Abstract In this paper we analyze connections between Wold-type decompositions of bi/isometries and pairs of semigroups of isometries, where at least one of semigroups is a product semigroup generated by two isometries.

Abstract In this paper we study relationships between some topological and analytic invariants of zero-dimensional germs, or multiple points. Among other things, it is shown that there exist no rigid zero-dimensional Gorenstein singularities and rigid almost complete intersections. In the proof of the first result we exploit the canonical duality between homology and cohomology of the cotangent complex

Abstract Let \(\mathfrak{D}_\mathbf{A}(N)\) be the set of all integers not exceeding \(N\) and equal to irreducible denominators of positive rational numbers with finite continued fraction expansions in which all partial quotients belong to a finite number alphabet \(\mathbf{A}\). A new lower bound for the cardinality \(|\mathfrak{D}_\mathbf{A}(N)|\) is obtained, whose nontrivial part improves that

Abstract A two-sided estimate is proposed for the \(K\)-functional of the pair \((C[0,1], BV(X))\), where \(BV(X)\) is the space of functions of generalized bounded variation constructed from a symmetric sequence space \(X\). The application of this estimate to various sequence spaces \(X\) yields new interpolation theorems for spaces of finite Wiener–Young \(h\)-variation, of finite Waterman \(\Lambda\)-variation

Abstract Let \(P_n(x)\) be any polynomial of degree \(n\geq 2\) with real coefficients such that \(P_n(k)\ne 0\) for \(k\in\mathbb{Z}\). In the paper, in particular, the sum of a series of the form \(\sum_{k=-\infty}^{+\infty}1/P_n(k)\) is expressed as the value at \((0,0)\) of the Green function of the self-adjoint problem generated by the differential expression \(l_n[y]=P_n(i\,d/dx) y\) and the

Abstract We construct an example of a Hilbert \(C^*\)-module which shows that Troitsky’s theorem on the geometric essence of \( {\mathcal A} \)-compact operators between Hilbert \(C^*\)-modules cannot be extended to modules which are not countably generated case (even in the case of a stronger uniform structure, which is also introduced). In addition, the constructed module admits no frames.

Abstract We introduce a notion of averaged mappings in the broader class of \(\operatorname{CAT}(0)\) spaces. We call these mappings \(\alpha\)-firmly nonexpansive and develop basic calculus rules for ones that are quasi-\(\alpha\)-firmly nonexpansive and have a common fixed point. We show that the iterates \(x_n:=Tx_{n-1}\) of a nonexpansive mapping \(T\) converge weakly to an element in \(\operatorname{Fix}

Abstract We consider the inverse problem for a massless Dirac operator on the half-line such that the support of its potential has fixed upper boundary and solve it in terms of a Jost function and a scattering matrix. We prove that the potential of such an operator is uniquely determined by its resonances.

Abstract A subspace lattice \(\{(0), M, N, H\}\) of a Hilbert space \(H\) is called a generalized generic lattice if \(M\cap N =M^\perp\cap N^\perp =(0)\) and \(\dim (M^\perp \cap N)=\dim (M\cap N^\perp)\). In this note, we show that each derivation of a generalized generic lattice algebra into itself is inner.

Abstract In this paper we will give some results on the support of Dunkl translations on compactly supported functions. Then we will define the Dunkl-type BMO space and Riesz transforms for the Dunkl transform on \(L^\infty\) and prove the boundedness of the Riesz transforms from \(L^\infty\) to the Dunkl-type BMO space under the assumption of the uniform boundedness of Dunkl translations. The proof

Abstract We study the analog of the classical infinitesimal center problem in the plane, but for zero cycles. We define the displacement function in this context and prove that it is identically zero if and only if the deformation has a composition factor. That is, we prove that here the composition conjecture is true, in contrast with the tangential center problem on zero cycles. Finally, we give

Abstract We study the stability of the single-valued extension property for operators on a Hilbert space. Further, relations between the stability of the single-valued extension property and of property \((\omega)\) are given.

Abstract Properties of the extreme points of families of concave measures on infinite-dimensional locally convex spaces are studied. The localization method is generalized to hyperbolic measures on Fréchet spaces.

Abstract In this note we discuss some necessary and some sufficient conditions for the relative injectivity of the \(C_0(S)\)-module \(C_0(S)\), where \(S\) is a locally compact Hausdorff space. We also give a Banach module version of Sobczyk’s theorem. The main result of the paper is as follows: if the \(C_0(S)\)-module \(C_0(S)\) is relatively injective, then \(S=\beta(S\setminus \{s\})\) for any

Abstract In a recent paper, given an arbitrary homogeneous cohomological field theory ( CohFT), Rossi, Shadrin, and the first author proposed a simple formula for a bracket on the space of local functionals, which conjecturally gives a second Hamiltonian structure for the double ramification hierarchy associated to the CohFT. In this paper we prove this conjecture in the approximation up to genus \(1\)

Abstract Following work by Casazza, Kutyniok, and Lammers and its development by Stoeva and Christensen, we provide some novel characterizations of R-dual sequences of type III in Hilbert spaces. We systematically extend the construction procedure by basing it on a choice of an antiunitary involution. For certain classes of R-duals of type III, we derive a representation of the associated frame operator

Abstract In a 2004 paper by V. M. Buchstaber and D. V. Leykin, published in “Functional Analysis and Its Applications,” for each \(g > 0\), a system of \(2g\) multidimensional heat equations in a nonholonomic frame was constructed. The sigma function of the universal hyperelliptic curve of genus \(g\) is a solution of this system. In our previous work, published in “Functional Analysis and Its Applications

Abstract We define a graded graph, called the Schur–Weyl graph, which arises naturally when one considers simultaneously the RSK algorithm and the classical duality between representations of the symmetric and general linear groups. As one of the first applications of this graph, we give a new proof of the completeness of the list of discrete indecomposable characters of the infinite symmetric group

Abstract Families of continuous mappings of the interval continuously depending on a parameter are considered. Any \(G_\delta\) set dense in the parameter space is realized as the set of continuity of topological entropy for a suitable family of continuous mappings.

Abstract Jacobian determines a bundle with total space consisting of orientation-preserving diffeomorphisms of a (connected) manifold over the space of positive functions on this manifold (with integral equal to volume for a compact manifold). It is proved that, for the \(n\)-sphere with standard metric, there is a unique connection on this bundle that is invariant with respect to all isometries of

Abstract We consider the self-adjointness and essential spectrum of 3D Dirac operators with bounded variable magnetic and electrostatic potentials and with singular delta-type potentials with supports on uniformly regular unbounded surfaces \(\Sigma\) in \(\mathbb{R}^{3}\).

Abstract A family of maximally monotone operators on a separable Hilbert space is considered. The domains of these operators depend on time ranging over an interval of the real line. The space of square-integrable functions on this interval taking values in the same Hilbert space is also considered. On the space of square-integrable functions a superposition (Nemytskii) operator is constructed based

Abstract A special singular limit \(\omega_1/\omega_2 \to 1\) is considered for the Faddeev modular quantum dilogarithm (hyperbolic gamma function) and the corresponding hyperbolic integrals. It brings a new class of hypergeometric identities associated with bilateral sums of Mellin–Barnes type integrals of particular Pochhammer symbol products.

Abstract We consider Borel measures on separable Banach spaces that are limits of their finite-dimensional images in the weak topology. The class of Banach spaces on which all measures have this property is introduced. The specified property is proved for all measures from the closure in variation of the linear span of the set of measures absolutely continuous with respect to Gaussian measures. Connections

Abstract Davies’ version of the Hardy inequality gives a lower bound for the Dirichlet integral of a function vanishing on the boundary of a domain in terms of the integral of the squared function with a weight containing the averaged distance to the boundary. This inequality is applied to easily derive two classical results of spectral theory, E. Lieb’s inequality for the first eigenvalue of the Dirichlet

Abstract The one-particle density matrix \(\gamma(x, y)\) is one of the key objects in quantum-mechanical approximation schemes. The self-adjoint operator \(\Gamma\) with kernel \(\gamma(x, y)\) is trace class, but no sharp results on the decay of its eigenvalues were previously known. The note presents the asymptotic formula \(\lambda_k \sim (Ak)^{-8/3}\), \(A \ge 0\), as \(k\to\infty\) for the eigenvalues

Abstract We find that, in the critical case \(2l= {\mathbf N} \), the eigenvalues of the problem \(\lambda(-\Delta)^{l}u=Pu\) with the singular measure \(P\) supported on a compact Lipschitz surface of an arbitrary dimension in \( {\mathbb R} ^{ {\mathbf N} }\) satisfy an asymptotic formula of the same order as in the case of an absolutely continuous measure.

Abstract In this note we discuss some recent results on extreme steady waves under gravity. They include the existence and regularity theorems for highest waves on finite depth with and without vorticity. Furthermore, we state new results concerning the asymptotic behavior of surface profiles near stagnation points. In particular, we find that the wave profile of an extreme wave is concave near each

Abstract We consider Hardy inequalities on antisymmetric functions. Such inequalities have substantially better constants. We show that they depend on the lowest degree of an antisymmetric harmonic polynomial. This allows us to obtain some Caffarelli–Kohn–Nirenberg-type inequalities that are useful for studying spectral properties of Schrödinger operators.

Abstract Hardy inequalities have been important topics of research for a century, and in the past twenty years or so, there has been a deluge of important papers on various versions, including discrete and fractional forms and extensions to Rellich and higher-order inequalities. This paper is a brief survey of known fractional and nonfractional forms of the Rellich inequality together with some new

Abstract We discuss the work of Birman and Solomyak on the singular numbers of integral operators from the point of view of modern approximation theory, in particular, with the use of wavelet techniques. We are able to provide a simple proof of norm estimates for integral operators with kernel in \(B^{1/p-1/2}_{p,p}(\mathbb R,L_2(\mathbb R))\). This recovers, extends, and sheds new light on a theorem

Abstract Orthogonal polynomials \(P_{n}(\lambda)\) are oscillating functions of \(n\) as \(n\to\infty\) for \(\lambda\) in the absolutely continuous spectrum of the corresponding Jacobi operator \(J\). We show that, irrespective of any specific assumptions on the coefficients of the operator \(J\), the amplitude and phase factors in asymptotic formulas for \(P_{n}(\lambda)\) are linked by certain universal

Abstract We consider a class of Jacobi matrices with unbounded entries in the so-called critical (double root, Jordan block) case. We prove a formula which relates the spectral density of a matrix to the asymptotics of orthogonal polynomials associated with it.

Abstract We study a nonstationary Maxwell system in \(\mathbb{R}^3\) with dielectric permittivity \(\eta(\varepsilon^{-1}{\mathbf x})\) and magnetic permeability \(\mu\). Here \(\eta(\mathbf{x})\) is a positive definite bounded symmetric \((3 \times 3)\)-matrix- valued function periodic with respect to some lattice and \(\mu\) is a constant positive \(3\times 3\) matrix. We obtain approximations for

Abstract Let \(K\) be a compact Hausdorff space, \(C(K)\) be the real Banach space of all continuous functions on \(K\) endowed with the supremum norm, and \(C(K)^+\) be the positive cone of \(C(K)\). A weak stability result for the symmetrization \(\Theta=(f(\,\boldsymbol\cdot\,)-f(-\;\boldsymbol\cdot\,)/2\) of a general \(\varepsilon\)-isometry \(f\) from \(C(K)^+\cup-C(K)^+\) to a Banach space \(Y\)

Abstract Algebraic characteristics of the polynomial Somos-\(k\) sequences for \(k=4, 5, 6, 7\) are calculated.

Abstract The asymptotic properties of sequences of partitions (\(\sigma\)-algebras) associated with the Robinson–Schensted–Knuth correspondence in spaces with Bernoulli measures are studied.

Abstract The norms of embedding operators \(\mathring{W}^n_2[0,1]\hookrightarrow\mathring{W}^k_\infty[0,1]\) (\(0\leqslant k\leqslant n-1\)) of Sobolev spaces are considered. The least possible values of \(A^2_{n,k}(x)\) in the inequalities \(|f^{(k)}(x)|^2\leqslant A^2_{n,k}(x)\|f^{(n)}\|^2_{L_2[0,1]}\) (\(f\in \mathring{W}^n_2[0,1]\)) are studied. On the basis of relations between the functions \(A^2_{n

Abstract V. I. Arnold classified simple (i.e., having no moduli for classification) singularities (function germs) and also simple boundary singularities, that is, function germs invariant with respect to the action \(\sigma(x_1; y_1,\dots, y_n)=(-x_1; y_1,\dots, y_n)\) of the group \({\mathbb Z}_2\). In particular, he showed that a function germ (a germ of a boundary singularity) is simple if and

Abstract We consider a dense network of elastic materials modeled by a dense network of elastic disks. More specifically, we consider a dense network of elastic disks in the unit disk \(D(0,1)\) of \(\mathbb{R}^{2}\) obtained from an Apollonian packing of elastic circular disks by removing disks of small sizes. We suppose that the disks are pressed against each other to form small rectilinear contact

Abstract We compute the multivariate signatures of any Seifert link (that is, a union of some fibers in a Seifert homology sphere), in particular, of the union of a torus link with one or both of its cores (cored torus link). The signatures of cored torus links are used in the Degtyarev–Florens–Lecuona splicing formula for computing multivariate signatures of cables over links. We use Neumann’s computation

Abstract A construction of numerically implementable explicit expressions for the solutions of the two- and three-dimensional equations \(\operatorname{div}(\alpha(w)\nabla w)=0\) and \(\operatorname{div}(\beta\nabla w)=0\) with Cauchy data on an analytic boundary is presented.

Abstract Relative projection constants of certain classes of cubspaces of codimension 2 in \(l_\infty^{2n}\) are found. The minimal projections under considerations are of two kinds, with unit norm and with norm larger than 1.

Abstract In continuous-discrete Sobolev spaces a generalized trace formula for orthogonal polynomials \(\{\widehat{q}_n\}_{n=0}^\infty\) is obtained. The proof of this formula is based on the representation of the Fejér kernel for the system \(\{\widehat{q}_n\}_{n=0}^\infty\). As a consequence, a generalized trace formula for Gegenbauer–Sobolev polynomials in a discrete Sobolev space is obtained.

Abstract In a 2004 paper by V. M. Buchstaber and D. V. Leikin, published in “Functional Analysis and Its Applications,” for each \(g > 0\), a system of \(2g\) multidimensional Schrödinger equations in magnetic fields with quadratic potentials was defined. Such systems are equivalent to systems of heat equations in a nonholonomic frame. It was proved that such a system determines the sigma function

Abstract Quite recently a criterion for the \(\mathcal{A}\)-compactness of an ajointable operator \(F\colon {\mathcal M} \to\mathcal{N}\) between Hilbert \(C^*\)-modules, where \(\mathcal{N}\) is countably generated, was obtained. Namely, a uniform structure (a system of pseudometrics) in \(\mathcal{N}\) was discovered such that \(F\) is \(\mathcal{A}\)-compact if and only if \(F(B)\) is totally bounded

Abstract A natural class of expansive endomorphisms \(G\in C^1\) of the infinite-dimensional torus \(\mathbb{T}^{\infty}\) (the Cartesian product of countably many circles with the product topology) is considered. The endomorphisms in this class can be represented in the form of the sum of a linear expansion and a periodic addition. The following standard facts of hyperbolic theory are proved: the

Abstract We consider the problem of the constancy of the minimizer in the fractional embedding theorem \(\mathcal{H}^s(\Omega) \hookrightarrow L_q(\Omega)\) for a bounded Lipschitz domain \(\Omega\), depending on the domain size. For the family of domains \(\varepsilon \Omega\), we prove that, for small dilation coefficients \(\varepsilon\), the unique minimizer is constant, whereas for large \(\varepsilon\)

Abstract Using a change of basis in the algebra of symmetric functions, we compute the moments of the Hermitian Jacobi process. After a careful arrangement of terms and the evaluation of the determinant of an “almost upper-triangular” matrix, we end up with a moment formula which is considerably simpler than the one derived in [8]. As an application, we propose the Hermitian Jacobi process as a dynamical

Abstract The classical Fourier transform on the line sends the operator of multiplication by \(x\) to \(i\frac{d}{d\xi}\) and the operator \(\frac{d}{d x}\) of differentiation to multiplication by \(-i\xi\). For the Fourier transform on the Lobachevsky plane, we establish a similar correspondence for a certain family of differential operators. It appears that differential operators on the Lobachevsky

Abstract This paper studies spaces of distributions on a dyadic half-line, which is the positive half-line equipped with bitwise binary addition and Lebesgue measure. We prove the nonexistence of a space of dyadic distributions which satisfies a number of natural requirements (for instance, the property of being invariant with respect to the Walsh–Fourier transform) and, in addition, is invariant with

Abstract A differential inclusion whose right-hand side is the sum of two set-valued mappings in a separable Banach space is considered. The values of the first mapping are bounded and closed but not necessarily convex, and this mapping is Lipschitz continuous in the phase variable. The values of the second one are closed, and this mapping has mixed semicontinuity properties: given any phase point

Abstract Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems related to integral estimates and regularity of solutions to elliptic and/or parabolic equations. The main purpose of our work is to provide an important observation concerning the \(L^p\)-boundedness property in the context of interpolation

Abstract We prove the polynomiality of the bigraded ring \(J_{*,*}^{w, W}(F_4)\) of weak Jacobi forms for the root system \(F_4\) which are invariant with respect to the corresponding Weyl group. This work is a continuation of a joint article with V. A. Gritsenko, where the structure of the algebras of weak Jacobi forms related to the root systems of \(D_n\) type for \(2\leqslant n \leqslant 8\) was

Abstract A complete description of indecomposable characters of the infinite symmetric inverse semigroup is given. The method essentially uses the decomposition of the elements of this semigroup into a product of independent quasi-cycles and the multiplicativity theorem. Realizations of all factor representations of finite type are constructed.

Abstract A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassiliev’s theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight system corresponding to the colored Jones polynomial. This weight system can be easily defined in terms of the Lie algebra \(\mathfrak{sl}_2\), but this definition

Abstract In this paper we propose a generalization of Krichever’s algebraic-geometric construction of orthogonal coordinate systems in a flat space. In the theory of integrable systems of hydrodynamic type a fundamental role is also played by orthogonal coordinates in some special nonflat spaces. The most important class of such spaces is given by metrics of submanifolds in flat spaces that have flat

Abstract An elliptic fourth-order differential operator \(A_\varepsilon\) on \(L_2(\mathbb{R}^d;\mathbb{C}^n)\) is studied. Here \(\varepsilon >0\) is a small parameter. It is assumed that the operator is given in the factorized form \(A_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})\), where \(g(\mathbf{x})\) is a Hermitian matrix-valued function periodic with respect to some

We consider the complex interpolation method in the context of protoquantum spaces. We apply it to construct an ℒp-protoquantization of an arbitrary normed space for 1 < p < ∞.

We describe the location of the essential spectrum of one-dimensional Dirac operators with singular potentials supported on discrete sets in ℝ in terms of limit operators.