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In the paper, we construct compact embedded convex λ -hypersurfaces which are diffeomorphic to a sphere and are not isometric to a standard sphere. As the special case of our result, we solve Sun's problem (Int. Math. Res. Not. (2021) 11818–11844). In this sense, one can not expect to have Alexandrov type theorem for λ -hypersurfaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, “NC ball maps” are very simple, in contrast to the classical result of D''Angelo on such analytic maps in . Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables. [Copyright &y& Elsevier]

Abstract: This is the final one in the series of papers where we introduce and study the -algebras associated with topological graphs. In this paper, we get a sufficient condition on topological graphs so that the associated -algebras are simple and purely infinite. Using this result, we give one method to construct all Kirchberg algebras as -algebras associated with topological graphs. [Copyright &y& Elsevier]

Abstract: Let be a class of unital C*-algebras. The class of C*-algebras which can be tracially approximated (in the Egorov-like sense first considered by Lin) by the C*-algebras in is studied (Lin considered the case that consists of finite-dimensional C*-algebras or the tensor products of such with ). In particular, the question is considered whether, for any simple separable , there is a C*-algebra B which is a simple inductive limit of certain basic homogeneous C*-algebras together with C*-algebras in , such that the Elliott invariant of A is isomorphic to the Elliott invariant of B. An interesting case of this question is answered. In the final part of the paper, the question is also considered which properties of C*-algebras are inherited by tracial approximation. (Results of this kind are obtained which are used in the proof of the main theorem of the paper, and also in the proof of the classification theorem of the second author given in [Z. Niu, A classification of tracially approximately splitting tree algebra, in preparation] and [Z. Niu, A classification of certain tracially approximately subhomogeneous C*-algebras, PhD thesis, University of Toronto, 2005]—which also uses the main result of the present paper.) [Copyright &y& Elsevier]

In a recent paper, Lagarias, Reeds and Wang established a characterization of spectra and tilings that can be used to prove a conjecture of Jorgensen and Pedersen by Keller''s criterion. Different techniques to prove these facts have also been developed by Kolountzakis, Iosevich and Pedersen. The primary aim of this paper is to present an elementary method of describing certain characterizations of spectra and tilings. To illustrate this method, we first give a simple proof of this characterization. We then use the method to derive some characteristic results connected with the dual Fuglede''s spectral-set conjecture. The results here extend several known conclusions in a simple manner. [Copyright &y& Elsevier]

For 2&les;m&les;l/2, let G be a simply connected Lie group with g0=so(2m,2l−2m) as Lie algebra, let g=k⊕p be the complexification of the usual Cartan decomposition, let K be the analytic subgroup with Lie algebra k∩g0, and let U(g) be the universal enveloping algebra of g. This work examines the unitarity and K spectrum of representations in the “analytic continuation” of discrete series of G, relating these properties to orbits in the nilpotent radical of a certain parabolic subalgebra of g.The roots with respect to the usual compact Cartan subalgebra are all ±ei±ej with 1&les;i. In the usual positive system of roots, the simple root em−em+1 is noncompact and the other simple roots are compact. Let q=l⊕u be the parabolic subalgebra of g for which em−em+1 contributes to u and the other simple roots contribute to l, let L be the analytic subgroup of G with Lie algebra l∩g0, let LC=Intg(l), let 2δ(u) be the sum of the roots contributing to u, and let q¯=l⊕u¯ be the parabolic subalgebra opposite to q.The members of u∩p are nilpotent members of g. The group LC acts on u∩p with finitely many orbits, and the topological closure of each orbit is an irreducible algebraic variety. If Y is one of these varieties, let R(Y) be the dual coordinate ring of Y; this is a quotient of the algebra of symmetric tensors on u∩p that carries a fully reducible representation of LC.For s∈Z, let λs=∑lower limit k=1, upper limit m (−l+s/2)ek. Then λs defines a one-dimensional (l,L) module Cλs. Extend this to a (q¯,L) module by having u¯ act by 0, and define N(λs+2δ(u))=U(g)⊗q¯Cλs+2δ(u). Let N′(λs+2δ(u)) be the unique irreducible quotient of N(λs+2δ(u)). The representations under study are πs=ΠS(N(λs+2δ(u))) and πs′=ΠS(N′(λs+2δ(u))), where S=dim(u∩k) and ΠS is the Sth derived Bernstein functor.For s>2l−2, it is known that πs=πs′ and that πs′ is in the discrete series. Enright, Parthsarathy, Wallach, and Wolf showed for m&les;s&les;2l−2 that πs=πs′ and that πs′ is still unitary. The present paper shows that πs′ is unitary for 0&les;s&les;m−1 even though πs≠πs′, and it relates the K spectrum of the representations πs′ to the representation of LC on a suitable R(Y) with Y depending on s. Use of a branching formula of D. E. Littlewood allows one to obtain an explicit multiplicity formula for each K type in πs′; the variety Y is indispensable in the proof. The chief tools involved are an idea of B. Gross and Wallach, a geometric interpretation of Littlewood's theorem, and some estimates of norms.It is shown further that the natural invariant Hermitian form on πs′ does not make πs′ unitary for s<0 and that the K spectrum of πs′ in these cases is not related in the above way to the representation of LC on any R(Y).A final section of the paper treats in similar fashion the simply connected Lie group with Lie algebra g0=so(2m,2l−2m+1), 2&les;m&les;l/2. [Copyright &y& Elsevier]

In our previous papers (Adv. in Math. 138 (1) (1998) 182; Potential Anal. 12 (2000) 419), we have obtained a decomposition of | f |, where f is a function defined on Rn, that is analogous to the one proved by H. Tanaka for martingales (the so-called “Tanaka formula”). More precisely, the decomposition has the form | f |=f˜+D&ast;0( f ), where D&ast;0( f ) is (a variant of ) the density of the area integral associated with f. This functional (introduced by R.F. Gundy in his 1983 paper (The density of area integral, Conference on Harmonic Analysis in Honor of Antoni Zygmund. Wadsworth, Belmont, CA, 1983, pp. 138–149.)) can be viewed as the counterpart of the local time in Euclidean harmonic analysis. In this paper, we are interested in boundedness and continuity properties of the mapping f↦f˜ (which we call the Le´vy transform in analysis) on some classical function or distribution spaces. As was shown in [4,5], the above (non-linear) decomposition is bounded in Lp for every p∈[1,+∞[, i.e. one has || f˜ ||p&les;Cp|| f||p, where Cp is a constant depending only on p. Nevertheless our methods (roughly speaking, the Caldero´n–Zygmund theory in [4], stochastic calculus and martingale inequalities in [5]) both gave constants Cp whose order of magnitude near 1 is O(1/(p−1)). The aim of this paper is two-fold: first, we improve the preceding result and we answer a natural question, by proving that the best constants Cp are bounded near 1. Second, we prove that the Le´vy transform f↦f˜ is continuous on the Hardy spaces Hp with p>n/(n+1). [Copyright &y& Elsevier]

The goal of harmonic analysis on a (noncommutative) group is to decompose the most “natural” unitary representations of this group (like the regular representation) on irreducible ones. The infinite-dimensional unitary group U(∞) is one of the basic examples of “big” groups whose irreducible representations depend on infinitely many parameters. Our aim is to explain what the harmonic analysis on U(∞) consists of.We deal with unitary representations of a reasonable class, which are in 1–1 correspondence with characters (central, positive definite, normalized functions on U(∞)). The decomposition of any representation of this class is described by a probability measure (called spectral measure) on the space of indecomposable characters. The indecomposable characters were found by Dan Voiculescu in 1976.The main result of the present paper consists in explicitly constructing a 4-parameter family of “natural” representations and computing their characters. We view these representations as a substitute of the nonexisting regular representation of U(∞). We state the problem of harmonic analysis on U(∞) as the problem of computing the spectral measures for these “natural” representations. A solution to this problem is given in the next paper (Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, math/0109194, to appear in Ann. Math.), joint with Alexei Borodin.We also prove a few auxiliary general results. In particular, it is proved that the spectral measure of any character of U(∞) can be approximated by a sequence of (discrete) spectral measures for the restrictions of the character to the compact unitary groups U(N). This fact is a starting point for computing spectral measures. [Copyright &y& Elsevier]

We consider the periodic weighted operator Ty=−ρ−2(ρ2y′)′+1/4 ρ−4 in L2(R,ρ2 dx) where ρ is a 1-periodic positive function satisfying q=ρ′/ρ∈L2(0,1). The spectrum of T consists of intervals separated by gaps. In the first part of the paper we construct the Marchenko–Ostrovski mapping q→h(q) and solve the corresponding inverse problem. For our approach it is essential that the mapping h has the factorization h(q)=h0(V(q)), where q→V(q) is a certain nonlinear mapping and V→h0(V) is the Marchenko–Ostrovski mapping for the Hill operator. Moreover, we solve the inverse problem for the gap length mapping. In the second part of this paper we derive the trace formula for T. [Copyright &y& Elsevier]

In this paper, we establish a moderate deviation principle for two-dimensional stochastic Navier–Stokes equations driven by multiplicative Lévy noises. The weak convergence method introduced by Budhiraja, Dupuis and Ganguly in [3] plays a key role. [ABSTRACT FROM AUTHOR]

In this paper, we give a formal solution of some nonlinear evolution equations. By the formal solution, we can obtain the blow-up solution of the heat equations, even in the supercritical case. [ABSTRACT FROM AUTHOR]

Abstract: We study the diffusive logistic equation with a free boundary in time-periodic environment. Such a model may be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. For time independent environment, in the cases of one space dimension, and higher space dimensions with radial symmetry, this free boundary problem has been studied in Du and Lin (2010) [12], Du and Guo (2011) [9]. In both cases, a spreading–vanishing dichotomy was established, and when spreading occurs, the asymptotic spreading speed was determined. In this paper, we show that the spreading–vanishing dichotomy is retained in time-periodic environment, and we also determine the spreading speed. The former is achieved by further developing the earlier techniques, and the latter is proved by introducing new ideas and methods. [Copyright &y& Elsevier]

In this paper, we study existence of isometric embedding of S q m into S p n , where 1 ≤ p ≠ q ≤ ∞ and n ≥ m ≥ 2. We show that for all n ≥ m ≥ 2 if there exists a linear isometry from S q m into S p n , where (q , p) ∈ (1 , ∞ ] × (1 , ∞) ∪ (1 , ∞) ∖ { 3 } × { 1 , ∞ } and p ≠ q , then we must have q = 2. This mostly generalizes a classical result of Lyubich and Vaserstein. We also show that whenever S q embeds isometrically into S p for (q , p) ∈ (1 , ∞) × [ 2 , ∞) ∪ [ 4 , ∞) × { 1 } ∪ { ∞ } × (1 , ∞) ∪ [ 2 , ∞) × { ∞ } with p ≠ q , we must have q = 2. Thus, our work complements work of Junge, Parcet, Xu and others on isometric and almost isometric embedding theory on non-commutative L p -spaces. Our methods rely on several new ingredients related to perturbation theory of linear operators, namely Kato-Rellich theorem, theory of multiple operator integrals and Birkhoff-James orthogonality, followed by thorough and careful case by case analysis. The question whether for m ≥ 2 and 1 < q < 2 , S q m embeds isometrically into S ∞ n , was left open in Bull. London Math. Soc. 52 (2020) 437-447. [ABSTRACT FROM AUTHOR]

Abstract: The asymptotic behavior of the heat kernel of a Riemannian manifold gives rise to the classical concepts of parabolicity, stochastic completeness (or conservative property) and Feller property (or -diffusion property). Both parabolicity and stochastic completeness have been the subject of a systematic study which led to discovering not only sharp geometric conditions for their validity but also an incredible rich family of tools, techniques and equivalent concepts ranging from maximum principles at infinity, function theoretic tests (Khasʼminskii criterion), comparison techniques etc. The present paper aims to move a number of steps forward in the development of a similar apparatus for the Feller property. [Copyright &y& Elsevier]

Abstract: In the paper we generalize the main results presented in Bentkus and Paulauskas (2004) by giving rates of approximation of some semigroups of operators of the order , . Also two classes of operators, generalizing sectorial and quasi-sectorial operators, are introduced and their properties are studied. [Copyright &y& Elsevier]

Abstract: In the paper, we first show the existence of global periodic conservative solutions to the Cauchy problem for a periodic modified two-component Camassa–Holm equation. Then we prove that these solutions, which depend continuously on the initial data, construct a semigroup. [Copyright &y& Elsevier]

Abstract: A well-known conjecture of R. Laugesen and C. Morpurgo asserts that the diagonal of the Neumann heat kernel of the unit ball is a strictly increasing radial function. In this paper we use probabilistic arguments to settle this conjecture and to prove some inequalities for the Neumann heat kernel in the ball. [ABSTRACT FROM AUTHOR]

Abstract: Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the persistence properties of such eigenvalues are linked intimately to the multiplicity of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold with infinite codimension in an appropriate space of potentials. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a locally compact group of type I and its dual space. Roughly speaking, qualitative uncertainty principles state that the concentration of a nonzero integrable function on G and of its operator-valued Fourier transform on is limited. Such principles have been established for locally compact abelian groups and for compact groups. In this paper we prove generalizations to the considerably larger class of groups with finite dimensional irreducible representations. [Copyright &y& Elsevier]

Abstract: The paper is devoted to the isomorphic structure of symmetrizations of quasi-Banach ideal function or sequence lattices. The symmetrization of a quasi-Banach ideal lattice E of measurable functions on , , or , consists of all functions with decreasing rearrangement belonging to E. For an order continuous E we show that every subsymmetric basic sequence in which converges to zero in measure is equivalent to another one in the cone of positive decreasing elements in E, and conversely. Among several consequences we show that, provided E is order continuous with Fatou property, contains an order isomorphic copy of if and only if either E contains a normalized -basic sequence which converges to zero in measure, or contains the function . We apply these results to the family of two-weighted Orlicz–Lorentz spaces defined on or , . This family contains usual Orlicz–Lorentz spaces when and Orlicz–Marcinkiewicz spaces when . We show that for a large class of weights , it is equivalent for the space , and for the non-weighted Orlicz space to contain a given sequential Orlicz space isomorphically as a sublattice in their respective order continuous parts. We provide a complete characterization of order isomorphic copies of in these spaces over or exclusively in terms of the indices of φ. If we show that the set of exponents p for which lattice embeds in the order continuous part of is the union of three intervals determined respectively by the indices of φ and by the condition that the function belongs to the space. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss a new method for computing the first Dirichlet eigenvalue of the p-Laplacian inspired by the inverse power method in finite dimensional linear algebra. The iterative technique is independent of the particular method used in solving the p-Laplacian equation and therefore can be made as efficient as the latter. The method is validated theoretically for any ball in if and for any bounded domain in the particular case . For the method is validated numerically for the square. [Copyright &y& Elsevier]

Abstract: Let be an elliptic operator with unbounded and sufficiently smooth coefficients and let μ be a (sub)-invariant measure of the operator . In this paper we give sufficient conditions guaranteeing that the closure of the operator generates a sub-Markovian strongly continuous semigroup of contractions in . Applications are given in the case when is a generalized Schrödinger operator. [Copyright &y& Elsevier]

Abstract: Let , , and . In this paper, we introduce a new class of function spaces which unify and generalize the Triebel–Lizorkin spaces with both and and Q spaces. By establishing the Carleson measure characterization of Q space, we then determine the relationship between Triebel–Lizorkin spaces and Q spaces, which answers a question posed by Dafni and Xiao in [G. Dafni, J. Xiao, Some new tent spaces and duality theorem for fractional Carleson measures and , J. Funct. Anal. 208 (2004) 377–422]. Moreover, via the Hausdorff capacity, we introduce a new class of tent spaces and determine their dual spaces , where , , , , and denotes the conjugate index of ; as an application of this, we further introduce certain Hardy–Hausdorff spaces and prove that the dual space of is just when . [Copyright &y& Elsevier]