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It is shown that an approach earlier used by the author for classification of closed simply connected 6-manifolds (reduction to the problem of calculating certain bordism groups) can also be [...]

Congruences mod p for a prime p and partial products of the numbers 1, ..., p - 1 are obtained. Bibliography: 2 titles., UDC 511 1. FULL FACTORIALS In accordance with the Wilson theorem, for a positive rational prime number p, the factorial of p - 1 satisfies the congruence (p - 1)! [...]

UDC 515.164 (Journal of Mathematical Sciences, Vol. 159, No. 6, pp. 753-760 (2009)) There is a misprint in the formulation of Lemma 3. Instead of the dimension constraints n-k = [...]

This paper is concerned with the theory of paragraded rings, which begins with a series of Krasner and Vukovic's notes in Proceedings of the Japan Academy, which first appeared in late 1980s. We present prime and Jacobson radicals, discuss the general Kurosh-Amitsur theory of radicals of paragraded rings, establish that the theorem of Anderson, Divinsky, and Sulinski holds for paragraded rings, characterize paragraded normal radicals, and prove that all special paragraded radicals of paragraded rings can be described by appropriate classes of their graded modules. I wholeheartedly dedicate this paper to one of the greatest Lomonosov algebraists A. V. Mikhalev in honor of his 80th birthday, 1. Introduction In this paper, we are going to present some results on the different types of radicals introduced and studied in joint papers with my student E. Ilic-Georgijevic [13,14] [...]

In this paper we introduce the notion of congruence for regions in lattice path models. This turns out to be useful for deriving a path counting formula for the auxiliary lattice path model in the presence of long steps, source and target points of which are situated near the filter restrictions. This problem was motivated by the fact, that weighted numbers of paths in such model mimic multiplicities in tensor power decomposition of [U.sub.q](s[l.sub.2])-module T[(1).sup.[cross product][N at roots of unity. We expand on combinatorial properties of such model and introduce the punchline of a proof for explicit path counting formula. 14 titles., Introduction The problem of tensor power decomposition can be considered from the combinatorial perspective as a problem of counting lattice paths in Weyl chambers [1-4]. In this paper we introduce [...]

The paper examines the question of an upper bound on the number of proper edge 3-colorings of a connected cubic graph with 2n vertices. For this purpose, the Karpov method is developed with the help of which a weaker version of the bound was previously obtained. Then the bound [2.sup.n] + 8 for even n and [2.sup.n] + 4 for odd n is proved. Moreover, a unique example is found, for which the upper bound is exact. Bibliography: 2 titles., 1. INTRODUCTION In paper [1], Karpov studies proper 3-edge-colorings of a connected cubic graph (in such a coloring each edge is colored in one of three fixed colors so that [...]

We consider three independent objects: a two-sided wide-sense stationary random sequence [xi] := (***, [[xi].sub.-1], [[xi].sub.-0], [[xi].sub.-1],...) with zero mean and finite variance, a standard Poisson process [PI] and a subordinator S, that is a nondecreasing Levy process. By means of reflection about zero we extend [PI] and S to the negative semi-axis and define a random time change [PI](S(t)), t [member of] R. Then we define a so-called PSI-process [psi](t):= [[xi].sub.[pi]](s(t)), t [member of] R, which is wide-sense stationary. Notice that PSI-processes generalize pseudo-Poisson processes. The main aim of the paper is to express spectral properties of the process [psi] in terms of spectral characteristics of the sequence [xi] and the Levy measure of the subordinator S. Using complex analytic techniques, we derive a general formula for the spectral measure G of the process [psi]. We also determine exact spectral characteristics of [psi] for the following examples of [xi]: almost periodic sequence; finite-order moving average; finite order autoregression. These results can find their applications in all areas where [L.sup.2]-theory of stationary processes is used. Bibliography:17 titles., INTRODUCTION Let [mu] be a random nonnegative measure with independent values on pairwise disjoint Borel subsets of R whose distribution is shift invariant (see [7, Chap. 8]). (In the paper, [...]

The present paper considers a discrete analog of the parking problem. Let n be an integer. If n> 1, then we randomly locate an interval (t,t + 1) with integer endpoints on a segment [0,n]. Thus, the original segment is divided into two: [0,t] and [t + 1, n], and each of them is further considered separately likewise the original one. The phrase 'randomly' in this problem means that t is a uniformly distributed on a set {1,..., n - 1} random variable. The process of location of the intervals finishes when the lengths of all the remaining intervals are less than 2. Define as [X.sub.n] the total amount of the located intervals. In the present paper, the expectations E{[X.sub.n]} are calculated. The process described above can be interpreted as a parking process of cars with handlebars on the left. Hence, the driver is able to leave his car only if the place on his left is free. This is exactly the case when the driver cannot take the left end place of any free segment. In this case, [X.sub.n] stands for the amount of the parked cars. Bibliography: 13 titles., 1. INTRODUCTION The problem of random interval filling was first formulated by Renyi in [1]. The research work considered the following problem. An interval (t, t + 1) is randomly [...]

The present work is aimed at supplementing the authors' paper (2018). Our results on the approximation of distributions of sums of independent summands by the accompanying compound Poisson laws and the estimates of the proximity of sequential convolutions of multidimensional distributions on convex polyhedra are shown to be applicable almost automatically to the infinite-dimensional case. Bibliography: 14 titles., The aim of the present work is to provide a supplement to the authors' paper [4]. It is shown that our results on the approximation of distributions of sums of [...]

In this paper, we construct and examine the time-discretization scheme for the Cauchy problem for a linear homogeneous differential equation with the Caputo fractional derivative of order [alpha] [member of] (0,1) in time and containing the sectorial operator in a Banach space in the spatial part. The convergence of the scheme is established and error estimates are obtained in terms of the step of discretization. Properties of the Mittag-Leffler function, hypergeometric functions, and the calculus of sectorial operators in Banach spaces are used. Results of numerical experiments that confirm theoretical conclusions are presented. Keywords and phrases: Cauchy problem, Caputo derivative, Banach space, finite-difference scheme, error estimate, Mittag-Leffler function, hypergeometric function, sectorial operator. AMS Subject Classification: 47N40, 65J08, 35R11, 1. Statement of the problem. The papers [13, 15-17] are devoted to the study of the class of finite-difference methods of the form [Please download the PDF to view the [...]

The Fermat-Steiner problem is the problem of finding all points of a metric space Y such that the sum of the distances from them to points of a certain fixed finite subset A of the space Y is minimal. In this paper, we examine the Fermat-Steiner problem in the case where Y is the space of compact subsets of the Euclidean plane endowed with the Hausdorff metric, and points of A are finite pairwise disjoint compact sets. Keywords and phrases: Fermat-Steiner problem, Hausdorff distance, compact subset, Euclidean space, Steiner compact. AMS Subject Classification: 51E99, 1. Introduction. The main results of this paper are the Steiner criterion of a minimum compact set in the class of solutions for finding such compact sets (Theorem 4.1) and [...]

In this paper, we develop an approach to the presentation of set-theoretic topology based on the systematic use of two standard operations: the union operation and the closure operation of a collection of sets (i.e., the closure with respect to the union operation). Definitions of these operations are given and their basic properties are formulated and proved. Based on the results obtained, we present the foundations of elementary topology. Keywords and phrases: set, topological space, base of topology, induced topology, product of topo-logical spaces. AMS Subject Classification: 54A05, 1. Introduction. This paper is based on the course on tensor analysis, which one of the authors (A. Badin) gives at the Faculty of Physics of the M. V. Lomonosov [...]

The paper suggests a general approach to deriving upper bounds for [[parallel][A.sup.-1]Q[parallel].sub.[infinity]] from those for [[parallel][A.sup.-1][parallel].sub.[infinity]] for matrices A belonging to different subclasses of the class of nonsingular [??]-matrices. The approach is applied to SDD, S-SDD, OBS, OB, and Nekrasov matrices. Bibliography: 20 titles., 1. INTRODUCTION AND PRELIMINARIES Lately, in a number of papers, some known upper bounds of [[parallel][A.sup.-1][parallel].sub.[infinity]] for n x n matrices A from certain subclasses of nonsingular [??]-matrices were generalized [...]

Let [I.sub.p,q] = [I.sub.p] [direct sum] - [I.sub.q]. Pseudounitary eigenvalues of a positive definite matrix A are the moduli of conventional eigenvalues of the matrix [I.sub.p,q]A. They are invariants of pseudounitary *-congruences performed with A. For a fixed n = p+q, the sum of the squares [[sigma].sub.p,q] of these numbers is a function of the parameter p. In general, its values for different p can differ very significantly. However, if A is the tridiagonal Toeplitz matrix with an entry a [greater than or equal to] 2 on the principal diagonal and the entry - 1 on the two neighboring diagonals, then [[sigma].sub.p,q] has the same value for all p. This nontrivial fact is explained in the paper. Bibliography: 1 title., 1. This paper considers reduction of positive definite n x n matrices to diagonal form via Hermitian congruences of a special class. Hermitian congruences are transformations of the type A [...]

The theory of commutative rings of differential operators relating the completely integrable systems with the geometry of algebraic curves was constructed several decades ago. It was especially complete in the case of rings, generated by two operators of the coprime order, usually of order 2 and of some odd order; the theory of such rings turned out to be equivalent to the theory of KdV hierarchy. However, the corresponding algebraic curves were always hyperelliptic. In order to handle the general (canonical curves), one should consider the rings, generated by more than two operators. In the previous paper of 1980, the author considered the simplest possible case of this kind-that of generators of orders 3, 4, 5. The goal of the present paper is to give the details of the calculations in that paper and to explain the conjectural geometry underlying some enigmatic phenomena that were used in 1980 to complete the calculations and give some algebro-geometric applications., Author(s): George B. Shabat [sup.1] Author Affiliations: (1) https://ror.org/0473ch268, grid.446275.6, 0000 0001 2162 6510, Russian State University for the Humanities, , Moskow, Russia Introduction and history The theory of commuting [...]

In the paper, we prove probabilistic analogs of the universal approximation theorems and link continuous random variables of a certain type and monotonic feedforward artificial neural networks with one-dimensional input, output and one hidden layer. In particular, we show that any continuous infinitely divisible random variable can be successfully approximated with a mix of logistic distributions. Based on the theorems proved in the current paper, we develop a new approach for developing Monte Carlo methods combined with artificial neural networks for pricing options in Lévy models. In contrast to straightforward incorporation of neural networks into Monte Carlo methods, we approximate the cumulative distribution function, but not its inverse. Moreover, we give a clear probabilistic interpretation of the constructed approximator that helps us simulate the Lévy process by using only separate components of our neural network., Author(s): Oleg Kudryavtsev [sup.1] [sup.2], Natalia Danilova [sup.1] [sup.2] Author Affiliations: (1) InWise Systems, LLC, , Rostov-on-Don, Russia (2) https://ror.org/01tv9ph92, grid.182798.d, 0000 0001 2172 8170, I.I.Vorovich Institute of Mathematics, Mechanics [...]

We consider the discrete Dirichlet boundary value problem for a discrete elliptic pseudodifferential equation in the quadrant and study its solvability in discrete counterparts of the Sobolev-Slobodetskii space. The study is based on a special factorization of the elliptic symbol. We compare the solutions to the discrete Dirichlet problem and its continuous counterpart. Bibliography: 10 titles. In this paper, based on the ideas and methods of [1, 2] (cf. also [3] [7]), we compare discrete and continuous elliptic boundary value problems in the quadrant for the simplest pseudodifferential operators. We emphasize that, in the case of a quadrant, there are principal differences from the case of a half-space, and new analytic tools are required., 1 Preliminaries We recall the main notions and results which will be used throughout the paper (we refer to [5] for details). Let 1? be an integer lattice in the [...]

The paper extends an earlier result of G.V. Kalachev et al. (Sb. Math. 210(8):1129-1147, 2019) on the existence of a maximizer of convolution operator acting between two Lebesgue spaces on [Formula omitted] with kernel from some [Formula omitted], [Formula omitted]. On the other hand, E. Lieb (Ann. of Math. 118:(2):349-374, 1983) proved the existence of a maximizer for the Hardy-Littlewood-Sobolev inequality and remarked that in general a convolution maximizer for a kernel from weak [Formula omitted] may not exist. In this paper we axiomatize some properties used in the proof of the Kalachev-Sadov 2019 theorem and obtain a more general result. As a consequence, we prove that the convolution maximizer always exists for kernels from a slightly more narrow class than weak [Formula omitted], which contains all Lorentz spaces [Formula omitted] with [Formula omitted]., Author(s): Sergey Sadov [sup.1] Author Affiliations: (1) Moscow, Russia Introduction In this paper we extend the main result of [1]. We will mostly follow [1] in notation and terminology. Following [...]

Recently the author proved that the Hummel-Scheinberg-Zalcman conjecture of 1977 on coefficients of nonvanishing [H.sup.p] functions is true for all p = 2m, m [member of] N, i.e., for the Hilbertian Hardy spaces [H.sup.2m]. As a consequence, this also implies the proof of the Krzyz conjecture for bounded nonvanishing functions, which originated this direction. In the present paper, we solve the problem for all spaces [H.sup.p] with p [greater than or equal to] 2. Keywords. Nonvanishing holomorphic functions, the Hardy spaces, the Hummel-Scheinberg-Zalcman conjecture, Schwarzian derivative, quasiconformal extension, the Teichmuller spaces, Bers' isomorphism theorem., 1. Introductory remarks ad main result This paper is devoted to construction of special quasiconformal deformations of nonvanishing Hardy functions with prescribed distortion properties and their application to proof of [...]

In this paper, we consider a variant of the Lambek calculus allowing empty antecedents. This variant uses two connectives: the left division and a unary modality that occurs only with negative polarity and allows weakening in antecedents of sequents. We define the notion of a proof net for this calculus, which is similar to those for the ordinary Lambek calculus and multiplicative linear logic. We prove that a sequent is derivable in the calculus under consideration if and only if there exists a proof net for it. We present a polynomial-time algorithm for deciding whether an arbitrary given sequent is derivable in this calculus., Introduction The aim of this paper is to find an efficient algorithm for deciding whether an arbitrary given sequent is derivable in a variant of the Lambek calculus allowing empty [...]

In this paper, Jordan-Kronecker invariants are calculated for all nilpotent 6- and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space g proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A - [lambda][BETA])([chi]), we consider Jordan-Kronecker invariants for this pencil and how they change with [chi] (the latter is done for 6-dimensional Lie algebras)., 1. Basic Definitions and Theorems Definition 1. A Poisson bracket is a bilinear skew-symmetric operation over functions f,g [right arrow] {f,g} satisfying the Jacobi identity and the Leibniz rule. The [...]

This paper is devoted to the generalization of the construction of a Jordan algebra of a Lie algebra and the known theorems on the local finite-dimensionality of Lie PI-algebras with an algebraic adjoint representation to Mal'tsev algebras., Introduction Jordan algebras of Lie algebras are defined in [9] by analogy with local algebras of associative algebras and triple Jordan systems. In the paper their generalized version for Mal'tsev [...]

This paper is devoted to quotient rings of graded rings with finiteness conditions for ideals. The analogs of Goldie, Shock, and Small theorems are proved. We find criteria for a graded ring R to admit a gr-semisimple and gr-Artinian (the Goldie theorem) and gr-Artinian (the Small theorem) classical graded quotient ring., Throughout the paper, [member of] is a group with neutral element e, R is an associative ring graded by G, i.e., [Please download the PDF to view the mathematical expression], [...]

Semirings of skew polynomials such as invariant, without nilpotent elements, Abelian, and Rickart without nilpotent elements are considered in this paper. Properties and characterizations of these semirings are obtained., This paper studies semirings of skew polynomials. The main task is to find connections of properties of the semiring of polynomials and the semiring of coefficients. A semiring is a [...]

The fourth-order partial differential equation for the biharmonic Poisson integral is presented in the case of the upper half-plane (y > 0). To solve this equation, two boundary conditions must be taken into account. The boundary-value problem is solved by transforming the presented boundary-value problem for the biharmonic Poisson integral into two boundary-value problems for some two-dimensional functions [??] (q, y) and [??] (q, y). After that, the biharmonic Poisson integral for the upper half-plane is obtained. It was found that the derived Taylor series of biharmonic Poisson integral for the upper half-plane contains the remainder in the integral form. Keywords. Biharmonic Poisson integral, Taylor series, integral kernel., 1. Introduction The properties of the biharmonic Poisson integral for the unit disk were first studied in the paper [1]. Further investigations were performed in the papers [2-10]. It is [...]

This is the first English translation of the paper originally published in the first issue of 'Problems in Mathematical Analysis' in 1966 and reproduced to the 85th birthday of V. Maz'ya. Bibliography: 15 titles. Illustrations: 3 figures., 1. In this paper, we consider the Dirichlet problem with zero boundary condition for the elliptic equation [Please download the PDF to view the mathematical expression] (1) in a finite [...]

In this paper, we apply methods of resurgent analysis (including the requantization method) to the construction of asymptotics for solutions of linear ordinary differential equations with holomorphic coefficients. We provide a classification of various types of asymptotics depending on the principal symbol of the differential operator. Using the requantization method, we construct asymptotics for solutions of an ordinary differential equation with holomorphic coefficients in a neighborhood of infinity. Keywords and phrases: Fuchsian linear differential equation, irregular singular point, asymptotics, resurgent function, Laplace--Borel transform, principal symbol of a differential operator, requantization method. AMS Subject Classification: 34E99, Introduction. In this paper, we consider methods for constructing asymptotics for solutions of degenerate ordinary differential equations with holomorphic coefficients, ordinary differential equations with holomorphic coefficients [Please download the PDF [...]

Extreme values are considered in samples with random size that have a mixed Poisson distribution being generated by a doubly stochastic Poisson process. We prove some inequalities providing bounds on the rate of convergence in limit theorems for the distributions of max-compound Cox processes., 1. Introduction In this paper we continue the research we started in [14, 15]. The aim of these papers was to study the analytic properties and asymptotic behavior of max-compound [...]

In this paper a retrial queuing system is considered with the help of two-way communication where the server is subject to random breakdowns. This is a M/M/1//N type of system so the population of the source is finite. The server becoming idle enables calls the customers in the orbit (outgoing call or secondary customers). The service time of the primary and secondary customers follows exponential distribution with different rates [[mu].sub.1] and [[mu].sub.2] respectively. All the random variables included in the model construction are assumed to be totally independent of each other. The novelty of this paper is to show the effect of the different distributions of failure time on the main performance measures such as the mean waiting time of an arbitrary customer or the utilization of the service unit. In order to achieve a valid comparison a fitting process is done; thus, in case of every distribution the mean value and dispersion is the same. Graphical illustrations are given with the help of the self-developed simulation program., 1. Introduction Nowadays, it is a very difficult task to analyze communications systems or create optimal designing patterns of this type of schemes owing to traffic growth and the rapidly [...]

In the paper, we obtain lower estimates for decreasing rearrangements of the convolutions through decreasing rearrangements of kernels and functions to be convolved. These estimates show the exactness of some corollaries of O'Neil's upper estimates for convolutions. The results are applied for equivalent descriptions of the cones of decreasing rearrangements for generalized Bessel and Riesz potentials. These are the key results for study of integral properties of potentials., Author(s): Mikhail L. Goldman [sup.1] Author Affiliations: (1) grid.77642.30, 0000 0004 0645 517X, People's Friendship University of Russia, , Moscow, Russia Introduction In this paper, decreasing rearrangements of generalized Bessel-Riesz [...]

In this paper, we first provide a short summary of the main properties of the so-called general fractional derivatives with the Sonin kernels introduced so far. These are integro-differential operators defined as compositions of the first order derivative and an integral operator of convolution type. Depending on succession of these operators, the general fractional derivatives of the Riemann-Liouville and of the Caputo types were defined and studied. The main objective of this paper is a construction of the 1st level general fractional derivatives that comprise both the general fractional derivative of the Riemann-Liouville type and the general fractional derivative of the Caputo type. We also provide some of their properties including the 1st and the 2nd fundamental theorems of Fractional Calculus for these derivatives and the suitably defined general fractional integrals., Author(s): Yuri Luchko [sup.1] Author Affiliations: (1) Department of Mathematics, Physics, and Chemistry, Berlin University of Applied Sciences and Technology, , Luxemburger Str. 10, 13353, Berlin, Germany Introduction In the [...]

In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the case when, for the two boundary value problems, some coefficients of the differential expressions and of the boundary conditions coincide. We estimate the difference of their spectral data in this case. Although the asymptotic behaviour of spectral data is well-studied for differential operators with regular (integrable) coefficients, to the best of the author's knowledge, there were no results in this direction for the higher-order differential operators with distribution coefficients (generalized functions) in a general form. The technique of this paper relies on the recently obtained regularization and the Birkhoff-type solutions for differential operators with distribution coefficients. Our results have applications to the theory of inverse spectral problems as well as a separate significance., Author(s): Natalia P. Bondarenko [sup.1] [sup.2] Author Affiliations: (1) grid.79011.3e, 0000 0004 0646 1422, Department of Applied Mathematics and Physics, Samara National Research University, , Moskovskoye Shosse 34, 443086, Samara, [...]

The present paper is devoted to the investigation of uniform stabilities of positive [Formula omitted]-semigroups defined on abstract state spaces by means of a generalized Dobrushin ergodicity coefficient. The most known results in the literature were obtained for Markov semigroups acting on Banach lattices having unique invariant states. The essence of the present paper is that the considered Markov semigroups (acting on abstract state spaces) do not generally have invariant states and, moreover, abstract state spaces need not necessarily be lattices, implying that the results of the paper are principal in this direction., Author(s): Nazife Erkursun-Özcan [sup.1], Farrukh Mukhamedov [sup.2] [sup.3] Author Affiliations: (1) https://ror.org/04kwvgz42, grid.14442.37, 0000 0001 2342 7339, Department of Mathematics, Faculty of Science, Hacettepe University, , 06800, Ankara, Turkey (2) [...]

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can fail near the boundary of these subdomains even for an infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary [partial derivative]Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one-dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. Also there was obtained the smoothness (in Sobolev spaces [W.sup.k.sub.2]) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding [epsilon]-neighborhoods of certain points. However, the smoothness (in Holder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Holder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Holder space., CONTENTS 1. Introduction 970 2. Geometric Questions and Auxiliary Questions 971 3. Difference Operators 973 4. Smoothness of Generalized Solutions on the Boundary of Adjacent 975 Subdomains in Holder Spaces [...]

In this paper, we investigate qualitative properties of solutions of boundary-value problems for strongly elliptic differential-difference equations. Earlier results establish the existence of generalized solutions of these problems. It was proved that smoothness of such solutions is preserved in some subdomains but can be violated on their boundaries even for infinitely smooth function on the right-hand side. For differential-difference equations on a segment with continuous right-hand sides and boundary conditions of the first, second, or the third kind, earlier we had obtained conditions on the coefficients of difference operators under which there is a classical solution of the problem that coincides with its generalized solution. Also, for the Dirichlet problem for strongly elliptic differential-difference equations, the necessary and sufficient conditions for smoothness of the generalized solution in Holder spaces on the boundaries between subdomains were obtained. The smoothness of solutions inside some subdomains except for [epsilon]-neighborhoods of angular points was established earlier as well. However, the problem of smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations remained uninvestigated. In this paper, we use approximation of the differential operator by finite-difference operators in order to increase the smoothness of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in the scale of Sobolev spaces inside subdomains. We prove the corresponding theorem., CONTENTS 1. Introduction 823 2. Geometry Properties 824 3. Difference Operators 827 4. Generalized and Classical Solutions 830 5. Smoothness of Generalized Solution Near the Segment Boundary in Terms of [...]

In this paper, we consider questions of inner regularity of weak solutions of initial-boundary value problems for the Zakharov -- Kuznetsov equation with two spatial variables. The initial function is assumed to be irregular, and the main parameter governing the regularity is the decay rate of the initial function at infinity. The main results of the paper are obtained for the problem on a half-strip. In this problem, different types of initial conditions (e.g., Dirichlet or Neumann conditions) influence the inner regularity. We also give a survey of earlier results for other types of domains: a plane, a half-plane, and a strip., CONTENTS 1. Introduction 313 2. Half-Strip Initial-Boundary Value Problems 317 3. Half-Plane Initial-Boundary Value Problems 335 4. Cauchy Problem 339 5. Strip Initial-Boundary Value Problems 341 References 342 1. Introduction [...]

The subject of this paper is the geometry of orbits of a family of smooth vector fields defined on a smooth manifold and singular foliations generated by the orbits. As is well known, the geometry of orbits of vector fields is one of the main subjects of investigation in geometry and control theory. Here we propose several author's results on this problem. Throughout this paper, smoothness means [C.sup.[infinity]]-smoothness., CONTENTS 1. Introduction 52 2. Preliminaries 53 3. Orbits of Families of Vector Fields 54 4. Singular Foliations 60 References 66 1. Introduction Structure orbits of families of smooth vector [...]

The paper considers karyon tilings T of the torus Td of an arbitrary dimension d. The prototypes of such tilings are the one-dimensional Fibonacci tilings and their two-dimensional counterpart, the Rauzy tilings. Karyon tilings T are important for applications to multidimensional continued fractions. In this paper, local properties of karyon tilings T are considered. Bibliography: 17 titles., Introduction The paper considers karyon tilings T of the torus Td of an arbitrary dimension d, which are important for applications to multidimensional continued fractions [1-4]. The prototypes of karyon [...]

Let K be a field with at least 19 elements. It is proved that any subgroup of GL(4, K) generated by a pair of 2-tori contains unipotent elements of rank 1 or 2. Taking into account previous papers of N. A. Vavilov and the author, this result is valid for any general linear group. It is one of the first steps in studying subgroups generated by a pair of microweight tori in Chevalley groups. Bibliography: 12 titles., 1. Introduction In the present paper we prove that in the group GL(4, K) over a field K with at least 19 elements, any subgroup generated by a pair of [...]

The paper is devoted to studying of subsets of multidimensional local fields such that any power series with coefficients from this subset converges when a maximal ideal element is substituted for a variable. Bibliography: 12 titles., 1. Structure of multidimensional complete fields Multidimensional local fields and multidimensional complete fields generalizing them first arose in papers of A. N. Parshin (see [7,8]). Recall (see [2]) that the [...]

In 2013, A. Thom proved that for any standard unitary group SU(C) (the compact form) and for any real number e > 0, there is a nontrivial word w(x, y) in two variables such that the image of the word map [Please download the PDF to view the mathematical expression] is contained in [member of]-neighborhood of the identity of the group SUn(C). Actually, in Thom's paper there is a construction of a sequence [Please download the PDF to view the mathematical expression], where [Please download the PDF to view the mathematical expression], that converges uniformly on a compact group to the identity. In the present paper, a method for constructing of such sequences is proposed. Also, with the help of results obtained by T Bandman et.al (2006), a sequence of surjective word maps [Please download the PDF to view the mathematical expression] is constructed, where each word wj is contained in the corresponding member [Please download the PDF to view the mathematical expression] of the derived series of the free group Fn. Some comments and remarks related to such results and general properties of word maps of compact groups are given. Bibliography: 21 titles. Dedicated to the 80th jubilee of A. V. Yakovlev, Introduction For any group G and any word w [member of] Fn, where Fn is a free group of rank n, a word map [Please download the PDF to view [...]

The paper continues the study of images of word maps with constants [Please download the PDF to view the mathematical expression] of a simple algebraic group G, started by F. Gnutov and N. Gordeev (2019). It is proved that for adjoint simple algebraic groups of type Bl, Cl, F4, G2 over a field of characteristic = 2, 3, the map [pi]ow, where [w.sub.[SIGMA]] is word map without small constants and[pi]:G[right arrow] T/W is the factorization map, is a constant map if and only if [Please download the PDF to view the mathematical expression], where g [member of] G and v is a word with constants. Bibliography: 9 titles., Introduction In the present paper, we consider a word map with constants [Please download the PDF to view the mathematical expression] of a simple algebraic group G. Here, [Please download [...]

The paper continues a series of papers devoted to study of the connection between two approaches to classification of complete discrete valuation fields with imperfect residue fields and, in particular, 2-dimensional local fields in the case of mixed characteristic. One of these approaches was introduced by Masato Kurihara (1987) in terms of the module of differentials. Another one is based on Epp's theory of elimination of wild ramification. A lower bound for the degree of a constant field extension that makes a given field into an almost standard one is established. This bound is expressed in terms of the invariant introduced in Kurihara's paper. Bibliography: 9 titles Dedicated to the 80th jubilee of A. V. Yakovlev, 1. Notation and basic definitions We use the following notation: p is a fixed prime, p > 2; [v.sub.p](x) is a p-adic valuation of a p-adic number x. 1.1. Discrete [...]

In this paper, we consider homogeneous spaces of functions defined on the whole real axis with values in a complex Banach space. A new class of functions from a homogeneous space that are almost uniformly periodic at infinity is introduced and examined. Four definitions of such functions are proposed and their equivalence is proved. Fourier series of functions almost periodic at infinity are constructed and their properties are analyzed. In this paper, we essentially used results of the theory of isometric representations and the theory of Banach modules. Keywords and phrases: function almost periodic at infinity, function slowly varying at infinity, homogeneous space, Banach module, almost periodic vector, Fourier series. AMS Subject Classification: 33E30, 43A60, 1. Homogeneous spaces of functions. Let X be a complex Banach space, End X be the Banach algebra of linear bounded operators acting in X. We denote by [L.sup.1.sub.loc](R, X) [...]

In this paper, we obtain optimality conditions in an optimal control problem with pointwise phase constraints of the equality and inequality types treated as constraints in a Hilbert space. The main results of this work are the regularized Lagrange principle stable under errors of source data and the pointwise Pontryagin maximum principle in the iterative form, which, in turn, yield a functional method of constructing a minimizing approximate solution to the problem considered. Keywords and phrases: optimal control, ill-posed problem, dual regularization, iterative dual regularization. AMS Subject Classification: 47A52, 93C15, This paper is devoted to the derivation of optimality conditions in the optimal control problem for a system of ordinary differential equations with pointwise phase constraints of equality and inequality [...]

In this paper, we study the system of equations of a boundary layer for a nonlinearly viscous, electrically conductive liquid described by a rheological law proposed by O. A. Ladyzhenskaya for incompressible media. The boundary-layer equations for the Ladyzhenskaya model were first obtained from Prandtl's axioms. By the Mises transform, the system of boundary-layer equations can be reduced to a single quasilinear equation. The main method used in this paper is the Crocco transform, which turns the system of boundary-layer equations into a quasilinear degenerate parabolic equation. In contrast to the Mises variables, the Crocco substitution allows one to study both stationary and nonstationary equations. Keywords and phrases: symmetric boundary layer, O. A. Ladyzhenskaya's equations, Crocco transform, electrically conductive liquid. AMS Subject Classification: 35K55, 1. Introduction. Various problems of hydrodynamics, mechanics, and physics are related to the motions electrically conductive liquids. For example, such problems appear in aviation and shipbuilding engineering and development of [...]

This paper is devoted to the study of slowly varying and almost periodic at infinity distributions from harmonic spaces; a number of spaces of homogeneous functions are considered. The notion of a harmonic space of distributions is introduced; this space is constructed by a homogeneous functional spaces. Properties of harmonic spaces of distributions endowed with the structure of Banach modules are studied. We prove that such spaces are isometrically isomorphic to the corresponding homogeneous functional spaces. Based on the definitions of slowly varying and almost periodic at infinity functions from a homogeneous space, we introduce the notions of slowly varying and almost periodic at infinity distributions from a harmonic space. Using methods of abstract harmonic analysis, we construct Fourier series of almost periodic distributions at infinity and examine their properties. In this paper, we essentially used results of the theory of isometric representations and the theory of Banach modules. Keywords and phrases: distribution of slow growth, distribution almost periodic at infinity, distribution slowly varying at infinity, homogeneous space, Banach module, almost periodic vector, Fourier series. AMS Subject Classification: 33E30, 43A60, 1. Homogeneous spaces of functions. Let X be a complex Banach space and End X be the Banach algebra of linear bounded operators (endomorphisms) acting in X. We denote by [...]

In this paper, we obtain a stochastic criterion for the k-motion of a regular two-dimensional surface in three-dimensional Euclidean space--a stochastic analog of the main theorem of bending theory . Keywords and phrases: surface bending, stochastic process. AMS Subject Classification: 31A10, 60J60 Dedicated to the 80th Anniversary of Professor V. T. Fomenko, In this paper, we obtain a stochastic criterion for the k-motion of a regular surface of nonzero mean curvature and constant-sign Gaussian curvature in three-dimensional Euclidean space. This paper continues [...]

In this paper, we study hybrid linear systems of functional differential equations with aftereffect using the W-method proposed by N. V. Azbelev. Two model equations are considered. We examine Banach spaces of right-hand sides and solutions of the equations considered; these spaces consist of asymptotically periodic functions. Analogs of the Bohl-Perron theorem on the asymptotic stability and on the existence of limits of solutions are obtained. Keywords and phrases: Bohl-Perron theorem, asymptotically periodic function, hybrid system, functional differential equation, equation with aftereffect, stability, method of model equations. AMS Subject Classification: 34K20, 34K25, 1. Introduction. In this paper, we continue to study hybrid linear systems of functional differential equations with aftereffect (HLSFDEA). The term 'hybrid system' is treated in the following sense: a [...]

In this paper, we consider the generating function of skew domino tableaux and conditions of decomposing it using fundamental quasisymmetric functions of type B. This question, as showed earlier by Mayorova and Vassilieva, is related to the descent statistics for elements of Coxeter groups and tableaux for Coxeter groups. In this paper, we obtain estimates of the number of diagrams that fit the conditions., 1. Introduction and Basic Definitions Definition 1. A Young diagram with n boxes filled with nonnegative integers is called a semistandard Young tableau if * each row is a nondecreasing [...]