pinecone/core-2020-05-10-deduplication huggingface dataset:
Ground States and Free Boundary Value Problems for the n-Laplacian in n Dimensional Space

AbstractUsing a new gradient estimate, we prove several theorems on the existence of radial ground states for the n-Laplace equation div(|∇u|n−2∇u)+f(u)=0 in Rn, n>1, and the existence of positive radial solutions for the associated Dirichlet–Neumann free boundary value problem in a ball. We treat exponentially subcritical, critical, and supercritical nonlinearities f(u), which also are allowed to have singularities at zero. Moreover, we show that the local behavior of f at zero affects the existence in a crucial way: this allows us to prove the existence of ground states for a large class of functions f(u) without imposing any restriction on their growth for large u
Amenable and Weakly Amenable Banach Algebras with Compact Multiplication

AbstractWe investigate amenable and weakly amenable Banach algebras with compact multiplication. Any amenable Banach algebra with compact multiplication is biprojective. As a consequence, every semisimple such algebra which has the approximation property is a topological direct sum of full matrix algebras. In the radical case no such structure theorem is at hand. We also investigate Banach algebras which have a bounded approximate identity consisting of normalized powers of an element x. Any such Banach algebra is either unital or radical; if the algebra is also generated by x, it is weakly amenable. We construct a radical example with compact multiplication which moreover is an integral domain. This furnishes a new example of a commutative, weakly amenable, non-amenable, radical Banach algebra
Morita–Rieffel Equivalence and Spectral Theory for Integrable Automorphism Groups of C*-Algebras

AbstractGiven a C*-dynamical system (A, G, α), we discuss conditions under which subalgebras of the multiplier algebra M(A) consisting of fixed points for α are Morita–Rieffel equivalent to ideals in the crossed product of A by G. In case G is abelian we also develop a spectral theory, giving a necessary and sufficient condition for α to be equivalent to the dual action on the cross-sectional C*-algebra of a Fell bundle. In our main application we show that a proper action of an abelian group on a locally compact space is equivalent to a dual action
Toeplitz Matrices with an Exponential Growth of Entries and the First Szegö Limit Theorem

AbstractThe Toeplitz (or block Toeplitz) matrices S(r)={sj−k}rk, j=1, generated by the Taylor coefficients at zero of analytic functions ϕ(λ)=s02+∑∞p=1s−pλp and ψ(μ)=s02+∑∞p=1spμp, are considered. A method is proposed for removing the poles of ϕ and ψ or, in other words, for replacing S(∞), whose entries grow exponentially, by a matrix Ŝ(∞)={ŝj−k}∞k, j=1 with better behaviour and the same asymptotics of Δ(r)=detŜ(r) (r→∞) as the sequence Δr=detS(r). A Szegö-type limit formula for the case when S(r)=S(r)* (r⩾n0) have a fixed number of negative eigenvalues is obtained
Uniformly Integrable Operators and Large Deviations for Markov Processes

AbstractIn this paper we introduce and investigate the notion of uniformly integrable operators on Lp(E, μ). Its relations to classical compactness and hypercontractivity are exhibited. Several consequences of this notion are established, such as Perron– Frobenius type theorems, independence on p of the spectral radius in Lpp, continuity of spectral radius, and especially the existence of spectral gap in the irreducible case. We also present some infinitesimal criteria ensuring the uniform integrability of a positive semigroup or of its resolvent. For a μ-essentially irreducible Markov process, we show that the uniform integrability in Lp(μ) of some type of resolvent associated with the transition semigroup implies the large deviation principle of level-3 with some rate function given by a modified Donsker–Varadhan entropy functional. We also prove that the uniform integrability condition becomes even necessary in the symmetric case. Finally, we present several applications of our results to Feynman–Kac semigroups, to the thermodynamical limits of grand ensembles, and to (non-symmetric) Markov processes given by Girsanov's formula
Approximations of Laplace Transforms and Integrated Semigroups

AbstractLet {fm; m∈N} be a sequence of functions from [0, ∞) to a Banach space E. We give a new and essential condition on fm, which is weaker than the usual “local Lipschitz continuity” condition, ensuring that the convergence of fm is equivalent to the convergence of their Laplace transforms. This enables us to establish new approximation theorems for r-times integrated semigroups on E, for all r⩾0. As a consequence, an open problem for the convergence of integrated semigroups on the whole space E, is solved in essence. Moreover, we present an application to nonhomogeneous Cauchy problems
Indefinite Hardy Spaces on Finite Bordered Riemann Surfaces

AbstractIt is well known that one may study Hardy spaces with the domain a finite bordered Riemann surface rather than the unit disk. However, for domains with more than one boundary component it is natural to consider, besides the usual positive definite inner product on H2, indefinite inner products obtained by picking up different signs (or in the vector-valued case, different signature matrices) when integrating over different components. In this paper we obtain a necessary and sufficient condition for such an indefinite inner product to be nondegenerate and show that when this condition is satisfied we actually get a Kreın space. Furthermore, each holomorphic mapping of the finite bordered Riemann surface onto the unit disk (which maps boundary to boundary) determines an explicit isometric isomorphism between this space and a usual vector-valued Hardy space on the unit disk with an indefinite inner product defined by an appropriate hermitian matrix. As is usual when studying Hardy spaces on a multiply connected domain, the elements of the space are sections of a vector bundle rather than functions. The main point is to construct an appropriate extension of this bundle to the double of the finite bordered Riemann surface and to use Cauchy kernels for certain vector bundles on a compact Riemann surface
Remark on the Rate of Decay of Higher Order Derivatives for Solutions to the Navier–Stokes Equations in Rn

AbstractWe present a new derivation of upper bounds for the decay of higher order derivatives of solutions to the unforced Navier–Stokes equations in Rn. The method, based on so-called Gevrey estimates, also yields explicit bounds on the growth of the radius of analyticity of the solution in time. Moreover, under the assumption that the Navier–Stokes solution stays sufficiently close to a solution of the heat equation in the L2 norm—a result known to be true for a large class of initial data—lower bounds on the decay of higher order derivatives can be obtained
An Abstract Interpretation of the Wavelet Dimension Function Using Group Representations

AbstractMethods from abstract harmonic analysis are used to derive a new formulation of the wavelet dimension function and its natural generalizations to higher dimensions. By means of this abstract description, necessary and sufficient conditions are derived for a multiwavelet in N dimensions, relative to an arbitrary expansive integral matrix A, to be a multiwavelet that arises from a multiresolution analysis (MRA), i.e., is an MRA wavelet. Even in the classical case, it is shown that this abstract approach gives new results
Endomorphisms of Banach Algebras of Infinitely Differentiable Functions on Compact Plane Sets

AbstractLet X be a perfect, compact subset of the complex plane, let (Mn) be a sequence of positive numbers satisfying M0=1 and ([formula])⩽Mm+n/MmMn, and let D(X,M)={f ∈D∞(X): ‖f‖ =∑∞n=0(‖f(n)‖∞/Mn)<∞{. With pointwise addition and multiplication, D(X,M) is a commutative normed algebra. In this note we study the endomorphisms of such algebras. In a previous paper this problem was solved when X=[0,1] for many weights (Mn). Here we investigate the extent to which the methods used previously apply to general X and introduce some new methods. In particular, we obtain many results when X=Δ where Δ is the open unit disc
The Hardy Inequality and the Asymptotic Behaviour of the Heat Equation with an Inverse-Square Potential

AbstractWe study the well-posedness and describe the asymptotic behavior of solutions of the heat equation with inverse-square potentials for the Cauchy–Dirichlet problem in a bounded domain and also for the Cauchy problem in RN. In the case of the bounded domain we use an improved form of the so-called Hardy–Poincaré inequality and prove the exponential stabilization towards a solution in separated variables. In RN we first establish a new weighted version of the Hardy–Poincaré inequality, and then show the stabilization towards a radially symmetric solution in self-similar variables with a polynomial decay rate. This work complements and explains well-known work by Baras and Goldstein on the existence of global solutions and blow-up for these equations. In the present article the sign restriction on the data and solutions is removed, the functional framework for well-posedness is described, and the asymptotic rates calculated. Examples of non-uniqueness are also given
Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

AbstractWe show that transport inequalities, similar to the one derived by M. Talagrand (1996, Geom. Funct. Anal.6, 587–600) for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately log-concave, in a precise sense. All constants are independent of the dimension and optimal in certain cases. The proofs are based on partial differential equations and an interpolation inequality involving the Wasserstein distance, the entropy functional, and the Fisher information
Application of log-Sobolov Inequality to the Stochastic Dynamics of Unbounded Spin Systems on the Lattice

AbstractWe consider a ferromagnetic lattice spin system with unbounded spins and investigate the relaxation property for the associated stochastic dynamics (the Glauber dynamics) in the finite volume case. We prove that the following two conditions are equivalent: (1)The log-Sobolev inequality for the finite volume Gibbs states holds uniformly in both the volume and the boundary condition. (2)The finite volume Glauber dynamics relaxes to equilibrium exponentially fast, uniformly in the volume whenever it starts from a tempered configuration. This can be considered as a complementary result to the ones previously obtained for infinite volume Glauber dynamics by B. Zegarlinski (1996, Comm. Math. Phys.175, 401–432). Our result can also be viewed as an extension of the equivalence theorem known for compact spin space settings
The Local Multiplier Algebra of a C*-Algebra, II

AbstractLet A be a separable C*-algebra and let Mloc(A) be the local multiplier algebra of A. It is shown that every minimal closed prime ideal of Mloc(A) is primitive. If Prim(A) has a dense Gδ consisting of closed points (for instance, if Prim(A) is a T1-space) and A is unital, then Mloc(A) is its own local multiplier algebra and has only inner derivations. The same is true for Mloc(Mloc(A)) if A is non-unital. If A is postliminal then Mloc(Mloc((A)) is the regular σ-completion of A, which is an AW*-algebra
Notes on the Wave Equation on Asymptotically Euclidean Manifolds

AbstractThe primary object of this paper is to discuss the asymptotics of solutions of the wave equation on an asymptotically Euclidean manifold, when the initial data have compact support. By going over to an appropriate conformal metric, it is shown that (just as for the ordinary wave equation) such a solution has a forward (“future”) and a backward (“past”) radiation field. The same method is then used to define “end points” of bicharacteristics that begin and end above the boundary (the analogue of the sphere at infinity in the Euclidean case), and to derive a relation between the wave front sets of the two radiation fields. The extension of these results to solutions with finite energy is also briefly discussed
Decay of Heat Semigroups in L∞ and Applications to Nonlinear Parabolic Problems in Unbounded Domains

AbstractWe characterize all domains Ω of RN such that the heat semigroup decays in L(L∞(Ω)) or L(L1(Ω)) as t→∞. Namely, we prove that this property is equivalent to the Poincaré inequality, and that it is also equivalent to the solvability of −Δu=f in L∞(Ω) for all f∈L∞(Ω). In particular, under mild regularity assumptions on Ω, these properties are equivalent to the geometric condition that Ω has finite inradius. Next, we give applications of this linear result to the study of two nonlinear parabolic problems in unbounded domains. First, we consider the quenching problem for singular parabolic equations. We prove that the solution in Ω quenches in finite time no matter how small the nonlinearity is, if and only if Ω does not fulfill the Poincaré inequality. Second, for the semilinear heat equation with a power nonlinearity, we prove, roughly speaking, that the trivial solution is stable in L∞ or in L1 if and only if Ω has finite inradius
A C*-Dynamical Entropy and Applications to Canonical Endomorphisms

AbstractFor an automorphism α of a unital C*-algebra A, we give a definition of an entropy htφ(α) with respect to an α-invariant state φ of A. For Connes– Narnhofer–Thirring entropy hφ(α) and Voiculescu's topological entropy ht(α), in general hφ(α)⩽htφ(α)⩽ht(α), but the equalities do not always hold. We compute entropies of an endomorphism ρ with respect to the state ϕ defined from a left inverse of ρ. Cuntz's canonical inner endomorphism Φ of On satisfies hϕ(Φ)=htϕ(Φ), which is determined by the mean entropy of ϕ on the UHF (uniformly hyperfinite) algebra. If γ is Longo's canonical endomorphism for an irreducible graded standard AFD (approximately finite dimensional) inclusion N⊂M of infinite factors with finite index, then hϕ(γ)=(1/2)log(IndEγ), for the conditional expectation Eγ on γ(M)
Pure Point Spectrum for the Laplacian on Unbounded Nested Fractals

AbstractWe prove for the class of nested fractals introduced by T. Lindstrøm (1990, Memoirs Amer. Math. Soc.420) that the integrated density of states is completely created by the so-called Neuman–Dirichlet eigenvalues. The corresponding eigenfunctions lead to eigenfunctions with compact support on the unbounded set and we prove that for a large class of blow-ups the set of Neuman–Dirichlet eigenfunctions is complete, leading to a pure point spectrum with compactly supported eigenfunctions. This generalizes previous results of H. Teplyaev (1998, J. Funct. Anal.159, 537–567) on the Sierpinski Gasket. Our methods are elementary and use only symmetry arguments via the representations of the symmetry group of the set
Symbolic Functional Calculus and N-Body Resolvent Estimates

AbstractWe construct a functional calculus for symbolic functions of “scattering pseudodifferential operators” on manifolds with boundary. This is applied to obtain mapping results on the scattering wavefront set of the B operator used by Gérard et al. (1994, “Commutator Algebra and Resolvent Estimates,” Advanced Studies in Pure Mathematics, Vol. 23, Math. Soc. Japan, Tokyo) to obtain N-body resolvent estimates. As an application, we construct the plane waves for an N-body, short range Schrödinger operator and obtain a scattering wavefront set estimate
Two Weight Extrapolation via the Maximal Operator

AbstractWe give several extrapolation theorems for pairs of weights of the form (w, Mkw) and (w, (Mw/w)rw), where w is any non-negative function, r>1, and Mk is the kth iterate of the Hardy–Littlewood maximal operator. As an application we show that our results can be used to extend and sharpen results for square functions and singular integral operators by Chang et al. (1985, Comment. Math. Helv.60, 217–246), Chanillo and Wheeden (1987, Indiana Univ. Math. J.36, 277–294), Wilson (1987, Duke Math. J.55, 879–887; 1989, Trans. Amer. Math. Soc.314, 661–692; 1989, Illinois J. Math.33, 361–366), and Uchiyama (1995, Studia Math.115, 135–149). In the process we prove a conjecture due to Wilson
Fourier Integral Operators on Noncompact Symmetric Spaces of Real Rank One

AbstractLet X=G/K be a noncompact symmetric space of real rank one. The purpose of this paper is to investigate Lp boundedness properties of a certain class of radial Fourier integral operators on the space X. We will prove that if uτ is the solution at some fixed time τ of the natural wave equation on X with initial data f and g and 1<p<∞, then ‖uτ‖Lp(X)⩽Cp(τ)(‖f‖Lpbp(X)+(1+τ)‖g‖Lpbp−1(X)). We will obtain both the precise behavior of the norm Cp(τ) and the sharp regularity assumptions on the functions f and g (i.e., the exponent bp) that make this inequality possible. In the second part of the paper we deal with the analog of E. M. Stein's maximal spherical averages and prove exponential decay estimates (of a highly non-euclidean nature) on the Lp norm of supT⩽τ⩽T+1|f*dστ(z)|, where dστ is a normalized spherical measure
Euclidean Quantum Vector Fields

AbstractWe study quantum vector fields in Euclidean space-time. These fields can be identified with generalized random vector fields, which we study in terms of their covariance. We prove that the conditions of translational invariance, covariance with respect to some representation τ=⊕τj of the orthogonal group O(n), where none of the irreducible components τj is trivial, and the condition of reflection positivity cannot be fulfilled at the same time unless the test function space is restricted by some gauge condition. However, if the representation τ is trivial, i.e., if every matrix in O(n) is mapped to the identity, we can explicitly write down covariance matrices which lead to Gaussian fields which fulfill all conditions in the axiomatic framework
BV Functions and Distorted Ornstein Uhlenbeck Processes over the Abstract Wiener Space

AbstractLet (E, H, μ) be an abstract Wiener space and H be the class of functions ρ∈L1+(E; μ) satisfying the ray Hamza condition in every direction ℓ∈E*. For ρ∈H, the closure (Eρ, Fρ) of the symmetric form Eρ(u, v)=∫E〈∇u(z), ∇v(z)〉Hρ(z)μ(dz), u, v∈FC1b, is a quasi-regular Dirichlet form on L2(F, ρdμ) (F=Supp[ρμ]), yielding an associated diffusion Mρ=(Xt, Pz) on F called a distorted Ornstein Uhlenbeck process. A function ρ on E is called a BV function (ρ∈BV(E) in notation) if ρ∈∪p>1Lp(E; μ) and V(ρ)=supG∈(FC1b)E*, ‖G‖H(z)⩽1∫E∇*G(z)ρ(z)μ(dz) is finite. For ρ∈H∩BV(E), there exist a positive finite measure ‖Dρ‖ on F and a weakly measurable function σρ:F→H such that ‖σρ(z)‖H=1‖Dρ‖-a.e. and ∫F∇*G(z)×ρ(z)μ(dz)=∫F〈G(z), σρ(z)〉H‖Dρ‖(dz), ∀G∈(FC1b)E*. Further, the sample path of Mρ admits an expression as a sum of E-valued CAFs, Xt−X0=Wt−12∫t0Xsds+12∫t0σρ(Xs)dL‖Dρ‖s, where Wt is an E-valued Brownian motion and L‖Dρ‖t is a PCAF of Mρ with Revuz measure ‖Dρ‖. A measurable set Γ⊂E is called Caccioppoli if IΓ∈BV(E). In this case, the support of the measure ‖DIΓ‖ is concentrated in ∂Γ and the above equations reduce to the Gauss formula and the Skorohod equation for the modified reflecting Ornstein Uhlenbeck process, respectively. A related coarea formula is also presented
Inner Ideals and Facial Structure of the Quasi-State Space of a JB-Algebra

AbstractIn this paper we relate the geometric structure of the quasi-state space A*1, + of a JB-algebra A to the algebraic structure of A. We prove that each non-empty w*-closed face F of A*1, + coincides with the set of elements of A* that attain their norms at a closed projection p in A**, uniquely determined by F. When A is unital, we establish an order anti- isomorphism between the complete lattice of w*-closed faces of A*1, + and the complete lattice of norm-closed inner ideals of A. These results are known for the special case when A is the self-adjoint part of a C*-algebra. The proofs use associative techniques and the Gelfand–Naimark representation. In the general case such techniques are not available, and it is necessary to introduce new techniques including the use of a new inequality of Cauchy–Schwarz type for elements of A*
Representations of Starshaped Sets in Normed Linear Spaces

AbstractIf S is a closed connected nonconvex locally compact and bounded subset of a real normed linear space or a closed connected nonconvex and bounded subset of a real reflexive Banach space, then kerS=∩{clconvSz:z∈D∩regS}, where regS denotes the set of regular points of S, D is a relatively open subset of S containing the set lncS of local nonconvexity points of S, and Sz={s∈S:z is visible from s via S}. An analogous intersection formula, with the set sphS of spherical points of S in place of regS is shown to hold for a closed connected nonconvex and bounded subset S of a real Banach space which is uniformly convex and uniformly smooth. If the assumption of boundedness of S is dropped, then in all specified settings the above representations hold with intersections of convSz in place of clconvSz. This strengthens and complements results of Borwein and Strojwas, Stavrakas, and the author. Finally, the possibility of generating similar intersection formulae in other configuration set-space is discussed
Quantum Stochastic Calculus on Full Fock Modules

AbstractWe develop a quantum stochastic calculus on full Fock modules over arbitrary Hilbert B–B-modules. We find a calculus of bounded operators where all quantum stochastic integrals are limits of Riemann–Stieltjes sums. After having estalished existence and uniqueness of solutions of a large class of quantum stochastic differential equations, we find necessary and sufficient conditions for unitarity of a subclass of solutions. As an application we find dilations of a conservative CP-semigroup (quantum dynamical semigroup) on B with arbitrary bounded (Christensen–Evans) generator. We point out that in the case B=B(G) the calculus may be interpreted as a calculus on the full Fock space tensor initial space G with arbitrary degree of freedom dilating CP-semigroups with arbitrary Lindblad generator. Finally, we show how a calculus on the boolean Fock module reduces to our calculus. As a special case this includes a calculus on the boolean Fock space
Taylor Approximations on Sierpinski Gasket Type Fractals

AbstractFor a class of fractals that includes the familiar Sierpinski gasket, there is now a theory involving Laplacians, Dirichlet forms, normal derivatives, Green's functions, and the Gauss–Green integration formula, analogous to the theory of analysis on manifolds. This theory was originally developed as a by-product of the construction of stochastic processes analogous to Brownian motion, but has been given by a direct analytic construction in the work of Kigami. Until now, this theory has not provided anything analogous to the gradient of a function, or a local Taylor approximation. In this paper we construct a family of derivatives, which includes the known normal derivative, at vertex points in the graphs that approximate the fractal, and obtain Taylor approximations at these points. We show that a function in the domain of Δn can be locally well approximated by an n-harmonic function (solution of Δnu=0). One novel feature of this result is that it requires several different estimates to describe the optimal rate of approximation