pinecone/core-2020-05-10-deduplication huggingface dataset:
An Isoperimetric Inequality and the First Steklov Eigenvalue

AbstractLet (Mn, g) be a compact Riemannian manifold with boundary. In this paper we give upper and lower estimates for the first nonzero Steklov eigenvalueΔϕ=0inM,∂ϕ∂η=ν1ϕon∂M,where ν1 is a positive real number. The estimate from below is for a star-shaped domain on a manifold whose Ricci curvature is bounded from below. The upper estimate is for a convex manifold with nonnegative Ricci curvature and is given in terms of the first nonzero eigenvalue for the Laplacian on the boundary. We prove a comparison theorem for simply connected domains in a simply connected manifold. We exhibit annuli domains for which the comparison theorem fails to be true. In (J. F. Escobar, J. Funct. Anal.60 (1997), 544–556) we introduced the isoperimetric constant I(M) defined asI(M)=infΩ⊂MVol(Σ)min{Vol(Ω1), Vol(Ω2)}, where Ω1=Ω∩∂M is a nonempty domain with boundary in the manifold ∂M, Ω2=∂M−Ω1, and Σ=∂Ω∩int(M), where int(M) is the interior of M. We proved a Cheeger's type inequality for ν1 using the constant I(M). In this paper we give upper and lower estimates for the constant I in terms of isoperimetric constants of the boundary of M
Short Time Behavior of Hermite Functions on Compact Lie Groups

AbstractLetpt(x) be the (Gaussian) heat kernel on Rnat timet. The classical Hermite polynomials at timetmay be defined by a Rodriguez formula, given byHα(−x, t)pt(x)=αpt(x), whereαis a constant coefficient differential operator on Rn. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new class of functions on a general compact Lie group,G. In analogy with the Rncase, these “Hermite functions” onGare obtained by the same formula, whereinpt(x) is now the heat kernel on the group, −xis replaced byx−1, andαis a right invariant differential operator. Let g be the Lie algebra ofG. Composing a Hermite function onGwith the exponential map produces a family of functions on g. We prove that these functions, scaled appropriately int, approach the classical Hermite polynomials at time 1 asttends to 0, both uniformly on compact subsets of g and inLp(g, μ), where 1⩽p<∞, andμis a Gaussian measure on g. Similar theorems are established whenGis replaced byG/K, whereKis some closed, connected subgroup ofG
Transference in Spaces of Measures

AbstractThe transference theory for Lp spaces of Calderón, Coifman, and Weiss is a powerful tool with many applications to singular integrals, ergodic theory, and spectral theory of operators. Transference methods afford a unified approach to many problems in diverse areas, which previously were proved by a variety of methods. The purpose of this paper is to bring about a similar approach to the study of measures. Specifically, deep results in classical harmonic analysis and ergodic theory, due to Bochner, de Leeuw and Glicksberg, Forelli, and others are all extensions of the classical F. & M. Riesz Theorem. We show that all these extensions are obtainable via our new transference principle for spaces of measures
Extensions of Pure States and Projections of Norm One

AbstractWe show that the extension property for pure states of a C*-subalgebra B of a C*-algebra A leads to the existence of a projection of norm one R:A→B in the case where B is liminal with Hausdorff primitive ideal space. Furthermore, R is given by a “Dixmier process” in which the averaging is effected by a group of unitary elements in the centre of the multiplier algebra M(B). These results generalize earlier work of J. Anderson and the author for the case when B is a masa of A. Various applications are given in the context of inductive limit algebras such as AF algebras and, more generally, Kumjian's ultraliminary C*-algebras
Finite Dimensional Approximations to Wiener Measure and Path Integral Formulas on Manifolds

AbstractCertain natural geometric approximation schemes are developed for Wiener measure on a compact Riemannian manifold. These approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi-group on Riemannian manifolds. The path space is approximated by finite dimensional manifolds HP(M) consisting of piecewise geodesic paths adapted to partitions P of [0, 1]. The finite dimensional manifolds HP(M) carry both an H1 and a L2 type Riemannian structures, G1P and G0P, respectively. It is proved that (1/ZiP)e−(1/2)E(σ)dVolGiP(σ)→ρi(σ)dν(σ) as mesh (P)→0, where E(σ) is the energy of the piecewise geodesic path σ∈HP(M), and for i=0 and 1, ZiP is a “normalization” constant, VolGiP is the Riemannian volume form relative to GiP, and ν is Wiener measure on paths on M. Here ρ1(σ)≡1 and ρ0(σ)=exp(−16∫10Scal(σ(s))ds) where Scal is the scalar curvature of M. These results are also shown to imply the well known integration by parts formula for the Wiener measure
A-Valued Semicircular Systems

AbstractTo a von Neurnann algebra A and a set of linear maps ηij:A→A, i, j∈I such that a↦(ηij)ij∈I:A→A⊗B(l2(I)) is normal and completely positive, we associate a von Neumann algebra Φ(A, η). This von Neumann algebra is generated by A and an A-valued semicircular system Xi, i∈I, associated to η. In many cases there is a faithful conditional expectation E:Φ(A, η)→A; if A is tracial, then under certain assumptions on η, Φ(A, η) also has a trace. One can think of the construction Φ(A, η) as an analogue of a crossed product construction. We show that most known algebras arising in free probability theory can be obtained from the complex field by iterating the construction Φ. Of a particular interest are free Krieger algebras, which, by analogy with crossed products and ordinary Krieger factors, are defined to be algebras of the form Φ(L∞[0, 1], η). The cores of free Araki–Woods factors are free Krieger algebras. We study the free Krieger algebras and as a result obtain several non-isomorphism results for free Araki–Woods factors. As another source of classification results for free Araki–Woods factors, we compute the τ invariant of Connes for free products of von Neumann algebras. This computation generalizes earlier work on computation of T, S, and Sd invariants for free product algebras
Absolutely Continuous Flows Generated by Sobolev Class Vector Fields in Finite and Infinite Dimensions

AbstractWe prove the existence of the global flow {Ut} generated by a vector field A from a Sobolev class W1, 1(μ) on a finite- or infinite-dimensional space X with a measure μ, provided μ is sufficiently smooth and that a ∇A and |δμA| (where δμA is the divergence with respect to μ) are exponentially integrable. In addition, the measure μ is shown to be quasi-invariant under {Ut}. In the case X=Rn and μ=pdx, where p∈W1, 1loc(Rn) is a locally uniformly positive probability density, a sufficient condition is exp(c‖∇A‖)+exp(c|(A, (∇p/p))|)∈L1(μ) for all c. In the infinite-dimensional case we get analogous results for measures differentiable along sufficiently many directions. Examples of measures which fit our framework, important for applications, are symmetric invariant measures of infinite-dimensional diffusions and Gibbs measures. Typically, in both cases such measures are essentially non-Gaussian. Our result in infinite dimensions significantly extends previously studied cases where μ was a Gaussian measure. Finally, we study flows generated by vector fields whose values are not necessarily in the Cameron–Martin space
What Is Not in the Domain of the Laplacian on Sierpinski Gasket Type Fractals

AbstractWe consider the analog of the Laplacian on the Sierpinski gasket and related fractals, constructed by Kigami. A function f is said to belong to the domain of Δ if f is continuous and Δf is defined as a continuous function. We show that if f is a nonconstant function in the domain of Δ, then f2 is not in the domain of Δ. We give two proofs of this fact. The first is based on the analog of the pointwise identity Δf2−2fΔf=|∇f|2, where we show that |∇f|2 does not exist as a continuous function. In fact the correct interpretation of Δf2 is as a singular measure, a result due to Kusuoka; we give a new proof of this fact. The second is based on a dichotomy for the local behavior of a function in the domain of Δ, at a junction point x0 of the fractal: in the typical case (nonvanishing of the normal derivative) we have upper and lower bounds for |f(x)−f(x0)| in terms of d(x, x0)β for a certain value β, and in the nontypical case (vanishing normal derivative) we have an upper bound with an exponent greater than 2. This method allows us to show that general nonlinear functions do not operate on the domain of Δ
Jacobi Matrices with Power-like Weights—Grouping in Blocks Approach

AbstractThe paper deals with Jacobi matrices with weights λk given by λk=kα(1+Δk), where α∈(12, 1) and limkΔk=0. The main question studied here concerns when the spectrum of the operator J defined by the Jacobi matrix has absolutely continuous component covering the real line. A sufficient condition is given for a positive answer to the above question. The method used in the paper is based on a detailed analysis of generalized eigenvectors of J. In turn this analysis relies on the so-called grouping in blocks approach to a large product of the transfer matrices associated to J
On the Approximation of Quasidiagonal C*-Algebras

AbstractLet A be a separable exact quasidiagonal C*-algebra. Suppose that π:A→L(H) is a faithful representation whose image does not contain nonzero compact operators. Then there exists a sequence ϕn:A→L(H) of completely positive contractions such that ‖π(a)−ϕn(a)‖→0 for all a∈A, and the C*-algebra generated by ϕn(A) is finite dimensional for each n. As an application it is shown that if the C*-algebra generated by a quasidiagonal operator T is exact and does not contain any nontrivial compact operator, then T is norm-limit of block-diagonal operators D=D1 ⊕D2⊕… with supirank(Di)<∞
Hereditarily Hypercyclic Operators

AbstractWe show that a continuous linear operator T on a Fréchet space satisfies the so-called Hypercyclicity Criterion if and only if it is hereditarily hypercyclic, and if and only if the direct sum T⊕T is hypercyclic. In particular, hypercyclic operators with either a dense generalized kernel or a dense set of periodic points (i.e., chaotic in the sense of R. L. Devaney (1989, “An Introduction to Chaotic Dynamical Systems,” Addison–Wesley, Reading, MA)) must satisfy the Criterion. Finally, we provide a characterization of those weighted shifts T that are hereditarily hyper-cyclic with respect to a given sequence (nk) of positive integers, as well as conditions under which T and {Tnk}k⩾1 share the same set of hypercyclic vectors
Discrete Morse Theory and Extended L2 Homology

AbstractA brief overview of Forman's discrete Morse theory is presented, from which analogues of the main results of classical Morse theory can be derived for discrete Morse functions, these being functions mapping the set of cells of a CW complex to the real numbers satisfying some combinatorial relations. The discrete analogue of the strong Morse inequality was proved by Forman for finite CW complexes using a Witten deformation technique. This deformation argument is adapted to provide strong Morse inequalities for infinite CW complexes which have a finite cellular domain under the free cellular action of a discrete group. The inequalities derived are analogous to the L2 Morse inequalities of Novikov and Shubin and the asymptotic L2 Morse inequalities of an inexact Morse 1-form as derived by Mathai and Shubin. We also obtain quantitative lower bounds for the Morse numbers whenever the spectrum of the Laplacian contains zero, using the extended category of Farber
Explicit Stochastic Analysis of Brownian Motion and Point Measures on Riemannian Manifolds

AbstractThe gradient and divergence operators of stochastic analysis on Riemannian manifolds are expressed using the gradient and divergence of the flat Brownian motion. By this method we obtain the almost-sure version of several useful identities that are usually stated under expectations. The manifold-valued Brownian motion and random point measures on manifolds are treated successively in the same framework, and stochastic analysis of the Brownian motion on a Riemannian manifold turns out to be closely related to classical stochastic calculus for jump processes. In the setting of point measures we introduce a damped gradient that was lacking in the multidimensional case
Continuous Domain Dependence of the Eigenvalues of the Dirichlet Laplacian and Related Operators in Hilbert Space

AbstractIn a Hilbert space (H, ‖·‖) is given a dense subspace W and a closed positive semidefinite quadratic form Q on W×W. Thus W is a Hilbert space with the norm ‖u‖1=(‖u‖2+Q(u))1/2. For any closed subspace D of (W, ‖·‖1) let A(D) denote the selfadjoint operator in the closure of D in H such that ‖A(D)1/2u‖2=Q(u) for every u∈D. For any decreasing sequence of closed subspaces Di of W with intersection ∩Di=D such that each A(Di) has compact resolvent it is shown that, for every n, the nth eigenvalue λn(Di) of A(Di) converges to that of A(D), and that A(Di)−1→A(D)−1 in operator norm. Similar results are obtained for any order convergent sequence in the conditionally complete lattice of all closed subspaces D of W such that A(D)−1 has compact resolvent. Next, these results are applied to the Dirichlet laplacian, and more generally to the Dirichlet poly-laplacian, on a sequence of bounded open subsets of RN
L1-Theory of Scalar Conservation Law with Continuous Flux Function

AbstractUniqueness of a generalized entropy solution (g.e.s.) to the Cauchy problem for N-dimensional scalar conservation laws ut+divxφ(u)=g, u(0, ·)=f with continuous flux function φ is still an open problem. For data (f, g) vanishing at infinity, we show that there exist a maximal and a minimal g.e.s. to the Cauchy problem and to the associated stationary problem u+divxφ(u)=f. In the case of L1 data, using the nonlinear semigroup theory, we prove that there is uniqueness for all data of a g.e.s. to the Cauchy problem if and only if there is uniqueness for all data of a g.e.s. to the related stationary problem. Applying this result and an induction argument on the dimension N, we extend uniqueness results of Bénilan, Kruzhkov (1996, Nonlinear Differential Equations Appl.3, 395–419) for flux having some monotonicity properties
Uniqueness of Symmetric Vortex Solutions in the Ginzburg–Landau Model of Superconductivity

AbstractSymmetric vortices are finite energy solutions ψ, A to the Ginzburg–Landau equations of superconductivity with the form ψ=f(r)eidθ, A=S(r)/r2(−y, x). The existence, regularity, and asymptotic form of the solutions f(r), S(r) for any d∈Z\{0} have been established by Plohr and by Burger and Chen. In this paper we prove the uniqueness of these solutions when the Ginzburg–Landau parameter κ satisfies κ2⩾2d2, for any fixed d∈Z\{0}. To do this, we show that any such solution is a non-degenerate relative minimizer of the free energy functional constrained to a convex set, then use a version of the Mountain Pass Theorem to derive a contradiction, should there be more than one solution
Restrictions of Quaternionic Representations

AbstractIn a paper by Gross and Wallach [1996, J. Reine Angew. Math.481, 73–123] the K-types of the continuations of the quaternionic discrete series of a quaternionic Lie group G are associated with projective orbits O of certain subgroups in G(C). In this paper, we will show that the restrictions of the representations to quaternionic subgroups are closely related with the intersection of the Zariski closure of O with hyperplanes. We apply this to the minimal representations of the exceptional groups of real rank 4 and investigate the correspondences of certain compact dual pairs
Quantum Galois Correspondence for Subfactors

AbstractOcneanu has obtained a certain type of quantized Galois correspondence for the Jones subfactors of type An and his arguments are quite general. By making use of them in a more general context, we define a notion of a subequivalent paragroup and establish a bijective correspondence between generalized intermediate subfactors in the sense of Ocneanu and subequivalent paragroups for a given strongly amenable subfactors of type II1 in the sense of Popa, by encoding the subequivalence in terms of a commuting square. For this encoding, we generalize Sato's construction of equivalent subfactors of finite depth from a single commuting square, to strongly amenable subfactors. We also explain a relation between our notion of subequivalent paragroups and sublattices of a Popa system, using open string bimodules
Completely Indecomposable Operators and a Uniqueness Theorem of Cartwright–Levinson Type

AbstractA bounded linear operator T on a complex Hilbert space will be called completely indecomposable if its spectrum is not a singleton, and is included in the spectrum of the restrictions of T and T* to any of their nonzero invariant subspaces. Two classes of completely indecomposable operators are constructed. The first consists of essentially selfadjoint operators with spectrum [−2, 2], and the second of bilateral weighted shifts whose spectrum is the unit circle. We do not know whether any of the operators in the first class has a proper invariant subspace and if any of the operators in the second class has a proper hyperinvariant subspace. We also establish a new uniqueness theorem of Cartwright–Levinson type which is the main ingredient in our proofs of complete indecomposability
On Fundamental Solutions of Generalized Schrödinger Operators

AbstractWe consider the generalized Schrödinger operator −Δ+μ, where μ is a nonnegative Radon measure in Rn, n⩾3. Assuming that μ satisfies certain scale-invariant Kato conditions and doubling conditions we establish the following bounds for the fundamental solution of −Δ+μ in Rn,ce−ε2d(x, y, μ)|x−y|n−2⩽Γμ(x, y)⩽Ce−ε1d(x, y, μ)|x−y|n−2, where d(x, y, μ) is the distance function for the modified Agmon metric m(x, μ)dx2 associated with μ. We also study the boundedness of the corresponding Riesz transforms ∇(−Δ+μ)−1/2 on Lp(Rn, dx)
Pullback and Pushout Constructions in C*-Algebra Theory

AbstractA systematic study of pullback and pushout diagrams is conducted in order to understand restricted direct sums and amalgamated free products of C*-algebras. Particular emphasis is given to the relations with tensor products (both with the minimal and the maximal C*-tensor norm). Thus it is shown that pullback and pushout diagrams are stable under tensoring with a fixed algebra and stable under crossed products with a fixed group. General tensor products between diagrams are also investigated. The relations between the theory of extensions and pullback and pushout diagrams are explored in some detail. The crowning result is that if three short exact sequences of C*-algebras are given, with appropriate morphisms between the sequences allowing for pullback or pushout constructions at the levels of ideals, algebras and quotients, then the three new C*-algebras will again form a short exact sequence under some mild extra conditions. As a generalization of a theorem of T. A. Loring it is shown that each morphism between a pair of C*-algebras, combined with its extension to the stabilized algebras, gives rise to a pushout diagram. This result has applications to corona extendibility and conditional projectivity. Finally the pullback and pushout constructions are applied to the class of noncommutative CW complexes defined by (S. Eilers, T. A Loring, and G. K. Pedersen J. Reine Angew. Math., 1998, 499, 101–143) to show that this category is stable under tensor products and under restricted direct sums
Integral Geometry on Grassmann Manifolds and Calculus of Invariant Differential Operators

AbstractIn this paper, we mainly deal with two problems in integral geometry, the range characterizations and construction of inversion formulas for Radon transforms on higher rank Grassmann manifolds. The results will be described explicitly in terms of invariant differential operators. We will also study the harmonic analysis on Grassmann manifolds, using the method of integral geometry. In particular, we will give eigenvalue formulas and radial part formulas for invariant differential operators
Microlocal WKB Expansions

AbstractIn relation with microlocal tunneling, we investigate the existence of micro-local WKB expansions for the eigenfunctions of semiclassical pseudodifferential operators whose symbol admits a non degenerate minimum at some point (x0, ξ0) of R2n. More precisely, under assumptions of analyticity, we prove that the FBI (or Bargman) transform of the first eigenfunction of such an operator admits near (x0, ξ0) an expansion of the form e−Φ(x, ξ)/h∑j⩾0hjaj(x, ξ) as the semiclassical parameter h tends to 0. Here Φ and the aj's are smooth (complex) functions, and the domain in which the expansion is valid can be described in terms of deformation properties. Our result can be applied e.g. to magnetic Schrödinger operators with electric potential admitting a non degenerate well
Commutator Algebra and Abstract Smoothing Effect

AbstractWe consider a “dispersive” evolution equation in a Hilbert space and prove abstract smoothing effects “in an incoming region” under a Mourre-type condition “near infinity.” For this purpose, we introduce commutator algebras acting on weighted Sobolev spaces associated with two self-adjoint operators and construct various time-dependent nonnegative observables with nonpositive Heisenberg derivative. Our approach is applicable to Schrödinger evolution equations on complete Riemannian manifolds with suitable strictly convex functions near infinity: (i) asymptotically Euclidean metric with long-range metric perturbation, (ii) conformally compact metric, (iii) generalized scattering metric, (iv) metric of separation of variables near infinity, etc
Universal Lower Bounds for Quantum Diffusion

AbstractWe study the connections between dynamical properties of Schrödinger operators H on separable Hilbert space H and the properties of corresponding spectral measures. Our main result establishes a relation for the moment of order p of the form〈〈|X|p〉ψ(t)〉(T)≡T−1∫T0‖|X|p/2e−itHψ‖2Hdt⩾Lψ, p/d(T). (1) Here Lψ, p/d(T) is a function connected to the behavior of the Fourier transform of measures in the subclass of measures absolutely continuous with respect to the spectral measure μψ. Beyond the intrinsic interest of the general formulation (1), this result allows us to derive necessary conditions for dynamical localization in the presence of a pure point spectrum. On the other hand, if we focus on subsequences of time Tk↗+∞, we can exhibit lower bounds which are, in certain cases, strictly larger than the well-known power-law lower bound for all T expressed in terms of the Hausdorff dimension of spectral measures
Rademacher's Theorem on Configuration Spaces and Applications

AbstractWe consider an L2-Wasserstein type distance ρ on the configuration space ΓX over a Riemannian manifold X, and we prove that ρ-Lipschitz functions are contained in a Dirichlet space associated with a measure on ΓX satisfying certain natural assumptions. These assumptions are in particular fulfilled by the classical Poisson measures and by a large class of tempered grandcanonical Gibbs measures with respect to a superstable lower regular pair potential. As an application we prove a criterion in terms of ρ for a set to be exceptional. This result immediately implies, for instance, a quasi-sure version of the spatial ergodic theorem. We also show that ρ is optimal in the sense that it is the intrinsic metric of our Dirichlet form