graphon Neural Tangent Kernel\",\n \"content\": \"To derive the WNTK, we will also consider a WNN with only one feature per layer for simplicity.Recall that the L-layer WNN associated with the graphon W is given by\\nTherefore, the WNTK is given by\\nCalculating the derivative with respect to the kth weight in layer l j = L − j, we get\\nfor j = 0, and similarly for 0 < j < L,\\nHence, (11) has L terms in total, explicitly\\nwhere σ ′ (U L ) is now evaluated and multiplied pointwise.Note that Θ is a linear operator on graphon signals, i.e., given a signal Y ∈ L 2 ([0, 1]), the WNTK applied to this signal yields\\nwhere the first term corresponds to the last layer, the second corrresponds to the second-last layer and so on.\\nFor a graph A n sampled from W with associated induced graphon W n , and for input features x n sampled from a graphon signal X and associated induced graphon signal X n [cf.( 8)], the induced WNTK is given by the same formula as (11) but with W, X replaced by W n , X n .\"\n },\n {\n \"title\": \"Graph NTK Converges to Graphon NTK\",\n \"content\": \"Next, we consider a sequence of graph signals {(G n , x n )} converging to a graphon signal (W, X) to evaluate the convergence of the GNTK Θ(x n , x ′ n ; A n , H) to the WNTK Θ(X, X ′ ; W, H).First, for each pair of signals X, X ′ , the WNTK Θ(X, X ′ ; W, H)-which we denote Θ(X, X ′ ) to simplify notation-is a linear operator on functions in L 2 .Hence, to prove convergence to this operator we introduce the operator norm\\nLet W be a graphon and X, X ′ be fixed graphon signals.\\nConsider a sequence of graph signals (Ruiz et al., 2021a)[Def. 2].The notion of convergence of graph signals on a sequence of dense graphs is simple; G n converges to W in the homomorphism density sense [cf.(4)], and X n , the graphon signal induced by x n , converges to X in the L 2 norm, i.e.\\n∥X n − X∥ → 0 (13) up to node relabelings (see (Ruiz et al., 2021a) for further details, and in particular Lemma 2 for the relationship between (G n , x n ) and the induced signal (W n , X n )).\\nLet (W n , X n ) be the graphon signal induced by the graph signal\\nThe main result of this paper is the following theorem that shows that the WNTK induced by the GNTK converges to the limiting WNTK.The proof is deferred to Appendix A. Theorem 5.1.Let W be a graphon and X, X ′ ∈ L 2 ([0, 1]) be arbitrary graphon signals.Suppose that {G n } is a sequence of graphs converging to W [cf. (4)] and {x n } , {x ′ n } are sequences of graph signals converging to X [cf.(13)].Then, for any L-layer GNN with finite K, fixed weights H and 1-Lipschitz nonlinearity σ, the associated GNTKs Θ(x n , x ′ n ; A n , H) converge in the operator norm:\\nThe convergence of the GNTK to a limit object-the WNTK-has two important implications for machine learning on large-scale graphs.First, consider that we have a large graph G N on which we want to predict signals y N using the GNTK, but that we do not have enough computational resources to compute the full-sized GNTK Θ(x N , x ′ N ; A N , H).This is expected, since calculating the kernel regression weights on this graph requires inverting a matrix with dimension proportional to M , the number of training samples, and the graph size N , see Sec. 4. Thm.5.1 implies that we can subsample the GNTK and the labels Y N on a smaller graph G n , n ≪ N -as Θ(x n , x ′ n ; A n , H) and y n respectively-and fit the data to the GNTK on this smaller graph.Once the kernel regression weights are obtained, they can then be transferred to make predictions on G N .Naturally, there will be an error associated with this transference; however, due to convergence, this error vanishes asymptotically in n.\\nThe precise convergence bounds for different graphs can be computed by plugging into (26).For instance, for graphs sampled from the graphon as in (Ruiz et al., 2021b)\\nwhere C is a constant, and A w and A x are the Lipschitz constants of the graphon and the graphon signal respectively.\\nIn practice, transferability is useful in a variety of real-world applications, especially on large graphs.For instance, one may be interested in the learning dynamics of a GNN trained to provide ad recommendations on a growing, dense social media network.Our theoretical results show that one can calculate the GNTK on a smaller subgraph, e.g., the same social media network at an earlier time, and then transfer it to the large target graph with theoretical performance guarantees.This works because dense networks growing under the same underlying 'rule' or 'process' should have similar homomorphism densities, and converge to similar graphons; or, inversely, they can be thought of as being sampled from similar graphons.The more similar these limiting graphons are (under a metric such as the cut metric or the L 2 norm), the lower the error when transferring the GNTK across graphs.\\nA second implication of Thm.5.1, which is perhaps more important, is that GNTK convergence implies that a GNTK fitted on a smaller graph can be used to infer details about the training dynamics of wide/overparametrized GNNs on larger graphs.This follows from Thm. 5.1 in conjunction with theoretical results by Liu et al. (2020), which proves that in the infinite-width limit the training dynamics of general convolutional models (including GNNs) reduces to kernel ridge regression on the corresponding NTK; and empirical results by Sabanayagam et al. (2021) showing that even when the output layer is nonlinear, constancy of the GNTK can still be observed.\\nAn example of property of a large-graph GNTK Θ(x N , x ′ N ; A N , H) that can be estimated from a smallgraph GNTK Θ(x n , x ′ n ; A n , H) is its eigenvalue spectrum.As we show in Thm.5.2, the spectrum of Θ(X n , X ′ n ; W n , H) converges to the the spectrum of Θ(X, X ′ ; W, H).The proof is deferred to Appendix B.\\nTheorem 5.2.Let {x i,n } M i=1 be a set of M signals on the graph G n , and {X i,n } M i=1 the corresponding induced signals on the induced graphon W n .Assume that G n → W [cf. (4)] and X i,n → X [cf.( 13)] for all i.Define the operators Θ n and Θ where\\nMoreover, the corresponding eigenspaces converge in the sense that the spectral projectors converge.\\nThis is an important result because, as proved by Jacot et al. (2018), the convergence of kernel gradient descent follows the kernel principal components.Thus, if the GNN f (H) is sufficiently wide, and if f (H 0 ) − f (H ⋆ ) is aligned with the ith GNTK principal component, we can expect gradient descent to converge with rate proportional to λ i , the ith eigenvalue of the GNTK.Thm.5.2 shows that the GNTK eigenvalues converge to the eigenvalues of the WNTK.As such, we can estimate the speed of convergence of a largegraph GNN f (x N ; A N , H) from the eigenvalues of a smallgraph GNTK Θ(x n , x ′ n ; A n , H).A numerical example of this application of Thms.5.1 and 5.2 is given in Sec.6.3.Spectral convergence of the GNTK is particularly relevant for practical applications on large graphs.Convergence of the spectrum implies convergence of the directions of learning.Intuitively, one can calculate the GNTK on the smaller graph, then transfer it to the large graph with the guarantee that the dominant directions of fastest convergence are being captured.We can learn GNN architectures that harness this property by pretraining on a smaller graph, and making adjustments on the large graph; or, by iterating between different resolutions of graph size.\"\n },\n {\n \"title\": \"Results\",\n \"content\": \"In the following, we illustrate the convergence of the GNTK to the WNTK on a simulated opinion dynamics model on random graphs; on a movie recommendation graph; and on citation networks (Sec.6.1).In the opinion dynamics problem, we further vary the width of the network to analyze the large width behavior of the GNTK and how it relates to the behavior of the GNN (Sec.6.2).We conclude with an example of how the convergence of the GNTK can be used in practice to estimate the eigenvalues of the GNTK, and thus the speed of GNN training along its principal components, on large-scale graphs (Sec.6.3).\\nOpinion dynamics.This is a node-level task modeling the outcomes of a mathematical model for studying the evolution of ideologies, affiliations, and opinions in society (Lorenz, 2007), including topics of important practical interest such as political ideologies and misinformation spread.On an undirected n-node graph G = (V, E), we consider an opinion dynamics process x t ∈ R n .The node data [x t ] i is the opinion of individual i under a standard boundedconfidence model described by Hegselmann et al. (2002):\\nwhere c > 0 is the so-called influence parameter and After drawing the initial opinions x 0 from a Gaussian distribution with µ = 0 and Σ = 2I, we run the process (14) until convergence (for at most T = 1000 iterations) to obtain the final opinions x T .The goal of the learning problem is then to predict y = x T from x = x 0 .We fix the training set size to 300 samples, and use 30 samples for both validation and testing.\\nMovie recommendation.In this problem, the graph is a movie similarity network, the data are existing user ratings (ranging from 1 to 5), and the task is to use the existing ratings to fill in the missing ratings for movie recommendation.The movie similarity graph is a dense correlation graph built by computing pairwise correlations between the rating vectors of the movies rated in the MovieLens-100k dataset (Harper & Konstan, 2016).We only consider movies with at least 10 ratings, and the large graph has N = 1000 nodes sampled at random from the set of admissible movies.\\nThe learning problem is to predict the ratings of the movie \\\"Contact\\\".We use a 90-10 train-test split of the data, and build the graph using only the training samples.Due to memory limitations, we only use 25% of the users (samples) to compute the GNTK.\\nCitation networks.The third problem we consider is a node classification problem on the Cora, CiteSeer and PubMed networks, where nodes represent documents and undirected edges represent citations between papers in either direction.\\nThe nodes a bag-of-words representations are grouped in X ∈ {0, 1} n×F and each node is associated with one of C classes.We consider the networks and train-test-splits from the full distribution (Bojchevski & Günnemann, 2017) available in PyTorch Geometric, but only sample F = 1000 features for CiteSeer and N = 10000 nodes for PubMed due to memory limitations.Through studying citation networks, we illustrate the empirical manifestation of our results on real-world graphs.Although citation networks are not best modeled by graphon limits, they certainly have limits, and our results suggest that convergence is a global property of GNTKs and graph limits.\\nArchitectures and experiment details.In all experiments, the GNNs have L = 1 layer (3) with ReLU nonlinearity followed by a perceptron layer.For opinion dynamics, K = 2, and for movie recommendation, K = 5.In both cases, we consider the MSE and fit the GNTK using linear regression.In the citation network experiment, the GNN architecture has K = 2 and includes a softmax layer followed by argmax; and we consider the cross-entropy (CE) and fit the GNTK using logistic regression.All reported results are averaged over 5 and 10 realizations for opinion dynamics and movie recommendation/citation networks respectively.Additional architecture and experiment details are listed in each subsection.The code can be found in this repository.\\nAll experiments were run on a NVIDIA RTX A6000 GPU.\"\n },\n {\n \"title\": \"Convergence\",\n \"content\": \"To visualize the convergence of the GNTK, we fit a GNTK with F 1 = 10 in the opinon dynamics and citation network experiments, and F 1 = 16 in the recommendation experiment, on the training set of a small n-node graph; and transfer this GNTK, without retraining, to predict the outputs on the test set of both the same n-node graph and a larger N -node graph.To visualize convergence, we plot in Fig. 1 the absolute difference between the test errors attained on the n-node and the N -node graph (normalized by the test error on the N -node graph) as a function of the size of the smaller graph n, for both the opinion dynamics (left) and node classification (right) experiments.The results of the recommendation experiment are reported in Table 2.\\nIn opinion dynamics, N = 300.We consider two types of graphs, corresponding to two graphon families: (a) a symmetrized geometric k-nearest neighbor graph where nodes are drawn at random from a 50 × 50 square and k = n/10; and (b) a stochastic block model (SBM) graph with intra-community probability p = 0.1 and inter-community probability q = 0.05.As n increases from 20 to 100, we observe that the difference between the MSEs for the n-node and the N -node graphs decrease steadily from ∼ 70% to ∼ 20% on the geometric graphs and from ∼ 90% to ∼ 40% on the SBM graphs.\\nIn the recommendation experiment, N = 1000.As n increases from 100 to 500, we see from Table 2 that the average test MSE difference on the large graph decreases from 6.8% to 0.3%, and so does the standard deviation, thus confirming our theoretical findings.In particular, n = 300 is enough to achieve an MSE difference of less than 1%.\\nIn the node classification experiment, N = 2708 for Cora, N = 3327 for CiteSeer, and N = 10000 for PubMed (sampled from the 19717 nodes in the original due to memory limitations).For all datasets, the difference between the CE loss for the n-node and the N -node graph also decreases as n increases.This can be interpreted to mean that node classi-fication on these citation networks depends largely on local information, so beyond a critical size n, the transference error saturates.An interesting point to make is that although these citation networks are not dense (i.e., are not best modeled by a graphon), we still see the empirical manifestation of our theoretical results.\"\n },\n {\n \"title\": \"Wide Network Behavior\",\n \"content\": \"Next, we analyze the effect of width on both the GNN and the GNTK when they are trained/fitted on a small graph and transferred to a large graph.From the NTK analysis by Jacot et al. (2018), as the GNN width increases we expect (i) the GNTK and the GNN to exhibit less variance over multiple weight initializations and (ii) the GNN outputs to approach those of kernel regression with the GNTK.\\nFor the opinion dynamics experiment on SBMs, we consider three GNNs with widths (number of features) F\\n(1) 1 = 10, F\\n(2) 1 = 50, and F\\n(3) 1 = 250 supported on the same 80-node graph.Using their initial weights, we construct the corresponding GNTKs.We train the three GNNs by minimizing the MSE loss over 20 epochs and with batch size 32, using ADAM (Kingma & Ba, 2015) with learning rate 1e −3 and weight decay 5e −3 .Simultaneously, we fit the GNTKs to the training set, and then transfer both the GNNs and the corresponding GNTKs to the N -node graph and compute their outputs on the test set.\\nIn Fig. 2, we plot the projection of the outputs y onto the second graph eigenvector v 2 , [ŷ] 2 = v T 2 y against the projections of the inputs onto the same vector [x] 2 = v T 2 x (sorted in ascending order) for both the GNNs and the GNTKs for F\\n(1) 1 = 10 (left), F\\n(2) 1 = 50 (center), and F\\n(3) 1 = 250 (right).The second graph eigenvector was chosen as it captures the community structure, however, note that the same behavior was observed upon projecting onto other eigenvectors, as shown in Appendix C. The results are averaged over 5 GNN initializations, with the solid lines representing the mean and the shaded areas representing the standard deviation.The behavior is as expected: as the width increases, the GNN and the GNTK have smaller variance in behavior for different weight initializations, and the GNN and GNTK curves align.In Appendix C, we include additional plots displaying the mean and variance over different initializations as we vary the graph size.\"\n },\n {\n \"title\": \"Application: Eigenvalue Convergence\",\n \"content\": \"In this section, we elucidate an important application of the convergence of the GNTK.Jacot et al. (2018) shows that the convergence of kernel gradient descent follows the kernel principal components.Therefore, if the GNN is sufficiently wide, and f (X n ; A n , H 0 ) − f (X n ; S n , H ⋆ ) is aligned with the ith GNTK principal component, we can expect gradient descent to converge with rate proportional to λ i , the ith eigenvalue of the GNTK2 .\\nRecall that we prove, in Thm. 2, that the spectrum of the GNTKs converge in the graphon limit.In this context, a simple application of the convergence of the GNTK spectrum is as follows: say that we know that f (X N ; A N , H 0 ) − f (X N ; A N , H ⋆ ) is aligned with the pth GNTK eigenvector, and we want to estimate its speed of convergence, but the kernel Θ(X N , X ′ N ; A N , H) is too expensive to compute.We could in this case sample a graph G n , n ≪ N from the induced graphon W N and calculate Θ(X n , X ′ n ; A n , H).From Thm. 5.2, λ n,p converges to λ N,p .Hence, for large enough n, λ n,p should provide a good approximation of λ N,p .\\nTo illustrate this empirically, in Fig. 3 we plot the dominant eigenvalues of the GNTK associated with a GNN with F 1 = 10 features supported on graphs of increasing size n sampled from the Cora, CiteSeer, and PubMed networks (averaged over 10 initializations).As n grows, we see that the difference between the dominant eigenvalues of consecutive GNTKs reduces.I.e., for each citation dataset, the value of the dominant eigenvalue converges as n increases, indicating that the speed of convergence of the GNN along the dominant GNTK eigenvector converges as the graph grows.For all but the PubMed dataset, the 1600-node graph gives an approximation of λ N,1 that is over 70% good.\"\n },\n {\n \"title\": \"Conclusions\",\n \"content\": \"In this paper, we define WNTKs as the limiting objects of GNTKs, and study how the GNTK evolves as the underlying graph of the corresponding GNN grows.We show that GNTKs converge to the WNTK and that their spectra also converges, thus providing theoretical insight into how the learning dynamics of a GNN evolves as an important dimension-the graph size-grows.In practice, these convergences imply that one can transfer the GNTK of a smaller graph to solve the same task on a larger graph without any further optimization, with theoretical guarantees of performance and insight on the rates of learning along eigendirections that converge as the graph grows.These results were demonstrated through simulations on synthetic and real-world graphs.To conclude, a limitation of this work is that graphons are only good models for dense graphs.In future work, we plan to extend our results to more general graph models, and to better understand the relationship between the spectra of the graph and of the GNTK.\"\n },\n {\n \"title\": \"A. Proof of GNTK Convergence\",\n \"content\": \"Proof of Thm.5.1.To simplify notation, we will write Θ(X, X ′ ; W, H) = Θ(X, X ′ ) and Θ(X n , X ′ n ; W n , H) = Θ(X n , X ′ n ).Let Y be an arbitrary L 2 signal.Using the triangle inequality,\\nwhere there are L terms for each k.\\nSince σ is Lipschitz, from the formula we see that it suffices to prove the following convergence statements for all k ∈ 1, ..., K − 1 as n → ∞:\\nas well as\\nHere, we prove these statements for each finite K using mathematical induction.For fixed, finite K, (18) implies ( 19).The convergence in ( 16) and ( 17) are essentially the same, so let us focus on the first without loss of generality.Expanding the first layer, we have\\nTherefore, by the triangle inequality and the definition of the operator norm, we have From 21, using mathematical induction, we have, for arbitrary l,\\nwhere h is the max of h l,k over all l and k, and so convergence of T (k)\\nW as in (18) and X n → X implies convergence of U 1,n → U 1 and U l,n → U l .Since the nonlinearity σ is Lipschitz, convergence of (17) follows from convergence of (16).\\nWe now focus on proving (18).By Cauchy-Schwarz,\"\n },\n {\n \"title\": \"||(T\",\n \"content\": \"\"\n },\n {\n \"title\": \"Figure 1 .\",\n \"content\": \"Figure 1.Difference between the test error achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph.(left) Opinion dynamics on geometric and SBM graphs, where the test error is the MSE N = 300.(right) Node classification on Cora, CiteSeer and Pubmed, where the test error is the CE and N = 2708, 3327, and 10000 respectively.\\nthe set of nodes whose opinion at time t diverges by at most 0 ≤ ϵ ≤ 1 of the opinion of node i.This model reduces to the classicDe-Groot (DeGroot, 1974) opinion model for ϵ = 1.We fix ϵ = 0.3 and c = 0.1.\"\n },\n {\n \"title\": \"Figure 2 .\",\n \"content\": \"Figure 2. Projections of the inputs and outputs of the GNTK and GNN, and of the target or true labels onto the second eigenvector of the adjacency matrix of a 80-node SBM graph for widths F = 10 (left), F = 50 (center) and F = 250 (right).\"\n },\n {\n \"title\": \"Figure 3 .\",\n \"content\": \"Figure 3.The leading eigenvalue of the GNTK with width F = 10 for Cora, CiteSeer and PubMed as a function of the graph size n.\"\n },\n {\n \"title\": \"W\",\n \"content\": \"∥∥X n − X∥. (21)Looking at layer l, we haveU l,n − U l = k h l,k (T (k) Wn σ(U l−1,n ) − T (k) W σ(U l−1 )).\\nWn − T W )Y || 2 = n (u, v) − W(u, v))Y (v)dv n (u, v) − W(u, v)) 2 dvdu ||Y || 2 .\"\n },\n {\n \"title\": \"Table 1 .\",\n \"content\": \"Description of variables.\\nThe graphon W n is induced by the graph G n , and the graphon signals X n and Y n are induced by the graph signals x n and y n .A succinct description of all graph and graphon variables is provided in Table 1.\"\n },\n {\n \"title\": \"Table 2 .\",\n \"content\": \"Movie recommendation results.Relative difference between the test MSE achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph, where N = 1000.\"\n }\n]","isConversational":false},"figures":{"kind":"list like","value":[{"id":"","caption":"Figure 1.Difference between the test error achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph.(left) Opinion dynamics on geometric and SBM graphs, where the test error is the MSE N = 300.(right) Node classification on Cora, CiteSeer and Pubmed, where the test error is the CE and N = 2708, 3327, and 10000 respectively.","content":"Figure 1 .\n1\nFigure 1.Difference between the test error achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph.(left) Opinion dynamics on geometric and SBM graphs, where the test error is the MSE N = 300.(right) Node classification on Cora, CiteSeer and Pubmed, where the test error is the CE and N = 2708, 3327, and 10000 respectively."},{"id":"","caption":"the set of nodes whose opinion at time t diverges by at most 0 ≤ ϵ ≤ 1 of the opinion of node i.This model reduces to the classicDe-Groot (DeGroot, 1974) opinion model for ϵ = 1.We fix ϵ = 0.3 and c = 0.1.","content":"the set of nodes whose opinion at time t diverges by at most 0 ≤ ϵ ≤ 1 of the opinion of node i.This model reduces to the classicDe-Groot (DeGroot, 1974) opinion model for ϵ = 1.We fix ϵ = 0.3 and c = 0.1."},{"id":"","caption":"Figure 2. Projections of the inputs and outputs of the GNTK and GNN, and of the target or true labels onto the second eigenvector of the adjacency matrix of a 80-node SBM graph for widths F = 10 (left), F = 50 (center) and F = 250 (right).","content":"Figure 2 .\n2\nFigure 2. Projections of the inputs and outputs of the GNTK and GNN, and of the target or true labels onto the second eigenvector of the adjacency matrix of a 80-node SBM graph for widths F = 10 (left), F = 50 (center) and F = 250 (right)."},{"id":"","caption":"Figure 3.The leading eigenvalue of the GNTK with width F = 10 for Cora, CiteSeer and PubMed as a function of the graph size n.","content":"Figure 3 .\n3\nFigure 3.The leading eigenvalue of the GNTK with width F = 10 for Cora, CiteSeer and PubMed as a function of the graph size n."},{"id":"","caption":"∥∥X n − X∥. (21)Looking at layer l, we haveU l,n − U l = k h l,k (T (k) Wn σ(U l−1,n ) − T (k) W σ(U l−1 )).","content":"W\n\n∥∥X n − X∥. (21)Looking at layer l, we haveU l,n − U l = k h l,k (T (k) Wn σ(U l−1,n ) − T (k) W σ(U l−1 ))."},{"id":"","caption":"Wn − T W )Y || 2 = n (u, v) − W(u, v))Y (v)dv n (u, v) − W(u, v)) 2 dvdu ||Y || 2 .","content":"Wn − T W )Y || 2 = n (u, v) − W(u, v))Y (v)dv n (u, v) − W(u, v)) 2 dvdu ||Y || 2 ."},{"id":"","caption":"Description of variables.","content":"Table 1 .\n1\nDescription of variables.\nVariableDescriptionA nAdjacency matrix of graph G nx nGraph signal on G nW nGraphon induced by G nX nGraphon signal induced by x nWLimiting graphonXLimiting graphon signalworks (MPNNs) (Gilmer et al., 2017).More generally, let X n ∈ R n×F and Y n ∈ R n×G be F -and G-dimensional signals respectively, where the f th(gth) column is a feature x f n (y g n ). In this case, the graphconvolution generalizes to"},{"id":"","caption":"The graphon W n is induced by the graph G n , and the graphon signals X n and Y n are induced by the graph signals x n and y n .A succinct description of all graph and graphon variables is provided in Table 1.","content":"The graphon W n is induced by the graph G n , and the graphon signals X n and Y n are induced by the graph signals x n and y n .A succinct description of all graph and graphon variables is provided in Table 1."},{"id":"","caption":"Movie recommendation results.Relative difference between the test MSE achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph, where N = 1000.","content":"Table 2 .\n2\nMovie recommendation results.Relative difference between the test MSE achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph, where N = 1000.\nNumber of nodes 100 200 300 400 500Mean (%)6.80 0.78 0.09 0.03 0.03Std. dev. (p.p.)3.51 1.16 0.03 0.01 0.02weights in H are bounded, then, for all p,"}],"string":"[\n {\n \"id\": \"\",\n \"caption\": \"Figure 1.Difference between the test error achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph.(left) Opinion dynamics on geometric and SBM graphs, where the test error is the MSE N = 300.(right) Node classification on Cora, CiteSeer and Pubmed, where the test error is the CE and N = 2708, 3327, and 10000 respectively.\",\n \"content\": \"Figure 1 .\\n1\\nFigure 1.Difference between the test error achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph.(left) Opinion dynamics on geometric and SBM graphs, where the test error is the MSE N = 300.(right) Node classification on Cora, CiteSeer and Pubmed, where the test error is the CE and N = 2708, 3327, and 10000 respectively.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"the set of nodes whose opinion at time t diverges by at most 0 ≤ ϵ ≤ 1 of the opinion of node i.This model reduces to the classicDe-Groot (DeGroot, 1974) opinion model for ϵ = 1.We fix ϵ = 0.3 and c = 0.1.\",\n \"content\": \"the set of nodes whose opinion at time t diverges by at most 0 ≤ ϵ ≤ 1 of the opinion of node i.This model reduces to the classicDe-Groot (DeGroot, 1974) opinion model for ϵ = 1.We fix ϵ = 0.3 and c = 0.1.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 2. Projections of the inputs and outputs of the GNTK and GNN, and of the target or true labels onto the second eigenvector of the adjacency matrix of a 80-node SBM graph for widths F = 10 (left), F = 50 (center) and F = 250 (right).\",\n \"content\": \"Figure 2 .\\n2\\nFigure 2. Projections of the inputs and outputs of the GNTK and GNN, and of the target or true labels onto the second eigenvector of the adjacency matrix of a 80-node SBM graph for widths F = 10 (left), F = 50 (center) and F = 250 (right).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Figure 3.The leading eigenvalue of the GNTK with width F = 10 for Cora, CiteSeer and PubMed as a function of the graph size n.\",\n \"content\": \"Figure 3 .\\n3\\nFigure 3.The leading eigenvalue of the GNTK with width F = 10 for Cora, CiteSeer and PubMed as a function of the graph size n.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"∥∥X n − X∥. (21)Looking at layer l, we haveU l,n − U l = k h l,k (T (k) Wn σ(U l−1,n ) − T (k) W σ(U l−1 )).\",\n \"content\": \"W\\n\\n∥∥X n − X∥. (21)Looking at layer l, we haveU l,n − U l = k h l,k (T (k) Wn σ(U l−1,n ) − T (k) W σ(U l−1 )).\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Wn − T W )Y || 2 = n (u, v) − W(u, v))Y (v)dv n (u, v) − W(u, v)) 2 dvdu ||Y || 2 .\",\n \"content\": \"Wn − T W )Y || 2 = n (u, v) − W(u, v))Y (v)dv n (u, v) − W(u, v)) 2 dvdu ||Y || 2 .\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Description of variables.\",\n \"content\": \"Table 1 .\\n1\\nDescription of variables.\\nVariableDescriptionA nAdjacency matrix of graph G nx nGraph signal on G nW nGraphon induced by G nX nGraphon signal induced by x nWLimiting graphonXLimiting graphon signalworks (MPNNs) (Gilmer et al., 2017).More generally, let X n ∈ R n×F and Y n ∈ R n×G be F -and G-dimensional signals respectively, where the f th(gth) column is a feature x f n (y g n ). In this case, the graphconvolution generalizes to\"\n },\n {\n \"id\": \"\",\n \"caption\": \"The graphon W n is induced by the graph G n , and the graphon signals X n and Y n are induced by the graph signals x n and y n .A succinct description of all graph and graphon variables is provided in Table 1.\",\n \"content\": \"The graphon W n is induced by the graph G n , and the graphon signals X n and Y n are induced by the graph signals x n and y n .A succinct description of all graph and graphon variables is provided in Table 1.\"\n },\n {\n \"id\": \"\",\n \"caption\": \"Movie recommendation results.Relative difference between the test MSE achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph, where N = 1000.\",\n \"content\": \"Table 2 .\\n2\\nMovie recommendation results.Relative difference between the test MSE achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph, where N = 1000.\\nNumber of nodes 100 200 300 400 500Mean (%)6.80 0.78 0.09 0.03 0.03Std. dev. (p.p.)3.51 1.16 0.03 0.01 0.02weights in H are bounded, then, for all p,\"\n }\n]","isConversational":false},"references":{"kind":"list like","value":[{"text":"A convergence theory for deep learning via over-parameterization. Z Allen-Zhu, Y Li, Z Song, Int. Conf. Mach. Learning. PMLR2019","doi":""},{"text":"Spectral analysis of collectively compact, strongly convergent operator sequences. P Anselone, T Palmer, Pacific Journal of Mathematics. 2531968","doi":""},{"text":"On the optimization of deep networks: Implicit acceleration by overparameterization. S Arora, N Cohen, E Hazan, Int. Conf. Mach. Learning. PMLR2018","doi":""},{"text":"On exact computation with an infinitely wide neural net. S Arora, S S Du, W Hu, Z Li, R R Salakhutdinov, R Wang, Neural Inform. Process. Syst. 322019a","doi":""},{"text":"Harnessing the power of infinitely wide deep nets on small-data tasks. S Arora, S S Du, Z Li, R Salakhutdinov, R Wang, D Yu, arXiv:1910.016632019barXiv preprint","doi":""},{"text":"A Bojchevski, S Günnemann, arXiv:1707.03815[cs.LGDeep gaussian embedding of graphs: Unsupervised inductive learning via ranking. 2017","doi":""},{"text":"Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. C Borgs, J T Chayes, L Lovász, V T Sós, K Vesztergombi, Adv. Math. 21962008","doi":""},{"text":"Generalization bounds of stochastic gradient descent for wide and deep neural networks. Y Cao, Q Gu, Neural Inform. Process. Syst. 322019","doi":""},{"text":"Learning by transference: Training graph neural networks on growing graphs. J Cerviño, L Ruiz, A Ribeiro, 10.1109/TSP.2023.3242374IEEE Trans. Signal Process. 712023","doi":""},{"text":"Convolutional neural networks on graphs with fast localized spectral filtering. M Defferrard, X Bresson, P Vandergheynst, NIPS Foundation. Barcelona, Spain10 Dec. 20165","doi":""},{"text":"Reaching a consensus. M H Degroot, Journal of the American Statistical association. 693451974","doi":""},{"text":"Graph neural tangent kernel: Fusing graph neural networks with graph kernels. S S Du, K Hou, R R Salakhutdinov, B Poczos, R Wang, K Xu, In Neural Inform. Process. Syst. 322019","doi":""},{"text":"Convolutional neural network architectures for signals supported on graphs. F Gama, G Marques, A Leus, G Ribeiro, A , IEEE Trans. Signal Process. 672018","doi":""},{"text":"Neural message passing for quantum chemistry. J Gilmer, S S Schoenholz, P F Riley, O Vinyals, G E Dahl, Int. Conf. Mach. Learning. PMLR2017","doi":""},{"text":"Inductive representation learning on large graphs. W Hamilton, Z Ying, J Leskovec, Neural Inform. Process. Syst. 302017","doi":""},{"text":"The movielens datasets: History and context. F M Harper, J A Konstan, ACM Trans. Interactive Intell. Syst. 5Jan. 2016","doi":""},{"text":"Opinion dynamics and bounded confidence models, analysis, and simulation. R Hegselmann, U Krause, Journal of artificial societies and social simulation. 532002","doi":""},{"text":"Why do deep residual networks generalize better than deep feedforward networks?-a neural tangent kernel perspective. K Huang, Y Wang, M Tao, T Zhao, Neural Inform. Process. Syst. 332020","doi":""},{"text":"Neural tangent kernel: Convergence and generalization in neural networks. A Jacot, F Gabriel, C Hongler, Neural Inform. Process. Syst. 312018","doi":""},{"text":"Convergence and stability of graph convolutional networks on large random graphs. N Keriven, A Bietti, S Vaiter, Neural Inform. Process. Syst. 332020NeurIPS Foundation","doi":""},{"text":"A method for stochastic optimization. D P Kingma, J L Ba, Adam, 3rd Int. Conf. Learning Representations. San Diego, CAMay 2015","doi":""},{"text":"Semi-supervised classification with graph convolutional networks. T N Kipf, M Welling, 5th Int. Conf. Learning Representations. Toulon, FranceApr. 2017","doi":""},{"text":"Wide neural networks of any depth evolve as linear models under gradient descent. J Lee, L Xiao, S Schoenholz, Y Bahri, R Novak, J Sohl-Dickstein, J Pennington, Neural Inform. Process. Syst. 322019","doi":""},{"text":"Transferability of spectral graph convolutional neural networks. R Levie, W Huang, L Bucci, M Bronstein, G Kutyniok, J. Mach. Learning Res. 222722021","doi":""},{"text":"Z Li, R Wang, D Yu, S S Du, W Hu, R Salakhutdinov, S Arora, arXiv:1911.00809Enhanced convolutional neural tangent kernels. 2019arXiv preprint","doi":""},{"text":"On the linearity of large non-linear models: when and why the tangent kernel is constant. C Liu, L Zhu, M Belkin, Neural Inform. Process. Syst. 332020","doi":""},{"text":"Continuous opinion dynamics under bounded confidence: A survey. J Lorenz, International Journal of Modern Physics C. 18122007","doi":""},{"text":"Large Networks and Graph Limits. L Lovász, 2012American Mathematical Society60","doi":""},{"text":"Limits of dense graph sequences. L Lovász, B Szegedy, J. Comb. Theory, Series B. 9662006","doi":""},{"text":"Generalization analysis of message passing neural networks on large random graphs. S Maskey, R Levie, Y Lee, G Kutyniok, Advances in Neural Information Processing Systems. 2022","doi":""},{"text":"Transferability of graph neural networks: an extended graphon approach. S Maskey, R Levie, G Kutyniok, Applied and Computational Harmonic Analysis. 632023","doi":""},{"text":"Priors for infinite networks. Bayesian learning for neural networks. R M Neal, R M Neal, 1996","doi":""},{"text":"Graphon neural networks and the transferability of graph neural networks. L Ruiz, L F O Chamon, A Ribeiro, Neural Inform. Process. Syst. Dec. 2020NeurIPS Foundation","doi":""},{"text":"Graphon signal processing. L Ruiz, L F O Chamon, A Ribeiro, IEEE Trans. Signal Process. 692021a","doi":""},{"text":"L Ruiz, L F O Chamon, A Ribeiro, arXiv:2112.04629Transferability properties of graph neural networks. 2021bSubmitted to IEEE TSP","doi":""},{"text":"M Sabanayagam, P Esser, D Ghoshdastidar, arXiv:2110.04060New insights into graph convolutional networks using neural tangent kernels. 2021arXiv preprint","doi":""},{"text":"The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. D I Shuman, S K Narang, P Frossard, A Ortega, P Vandergheynst, IEEE Signal Process. Mag. 303May 2013","doi":""},{"text":"Computing with infinite networks. C Williams, Neural Inform. Process. Syst. 91996","doi":""},{"text":"X Xia, G Mishne, Y Wang, arXiv:2211.03329Implicit graphon neural representation. 2022arXiv preprint","doi":""},{"text":"How powerful are graph neural networks?. K Xu, W Hu, J Leskovec, S Jegelka, 7th Int. Conf. Learning Representations. New Orleans, LA6-9 May 2019","doi":""},{"text":"From local structures to size generalization in graph neural networks. G Yehudai, E Fetaya, E Meirom, G Chechik, H Maron, Int. Conf. Mach. Learning. PMLR2021","doi":""}],"string":"[\n {\n \"text\": \"A convergence theory for deep learning via over-parameterization. Z Allen-Zhu, Y Li, Z Song, Int. Conf. Mach. Learning. PMLR2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Spectral analysis of collectively compact, strongly convergent operator sequences. P Anselone, T Palmer, Pacific Journal of Mathematics. 2531968\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the optimization of deep networks: Implicit acceleration by overparameterization. S Arora, N Cohen, E Hazan, Int. Conf. Mach. Learning. PMLR2018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On exact computation with an infinitely wide neural net. S Arora, S S Du, W Hu, Z Li, R R Salakhutdinov, R Wang, Neural Inform. Process. Syst. 322019a\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Harnessing the power of infinitely wide deep nets on small-data tasks. S Arora, S S Du, Z Li, R Salakhutdinov, R Wang, D Yu, arXiv:1910.016632019barXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A Bojchevski, S Günnemann, arXiv:1707.03815[cs.LGDeep gaussian embedding of graphs: Unsupervised inductive learning via ranking. 2017\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing. C Borgs, J T Chayes, L Lovász, V T Sós, K Vesztergombi, Adv. Math. 21962008\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Generalization bounds of stochastic gradient descent for wide and deep neural networks. Y Cao, Q Gu, Neural Inform. Process. Syst. 322019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Learning by transference: Training graph neural networks on growing graphs. J Cerviño, L Ruiz, A Ribeiro, 10.1109/TSP.2023.3242374IEEE Trans. Signal Process. 712023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convolutional neural networks on graphs with fast localized spectral filtering. M Defferrard, X Bresson, P Vandergheynst, NIPS Foundation. Barcelona, Spain10 Dec. 20165\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Reaching a consensus. M H Degroot, Journal of the American Statistical association. 693451974\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Graph neural tangent kernel: Fusing graph neural networks with graph kernels. S S Du, K Hou, R R Salakhutdinov, B Poczos, R Wang, K Xu, In Neural Inform. Process. Syst. 322019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convolutional neural network architectures for signals supported on graphs. F Gama, G Marques, A Leus, G Ribeiro, A , IEEE Trans. Signal Process. 672018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Neural message passing for quantum chemistry. J Gilmer, S S Schoenholz, P F Riley, O Vinyals, G E Dahl, Int. Conf. Mach. Learning. PMLR2017\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Inductive representation learning on large graphs. W Hamilton, Z Ying, J Leskovec, Neural Inform. Process. Syst. 302017\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The movielens datasets: History and context. F M Harper, J A Konstan, ACM Trans. Interactive Intell. Syst. 5Jan. 2016\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Opinion dynamics and bounded confidence models, analysis, and simulation. R Hegselmann, U Krause, Journal of artificial societies and social simulation. 532002\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Why do deep residual networks generalize better than deep feedforward networks?-a neural tangent kernel perspective. K Huang, Y Wang, M Tao, T Zhao, Neural Inform. Process. Syst. 332020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Neural tangent kernel: Convergence and generalization in neural networks. A Jacot, F Gabriel, C Hongler, Neural Inform. Process. Syst. 312018\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Convergence and stability of graph convolutional networks on large random graphs. N Keriven, A Bietti, S Vaiter, Neural Inform. Process. Syst. 332020NeurIPS Foundation\",\n \"doi\": \"\"\n },\n {\n \"text\": \"A method for stochastic optimization. D P Kingma, J L Ba, Adam, 3rd Int. Conf. Learning Representations. San Diego, CAMay 2015\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Semi-supervised classification with graph convolutional networks. T N Kipf, M Welling, 5th Int. Conf. Learning Representations. Toulon, FranceApr. 2017\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Wide neural networks of any depth evolve as linear models under gradient descent. J Lee, L Xiao, S Schoenholz, Y Bahri, R Novak, J Sohl-Dickstein, J Pennington, Neural Inform. Process. Syst. 322019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transferability of spectral graph convolutional neural networks. R Levie, W Huang, L Bucci, M Bronstein, G Kutyniok, J. Mach. Learning Res. 222722021\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Z Li, R Wang, D Yu, S S Du, W Hu, R Salakhutdinov, S Arora, arXiv:1911.00809Enhanced convolutional neural tangent kernels. 2019arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"On the linearity of large non-linear models: when and why the tangent kernel is constant. C Liu, L Zhu, M Belkin, Neural Inform. Process. Syst. 332020\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Continuous opinion dynamics under bounded confidence: A survey. J Lorenz, International Journal of Modern Physics C. 18122007\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Large Networks and Graph Limits. L Lovász, 2012American Mathematical Society60\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Limits of dense graph sequences. L Lovász, B Szegedy, J. Comb. Theory, Series B. 9662006\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Generalization analysis of message passing neural networks on large random graphs. S Maskey, R Levie, Y Lee, G Kutyniok, Advances in Neural Information Processing Systems. 2022\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Transferability of graph neural networks: an extended graphon approach. S Maskey, R Levie, G Kutyniok, Applied and Computational Harmonic Analysis. 632023\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Priors for infinite networks. Bayesian learning for neural networks. R M Neal, R M Neal, 1996\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Graphon neural networks and the transferability of graph neural networks. L Ruiz, L F O Chamon, A Ribeiro, Neural Inform. Process. Syst. Dec. 2020NeurIPS Foundation\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Graphon signal processing. L Ruiz, L F O Chamon, A Ribeiro, IEEE Trans. Signal Process. 692021a\",\n \"doi\": \"\"\n },\n {\n \"text\": \"L Ruiz, L F O Chamon, A Ribeiro, arXiv:2112.04629Transferability properties of graph neural networks. 2021bSubmitted to IEEE TSP\",\n \"doi\": \"\"\n },\n {\n \"text\": \"M Sabanayagam, P Esser, D Ghoshdastidar, arXiv:2110.04060New insights into graph convolutional networks using neural tangent kernels. 2021arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. D I Shuman, S K Narang, P Frossard, A Ortega, P Vandergheynst, IEEE Signal Process. Mag. 303May 2013\",\n \"doi\": \"\"\n },\n {\n \"text\": \"Computing with infinite networks. C Williams, Neural Inform. Process. Syst. 91996\",\n \"doi\": \"\"\n },\n {\n \"text\": \"X Xia, G Mishne, Y Wang, arXiv:2211.03329Implicit graphon neural representation. 2022arXiv preprint\",\n \"doi\": \"\"\n },\n {\n \"text\": \"How powerful are graph neural networks?. K Xu, W Hu, J Leskovec, S Jegelka, 7th Int. Conf. Learning Representations. New Orleans, LA6-9 May 2019\",\n \"doi\": \"\"\n },\n {\n \"text\": \"From local structures to size generalization in graph neural networks. G Yehudai, E Fetaya, E Meirom, G Chechik, H Maron, Int. Conf. Mach. Learning. PMLR2021\",\n \"doi\": \"\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":5,"numItemsPerPage":100,"numTotalItems":2579762,"offset":500,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sInJlcG9JZHMiOlsiNjc3NDI4YmI1YThjMWM5YzgyNWVmYTYwIl0sImlhdCI6MTc5NDUwNjI1NywianRpIjoiN2ZjMTg1MjYtZmY1OC00ZjgwLTk2MDctYWU3ZTI1NDEzZTUzIiwic3ViIjoiL2RhdGFzZXRzL0FsZ29yaXRobWljUmVzZWFyY2hHcm91cC9zMm9yY19hcnhpdiIsImV4cCI6MTc5NDUwOTg1NywiaXNzIjoiaHR0cHM6Ly9odWdnaW5nZmFjZS5jbyJ9.ampOlE5npnZiOAF5SUwllkbAhbDuaM-D7XF5wVOLdWkyptroFjOnbqazZJKIgWdEA4aJ_w5Rwt5oLa8CAxsfBA","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 |
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CRUSH4SQL: Collective Retrieval Using Schema Hallucination For Text2SQL
Mayank Kothyari
Department of Computer Science and Engineering
Indian Institute of Technology Bombay
MumbaiIndia
Dhruva Dhingra
Department of Computer Science and Engineering
Indian Institute of Technology Bombay
MumbaiIndia
Sunita Sarawagi ... | 264,935,600 | {
"arxiv": "2311.01173",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/emnlp/KothyariDSC23",
"doi": "10.18653/v1/2023.emnlp-main.868"
} | {
"pdf_urls": [
"https://aclanthology.org/2023.emnlp-main.868.pdf"
],
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"open_access": {
"license": "CCBY",
"url": "https://aclanthology.org/2023.emnlp-main.868.pdf",
"status": "HYBRID"
}
} | CRUSH4SQL: Collective Retrieval Using Schema Hallucination For Text2SQL | [
"Mayank Kothyari \nDepartment of Computer Science and Engineering\nIndian Institute of Technology Bombay\nMumbaiIndia",
"Dhruva Dhingra \nDepartment of Computer Science and Engineering\nIndian Institute of Technology Bombay\nMumbaiIndia",
"Sunita Sarawagi sunita@iitb.ac.in \nDepartment of Computer Science and E... | Existing Text-to-SQL generators require the entire schema to be encoded with the user text.This is expensive or impractical for large databases with tens of thousands of columns.Standard dense retrieval techniques are inadequate for schema subsetting of a large structured database, where the correct semantics of retrie... | [
{
"title": "Introduction",
"content": "State-of-the-art language model based Text-to-SQL generators provide impressive accuracies on well known benchmarks, but they require the entire DB schema as input, along with the user question text (Scholak et al., 2021;Li et al., 2023a;Liu et al., 2023;Rajkumar et al... | [
{
"id": "",
"caption": "Examples of in-context training examples given to the LLM to prompt it to hallucinate a minimal schema of a database that can be used to answer the given question.",
"content": "Table 1 :\n1\nExamples of in-context training examples given to the LLM to prompt it to hallucinate a ... | [
{
"text": "Chat-rec: Towards interactive and explainable llms-augmented recommender system. Yunfan Gao, Tao Sheng, Youlin Xiang, Yun Xiong, Haofen Wang, Jiawei Zhang, 2023",
"doi": ""
},
{
"text": "Information retrieval with verbose queries. Manish Gupta, Michael Bendersky, SIGIR. 2015",
"doi": ... |
Fairness with Continuous Optimal Transport
Silvia Chiappa
Berkeley
Aldo Pacchiano pacchiano@berkeley.edu
Berkeley
Fairness with Continuous Optimal Transport
Whilst optimal transport (OT) is increasingly being recognized as a powerful and flexible approach for dealing with fairness issues, current OT fairness me... | 230,770,345 | {
"arxiv": "2101.02084",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2101-02084",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2101.02084v1.pdf"
],
"pdf_sha": "068cf004ecfb43728c1748b8c992b0a9a88bdbfa",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Fairness with Continuous Optimal Transport | [
"Silvia Chiappa \nBerkeley",
"Aldo Pacchiano pacchiano@berkeley.edu \nBerkeley"
] | Whilst optimal transport (OT) is increasingly being recognized as a powerful and flexible approach for dealing with fairness issues, current OT fairness methods are confined to the use of discrete OT. In this paper, we leverage recent advances from the OT literature to introduce a stochasticgradient fairness method bas... | [
{
"title": "Introduction",
"content": "ML fairness techniques aim at ensuring that models do not encode nor amplify human and societal biases contained in the training data, and therefore that individuals are not treated unfavorably on the basis of race, gender, disabilities, or other sensitive attributes (... | [
{
"id": "",
"caption": "Following Genevay et al. (2016); Seguy et al. (2018); Genevay (2019); Lee, Pacchiano, and Jordan (2020), we consider 1",
"content": "Following Genevay et al. (2016); Seguy et al. (2018); Genevay (2019); Lee, Pacchiano, and Jordan (2020), we consider 1"
},
{
"id": "",
... | [
{
"text": "Wasserstein Generative Adversarial Networks. M Arjovsky, S Chintala, L Bottou, Proceedings of the 34th International Conference on Machine Learning. the 34th International Conference on Machine LearningArjovsky, M.; Chintala, S.; and Bottou, L. 2017. Wasserstein Generative Adversarial Networks. In Pr... |
Unsupervised organization of cervical cells using high resolution digital holographic microscopy
Jyoti Mangal
Department of Physics
Indian Institute of Technology
110016DelhiNew DelhiIndia
Rashi Monga
Department of Physics
Indian Institute of Technology
110016DelhiNew DelhiIndia
Sandeep R Mathur
Department of P... | 53,677,613 | {
"arxiv": "1811.05214",
"mag": 2900641222,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1811.05214v1.pdf"
],
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"open_access": {
"license": null,
"url": null,
"status": null
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} | Unsupervised organization of cervical cells using high resolution digital holographic microscopy | [
"Jyoti Mangal \nDepartment of Physics\nIndian Institute of Technology\n110016DelhiNew DelhiIndia",
"Rashi Monga \nDepartment of Physics\nIndian Institute of Technology\n110016DelhiNew DelhiIndia",
"Sandeep R Mathur \nDepartment of Pathology\nAll India Institute of Medical Sciences\n110039New Delhi, India",
"A... | We report results on unsupervised organization of cervical cells using microscopy of Pap-smear samples in brightfield (3-channel colour) as well as high resolution quantitative phase imaging modalities. A number of morphological parameters are measured for each of the 1450 cell nuclei (from 10 woman subjects) imaged in... | [
{
"title": "I. INTRODUCTION",
"content": "Visual examination of cell samples using a microscope is a commonly used diagnostic methodology world over. The pathologists typically rely on their subjective interpretation of various cell features or other diagnostic markers that they learn over years of practice... | [
{
"id": "",
"caption": "Digital Holographic Microscope (DHM) setup depicting the Mach-Zehnder configuration used with balanced path length in the reference and object arms.",
"content": "FIG. 1 .\n1Digital Holographic Microscope (DHM) setup depicting the Mach-Zehnder configuration used with balanced pat... | [
{
"text": ". The National Cervical Cancer Coalition. The National Cervical Cancer Coalition, http://www.nccc-online.org/hpvcervical-cancer/ cervical-cancer-overview/",
"doi": ""
},
{
"text": "Screening and cervical cancer cure: Population based cohort study. B Andrae, T M L Andersson, P C Lambert, L... |
Structure of road networks and the shape of the macroscopic fundamental diagram
10 Oct 2023
Erwan Taillanter erwan.taillanter@ipht.fr
Institut de Physique Théorique
Université Paris-Saclay
CNRS
CEA
91191Gif-sur-YvetteFrance
Andreas Schadschneider
Institut für Theoretische Physik
Universität zu Köln
50937KölnGerman... | 263,829,297 | {
"arxiv": "2310.06531",
"mag": null,
"acl": null,
"pubmed": "38366531v1",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physreve.109.014314"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2310.06531v1.pdf"
],
"pdf_sha": "6d7982350b6698e14167ec3ced55b67143729da9",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Structure of road networks and the shape of the macroscopic fundamental diagram | [
"Erwan Taillanter erwan.taillanter@ipht.fr \nInstitut de Physique Théorique\nUniversité Paris-Saclay\nCNRS\nCEA\n91191Gif-sur-YvetteFrance",
"Andreas Schadschneider \nInstitut für Theoretische Physik\nUniversität zu Köln\n50937KölnGermany",
"Marc Barthelemy marc.barthelemy@ipht.fr \nInstitut de Physique Théoriq... | The macroscopic fundamental diagram (MFD) is a large scale description of the traffic in a urban area and relates the average car flow to the average car density.This MFD has been observed empirically in several cities but how its properties are related to the structure of the road network has remained unclear so far.T... | [
{
"title": "I. INTRODUCTION",
"content": "Traffic congestion in large urban areas is a crucial problem as a source of pollution and loss of time.Congestion is not only a product of insufficient road capacity [1] and it is clear now that the structure of the road network has to be carefully planned in order ... | [
{
"id": "",
"caption": "FIG. 1. Macroscopic Fundamental Diagram of Paris (France).Each point represents the traffic state of the network at a given date.The vertical line indicates the optimal density k * and the horizontal dashed line the optimal flow q * .This MFD was obtained for the period April-May 201... | [
{
"text": ". G Duranton, M Turner, 10.1257/aer.101.6.2616American Economic Review. 10126162011",
"doi": ""
},
{
"text": ". B D Greenshield, Highway Research Board Proceedings. 133821933",
"doi": ""
},
{
"text": ". C Daganzo, 10.1016/j.trb.2006.03.001Transp. Res. B. 41492007",
"doi": ... |
Parallelepipeds obtaining HBL lower bounds
November 21, 2016
James Demmel demmel@eecs.berkeley.edu
EECS, Mathematics
University of California
BerkeleyCA
Alex Rusciano rusciano@math.berkeley.edu
Mathematics
University of California
BerkeleyCA
Parallelepipeds obtaining HBL lower bounds
November 21, 20160E7D4FAA826B... | 10,306,307 | {
"arxiv": "1611.05944",
"mag": 2554941931,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/DemmelR16",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1611.05944v1.pdf"
],
"pdf_sha": "15eb97d58b9bee2d93eccde9ec8da2c56aa21c4e",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Parallelepipeds obtaining HBL lower bounds | [
"James Demmel demmel@eecs.berkeley.edu \nEECS, Mathematics\nUniversity of California\nBerkeleyCA",
"Alex Rusciano rusciano@math.berkeley.edu \nMathematics\nUniversity of California\nBerkeleyCA"
] | This work studies the application of the discrete Hölder-Brascamp-Lieb (HBL) inequalities to the design of communication optimal algorithms.In particular, it describes optimal tiling (blocking) strategies for nested loops that lack data dependencies and exhibit linear memory access patterns.We attain known lower bounds... | [
{
"title": "Background: Solving for s HBL through an LP",
"content": "For the maps φ, there is a natural necessary condition for the s: Proposition 4. If s i are to satisfy inequality (1.1), then it is necessary that for any subgroup H of G we have the following\nProof.To prove this fact, it suffices to use... | [
{
"id": "",
"caption": "The family of sets S(M ), parametrized by integer M , form asymptotically optimal tilings if translations of the set S(M ) can tile Z d and S(M ) satisfies |S(M )| = Θ(M s HBL ) (1.3) as well as ∀i, |φ i (S(M ))| = O(M ) (1.4)",
"content": "Definition 2 .\n2\nThe family of sets S... | [
{
"text": "Finite bounds for Holder-Brascamp-Lieb multilinear inequalities. A Bennett, M Carbery, T Christ, Tao, Mathematical Research Letters. 5542010",
"doi": ""
},
{
"text": "Communication Lower Bounds and Optimal Algorithms for Programs That Reference Arrays Part 1. M Christ, J Demmel, N Knight,... |
Low Scale Seesaw with Local Lepton Number
7 Sep 2023
Hridoy Debnath
Physics Department and Center for Education and Research in Cosmology and Astrophysics (CERCA)
Case Western Reserve University
44106ClevelandOHUSA
Pavel Fileviez Pérez
Physics Department and Center for Education and Research in Cosmology and Astro... | 259,375,772 | {
"arxiv": "2307.03646",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevd.108.075009"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2307.03646v2.pdf"
],
"pdf_sha": "83cda8935bd43da20db25155bbfc0426eacf9e7a",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Low Scale Seesaw with Local Lepton Number | [
"Hridoy Debnath \nPhysics Department and Center for Education and Research in Cosmology and Astrophysics (CERCA)\nCase Western Reserve University\n44106ClevelandOHUSA",
"Pavel Fileviez Pérez \nPhysics Department and Center for Education and Research in Cosmology and Astrophysics (CERCA)\nCase Western Reserve Univ... | We discuss a class of theories for Majorana neutrinos where the total lepton number is a local gauge symmetry. These theories predict a dark matter candidate from anomaly cancellation. We discuss the properties of the dark matter candidate and using the cosmological bounds, we obtain the upper bound on the lepton numbe... | [
{
"title": "INTRODUCTION",
"content": "The origin of the charged fermion and gauge bosons masses in the Standard Model (SM) of particle physics is well-understood. After the discovery of the Brout-Englert-Higgs (BEH) boson at the Large Hadron Collider we know that spontaneous symmetry breaking plays a funda... | [
{
"id": "",
"caption": "where Y ν is the Yukawa coupling between the SM neutrinos and the new right-handed neutrinos, M N defines a new Majorana mass scale and v 0 = 246 GeV is the constant value of the BEH field. This simple mechanism does not predict the scale where the new mass scale M N is generated, it... | [
{
"text": "µ → eγ at a Rate of One Out of 10 9 Muon Decays?. Peter Minkowski, 10.1016/0370-2693(77)90435-XPhys. Lett. B. 67Peter Minkowski, \"µ → eγ at a Rate of One Out of 10 9 Muon Decays?\" Phys. Lett. B 67, 421-428 (1977).",
"doi": "10.1016/0370-2693(77)90435-X"
},
{
"text": "Complex Spinors and... |
The authors would like to thank Drs. Hengyong Yu, Jiang Hiesh and Ge Wang for the rights of using the real data. This work was supported in part by NIH (5R01 CA176553). Q. Xu and L. Xing are with the
J Wang
H Shirato
Department of Radiation Oncology
School of Medicine
Stanford University
94305StanfordCAUSA
Glob... | 224,803,754 | {
"arxiv": "2010.09953",
"mag": 3093814631,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2010-09953",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2010.09953v1.pdf"
],
"pdf_sha": "e95fbb2ffd3827c5afac792e1955c8838c6cfa47",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | The authors would like to thank Drs. Hengyong Yu, Jiang Hiesh and Ge Wang for the rights of using the real data. This work was supported in part by NIH (5R01 CA176553). Q. Xu and L. Xing are with the | [
"J Wang",
"H Shirato",
"Department of Radiation Oncology\nSchool of Medicine\nStanford University\n94305StanfordCAUSA",
"Global Station for Quantum Medical Science and Engineering\nGlobal Institution for Collaborative Research and Education (GI-CoRE)\nHokkaido University\n060-8648SapporoJapan",
"Global Stat... | [
{
"title": "I. INTRODUCTION",
"content": "T is widely used in diagnosis of diseases in the thoracic region, such as lung tumor and cardiac disease.It is also indispensable for treatment planning, interventional guidance, and therapeutic assessment.However, X-ray radiation dose is an important concern since ... | [
{
"id": "",
"caption": "Fig. 1.Schematic diagram of region segmentation and patch classification.Juncture patch is the patch on the boundary of regions, where not all pixels belong to one region.",
"content": "Fig. 1 .\n1\nFig. 1.Schematic diagram of region segmentation and patch classification.Juncture... | [
{
"text": "Low-dose CT of the lungs: preliminary observations. D P Naidich, C H Marshall, C Gribbin, R S Arams, D I Mccauley, Radiology. 1751990",
"doi": ""
},
{
"text": "Pulmonary nodule detection: low-dose versus conventional CT. H Rusinek, D P Naidich, G Mcguinness, B S Leitman, D I Mccauley, G A... | |
Charge fluctuations above T CDW revealed by glasslike thermal transport in kagome metals AV 3 Sb 5 (A = K, Rb, Cs)
Kunya Yang
College of Physics & Center of Quantum Materials and Devices
Low Temperature Physics Lab
Chongqing University
401331ChongqingChina
Wei Xia
School of Physical Science and Technology
Shangha... | 253,397,593 | {
"arxiv": "2211.04216",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.107.184506"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2211.04216v2.pdf"
],
"pdf_sha": "d6134c716b18bd2c2ff941af4dce47220a7f99cd",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Charge fluctuations above T CDW revealed by glasslike thermal transport in kagome metals AV 3 Sb 5 (A = K, Rb, Cs) | [
"Kunya Yang \nCollege of Physics & Center of Quantum Materials and Devices\nLow Temperature Physics Lab\nChongqing University\n401331ChongqingChina",
"Wei Xia \nSchool of Physical Science and Technology\nShanghaiTech University\n201210ShanghaiChina\n\nShanghaiTech Laboratory for Topological Physics\n201210Shangha... | We present heat capacity, electrical and thermal transport measurements of kagome metals AV3Sb5 (A = K, Rb, Cs). In the all three compounds, development of short-range charge fluctuations above the charge density wave (CDW) transition temperature TCDW strongly scatters phonons via the electron-phonon coupling, leading ... | [
{
"title": "I. INTRODUCTION",
"content": "The advent of kagome metals AV 3 Sb 5 (A = K, Rb, Cs) possessing intertwined charge density wave (CDW) and superconductivity have attracted extensive research interest lately [1][2][3][4]. The particular kagome lattice formed by V atoms in AV 3 Sb 5 gives rise to va... | [
{
"id": "",
"caption": "Temperature dependence of the specific heat Cp of (a) KV3Sb5, (b) RbV3Sb5, and (c) CsV3Sb5. The arrows mark out the CDW transitions. Insets in (a-c) plot the linear fitting (black dash lines) of the specific heat according to Cp/T = γ + βT 2 within a temperature range of 2",
"con... | [
{
"text": ". B R Ortiz, L C Gomes, J R Morey, M Winiarski, M Bordelon, J S Mangum, I W Oswald, J A Rodriguez-Rivera, J R Neilson, S D Wilson, E Ertekin, T M Mcqueen, E S Toberer, 10.1103/PhysRevMaterials.3.094407Phys. Rev. Materials. 394407B. R. Ortiz, L. C. Gomes, J. R. Morey, M. Winiarski, M. Bordelon, J. S. ... |
Koopman-based generalization bound: New aspect for full-rank weights
16 Mar 2024
Yuka Hashimoto
NTT Corporation
Center for Advanced Intelligence Project
RIKEN
Sho Sonoda
Center for Advanced Intelligence Project
RIKEN
Isao Ishikawa
Center for Advanced Intelligence Project
RIKEN
Center for Data Science
Ehime Un... | 256,827,166 | {
"arxiv": "2302.05825",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/iclr/HashimotoSINS24",
"doi": "10.48550/arxiv.2302.05825"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2302.05825v3.pdf"
],
"pdf_sha": "b8208baaf0a19258052c88799d06d26aeca4ff64",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Koopman-based generalization bound: New aspect for full-rank weights | [
"Yuka Hashimoto \nNTT Corporation\n\n\nCenter for Advanced Intelligence Project\nRIKEN",
"Sho Sonoda \nCenter for Advanced Intelligence Project\nRIKEN",
"Isao Ishikawa \nCenter for Advanced Intelligence Project\nRIKEN\n\nCenter for Data Science\nEhime University",
"Atsushi Nitanda",
"Taiji Suzuki \nCenter f... | We propose a new bound for generalization of neural networks using Koopman operators.Whereas most of existing works focus on low-rank weight matrices, we focus on full-rank weight matrices.Our bound is tighter than existing norm-based bounds when the condition numbers of weight matrices are small.Especially, it is comp... | [
{
"title": "Introduction",
"content": "Understanding the generalization property has been one of the biggest topics for analyzing neural networks.A major approach for theoretical investigation of this topic is bounding some complexity of networks (Bartlett & Mendelson, 2002;Mohri et al., 2018).Intuitively, ... | [
{
"id": "",
"caption": "Figure 1: (a) Scatter plot of the generalization error versus our bound (for 5 independent runs).The color is set to get dark as the epoch proceeds.(b) Test accuracy with and without the regularization based on our bound.(c) The condition number r d,j = η1,j/η d,j of the weight matri... | [
{
"text": "Martin References, Peter L Anthony, Bartlett, Neural network learning: Theoretical foundations. Cambridge University Press2009",
"doi": ""
},
{
"text": "Stronger generalization bounds for deep nets via a compression approach. Sanjeev Arora, Rong Ge, Behnam Neyshabur, Yi Zhang, Proceedings... |
N eutrino B eam s from M uon Storage R ings: C haracteristics and P hysics P otential
7 Dec 1997 February 6,2022
S G Eer
Ferm iN ationalAccel erator Laboratory
P.O .Box 50060510BataviaIll inois
N eutrino B eam s from M uon Storage R ings: C haracteristics and P hysics P otential
7 Dec 1997 February 6,2022Ferm i l a... | 16,872,161 | {
"arxiv": "hep-ph/9712290",
"mag": 2238120085,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevd.57.6989 10.1103/physrevd.59.039903"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/hep-ph/9712290v1.pdf"
],
"pdf_sha": "ef1f340d0e3cb3b2493c4b1cd4b54ff4022ab911",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | N eutrino B eam s from M uon Storage R ings: C haracteristics and P hysics P otential | [
"S G Eer \nFerm iN ationalAccel erator Laboratory\nP.O .Box 50060510BataviaIll inois"
] | A bstract H i gh-i ntensi ty hi gh-energy neutri no beam s coul d be produced by expl oi ti ng a very i ntense future m uon source,and al l ow i ng the m uons to decay i n a storage ri ng contai ni ng a l ong strai ghtsecti on. Taki ng the param eters ofm uon sourcedesi gnsthatarecurrentl y understudy,thecharacteri sti... | [
{
"title": "Introduction",
"content": "H i gh energy neutri no beam shave pl ayed an i m portantrol e i n the devel opm ent of parti cl e physi cs. Experi m ents usi ng neutri no and anti neutri no beam s produced from charged m eson decays have,for exam pl e,dem onstrated that m uon neutri nos are di erent... | [
{
"id": "",
"caption": "(i i ) T he sm al l e contam i nati on i n the otherw i se pure beam i s a nui sance for experi m ents searchi ng for e { osci l l ati ons. (i i i )T he sm al l nessofthe e ux m akes e { osci l l ati on searchesdi cul t,and (i v)T hesm al l contam i nati on i n thebeam w i l leventua... | [
{
"text": "T he M uon C ol l i der C ol l aborati on, Ferm i l ab-C onf-96/092. \\ + C Ol L I Der Feasi Bi L I Ty Study, unpubl i shed\\ + C ol l i der Feasi bi l i ty Study\", T he M uon C ol l i der C ol l aborati on, Ferm i l ab-C onf-96/092,Jul y 1996,unpubl i shed.",
"doi": ""
},
{
"text": "Ses... |
Visual Tracking with Intermittent Visibility: Switched Control Design and Implementation
12 Nov 2024
Yangge Li
Yangge Li
University of Illinois Urbana-Champaign
Benjamin C Yang bcyang2@illinois.edu
Benjamin C Yang
University of Illinois Urbana-Champaign
Sayan Mitra mitras@illinois.edu
Sayan Mitra
University ... | 273,993,579 | {
"arxiv": "2411.08144",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2411.08144v1.pdf"
],
"pdf_sha": "8133318a2db193043258dfacbdc480b5ad9402c4",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Visual Tracking with Intermittent Visibility: Switched Control Design and Implementation | [
"Yangge Li \nYangge Li\n\n\nUniversity of Illinois Urbana-Champaign",
"Benjamin C Yang bcyang2@illinois.edu \nBenjamin C Yang\nUniversity of Illinois Urbana-Champaign",
"Sayan Mitra mitras@illinois.edu \nSayan Mitra\nUniversity of Illinois Urbana-Champaign"
] | This paper addresses the problem of visual target tracking in scenarios where a pursuer may experience intermittent loss of visibility of the target.The design of a Switched Visual Tracker (SVT) is presented which aims to meet the competing requirements of maintaining both proximity and visibility.SVT alternates betwee... | [
{
"title": "I. INTRODUCTION",
"content": "The visual tracking task requires a pursuer to follow a moving target using only camera as a sensor.This capability is relevant for search and rescue, delivery, spacecraft docking, in-air-refuelling, navigation, and other autonomous missions.Visual tracking, also ca... | [
{
"id": "",
"caption": "Fig. 2: Computation of the recovery pose x R .The pursuer (black) at x R can include the whole reachable set of target (blue) in it's camera range.",
"content": "Fig. 2 :\n2\nFig. 2: Computation of the recovery pose x R .The pursuer (black) at x R can include the whole reachable ... | [
{
"text": "Agilicious: Open-source and open-hardware agile quadrotor for visionbased flight. P Foehn, E Kaufmann, A Romero, R Penicka, S Sun, L Bauersfeld, T Laengle, G Cioffi, Y Song, A Loquercio, D Scaramuzza, AAAS Science Robotics. 2022",
"doi": ""
},
{
"text": "A vision-based target tracking con... |
Significant loophole-free test of Kochen-Specker contextuality using two species of atomic ions
2022. 9 February 2022
Pengfei Wang
Junhua Zhang
Chun-Yang Luan
Mark Um
Ye Wang
Mu Qiao
Tian Xie
Jing-Ning Zhang
Adán Cabello
Kihwan Kim
Significant loophole-free test of Kochen-Specker contextuality using two spe... | 245,502,587 | {
"arxiv": "2112.13612",
"mag": null,
"acl": null,
"pubmed": "35138888",
"pubmedcentral": 8827658,
"dblp": null,
"doi": "10.1126/sciadv.abk1660"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2112.13612v1.pdf"
],
"pdf_sha": "e2e193fe8ed41a76aee70213b3d1fc086ba7fad3",
"open_access": {
"license": "CCBYNC",
"url": null,
"status": null
}
} | Significant loophole-free test of Kochen-Specker contextuality using two species of atomic ions | [
"Pengfei Wang",
"Junhua Zhang",
"Chun-Yang Luan",
"Mark Um",
"Ye Wang",
"Mu Qiao",
"Tian Xie",
"Jing-Ning Zhang",
"Adán Cabello",
"Kihwan Kim"
] | Quantum measurements cannot be thought of as revealing preexisting results, even when they do not disturb any other measurement in the same trial. This feature is called contextuality and is crucial for the quantum advantage in computing. Here, we report the observation of quantum contextuality simultaneously free of t... | [
{
"title": "INTRODUCTION",
"content": "In everyday life, whenever the measurements of two observables A and B yield the same values (a for A and b for B) when the measurements are repeated in any order, we attribute it to the measured system having preexisting values revealed by every measurement and which ... | [
{
"id": "",
"caption": "The four observables and their compatibility relations. O 0 and O 2 are measured on the 171 Yb + ion, and O 1 and O 3 are measured on the 138 Ba + ion. Connected observables are compatible (jointly measurable). Here, ˆ x , ˆ y are Pauli operators, and I 2 is the identity operator... | [
{
"text": "Simple test for hidden variables in spin-1 systems. A A Klyachko, M A Can, S Binicioğlu, A S Shumovsky, Phys. Rev. Lett. 10120403A. A. Klyachko, M. A. Can, S. Binicioğlu, A. S. Shumovsky, Simple test for hidden variables in spin-1 systems. Phys. Rev. Lett. 101, 020403 (2008).",
"doi": ""
},
{... |
**Volume Title** ASP Conference Series, Vol. **Volume Number** **Author** Dynamics of extended AGB star envelopes
Claudia Dreyer
Zentrum für Astronomie und Astrophysik
Technische Universität Berlin
BerlinGermany
Michael Hegmann
Zentrum für Astronomie und Astrophysik
Technische Universität Berlin
BerlinGermany
Er... | 117,822,683 | {
"arxiv": "1011.0948",
"mag": 1491110926,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1011.0948v1.pdf"
],
"pdf_sha": "d62f5f98a14480a62eb00f1d82591e9efb1b1f65",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | **Volume Title** ASP Conference Series, Vol. **Volume Number** **Author** Dynamics of extended AGB star envelopes | [
"Claudia Dreyer \nZentrum für Astronomie und Astrophysik\nTechnische Universität Berlin\nBerlinGermany",
"Michael Hegmann \nZentrum für Astronomie und Astrophysik\nTechnische Universität Berlin\nBerlinGermany",
"Erwin Sedlmayr \nZentrum für Astronomie und Astrophysik\nTechnische Universität Berlin\nBerlinGerman... | The dust formed in extended circumstellar envelopes of long-period variables and Miras has a strong influence on the envelope dynamics. A radiatively driven instability caused by the formation of dust leads to the development of an autonomous dynamics characterised by a set of distinct frequencies. We study the interpl... | [
{
"title": "Introduction",
"content": "Long-period variables (LPVs) and Miras are radially pulsating, highly evolved stars on the asymptotic giant branch (AGB). Their cool, extended atmospheres are excellent sites for the formation of complex molecules and dust particles. The interplay between dust formatio... | [
{
"id": "",
"caption": "Power spectra of the radial outflow velocity u for a high luminous LPV purely caused by the exterior κ-mechanism (model A, left) and caused by a stochastic excitation for a standard luminous LPV (model B, right). The position of the dust formation zone in the envelope is highlighted.... | [
{
"text": ". C Dreyer, M Hegmann, E Sedlmayr, A&A. 499765Dreyer C., Hegmann M., Sedlmayr E.: 2009, A&A, 499, 765",
"doi": ""
},
{
"text": ". C Dreyer, M Hegmann, E Sedlmayr, A&A. Fleischer A.J., Gauger A., Sedlmayr E.266321A&ADreyer C., Hegmann M., Sedlmayr E.: 2010, A&A, accepted Fleischer A.J., Ga... |
A phase-field model for thermo-mechanical fracture with an open-source implementation of it using Gridap in Julia
Ved Prakash
Akash Kumar Behera
Mohammad Masiur Rahaman
School of Infrastructure
School of Infrastructure Indian Institute of Technology Bhubaneswar Odisha-752050
Indian Institute of Technology Bhuban... | 239,768,488 | {
"arxiv": "2110.12284",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2110-12284",
"doi": "10.1177/10812865221133860"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2110.12284v1.pdf"
],
"pdf_sha": "c09bfc854dbdb4beebe91d7c4c7f3fcf639971dc",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | A phase-field model for thermo-mechanical fracture with an open-source implementation of it using Gridap in Julia | [
"Ved Prakash",
"Akash Kumar Behera",
"Mohammad Masiur Rahaman",
"School of Infrastructure\nSchool of Infrastructure Indian Institute of Technology Bhubaneswar Odisha-752050\nIndian Institute of Technology Bhubaneswar Odisha-752050\nIndia, India",
"School of Infrastructure Indian Institute of Technology Bhub... | In this article, we propose a thermodynamically consistent phase-field model for thermo-mechanical fracture and provide an open-source implementation of the proposed model using a recently developed finite element toolbox, Gridap in Julia. Here, we have derived the balance equations for the thermo-mechanical fracture b... | [
{
"title": "Introduction",
"content": "Analysis of thermally induced stresses and cracks are of great importance in the design of engineering materials. * Corresponding Author When materials are exposed to a sudden change in ambient temperature, it can result in a non-uniform change of volume that induces t... | [
{
"id": "",
"caption": "In this work, we have used a non-linear smooth increment of temperature by choosing option one of the two functions as given below.uAppVec = range(0,uMax,length = Int64( floor(tMax/delt))) AppTOption = 1 ## 1 for smooth and otherwise linear than constant if AppTOption == 1 TAppVec = ... | [
{
"text": "A phase-field damage model for orthotropic materials and delamination in composites. B Dhas, M Rahaman, K Akella, D Roy, J Reddy, Journal of Applied Mechanics. 185Dhas, B., Rahaman, M., Akella, K., Roy, D., Reddy, J., et al., 2018. \"A phase-field damage model for or- thotropic materials and delamina... |
Bootstrapping Semi-supervised Medical Image Segmentation with Anatomical-aware Contrastive Distillation
Chenyu You chenyu.you@yale.edu
Department of Electrical Engineering
Yale University
Weicheng Dai
Department of Computer Science and Engineering
New York University
Yifei Min
Department of Statistics and Dat... | 249,394,599 | {
"arxiv": "2206.02307",
"mag": null,
"acl": null,
"pubmed": "37409056",
"pubmedcentral": null,
"dblp": "conf/ipmi/YouDMSD23",
"doi": "10.1007/978-3-031-34048-2_49"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2206.02307v4.pdf"
],
"pdf_sha": "a032acfc0f242cff298e52561372d2b0f9a05c36",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Bootstrapping Semi-supervised Medical Image Segmentation with Anatomical-aware Contrastive Distillation | [
"Chenyu You chenyu.you@yale.edu \nDepartment of Electrical Engineering\nYale University",
"Weicheng Dai \nDepartment of Computer Science and Engineering\nNew York University",
"Yifei Min \nDepartment of Statistics and Data Science\nYale University",
"Lawrence Staib \nDepartment of Electrical Engineering\nYale... | Contrastive learning has shown great promise over annotation scarcity problems in the context of medical image segmentation. Existing approaches typically assume a balanced class distribution for both labeled and unlabeled medical images. However, medical image data in reality is commonly imbalanced (i.e., multi-class ... | [
{
"title": "Introduction",
"content": "Manually labeling sufficient medical data with pixel-level accuracy is timeconsuming, expensive, and often requires domain-specific knowledge. To bypass arXiv:2206.02307v4 [cs.CV] 10 Mar 2023 the cost for labeled data, semi-supervised learning (SSL) is one of the promi... | [
{
"id": "",
"caption": "Examples of two benchmarks (i.e., ACDC and LiTS) showing the large variations of class distribution.",
"content": "Fig. 1 .\n1Examples of two benchmarks (i.e., ACDC and LiTS) showing the large variations of class distribution."
},
{
"id": "",
"caption": "Overview of t... | [
{
"text": "Deep learning techniques for automatic MRI cardiac multi-structures segmentation and diagnosis: Is the problem solved?. O Bernard, A Lalande, C Zotti, F Cervenansky, X Yang, P A Heng, I Cetin, K Lekadir, O Camara, M A G Ballester, IEEE Transactions on Medical Imaging. Bernard, O., Lalande, A., Zotti,... |
On Distance Preserving and Sequentially Distance Preserving Graphs
January 24, 2017
Jason P Smith jason.p.smith@strath.ac.uk
Department of Computer and Information Sciences
University of Strathclyde
GlasgowUK
Emad Zahedi zahediem@msu.edu
Department of Mathematics
Michigan State University
48824-1027East LansingMIU... | 15,579,858 | {
"arxiv": "1701.06404",
"mag": 2582271637,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/SmithZ17",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1701.06404v1.pdf"
],
"pdf_sha": "2b781303eecdb93e50525d07821c96d18f7dd452",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | On Distance Preserving and Sequentially Distance Preserving Graphs | [
"Jason P Smith jason.p.smith@strath.ac.uk \nDepartment of Computer and Information Sciences\nUniversity of Strathclyde\nGlasgowUK",
"Emad Zahedi zahediem@msu.edu \nDepartment of Mathematics\nMichigan State University\n48824-1027East LansingMIU.S.A\n\nDepartment of Computer Science and Engineering\nMichigan State ... | A graph is distance preserving if it has an isometric subgraph of every possible order.A graph is sequentially distance preserving if its vertices can be ordered such that deleting the first i vertices results in an isometric subgraph, for all i ≥ 1.We give an equivalent condition to sequentially distance preserving ba... | [
{
"title": "Introduction",
"content": "The distance between two vertices in a graph plays an important role in many areas of graph theory.Moreover, computing distances in graphs is integral to many real-world applications, especially in areas such as network and optimisation theory.Computing distances betwe... | [
{
"id": "",
"caption": "[9, Theorem 1] Consider a graph G and integer k ≥ 3. The graph G is k-chordal if and only if G has a k-simplicial ordering.",
"content": "Theorem 3 . 2 .\n32\n[9, Theorem 1] Consider a graph G and integer k ≥ 3. The graph G is k-chordal if and only if G has a k-simplicial ord... | [
{
"text": "Distance-hereditary graphs. H.-J Bandelt, H M Mulder, Journal of Combinatorial Theory, Series B. 4121986",
"doi": ""
},
{
"text": "Graph theory with applications. J A Bondy, U S R Murty, 1976Macmillan London6",
"doi": ""
},
{
"text": "A simple paradigm for graph recognition: a... |
Design of a tabletop interferometer with quantum amplification
Jiri Smetana
Institute for Gravitational Wave Astronomy
School of Physics and Astronomy
University of Birmingham
B15 2TTBirminghamUnited Kingdom
Artemiy Dmitriev
Institute for Gravitational Wave Astronomy
School of Physics and Astronomy
University of ... | 252,780,854 | {
"arxiv": "2210.04566",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physreva.107.043701"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2210.04566v1.pdf"
],
"pdf_sha": "1874ed69cc50994f6a19f56ae6ddfee147ba08aa",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Design of a tabletop interferometer with quantum amplification | [
"Jiri Smetana \nInstitute for Gravitational Wave Astronomy\nSchool of Physics and Astronomy\nUniversity of Birmingham\nB15 2TTBirminghamUnited Kingdom",
"Artemiy Dmitriev \nInstitute for Gravitational Wave Astronomy\nSchool of Physics and Astronomy\nUniversity of Birmingham\nB15 2TTBirminghamUnited Kingdom",
"C... | The sensitivity of laser interferometers is fundamentally limited by the quantum nature of light. Recent theoretical studies have opened a new avenue to enhance their quantum-limited sensitivity by using active paritytime-symmetric and phase-insensitive quantum amplification. These systems can enhance the signal respon... | [
{
"title": "I. INTRODUCTION",
"content": "Improvements in the performance of gravitational-wave (GW) detectors continue to stretch the known boundaries of precision measurement. Ever since the first discovery of gravitational waves [1], there has been a concerted effort to enhance both the sensitivity and b... | [
{
"id": "",
"caption": "Layout of the proposed experiment. It consists of the two coupled main and filter cavities. The setup includes control of the pump power for the filter cavity. The silicon nitride membrade embedded in the filter cavity implements the quantum amplification. The pump field is derived f... | [
{
"text": ". B P Abbott, LIGO Scientific Collaboration ; Virgo CollaborationR Abbott, LIGO Scientific Collaboration ; Virgo CollaborationT D Abbott, LIGO Scientific Collaboration ; Virgo Collaboration10.1103/PhysRevLett.116.061102Phys. Rev. Lett. 11661102B. P. Abbott, R. Abbott, T. D. Abbott, et al. (LIGO Scien... |
Ab initio downfolding based on the GW approximation for infinite-layer nickelates
8 Feb 2022
Motoaki Hirayama hirayama@ap.t.u-tokyo.ac.jp
QPEC
The University of Tokyo
113-8656Hongo, TokyoJapan
RIKEN Center for Emergent Matter Sciences (CEMS)
351-0198WakoSaitamaJapan
Yusuke Nomura
RIKEN Center for Emergent Matter ... | 246,652,185 | {
"arxiv": "2202.03661",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.3389/fphy.2022.824144"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2202.03661v1.pdf"
],
"pdf_sha": "3d5a42839a947fce0c1bd7dc2b5994cf610e75ee",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Ab initio downfolding based on the GW approximation for infinite-layer nickelates | [
"Motoaki Hirayama hirayama@ap.t.u-tokyo.ac.jp \nQPEC\nThe University of Tokyo\n113-8656Hongo, TokyoJapan\n\nRIKEN Center for Emergent Matter Sciences (CEMS)\n351-0198WakoSaitamaJapan",
"Yusuke Nomura \nRIKEN Center for Emergent Matter Sciences (CEMS)\n351-0198WakoSaitamaJapan",
"Ryotaro Arita \nRIKEN Center for... | We derive an effective three-orbital model for the infinite-layer nickelates based on the band structure obtained by the GW approximation (GWA), where we consider the Ni 3d x 2 −y 2 and O 2p orbitals forming the σ-bond. In the GWA, the self-energy correction to the local density approximation (LDA) increases the energy... | [
{
"title": "INTRODUCTION",
"content": "The discovery of nickel superconductors [1] has attracted renewed attention to superconductivity in strongly correlated electron systems [2,3,4,5,6,7]. So far, superconductivity has been found in film samples of doped infinite-layer nickelates RNiO 2 (R=Nd, Pr, and La)... | [
{
"id": "",
"caption": "Crystal structure of YNiO 2 .",
"content": "Figure 1 .\n1Crystal structure of YNiO 2 ."
},
{
"id": "",
"caption": "(a) DFT/LDA band structure for YNiO 2 and (b) its magnified figure. The zero energy corresponds to the Fermi level.",
"content": "Figure 2 .\n2(a) DF... | [
{
"text": "Superconductivity in an infinite-layer nickelate. D Li, K Lee, B Y Wang, M Osada, S Crossley, H R Lee, Nature. 572Li D, Lee K, Wang BY, Osada M, Crossley S, Lee HR, et al. Superconductivity in an infinite-layer nickelate. Nature 572 (2019) 624-627.",
"doi": ""
},
{
"text": "Entering the N... |
DIAMETRAL DIAMETER TWO PROPERTIES IN BANACH SPACES
15 Apr 2016
Julio Becerra Guerrero
Ginés López-Pérez
Abraham Rueda Zoca
DIAMETRAL DIAMETER TWO PROPERTIES IN BANACH SPACES
15 Apr 2016arXiv:1509.02061v4 [math.FA]
The aim of this note is to provide several variants of the diameter two properties for Banach spaces.... | 119,154,854 | {
"arxiv": "1509.02061",
"mag": 2337734105,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1509.02061v4.pdf"
],
"pdf_sha": "2de2e57ab5f20cfe48ff2d990b909547d67f48f8",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | DIAMETRAL DIAMETER TWO PROPERTIES IN BANACH SPACES | [
"Julio Becerra Guerrero",
"Ginés López-Pérez",
"Abraham Rueda Zoca"
] | The aim of this note is to provide several variants of the diameter two properties for Banach spaces. We study such properties looking for the abundance of diametral points, which holds in the setting of Banach spaces with the Daugavet property, for example, and we introduce the diametral diameter two properties in Ban... | [
{
"title": "Introduction",
"content": "We recall that a Banach space X satisfies the strong diameter two property (SD2P), respectively diameter two property (D2P), slice-diameter two property (LD2P), if every convex combination of slices, respectively every nonempty relatively weakly open subset, every slic... | [
{
"id": "",
"caption": "Now we shall argue by cases:(1) If p = ∞ define the weak open subset of B Z given by",
"content": "Now we shall argue by cases:(1) If p = ∞ define the weak open subset of B Z given by"
},
{
"id": "",
"caption": "so we are done. Even though Example 2.2 shows that Dauga... | [
{
"text": "By Example 2.2 or Theorem 2.11, neither the converse implication of (2) nor the one of (7) holds. In addition, there are Banach spaces with the Daugavet property whose dual unit ball have denting points (e.g. C([0, 1])). Consequently, the converse of (4),(5) or (6) is not true. However. ) or (8) hold... |
21 Oct 2024
Ekaterina Artemova
Vladislav Mikhailov
Zain Muhammad Mujahid
Akim Tsvigun
10 Alham Fikri Aji, 1 Artem Shelmanov, 1 Nizar Habash, 1
Tarek Mahmoud
Rui Xing
1 Jiahui Geng, 1 Hasan Iqbal
Iryna Gurevych
1 Preslav Nakov 1 1 MBZUAI
Alexandria University
Zewail City of Science and Technology
Univ... | 271,768,972 | {
"arxiv": "2408.04284",
"mag": null,
"acl": "2024.emnlp-demo.35",
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2408-04284",
"doi": "10.18653/v1/2024.emnlp-demo.35"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2408.04284v2.pdf"
],
"pdf_sha": "d8219ece8679a23cbe80d601291cdecbdd4d43ce",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | [
"Ekaterina Artemova",
"Vladislav Mikhailov",
"Zain Muhammad Mujahid \nAkim Tsvigun\n10 Alham Fikri Aji, 1 Artem Shelmanov, 1 Nizar Habash, 1",
"Tarek Mahmoud",
"Rui Xing\n1 Jiahui Geng, 1 Hasan Iqbal",
"Iryna Gurevych\n1 Preslav Nakov 1 1 MBZUAI",
"Alexandria University",
"Zewail City of Science and T... | The ease of access to large language models (LLMs) has enabled a widespread of machinegenerated texts, and now it is often hard to tell whether a piece of text was human-written or machine-generated.This raises concerns about potential misuse, particularly within educational and academic domains.Thus, it is important t... | [
{
"title": "Dataset",
"content": "To collect the dataset for our multi-way finegrained detector, we first gathered datasets that were curated for binary machine-generated text detection from previous work, and then we extended the data into our four labels by introducing new corresponding generations.Sectio... | [
{
"id": "",
"caption": "Figure 1: LLM-DetectAIve interface: automatic text detection (top) and human detector playground (bottom).",
"content": "Figure 1 :\n1\nFigure 1: LLM-DetectAIve interface: automatic text detection (top) and human detector playground (bottom)."
},
{
"id": "",
"caption"... | [
{
"text": "Anthropic, The Claude 3 model family: Opus, Sonnet. Haiku2024",
"doi": ""
},
{
"text": "Machine-generated text: A comprehensive survey of threat models and detection methods. Evan N Crothers, Nathalie Japkowicz, Herna, Viktor, IEEE Access. 112023",
"doi": ""
},
{
"text": "Doma... | |
Improving image watermarking based on Tabu search by Chaos
Mohammad Tafaghodi m.tafaghodi@ec.iut.ac.ir
School of Electrical and Computer Engineering
Isfahan University of Technology
IsfahanIran
Meysam Ghaffari meysam.ghafari@ec.iut.ac.ir
Isfahan University of Technology
IsfahanIran
Alimohammad Latif alatif@yazd.... | 9,517,975 | {
"arxiv": "1501.01576",
"mag": 1927889274,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/TafaghodiGLM15",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1501.01576v1.pdf"
],
"pdf_sha": "81c8e7439331b77b2b91dd48c29a58200dca7af5",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Improving image watermarking based on Tabu search by Chaos | [
"Mohammad Tafaghodi m.tafaghodi@ec.iut.ac.ir \nSchool of Electrical and Computer Engineering\nIsfahan University of Technology\nIsfahanIran",
"Meysam Ghaffari meysam.ghafari@ec.iut.ac.ir \nIsfahan University of Technology\nIsfahanIran",
"Alimohammad Latif alatif@yazd.ac.ir \nYazd University\nYazdIran",
"Seyed... | With the fast development of communication and multimedia technology, the rights of the owners of multimedia products is vulnerable to the unauthorized copies and watermarking is one of the best known methods for proving the ownership of a product. In this paper we prosper the previous watermarking method which was bas... | [
{
"title": "I. INTRODUCTION",
"content": "The fast advancement in information and communication technology which cause easier access to multimedia contents, new challenges has appears. Digital multimedia suffer from copyright violation, unlimited duplication, modification and easily transfer over the Intern... | [
{
"id": "",
"caption": "the coefficient of the DCT in the row u and column v of the image matrix. I(m,n) is the intensity of the pixel and if u and v are zero, 1. The image could be reconstructed by using IDCT based on equation 2.",
"content": "the coefficient of the DCT in the row u and column v of the... | [
{
"text": "A novel bat algorithm based on chaos for optimization tasks. Homayun Afrabandpey, Meysam Ghaffari, Abdolreza Mirzaei, Mehran Safayani, Intelligent Systems (ICIS), 2014 Iranian Conference on. IEEEHomayun Afrabandpey, Meysam Ghaffari, Abdolreza Mirzaei, and Mehran Safayani. A novel bat algorithm based ... |
SANTIAGO: Spine Association for Neuron Topology Improvement and Graph Optimization
August 9, 2016
William Gray Roncal
Department of Computer Science
Johns Hopkins University
BaltimoreMD
JHU Applied Physics Laboratory
Research and Exploratory Development
LaurelMD
Colin Lea
Department of Computer Science
Johns Hopk... | 14,886,098 | {
"arxiv": "1608.02307",
"mag": 2481877995,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/RoncalLBH16",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1608.02307v1.pdf"
],
"pdf_sha": "ad9d8798a618966d4aaacd0d537b9ba087bbb9d2",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | SANTIAGO: Spine Association for Neuron Topology Improvement and Graph Optimization | [
"William Gray Roncal \nDepartment of Computer Science\nJohns Hopkins University\nBaltimoreMD\n\nJHU Applied Physics Laboratory\nResearch and Exploratory Development\nLaurelMD",
"Colin Lea \nDepartment of Computer Science\nJohns Hopkins University\nBaltimoreMD",
"Akira Baruah \nColumbia University\nNew YorkNY",
... | Developing automated and semi-automated solutions for reconstructing wiring diagrams of the brain from electron micrographs is important for advancing the field of connectomics.While the ultimate goal is to generate a graph of neuron connectivity, most prior automated methods have focused on volume segmentation rather ... | [
{
"title": "Introduction",
"content": "In adult vertebrate brains, each spine is typically associated with an excitatory synaptic connection, making their detection and association a critical task for building brain graphs [1,2].Many current algorithms in connectomics are designed and evaluated using surrog... | [
{
"id": "",
"caption": "Figure 1: (Left) Examples of spine fragmentation.Images illustrate typical split errors made in reconstructing spines by superimposing the automated segmentation labels on the ground truth for individual neurites.If reconstructed correctly, each object should be only a single color.T... | [
{
"text": "Ultrastructure of Dendritic Spines: Correlation Between Synaptic and Spine Morphologies. J I Arellano, R Benavides-Piccione, J Defelipe, R Yuste, nov 2007",
"doi": ""
},
{
"text": "Dendritic Spines. R Yuste, 2010MIT Press",
"doi": ""
},
{
"text": "Estructura de los centros ner... |
Segmentation Style Discovery: Application to Skin Lesion Images
5 Aug 2024
Kumar Abhishek kabhishe@sfu.ca
School of Computing Science
Simon Fraser University
Canada
Jeremy Kawahara jeremy@aip.ai
AIP Labs
BudapestHungary
Ghassan Hamarneh hamarneh@sfu.ca
School of Computing Science
Simon Fraser University
Canada
... | 271,720,088 | {
"arxiv": "2408.02787",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2408-02787",
"doi": "10.48550/arxiv.2408.02787"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2408.02787v1.pdf"
],
"pdf_sha": "d7d1409824a14b6ca10f098584add649476b2715",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Segmentation Style Discovery: Application to Skin Lesion Images | [
"Kumar Abhishek kabhishe@sfu.ca \nSchool of Computing Science\nSimon Fraser University\nCanada",
"Jeremy Kawahara jeremy@aip.ai \nAIP Labs\nBudapestHungary",
"Ghassan Hamarneh hamarneh@sfu.ca \nSchool of Computing Science\nSimon Fraser University\nCanada"
] | Variability in medical image segmentation, arising from annotator preferences, expertise, and their choice of tools, has been well documented.While the majority of multi-annotator segmentation approaches focus on modeling annotator-specific preferences, they require annotator-segmentation correspondence.In this work, w... | [
{
"title": "Introduction",
"content": "Medical image segmentation is a critical component in medical image analysis pipelines, either as a preprocessing step for subsequent analyses or for treatment planning and image-guided human or robotic intervention.Following the seminal works of Long et al. [15] and R... | [
{
"id": "",
"caption": "StyleSeg overview with M = 3: the segmentation model fs predicts 3 plausible segmentations of different styles, while the style classifier fc predicts the style that is the most similar to the ground truth.(b) Sample training images showing variability of segmentations.",
"conten... | [
{
"text": "International Skin Imaging Collaboration: Melanoma Project. February 01, 2024",
"doi": ""
},
{
"text": "Agreement among ophthalmologists in marking the optic disc and optic cup in fundus images. A Almazroa, S Alodhayb, E Osman, E Ramadan, M Hummadi, M Dlaim, M Alkatee, K Raahemifar, V Lak... |
The n:m resonance dual pair
22 Feb 2011
Darryl D Holm
Cornelia Vizman
The n:m resonance dual pair
22 Feb 2011In honor of Tudor Ratiu's sixtieth birthday.AMS Classification: 53D1753D20 Keywords: dual pairsPoisson brackets
Holm and VizmanThe n : m resonance dual pair 2Dual pairsLet (M, ω) be a symplectic manifold and... | 119,165,014 | {
"arxiv": "1102.4377",
"mag": 1824851704,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1102.4377v1.pdf"
],
"pdf_sha": "b579a2b117899ebcd521f4d94f428a1b308c899a",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | The n:m resonance dual pair | [
"Darryl D Holm",
"Cornelia Vizman"
] | Holm and VizmanThe n : m resonance dual pair 2Dual pairsLet (M, ω) be a symplectic manifold and P 1 , P 2 be two Poisson manifolds. A pair of Poisson mappingsis called a dual pair[Weinstein(1983)] if ker T J 1 and ker T J 2 are symplectic orthogonal complements of one another. That isA systematic treatment of dual pair... | [
{
"title": "Introduction",
"content": "The dual pairs for 1:1 and 1:-1 resonance are presented in [Marsden(1987)], reformulating results from [Cushman and Rod(1982)] and [Iwai(1985)]. The dual pair for 1:1 resonance is the pair of momentum maps associated to the commuting Hamiltonian actions of the Lie grou... | [
{
"id": "",
"caption": "The vector field v = f ∇C, where f is a nonvanishing function on R 3 , determines a Poisson structure π v on R 3 with symplectic leaves the level surfaces of the function C.",
"content": "Proposition 4. 1 .\n1The vector field v = f ∇C, where f is a nonvanishing function on R 3 , ... | [
{
"text": "Complete reduction of oscillators in resonance p : q. ; R Rod, D L Cushman, Rod, Reduction of the semisimple 1:1 resonance. 6Physica Dand Rod(1982)] R. Cushman and D. L. Rod [1982], Reduction of the semisimple 1:1 resonance, Physica D 6, 105-112. [Elipe(2000)] A. Elipe [2000], Complete reduction of o... |
Monotone Complexity of Spanning Tree Polynomial Re-visited
14 Sep 2021 September 16, 2021
Arkadev Chattopadhyay
Rajit Datta
Utsab Ghosal
Partha Mukhopadhyay
Monotone Complexity of Spanning Tree Polynomial Re-visited
14 Sep 2021 September 16, 2021
We prove two results that shed new light on the monotone complexity... | 237,513,515 | {
"arxiv": "2109.06941",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/innovations/ChattopadhyayDG22",
"doi": "10.4230/lipics.itcs.2022.39"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2109.06941v1.pdf"
],
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"open_access": {
"license": null,
"url": null,
"status": null
}
} | Monotone Complexity of Spanning Tree Polynomial Re-visited | [
"Arkadev Chattopadhyay",
"Rajit Datta",
"Utsab Ghosal",
"Partha Mukhopadhyay"
] | We prove two results that shed new light on the monotone complexity of the spanning tree polynomial, a classic polynomial in algebraic complexity and beyond.First, we show that the spanning tree polynomials having n variables and defined over constantdegree expander graphs, have monotone arithmetic complexity 2 Ω(n) . ... | [
{
"title": "Introduction",
"content": "A central problem in complexity theory aims to understand the following question: How much more powerful are computations with access to negations than computations that are monotone? While strong limitations have been known since almost three decades on monotone compu... | [
{
"id": "",
"caption": "Let us very briefly first recall the basic notions in the 2-party communication model of Yao. The joint input space of Alice and Bob is {0, 1} m × {0, 1} m with each player receiving an m-bit Boolean string, and they want to evaluate a Boolean function F : {0, 1} m × {0, 1} m → {0, 1... | [
{
"text": "Lower bounds for monotone arithmetic circuits via communication complexity. Arkadev Chattopadhyay, Rajit Datta, Partha Mukhopadhyay, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event. Samir Khuller and Virginia Vassilevska WilliamsItalyACMArkadev Chattopadhyay, Rajit Da... |
A Game Theoretical Error-Correction Framework for Secure Traffic-Sign Classification
Muhammed O Sayin
Chung-Wei Lin
Eunsuk Kang
Shinichi Shiraishi
Life Fellow, IEEETamer Başar
A Game Theoretical Error-Correction Framework for Secure Traffic-Sign Classification
1Index Terms Game theoryAutonomous drivingTraffic s... | 59,413,764 | {
"arxiv": "1901.10622",
"mag": 2919666072,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-1901-10622",
"doi": null
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"open_access": {
"license": null,
"url": null,
"status": null
}
} | A Game Theoretical Error-Correction Framework for Secure Traffic-Sign Classification | [
"Muhammed O Sayin",
"Chung-Wei Lin",
"Eunsuk Kang",
"Shinichi Shiraishi",
"Life Fellow, IEEETamer Başar"
] | We introduce a game theoretical error-correction framework to design classification algorithms that are reliable even in adversarial environments, with a specific focus on traffic-sign classification. Machine learning algorithms possess inherent vulnerabilities against maliciously crafted inputs especially at high dime... | [
{
"title": "I. INTRODUCTION",
"content": "M ACHINE learning is one of the key enabling technologies for autonomous vehicles. An autonomous vehicle can learn how to recognize the surroundings, e.g., traffic-signs, and can base its strategic decisions on this information. Along with the development in such te... | [
{
"id": "",
"caption": "Example implementations on the stop sign, which is only limited by imagination and creativity of the designer.",
"content": "Fig. 1 :\n1Example implementations on the stop sign, which is only limited by imagination and creativity of the designer."
},
{
"id": "",
"capt... | [
{
"text": "Vision-based traffic sign detection and analysis for intelligent driver assistance systems: Perspectives and survey. A Mogelmose, M M Trivedi, T B Moeslund, IEEE Transactions on Intelligent Transportation Systems. 134A. Mogelmose, M. M. Trivedi, and T. B. Moeslund, \"Vision-based traffic sign detecti... |
SciPost Physics Submission Boundary Chaos: Exact Entanglement Dynamics
June 2, 2023
F Fritzsch *felix.fritzsch@fmf.uni-lj.si
Physics Department
Faculty of Mathematics and Physics
University of Ljubljana
LjubljanaSlovenia
R Ghosh
Physics Department
Faculty of Mathematics and Physics
University of Ljubljana
Ljubljan... | 255,999,902 | {
"arxiv": "2301.08168",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.21468/scipostphys.15.3.092"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2301.08168v3.pdf"
],
"pdf_sha": "cb79541974a1b408f98c59f634e2db85de42285f",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | SciPost Physics Submission Boundary Chaos: Exact Entanglement Dynamics | [
"F Fritzsch *felix.fritzsch@fmf.uni-lj.si \nPhysics Department\nFaculty of Mathematics and Physics\nUniversity of Ljubljana\nLjubljanaSlovenia",
"R Ghosh \nPhysics Department\nFaculty of Mathematics and Physics\nUniversity of Ljubljana\nLjubljanaSlovenia\n\nDepartment of Physics and Astronomy\nUniversity College ... | We compute the dynamics of entanglement in the minimal setup producing ergodic and mixing quantum many-body dynamics, which we previously dubbed boundary chaos. This consists of a free, non-interacting brickwork quantum circuit, in which chaos and ergodicity is induced by an impurity interaction, i.e., an entangling tw... | [
{
"title": "Introduction",
"content": "One of the hallmark features of quantum mechanics which has no counterpart in classical physics is quantum entanglement. With the advent of quantum computing and communication, quantum entanglement has become an important resource to overcome limitations of classical c... | [
{
"id": "",
"caption": "Second Rényi entropy for q = 2 and l = 2 for an impurity interaction with a vacuum state for various system sizes in (a) linear and (b) semi-logarithmic scale. (a) The dashed line corresponds to the maximum entropy given by (l + 1) ln(q). Orange dots depict R 2 obtained from a direct... | [
{
"text": "M A Nielsen, I L Chuang, 10.1017/CBO9780511976667Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University PressM. A. Nielsen and I. L. Chuang, Quantum Computation and Quan- tum Information: 10th Anniversary Edition, Cambridge University Press, doi:10.1017/CBO9780511... |
ROBUST NEAR-DIAGONAL GREEN FUNCTION ESTIMATES
25 Apr 2023
Moritz Kassmann
ANDMinhyun Kim
Ki-Ahm Lee
ROBUST NEAR-DIAGONAL GREEN FUNCTION ESTIMATES
25 Apr 2023arXiv:2111.05768v2 [math.AP]
We prove sharp near-diagonal pointwise bounds for the Green function G Ω (x, y) for nonlocal operators of fractional order α ∈ (0... | 243,938,310 | {
"arxiv": "2111.05768",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1093/imrn/rnad106"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2111.05768v2.pdf"
],
"pdf_sha": "c0ed52fc6b3e3b8afaa4ed5e347aa4118979ed9c",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | ROBUST NEAR-DIAGONAL GREEN FUNCTION ESTIMATES | [
"Moritz Kassmann",
"ANDMinhyun Kim",
"Ki-Ahm Lee"
] | We prove sharp near-diagonal pointwise bounds for the Green function G Ω (x, y) for nonlocal operators of fractional order α ∈ (0, 2). The novelty of our results is two-fold: the estimates are robust as α → 2− and we prove the bounds without making use of the Dirichlet heat kernel p Ω (t; x, y). In this way we can cove... | [
{
"title": "Introduction",
"content": "An important object in Potential Analysis is the Green function. Explicit formulas for the Green function of the fractional Laplace operator for 0 < α < 2 are complicated, even for domains like a ball in the Euclidean space, see [Rie38,BGR61] or more recently [Buc16]. ... | [
{
"id": "",
"caption": "Let Ω be a bounded open set inR d . A measurable function G : R d × R d → [0, +∞] is called a Green function of L on Ω if G(·, y) ∈ L 1 (Ω) for each y ∈ Ω, G = 0 a.e. on (R d × R d ) \\ (Ω × Ω), andˆR d G(x, y)ψ(x) dx = ϕ(y) (1.3)for every y ∈ Ω, ϕ ∈ H Ω (R d ; k) ∩ C(Ω), and ψ ∈ L ∞... | [
{
"text": "Limiting embedding theorems for W s,p when s ↑ 1 and applications. Jean Bourgain, Haïm Brezis, Petru Mironescu, J. Anal. Math. Thomas H. Wolff87Jean Bourgain, Haïm Brezis, and Petru Mironescu. Limiting embedding theorems for W s,p when s ↑ 1 and applications. J. Anal. Math., 87:77-101, 2002. Dedicate... |
Game-driven random walks: Survival time statistics
M I Krivonosov krivonosov@itmm.unn.rubelectronicmail:sergiyde@oslomet.no
Department of Applied Mathematics
Lobachevsky University
603022Nizhny NovgorodRussia
Mathematical Center
Lobachevsky University
603022Nizhny NovgorodRussia
S N Tikhomirov
Department of Appl... | 242,757,083 | {
"arxiv": "2111.02785",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2111.02785v2.pdf"
],
"pdf_sha": "e823a35e2d9406f31ce91bb42d4941bc98959e7b",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Game-driven random walks: Survival time statistics | [
"M I Krivonosov krivonosov@itmm.unn.rubelectronicmail:sergiyde@oslomet.no \nDepartment of Applied Mathematics\nLobachevsky University\n603022Nizhny NovgorodRussia\n\nMathematical Center\nLobachevsky University\n603022Nizhny NovgorodRussia",
"S N Tikhomirov \nDepartment of Applied Mathematics\nLobachevsky Universi... | Random walks are powerful tools to analyze spatial-temporal patterns produced by living organisms ranging from cells to humans. At the same time, it is evident that these patterns are not completely random but are results of a convolution of organisms' sensor-based information processing and motility. The complexity of... | [
{
"title": "I. INTRODUCTION",
"content": "From the very start, random walks were interpreted in biological and behavioral terms, and now Pearson's drunkard 9 is no less famous than Schrödinger's cat. Different type of random walks 10,11 are used as foundations to build optimal search theories and explain th... | [
{
"id": "",
"caption": "(Color online) Random Walk Game app screenshot. The header line consists of the current number of turns, their goals and the number of turns in the longest game. Player see the game field, the piece position and the trajectory of piece. At the bottom of the screen shown the control m... | [
{
"text": "The gambler's ruin. J L Coolidge, Ann. Math. 10181J. L. Coolidge, The gambler's ruin, Ann. Math. 10, 181 (1909).",
"doi": ""
},
{
"text": "W D Feller, An Introduction to Probability Theory and Its Applications. New YorkWileyW. D. Feller, An Introduction to Probability Theory and Its Appli... |
Continuous Choreographies as Limiting Solutions of N -body Type Problems with Weak Interaction
2016
Integrability Symmetry
Geometry
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas
UNAM
D.F. 04510México, México
Continuous Choreographies as Limiting Solutions of N -body Type Problems with Weak I... | 1,532,376 | {
"arxiv": "1510.05979",
"mag": 2543484957,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.3842/sigma.2016.104"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1510.05979v5.pdf"
],
"pdf_sha": "b7e68aedb2e0d67447792ef1c0a45d485af0a75a",
"open_access": {
"license": "CCBYSA",
"url": "http://www.emis.de/journals/SIGMA/2016/104/sigma16-104.pdf",
"status": "GOLD"
}
} | Continuous Choreographies as Limiting Solutions of N -body Type Problems with Weak Interaction | [
"Integrability Symmetry",
"Geometry",
"Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas\nUNAM\nD.F. 04510México, México"
] | We consider the limit N → +∞ of N -body type problems with weak interaction, equal masses and −σ-homogeneous potential, 0 < σ < 1. We obtain the integro-differential equation that the motions must satisfy, with limit choreographic solutions corresponding to travelling waves of this equation. Such equation is the Euler-... | [
{
"title": "Introduction",
"content": "It is well known that the planar regular N -gon relative equilibrium is a solution of the equations of motion for the Newtonian N -body problem with N ≥ 3 and equal mass bodies [1,3,6]. We consider N -body type problems with weak interaction, equal masses in R d , d ≥ ... | [
{
"id": "",
"caption": "For y ∈ Λ, we have that y L 2 [0,1] ≤ ẏ L 2 [0,1] and therefore y H 1 [0,1] is equivalent to y L 2 [0,1] . Proposition 2.2. Assume y ∈ C 2 1 (R, R d ) defines a regular simple closed curve. Then y is an extremal of the functional (2.6) if and only if it satisfies (2.5).",
"conten... | [
{
"text": "Action minimizing orbits in the n-body problem with simple choreography constraint. V Barutello, S Terracini, 10.1088/0951-7715/17/6/002math.DS/0307088Nonlinearity. 17Barutello V., Terracini S., Action minimizing orbits in the n-body problem with simple choreography con- straint, Nonlinearity 17 (200... |
Towards a Smaller Student: Capacity Dynamic Distillation for Efficient Image Retrieval
31 May 2023
Yi Xie
South China University of Technology
Huaidong Zhang huaidongz@scut.edu.cn
South China University of Technology
Xuemiao Xu
South China University of Technology
State Key Laboratory of Subtropical Building... | 257,557,237 | {
"arxiv": "2303.09230",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/cvpr/XieZXZH23",
"doi": "10.1109/cvpr52729.2023.01536"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2303.09230v2.pdf"
],
"pdf_sha": "d4a4616ad77364acac8c4d471188ee6d0b2fb616",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2303.09230",
"status": "GREEN"
}
} | Towards a Smaller Student: Capacity Dynamic Distillation for Efficient Image Retrieval | [
"Yi Xie \nSouth China University of Technology",
"Huaidong Zhang huaidongz@scut.edu.cn \nSouth China University of Technology",
"Xuemiao Xu \nSouth China University of Technology\n\n\nState Key Laboratory of Subtropical Building Science 5 Ministry of Education Key Laboratory of Big Data and Intelligent Robot 6 ... | Previous Knowledge Distillation based efficient image retrieval methods employs a lightweight network as the student model for fast inference.However, the lightweight student model lacks adequate representation capacity for effective knowledge imitation during the most critical early training period, causing final perf... | [
{
"title": "Introduction",
"content": "Image retrieval [4] aims to rank all the instances in a retrieval set based on their relevance to the query image.However, many image retrieval methods [45,50] use heavy networks to acquire a high accuracy, causing a low inference speed and hindering practical applicat... | [
{
"id": "",
"caption": "Figure 2. The capacity dynamic distillation framework.The highcapacity student can comprehensively understand teachers' knowledge in the early KD iterations.After training, the student can be converted into a light network to acquire a fast inference speed.",
"content": "Figure 2... | [
{
"text": ". Vit-Base, 16] * ViT [16] 86.7 55.6 78.92 95.84",
"doi": ""
},
{
"text": "The * represents the result is re-implemented. Table 4. Performance comparison on MSMT17. METHOD 1 TEACHER STUDENT MP FLOPs mAP R1 IANet. ResNet50 25.54 4.07 46.8 75.525",
"doi": ""
},
{
"text": "AT [30... |
mRUBiS: An Exemplar for Model-Based Architectural Self-Healing and Self-Optimization
3 Apr 2018
Thomas Vogel thomas.vogel@cs.hu-berlin.de
Humboldt-Universität zu Berlin
Germany
mRUBiS: An Exemplar for Model-Based Architectural Self-Healing and Self-Optimization
3 Apr 20184B31DA78750E9A725F6257D16576E39710.1145/3194... | 4,570,446 | {
"arxiv": "1804.00954",
"mag": 3099400957,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/icse/Vogel18",
"doi": "10.1145/3194133.3194161"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1804.00954v1.pdf"
],
"pdf_sha": "ae284d18ca2c9f65c6ecce4b8a6d47587d0fb67d",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/1804.00954",
"status": "GREEN"
}
} | mRUBiS: An Exemplar for Model-Based Architectural Self-Healing and Self-Optimization | [
"Thomas Vogel thomas.vogel@cs.hu-berlin.de \nHumboldt-Universität zu Berlin\nGermany"
] | Self-adaptive software systems are often structured into an adaptation engine that manages an adaptable software by operating on a runtime model that represents the architecture of the software (model-based architectural self-adaptation).Despite the popularity of such approaches, existing exemplars provide application ... | [
{
"title": "INTRODUCTION",
"content": "Self-adaptive software is a closed-loop system that implements a feedback loop to autonomously adapt to changes at runtime [11].Engineering such systems is challenging as it involves domain and adaptation concerns.Thus, a popular engineering approach is to split the sy... | [
{
"id": "",
"caption": "Figure 1: Self-adaptive system with a runtime model.",
"content": "Figure 1 :\n1\nFigure 1: Self-adaptive system with a runtime model."
},
{
"id": "",
"caption": "Figure 2: Generic simulation framework.To enable the simulation, the simulator must be configured with in... | [
{
"text": "Requirements and Architectural Approaches to Adaptive Software Systems: A Comparative Study. Konstantinos Angelopoulos, E Silva Vítor, João Souza, Pimentel, 10.1109/SEAMS.2013.6595489Intl. Symposium on Software Engineering for Adaptive and Self-Managing Systems (SEAMS). IEEE2013",
"doi": ""
},
... |
A semi-supervised approach to message stance classification
February 11, 2019 29 Jan 2019
Georgios Giasemidis 1georgios@countinglab.co.uk
Nikolaos Kaplis
Ioannis Agrafiotis 2ioannis.agrafiotis@cs.ox.ac.uk
Jason R C Nurse
Department of Computer Science
CountingLab Ltd
ReadingUK
School of Computing
University of... | 59,842,982 | {
"arxiv": "1902.03097",
"mag": 3103344055,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/tkde/GiasemidisKAN20",
"doi": "10.1109/tkde.2018.2880192"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1902.03097v1.pdf"
],
"pdf_sha": "81da79294781a1a5252cab71dfe2e4815dfb645a",
"open_access": {
"license": null,
"url": "http://centaur.reading.ac.uk/81213/1/main.pdf",
"status": "GREEN"
}
} | A semi-supervised approach to message stance classification | [
"Georgios Giasemidis 1georgios@countinglab.co.uk",
"Nikolaos Kaplis",
"Ioannis Agrafiotis 2ioannis.agrafiotis@cs.ox.ac.uk",
"Jason R C Nurse",
"Department of Computer Science\nCountingLab Ltd\nReadingUK",
"School of Computing\nUniversity of Oxford\nOxfordUK",
"University of Kent\nCanterburyUK"
] | Social media communications are becoming increasingly prevalent; some useful, some false, whether unwittingly or maliciously. An increasing number of rumours daily flood the social networks. Determining their veracity in an autonomous way is a very active and challenging field of research, with a variety of methods pro... | [
{
"title": "Introduction",
"content": "Online content is at the centre of today's information world. A primary source of this content is social media, with the public acting as a major contributor on everything from election discussions to reports on ongoing crisis events. This level of free engagement has ... | [
{
"id": "",
"caption": "Distribution of the number of tweets in rumours: total (left) and original English only (right).",
"content": "Figure 1 :\n1Distribution of the number of tweets in rumours: total (left) and original English only (right)."
},
{
"id": "",
"caption": "Accuracy of the LS ... | [
{
"text": "Hurricane Harvey Victims Turn to Social Media for Assistance. 15The Wall Street Journal, Hurricane Harvey Victims Turn to Social Media for Assistance, https://www.wsj.com/articles/hurricane-harvey- victims-turn-to-social-media-for-assistance-1503999001 (Accessed on 15- Dec-2017) (2017).",
"doi": ... |
MAXIMAL BREAKING OF SYMMETRY AT MAXIMAL BREAKING OF SYMMETRY AT CRITICAL ANGLE AND CLOSED FORM CRITICAL ANGLE AND CLOSED FORM EXPRESSION FOR ANGULAR DEVIATIONS EXPRESSION FOR ANGULAR DEVIATIONS OF THE SNELL LAW OF THE SNELL LAW
21 Oct 2014
Manoel P Araújo mparaujo@ifi.unicamp.br
Department of Applied Mathematics Sta... | 119,268,123 | {
"arxiv": "1410.5724",
"mag": 2316733863,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physreva.90.033844"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1410.5724v1.pdf"
],
"pdf_sha": "2e772d3297c9e0cfe27635ab430265470bca4219",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1410.5724",
"status": "GREEN"
}
} | MAXIMAL BREAKING OF SYMMETRY AT MAXIMAL BREAKING OF SYMMETRY AT CRITICAL ANGLE AND CLOSED FORM CRITICAL ANGLE AND CLOSED FORM EXPRESSION FOR ANGULAR DEVIATIONS EXPRESSION FOR ANGULAR DEVIATIONS OF THE SNELL LAW OF THE SNELL LAW | [
"Manoel P Araújo mparaujo@ifi.unicamp.br \nDepartment of Applied Mathematics State\nInstitute of Physics \"Gleb Wataghin\" State University of Campinas\nBrazil",
"Silvânia A Carvalho \nDepartment of Applied Mathematics State\nUniversity of Campinas\nBrazil",
"Stefano De Leo deleo@ime.unicamp.br \nUniversity of ... | Physical Review A 90, 033844-11 (2014) Physical Review A 90, 033844-11 (2014) • • Abstract. A detailed analysis of the propagation of laser gaussian beams at critical angles shows in which conditions it is possible to maximize the breaking of symmetry in the angular distribution and for which values of the laser wave... | [
{
"title": "II. ASYMMETRICALLY MODELED BEAMS",
"content": "As anticipated in the introduction, the breaking of symmetry [26] in the angular distribution of optical beams plays a fundamental role in the angular deviation from the optical path predicted by the Snell law. In this section, to understand why the... | [
{
"id": "",
"caption": "ACKNOWLEDGMENTS. The authors gratefully acknowledge Capes (M.P.A.), CNPq (S.D.L.), and Fapesp (S.A.C.) for the the financial support and the referee for his comments and suggestions, in particular for drawing the attention to the misalignment effect which stimulated the discussion wh... | [
{
"text": "M Born, E Wolf, Principles of optics. Cambridge UP, CambridgeM. Born and E. Wolf, Principles of optics (Cambridge UP, Cambridge, 1999).",
"doi": ""
},
{
"text": "B E A Saleh, M C Teich, Fundamentals of Photonics. New JerseyWiley & SonsB. E. A. Saleh and M. C. Teich, Fundamentals of Photon... |
QCD equation of state in a virial expansion
4 May 2009
S Mattiello
Institute for Theoretical Physics
University of Gießen
Germany
W Cassing
Institute for Theoretical Physics
University of Gießen
Germany
QCD equation of state in a virial expansion
4 May 2009numbers: 1238Mh2575Nq2165Qr1238Aw
We describe recent thre... | 119,235,780 | {
"arxiv": "0905.0364",
"mag": 1976486494,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/0954-3899/36/12/125003"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0905.0364v1.pdf"
],
"pdf_sha": "16328263d8ca89fd46b63db7266c06096f8857db",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/0905.0364",
"status": "GREEN"
}
} | QCD equation of state in a virial expansion | [
"S Mattiello \nInstitute for Theoretical Physics\nUniversity of Gießen\nGermany",
"W Cassing \nInstitute for Theoretical Physics\nUniversity of Gießen\nGermany"
] | We describe recent three-flavor QCD lattice data for the pressure, speed of sound and interaction measure at nonzero temperature and vanishing chemical potential within a virial expansion. For the deconfined phase we use a phenomenological model which includes non-perturbative effects from dimension two gluon condensat... | [
{
"title": "I. INTRODUCTION",
"content": "The exploration of the phase structure of quantum chromodynamics (QCD) for the whole densitytemperature plane is a challenging task for the standard model. Lattice Monte Carlo simulations have revealed several exciting results over the past decade [1,2,3,4,5,6,7,8].... | [
{
"id": "",
"caption": "(Color online) Pressure P as a function of the temperature divided by T 4 forg = 1.30, TH = 205 MeV and a = 9.17 MeV 1/2 in comparison to the lattice data from Ref.[8].",
"content": "FIG. 1 :\n1(Color online) Pressure P as a function of the temperature divided by T 4 forg = 1.30,... | [
{
"text": ". M J Tannenbaum, Int. J. Mod. Phys. E. 17771M. J. Tannenbaum, Int. J. Mod. Phys. E 17, 771 (2008).",
"doi": ""
},
{
"text": ". Z Fodor, S D Katz, K K Szabo, Phys. Lett. B. 56873Z. Fodor, S. D. Katz and K. K. Szabo, Phys. Lett. B 568, 73 (2003).",
"doi": ""
},
{
"text": ". Z F... |
Draft version Is superorbital modulation in SMC X-1 caused by absorption in warped precessing accretion disc?
April 3, 2020
Pragati Pradhan
Massachusetts Institute of Technology
Kavli Institute for Astrophysics and Space Research
70 Vassar St02139CambridgeMAUSA
Chandreyee Maitra
Max Planck Institute For Extraterre... | 214,775,205 | {
"arxiv": "2004.00664",
"mag": 3105465488,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.3847/1538-4357/ab8224"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2004.00664v1.pdf"
],
"pdf_sha": "a505cc94c1272bed765171155ef149564fd6f543",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2004.00664",
"status": "GREEN"
}
} | Draft version Is superorbital modulation in SMC X-1 caused by absorption in warped precessing accretion disc? | [
"Pragati Pradhan \nMassachusetts Institute of Technology\nKavli Institute for Astrophysics and Space Research\n70 Vassar St02139CambridgeMAUSA",
"Chandreyee Maitra \nMax Planck Institute For Extraterrestrial Physics\n85748GarchingGermany",
"Biswajit Paul \nRaman Research Institute, Astronomy and Astrophysics\nC... | We present a broadband spectral-timing analysis of SMC X-1 at different intensity states of its superorbital variation using 10 Suzaku and 6 NuSTAR observations. The spectrum in all the states can be described by an absorbed powerlaw with a high energy cutoff and a black-body component along with an iron emission line.... | [
{
"title": "INTRODUCTION",
"content": "Super-orbital periods are long term periodic/quasi-periodic intensity variations at timescale often several times the orbital period seen in X-ray binaries (an optical companion with a compact object). For X-ray binary systems with an accretion disk (e.g., LMC X-4), th... | [
{
"id": "",
"caption": "SMC X-1 was observed with NuSTAR (The Nuclear Spectroscopic Telescope Array ; Harrison et al. 2013) six times during 2012-2016 at various superorbital phases (reported in",
"content": "SMC X-1 was observed with NuSTAR (The Nuclear Spectroscopic Telescope Array ; Harrison et al. 2... | [
{
"text": ". L Bildsten, 10.1086/313060ApJS. 113367Bildsten L, et al. 1997 ApJS 113 367.",
"doi": ""
},
{
"text": ". E Bozzo, 10.1051/0004-6361/201731930A&A. 60610Bozzo E, et al. 2017 A&A 606 L10.",
"doi": ""
},
{
"text": ". M C Brumback, 10.3847/1538-4357/ab5b04The Astrophysical Journal... |
Shifting Opinions in a Social Network Through Leader Selection
28 Oct 2019
Yuhao Yi
Timothy Castiglia
Stacy Patterson
Shifting Opinions in a Social Network Through Leader Selection
28 Oct 20191Index Terms-Social NetworkFrench-DeGroot modelBal- ance of OpinionsOptimizationApproximation Algorithm
We study French-Deg... | 204,949,891 | {
"arxiv": "1910.13009",
"mag": 3133146637,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/tcns/YiCP21",
"doi": "10.1109/tcns.2021.3058620"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1910.13009v1.pdf"
],
"pdf_sha": "05cb244441e73b2717f43c02789771bde6139f69",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1910.13009",
"status": "GREEN"
}
} | Shifting Opinions in a Social Network Through Leader Selection | [
"Yuhao Yi",
"Timothy Castiglia",
"Stacy Patterson"
] | We study French-Degroot opinion dynamics in a social network with two polarizing parties. We consider a network in which the leaders of one party are given, and we pose the problem of selecting the leader set of the opposing party so as to shift the average opinion to a desired value. This problem generalizes the inten... | [
{
"title": "I. INTRODUCTION",
"content": "Social networks have become increasingly influential in shaping people's opinions. Within this field, the problem of effectively shifting opinions in a social network has received great interest in last two decades [1], [2], [3], [4], [5], [6], [7], [8], [9]. Much o... | [
{
"id": "",
"caption": "x F = diag ([6, . . . , 6]) −1 [3, . . . , 3] ⊤ = [ 1 2 , . . . , 1 2 ] ⊤ . There are n − 1 − k follower nodes; thus, we have µ(S (n − 1 − k) + k = n−1+k 2n . Soundness: If S 1 is not a vertex cover of graph G, then the follower node set is not an independent set. So, the matrix L F,... | [
{
"text": "Maximizing the spread of influence through a social network. D Kempe, J M Kleinberg, Tardos, Proc. of 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. of 9th ACM SIGKDD International Conference on Knowledge Discovery and Data MiningD. Kempe, J. M. Kleinberg, andÉ. Tardo... |
arXiv:1207.6719v1 [math-ph] Mean Field Asymptotics of Generalized Quantum Kinetic Equation
28 Jul 2012
V I Gerasimenko
Zh A Tsvir
Academician Glushkov Av
03187KyivUkraine
Institute of Mathematics
NAS of Ukraine
3, Tereshchenkivs'ka Str., Kyiv-401601Ukraine
Department of Mechanics and Mathematics
Taras Shevchenk... | 119,708,508 | {
"arxiv": "1207.6719",
"mag": 2155847508,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/s0034-4877(12)60034-2"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1207.6719v1.pdf"
],
"pdf_sha": "0551c261f69ba3fee365a2833496303b31f2b6ea",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1207.6719",
"status": "GREEN"
}
} | arXiv:1207.6719v1 [math-ph] Mean Field Asymptotics of Generalized Quantum Kinetic Equation | [
"V I Gerasimenko",
"Zh A Tsvir \nAcademician Glushkov Av\n03187KyivUkraine",
"Institute of Mathematics\nNAS of Ukraine\n3, Tereshchenkivs'ka Str., Kyiv-401601Ukraine",
"Department of Mechanics and Mathematics\nTaras Shevchenko National University of Kyiv"
] | We construct the mean field asymptotics of a solution of initial-value problem of the generalized quantum kinetic equation and a sequence of explicitly defined functionals of a solution of stated kinetic equation. As a result the quantum Vlasov kinetic equation is rigorously derived. Moreover, in case of the presence o... | [
{
"title": "Introduction",
"content": "During the last decade the considerable progress in the rigorous derivation in scaling limits of quantum kinetic equations, in particular the nonlinear Schrödinger equation [1]- [16] and the Gross-Pitaevskii equation [17]- [19] as well as the quantum Boltzmann equation... | [
{
"id": "",
"caption": "(t, {Y }) − I f s L 1 (Hs) = 0, and in the general case the following equalities hold lim ǫ→0 1 ǫ n V 1+n (t, {Y }, X \\ Y )f s+n L 1 (H s+n ) = 0, n ≥ 1.",
"content": "lim ǫ→0 V 1\nǫ→01(t, {Y }) − I f s L 1 (Hs) = 0, and in the general case the following equalities hold lim ǫ→0 ... | [
{
"text": "Rigorous derivation of the cubic NLS in dimension one. R Adami, F Golse, A Teta, J. Stat. Phys. 1276R. Adami, F. Golse and A. Teta, Rigorous derivation of the cubic NLS in dimension one. J. Stat. Phys. 127, (6), (2007), 1193-1220.",
"doi": ""
},
{
"text": "Mean field limit for bosons and ... |
Characterization of New Silicon Photomultipliers with Low Dark Noise at Low Temperature
27 Jul 2020
K Ozaki
Institute for Space-Earth Environmental Research
Nagoya University
464-8601NagoyaAichiJapan
S Kazama kazama@isee.nagoya-u.ac.jp
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe
Nagoya... | 220,793,572 | {
"arxiv": "2007.13537",
"mag": 3134028672,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/1748-0221/16/03/p03014"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2007.13537v1.pdf"
],
"pdf_sha": "a9f003aa12d7ac6bc5a67b35effef7ef4e5e4050",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2007.13537",
"status": "GREEN"
}
} | Characterization of New Silicon Photomultipliers with Low Dark Noise at Low Temperature | [
"K Ozaki \nInstitute for Space-Earth Environmental Research\nNagoya University\n464-8601NagoyaAichiJapan",
"S Kazama kazama@isee.nagoya-u.ac.jp \nKobayashi-Maskawa Institute for the Origin of Particles and the Universe\nNagoya University\n464-8601NagoyaAichiJapan\n\nInstitute for Advanced Research\nNagoya Univers... | A: Silicon photomultipliers (SiPMs) have a low radioactivity, compact geometry, low operation voltage, and reasonable photo-detection efficiency for vacuum ultraviolet light (VUV). Therefore it has the potential to replace photomultiplier tubes (PMTs) for future dark matter experiments with liquid xenon (LXe). However,... | [
{
"title": "Introduction",
"content": "In the past three decades, numerous terrestrial experiments have been conducted to search for a faint interaction between weakly interacting massive particles (WIMPs) and ordinary matter. Among them, experiments using dual-phase (liquid/gas) xenon time projection chamb... | [
{
"id": "",
"caption": "Block diagram of the setup of this experiment.",
"content": "Figure 1 .\n1Block diagram of the setup of this experiment."
},
{
"id": "",
"caption": "(Left): Readout schematic of SiPMs. It is possible to readout two SiPMs using a common bias voltage. (Right): Typical w... | [
{
"text": "Dark Matter Search Results from a One Ton-Year Exposure of XENON1T. E Aprile, XENON CollaborationPhys. Rev. Lett. 121111302E. Aprile et al. (XENON Collaboration), Dark Matter Search Results from a One Ton-Year Exposure of XENON1T, Phys. Rev. Lett. 121 111302 (2018).",
"doi": ""
},
{
"text... |
The sh-Lie algebra perturbation Lemma
2000
J Huebschmann johannes.huebschmann@math.univ-lille1.fr
UMR 8524
USTL
UFR de Mathématiques CNRS
59655Villeneuve d'Ascq CedexFrance
The sh-Lie algebra perturbation Lemma
Mathematics Subject Classification: 16E45, 16W30, 17B55, 17B56, 17B65, 17B70
181020001
Let R be a commut... | 473,344 | {
"arxiv": "0710.2070",
"mag": 2963818658,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1515/form.2011.023"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0710.2070v3.pdf"
],
"pdf_sha": "81135948ad517cc73d1853e2bcb9368e018f62b8",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/0708.3977",
"status": "GREEN"
}
} | The sh-Lie algebra perturbation Lemma | [
"J Huebschmann johannes.huebschmann@math.univ-lille1.fr \nUMR 8524\nUSTL\nUFR de Mathématiques CNRS\n59655Villeneuve d'Ascq CedexFrance"
] | Let R be a commutative ring with 1 which contains the rationals as a subring andbe a contraction of chain complexes (over R). We denote the symmetric coalgebra functor by S c , the loop Lie algebra functor by L, the classifying coalgebra functor by C, and the suspension operator by s. We shall establish the following. ... | [
{
"title": "Introduction",
"content": "Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. Higher homotopies often arise as follows: Suppose we are given a huge object, e. g. a chain complex, whose homology includes invariants of a ce... | [
{
"id": "",
"caption": "The coaugmented differential graded coalgebra C being assumed to be graded cocommutative, the values of the Lie algebra perturbation∂ = ∂ ∆ , restricted to L[s −1 JC], lie in L[s −1 JC]. Proof. Write Y = s −1 JC, so that L[s −1 JC] = L[Y ]⊆ ΩC, and so that the augmented graded algebr... | [
{
"text": "Homological Algebra. H Cartan, S Eilenberg, Princeton University PressPrincetonH. Cartan and S. Eilenberg: Homological Algebra. Princeton University Press, Princeton (1956).",
"doi": ""
},
{
"text": "Torsion in homotopy groups. F R Cohen, J C Moore, J A Neisendorfer, Annals of Math. 109F.... |
Millimeter Wave Localization with Imperfect Training Data using Shallow Neural Networks
Anish Shastri anish.shastri@unitn.it
DISI
University of Trento Trento
Italy
Joan Palacios
North Carolina State University Raleigh
NCUSA
Paolo Casari paolo.casari@unitn.it
DISI
University of Trento Trento
Italy
Millimeter Wa... | 245,005,906 | {
"arxiv": "2112.05008",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/wcnc/ShastriPC22",
"doi": "10.1109/wcnc51071.2022.9771668"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2112.05008v2.pdf"
],
"pdf_sha": "6b271fabbf9bf6a410f0e7853a0736da8071ba3d",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2112.05008",
"status": "GREEN"
}
} | Millimeter Wave Localization with Imperfect Training Data using Shallow Neural Networks | [
"Anish Shastri anish.shastri@unitn.it \nDISI\nUniversity of Trento Trento\nItaly",
"Joan Palacios \nNorth Carolina State University Raleigh\nNCUSA",
"Paolo Casari paolo.casari@unitn.it \nDISI\nUniversity of Trento Trento\nItaly"
] | Millimeter wave (mmWave) localization algorithms exploit the quasi-optical propagation of mmWave signals, which yields sparse angular spectra at the receiver. Geometric approaches to angle-based localization typically require to know the map of the environment and the location of the access points. Thus, several works ... | [
{
"title": "I. INTRODUCTION",
"content": "Millimeter wave (mmWave) communication technology in the 30-300 GHz band is showing great promise for high-data rate, short-range wireless communications [1]. A number of applications are starting to benefit from the capabilities of mmWave technology: these applicat... | [
{
"id": "",
"caption": "Illustration of the concept of virtual anchor and first-order reflections.",
"content": "Fig. 1 .\n1Illustration of the concept of virtual anchor and first-order reflections."
},
{
"id": "",
"caption": "Structure of our proposed NN model.",
"content": "Fig. 2 .\n2... | [
{
"text": "Millimeter-wave cellular wireless networks: Potentials and challenges. S Rangan, T S Rappaport, E Erkip, Proceedings of the IEEE. 1023S. Rangan, T. S. Rappaport, and E. Erkip, \"Millimeter-wave cellular wireless networks: Potentials and challenges,\" Proceedings of the IEEE, vol. 102, no. 3, pp. 366-... |
Effective Equations for Repulsive Quasi-One Dimensional Bose-Einstein Condensates Trapped with Anharmonic Transverse Potentials
Hugo L C Couto
Ardiley T Avelar
Wesley B Cardoso
Effective Equations for Repulsive Quasi-One Dimensional Bose-Einstein Condensates Trapped with Anharmonic Transverse Potentials
10.1002/a... | 119,461,624 | {
"arxiv": "1707.03260",
"mag": 2792872832,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1002/andp.201700352"
} | {
"pdf_urls": [],
"pdf_sha": "b0138c696d806f19c4793cb745322467852c2772",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1707.03260",
"status": "GREEN"
}
} | Effective Equations for Repulsive Quasi-One Dimensional Bose-Einstein Condensates Trapped with Anharmonic Transverse Potentials | [
"Hugo L C Couto",
"Ardiley T Avelar",
"Wesley B Cardoso"
] | One-dimensional nonlinear Schrödinger equations are derived to describe the axial effective dynamics of cigar-shaped atomic repulsive Bose-Einstein condensates trapped with anharmonic transverse potentials. The accuracy of these equations in the perturbative, Thomas-Fermi, and crossover regimes were verified numericall... | [
{
"title": "Introduction",
"content": "It is unanimously recognized the importance of the experimental realization of Bose-Einstein condensates (BECs) of atomic dilute gases confined in optical and magnetic traps, [1,2] which have sparked many theoretical and experimental studies of coherent atomic matter. ... | [
{
"id": "",
"caption": "H. L. C. Couto, Dr. A. T. Avelar, Dr. W. B. Cardoso Instituto de Física Universidade Federal de Goiás 74.690-900, Goiânia, Goiás, Brazil E-mail: huoguh@gmail.com The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/andp.20170... | [
{
"text": ". M H Anderson, J R Ensher, M R Matthews, C E Wieman, E A Cornell, Science. 269M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, E. A. Cornell, Science 1995, 269, 198.",
"doi": ""
},
{
"text": ". K B Davis, M O Mewes, M R Andrews, N J Van Druten, D S Durfee, D M Kurn, W Ketterle... |
Effect of Spin-Orbit Coupling on Kondo Phenomena in f 7 -Electron Systems
20 Jul 2015
Takashi Hotta
Department of Physics
Tokyo Metropolitan University
192-0397HachiojiTokyoJapan
Effect of Spin-Orbit Coupling on Kondo Phenomena in f 7 -Electron Systems
Journal of the Physical Society of Japan
20 Jul 2015(Received ... | 119,278,011 | {
"arxiv": "1507.05402",
"mag": 2270485321,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.7566/jpsj.84.114707"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1507.05402v1.pdf"
],
"pdf_sha": "fe5a3547841c05d0d26e0dd42cd5be751daa03c3",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1507.05402",
"status": "GREEN"
}
} | Effect of Spin-Orbit Coupling on Kondo Phenomena in f 7 -Electron Systems | [
"Takashi Hotta \nDepartment of Physics\nTokyo Metropolitan University\n192-0397HachiojiTokyoJapan"
] | In order to promote our basic understanding on the Kondo behavior recently observed in europium compounds, we analyze an impurity Anderson model with seven f electrons at an impurity site by employing a numerical renormalization group method. The local part of the model consists of Coulomb interactions among f electron... | [
{
"title": "Introduction",
"content": "Research on heavy-electron systems has been one of central topics in the field of strongly correlated electron physics. [1][2][3] The origin of such heavy-electron state has been understood on the basis of quantum critical phenomena, [10][11][12][13][14][15][16] emergi... | [
{
"id": "",
"caption": "Fig. 2. Curie constant C for n = 7 as a function of λ/U for the case of W = 0. We find the transition from the value for the LS coupling limit to that for the j-j coupling limit in the region pf 0.1 < λ/U < 1, which includes the parameters for actual materials.",
"content": "Fig.... | [
{
"text": "Kondo effect and its related phenomena have been reviewed in. J. Phys. Soc. Jpn. 74Kondo effect and its related phenomena have been reviewed in J. Phys. Soc. Jpn. 74 (2005) 1-238.",
"doi": ""
},
{
"text": "Proc. Int. Conf. Heavy Electrons (ICHE2010). Int. Conf. Heavy Electrons (ICHE2010)8... |
Real eigenvalues of non-Gaussian random matrices and their products
24 Apr 2015
Sajna Hameed hamee007@umn.edu
Kavita Jain
Arul Lakshminarayan
School of Physics and Astronomy
Department of Physics
Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O
University of Minnesot... | 117,501,747 | {
"arxiv": "1504.06256",
"mag": 3098759207,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/1751-8113/48/38/385204"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1504.06256v4.pdf"
],
"pdf_sha": "53d8974ba5cb5dfe80e397559be57304cbda5b6f",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1504.06256",
"status": "GREEN"
}
} | Real eigenvalues of non-Gaussian random matrices and their products | [
"Sajna Hameed hamee007@umn.edu",
"Kavita Jain",
"Arul Lakshminarayan",
"School of Physics and Astronomy\nDepartment of Physics\nTheoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O\nUniversity of Minnesota\n55455, 560064Minneapolis, BangaloreMNU.S.A.., India",
"and... | We study the properties of the eigenvalues of real random matrices and their products. It is known that when the matrix elements are Gaussian-distributed independent random variables, the fraction of real eigenvalues tends to unity as the number of matrices in the product increases. Here we present numerical evidence t... | [
{
"title": "I. INTRODUCTION",
"content": "The problem of the number of real roots of algebraic equations has a long history, but continues to attract attention, from the early works of Littlewood and Offord [1] to more recent developments [2,3] (see the latter article for more related history and references... | [
{
"id": "",
"caption": "dz q(z)p(x)p(y) = 1 − 4 q(z)p(x)p(y).",
"content": "dz q(z)p(x)p(y) = 1 − 4 q(z)p(x)p(y)."
},
{
"id": "",
"caption": "Comparison of probability that all eigenvalues are real for a product of K random matrices with different symmetric distributions and the dimensionali... | [
{
"text": ". J E Littlewood, A C Offord, Proc. Cambridge Philos. Soc. 35133J. E. Littlewood and A. C. Offord, Proc. Cambridge Philos. Soc. 35, 133 (1939).",
"doi": ""
},
{
"text": ". A Edelman, E Kostlan, Bull. American Math. Soc. 321A. Edelman and E. Kostlan, Bull. American Math. Soc. 32, 1 (1995).... |
Testing scalar-tensor gravity with gravitational-wave observations of inspiralling compact binaries
Jun 1994
Clifford M Will
Department of Physics
McDonnell Center for the Space Sciences
Washington University
63130St. LouisMissouri
Testing scalar-tensor gravity with gravitational-wave observations of inspiralling c... | 39,588,218 | {
"arxiv": "gr-qc/9406022",
"mag": 2118237158,
"acl": null,
"pubmed": "10017576",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevd.50.6058"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/gr-qc/9406022v1.pdf"
],
"pdf_sha": "79373394f0e9eb4508b9601b568571e457a910f7",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/gr-qc/9406022",
"status": "GREEN"
}
} | Testing scalar-tensor gravity with gravitational-wave observations of inspiralling compact binaries | [
"Clifford M Will \nDepartment of Physics\nMcDonnell Center for the Space Sciences\nWashington University\n63130St. LouisMissouri"
] | Observations of gravitational waves from inspiralling compact binaries using laser-interferometric detectors can provide accurate measures of parameters of the source. They can also constrain alternative gravitation theories. We analyse inspiralling compact binaries in the context of the scalar-tensor theory of Jordan,... | [
{
"title": "I. INTRODUCTION AND SUMMARY",
"content": "The regular detection of gravitational radiation from astrophysical sources by large-scale laser-interferometer systems as the US LIGO or the French-Italian VIRGO projects will usher in a new era of gravitational-wave astronomy [1]. One of the most promi... | [
{
"id": "",
"caption": ") into the forṁ u = M −1 (96/5)u 11/3 (1 + bη 2/5 u −2/3 ) .",
"content": "(3. 1\n1) into the forṁ u = M −1 (96/5)u 11/3 (1 + bη 2/5 u −2/3 ) ."
},
{
"id": "",
"caption": "t c is the time at which u → ∞. We have expanded the expression (3.4) to first order in bη 2/5 u... | [
{
"text": ". A Abramovici, W E Althouse, R W P Drever, Y Gürsel, S Kawamura, F J Raab, D Shoemaker, L Sievers, R E Spero, K S Thorne, R E Vogt, R Weiss, S E Whitcomb, M E Zucker, Science. 256325A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Siev- ers, R. E. ... |
spectrum from a Discrete Scale Invariance
10 May 2014
A Kievsky
Istituto Nazionale di Fisica Nucleare
Largo Pontecorvo 356100PisaItaly
N K Timofeyuk
Department of Physics
University of Surrey
GU2 7XHGuildfordSurreyUnited Kingdom
M Gattobigio
Institut Non-Linéaire de Nice
Université de Nice-Sophia Antipolis
CNRS
... | 2,921,090 | {
"arxiv": "1405.2371",
"mag": 2029849925,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physreva.90.032504"
} | {
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"https://arxiv.org/pdf/1405.2371v1.pdf"
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"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1405.2371",
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} | spectrum from a Discrete Scale Invariance | [
"A Kievsky \nIstituto Nazionale di Fisica Nucleare\nLargo Pontecorvo 356100PisaItaly",
"N K Timofeyuk \nDepartment of Physics\nUniversity of Surrey\nGU2 7XHGuildfordSurreyUnited Kingdom",
"M Gattobigio \nInstitut Non-Linéaire de Nice\nUniversité de Nice-Sophia Antipolis\nCNRS\n1361, 06560route des Lucioles, Val... | We present the analysis of the N -boson spectrum computed using a soft two-body potential the strength of which has been varied in order to cover an extended range of positive and negative values of the two-body scattering length a close to the unitary limit. The spectrum shows a tree structure of two states, one shall... | [
{
"title": "I. INTRODUCTION.",
"content": "Physical systems present universal behavior when specific details of the interaction between their constituents are suppressed in favor of a few control parameters which determine the dynamics. Well-known examples of this kind are critical phenomena in which the sy... | [
{
"id": "",
"caption": "which, as in the N = 3 case, can be fitted by straight lines shown as solid lines in this figure. Panels (a) and (b) show the results for the ground state (m = 0) and the first excited state (m = 1) respectively. Panels (c) and (d) zoom the area around the threshold y(−π) = −1.56 (sh... | [
{
"text": ". E Braaten, H Hammer, Phys. Rep. 428259E. Braaten and H. Hammer, Phys. Rep. 428, 259 (2006)",
"doi": ""
},
{
"text": ". V Efimov, Phys. Lett. B. 33563V. Efimov, Phys. Lett. B 33, 563 (1970)",
"doi": ""
},
{
"text": ". V Efimov, Sov J , Nucl.Phys. 121080Yad.Fiz.V. Efimov, Sov.... |
Brane Surgery with Thom Classes
May 1999 May, 1999
G Papadopoulos
University of Cambridge
CB3 9EWCambridge
⋆ Damtp
University of Cambridge
CB3 9EWCambridge
Silver Street
University of Cambridge
CB3 9EWCambridge
Brane Surgery with Thom Classes
May 1999 May, 1999arXiv:hep-th/9905073v1 11 ⋆ Address from 1st June 1... | 14,464,182 | {
"arxiv": "hep-th/9905073",
"mag": 2027794049,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/1126-6708/1999/05/020"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/hep-th/9905073v1.pdf"
],
"pdf_sha": "e5096c6578062113b8cef951e648cf1b60b52551",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/hep-th/9905073",
"status": "GREEN"
}
} | Brane Surgery with Thom Classes | [
"G Papadopoulos \nUniversity of Cambridge\nCB3 9EWCambridge",
"⋆ Damtp \nUniversity of Cambridge\nCB3 9EWCambridge",
"Silver Street \nUniversity of Cambridge\nCB3 9EWCambridge"
] | We propose a method to investigate the conservation of brane charges at the intersection of two or more branes using the Thom classes of their normal bundles.In particular we find a relation between the charge of the branes involved in the configuration and the charge of the defects on the branes due to the intersectio... | [
{
"title": "Introduction",
"content": "One of the most remarkable properties of branes is that they can end or intersect with others [1,2,3,4]. The arguments that lead us to believe that this is the case are either based on brane charge conservation [2,5] or the analysis of the couplings of brane effective ... | [
{
"id": "",
"caption": "the transverse spatial infinity, N (p) = R (1,p) × R d−p−1 . The form dF can be thought of as a (d − p − 1)-form on N (p) which has support at the zero section of N (p) . In addition, the computation of charge of the p-brane above can be thought of as the integration of dF along a fi... | [
{
"text": "New Connections between String Theories. J Dai, R G Leigh, J Polchinski, Mod. Phys. Lett. 42073J. Dai, R.G. Leigh, and J. Polchinski, New Connections between String Theories, Mod. Phys. Lett. A4 (1989) 2073.",
"doi": ""
},
{
"text": "Open p-branes. A Strominger, hep-th/9512059Phys. Lett. ... |
Sound velocity and dimensional crossover in a superfluid Fermi gas in an optical lattice
27 Mar 2006
T Koponen
Department of Physics
Nanoscience Center
University of Jyväskylä
P.O. Box 35FI-40014Finland
J.-P Martikainen
Department of Physics
Nanoscience Center
University of Jyväskylä
P.O. Box 35FI-40014Finland
De... | 119,370,797 | {
"arxiv": "cond-mat/0512010",
"mag": 2001638356,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physreva.73.033620"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/cond-mat/0512010v3.pdf"
],
"pdf_sha": "a7571235e18a39c6afb8b5db5b4395160e6455b7",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/cond-mat/0512010",
"status": "GREEN"
}
} | Sound velocity and dimensional crossover in a superfluid Fermi gas in an optical lattice | [
"T Koponen \nDepartment of Physics\nNanoscience Center\nUniversity of Jyväskylä\nP.O. Box 35FI-40014Finland",
"J.-P Martikainen \nDepartment of Physics\nNanoscience Center\nUniversity of Jyväskylä\nP.O. Box 35FI-40014Finland\n\nDepartment of Physical Sciences\nUniversity of Helsinki\nP.O. Box 64FI-00014Finland",
... | We study the sound velocity in cubic and non-cubic three-dimensional optical lattices. We show how the van Hove singularity of the free Fermi gas is smoothened by interactions and eventually vanishes when interactions are strong enough. For non-cubic lattices, we show that the speed of sound (Bogoliubov-Anderson phonon... | [
{
"title": "II. BCS-THEORY",
"content": "The single-band Hubbard Hamiltonian in the context of ultracold gases can be derived from the full microscopic Hamiltonian [45]. To achieve this, one must expand the field operators of the microscopic Hamiltonian in the basis of the well localized Wannier functions, ... | [
{
"id": "",
"caption": "elements can be denoted as a = L 1111 = L 2222 b = L 1212 = −L 1221 = −L 2112 = L 2121 c = L 1112 = −L 1121 = L 1222 = −L 2122 c = L 1211 = −L 2111 = L 2212 = −L 2221 d = L 1122 d = L 2211 .",
"content": "elements can be denoted as a = L 1111 = L 2222 b = L 1212 = −L 1221 = −L 21... | [
{
"text": ". K B Davis, M.-O Mewes, M R Andrews, N J Van Druten, D S Durfee, D M Kurn, W Ketterle, Phys. Rev. Lett. 753969K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ket- terle, Phys. Rev. Lett. 75, 3969 (1995).",
"doi": ""
},
{
"text": ". C C Bradley,... |
Perturbative Calculation of the Adiabatic Geometric Phase and Particle in a Well with Moving Walls
25 Oct 1999
Ali Mostafazadeh
Department of Mathematics
Koç University
80860Istinye, IstanbulTURKEY
Perturbative Calculation of the Adiabatic Geometric Phase and Particle in a Well with Moving Walls
25 Oct 1999
We use ... | 17,025,222 | {
"arxiv": "quant-ph/9910104",
"mag": 2059950988,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/0305-4470/32/47/311"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/quant-ph/9910104v1.pdf"
],
"pdf_sha": "c7b597d33c86e8f28192cb4ca3c04055739b8655",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/quant-ph/9910104",
"status": "GREEN"
}
} | Perturbative Calculation of the Adiabatic Geometric Phase and Particle in a Well with Moving Walls | [
"Ali Mostafazadeh \nDepartment of Mathematics\nKoç University\n80860Istinye, IstanbulTURKEY"
] | We use the Rayleigh-Schrödinger perturbation theory to calculate the corrections to the adiabatic geometric phase due to a perturbation of the Hamiltonian. We show that these corrections are at least of second order in the perturbation parameter. As an application of our general results we address the problem of the ad... | [
{
"title": "Introduction",
"content": "In Ref. [1], Pereshogin and Pronin consider the problem of the calculation of the adiabatic geometric phase [2] for a free particle confined between moving walls. The quantum dynamics of this system has been studied by Doescher and Rice [3], Munier et al [4], Berry and... | [
{
"id": "",
"caption": "and (73), therefore, maps the dynamics of the system with a time-dependent configuration space, i.e., [0, L(t)], to a system with a constant configuration space, i.e., [0, L 0 ]. The idea of transforming the problem with moving boundaries",
"content": "and (73), therefore, maps t... | [
{
"text": ". P Pereshogin, P Pronin, Phys. Lett. A. 15612P. Pereshogin and P. Pronin, Phys. Lett. A 156, 12 (1991).",
"doi": ""
},
{
"text": ". M V Berry, Proc. Roy. Soc. London A. 39245M. V. Berry, Proc. Roy. Soc. London A 392, 45 (1984).",
"doi": ""
},
{
"text": ". S W Doescher, M H Ri... |
Randomization Inference for Cluster-Randomized Test-Negative Designs with Application to Dengue Studies Unbiased estimation, Partial compliance, and Stepped-wedge design
7 Feb 2022
Bingkai Wang
Department of Statistics and Data Science
the Wharton School
University of Pennsylvania
PAU.S.A
Suzanne M Dufault
Divisio... | 246,634,857 | {
"arxiv": "2202.03379",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1214/22-aoas1684"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2202.03379v1.pdf"
],
"pdf_sha": "880c1abf0aa36a137c6cd7099c420501502a9df3",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2202.03379",
"status": "GREEN"
}
} | Randomization Inference for Cluster-Randomized Test-Negative Designs with Application to Dengue Studies Unbiased estimation, Partial compliance, and Stepped-wedge design | [
"Bingkai Wang \nDepartment of Statistics and Data Science\nthe Wharton School\nUniversity of Pennsylvania\nPAU.S.A",
"Suzanne M Dufault \nDivision of Pulmonary and Critical Care Medicine\nUniversity of California\nSan FranciscoCAU.S.A",
"Dylan S Small \nDepartment of Statistics and Data Science\nthe Wharton Sch... | In 2019, the World Health Organization identified dengue as one of the top ten global health threats.For the control of dengue, the Applying Wolbachia to Eliminate Dengue (AWED) study group conducted a cluster-randomized trial in Yogyakarta, Indonesia, and used a novel design, called the cluster-randomized test-negativ... | [
{
"title": "Cluster-randomized studies with a TND",
"content": "In cluster-randomized trials where data are passively collected, a CR-TND borrows methods from a standard TND to yield valid inferences.A TND is a modification of a case-control study popularly used for studying vaccine effectiveness in observa... | [
{
"id": "",
"caption": "Jewell et al. (2019) proposed a test-positive fraction estimator for λ, based on the following statistic:",
"content": "Jewell et al. (2019) proposed a test-positive fraction estimator for λ, based on the following statistic:"
},
{
"id": "",
"caption": "The unbiased l... | [
{
"text": "Cluster-randomized test-negative design trials: a novel and efficient method to assess the efficacy of community-level dengue interventions. K L Anders, Z Cutcher, I Kleinschmidt, C A Donnelly, N M Ferguson, C Indriani, P A Ryan, S L O'neill, N P Jewell, C P Simmons, American Journal of Epidemiology.... |
Optimal Voltage for Nanoparticle Detection with Thin Nanopores
Yinghua Qiu y.qiu@neu.edu
Department of Physics
Northeastern University
02115BostonMA
Optimal Voltage for Nanoparticle Detection with Thin Nanopores
508D402B9C996A5DFB728239498EBA93
The resistive-pulse technique provides a fast and label-free method fo... | 52,135,111 | {
"arxiv": "1808.09498",
"mag": 3098849621,
"acl": null,
"pubmed": "30167601",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1039/c8an01270a"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1808.09498v2.pdf"
],
"pdf_sha": "03de42f2565a38d5fda974a2d24078a9c5437945",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/1808.09498",
"status": "GREEN"
}
} | Optimal Voltage for Nanoparticle Detection with Thin Nanopores | [
"Yinghua Qiu y.qiu@neu.edu \nDepartment of Physics\nNortheastern University\n02115BostonMA"
] | The resistive-pulse technique provides a fast and label-free method for nanoparticle detection.In order to achieve a higher sensitivity, thin nanopores, such as silicon nitride pores, are usually considered.In this paper, nanoparticle detection has been mimicked with simulations.We found the surface charges of the part... | [
{
"title": "TOC Introduction:",
"content": "Fast and label-free nanoparticle detection has become of great importance due to its application in virus sensing, disease diagnosis, and drug delivery. 1,2 esistive-pulse technique originated from Coulter counter 3 provides a perfect candidate for nanoparticle de... | [
{
"id": "",
"caption": "Figure 1 Scheme of the simulation.Zoomed-in pore region is shown on the right.Yellow part shows the particle.",
"content": "Figure 1\n1\nFigure 1 Scheme of the simulation.Zoomed-in pore region is shown on the right.Yellow part shows the particle."
},
{
"id": "",
"capt... | [
{
"text": ". L Luo, S R German, W.-J Lan, D A Holden, T L Mega, H S White, Annu. Rev. Anal. Chem. 72014",
"doi": ""
},
{
"text": ". H Bayley, C R Martin, Chem. Rev. 1002000",
"doi": ""
},
{
"text": ". U S Pat, 1953656508",
"doi": ""
},
{
"text": ". J J Kasianowicz, E Brandin,... |
A proposal of adaptive parameter tuning for robust stabilizing control of N -level quantum angular momentum systems
Shoju Enami
Department of Applied Mathematics and Physics
Graduate School of Informatics
Kyoto University
Sakyo-ku606-8501Yoshida-Honmachi, KyotoJapan
Kentaro Ohki
Department of Applied Mathematics ... | 238,253,010 | {
"arxiv": "2110.00172",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/cdc/EnamiO21",
"doi": "10.1109/cdc45484.2021.9683605"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2110.00172v1.pdf"
],
"pdf_sha": "8d2c00a24b4f0408730a6d1201269f109af3f1d0",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2110.00172",
"status": "GREEN"
}
} | A proposal of adaptive parameter tuning for robust stabilizing control of N -level quantum angular momentum systems | [
"Shoju Enami \nDepartment of Applied Mathematics and Physics\nGraduate School of Informatics\nKyoto University\nSakyo-ku606-8501Yoshida-Honmachi, KyotoJapan",
"Kentaro Ohki \nDepartment of Applied Mathematics and Physics\nGraduate School of Informatics\nKyoto University\nSakyo-ku606-8501Yoshida-Honmachi, KyotoJap... | Stabilizing control synthesis is one of the central subjects in control theory and engineering, and it always has to deal with unavoidable uncertainties in practice.In this study, we propose an adaptive parameter tuning algorithm for robust stabilizing quantum feedback control of N -level quantum angular momentum syste... | [
{
"title": "I. INTRODUCTION",
"content": "Stabilizing controller synthesis is one of the central problems in control systems, even if systems are described by quantum mechanics [1]- [11].Unfortunately, stabilizing control is vulnerable to failure due to the existence of uncertainties in practice.There are t... | [
{
"id": "",
"caption": "Fig. 1.The proposed adaptive controller",
"content": "Fig. 1 .\n1\nFig. 1.The proposed adaptive controller"
},
{
"id": "",
"caption": "represents the distance d(t) = d B ((ρ(t), ρ(t)), (ρ n, ρ n)) [21], d B ((ρ, ρ), (ρ n , ρ m )) = 2 − 2 Tr[ρρ n ] + 2 − 2 Tr[ρρ m ].",... | [
{
"text": "Feedback control of quantum state reduction. R Van Handel, J K Stockton, H Mabuchi, IEEE Transactions on Automatic Control. 5062005",
"doi": ""
},
{
"text": "Stabilizing Feedback Controls for quantum systems. M Mirrahimi, R Van Handel, SIAM Journal on Control and Optimization. 4622007",
... |
Performance of Graph Database Management Systems as route planning solutions for different data and usage characteristics
12 Jun 2023
Karin Festl karin.festl@v2c2.at
Virtual Vehicle Research GmbH
GrazAustria
Patrick Promitzer
Virtual Vehicle Research GmbH
GrazAustria
Daniel Watzenig
Virtual Vehicle Research GmbH... | 259,138,273 | {
"arxiv": "2306.07084",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2306-07084",
"doi": "10.1109/iavvc57316.2023.10328126"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2306.07084v1.pdf"
],
"pdf_sha": "2f2dcd17b3ea4a5393bed6033a90ae04580ff150",
"open_access": {
"license": null,
"url": "https://arxiv.org/pdf/2306.07084",
"status": "GREEN"
}
} | Performance of Graph Database Management Systems as route planning solutions for different data and usage characteristics | [
"Karin Festl karin.festl@v2c2.at \nVirtual Vehicle Research GmbH\nGrazAustria",
"Patrick Promitzer \nVirtual Vehicle Research GmbH\nGrazAustria",
"Daniel Watzenig \nVirtual Vehicle Research GmbH\nGrazAustria",
"Huilin Yin \nVirtual Vehicle Research GmbH\nGrazAustria"
] | Graph databases have grown in popularity in recent years as they are able to efficiently store and query complex relationships between data.Incidentally, navigation data and road networks can be processed, sampled or modified efficiently when stored as a graph.As a result, graph databases are a solution for solving rou... | [
{
"title": "I. INTRODUCTION",
"content": "With the increasing amount of data that is availalbe and processed in different areas of applications, Graph Database Management systems (GDBMSs) have become more popular than ever.More and more database vendors developed GDBMSs, among others also Oracle Property Gr... | [
{
"id": "",
"caption": "Fig. 1: Road network of the city Amberg.In red is an exemplary shortest route from one position to another.",
"content": "Fig. 1 :\n1\nFig. 1: Road network of the city Amberg.In red is an exemplary shortest route from one position to another."
},
{
"id": "",
"caption"... | [
{
"text": "Oracle Property Graph.",
"doi": ""
},
{
"text": "A Centralized Route-Management Solution for Autonomous Vehicles in Urban Areas. J L Zambrano-Martinez, C T Calafate, D Soler, L G Lemus-Zúñiga, J C Cano, P Manzoni, T Gayraud, Electronics. 877222019. jun 2019",
"doi": ""
},
{
"t... |
EFFECTS OF SHORT RANGE CORRELATIONS ON Ca ISOTOPES
arXiv:nucl-th/9601014v1 10 Jan 1996
G A Lalazissis
Physik Department
Technische Universität München
D 85747GarchingGermany
S E Massen
Department of Physics
Nuclear Physics Laboratory
University of Oxford
OX1-3RHOxfordUnited Kingdom
Department of Theoretical Physi... | 37,780,537 | {
"arxiv": "nucl-th/9601014",
"mag": 2023527516,
"acl": null,
"pubmed": "9971108",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevc.53.1599"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/nucl-th/9601014v1.pdf"
],
"pdf_sha": "89d66aa4cf221c7cbdf808daae1d1ddd0d05a63f",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/nucl-th/9601014",
"status": "GREEN"
}
} | EFFECTS OF SHORT RANGE CORRELATIONS ON Ca ISOTOPES | [
"G A Lalazissis \nPhysik Department\nTechnische Universität München\nD 85747GarchingGermany",
"S E Massen \nDepartment of Physics\nNuclear Physics Laboratory\nUniversity of Oxford\nOX1-3RHOxfordUnited Kingdom\n\nDepartment of Theoretical Physics\nAristotle University of Thessaloniki\n54006ThessalonikiGRGreece"
] | The effect of Short Range Correlations (SRC) on Ca isotopes is studied using a simple phenomenological model. Theoretical expressions for the charge (proton) form factors, densities and moments of Ca nuclei are derived. The role of SRC in reproducing the empirical data for the charge density differences is examined. Th... | [
{
"title": "I. INTRODUCTION",
"content": "Calcium nuclei have been of great experimental as well as theoretical interest. It is the only magic element for which precision measurements on isotope shifts [1,2] (see also [3]) have been carried out over a full neutron shell, namely the 1f 7/2 shell between the ... | [
{
"id": "",
"caption": "The isotopic change δµ of the correlation parameter µ as function of the mass number A.",
"content": "Fig. 1 .\n1The isotopic change δµ of the correlation parameter µ as function of the mass number A."
},
{
"id": "",
"caption": "The difference of the charge distributi... | [
{
"text": "E W Otten, Treatise on Heavy-Ion Science. D.A. BromleyNew YorkPlenum7515E.W. Otten, in Treatise on Heavy-Ion Science, edited by D.A. Bromley (Plenum, New York, 1989) Vol 7, p. 515",
"doi": ""
},
{
"text": ". J Billowes, P Campbell, J. Phys. G: Nucl. Part. Phys. 21707J. Billowes and P. Cam... |
Fluctuations and symmetry in the speed and direction of the jets of SS 433 on different timescales
February 5, 2008
Katherine M Blundell
Department of Physics
University of Oxford
Keble RoadOX1 3RHOxfordUK
Michael G Bowler
Department of Physics
University of Oxford
Keble RoadOX1 3RHOxfordUK
Linda Schmidtobreick
... | 18,787,362 | {
"arxiv": "0708.2930",
"mag": 3102730500,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1051/0004-6361:20077924"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0708.2930v1.pdf"
],
"pdf_sha": "2bce2c32b30f943851316b1763124e529549bfe8",
"open_access": {
"license": null,
"url": "https://www.aanda.org/articles/aa/pdf/2007/42/aa7924-07.pdf",
"status": "BRONZE"
}
} | Fluctuations and symmetry in the speed and direction of the jets of SS 433 on different timescales | [
"Katherine M Blundell \nDepartment of Physics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK",
"Michael G Bowler \nDepartment of Physics\nUniversity of Oxford\nKeble RoadOX1 3RHOxfordUK",
"Linda Schmidtobreick \nEuropean Southern Observatory\nVitacura\n\nAlonso de Cordova\nSantiagoChile"
] | Context. The Galactic microquasar SS 433 launches oppositely-directed plasma jets at speeds approximately a quarter of the speed of light along an axis which precesses, tracing out a cone of polar angle 20 degrees. Both the speed and direction of launch of the bolides that comprise the jets exhibit small fluctuations. ... | [
{
"title": "Introduction",
"content": "Object number 433 in the catalogue of Stephenson & Sanduleak (1977) was the first Galactic microquasar to be discovered (Margon, 1984). It is unusual in that it ejects, more or less continually, oppositely directed jets of gas (radiating particularly strongly in Balmer... | [
{
"id": "",
"caption": "Optical spectrum of SS 433 on Julian day +260.5. The upper panel shows the spectrum after wavelength calibration and removal of the atmospheric telluric features and, in addition to the \"moving\" Balmer Hα lines, shows the \"stationary\" Hα line, as well as HeI and CII lines. The lo... | [
{
"text": ". G O Abell, B Margon, Nature. 279701Abell, G. O., & Margon, B. 1979, Nature, 279, 701",
"doi": ""
},
{
"text": ". M C Begelman, A R King, J E Pringle, MNRAS. 370399Begelman, M. C., King, A. R., & Pringle, J. E. 2006, MNRAS, 370, 399",
"doi": ""
},
{
"text": ". K M Blundell, M... |
STOCHASTIC TURBULENT ACCELERATION IN A FRACTAL ENVIRONMENT
May 7, 2020
Nikos Sioulas
Department of Physics
Aristotle University of Thessaloniki
GR-52124ThessalonikiGreece
Heinz Isliker
Department of Physics
Aristotle University of Thessaloniki
GR-52124ThessalonikiGreece
Loukas Vlahos
Department of Physics
Aristo... | 218,517,010 | {
"arxiv": "2005.02668",
"mag": 3098162999,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.3847/2041-8213/ab9092"
} | {
"pdf_urls": [],
"pdf_sha": "30c9a5f8453ae1f1b98a2533afd53c5505e5ea44",
"open_access": {
"license": null,
"url": "https://iopscience.iop.org/article/10.3847/2041-8213/ab9092/pdf",
"status": "BRONZE"
}
} | STOCHASTIC TURBULENT ACCELERATION IN A FRACTAL ENVIRONMENT | [
"Nikos Sioulas \nDepartment of Physics\nAristotle University of Thessaloniki\nGR-52124ThessalonikiGreece",
"Heinz Isliker \nDepartment of Physics\nAristotle University of Thessaloniki\nGR-52124ThessalonikiGreece",
"Loukas Vlahos \nDepartment of Physics\nAristotle University of Thessaloniki\nGR-52124Thessaloniki... | We analyze the stochastic acceleration of particles inside a fully developed turbulent plasma. It is well known that large-amplitude magnetic fluctuations and coherent structures in such an environment obey a fractal scaling, and our specific aim is to study for the first time the effects of the fractality of these env... | [
{
"title": "INTRODUCTION",
"content": "Magnetic reconnection, weak turbulence, and shock waves surrounded by passive scattering centers upstream and downstream were for years the prominent acceleration mechanisms in most astrophysical and laboratory plasmas (Melrose 2009(Melrose , 1994. Recent magnetohydrod... | [
{
"id": "",
"caption": "Illustration of the random walk through a fractal environment: Part of the fractal, with its constituent elementary volumes in blue color, and the orbit of a particle in red color, moving along straight paths and occasionally scattering off elementary volumes of the fractal.",
"c... | [
{
"text": ". K Arzner, B Knaepen, D Carati, N Denewet, L Vlahos, Astrophys. J. 637322Arzner, K., Knaepen, B., Carati, D., Denewet, N., & Vlahos, L. 2006, Astrophys. J., 637, 322",
"doi": ""
},
{
"text": ". K Arzner, L Vlahos, Astrophys. J. Lett. 60569Arzner, K., & Vlahos, L. 2004, Astrophys. J. Lett... |
A SEARCH FOR DIFFUSE NEUTRAL HYDROGEN AND H i CLOUDS IN THE NGC 2403 GROUP
6 May 2009
Katie M Chynoweth
Glen I Langston
Kelly Holley-Bockelmann
Felix J Lockman
Physics and Astronomy Department
Vanderbilt University
1807 Station B37235NashvilleTN
National Radio Astronomy Observatory
24944Green BankWV
National... | 16,732,414 | {
"arxiv": "0905.0906",
"mag": 2127052401,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/0004-6256/138/1/287"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0905.0906v1.pdf"
],
"pdf_sha": "f67673182bbf8383543b2b887f24c3a7bf9c9d25",
"open_access": {
"license": null,
"url": "https://iopscience.iop.org/article/10.1088/0004-6256/138/1/287/pdf",
"status": "BRONZE"
}
} | A SEARCH FOR DIFFUSE NEUTRAL HYDROGEN AND H i CLOUDS IN THE NGC 2403 GROUP | [
"Katie M Chynoweth",
"Glen I Langston",
"Kelly Holley-Bockelmann",
"Felix J Lockman",
"Physics and Astronomy Department\nVanderbilt University\n1807 Station B37235NashvilleTN",
"National Radio Astronomy Observatory\n24944Green BankWV",
"National Radio Astronomy Observatory\n24944Green BankWV",
"Physic... | We have observed the NGC 2403 group of galaxies using the Robert C. Byrd Green Bank Telescope (GBT) in a search for faint, extended neutral hydrogen clouds similar to the clouds found around the M81/M82 group, which is located approximately 250 kpc from the NGC 2403 group along the same filament of galaxies.For an H i ... | [
{
"title": "INTRODUCTION",
"content": "The neutral hydrogen in the environment of galaxies and galaxy groups is morphologically and kinematically complex.Surrounding the Milky Way are the High Velocity Clouds (HVCs), which are observed in both ionized and neutral states in large complexes including the Mage... | [
{
"id": "",
"caption": "Fig. 1.-Masses of known H i clouds associated with nearby galaxies.Our mass detection threshold is indicated with a dotted line.If this population of clouds is representative of the NGC 2403 group, our detection limit allows us to detect the most massive 23% of H i clouds in the grou... | [
{
"text": ". A Begum, J N Chengalur, I D Karachentsev, S S Kaisin, M E Sharina, arXiv:astro-ph/0511253MNRAS. 36512202006",
"doi": ""
},
{
"text": ". Ben Bekhti, N Brüns, C Kerp, J Westmeier, T , arXiv:astro-ph/0607383A&A. 4579172006",
"doi": ""
},
{
"text": ". R Braun, W B Burton, arXiv:... |
The formation and evolution of planetary systems: Grain growth and chemical processing of dust in T Tauri systems
21 Feb 2008
J Bouwman
Th Henning
L A Hillenbrand
M R Meyer
I Pascucci
J Carpenter
D Hines
J S Kim
M D Silverstone
D Hollenbach
S Wolf
Max Planck Institute for Astronomy
Königstuhl 17D-69117Hei... | 7,806,001 | {
"arxiv": "0802.3033",
"mag": 3101545078,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1086/587793"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0802.3033v1.pdf"
],
"pdf_sha": "46b50c55da42acac9cb63e22f9960d0bdd69e866",
"open_access": {
"license": null,
"url": "http://iopscience.iop.org/article/10.1086/587793/pdf",
"status": "BRONZE"
}
} | The formation and evolution of planetary systems: Grain growth and chemical processing of dust in T Tauri systems | [
"J Bouwman",
"Th Henning",
"L A Hillenbrand",
"M R Meyer",
"I Pascucci",
"J Carpenter",
"D Hines",
"J S Kim",
"M D Silverstone",
"D Hollenbach",
"S Wolf",
"Max Planck Institute for Astronomy\nKönigstuhl 17D-69117HeidelbergGermany",
"Department of Astronomy\nCalifornia Institute of Technology... | This paper is one in a series presenting results obtained within the Formation and Evolution of Planetary Systems (FEPS) Legacy Science Program on the Spitzer Space Telescope.Here we present a study of dust processing and growth in seven protoplanetary disks.Our spectra indicate that the circumstellar silicate dust gra... | [
{
"title": "Introduction",
"content": "The circumstellar disks surrounding the pre-main-sequence Brown Dwarfs (BD), T Tauri stars (TTS) and Herbig Ae/Be (HAEBE) systems are believed to be the sites of ongoing planet formation (e.g.Luhman et al. 2007;Natta et al. 2007, from here on we will refer to these dis... | [
{
"id": "",
"caption": "results in τ = 0.81 with P = 0.011 for the 11.3 µm versus 10.1 µm band strength, τ = 0.90 with P = 0.004 for the 33.5 µm versus 27.8 µm band strength, and τ = 0.24 with P = 0.45 for the 23.4 µm versus 27.8 µm band strength.The apparent lack of correlation between the 23.4 µm and 27.8... | [
{
"text": ". B Acke, M E Van Den Ancker, A&A. 4261512004",
"doi": ""
},
{
"text": ". B Acke, M E Van Den Ancker, C P Dullemond, R Van Boekel, L B F M Waters, A&A. 4226212004",
"doi": ""
},
{
"text": ". D Apai, I Pascucci, M F Sterzik, A&A. 426L532004",
"doi": ""
},
{
"text": ... |
Bundled References: An Abstraction for Highly-Concurrent Linearizable Range Queries
Jacob Nelson
Computer Science and Engineering
Computer Science and Engineering
Lehigh University
USA
Ahmed Hassan
Computer Science and Engineering
Lehigh University
USA
Roberto Palmieri palmieri@lehigh.edu
Lehigh University
USA
... | 229,924,097 | {
"arxiv": "2012.15438",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/ppopp/NelsonHP21",
"doi": "10.1145/3437801.3441614"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2012.15438v1.pdf"
],
"pdf_sha": "fa21388d661004f322210547b5b2a5464eca5d00",
"open_access": {
"license": null,
"url": null,
"status": "CLOSED"
}
} | Bundled References: An Abstraction for Highly-Concurrent Linearizable Range Queries | [
"Jacob Nelson \nComputer Science and Engineering\nComputer Science and Engineering\nLehigh University\nUSA",
"Ahmed Hassan \nComputer Science and Engineering\nLehigh University\nUSA",
"Roberto Palmieri palmieri@lehigh.edu \nLehigh University\nUSA"
] | We present bundled references, a new building block to provide linearizable range query operations for highly concurrent linked data structures. Bundled references allow range queries to traverse a path through the data structure that is consistent with the target atomic snapshot and is made of the minimal amount of no... | [
{
"title": "Introduction",
"content": "Iterating over a collection of elements to return those that fall within a contiguous range (also known as a range query operation) is without contest an essential feature for data repositories. In addition to database management systems, which historically deploy supp... | [
{
"id": "",
"caption": "An example of using bundled references in a linked list. The path made of solid lines represents the state of the linked list after all update operations take place. Edges are labeled with their respective timestamps.",
"content": "Figure 1 .\n1An example of using bundled referen... | [
{
"text": "StackTrack: an automated transactional approach to concurrent memory reclamation. Dan Alistarh, Patrick Eugster, Maurice Herlihy, Alexander Matveev, Nir Shavit, 10.1145/2592798.2592808Ninth Eurosys Conference. Amsterdam, The NetherlandsACM25Dan Alistarh, Patrick Eugster, Maurice Herlihy, Alexander Ma... |
Investigation of phonons and magnons in [Ni80Fe20/Au/Co/Au]N multilayers
M Zdunek
Faculty of Physics
Adam Mickiewicz University in Poznań
PoznańPoland
S Shekhar
Faculty of Physics
Adam Mickiewicz University in Poznań
PoznańPoland
S Mielcarek
Faculty of Physics
Adam Mickiewicz University in Poznań
PoznańPoland
... | 269,292,977 | {
"arxiv": "2404.13052",
"mag": null,
"acl": null,
"pubmed": "38838700v1",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/1361-648x/ad5486"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2404.13052v1.pdf"
],
"pdf_sha": "7b3b6fd6971828aa6ce6a7610fadecd4098a1f0e",
"open_access": {
"license": "CCBY",
"url": "https://iopscience.iop.org/article/10.1088/1361-648X/ad5486/pdf",
"status": "HYBRID"
}
} | Investigation of phonons and magnons in [Ni80Fe20/Au/Co/Au]N multilayers | [
"M Zdunek \nFaculty of Physics\nAdam Mickiewicz University in Poznań\nPoznańPoland",
"S Shekhar \nFaculty of Physics\nAdam Mickiewicz University in Poznań\nPoznańPoland",
"S Mielcarek \nFaculty of Physics\nAdam Mickiewicz University in Poznań\nPoznańPoland",
"A Trzaskowska \nFaculty of Physics\nAdam Mickiewic... | The interaction between phonons and magnons is a widely developing topic, especially in the field of acoustic spintronics.To discuss this interaction, it is necessary to observe two different waves (acoustic and spin waves) with the same frequency and wavelength.In the Ni80Fe20/Au/Co/Au system deposited on a silicon su... | [
{
"title": "Introduction",
"content": "In conventional electronic devices, charge current is used for transmitting and processing information.To meet the increasing demand for smaller, more intelligent, and energy-efficient microwave devices, an alternative technology is urgently needed that goes beyond the... | [
{
"id": "",
"caption": "Fig. 1.(a) Schematic sketch of the considered system, composed of thick silicon (100) substrate with the gold buffer layer (30 nm) and magnetic multilayer.(b) Schematic representation of backward volume (BV) geometry.The magnetic field H (blue arrow) is applied in the plane of the sa... | [
{
"text": "Kagome magnets: The emerging materials for spintronic memories. N Chowdhury, 10.1007/s40010-023-00823-1Proceedings of the National Academy of Sciences. 933Sep. 2023India Section A: Physical Sciences",
"doi": ""
},
{
"text": "Magnon spintronics. A V Chumak, V I Vasyuchka, A A Serga, B Hill... |
HIGHER HÖLDER REGULARITY FOR NONLOCAL EQUATIONS WITH IRREGULAR KERNEL
5 Jul 2020
Simon Nowak
HIGHER HÖLDER REGULARITY FOR NONLOCAL EQUATIONS WITH IRREGULAR KERNEL
5 Jul 2020
We study the higher Hölder regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a m... | 220,250,359 | {
"arxiv": "2006.15906",
"mag": 3037585793,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/s00526-020-01915-1"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2006.15906v2.pdf"
],
"pdf_sha": "18858d9fd2b8933604d40e2d39b4922747828579",
"open_access": {
"license": "CCBY",
"url": "https://link.springer.com/content/pdf/10.1007/s00526-020-01915-1.pdf",
"status": "HYBRID"
}
} | HIGHER HÖLDER REGULARITY FOR NONLOCAL EQUATIONS WITH IRREGULAR KERNEL | [
"Simon Nowak"
] | We study the higher Hölder regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained regularity is better than one might expect when considering corresponding results for l... | [
{
"title": "Definition.",
"content": "Let Ω be a domain and λ ≥ 1. We say that a kernel coefficient A 0 ∈ L 0 (λ) belongs to the class L 1 (λ, Ω), if there exists a measurable function a : R n → R such that A 0 (x, y) = a(x − y) for all x, y ∈ Ω.\nA kernel coefficient that belongs to the class L 1 (λ, Ω) ca... | [
{
"id": "",
"caption": "Basic setting and main result. In this work, we study the higher Hölder regularity of solutions to nonlinear nonlocal equations of the form(1) L Φ A u = f in Ω ⊂ R ndriven by a kernel that potentially exhibits a very irregular behaviour. More precisely, by modifying an approach intro... | [
{
"text": "Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case. L Brasco, E Lindgren, Adv. Math. 304L. Brasco and E. Lindgren, Higher Sobolev regularity for the fractional p-Laplace equation in the superquadratic case, Adv. Math. 304 (2017), 300-354.",
"doi": ""
},
... |
Lifetimes of Metal Nanowires with Broken Axial Symmetry
8 Oct 2014
Lan Gong
J Bürki
Charles A Stafford
Daniel L Stein
Department of Physics
Department of Physics and Astronomy
New York University
10003New YorkNY
Department of Physics
California State University
95819-6041Sacramento, SacramentoCA
Department o... | 5,088,360 | {
"arxiv": "1410.1949",
"mag": 1992144945,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.91.035401"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1410.1949v1.pdf"
],
"pdf_sha": "8a78fb6d9f679a22b6a71995b7996647f7721d94",
"open_access": {
"license": "publisher-specific, author manuscript",
"url": "https://link.aps.org/accepted/10.1103/PhysRevB.91.035401",
"status": "HYBRID"
}
} | Lifetimes of Metal Nanowires with Broken Axial Symmetry | [
"Lan Gong",
"J Bürki",
"Charles A Stafford",
"Daniel L Stein",
"Department of Physics\nDepartment of Physics and Astronomy\nNew York University\n10003New YorkNY",
"Department of Physics\nCalifornia State University\n95819-6041Sacramento, SacramentoCA",
"Department of Physics\nUniversity of Arizona\n1118... | We present a theoretical approach for understanding the stability of simple metal nanowires, in particular monovalent metals such as the alkalis and noble metals. Their cross sections are of order one nanometer so that small perturbations from external (usually thermal) noise can cause large geometrical deformations. T... | [
{
"title": "I. INTRODUCTION",
"content": "Nanowires made of monovalent metals, such as sodium, copper, and gold, live at the boundary between classical and quantum mechanics and exhibit some of the behavior of each. They are therefore of great interest from both a fundamental physics perspective as well as ... | [
{
"id": "",
"caption": "Contour plot of the electron-shell potential V shell as a function of the Sharvin conductance G s defined by Eq. (1) and the quadrupolar deformation parameter λ 2 .",
"content": "FIG. 1 :\n1Contour plot of the electron-shell potential V shell as a function of the Sharvin conducta... | [
{
"text": "On the stability and structural dynamics of metal nanowires. J Bürki, C A Stafford, Appl. Phys. A. 811519J. Bürki and C. A. Stafford, \"On the stability and structural dynamics of metal nanowires,\" Appl. Phys. A, vol. 81, p. 1519, 2005.",
"doi": ""
},
{
"text": "Statique Expérimentale et... |
Microscopic Theory of Equilibrium Polariton Condensates
Fei Xue
Department of Physics
The University of Texas at Austin
78712AustinTXUSA
Fengcheng Wu
Department of Physics
The University of Texas at Austin
78712AustinTXUSA
Ming Xie
Department of Physics
The University of Texas at Austin
78712AustinTXUSA
Jung-J... | 118,646,545 | {
"arxiv": "1608.06916",
"mag": 2515496519,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.94.235302"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1608.06916v1.pdf"
],
"pdf_sha": "71fea2781d5ad084a3714c592087ff7214fee297",
"open_access": {
"license": "publisher-specific, author manuscript",
"url": "https://link.aps.org/accepted/10.1103/PhysRevB.94.235302",
"status": "HYBRID"
}
} | Microscopic Theory of Equilibrium Polariton Condensates | [
"Fei Xue \nDepartment of Physics\nThe University of Texas at Austin\n78712AustinTXUSA",
"Fengcheng Wu \nDepartment of Physics\nThe University of Texas at Austin\n78712AustinTXUSA",
"Ming Xie \nDepartment of Physics\nThe University of Texas at Austin\n78712AustinTXUSA",
"Jung-Jung Su \nDepartment of Electrophy... | We present a microscopic theory of the equilibrium polariton condensate state of a semiconductor quantum well in a planar optical cavity. The theory accounts for the adjustment of matter excitations to the presence of a coherent photon field, predicts effective polariton-polariton interaction strengths that are weaker ... | [
{
"title": "I. INTRODUCTION",
"content": "A polariton is a quantum state in which a photon is coherently mixed with an elementary excitation of condensed matter, for example an exciton in a semiconductor or a surface plasmon in a metal. Two-dimensional polariton condensate states can be formed 1-12 when sem... | [
{
"id": "",
"caption": "(Color online) (a): Typical polariton condensate geometry. One or several quantum wells are placed between a pair of distributed Bragg reflectors (DBRs). The twodimensional quantum well conduction and valence band states are dressed by their interactions with condensed cavity photons... | [
{
"text": ". F Tassone, Y Yamamoto, 10.1103/PhysRevB.59.10830Phys. Rev. B. 5910830F. Tassone and Y. Yamamoto, Phys. Rev. B 59, 10830 (1999).",
"doi": ""
},
{
"text": ". H Deng, G Weihs, C Santori, J Bloch, Y Yamamoto, 10.1126/science.1074464Science. 298199H. Deng, G. Weihs, C. Santori, J. Bloch, and... |
Solar sail with inflatable toroidal shell
V Ya Kezerashvili
Physics Department
New York City College of Technology
The City University of New York
11201BrooklynNYUSA
R Ya Kezerashvili
The Graduate School and University Center
The City University of New York
10016New YorkNYUSA
O L Starinova
Physics Department
Ne... | 251,468,011 | {
"arxiv": "2208.05255",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.actaastro.2022.09.039"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2208.05255v2.pdf"
],
"pdf_sha": "dd84bdb7a4afd2f2b813ec8d6c4e64516a79ef37",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Solar sail with inflatable toroidal shell | [
"V Ya Kezerashvili \nPhysics Department\nNew York City College of Technology\nThe City University of New York\n11201BrooklynNYUSA",
"R Ya Kezerashvili \nThe Graduate School and University Center\nThe City University of New York\n10016New YorkNYUSA",
"O L Starinova \nPhysics Department\nNew York City College of ... | In the framework of a strict mathematical approach based on classical theory of elasticity we present an idea of the deployment and stretching of the circular solar sail attached to the inflatable toroidal shell. It is predicted that by introducing the gas into the inflatable toroidal shell one can deploy and stretch a... | [
{
"title": "I. INTRODUCTION",
"content": "Currently the main space exploration vehicle relies on chemical propulsion system and most missions provide energy by means of prelaunch onboard fuel. A solar sail is propellant-less propulsion system for space exploration that uses the Sun radiation as a propulsion... | [
{
"id": "",
"caption": "(Color online) (a) Torus-shaped solar sail and its geometric characteristics. (b) R and r are the radii of the torus and the toroidal shell cross-section, respectively. Throughout of this work we assume that the radius of the torus is much greater than the radius of the toroidal shel... | [
{
"text": "Kosmicheskii Polet s Solnechnim Parusom (Space Solar Sailing) (in Russian) Moscow, Nauka. E N Polyakhova, E. N. Polyakhova, Kosmicheskii Polet s Solnechnim Parusom (Space Solar Sailing) (in Russian) Moscow, Nauka, 1986.",
"doi": ""
},
{
"text": "Solar Sailing -Technology, Dynamics and Mis... |
Collaborative Bearing Estimation Using Set Membership Methods
Mohammad Zamani mohammad.zamani@unimelb.edu.au
Jochen Trumpf jochen.trumpf@anu.edu.au
School of Engineering
The Australian National University Canberra
Australia
Chris Manzie manziec@unimelb.edu.au
Department of Electrical and Electronic Engineering
... | 258,179,049 | {
"arxiv": "2304.07996",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/fusion/ZamaniTM23",
"doi": "10.23919/fusion52260.2023.10224173"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2304.07996v1.pdf"
],
"pdf_sha": "1367f4e4fe4415790f406fce574a0fad4ff75108",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Collaborative Bearing Estimation Using Set Membership Methods | [
"Mohammad Zamani mohammad.zamani@unimelb.edu.au",
"Jochen Trumpf jochen.trumpf@anu.edu.au \nSchool of Engineering\nThe Australian National University Canberra\nAustralia",
"Chris Manzie manziec@unimelb.edu.au",
"Department of Electrical and Electronic Engineering\nUniversity of Melbourne\nMelbourneAustralia"
... | We consider the problem of collaborative bearing estimation using a method with historic roots in set theoretic estimation techniques. We refer to this method as the Convex Combination Ellipsoid (CCE) method and show that it provides a less conservative covariance estimate than the well known Covariance Intersection (C... | [
{
"title": "I. INTRODUCTION",
"content": "In this work we revisit the concept of distributed fusion in the context of collaborative networked bearing estimation. Consider a network of estimators each independently estimating the position of a target using information from their onboard bearing sensor and in... | [
{
"id": "",
"caption": "A counterexample where (optimal) CI [13], [16] and (optimal) ICI",
"content": "Fig. 1 .\n1A counterexample where (optimal) CI [13], [16] and (optimal) ICI"
},
{
"id": "",
"caption": "The 2-agent bearing estimation scenario where the initial estimated uncertainty sets ... | [
{
"text": "Localization and circumnavigation of a slowly moving target using bearing measurements. M Deghat, I Shames, B D Anderson, C Yu, IEEE Transactions on Automatic Control. 598M. Deghat, I. Shames, B. D. Anderson, and C. Yu, \"Localization and cir- cumnavigation of a slowly moving target using bearing mea... |
A framework for optimisation based stochastic process discovery ⋆
2 Jul 2024
Pierre Cry pierre.cry@centralesupelec.fr
CentraleSupélec
Université Paris Saclay
Gif-sur-YvetteFrance
András Horváth horvath@di.unito.it
Università di Torino
TorinoItaly
Paolo Ballarini paolo.ballarini@centralesupelec.fr
CentraleSupélec... | 270,560,592 | {
"arxiv": "2406.10817",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/qestformats/CryHBG24",
"doi": "10.48550/arxiv.2406.10817"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2406.10817v3.pdf"
],
"pdf_sha": "5698a2d5b1274ab7ecbc3f0b48565e9a64de35a0",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | A framework for optimisation based stochastic process discovery ⋆ | [
"Pierre Cry pierre.cry@centralesupelec.fr \nCentraleSupélec\nUniversité Paris Saclay\nGif-sur-YvetteFrance",
"András Horváth horvath@di.unito.it \nUniversità di Torino\nTorinoItaly",
"Paolo Ballarini paolo.ballarini@centralesupelec.fr \nCentraleSupélec\nUniversité Paris Saclay\nGif-sur-YvetteFrance",
"Pascale... | Process mining is concerned with deriving formal models capable of reproducing the behaviour of a given organisational process by analysing observed executions collected in an event log.The elements of an event log are finite sequences (i.e., traces or words) of actions.Many effective algorithms have been introduced wh... | [
{
"title": "Introduction",
"content": "A business process consists in a set of activities performed so to accomplish a specific organisational goal.Understanding business processes is of prime importance for organisations in many respects: firstly, to obtain a formal description of often undocumented proces... | [
{
"id": "",
"caption": "Fig.1: An event log L 1 with corresponding workflow net (left) and stochastic workflow net (right) whose transitions' weights are depicted above each transition: notice the stochastic language M 1 induced by the sWN conforms that of L 1 .",
"content": "Fig. 1 :\n1\nFig.1: An even... | [
{
"text": "Split miner: automated discovery of accurate and simple business process models from event logs. Adriano Augusto, Raffaele Conforti, Marlon Dumas, Marcello La Rosa, Artem Polyvyanyy, Knowl. Inf. Syst. 592may 2019",
"doi": ""
},
{
"text": "Stochastic process discovery by weight estimation.... |
GPRAR: Graph Convolutional Network based Pose Reconstruction and Action Recognition for Human Trajectory Prediction
Manh Huynh manh.huynh@ucdenver.edu
University of Colorado Denver
Gita Alaghband gita.alaghband@ucdenver.edu
University of Colorado Denver
GPRAR: Graph Convolutional Network based Pose Reconstruct... | 232,380,372 | {
"arxiv": "2103.14113",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/bmvc/HuynhA21",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2103.14113v1.pdf"
],
"pdf_sha": "5c46c9bc7c542cf35645708b2315c64d00dfae71",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | GPRAR: Graph Convolutional Network based Pose Reconstruction and Action Recognition for Human Trajectory Prediction | [
"Manh Huynh manh.huynh@ucdenver.edu \nUniversity of Colorado Denver",
"Gita Alaghband gita.alaghband@ucdenver.edu \nUniversity of Colorado Denver"
] | Prediction with high accuracy is essential for various applications such as autonomous driving. Existing prediction models are easily prone to errors in real-world settings where observations (e.g. human poses and locations) are often noisy. To address this problem, we introduce GPRAR, a graph convolutional network bas... | [
{
"title": "Introduction",
"content": "Accurate prediction of human trajectory, i.e., forecasting pedestrians' future locations given their past (observed) frames in dynamic scenes, is critical for various applications such as autonomous driving [16], robotic navigation systems [18], and pedestrian tracking... | [
{
"id": "",
"caption": "GPRAR System Overview. Our prediction model consists of two sub-networks: PRAR and FA. Given the sequence of observed noisy human skeletons K i obs of the target pedestrian i as input, PRAR reconstructs (denoises) the noisy human skeletons and recognizes the human action. In the late... | [
{
"text": "Social lstm: Human trajectory prediction in crowded spaces. Alexandre Alahi, Kratarth Goel, Vignesh Ramanathan, Alexandre Robicquet, Li Fei-Fei, Silvio Savarese, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recogniti... |
A quasi-energy function for Pixton diffeomorphisms defined by generalized Mazur knots
6 Jan 2023
Timur Medvedev
Olga Pochinka
O Pochinka
grant of Government of Russian Federation
Laboratory of Dynamical Systems and Applications of National Research University Higher School of Economics
075-152022-1101
T. Medved... | 255,522,546 | {
"arxiv": "2301.02405",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/s11856-024-2697-7"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2301.02405v1.pdf"
],
"pdf_sha": "cb26f0a9f85c3226c7ea0b189eabdb0dfaf21f6e",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | A quasi-energy function for Pixton diffeomorphisms defined by generalized Mazur knots | [
"Timur Medvedev",
"Olga Pochinka",
"O Pochinka",
"grant of Government of Russian Federation\nLaboratory of Dynamical Systems and Applications of National Research University Higher School of Economics\n075-152022-1101",
"T. Medvedev Laboratory of Algorithms and Technologies for Network Analysis\nHSE Univers... | In this paper we give a lower estimate for the number of critical points of the Lyapunov function for Pixton diffeomorphisms (i.e. Morse-Smale diffeomorphisms in dimension 3 whose chain recurrent set consists of four points: one source, one saddle and two sinks). Ch. Bonatti and V. Grines proved that the class of topol... | [
{
"title": "Introduction and the main results",
"content": "Let M n be a smooth closed n-manifold with a metric d and let f : M n → M n be a diffeomorphism. For two given points x, y ∈ M n a sequence of points x = x 0 , . . . , x m = y is called an ε-chain of length m ∈ N connecting x to y if d(f (x i−1 ), ... | [
{
"id": "",
"caption": "An ε-chain of length m ∈ N",
"content": "Fig. 1\n1An ε-chain of length m ∈ N"
},
{
"id": "",
"caption": "Two non-isotopic and non equivalent Hopf knots L 0 and L M : a) the standard Hopf knot L 0 ; b) the Mazur knot L M non-equivalent Hopf knots L n , n ∈ N which are ... | [
{
"text": "On the number of the classes of topological conjugacy of Pixton diffeomorphisms. P Akhmetiev, T Medvedev, O Pochinka, Qualitative Theory of Dynamical Systems. 203Akhmetiev, P., Medvedev, T., Pochinka, O.: On the number of the classes of topological conjugacy of Pixton diffeomorphisms. Qualitative The... |
DNA Sequencing via Quantum Mechanics and Machine Learning
Henry Yuen hyuen@usc.edu
Collaboratory for Advanced Computing and Simulations
Department of Computer Science
Department of Physics & Astronomy
Department of Chemical Engineering & Materials Science
University of Southern California
90089-0242Los AngelesCAUSA... | 15,673,808 | {
"arxiv": "1012.0900",
"mag": 1706845200,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-1012-0900",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1012.0900v1.pdf"
],
"pdf_sha": "7cb693ad49c8500cf9e32acc337d27f9f47bf7ac",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | DNA Sequencing via Quantum Mechanics and Machine Learning | [
"Henry Yuen hyuen@usc.edu \nCollaboratory for Advanced Computing and Simulations\nDepartment of Computer Science\nDepartment of Physics & Astronomy\nDepartment of Chemical Engineering & Materials Science\nUniversity of Southern California\n90089-0242Los AngelesCAUSA",
"Fuyuki Shimojo shimojo@kumamoto-u.ac.jp \nCo... | Rapid sequencing of individual human genome is prerequisite to genomic medicine, where diseases will be prevented by preemptive cures. Quantum-mechanical tunneling through single-stranded DNA in a solid-state nanopore has been proposed for rapid DNA sequencing, but unfortunately the tunneling current alone cannot disti... | [
{
"title": "Introduction",
"content": "DNA sequencing determines the order of four nucleotide bases-adenine (A), guanine (G), cytosine (C), and thymine (T)-that constitute a DNA molecule. State-of-the-art methods for DNA sequencing, while much improved from the technology used to first sequence the human ge... | [
{
"id": "",
"caption": "Schematic of traversal tunneling current-based sequencing of a single-stranded DNA translocating through a solid-state nanopore, where V is the applied voltage, and the arrow next to the ssDNA indicates the direction of translocation. At each time step t, the device measures the indu... | [
{
"text": "The sequence of the human genome. J C Venter, M D Adams, E W Myers, Science. 291Venter, J. C., Adams, M. D., Myers, E. W., et al.: The sequence of the human genome. Science, 291 (2001) 1304-1351.",
"doi": ""
},
{
"text": "Initial sequencing and analysis of the human genome. E S Lander, L ... |
EXTREME VALUES OF THE FIEDLER VECTOR ON TREES
10 Mar 2023
Roy R Lederman
Stefan Steinerberger
EXTREME VALUES OF THE FIEDLER VECTOR ON TREES
10 Mar 2023arXiv:1912.08327v3 [math.CO]
Let G be a tree on n vertices and let L = D − A denote the Laplacian matrix on G. The second-smallest eigenvalue λ 2 (G) > 0, also known... | 209,405,299 | {
"arxiv": "1912.08327",
"mag": 2995638642,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-1912-08327",
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/1912.08327v3.pdf"
],
"pdf_sha": "f5ae49d0906ec6fb1bbc3862b8a5ab7e05a4eaa8",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | EXTREME VALUES OF THE FIEDLER VECTOR ON TREES | [
"Roy R Lederman",
"Stefan Steinerberger"
] | Let G be a tree on n vertices and let L = D − A denote the Laplacian matrix on G. The second-smallest eigenvalue λ 2 (G) > 0, also known as the algebraic connectivity, as well as the associated eigenvector φ 2 have been of substantial interest. We investigate the question of when the maxima and minima of φ 2 are assume... | [
{
"title": "Introduction",
"content": "1.1. Introduction. Let G = (V, E) be a simple, undirected, unweighted, connected graph on n vertices V = {v 1 , . . . , v n }. The adjacency matrix A ∈ {0, 1} n×n encodes the connections between the vertices via A ij = 1 if the i and j are connected by an edge (i, j) ∈... | [
{
"id": "",
"caption": "The 'Fiedler rose' counterexample of Evans",
"content": "Figure 1 .\n1The 'Fiedler rose' counterexample of Evans"
},
{
"id": "",
"caption": "Sketch: proof of Corollary 1.",
"content": "Figure 2 .\n2Sketch: proof of Corollary 1."
},
{
"id": "",
"caption... | [
{
"text": "Old and new results on algebraic connectivity of graphs. N M M De Abreu, Linear Algebra Appl. 4231N. M. M. de Abreu, Old and new results on algebraic connectivity of graphs. Linear Algebra Appl. 423 (2007), no. 1, 53-73.",
"doi": ""
},
{
"text": "Reversible Markov Chains and Random Walks ... |
Mutual Information Based Knowledge Transfer Under State-Action Dimension Mismatch
2020
Michael Wan
Computer Science Dept. UIUC
Computer Science Dept. UIUC
Computer Science Dept. UIUC
Tanmay Gangwani gangwan2@illinois.edu
Computer Science Dept. UIUC
Computer Science Dept. UIUC
Computer Science Dept. UIUC
Jian Pe... | 219,635,852 | {
"arxiv": "2006.07041",
"mag": 3091062495,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/uai/WanG020",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/2006.07041v1.pdf"
],
"pdf_sha": "01777d6a3789549c6e953ddea910f7368cb1c3b5",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Mutual Information Based Knowledge Transfer Under State-Action Dimension Mismatch | [
"Michael Wan \nComputer Science Dept. UIUC\nComputer Science Dept. UIUC\nComputer Science Dept. UIUC",
"Tanmay Gangwani gangwan2@illinois.edu \nComputer Science Dept. UIUC\nComputer Science Dept. UIUC\nComputer Science Dept. UIUC",
"Jian Peng jianpeng@illinois.edu \nComputer Science Dept. UIUC\nComputer Science... | Deep reinforcement learning (RL) algorithms have achieved great success on a wide variety of sequential decision-making tasks. However, many of these algorithms suffer from high sample complexity when learning from scratch using environmental rewards, due to issues such as credit-assignment and high-variance gradients,... | [
{
"title": "INTRODUCTION",
"content": "Deep reinforcement learning (RL), which combines the rigor of RL algorithms with the flexibility of universal function approximators such as deep neural networks, has demonstrated a plethora of success stories in recent times. These include computer and board games , c... | [
{
"id": "",
"caption": "Mutual Information based Knowledge Transfer (MIKT) Input : θ , ψ fixed teacher policy and value networks θ, ψ: student policy and value networks {p}: set of coupling parameters for policy and value networks φ: encoder parameters ω: variational distribution parameters for each iterati... | [
{
"text": "Apprenticeship learning via inverse reinforcement learning. Pieter Abbeel, Y Andrew, Ng, Proceedings of the twenty-first international conference on Machine learning. the twenty-first international conference on Machine learningPieter Abbeel and Andrew Y Ng. Apprenticeship learn- ing via inverse rein... |
THE TORSION FLOW ON A CLOSED PSEUDOHERMITIAN 3-MANIFOLD
Shu-Cheng Chang
Otto Van Koert
Chin-Tung Wu
THE TORSION FLOW ON A CLOSED PSEUDOHERMITIAN 3-MANIFOLD
In this paper we define the torsion flow, a CR analogue of the Ricci flow. We show that the torsion flow on a closed, strictly pseudoconvex pseudohermitian m... | 119,309,615 | {
"arxiv": "1305.5391",
"mag": 2129172854,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1305.5391v2.pdf"
],
"pdf_sha": "e486d0cfa016026532801bab53c7e036ffb97df2",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | THE TORSION FLOW ON A CLOSED PSEUDOHERMITIAN 3-MANIFOLD | [
"Shu-Cheng Chang",
"Otto Van Koert",
"Chin-Tung Wu"
] | In this paper we define the torsion flow, a CR analogue of the Ricci flow. We show that the torsion flow on a closed, strictly pseudoconvex pseudohermitian manifold has short time existence for some favorable choice of initial conditions. For homogeneous CR manifolds we give explicit solutions to the torsion flow illus... | [
{
"title": "Introduction",
"content": "The Ricci flow, introduced by Hamilton, is a geometric flow for metrics on 3-manifolds, and has played a decisive role in the proof of the Poincaré conjecture and Thurston's geometrization conjecture for 3-manifolds. It is natural to then investigate a corresponding pr... | [
{
"id": "",
"caption": "Let (M, ξ = ker θ) be a contact 3-manifold. Then there is a global trivialization U, V of its contact structure if and only if c 1 (ξ) = 0.",
"content": "Lemma 4 . 1 .\n41Let (M, ξ = ker θ) be a contact 3-manifold. Then there is a global trivialization U, V of its contact structu... | [
{
"text": "D Blair, Riemannian geometry of contact and symplectic manifolds. Boston, MABirkhäuser Boston, Inc203D. Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203. Birkhäuser Boston, Inc., Boston, MA, 2002. xii+260 pp. ISBN: 0-8176-4261-7.",
"doi": ""
},
{
... |
Imaging, counting, and positioning single interstitial atoms in solids
Jizhe Cui
School of Materials Science and Engineering
Tsinghua University
100084BeijingChina
Key Laboratory of Advanced Materials of Ministry of Education
Tsinghua University
100084BeijingChina
State Key Laboratory of New Ceramics and Fine Pro... | 270,067,865 | {
"arxiv": "2405.18164",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2405.18164v1.pdf"
],
"pdf_sha": "54dfe98b9413890583e39b9594351d5ff61e468d",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Imaging, counting, and positioning single interstitial atoms in solids | [
"Jizhe Cui \nSchool of Materials Science and Engineering\nTsinghua University\n100084BeijingChina\n\nKey Laboratory of Advanced Materials of Ministry of Education\nTsinghua University\n100084BeijingChina\n\nState Key Laboratory of New Ceramics and Fine Processing\nTsinghua University\n100084BeijingChina",
"Haozhi... | Interstitial atoms are ubiquitous in solids and they are widely incorporated into materials to tune their lattice structure, electronic transportation, and mechanical properties.Because the distribution of interstitial atoms in matrix materials is usually disordered and most of them are light atoms with weak scattering... | [
{
"title": "Introduction",
"content": "Interstitial atoms are ubiquitous in materials and play important role in tuning the mechanical and physical properties of semiconductors, superconductors, and alloys [1][2][3][4] .Recently, incorporation of interstitial atoms in alloys has been increasingly employed t... | [
{
"id": "",
"caption": "Fig. 1.The schematic of the procedure of investigating single interstitial atoms.(a)",
"content": "Fig. 1 .\n1\nFig. 1.The schematic of the procedure of investigating single interstitial atoms.(a)"
},
{
"id": "",
"caption": "Fig.2compares the experimental and simulate... | [
{
"text": "Fundamentals of semiconductors. P Yu, 2005Springer",
"doi": ""
},
{
"text": "Superconductivity: fundamentals and applications. W Buckel, R Kleiner, 2008John Wiley & Sons",
"doi": ""
},
{
"text": "Interstitial alloys. H J Goldschmid, 2013Springer",
"doi": ""
},
{
"t... |
Experimental assessment of mixing layer scaling laws in Rayleigh-Taylor instability
Marco De Paoli marco.de.paoli@tuwien.ac.at
Institute of Fluid Mechanics and Heat Transfer
1060Wien, ViennaTUAustria
Physics of Fluids Group
University of Twente
7500AEEnschedeThe Netherlands
Diego Perissutti
Polytechnic Departmen... | 250,073,208 | {
"arxiv": "2206.13363",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevfluids.7.093503"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2206.13363v2.pdf"
],
"pdf_sha": "9cd766205ebe08bb057044bccd231e3f13e90777",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Experimental assessment of mixing layer scaling laws in Rayleigh-Taylor instability | [
"Marco De Paoli marco.de.paoli@tuwien.ac.at \nInstitute of Fluid Mechanics and Heat Transfer\n1060Wien, ViennaTUAustria\n\nPhysics of Fluids Group\nUniversity of Twente\n7500AEEnschedeThe Netherlands",
"Diego Perissutti \nPolytechnic Department\nUniversity of Udine\n33100UdineItaly",
"Cristian Marchioli \nPolyt... | We assess experimentally the scaling laws that characterize the mixing region produced by the Rayleigh-Taylor instability in a confined porous medium. In particular, we wish to assess experimentally the existence of a superlinear scaling for the growth of the mixing region, which was observed in recent two-dimensional ... | [
{
"title": "I. Introduction",
"content": "When two miscible fluids with different densities move under the action of gravity the relative acceleration between the fluids generates an instability at the separation interface. This instability, called Rayleigh-Taylor instability [1,2], grows in time and leads ... | [
{
"id": "",
"caption": "(a) Experimental setup. The Hele-Shaw cell is shown, with explicit indication of the position of the connection gates (top, bottom, left and right, indicated as T, B, L, and R respectively), sCMOS camera and backlight illumination. Fluids are supplied to the cell through a system (sk... | [
{
"text": "Uncertainties on the light intensity and mass fraction are also shown (see Sec. II B for further details). Finally, the mass fraction-light intensity space considered in each experiment, labelled as in Tab. I, is highlighted in red. The measurements are performed where the sensitivity of the method e... |
ORGANISATION EUROP ÉENNE POUR LA RECHERCHE NUCL ÉAIRE (CERN) Laboratoire Européen pour la Physique des Particules
18 April 2002
The Aleph Collaboration
A Heister
S Schael
R Barate
R Brunelière
I De Bonis
D Decamp
C Goy
S Jezequel
J.-P Lees
F Martin
E Merle
M.-N Minard
B Pietrzyk
B Trocmé
G Boix
Now at... | 262,630,558 | {
"arxiv": "hep-ex/0204036",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/hep-ex/0204036v1.pdf"
],
"pdf_sha": "a2da7c63135809292e1d910ae8bf04e170a5f98a",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | ORGANISATION EUROP ÉENNE POUR LA RECHERCHE NUCL ÉAIRE (CERN) Laboratoire Européen pour la Physique des Particules | [
"The Aleph Collaboration",
"A Heister",
"S Schael",
"R Barate",
"R Brunelière",
"I De Bonis",
"D Decamp",
"C Goy",
"S Jezequel",
"J.-P Lees",
"F Martin",
"E Merle",
"M.-N Minard",
"B Pietrzyk",
"B Trocmé",
"G Boix \nNow at McKinsey and Compagny\nAvenue Louis Casal 181203GenevaSwitzerla... | Searches for scalar top, scalar bottom and mass-degenerate scalar quarks are performed in the data collected by the ALEPH detector at LEP, at centre-of-mass energies up to 209 GeV, corresponding to an integrated luminosity of 675 pb −1 .No evidence for the production of such particles is found in the decay channels t →... | [
{
"title": "Introduction",
"content": "The results of searches for scalar quarks with the data collected in the year 2000 by the ALEPH detector at LEP are presented in this letter.The energies and integrated luminosities of the analysed data samples are given in Table 1.Previous results obtained with lower ... | [
{
"id": "",
"caption": "",
"content": "1\n1"
},
{
"id": "",
"caption": "Squark decay diagrams considered in this letter: (a) t → c/u χ; (b) t → bχf f′ via W exchange and (c) t → bχf f′ via sfermion exchange; (d) t → bℓν; (e) b → bχ.",
"content": "1 :\n1\nSquark decay diagrams considered ... | [
{
"text": "Searches for scalar top and scalar bottom quarks at LEP2. Phys. Lett. B. 4134311997",
"doi": ""
},
{
"text": "Scalar quark searches in e + e − collisions at √ s = 181 − 184 GeV. Phys. Lett. B. 4341891998",
"doi": ""
},
{
"text": "Searches for slepton and squarks in e + e − col... |
Cooper-Frye sampling with short-range repulsion
Volodymyr Vovchenko
Institute for Nuclear Theory
University of Washington
Box 35155098195SeattleWAUSA
Giersch Science Center
Frankfurt Institute for Advanced Studies
D-60438Frankfurt am MainGermany
and Nuclear Science Division
Lawrence Berkeley National Laboratory
1... | 251,903,578 | {
"arxiv": "2208.13693",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevc.106.064906"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2208.13693v2.pdf"
],
"pdf_sha": "bb9bd89a5500b58e789ed5a8d02ded13e49cd722",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Cooper-Frye sampling with short-range repulsion | [
"Volodymyr Vovchenko \nInstitute for Nuclear Theory\nUniversity of Washington\nBox 35155098195SeattleWAUSA\n\nGiersch Science Center\nFrankfurt Institute for Advanced Studies\nD-60438Frankfurt am MainGermany\n\nand Nuclear Science Division\nLawrence Berkeley National Laboratory\n1 Cyclotron Road94720BerkeleyCAUSA"
... | This work incorporates the effect of short-range repulsion between particles into the Cooper-Frye hadron sampling procedure. This is achieved by means of a rejection sampling step, which prohibits any pair of particles from overlapping in the coordinate space, effectively modeling the effect of hard-core repulsion. The... | [
{
"title": "I. INTRODUCTION",
"content": "The description of bulk observables in relativistic heavy-ion collisions is usually performed in the framework of relativistic hydrodynamics (see e.g. [1,2] for an overview). This modeling incorporates the Cooper-Frye particlization stage, where the expanding QCD fl... | [
{
"id": "",
"caption": "The dependence of the corrected scaled variancẽ ωα = ωα/(1 − α) on the acceptance fraction α, evaluated in a box with (bands) and without (dashed lines) periodic boundary conditions resulting from the sampling of particles with a hard-core repulsion for different values of total part... | [
{
"text": ". A Bzdak, S Esumi, V Koch, J Liao, M Stephanov, N Xu, 10.1016/j.physrep.2020.01.005arXiv:1906.00936Phys. Rept. 853nucl-thA. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov, and N. Xu, Phys. Rept. 853, 1 (2020), arXiv:1906.00936 [nucl-th].",
"doi": ""
},
{
"text": ". C Shen, L Yan, 10.100... |
FLSH -Friendly Library for the Simulation of Humans
Pablo Ramón
Universidad Rey Juan Carlos
MadridSpain
Cristian Romero
Universidad Rey Juan Carlos
MadridSpain
Javier Tapia
Universidad Rey Juan Carlos
MadridSpain
Miguel A Otaduy
Universidad Rey Juan Carlos
MadridSpain
FLSH -Friendly Library for the Simulatio... | 263,684,742 | {
"arxiv": "2310.18206",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2310-18206",
"doi": "10.15221/23.20"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2310.18206v1.pdf"
],
"pdf_sha": "b1df2310ea93d691633c683df4974d18a0a166a2",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | FLSH -Friendly Library for the Simulation of Humans | [
"Pablo Ramón \nUniversidad Rey Juan Carlos\nMadridSpain",
"Cristian Romero \nUniversidad Rey Juan Carlos\nMadridSpain",
"Javier Tapia \nUniversidad Rey Juan Carlos\nMadridSpain",
"Miguel A Otaduy \nUniversidad Rey Juan Carlos\nMadridSpain"
] | Computer models of humans are ubiquitous throughout computer animation and computer vision.However, these models rarely represent the dynamics of human motion, as this requires adding a complex layer that solves body motion in response to external interactions and according to the laws of physics.FLSH is a library that... | [
{
"title": "Introduction",
"content": "Parametric models of humans [2] have greatly simplified the modeling, animation, reconstruction and tracking of humans across computer vision and computer graphics applications.However, to date these parametric models are intrinsically geometric, and their shape, pose ... | [
{
"id": "",
"caption": "In this way, users can define externally arbitrary energy terms (), and obtain gradients",
"content": ".\n\nIn this way, users can define externally arbitrary energy terms (), and obtain gradients"
},
{
"id": "",
"caption": "Fig. 2. Comparison of deformations prod... | [
{
"text": "This can be used e.g. in line-search solvers. popState: resets the previous state from the stack. getStateJacobian: it gets the Jacobian of the skeleton pose, for external skeleton-based energies. getNodesStateJacobian: it gets the Jacobian of mesh nodes, for external mesh-based energies. Optimizatio... |
Fluid Antenna Multiple Access with Simultaneous Non-unique Decoding in Strong Interference Channel
28 Oct 2024
Member, IEEEFarshad Rostami Ghadi f.rostamighadi@ucl.ac.uk
Fellow, IEEEKai-Kit K Wong kai-kit.wong@ucl.ac.uk
Masoud Kaveh masoud.kaveh@aalto.fi
Member, IEEEH Xu
Member, IEEEW K New
Senior Member, IEEEF ... | 273,654,288 | {
"arxiv": "2410.20930",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/abs-2410-20930",
"doi": "10.48550/arxiv.2410.20930"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2410.20930v1.pdf"
],
"pdf_sha": "35efa64f1c804b92e5c64e7c1ce3fe9b1c4b814a",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Fluid Antenna Multiple Access with Simultaneous Non-unique Decoding in Strong Interference Channel | [
"Member, IEEEFarshad Rostami Ghadi f.rostamighadi@ucl.ac.uk",
"Fellow, IEEEKai-Kit K Wong kai-kit.wong@ucl.ac.uk",
"Masoud Kaveh masoud.kaveh@aalto.fi",
"Member, IEEEH Xu",
"Member, IEEEW K New",
"Senior Member, IEEEF Javier López-Martínez",
"Fellow, IEEEHyundong Shin hshin@khu.ac.kr",
"De-",
"F J L... | Fluid antenna system (FAS) is gaining attention as an innovative technology for boosting diversity and multiplexing gains.As a key innovation, it presents the possibility to overcome interference by position reconfigurability on one radio frequency (RF) chain, giving rise to the concept of fluid antenna multiple access... | [
{
"title": "I. INTRODUCTION A. Context and Literature Review",
"content": "I N THE RAPIDLY-EVOLVING landscape of wireless com- munication, the emergence of fluid antenna systems (FAS) represents a transformative leap forward [1], [2], [3].As a new form of reconfigurable antennas, FAS emphasizes on position ... | [
{
"id": "",
"caption": "Fig. 1.System model for a two-user FAMA-IC.",
"content": "Fig. 1 .\n1\nFig. 1.System model for a two-user FAMA-IC."
},
{
"id": "",
"caption": "exp(1).Hence, the PDF and CDF for γ ňr i ri and ζ ňr i ri are, respectively, expressed as",
"content": "nr i tiri 2 ∼\n2\... | [
{
"text": "W K New, arXiv:2407.03449A tutorial on fluid antenna system for 6G networks: Encompassing communication theory, optimization methods and hardware designs. Jul. 2024arXiv preprint",
"doi": ""
},
{
"text": "Bruce Leeinspired fluid antenna system: Six research topics and the potentials for 6... |
Hexagonal-Close-Packed Lattice: Phase Transition and Spin Transport
24 Dec 2011
Danh-Tai Hoang
Laboratoire de Physique Théorique et Modélisation
Université de Cergy-Pontoise
CNRS
Avenue Adolphe Chauvin8089 2, 95302Cergy-Pontoise CedexUMRFrance
H T Diep
Laboratoire de Physique Théorique et Modélisation
Université d... | 85,520,964 | {
"arxiv": "1112.5724",
"mag": 1540957907,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1112.5724v1.pdf"
],
"pdf_sha": "50c317a952dc3ee4b8e83d63b07756a7561df400",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Hexagonal-Close-Packed Lattice: Phase Transition and Spin Transport | [
"Danh-Tai Hoang \nLaboratoire de Physique Théorique et Modélisation\nUniversité de Cergy-Pontoise\nCNRS\nAvenue Adolphe Chauvin8089 2, 95302Cergy-Pontoise CedexUMRFrance",
"H T Diep \nLaboratoire de Physique Théorique et Modélisation\nUniversité de Cergy-Pontoise\nCNRS\nAvenue Adolphe Chauvin8089 2, 95302Cergy-Po... | We study the ground state (GS) and the phase transition in a hexagonal-close-packed lattice with both XY and Ising models by using extensive Monte Carlo simulation. We suppose the in-plane interaction J1 and inter-plane interaction J2, both antiferromagnetic. The system is frustrated with two kinds of GS configuration ... | [
{
"title": "I. INTRODUCTION",
"content": "Frustrated spin systems have been subject of intensive investigations during the last 30 years [1]. These systems are very unstable due to the competition between antagonist interactions or due to a geometrical frustration such as in the antiferromagnetic (AF) trian... | [
{
"id": "",
"caption": "HCP lattice. The in-plane and inter-plane interactions are indicated by J1 and J2.",
"content": "FIG. 1 .\n1HCP lattice. The in-plane and inter-plane interactions are indicated by J1 and J2."
},
{
"id": "",
"caption": "Ground state in the XY case. The tetrahedron is p... | [
{
"text": "Frustrated Spin Systems. World Scientific. H. T. DiepH. T. Diep (ed.), Frustrated Spin Systems, World Scien- tific, Singapor (2005).",
"doi": ""
},
{
"text": ". G Toulouse, Commun. Phys. 2115G. Toulouse, Commun. Phys. 2, 115 (1977).",
"doi": ""
},
{
"text": ". J Villain, J. Ph... |
Hardware-In-the-Loop Measurements of the Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) System
Carsten Bockelmann bockelmann@ant.uni-bremen.de
Universität Bremen Otto-Hahn-Alle
NW1 28359Bremen
Fabian Monsees monsees@ant.uni-bremen.de
Universität Bremen Otto-Hahn-Alle
NW1 28359Bremen
Matthias Wolter... | 10,927,775 | {
"arxiv": "1511.09278",
"mag": 2277299231,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/BockelmannMWD15",
"doi": null
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1511.09278v1.pdf"
],
"pdf_sha": "13b7000def6f40dc5da4107012539265b7ee0796",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Hardware-In-the-Loop Measurements of the Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) System | [
"Carsten Bockelmann bockelmann@ant.uni-bremen.de \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen",
"Fabian Monsees monsees@ant.uni-bremen.de \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen",
"Matthias Woltering woltering@ant.uni-bremen.de \nUniversität Bremen Otto-Hahn-Alle\nNW1 28359Bremen",
"Armin... | MCSM is a recently proposed novel system concept to solve the massive access problem envisioned in future communication systems like 5G and industry 4.0 systems. This work focuses on the practical verification of the theoretical gains that MCSM provides using a Hardware-In-the-Loop (HIL) measurement setup. We present r... | [
{
"title": "Introduction",
"content": "Massive Machine-to-Machine (M2M) communication is expected to be one of the major drivers for new radio access technologies in the development of the 5th generation mobile networks (5G) as well as industrial radio systems for industry 4.0 (I4.0). For cellular networks ... | [
{
"id": "",
"caption": "Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) concept.",
"content": "Figure 1 -\n1Multi-Carrier Compressed Sensing Multi-User Detection (MCSM) concept."
},
{
"id": "",
"caption": "Base Station processing.",
"content": "Figure 3 -\n3Base Station proc... | [
{
"text": "Compressive Sensing based Multi-User Detection for Machine-to-Machine. C Bockelmann, H Schepker, A Dekorsy, Transactions on Emerging Telecommunications Technologies: Special Issue on Machine-to-Machine: An emerging communication paradigm. 24Bockelmann, C.; Schepker, H.; Dekorsy, A.: Compressive Sensi... |
ANALYSIS OF THE TRANSPORT MODEL THAT APPROXIMATES DECISION TAKER'S PREFERENCES
V Ya
University of Technology
KorolevMoscow RegionRussia
Vilisov
University of Technology
KorolevMoscow RegionRussia
ANALYSIS OF THE TRANSPORT MODEL THAT APPROXIMATES DECISION TAKER'S PREFERENCES
1transport problemreverse problemdecis... | 13,275,516 | {
"arxiv": "1509.01815",
"mag": 2229806270,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "journals/corr/Vilisov15b",
"doi": "10.13140/rg.2.1.5085.6166"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1509.01815v1.pdf"
],
"pdf_sha": "c04f17b1c5f02e86e859b451171159a6b0980cd8",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | ANALYSIS OF THE TRANSPORT MODEL THAT APPROXIMATES DECISION TAKER'S PREFERENCES | [
"V Ya \nUniversity of Technology\nKorolevMoscow RegionRussia",
"Vilisov \nUniversity of Technology\nKorolevMoscow RegionRussia"
] | Paper provides a method for solving the reverse Monge-Kantorovich transport problem (TP). It allows to accumulate positive decision-taking experience made by decision-taker in situations that can be presented in the form of TP. The initial data for the solution of the inverse TP is the information on orders, inventorie... | [
{
"title": "Introduction",
"content": "Modern information technologies provide new possibilities for managing complicated social and economic objects. However, computers' computational abilities very often outpace technological capabilities of the existing management procedures. Thus, the tendency of \"burd... | [
{
"id": "",
"caption": "For the TPM, ̅ solution per each DMS can be represented as the R-values matrix ̅ = ‖ ‖ . In this case, the approximation of the decision taker's preferences performed by the transport problem model lies in the evaluation of coefficients, pertaining to the target function:",
"cont... | [
{
"text": "A concept of Corporate Planning. R L Ackoff, John WileyNew YorkR.L. Ackoff. A concept of Corporate Planning. -New York: John Wiley, 1970.",
"doi": ""
},
{
"text": "On Adaptive Management Methods in Transitional Economy // Economic Science of Modern Russia. K A Bagrinovskii, CEMI RSSI Publ... |
Properties of Laughlin states on fractal lattices
Mani Chandra
Max-Planck-Institut für Physik komplexer Systeme
D-01187DresdenGermany
Jha
Anne E B Nielsen
Department of Physics and Astronomy
Aarhus University
DK-8000Aarhus CDenmark
Properties of Laughlin states on fractal lattices
Laughlin states have recently... | 257,038,438 | {
"arxiv": "2302.10010",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/1742-5468/acd104"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/2302.10010v1.pdf"
],
"pdf_sha": "5a27cc07258fc1957a87ec0b476e2e3ea50d0bcc",
"open_access": {
"license": null,
"url": null,
"status": null
}
} | Properties of Laughlin states on fractal lattices | [
"Mani Chandra \nMax-Planck-Institut für Physik komplexer Systeme\nD-01187DresdenGermany",
"Jha",
"Anne E B Nielsen \nDepartment of Physics and Astronomy\nAarhus University\nDK-8000Aarhus CDenmark"
] | Laughlin states have recently been constructed on fractal lattices and have been shown to be topological in such systems. Some of their properties are, however, quite different from the two-dimensional case. On the Sierpinski triangle, for instance, the entanglement entropy shows oscillations as a function of particle ... | [
{
"title": "Introduction",
"content": "When a two-dimensional electron gas is placed in a strong magnetic field and at very low temperatures, the fractional quantum Hall (FQH) effect can appear [1]. The Hamiltonian describing this effect, which is basically the Coulomb interaction projected to the lowest La... | [
{
"id": "",
"caption": "b) shows n i r = n i − M/N of a generation g = 4 Sierpinski triangle for all values of the number of particles M in the range [1,80]. The data roughly form a thick central band and two sets of lines (blue and reddish) that are quite far from the central band for most values of M . Th... | [
{
"text": ". D C Tsui, H Stormer, A C Gossard, Phys. Rev. Lett. 481559Tsui D C, Stormer H L and Gossard A C 1982 Phys. Rev. Lett. 48 1559",
"doi": ""
},
{
"text": ". R Laughlin, Phys. Rev. Lett. 501395Laughlin R B 1983 Phys. Rev. Lett. 50 1395",
"doi": ""
},
{
"text": ". V Kalmeyer, R La... |
Non-local non-linear sigma models
June 2019 25 Jun 2019
Steven S Gubser ssgubser@princeton.edu
Joseph Henry Laboratories
Princeton University
08544PrincetonNJUSA
Christian B Jepsen cjepsen@princeton.edu
Joseph Henry Laboratories
Princeton University
08544PrincetonNJUSA
Ziming Ji
Joseph Henry Laboratories
Princet... | 195,584,174 | {
"arxiv": "1906.10281",
"mag": 3101387650,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1007/jhep09(2019)005"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/1906.10281v1.pdf"
],
"pdf_sha": "4e3df0cf3b274c5a4911d1f10ad54a28fa9039d3",
"open_access": {
"license": "CCBY",
"url": "https://link.springer.com/content/pdf/10.1007/JHEP09(2019)005.pdf",
"status": "GOLD"
}
} | Non-local non-linear sigma models | [
"Steven S Gubser ssgubser@princeton.edu \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA",
"Christian B Jepsen cjepsen@princeton.edu \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA",
"Ziming Ji \nJoseph Henry Laboratories\nPrinceton University\n08544PrincetonNJUSA",
... | We study non-local non-linear sigma models in arbitrary dimension, focusing on the scale invariant limit in which the scalar fields naturally have scaling dimension zero, so that the free propagator is logarithmic. The classical action is a bi-local integral of the square of the arc length between points on the target ... | [
{
"title": "Introduction",
"content": "Scalar field theories over the reals with bi-local kinetic terms were introduced in [1], and the recent work [2] provides a useful point of entry into the extensive literature. Similar field theories over the p-adic numbers were considered in [3] as a continuum descrip... | [
{
"id": "",
"caption": "(a) One-loop contribution to the 1PI two-point function. (b) A diagram where the loop consists of a single propagator closing on itself.",
"content": "Figure 1 :\n1(a) One-loop contribution to the 1PI two-point function. (b) A diagram where the loop consists of a single propagato... | [
{
"text": "Critical Exponents for Long-Range Interactions. M E Fisher, S Ma, B G Nickel, 10.1103/PhysRevLett.29.917Phys. Rev. Lett. 29M. E. Fisher, S.-k. Ma, and B. G. Nickel, \"Critical Exponents for Long-Range Interactions,\" Phys. Rev. Lett. 29 (1972) 917-920.",
"doi": "10.1103/PhysRevLett.29.917"
},
... |
An in vivo Comparison Study Between Strontium Nanoparticles and rhBMP2
June 2020
Silvia Minardi
Hakan Ceylan
Gloria Belen Ramirez
Beth Bragdon bragdon@bu.edu
Livia Visai livia.visai@unipv.it
Giulia Montagna
Department of Molecular Medicine (DMM)
Center for Health Technologies (CHT)
UdR INSTM
University of Pavia... | 219,689,268 | {
"arxiv": "2006.12490",
"mag": 3099288547,
"acl": null,
"pubmed": "32612980",
"pubmedcentral": 7308719,
"dblp": null,
"doi": "10.3389/fbioe.2020.00499"
} | {
"pdf_urls": [],
"pdf_sha": "987232bb8f65073911bf0ce49dc87ee8ddef54d8",
"open_access": {
"license": "CCBY",
"url": "https://www.frontiersin.org/articles/10.3389/fbioe.2020.00499/pdf",
"status": "GOLD"
}
} | An in vivo Comparison Study Between Strontium Nanoparticles and rhBMP2 | [
"Silvia Minardi",
"Hakan Ceylan",
"Gloria Belen Ramirez",
"Beth Bragdon bragdon@bu.edu",
"Livia Visai livia.visai@unipv.it",
"Giulia Montagna \nDepartment of Molecular Medicine (DMM)\nCenter for Health Technologies (CHT)\nUdR INSTM\nUniversity of Pavia\nPaviaItaly\n\nDepartment of Electrical, Computer and... | The osteoinductive property of strontium was repeatedly proven in the last decades. Compelling in vitro data demonstrated that strontium hydroxyapatite nanoparticles exert a dual action, by promoting osteoblasts-driven matrix secretion and inhibiting osteoclasts-driven matrix resorption. Recombinant human bone morphoge... | [
{
"title": "INTRODUCTION",
"content": "Bone tissue constitutes the rigid scaffold that is the skeleton, which provides structural support for vertebrates and confers protection to the most delicate vital organs. The extraordinary regenerative capability of bone tissue to repair and heal without the formatio... | [
{
"id": "",
"caption": "IR evaluation of sponge-nanoparticles interactions. FT-IR spectra of sponges and nanoparticles physical mixtures (Han_PM and SrHAn_PM) (C,D,G,H) were compared with spectra corresponding to the untreated and loaded sponges HAn and SrHAn at 28 days (E,F,I,J). In the panels a and b, FT-... | [
{
"text": "The transcription factor Sox9 has essential roles in successive steps of the chondrocyte differentiation pathway and is required for expression of Sox5 and Sox6. H Akiyama, M C Chaboissier, J F Martin, A Schedl, B De Crombrugghe, 10.1101/gad.1017802Genes Dev. 16Akiyama, H., Chaboissier, M. C., Martin... |
THE CONTINUOUS SPECTRUM IN DISCRETE SERIES BRANCHING LAWS
7 Aug 2013
HONGYU HE, AND GESTURÓLAFSSONBenjamin Harris
THE CONTINUOUS SPECTRUM IN DISCRETE SERIES BRANCHING LAWS
7 Aug 2013
If G is a reductive Lie group of Harish-Chandra class, H is a symmetric subgroup, and π is a discrete series representation of G, the ... | 119,609,556 | {
"arxiv": "1209.4125",
"mag": 2952238100,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1142/s0129167x13500493"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1209.4125v3.pdf"
],
"pdf_sha": "7df02ebc0e9382b0d3d0e2ef8aebe2a26d3c34b5",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1209.4125.pdf",
"status": "GREEN"
}
} | THE CONTINUOUS SPECTRUM IN DISCRETE SERIES BRANCHING LAWS | [
"HONGYU HE, AND GESTURÓLAFSSONBenjamin Harris"
] | If G is a reductive Lie group of Harish-Chandra class, H is a symmetric subgroup, and π is a discrete series representation of G, the authors give a condition on the pair (G, H) which guarantees that the direct integral decomposition of π| H contains each irreducible representation of H with finite multiplicity. In add... | [
{
"title": "Introduction",
"content": "If π is an irreducible representation of a Lie group G and H ⊂ G is a type 1 closed Lie subgroup, it is natural to consider the restriction π| H and to decompose this restriction into irreducible representations of H. This type of problem comes up naturally in physics;... | [
{
"id": "",
"caption": "Ørsted-Vargas). There exists a family of continuous, H equivariant maps",
"content": "Proposition 2 . 1 (\n21Ørsted-Vargas). There exists a family of continuous, H equivariant maps"
},
{
"id": "",
"caption": "The map r m restricts to an H equivariant map between the d... | [
{
"text": "A Andreotti, E Vesentini, Carleman Estimates for the Laplace-Beltrami Equations on Complex Manifolds, Institute Hautes Etudes Scientifique Publications Mathematiques. 25A. Andreotti, E. Vesentini, Carleman Estimates for the Laplace-Beltrami Equations on Com- plex Manifolds, Institute Hautes Etudes Sc... |
THE CHANDRA COSMOS LEGACY SURVEY: CLUSTERING OF X-RAY SELECTED AGN AT 2.9≤z≤5.5 USING PHOTOMETRIC REDSHIFT PROBABILITY DISTRIBUTION FUNCTIONS
16 Sep 2016
V Allevato
F Civano
A Finoguenov
S Marchesi
F Shankar
G Zamorani
G Hasinger
M Salvato
T Miyaji
R Gilli
N Cappelluti
M Brusa
H Suh
G Lanzuisi
B Trakhte... | 209,867,532 | {
"arxiv": "1609.02553",
"mag": 2953019277,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.3847/0004-637x/832/1/70"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/1609.02553v2.pdf"
],
"pdf_sha": "f73bef1233875dd4457983337863d0387d9eea9d",
"open_access": {
"license": null,
"url": "https://helda.helsinki.fi/bitstream/10138/187230/1/Allevato_2016_ApJ_832_70.pdf",
"status": "GREEN"
}
} | THE CHANDRA COSMOS LEGACY SURVEY: CLUSTERING OF X-RAY SELECTED AGN AT 2.9≤z≤5.5 USING PHOTOMETRIC REDSHIFT PROBABILITY DISTRIBUTION FUNCTIONS | [
"V Allevato",
"F Civano",
"A Finoguenov",
"S Marchesi",
"F Shankar",
"G Zamorani",
"G Hasinger",
"M Salvato",
"T Miyaji",
"R Gilli",
"N Cappelluti",
"M Brusa",
"H Suh",
"G Lanzuisi",
"B Trakhtenbrot",
"R Griffiths",
"C Vignali",
"K Schawinski",
"A Karim"
] | We present the measurement of the projected and redshift space 2-point correlation function (2pcf) of the new catalog of Chandra COSMOS-Legacy AGN at 2.9≤z≤5.5 ( L bol ∼10 46 erg/s) using the generalized clustering estimator based on phot-z probability distribution functions (Pdfs) in addition to any available spec-z. ... | [
{
"title": "INTRODUCTION",
"content": "The presence of a nuclear supermassive black hole (BH) in almost all galaxies in the present day Universe is an accepted paradigm in astronomy (e.g. Kormendy & Richstone 1995;Kormendy & Bender 2011). Despite major observational and theoretical efforts over the last two... | [
{
"id": "",
"caption": "-Left Panel:",
"content": "Fig. 1 .\n1-Left Panel:"
},
{
"id": "",
"caption": "-Redshift (Left Panel) and 2-10 keV X-ray luminosity (Right Panel) distribution for 107 CCL AGN with known spec-z (red dashed line), 212 AGN with known spec or best phot-z (solid grey line)... | [
{
"text": ". V Allevato, ApJ. 73699Allevato V., et al. 2011, ApJ, 736, 99",
"doi": ""
},
{
"text": ". V Allevato, ApJ. 7964Allevato V., et al. 2014, ApJ, 796, 4",
"doi": ""
},
{
"text": ". F Bournaud, A Dekel, R Teyssier, ApJ. 74133Bournaud, F., Dekel, A., Teyssier, R., et al. 2011, ApJ,... |
SMOOTH SQUAREFREE AND SQUARE-FULL INTEGERS IN ARITHMETIC PROGRESSIONS
5 Oct 2018
Marc Munsch
ANDIgor E Shparlinski
Kam Hung Yau
SMOOTH SQUAREFREE AND SQUARE-FULL INTEGERS IN ARITHMETIC PROGRESSIONS
5 Oct 2018
We obtain new lower bounds on the number of smooth squarefree integers up to x in residue classes modulo a... | 119,712,364 | {
"arxiv": "1810.02573",
"mag": 2990938197,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1112/mtk.12012"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1810.02573v2.pdf"
],
"pdf_sha": "e3d054f22d8c14ecb2c8705df24f5e5e603f56d3",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1810.02573",
"status": "GREEN"
}
} | SMOOTH SQUAREFREE AND SQUARE-FULL INTEGERS IN ARITHMETIC PROGRESSIONS | [
"Marc Munsch",
"ANDIgor E Shparlinski",
"Kam Hung Yau"
] | We obtain new lower bounds on the number of smooth squarefree integers up to x in residue classes modulo a prime p, relatively large compared to x, which in some ranges of p and x improve that of A.Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prim... | [
{
"title": "1.2.",
"content": "Results for squarefree numbers. We start with a lower bound on ψ ♯ (x, y; p, a) which holds for any prime p. Theorem 1.1. For any fixed real numbers α and β with β ∈ (23/24, 1] and α ∈ (9/2 − 3β, 3β], for x = p α+o(1) and y = p β+o(1) we have\nTaking y = p β with 23/24 < β ≤ 1... | [
{
"id": "",
"caption": "For any integer a and prime p with gcd(a, p) = 1 and reals h, D and L with",
"content": "Lemma 2. 4 .\n4For any integer a and prime p with gcd(a, p) = 1 and reals h, D and L with"
},
{
"id": "",
"caption": "For any integer K ≥ 1, we have χ∈Ω * p 1≤n≤Kχ(n) 4 ≤ K 2 p 1+... | [
{
"text": "The congruence x 1 x 2 ≡ x 3 x 4 mod p, the equation x 1 x 2 = x 3 x 4 and mean values of character sums. A Ayyad, T Cochrane, Z Zheng, J. Number Theory. 596A. Ayyad, T. Cochrane and Z. Zheng, 'The congruence x 1 x 2 ≡ x 3 x 4 mod p, the equation x 1 x 2 = x 3 x 4 and mean values of character sums', ... |
Probing high order dependencies with information theory
C Granero-Belinchón
Univ Lyon
Ens de Lyon
Laboratoire de Physique
Univ Claude Bernard
CNRS
F-69342LyonFrance
S G Roux
Univ Lyon
Ens de Lyon
Laboratoire de Physique
Univ Claude Bernard
CNRS
F-69342LyonFrance
IEEE Fellow) *P Abry
Univ Lyon
Ens de Lyon
Lab... | 119,402,330 | {
"arxiv": "1812.05325",
"mag": 2951885956,
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"doi": "10.1109/tsp.2019.2920472"
} | {
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"url": "http://arxiv.org/pdf/1812.05325",
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} | Probing high order dependencies with information theory | [
"C Granero-Belinchón \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nUniv Claude Bernard\nCNRS\nF-69342LyonFrance",
"S G Roux \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nUniv Claude Bernard\nCNRS\nF-69342LyonFrance",
"IEEE Fellow) *P Abry \nUniv Lyon\nEns de Lyon\n\nLaboratoire de Physique\nUniv C... | Information theoretic measures (entropies, entropy rates, mutual information) are nowadays commonly used in statistical signal processing for real-world data analysis. The present work proposes the use of Auto Mutual Information (Mutual Information between subsets of the same signal) and entropy rate as powerful tools ... | [
{
"title": "I. INTRODUCTION",
"content": "Context. To characterize a (stationary) process X(t), evolving along a given dimension t, e.g., time, the precise assessment of both its static properties (marginal distribution or one-point statistics) and temporal dynamics (joint distributions or multipoint statis... | [
{
"id": "",
"caption": "the (m + p)-dimensional time-embedded vector x (m+p)",
"content": "the (m + p)-dimensional time-embedded vector x (m+p)"
},
{
"id": "",
"caption": "Short-range dependencies. Classical analysis: histogram (a), power spectrum (b) and correlation function (c). Informatio... | [
{
"text": "Statistical signs of synaptic interaction in neurons. G P Moore, J P Segundo, D H Perkel, H Levitan, Biophysical Journal. 109G. P. Moore, J. P. Segundo, D. H. Perkel, and H. Levitan, \"Statistical signs of synaptic interaction in neurons,\" Biophysical Journal, vol. 10, no. 9, pp. 876 -900, 1970.",
... |
EFFECTS OF EARLY INTENSE BOMBARDMENT ON MEGAREGOLITH EVOLUTION AND ON LUNAR (AND PLANETARY) SURFACE SAMPLES
William K Hartmann
Planetary Science Institute
TucsonArizona
Alessandro Morbidelli
Université Côte d'Azur
CNRS-Lagrange Observatoiré de la Côte d'Azur
34229-F 06304, Cedex 4NiceCSFrance
EFFECTS OF EARLY IN... | 225,075,660 | {
"arxiv": "2010.14275",
"mag": 3111441048,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1111/maps.13595"
} | {
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],
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"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2010.14275",
"status": "GREEN"
}
} | EFFECTS OF EARLY INTENSE BOMBARDMENT ON MEGAREGOLITH EVOLUTION AND ON LUNAR (AND PLANETARY) SURFACE SAMPLES | [
"William K Hartmann \nPlanetary Science Institute\nTucsonArizona",
"Alessandro Morbidelli \nUniversité Côte d'Azur\nCNRS-Lagrange Observatoiré de la Côte d'Azur\n34229-F 06304, Cedex 4NiceCSFrance"
] | Impact rates in the first 500 Myr of the solar system are critical to an understanding of lunar geological history, but they have been controversial. The widely accepted, post-Apollo paradigm of early lunar impact cratering (ca. 1975-2014) proposed very low or negligible impact cratering in the period from accretion (... | [
{
"title": "ANALYSIS AND APPLICATION OF ACCRETION TAIL SCENARIO",
"content": ""
},
{
"title": "VIS-À-VIS IMPACT RATE VS. TIME",
"content": "As pointed out by Morbidelli et al. (2018), the \"accretion tail\" dynamical model creates a curve of declining impact rate vs. time, similar to -but indepe... | [
{
"id": "",
"caption": "model. See text. **The accumulated crater numbers in the last bins of the table should probably be higher. This involves intersection of two different processes of scattering interplanetary bodies in the Morbidelli et al. (2018) model, around 3 Ga. (See text for discussion.)",
"c... | [
{
"text": "Structure and evolution of the lunar Procellarum region as revealed by GRAIL gravity data. J C Andrews-Hanna, J Besserer, J Head, C Howett, W Kiefer, P Lucey, P Mcgovern, H Melosh, G Neumann, R Phillips, D Smith, S Solomon, M Zuber, Nature. 514Andrews-Hanna J.C., Besserer J., Head J., Howett C., Kief... |
Exploring the evidence for a large local void with supernovae Ia data
2019
Vladimir V Luković
Dipartimento di Fisica
Università di Roma "Tor Vergata"
Via della Ricerca Scientifica 1I-00133RomaItaly
Sezione INFN
Università di Roma "Tor Vergata"
Via della Ricerca Scientifica 1I-00133RomaItaly
Balakrishna S Haridasu ... | 198,897,423 | {
"arxiv": "1907.11219",
"mag": 2986977326,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1093/mnras/stz3070"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1907.11219v2.pdf"
],
"pdf_sha": "a00da862e84d207e8fcfb492e15e3abf0d9159e6",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1907.11219",
"status": "GREEN"
}
} | Exploring the evidence for a large local void with supernovae Ia data | [
"Vladimir V Luković \nDipartimento di Fisica\nUniversità di Roma \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomaItaly\n\nSezione INFN\nUniversità di Roma \"Tor Vergata\"\nVia della Ricerca Scientifica 1I-00133RomaItaly",
"Balakrishna S Haridasu \nDipartimento di Fisica\nUniversità di Roma \"Tor Vergat... | In this work we utilise the most recent publicly available type Ia supernova (SN Ia) compilations and implement a well formulated cosmological model based on Lemaître-Tolman-Bondi metric in presence of cosmological constant Λ (ΛLTB) to test for signatures of large local inhomogeneities at z ≤ 0.15. Local underdensities... | [
{
"title": "INTRODUCTION",
"content": "The most reliable standard candles and one of the fundamental low-redshift astrophysical observables (Betoule et al. 2014;Scolnic et al. 2018), type Ia supernovae (SN Ia), also provide a cosmologyindependent measurement of present expansion rate through the local dista... | [
{
"id": "",
"caption": "; Tokutake et al. (2018); Kenworthy et al. (2019); Boehringer et al. (",
"content": "; Tokutake et al. (2018); Kenworthy et al. (2019); Boehringer et al. ("
},
{
"id": "",
"caption": "The evolution of matter density contrast δρ/ρ is shown for ΛLTB model with present l... | [
{
"text": ". H Akaike, IEEE Transactions on Automatic Control. 19716Akaike H., 1974, IEEE Transactions on Automatic Control, 19, 716",
"doi": ""
},
{
"text": ". S Alam, 10.1093/mnras/stx721MNRAS. 4702617Alam S., et al., 2017, MNRAS, 470, 2617",
"doi": ""
},
{
"text": ". A H A Alfedeel, C... |
Photoacoustic imaging taking into account thermodynamic attenuation
Sebastián Acosta sacosta@bcm.edu
Department of Pediatrics -Cardiology
Baylor College of Medicine and Texas Children's Hospital
HoustonTXUSA
Carlos Montalto
Department of Mathematics
University of Washington
SeattleWAUSA
Photoacoustic imaging tak... | 16,564,308 | {
"arxiv": "1602.01872",
"mag": 3105025489,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/0266-5611/32/11/115001"
} | {
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"https://arxiv.org/pdf/1602.01872v1.pdf"
],
"pdf_sha": "d93a690d4e2c22aeff6d7ae51322744facd12f27",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1602.01872",
"status": "GREEN"
}
} | Photoacoustic imaging taking into account thermodynamic attenuation | [
"Sebastián Acosta sacosta@bcm.edu \nDepartment of Pediatrics -Cardiology\nBaylor College of Medicine and Texas Children's Hospital\nHoustonTXUSA",
"Carlos Montalto \nDepartment of Mathematics\nUniversity of Washington\nSeattleWAUSA"
] | In this paper we consider a mathematical model for photoacoustic imaging which takes into account attenuation due to thermodynamic dissipation. The propagation of acoustic (compressional) waves is governed by a scalar wave equation coupled to the heat equation for the excess temperature. We seek to recover the initial ... | [
{
"title": "Introduction",
"content": "Photoacoustic tomography is an imaging technique that takes advantage of the highcontrast exhibited by optical absorption and the high-resolution properties of wideband acoustic waves in soft biological tissue. Details concerning this type of imaging modalities are fou... | [
{
"id": "",
"caption": "Assumption 2.1 (Rapid Deposition of Heat). The initial conditions (3) satisfy, p 1 = 0 and θ 0 = p 0 .",
"content": "Assumption 2.1 (Rapid Deposition of Heat). The initial conditions (3) satisfy, p 1 = 0 and θ 0 = p 0 ."
},
{
"id": "",
"caption": "Main Result). Suppos... | [
{
"text": "Biomedical optics : principles and imaging. L V Wang, H Wu, Wiley-Interscience14Hoboken, N.J.L. V. Wang and H. Wu. Biomedical optics : principles and imaging. Hoboken, N.J. : Wiley- Interscience, 2007. 1, 4",
"doi": ""
},
{
"text": "Photoacoustic imaging and spectroscopy. Optical science ... |
Mixing Time and Stationary Expected Social Welfare of Logit Dynamics *
23 May 2011 May 24, 2011
Vincenzo Auletta auletta@dia.unisa.it
Dipartimento di Informatica "Renato M. Capocelli
Università di Salerno
Diodato Ferraioli ferraioli@dia.unisa.it
Dipartimento di Informatica "Renato M. Capocelli
Università di Saler... | 6,772,074 | {
"arxiv": "1002.3474",
"mag": 3044044770,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/sagt/AulettaFPP10",
"doi": "10.1007/978-3-642-16170-4_6"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1002.3474v4.pdf"
],
"pdf_sha": "d30d1a8bb05839eea36bd7c0d6c8db54c75d0ea1",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1002.3474",
"status": "GREEN"
}
} | Mixing Time and Stationary Expected Social Welfare of Logit Dynamics * | [
"Vincenzo Auletta auletta@dia.unisa.it \nDipartimento di Informatica \"Renato M. Capocelli\nUniversità di Salerno",
"Diodato Ferraioli ferraioli@dia.unisa.it \nDipartimento di Informatica \"Renato M. Capocelli\nUniversità di Salerno",
"Francesco Pasquale pasquale@dia.unisa.it \nDipartimento di Informatica \"Ren... | We study logit dynamics [3] for strategic games. This dynamics works as follows: at every stage of the game a player is selected uniformly at random and she plays according to a noisy best-response where the noise level is tuned by a parameter β. Such a dynamics defines a family of ergodic Markov chains, indexed by β, ... | [
{
"title": "Introduction",
"content": "The evolution of a system is determined by its dynamics and complex systems are often described by looking at the equilibrium states induced by their dynamics. Once the system reaches an equilibrium state it stays there, thus equilibrium states describe the long-term b... | [
{
"id": "",
"caption": "strategic game is a triple ([n], S, U), where [n] = {1, . . . , n} is a finite set of players, S = {S 1 , .. . , S n } is a family of non-empty finite sets (S i is the set of strategies for player i), and U = {u 1 , . . . , u n } is a family of utility functions (or payoffs), where u... | [
{
"text": "Tom Wexler, and Tim Roughgarden. The price of stability for network design with fair cost allocation. Elliot Anshelevich, Anirban Dasgupta, Jon M Kleinberg, Éva Tardos, SIAM Journal on Computing. 384Preliminary version in FOCS'04Elliot Anshelevich, Anirban Dasgupta, Jon M. Kleinberg,Éva Tardos, Tom W... |
LR Scalar Mixings and One-loop Neutrino Masses
Otto C W Kong
Institute of Physics
Academia Sinica
11529Nankang, TaipeiTAIWAN
LR Scalar Mixings and One-loop Neutrino Masses
arXiv:hep-ph/0004107v4 19 Oct 2000 HYPER VERSIONSolar and Atmospheric Neutrinos Supersymmetric Standard Model Neutrino Physics
Within the frame... | 1,522,414 | {
"arxiv": "hep-ph/0004107",
"mag": 2030303578,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/1126-6708/2000/09/037"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/hep-ph/0004107v4.pdf"
],
"pdf_sha": "c27c3157118ba74c2f6c012dfb39da193ecf6262",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/hep-ph/0004107",
"status": "GREEN"
}
} | LR Scalar Mixings and One-loop Neutrino Masses | [
"Otto C W Kong \nInstitute of Physics\nAcademia Sinica\n11529Nankang, TaipeiTAIWAN"
] | Within the framework of the complete theory of supersymmetry without R-parity, where all possible R-parity violating terms are admitted, we perform a systematic analytical study of all sources of neutrino masses up to "direct one-loop" (defined explicitly below) level. In the passing, we present the full result for squ... | [
{
"title": "Introduction",
"content": "The minimal supersymmetric standard model (MSSM) is no doubt the most popular candidate theory for physics beyond the Standard Model (SM). The alternative theory with a discrete symmetry called R-parity not imposed deserves no less attention. In particular, the latter ... | [
{
"id": "",
"caption": "(3.1), one neutrino state get a tree-level mass. The seesaw suppressed contribution [see Eqs.(3.2) and (3.3)] is given by",
"content": "(3.1), one neutrino state get a tree-level mass. The seesaw suppressed contribution [see Eqs.(3.2) and (3.3)] is given by"
},
{
"id": ""... | [
{
"text": "For an updated review, see V. hep-ph/0003212For an updated review, see V. Barger, hep-ph/0003212.",
"doi": "hep-ph/0003212"
},
{
"text": ". L J Hall, M Suzuki, Nucl. Phys. 231419L.J. Hall and M. Suzuki, Nucl. Phys. B231 (1984) 419.",
"doi": ""
},
{
"text": ". R Hempfling, Nucl... |
On Low Rank Classical Groups in String Theory, Gauge Theory and Matrix Models
Dec 2003
Ken Intriligator
Department of Physics
University of California
San Diego, La Jolla92093-0354CAUSA
Department of Physics and Astronomy
UCLA
90095-1547Los AngelesCAUSA
Anton V Ryzhov
Department of Physics
Brandeis University
024... | 119,068,847 | {
"arxiv": "hep-th/0311181",
"mag": 2171782575,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.nuclphysb.2003.12.030"
} | {
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"https://export.arxiv.org/pdf/hep-th/0311181v2.pdf"
],
"pdf_sha": "0078b4f507a921c898b1f97f6a28456164d57f21",
"open_access": {
"license": null,
"url": "https://dash.harvard.edu/bitstream/1/41385089/1/69919%200311181.pdf",
"status": "GREEN"
}
} | On Low Rank Classical Groups in String Theory, Gauge Theory and Matrix Models | [
"Ken Intriligator \nDepartment of Physics\nUniversity of California\nSan Diego, La Jolla92093-0354CAUSA\n\nDepartment of Physics and Astronomy\nUCLA\n90095-1547Los AngelesCAUSA",
"Anton V Ryzhov \nDepartment of Physics\nBrandeis University\n02454WalthamMAUSA",
"Masaki Shigemori \nDepartment of Physics and Astro... | We consider N = 1 supersymmetric U (N ), SO(N ), and Sp(N ) gauge theories, with two-index tensor matter and added tree-level superpotential, for general breaking patterns of the gauge group. By considering the string theory realization and geometric transitions, we clarify when glueball superfields should be included ... | [
{
"title": "Introduction",
"content": "Large N topological string duality [1] embedded in superstrings [2,3] has led to a new perspective on N = 1 supersymmetric gauge theories: that the exact effective superpotential can be efficiently computed by including glueball fields. For example, in a theory with ga... | [
{
"id": "",
"caption": "of [11] and sect. 3.3 relates this to Sp(1) → Sp(1) × Sp(0) N−2 , where the exact gauge theory result agrees with (6.4), up to the classical shift, upon using the relation (3.18). A priori, one might expect the string theory / matrix model result to disagree with (6.4), due to residu... | [
{
"text": "On the gauge theory/geometry correspondence. R Gopakumar, C Vafa, arXiv:hep-th/9811131Adv. Theor. Math. Phys. 31415R. Gopakumar and C. Vafa, \"On the gauge theory/geometry correspondence,\" Adv. Theor. Math. Phys. 3, 1415 (1999) [arXiv:hep-th/9811131].",
"doi": "arXiv:hep-th/9811131"
},
{
... |
ON A MIXED PROBLEM FOR THE PARABOLIC LAMÉ TYPE OPERATOR
15 Apr 2019
R Puzyrev
A Shlapunov
ON A MIXED PROBLEM FOR THE PARABOLIC LAMÉ TYPE OPERATOR
15 Apr 201910.1515/JIIP-2014-0043arXiv:1904.06797v1 [math.AP]
We consider a boundary value problem for the parabolic Lamé type operator being a linearization of the Navie... | 119,112,683 | {
"arxiv": "1904.06797",
"mag": 2333818229,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1515/jiip-2014-0043"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1904.06797v1.pdf"
],
"pdf_sha": "51a86184b95e0f61821791dd8f1c3134c77ec07f",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1904.06797",
"status": "GREEN"
}
} | ON A MIXED PROBLEM FOR THE PARABOLIC LAMÉ TYPE OPERATOR | [
"R Puzyrev",
"A Shlapunov"
] | We consider a boundary value problem for the parabolic Lamé type operator being a linearization of the Navier-Stokes' equations for compressible flow of Newtonian fluids. It consists of recovering a vector-function, satisfying the parabolic Lamé type system in a cylindrical domain, via its values and the values of the ... | [
{
"title": "Introduction",
"content": "Let, as usual, ∆ n be the Laplace operator, ∇ n be the gradient operator and div n be the divergence operator in R n , n ≥ 2. The Navier-Stokes' equations for compressible flow of Newtonian fluids over the four-dimensional domain D ⊂ R 3\nx ×R t under the action of a b... | [
{
"id": "",
"caption": "Find a vector-function u(x, t) ∈ [C 2,1 (Ω T )∩C 1,0 (Ω T ∪Γ T )] n satisfying the Lamé type equationL n+1 u = f in Ω T(14)",
"content": "Problem 1 .\n1Find a vector-function u(x, t) ∈ [C 2,1 (Ω T )∩C 1,0 (Ω T ∪Γ T )] n satisfying the Lamé type equationL n+1 u = f in Ω T(14)"
}... | [
{
"text": ". L D Landau, E M Lifshitz, A Course of Theoretical Physics. 6Pergamon PressFluid MechanicsLandau, L.D., Lifshitz, E.M. Fluid Mechanics (Volume 6 of A Course of Theoretical Physics), Pergamon Press, 1959.",
"doi": ""
},
{
"text": "Linear and Quasilinear Equations of Parabolic Type. O A La... |
Anisotropic magnetic properties of CeAg 2 Ge 2 single crystals
10 Jan 2007
A Thamizhavel
Department of Condensed Matter Physics and Materials Science
Tata Institute of Fundamental Research
Homi Bhabha Road400 005ColabaMumbaiIndia
R Kulkarni
Department of Condensed Matter Physics and Materials Science
Tata Institut... | 1,244,314 | {
"arxiv": "cond-mat/0611197",
"mag": 3021840504,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1016/j.physb.2006.01.483"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/cond-mat/0611197v2.pdf"
],
"pdf_sha": "a3cc9dc83e47764b5784372c5d6d438d9e8b2fc1",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/cond-mat/0611197",
"status": "GREEN"
}
} | Anisotropic magnetic properties of CeAg 2 Ge 2 single crystals | [
"A Thamizhavel \nDepartment of Condensed Matter Physics and Materials Science\nTata Institute of Fundamental Research\nHomi Bhabha Road400 005ColabaMumbaiIndia",
"R Kulkarni \nDepartment of Condensed Matter Physics and Materials Science\nTata Institute of Fundamental Research\nHomi Bhabha Road400 005ColabaMumbaiI... | In order to investigate the anisotropic magnetic properties of CeAg2Ge2, we have successfully grown the single crystals, for the first time, by high temperature solution growth (flux) method. We have performed a detailed study of the grown single crystals by measuring their electrical resistivity, magnetic susceptibili... | [
{
"title": "I. INTRODUCTION",
"content": "In the Ce-based intermetallic compounds, the competition between the RKKY interaction and the Kondo effect leads to diverse ground states. This competition can be readily studied in the multifarious CeT 2 X 2 compounds, where T is a transition metal and X is a group... | [
{
"id": "",
"caption": "(Color online) Powder X-ray diffraction pattern recorded for crushed single crystals of CeAg2Ge2 at room temperature. The solid line through the experimental data points is the Rietveld refinement profile calculated for the tetragonal CeAg2Ge2.",
"content": "FIG. 1 :\n1(Color onl... | [
{
"text": ". D Mond, J Jaccard, Flouquet, J. Low Temp. Phys. 122591mond, D. Jaccard and J. Flouquet, J. Low Temp. Phys. 122, 591 (2001).",
"doi": ""
},
{
"text": ". R Movshovich, T Graf, D Mandrus, J D Thompson, J L Smith, Z Fisk, Phys. Rev. B. 538241R. Movshovich, T. Graf, D. Mandrus, J. D. Thompso... |
arXiv:cond-mat/9409090v1 21 Sep 1994 Semi-classical description of the frustrated antiferromagnetic chain Typeset using REVT E X 1
D Allen
Centre de Recherche en Physique du Solide et Département de Physique
Université de Sherbrooke
J1K 2R1SherbrookeQuébecCanada
D Sénéchal
Centre de Recherche en Physique du Solid... | 9,529,653 | {
"arxiv": "cond-mat/9409090",
"mag": 2035979179,
"acl": null,
"pubmed": "9977179",
"pubmedcentral": null,
"dblp": null,
"doi": "10.1103/physrevb.51.6394"
} | {
"pdf_urls": [
"https://export.arxiv.org/pdf/cond-mat/9409090v1.pdf"
],
"pdf_sha": "3703467003d6d3e16a956152091d9c147a480ea7",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/cond-mat/9409090",
"status": "GREEN"
}
} | arXiv:cond-mat/9409090v1 21 Sep 1994 Semi-classical description of the frustrated antiferromagnetic chain Typeset using REVT E X 1 | [
"D Allen \nCentre de Recherche en Physique du Solide et Département de Physique\nUniversité de Sherbrooke\nJ1K 2R1SherbrookeQuébecCanada",
"D Sénéchal \nCentre de Recherche en Physique du Solide et Département de Physique\nUniversité de Sherbrooke\nJ1K 2R1SherbrookeQuébecCanada"
] | The antiferromagnetic Heisenberg model on a chain with nearest and next nearest neighbor couplings is mapped onto the SO(3) nonlinear sigma model in the continuum limit. In one spatial dimension this model is always in its disordered phase and a gap opens to excited states. The latter form a doubly degenerate spin-1 br... | [
{
"title": "I. INTRODUCTION",
"content": "Interest in low-dimensional magnetic systems has been great in recent years, partly because of the widespread belief that magnetism plays a key role in high-temperature superconductivity, but also because of the successful application of field-theoretic methods to t... | [
{
"id": "",
"caption": "AF) Heisenberg chain with nearest-neighbor (NN) coupling to the O(3)/O(2) sigma model has led Haldane to conjecture 1 that integer spin chains should exhibit a gap to a triplet of excited states, whereas half-integer spin chains should not. This conjecture has later been confirmed by... | [
{
"text": ". F D M Haldane, Phys. Lett. 93464F. D. M. Haldane, Phys. Lett. 93A, 464 (1983);",
"doi": ""
},
{
"text": ". Phys. Rev. Lett. 501153Phys. Rev. Lett. 50, 1153 (1983).",
"doi": ""
},
{
"text": ". M P Nightingale, H W J Blöte, Phys. Rev. B. 33659M. P. Nightingale, H. W. J. Blöte,... |
Sub-systems in nearby solar-type wide binaries 1
7 Jun 2010
Andrei Tokovinin atokovinin@ctio.noao.edu
Gemini Observatory
Cerro Tololo Inter-American Observatory
Casilla 603, Southern Operations Center, c/o AURA, Casilla 603, La SerenaLa SerenaChile, Chile
Markus Hartung mhartung@gemini.edu
Gemini Observatory
Cerro... | 73,629,363 | {
"arxiv": "1006.1253",
"mag": 2085464080,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1088/0004-6256/140/2/510"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/1006.1253v1.pdf"
],
"pdf_sha": "f7144aa52d1c979febd9b6c98f3be65b48ddc156",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/1006.1253",
"status": "GREEN"
}
} | Sub-systems in nearby solar-type wide binaries 1 | [
"Andrei Tokovinin atokovinin@ctio.noao.edu \nGemini Observatory\nCerro Tololo Inter-American Observatory\nCasilla 603, Southern Operations Center, c/o AURA, Casilla 603, La SerenaLa SerenaChile, Chile",
"Markus Hartung mhartung@gemini.edu \nGemini Observatory\nCerro Tololo Inter-American Observatory\nCasilla 603,... | We conducted a deep survey of resolved sub-systems among wide binaries with solar-type components within 67 pc from the Sun. Images of 61 stars in the K and H bands were obtained with the NICI adaptive-optics instrument on the 8-m Gemini-South telescope. Our maximum detectable magnitude difference is about 5 m and 7.8 ... | [
{
"title": "Introduction",
"content": "The quest for exo-planets has attracted attention to stars in the solar neighbourhood. A complete understanding of their formation requires a quantitative and predictive description of stellar multiplicity. Empirical data on stellar multiplicity serve as a benchmark fo... | [
{
"id": "",
"caption": "Based on observations obtained at the Gemini Observatory (Program ID GS-2009B-Q-49), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (... | [
{
"text": "E Artigau, B A Biller, Z Wahhaj, M Hartung, T L Hayward, L M Close, M R Chun, M C Liu, G Trancho, F Rigaut, Ground-based and Airborne Instrumentation for Astronomy II. Ed. McLean, I. S. & Casali, M. Proc. SPIE. 701466Artigau, E., Biller, B. A., Wahhaj, Z., Hartung, M., Hayward, T. L., Close, L. M., C... |
1943-2007
Olaf Post 1943-2007FIRST ORDER OPERATORS AND BOUNDARY TRIPLES 1
The aim of the present paper is to introduce a first order approach to the abstract concept of boundary triples for Laplace operators. Our main application is the Laplace operator on a manifold with boundary; a case in which the ordinary conce... | 17,085,199 | {
"arxiv": "0711.2218",
"mag": 2162102317,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": null,
"doi": "10.1134/s1061920807040139"
} | {
"pdf_urls": [
"https://arxiv.org/pdf/0711.2218v3.pdf"
],
"pdf_sha": "3eda14ebb2a7c1ce7cfd4d8819a9ff7175aa4357",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/0711.2218",
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} | [
"Olaf Post"
] | The aim of the present paper is to introduce a first order approach to the abstract concept of boundary triples for Laplace operators. Our main application is the Laplace operator on a manifold with boundary; a case in which the ordinary concept of boundary triples does not apply directly. In our first order approach, ... | [
{
"title": "Introduction",
"content": "The concept of boundary triples, originally introduced in [V63], has successfully be applied to the theory of self-adjoint extensions of symmetric operators, for example on quantum graphs, singular perturbations or point interactions on manifolds (see e.g. [BGP06]). Fo... | [
{
"id": "",
"caption": ". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . δ . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. ..... | [
{
"text": "Abstract boundary conditions for maximal sectorial extensions of sectorial operators. Y Arlinskii, Math. Nachr. 209Y. Arlinskii, Abstract boundary conditions for maximal sectorial extensions of sectorial operators, Math. Nachr. 209 (2000), 5-36.",
"doi": ""
},
{
"text": "Boundary value pr... | |
Graph Neural Tangent Kernel: Convergence on Large Graphs
31 May 2023
Sanjukta Krishnagopal
Luana Ruiz luanaruiz<ruizl@mit.edu>.
Graph Neural Tangent Kernel: Convergence on Large Graphs
31 May 2023D705755388A517EBAE3EC3909971A543arXiv:2301.10808v2[cs.LG]
Graph neural networks (GNNs) achieve remarkable performance in... | 256,274,591 | {
"arxiv": "2301.10808",
"mag": null,
"acl": null,
"pubmed": null,
"pubmedcentral": null,
"dblp": "conf/icml/KrishnagopalR23",
"doi": "10.48550/arxiv.2301.10808"
} | {
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"https://export.arxiv.org/pdf/2301.10808v2.pdf"
],
"pdf_sha": "4e0cef1f767e9a8a643eb529a6ba8584cbdd68c7",
"open_access": {
"license": null,
"url": "http://arxiv.org/pdf/2301.10808",
"status": "GREEN"
}
} | Graph Neural Tangent Kernel: Convergence on Large Graphs | [
"Sanjukta Krishnagopal",
"Luana Ruiz luanaruiz<ruizl@mit.edu>."
] | Graph neural networks (GNNs) achieve remarkable performance in graph machine learning tasks but can be hard to train on large-graph data, where their learning dynamics are not well understood.We investigate the training dynamics of largegraph GNNs using graph neural tangent kernels (GNTKs) and graphons.In the limit of ... | [
{
"title": "Introduction",
"content": "Several real-world systems such as social-interactions, brainconnectome, epidemic spread, recommender systems, and traffic patterns are best represented by structured data in the form of large graphs.Graph neural networks (GNNs) are deep neural network architectures th... | [
{
"id": "",
"caption": "Figure 1.Difference between the test error achieved by the GNTK fitted on the n-node graph when used for prediction the n-node graph, and the test error achived by the same GNTK when used for predicition on the N -node graph.(left) Opinion dynamics on geometric and SBM graphs, where ... | [
{
"text": "A convergence theory for deep learning via over-parameterization. Z Allen-Zhu, Y Li, Z Song, Int. Conf. Mach. Learning. PMLR2019",
"doi": ""
},
{
"text": "Spectral analysis of collectively compact, strongly convergent operator sequences. P Anselone, T Palmer, Pacific Journal of Mathematic... |