pinecone/core-2020-05-10-deduplication huggingface dataset:
Affine Systems inL2(Rd): The Analysis of the Analysis Operator

AbstractDiscrete affine systems are obtained by applying dilations to a given shift-invariant system. The complicated structure of the affine system is due, first and foremost, to the fact that it is not invariant under shifts. Affine frames carry the additional difficulty that they are “global” in nature: it is the entire interaction between the various dilation levels that determines whether the system is a frame, and not the behaviour of the system within one dilation level. We completely unravel the structure of the affine system with the aid of two new notions:the affine product, anda quasi-affine system. This leads to a characterization of affine frames; the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the Fourier domain. Several results, such as a general oversampling theorem follow from these characterizations. Most importantly, the affine product can be factored during a multiresolution analysis construction, and this leads to a complete characterization of all tight frames that can be constructed by such methods. Moreover, this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution. Of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions, and that the freedom offered by redundancy can be fully exploited in these constructions
Simplified Excision Techniques for Free Discontinuity Problems in Several Variables

AbstractOne of the methods proposed for the study of the problems with a free discontinuity set in two variables has been considered here under slightly different abstract assumptions from those in the previous papers. The technical difficulties are in this new setting simplified and less background is required to the reader. Moreover, applications to problems in several variables can be found thanks to a Sobolev-type theorem which ensures the Hölder continuity of certain functions out of a suitable neighborhood of their discontinuity set. Many of the results established so far for the case of two dimensions can be so extended to the general case. A new property is also proved
Equivariant Deformations of Homogeneous Spaces

AbstractC*-algebraic deformations of homogeneous spacesG/Γare constructed by completing dense subspaces ofC0(G/Γ) in a different multiplication andC*-norm; these deformations are equivariant in the sense that they still carry a natural action ofGby left translation. The motivating examples are the noncommutative Heisenberg manifolds of Rieffel, but the construction given here is based on the authors' twisted dual-group algebras, which are equivariant deformations ofC0(G). The procedure is general enough to give other recently studied families of noncommutative spaces and some interesting new examples of simpleC*-algebra
Fourier–Stieltjes Algebras of Locally Compact Groupoids

AbstractFor locally compact groups, Fourier algebras and Fourier–Stieltjes algebras have proven to be useful dual objects. They encode the representation theory of the group via the positive definite functions on the group: positive definite functions correspond to cyclic representations and span these algebras as linear spaces. They encode information about the algebra of the group in the geometry of the Banach space structure, and the group appears as a topological subspace of the maximal ideal space of the algebra. Because groupoids and their representations appear in studying operator algebras, ergodic theory, geometry, and the representation theory of groups, it would be useful to have a duality theory for them. This paper gives a first step toward extending the theory of Fourier–Stieltjes algebras from groups to groupoids. IfGis a locally compact (second countable) groupoid, we show that B(G), the linear span of the Borel positive definite functions onG, is a Banach algebra when represented as an algebra of completely bounded maps on aC*-algebra associated withG. This necessarily involves identifying equivalent elements of B(G). An example shows that the linear span of the continuous positive definite functions need not be complete. For groups, B(G) is isometric to the Banach space dual ofC*(G). For groupoids, the best analog of that fact is to be found in a representation of B(G) as a Banach space of completely bounded maps from aC*-algebra associated withGto aC*-algebra associated with the equivalence relation induced byG. This paper adds weight to the clues in earlier work of M. E. Walter on Fourier–Stieltjes algebras that there is a much more general kind of duality for Banach algebras waiting to be explored
Linear Structure of Hypercyclic Vectors

AbstractA vectorxin a Banach space B is called hypercyclic for a bounded linear operatorT:B→B if the orbit {Tnx:n⩾1} is dense in B. Our main result states that ifTis a compact perturbation of an operator of norm ⩽1 and satisfies an appropiate extra hypothesis, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors forT. In particular the result applies to compact perturbations of the identity. We also include applications to some weighted backward shifts and compact perturbations of the identity by weighted backward shifts. This last result in combination with a recent one that states that every Banach space admits an operator with a hypercyclic vector proves that in all Banach space there is an operatorTwith an infinite-dimensional closed subspace consisting, except for zero, of hypercyclic vectors. The main result also applies to the differentiation operator and the translation operatorT:f(z)→f(z+1) on certain Hilbert spaces consisting of entire functions
Semi-classical States for Nonlinear Schrödinger Equations

AbstractWe consider existence and asymptotic behavior of solutions for an equation of the formε2Δu−V(x)u+f(u)=0,u>0,u∈H10(Δ), ((*))whereΔis a smooth domain in RN, not necessarily bounded. We assume that the potentialVis positive and that it possesses atopologically nontrivialcritical valuec, characterized through a min–max scheme. The functionfis assumed to be locally Hölder continuous having a subcritical, superlinear growth. Further we assume thatfis such that the correspondinglimiting equation inRNhas a unique solution, up to translations. We prove that there existsε0so that for all 0<ε<ε0, Eq.(*) possesses a solution having exactly one maximum pointxε∈Δ, such thatV(xε)→cand ∇V(xε)→0 asε→0
An Inequality for Fourier–Laplace Transforms of Entire Functions, and the Existence of Exponential Frames in Fock Space

AbstractWe study mapping properties of the Fourier–Laplace transform between certain spaces of entire functions. We introduce a variant of the classical Fock space by integrating against the Monge–Ampère measure of the weight function and show that the norm of the Fourier–Laplace transform, in a dual Fock space, dominates the norm of the function. Equality holds when the weight function is an Hermitean form. As an application we get a criterium for the existence of frames of exponential functions in Fock space
Flow Equations on Spaces of Rough Paths

AbstractGiven an Itô vector fieldM, there is a unique solutionξt(h) to the differential equationdξt(h)dt=M(ξt(h)),ξ0(h)=hfor any continuous and piece-wisely smooth pathh. We show that for anyt∈R, the maph→ξt(h) is continuous in thep-variation topology for anyp⩾1, so that it uniquely extends to a solution flow on the space of all geometric rough paths. Applying this result to the Driver's geometric flow equation on the path space over a closed Riemannian manifolddζtdt=Xh(ζt),ξ0=id,whereXhis the vector field defined by parallel translatinghvia a connection, our result especially yields a deterministic construction of the Driver's flow
Weighted Stochastic Sobolev Spaces and Bilinear SPDEs Driven by Space–Time White Noise

AbstractIn this paper we develop basic elements of Malliavin calculus on a weightedL2(Ω). This class of generalized Wiener functionals is a Hilbert space. It turns out to be substantially smaller than the space of Hida distributions while large enough to accommodate solutions of bilinear stochastic PDEs. As an example, we consider a stochastic advection-diffusion equation driven by space-time white noise in Rd. It is known that ford>1, this equation has no solutions inL2(Ω). In contrast, it is shown in the paper that in an appropriately weightedL2(Ω) there is a unique solution to the stochastic advection-diffusion equation for anyd⩾1. In addition we present explicit formulas for the Hermite–Fourier coefficients in the Wiener chaos expansion of the solution
Laplace Transforms and Integrated, Regularized Semigroups in Locally Convex Spaces

AbstractIn this paper, we consider the vector-valued Laplace transforms,r-times (r∈[0,∞)) integrated semigroups and regularized semigroups in the context of sequentially complete locally convex spaces. Our theorems develop the corresponding results in [1,11], including the well known integrated version of the classical Widder's representation theorem of Laplace transforms for functions taking values in Banach spaces. Moreover, we study a class of differential operators on certain function spaces. Optimal conditions, making them the generators of integrated or regularized semigroups, are obtained. We finally show some applications to abstract Cauchy problems
On the Asymptotic Behavior of Counting Functions Associated to Degenerating Hyperbolic Riemann Surfaces

AbstractIn this article we will study what we call weighted counting functions on hyperbolic Riemann surfaces of finite volume. IfMis compact, then we define the weighted counting function forw⩾0 to beNM,w(T)=∑λn⩽T(T−λn)w,where {λn} is the set of eigenvalues of the Laplacian which acts on the space of smooth functions onM. IfMis non-compact, then we define the weighted counting functionNM,w(T) via the inverse Laplace transform from which one can express the weighted counting function in terms of spectral data associated toM(see Proposition 5.1 and Remark 5.2). Using the convergence results from [JL2] concerning the regularized heat trace on finite volume hyperbolic Riemann surfaces, we shall prove the following results. Theorem.Let Mldenote a degenerating family of compact or non-compact hyperbolic Riemann surfaces of finite volume which converges to the non-compact hyperbolic surface M0.LetSldenote the set of lengths of pinching geodesics on the family of surfaces Ml.For w⩾0and T⩾1/4,letGl,w(T)=2Γ(w+1)(16π)1/2∑n=1∞∑lk∈S1lksinfh(nlk/2)T−1/4(nlk/2)2(w+1/2)/2Jw+1/2(nlk(T−1/4)),where Js(x)is the J-Bessel function and Γ(x)is the standard gamma function. For0⩽T<1/4,defineGl,w(T)to be zero.Then: (a)Forw>3/2andT>0,we haveNMl,w(T)=Gl,w(T)+NM0,w(T)+o(1)asl→0;(b)For w⩾0andT>0,we haveNMl,w(T)=Gl,w(T)+o∑lk∈Sllog(1/lk)asl→0;:(c)For w⩾0andT⩾1/4,we haveGl,w(T)=2Γ(w+1)(T−1/4)w+1/2(4π)1/2Γ(w+3/2)∑lk∈Sllog(1/lk)+O(1)asl→0.Observe that by takingT<1/4, part (a) proves convergence of certain symmetric functions of the small eigenvalues, thus we prove the main result from [CC]. We discuss a method to improve the error terms in asymptotic expansion of counting functions for 0⩽w⩽3/2; see Remark 2.2 and Remark 3.6 for further discussion of the organization of this article
Convergence of the Normalized Spectral Counting Function on Degenerating Hyperbolic Riemann Surfaces of Finite Volume

AbstractIn this paper we study spectral asymptotics of degenerating families of hyperbolic Riemann surfaces, either compact or non-compact but always of finite volume. We prove that the second integral of the spectral counting function has an asymptotic expansion out too(ℓ), where ℓ is the degeneration parameter. The first term in the expansion is a diverging term which depends solely on the degeneration parameter and the counting parameters and the second term is the second integral of the spectral counting function of the limit surface
Ergodicity for the Stochastic Dynamics of Quasi-invariant Measures with Applications to Gibbs States

AbstractThe convex set Maof quasi-invariant measures on a locally convex spaceEwith given “shift”-Radon–Nikodym derivatives (i.e., cocycles)a=(atk)k∈K0,t∈Ris analyzed. The extreme points of Maare characterized and proved to be non-empty. A specification (of lattice type) is constructed so that Macoincides with the set of the corresponding Gibbs states. As a consequence, via a well known method due to Dynkin and Föllmer a unique representation of an arbitrary element in Main terms of extreme ones is derived. Furthermore, the corresponding classical Dirichlet forms (Eν,D(Eν)) and their associated semigroups (Tνt)t>0onL2(E;ν) are discussed. Under a mild positivity condition it is shown thatν∈Mais extreme if and only if (Eν,D(Eν)) is irreducible or equivalently, (Tνt)t>0is ergodic. This implies time-ergodicity of associated diffusions. Applications to Gibbs states of classical and quantum lattice models as well as those occuring in Euclidean quantum field theory are presented. In particular, it is proved that the stochastic quantization of a Guerra–Rosen–Simon Gibbs state on D′(R2) ininfinite volumewith polynomial interaction is ergodic if the Gibbs state is extreme (i.e., is a pure phase), which solves a long-standing open problem
Weighted Norm Inequalities for Some Singular Integral Operators

AbstractFor bounded Lebesgue measurable functionsα,βon the unit circle,Sα,β=αP++βP−is called a singular integral operator, whereP+is an analytic projection andP−is a co-analytic projection. We study one-weighted norm inequalities ofSα,βonL2(W). We introduce a classHSrof weights withr=|α−β|/‖α−β‖∞in order to characterize those weights. For example, we show thatSα,βis bounded with respect to a weightWif and only ifWbelongs toHSror |α−β|W≡0. Ifris a nonzero constant, thenHSris just a well-known class of weights due to H. Helson and G. Szegő (Ann. Mat. Pura Appl.51(1960), 107–138). Moreover we study the Koosis-type problem of two weights ofSα,βand get very simple necessary and sufficient conditions for such weights
Essentially Smooth Lipschitz Functions

AbstractIn this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability,D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipschitz functions that are integrable,D-representable and possess desirable differentiability properties. We also present some striking applications of our results to distance functions
Integration by Parts and Quasi-Invariance for Heat Kernel Measures on Loop Groups

AbstractIntegration by parts formulas are established both for Wiener measure on the path space of a loop group and for the heat kernel measures on the loop group. The Wiener measure is defined to be the law of a certain loop group valued “Brownian motion” and the heat kernel measures are timet,t>0, distributions of this Brownian motion. A corollary of either of these integrations by parts formulas is the closability of the pre-Dirichlet form considered by B. K. Driver and T. Lohrenz [1996,J. Functional Anal.140, 381–448]. We also show that the heat kernel measures are quasi-invariant under right under right and left translations by finite energy loops
Ergodic Theorems for Free Group Actions on von Neumann Algebras

AbstractWe extend a recent ergodic theorem of A. Nevo and E. Stein to the non-commutative case. Letρbe a faithful normal state on the von Neumann algebraA. Let {ai}ri=1generateFr, the free group onrgenerators, and let {αi}ri=1be *-automorphisms ofAwhich leaveρinvariant. Defineφto be the group homomorphism fromFrto the *-automorphisms ofAdefined on base elements byφ:ai↦αi. Definewnas the set of all reduced words inFrof lengthn(the identity is a neutral element), and |wn| as the number of elements ofwn. Letσn=(1/|wn|) ∑a∈wnφ(a) andSn=(1/n) ∑n−1k=0σk. We then show that ifxis inA, thenSn(x) converges almost uniformly to an element○∈A. To prove the above theorem, we prove an ergodic theorem involving completely positive maps, of which the free group situation is a special case. Roughly, ifp1⩾p2⩾0,p1+p2=1,σnpositive maps such thatσ1∘σn=p1σn+1+p2σn−1(n⩾1), withσ0(x)=xandσ1(1)=1 then, with a few technical assumptions, we show a convergence result for limn→∞σn(x) and show that limn→∞n−1∑n−1k=0σk(x) converges almost uniformly. In the casep2=0,σ1a *-automorphism, our theorems correspond to the non-commutative pointwise ergodic theorem of E. C. Lance. The results partially generalize a result of Kummerer. Our theorems also include results concerning normal operators on a Hilbert space which generalizes work of Guivarc'
Proximity to ℓ1and Distortion in Asymptotic L1Spaces

AbstractFor an asymptotic ℓ1spaceXwith a basis (xi) certain asymptotic ℓ1constants,δα(X) are defined forα<ω1.δα(X) measures the equivalence between all normalized block bases (yi)ki=1of (xi) which areSα-admissible with respect to (xi) (Sαis theαth-Schreier class of sets) and the unit vector basis of ℓk1. This leads to the concept of the delta spectrum ofX,Δ(X), which reflects the behavior of stabilized limits ofδα(X). The analogues of these constants under all renormings ofXare also defined and studied. We investigateΔ(X) both in general and for spaces of bounded distortion. We also prove several results on distorting the classical Tsirelson's spaceTand its relatives
Invariance Principles for Parabolic Equations with Random Coefficients

AbstractA general Hilbert-space-based stochastic averaging theory is brought forth herein for arbitrary-order parabolic equations with (possibly long range dependent) random coefficients. We use regularity conditions on∂tuε(t,x)=∑0⩽|k|⩽2pAk(t/ε,x,ω)∂kxuε(t,x),uε(0,x)=ϕ(x) ((1))which are slightly stronger than those required to prove pathwise existence and uniqueness for (1). Equation (1) can be obtained from the singularly perturbed system∂τvε(τ,x)=∑0⩽|k|⩽2pεAk(τ,x,ω)∂kxvε(τ,x),vε(0,x)=ϕ(x) ((2))through time change. Next, we impose on the coefficients of (1) a pointwise (inxandt) weak law of large numbers and a weak invariance principleεh∫tε−10Ak(x,s)−A0k(x)ds|k|⩽2p⇒{Θk}|k|⩽2p ((3))inC([0,T],H1), H1being a separable Hilbert space of functions andh∈(0,1) denoting a constant. (h>1/2 allows for long range time dependence.) Then, under these extraordinarily general conditions, we infer the weak invariance principleεh−1(uε−u)⇒ŷ.uis the non-random,ε-homogeneous solution of∂tu(t,x)=∑0⩽|k|⩽2pA0k(x)∂kxu(t,x),u(0,x)=ϕ(x) ((4))andŷ; mildly satisfies the linear stochastic partial differential equation∂tŷ(t,x)=∑|k|⩽2pA0k(x)∂kxŷ(t,x)dt+∑|k|⩽2pΘk(dt,x)∂kxu(t,x). ((5)
Functional Integration on Spaces of Connections

AbstractSuppose thatGis a compact connected Lie group andP→Mis a smooth principalG-bundle. We define a “cylinder function” on the space A of smooth connections onPto be a continuous complex function of the holonomies along finitely many piecewise smoothly immersed curves inM. Completing the algebra of cylinder functions in the sup norm, we obtain a commutative C*-algebra Fun(A). Let a “generalized measure” on A be a bounded linear functional on Fun(A). We construct a generalized measureμ0on A that is invariant under all automorphisms of the bundleP(not necessarily fixing the baseM). This result extends previous work which assumedMwas real-analytic and used only piecewise analytic curves in the definition of cylinder functions. As before, any graph withnedges embedded inMdetermines a C*-subalgebra of Fun(A) isomorphic toC(Gn), and the generalized measureμ0:Fun(A)→C restricts to the linear functional onC(Gn) given by integration against normalized Haar measure onGn. Our result implies that the group G of gauge transformations acts as unitary operators onL2(A), the Hilbert space completion of Fun(A) in the norm ‖F‖2=μ0(|F|2)1/2. Using “spin networks,” we construct explicit functions spanning the subspaceL2(A/G)⊆L2(A) consisting of vectors invariant under the action of G
Noncommutative Residues and Manifolds with Conical Singularities

AbstractFor every singular point on a manifold with conical singularities a trace can be constructed on the “Cone Algebra with Asymptotics” introduced by B.-W. Schulze. Each of them vanishes on operators supported in the interior and is therefore different from the noncommutative residue established by Wodzicki. On the ideal of operators with zero conormal symbol, however, we find another trace which coincides with the noncommutative residue in the interior. All these traces are shown to be essentially unique on a slightly extended version of the cone algebra
The Sharp Maximal Function on Spaces of Generalized Homogeneous Type

AbstractThe notion of spaces of a generalized homogeneous type is developed in [2]. In this paper, we introduce the sharp maximal function in this general setting, and establish the equivalence of theLpnorms between the sharp maximal function and the Hardy–Littlewood maximal function, as well the John–Nirenberg type inequalities. As applications, we discuss the concepts of the class of BMO functions and Hardy spaces, and apply our BMO–Hardy space theory to the study of singular integral operator
The Geometry of the First Non-zero Stekloff Eigenvalue

AbstractLet (Mn,g) be a compact Riemannian manifold with boundary and dimensionn⩾2. In this paper we discuss the first non-zero eigenvalue problem \begin{align}\Delta\varphi & = & 0\qquad & on\quad M,\\ \frac{\partial\varphi}{\partial \eta} & = & \ u_1\varphi\qquad & on\quad\partial M.\end{align}\eqno (1) Problem (1) is known as the Stekloff problem because it was introduced by him in 1902, for bounded domains of the plane. We discuss estimates of the eigenvalueν1in terms of the geometry of the manifold (Mn,g). In the two-dimensional case we generalize Payne's Theorem [P] for bounded domains in the plane to non-negative curvature manifolds. In this case we show thatν1⩾k0, wherekg⩾k0andkgrepresents the geodesic curvature of the boundary. In higher dimensionsn⩾3 for non-negative Ricci curvature manifolds we show thatν1>k0/2, wherek0is a lower bound for any eigenvalue of the second fundamental form of the boundary. We introduce an isoperimetric constant and prove a Cheeger's type inequality for the Stekloff eigenvalue
Hua Operators on Bounded Homogeneous Domains in Cnand Alternative Reproducing Kernels for Holomorphic Functions

AbstractLet D be a bounded homogeneous domain in Cn. (Note that D is not assumed to be Hermitian-symmetric.) In this work we are interested in studying various classes of “harmonic” functions on D and the possibility of representing them as “Poisson integrals” over the Bergman-Shilov boundary. One such class of harmonic functions is the “Hua-harmonic” functions. Specifically, by forming a contraction of ∂∂ with the holomorphic curvature tensor, we define a canonical system of differential operators which generalizes the classical Hua system. This system is invariant under all bi-holomorphisms of D. The Hua-harmonic functions are, by definition, the nullspace of this system. Our main result concerning this system is that every bounded Hua-harmonic function is the Poisson-integral over the Bergman-Shilov boundary of a uniqueL∞function against the Poisson kernel for the Laplace-Beltrami operator. We also consider spaces of harmonic functions defined as the kernel of a single real differential operator which is invariant under a particular solvable Lie group which acts transitively on D. We show that there exists such an operator which (a) annihilates holomorphic functions, (b) satisfies the Hormander condition, and (c) has the Bergman-Shilov boundary as its maximal boundary. It follows that the corresponding bounded harmonic functions are in one-to-one correspondence with theL∞functions on the Bergman-Shilov boundary under Poisson integration