pinecone/core-2020-05-10-deduplication huggingface dataset:
The Peierls–Nabarro and Benjamin–Ono Equations

AbstractAn intimate connection between the Peierls–Nabarro equation in crystal-dislocation theory and the travelling-wave form of the Benjamin–Ono equation in hydro-dynamics is uncovered. It is used to prove the essential uniqueness of Peierls' solution of the Peierls–Nabarro equation and to give, in closed form, all solutions of the analogous periodic problem. The latter problem is shown to be an example of global bifurcation with no secondary, symmetry-breaking, bifurcations for a nonlinear Neumann boundary-value problem or, equivalently, for an equation involving the conjugate operator, which is the Hilbert transform of functions on the unit circle
A Few Remarks about the Hilbert Transform

AbstractA generalized integral similar to integrability B is used to study the Hilbert transform onX=∑1⩽p<∞Lp(R), with a view to obtaining (i) a mollifier which commutes with the Hilbert transform onXand coincides with the Friedrichs mollifier onLloc1(R); (ii) estimates for nonlinear equations; (iii) an integral representation for the Hilbert transform of a regular Schwartz distribution; (iv) a generalized multiplier representation of the Hilbert transform onL1(R); (v) an elementary proof of the injectivity of H onX
Prime-like Elements and Semi-direct Products in Commutative Banach Algebras

AbstractWe develop results which show that elements in the radical of a commutative Banach algebra are often precluded from having prime-like properties if we avoid certain exceptional situations involving torsion elements. This makes the proof of the Singer-Wermer conjecture conceptually much clearer. It also motivates the definition of an element havingregular powersand allows us to strengthen our previous results concerning necessary conditions for a commutative Banach algebra A to be the semidirect product of some subalgebra together with a specified principal idealsA#, or, equivalently, concerning necessary conditions for there to be an algebraicsplittingof the short exact sequence0→sA#→A→(A/sA#)→0for some given elementsin A. In particular, we show that if A is aradicalBanach algebra andshas regular powers then no such splitting is possible
A Martingale Study of the Beurling–Ahlfors Transform inRn

AbstractWe apply the theory of martingale transforms to study the Beurling–Ahlfors transform,S, in dimensionsn⩾2. This operator reveals a rich structure through its representation as a martingale, and we obtain new results concerning the operator norm ofSacting on the class of differential forms havingLpcoefficients. In particular, we show that its norm is independent of the dimension when restricted tok-forms and we present new “Essén-type” norm inequalities related to this martingale structure. Finally, we suggest a purely analytic method to further investigate these norms which up to now has been lacking
A Quasi Maximum Principle for Holomorphic Solutions of Partial Differential Equations in Cn

AbstractWe present a quasi maximum principle stating roughly that holomorphic solutions of a given partial differential equation with constant coefficents in Cn,P(D)u=0,(†)achieve essentially their maximal growth on a certain algebraic hypersurfaceΓrelated to the operator. We prove it in the case wherePis homogeneous andΓis the conjugate dual cone, and also in the case whereP(D)=D21+…+D2nandΓis the complexified real sphere. We obtain a weak (semi-local) variant of the quasi maximum principle for certain non-homogeneous operatorsP(D), in which caseΓis the conjugate dual cone related to the principal part of the operator. This weaker variant is closely intertwined with several other notions. One of them is a quasi balayage principle for solutions of (†), involving the “sweeping” of measures in CnontoΓ
On the Attracting Orbit of a Nonlinear Transformation Arising from Renormalization of Hierarchically Interacting Diffusions

AbstractThis paper analyzes then-fold composition of a non-linear integral operator acting on a class of functions on [0, ∞). Attracting orbits and attracting fixed points are identified. Various results of convergence to these orbits and to these fixed points are derived. The proofs are based on order-preserving properties and comparison techniques. A key role is played by the eigenfunctions of the operator, which are used as comparison objects. The results imply that the space-time scaling limit of an infinite system of interacting diffusions has universal behavior independent of model parameters. The paper can be read independently of Part I
Endomorphisms of B(H). II. Finitely Correlated States on On

AbstractWe identify sets of conjugacy classes of ergodic endomorphisms of B(H) where H is a fixed separable Hilbert space. They correspond to certain equivalence classes of pure states on the Cuntz algebras Onwherenis the Powers index. These states, called finitely correlated states, and strongly asymptotically shift invariant states, are defined and characterized. The subsets of these states defining shifts will in general be identified in a later work, but here an interesting cross section for the conjugacy classes of shifts called diagonalizable shifts is introduced and studied
Sinclair Homomorphisms and Semigroups of Analytic Functions

AbstractIn this paper we study homomorphisms from the convolution algebraL1(R+) into certain Banach algebras of functions on the closed unit disc Δ. For the algebra A+of absolutely convergent Taylor series on Δ, we prove that every homomorphismΦis a “Sinclair map,” that is, of the formΦ(h)=∫∞0h(t)νtdt,h∈L1(R+)for some bounded, continuous semigroup (νt) in A+. A similar result holds for the algebra AC+of functions analytic inΔand absolutely continuous on the unit circle T. The result does not, however, hold in the disc algebra A(Δ), although we are able to represent homomorphisms into A(Δ) by means of semigroups in a certain weaker sense. Finally, we discuss the “Pisier algebra” P+defined in terms of random Taylor series on Δ. In particular, we prove that a homomorphism fromL1(R+) into P+need not be a Sinclair map, but that it can be represented by means of a semigroup which belongs to a certain larger Banach algebra of random Taylor series
A Semi-classical Trace Formula for Schrödinger Operators in the Case of a Critical Energy Level

AbstractLetĤ=−(ℏ2/2)Δ+V(x) be a Schrödinger operator on Rn, with smooth potentialV(x)→+∞ as |x|→+∞. The spectrum ofĤis discrete, and one can study the asymptotic of the smoothed spectral densityϒ(E,ℏ)=∑kϕEk(ℏ)−Eℏ,as ℏ→0. Here, {Ek(ℏ)}k∈Nis the spectrum ofĤand ϕ∈C∞0(R). We shall investigate the case whereEis a critical value of the symbolHofĤand, extending the work of Brummelhuis, Paul and Uribe in [3], we will prove the existence of a full asymptotic expansion forϒin terms ofℏand lnℏ and compute the leading coefficient. We will consider new Weyl-type estimates for the counting function:NEc,ρ(ℏ)=#{k∈N/|Ek(ℏ)−E|⩽ρℏ}
Upper Multiplicity and Bounded Trace Ideals inC*-Algebras

AbstractUpper and lower multiplicitiesMU(π,Ω) andML(π,Ω) for an irreducible representationπof aC*-algebraA, relative to a netΩ=(πα) inÂ, are shown to generalize the multiplicity numbers obtained by previous authors in trace formulae for (group)C*-algebras. This leads, in the presence of an auxiliary finiteness condition, to anuppersemi-continuity result in [0,∞] for trace functions onÂ:limsupTr(πα(a))⩽∑MU(π,Ω)Tr(π(a)) (a∈A+),where the summation is taken over the cluster points ofΩ. A characterization is given for the conditionMU(π,Ω)⩽k, wherekis a positive integer, from which it follows that aC*-algebra has all upper multiplicities finite if and only if it has bounded trace. More generally, the largest bounded trace idealJof aC*-algebraAis given byĴ={π∈Â:MU(π)<∞}
Quasi-Distribution Semigroups and Integrated Semigroups

AbstractSupposeXis a Banach space. We introduce the concept of quasi-distribution semigroups onX, as a generalization of the concept of distribution semigroups introduced by Lions in [19]. In our approach, the generatorAof a quasi-distribution semigroup may not be densely defined. Also introduced is a functional calculus forAin terms of the Fourier transform. For fixedr>0,k∈N∪{0}, define order (r,k) for a quasi-distribution semigroup and define an Fr,kfunctional calculus. We prove thatAgenerates a (k+1)-times integrated semigroup of exponential typerwith local Lipschitz continuity if and only ifAgenerates a quasi-distribution semigroup of order (r,k) if and only ifAhas an Fr,kfunctional calculus. WhenAis densely defined, a quasi-distribution semigroup reduces to a distribution semigroup in the sense of Lions
Localization for Schrödinger Operators with Effective Barriers

AbstractOur goal is to show that large classes of Schrödinger operatorsH=−Δ+VinL2(Rd) exhibit intervals of dense pure point spectrum, in any dimensiond. We approach this by assuming that the potentialV(x) coincides with a potentialV0(x) of a “comparison operator”H0=−Δ+V0on a sequence of ring shaped (but nog necessarily spherical) regionsUn,n=1,2,… . For energies in the resolvent setρ(H0) ofH0the regionsUnact as “effective barriers” in the sense of quantum mechanical scattering under the potentialV. Under certain assumptions on the geometry of theUnand their complements we show that (i)σac(H(λ))∩ρ(H0)=∅ for everyλ∈R, and (ii)σc(H(λ))∩ρ(H0)=∅ for almost everyλ∈Rwith respect to Lebesgue measure. Hereσacandσcdenote the absolutely continuous and continuous spectrum, respectively, andH(λ) is a “local randomization” ofH, i.e.,H(λ)=H+λW, whereWis any continuous and compactly supported perturbation of fixed sign. Our assumptions leave plenty of room for examples where the spectrum ofHfills entire spectral gaps ofH0. This leads to intervals of dense pure point spectrum forH(λ). We also give an explicit decay estimate for eigenfunctions, thus establishing localization forH(λ) in arbitrary spectral gaps ofH0
Spectral Asymptotics of Toeplitz Operators on Zoll Manifolds

AbstractWe prove a generalization of the Strong Szegö Limit Theorem for Zoll type operators on smooth compact closed manifolds. More precisely, letXbe a compact manifold, and letQ:C∞(X)→C∞(X) be a self-adjoint first order ellipticΨDOwhose spectrum is {1,2,3,…}. LetAbe a zeroth orderΨDO onXwhose spectrum does not contain zero in its closed convex hull. WritePnfor the projector onto the span of the eigenfunctions ofQwith eigenvalues in {1,…,n}. Our main result is a second order asymptotic formula for log detPnAPn, asn→∞
Uniform Estimates of Entire Functions by Logarithmic Sums

AbstractWe give uniform estimates in the whole complex plane of entire functions of exponential type less than a certain numerical constant (approximately equal to 0.44) having sufficiently small logarithmic sums. In these estimates the entire dependence on the function is through its type and logarithmic sum. This result extends a theorem of Koosis about polynomials and gives a new proof of that theorem. The proof is based on material related to multiplier theorems, first obtained by Beurling and Malliavin
A White Noise Approach to a Class of Non-linear Stochastic Heat Equations

AbstractCauchy problems for a class of non-linear stochastic evolution equations are studied. They are formulated as integral equations for generalized random fields. By methods of white noise analysis (S-transformation, characterization theorem, etc.) these problems are reduced to fixed point problems in appropriatly constructed Banach spaces. This technique provides a systematic treatment of existence and uniqueness questions for a variety of equations. The method is applied to non-linear heat equations, non-linear Volterra equations and non-linear ordinary differential equations, which may all be anticipating. For each case examples are given, such as the stochastic Burgers equation and stochastic reaction–diffusion equations
Ramanujan's Master Theorem and Duality of Symmetric Spaces

AbstractWe interpret Ramanujan's Master Theorem (B. Berndt, “Ramanujan's Notebooks, Part I,” Springer-Verlag, New York, 1985),∫∞0x−s−1∑k=0∞((−1)ka(k)xk)dx=−πsin(πs)a(s)(R)as a relation between the Fourier transforms of an analytic functionfwith respect to the real formsU(1) (compact) and R+(non-compact) of the multiplicative group of non-zero complex numbers, and we ask for a similar relation between the spherical Fourier transforms of an analytic function with respect to a compact real form and the non-compact dual real form of a complex symmetric space. We obtained results in the case of symmetric cones and in the rank-one case. Here we present the latter case in detail, describing features which will be also important for the general rank cas
Decay and Scattering of Small Solutions of a Generalized Boussinesq Equation

AbstractWe study the long-time behavior of small solutions of the initial-value problem for a generalized Boussinesq equation. We obtain a lower bound for the degrees of nonlinearity which allows us to establish a nonlinear scattering result for small perturbations; that is, the small solutions of the nonlinear problem behave asymptotically like the solution of the associated linear problem. Under certain hypotheses, we can construct a scattering operator for the Boussinesq equation which carries a neighborhood of 0 in the energy spaceXintoX
LowM*-Estimates on Coordinate Subspaces

AbstractLetKbe a symmetric convex body inRn. It is well-known that for everyθ∈(0,1) there exists a subspaceFofRnwith dimF=[(1−θ)n] such thatPF(K)⊇cθMKDn∩F, ((*))where PFdenotes the orthogonal projection ontoF. Consider a fixed coordinate system inRn. We study the question whether an analogue of (*) can be obtained when one is restricted to chooseFamong the coordinate subspacesRσ,σ⊆{1,…,n}, with |σ|=[(1−θ)n]. We prove several “coordinate versions” of (*) in terms of the cotype-2 constant, of the volume ratio and other parameters ofK. The basic source of our estimates is an exact coordinate analogue of (*) in the ellipsoidal case. Applications to the computation of the number of lattice points inside a convex body are considered throughout the paper
Lipschitz Continuity of the Absolute Value and Riesz Projections in Symmetric Operator Spaces

AbstractA principal result of the paper is that ifEis a symmetric Banach function space on the positive half-line with the Fatou property then, for all semifinite von Neumann algebras (M, τ), the absolute value mapping is Lipschitz continuous on the associated symmetric operator spaceE(M, τ) with Lipschitz constant depending only onEif and only ifEhas non-trivial Boyd indices. It follows that if M is any von Neumann algebra, then the absolute value map is Lipschitz continuous on the corresponding HaagerupLp-space, provided 1<p<∞
Amenable Groups and Generalized Cuntz Algebras

AbstractWithHa Hilbert space, OHthe generalized Cuntz algebra overHendowed with the canonical actionΛofU(H), we consider a natural family S0of product representations of the zero grade part O0H, and the family S obtained from S0by inducing to OHvia the natural conditional expectationP:OH→O0H. Each element of S0(and hence of S) is labelled by a sequence of unit vectors inH. LetGbe a non-compact locally compact group,λits regular representation onH=L2(G), and OHendowed with the actionΛ∘λofG; we prove thatGisσ-amenable iff there are ireducible covariant representations in S
Feynman–Kac Representation of Some Noncommutative Elliptic Operators

AbstractGaussian averages of automorphisms of a von Neumannn algebra yield Markov semigroups by the well-known procedure of subordination. We construct operator-valued martingales to realise perturbations of such semigroups through Feynman–Kac formulae. The perturbations are noncommutative vector fields, and the martingales are operator families, which are determined by an Itô equation on each vector and satisfy cocycle relations with respect to a randomised flow on the algebra. In particular this gives a probabilistic representation of some symmetric Markov semigroups considered by Davies and Lindsay
Dirichlet Forms and Markovian Semigroups on Standard Forms of von Neumann Algebras

AbstractWe characterize w*-continuous, Markovian semigroups on a von Neumann algebra M, which areφ0-symmetric w.r.t. a faithful, normal stateφ0in M*+, in terms of quadratic forms on the Hilbert space H of a standard form (M,H,P,J). We characterize also symmetric, strongly continuous, contraction semigroups on a real Hilbert space H which leave invariant a closed, convex set in H, in terms of a contraction property of the associated quadratic forms. We apply the results to give criteria of essential selfadjointness for quadratic form sums and to give a characterization of w*-continuous, Markovian semigroups on M, which commute with the modular automorphism groupσφ0t
The Pełczyński Property for Tight Subspaces

AbstractWe show that ifXis a tight subspace ofC(K) thenXhas the Pełczyński property andX* is weakly sequentially complete. We apply this result to the spaceUof uniformly convergent Taylor series on the unit circle and using a minimal amount of Fourier theory prove thatUhas the Pełczyński property andU* is weakly sequentially complete. Using separate methods, we proveUandU* have the Dunford–Pettis property. Some results concerning pointwise bounded approximation are proved for tight uniform algebras. We use tightness and the Pełczyński property to make a remark about inner functions on strictly pseudoconvex domains in Cn
Subharmonicity without Upper Semicontinuity

AbstractLetΩbe an open subset ofRd(d⩾2). Givenx∈Ω, a Jensenmeasureforxis a Borel probability measureμ, supported on a compact subset ofΩ, such that each subharmonic functionuonΩsatisfiesu(x)⩽∫udμ. ((*))A functionuonΩ(subject to certain measurability conditions, much weaker for example than upper semicontinuity) is calledquasi-subharmonicif it satisfies (*) for eachx∈Ωand each Jensen measureμforx. Our first result is thatuis quasi-subharmonic onΩif and only if its upper semicontinuous regularizationu* is subharmonic onΩand equal tououtside a set of capacity zero. This easily implies, for example, Cartan's classical theorem on the supremum of a locally bounded family of subharmonic functions. Our second result is a converse to Cartan's theorem: provided thatΩhas a Green's function, every quasi-subharmonic function 4 onΩis the supremum of some family of subharmonic functions. These ideas eventually lead to the following duality theorem. Ifϕis a Borel function locally bounded above onΩ, and ifΩhas a Green's function, then for each x∈Ω,sup{v(x):vsubharmoniconΩv⩽ϕ}=inf∫ϕdμ:μisaJensenmeasureforx.We also investigate to what extent these results carry over to plurisubharmonic functions
Holomorphic Discrete Series for Hyperboloids of Hermitian Type

AbstractIn the present paper we study the holomorphic discrete series (h.d.s.) for hyperboloids of Hermitian type. They are the spacesG/HwhereG=SO0(p,2),H=SO0(p,1). We find some complex hullsY±ofG/H(they correspond to minimalG-invariant cones in the Lie algebra ofG), consider the Hardy spacesH2(Y±), and give explicit expressions for the corresponding Cauchy–Szegö kernels. Earlier these expressions were known forp=2 (Gelfand–Gindikin) andp=1 (Molchanov). We compute the projectionsE±of theL2(G/H) onto the images of the Hardy spaces. The differenceQ=E+−E−is most interesting, it is an analogue of the Hilbert transform. We study the relations between the h.d.s. forGand the h.d.s. forG/H. It turns out that forp>2 there is a finite number of representations of the h.d.s. forG/Hwhich are not representations of the h.d.s. ofG. We give explicit expressions for the kernels of the projections on the subspaces of the Hardy spaces where the h.d.s. forGacts
Approximation by Analytic Matrix Functions: The Four Block Problem

AbstractThe four block problem is a generalization of Nehari's problem for matrix functions. It plays an important role inH∞-optimal control theory. It is well known that Nehari's problem for a continuous scalar function has a unique solution. However, in the matrix case the situation is quite different. V. V. Peller and N. J. Young (1994,J. Funct. Anal.120, 300–343) studiedsuperoptimal solutionsof Nehari's problem. They minimize not only theL∞-norm of the corresponding matrix function but also the essential suprema of all further singular values. It was shown that forH∞+Cmatrix functions Nehari's problem has a unique superoptimal solution. In this paper we study superoptimal solutions of the four block problem and we find a natural condition under which such a superoptimal solution is unique. Our result is new even in the case of Nehari's problem. We study some related problems such as thematic factorizations, invariance of indices, and inequalities between the singular values of the four block operator and the superoptimal singular values
Toeplitz Determinants with One Fisher–Hartwig Singularity

AbstractLetcbe a function defined on the unit circle with Fourier coefficients {cn}∞n=−∞. The Fisher–Hartwig conjecture describes the asymptotic behaviour of the determinants of then×nToeplitz matricesDn(c)=det[ci−j]n−1i, j=0for a certain class of functionsc. In this paper we prove this conjecture in the case of functions with one singularity. More precisely, we consider functions of the formc(eiθ)=b(eiθ)tβ(ei(θ−θ1))uα(ei(θ−θ1)).Heretβ(eiθ)=exp(iβ(θ−π)), 0<θ<2π, is a function with a jump discontinuity,uα(eiθ)=(2−2cosθ)αis a function which may have a zero, a pole, or a discontinuity of oscillating type, andbis a sufficiently smooth nonvanishing function with winding number equal to zero. The only restriction we impose on the parameters is that 2αis required not to be a negative integer. In the case where Reα⩽−1/2, i.e., where the corresponding functioncis not integrable, we identifycin an appropriate way with a distribution
Polylinear Additive Functionals of Superprocesses

AbstractLetXbe a superdiffusion in a domainEof Rd. A polylinear additive functional ofXcorresponding to a positive Borel functionρis given by the formulaA(B)=∫Bdt1, …, dtk∫Ekρ(t1, z1; …; tk, zk)Xt1(dz1)…Xtk(dzk).By a passage to the limit, we extend this definition to a certain class of generalized functionsρ. More precisely, we associate an additive functionalAνof (Xt, Pμ) with every measureνsubject to the condition∫ν(dt1, dz1; …; dtk, dzk)ν(dt′1, dz′1; …; dt′k, dz′k) emsp;×qμ(t1, z1; …; tk, dzk; t′1, z′1; …; t′k, dz′k)<∞for a certain kernelqμ. We prove that, ifν{t:ti=s}=0 fori=1, …, kand for alls, then measureAνhas, a.s., the same property. This result is applicable, in particular, to self-intersection local times ofX
On the Solvability of Homogeneous Left-Invariant Differential Operators on the Heisenberg Group

AbstractWe discuss the problem of solvability for the class of homogenoeus left-invariant operators □S,αon the Heisenberg groupHnintroduced by F. De Mari, M. M. Peloso, and F. Ricci. (1995,J. Reine Angew. Math.464, 67–96), whereS∈Sp(C,n) has a degenerated semi-definite real part, andα=±(n+2l),l=0,1,…. We completely characterize the solvable operators in this class. As a consequence we obtain that certain operators □S,αare solvable while their transposest□S,αare not. This is the case, for instance, for the operatorYZonH1, whereZ=X−iYis the unsolvable Lewy operator
Gabor Frames and Time-Frequency Analysis of Distributions

AbstractThis paper lays the foundation for a quantitative theory of Gabor expansionsf(x)=∑k,nck,ne2πinαxg(x−kβ). In analogy to wavelet expansions of Besov–Triebel–Lizorkin spaces, we show that the correct class of spaces which can be characterized by the magnitude of the coefficientsck,nis the class of modulation spaces. To analyze the behavior of the coefficients, it is necessary to invert the Gabor frame operator on these spaces. We show that the frame operator is invertible on modulation spaces if and only if it is invertible onL2and the atomgis in a suitable space of test functions. A similar statement for wavelet theory is false. The second part is devoted to Gabor analysis on general time–frequency lattices