Showing papers in "Mathematical Notes in 2020"
TL;DR: In this article, a brief proof of the zero set of a nontrivial real-analytic function in space has zero measure is provided. But the proof is limited to the case of real analytic functions.
Abstract: A brief proof of the statement that the zero-set of a nontrivial real-analytic function in $d$-dimensional space has zero measure is provided.
64 citations
14 citations
TL;DR: In this article, the authors studied the asymptotics of roots of a sequence of Bernstein polynomials approximating a piecewise linear function, and proved that the limiting curve for roots is the boundary of the domain of convergence of the Bernstein poynomials on the complex plane.
Abstract: This paper is devoted to the study of the asymptotics of roots of a sequence of Bernstein polynomials approximating a piecewise linear function. This sequence arises in the construction of modified compactly supported wavelets that, in contrast to classical Daubechies wavelets, preserve localization with the growth of smoothness. It is proved that the limiting curve for roots is the boundary of the domain of convergence of the Bernstein polynomials on the complex plane.
14 citations
TL;DR: In this article, a smooth solution of the eikonal equation is studied, and a relationship between the geometry of a hypersurface and the set of singular points of its metric function on both sides of this hypersuran surface is investigated.
Abstract: Smooth solutions of the eikonal equation are studied. A relationship between the geometry of a hypersurface and the set of singular points of its metric function on both sides of this hypersurface is investigated.
14 citations
TL;DR: In this article, the classical problem of estimating the number of edges in a subgraph of a special distance graph is considered, and the classical results are improved by a large margin.
Abstract: The classical problem of estimating the number of edges in a subgraph of a special distance graph is considered. Old results are significantly improved.
13 citations
13 citations
TL;DR: In this paper, the problem of determining the chromatic number of Kneser graphs is considered and an upper and lower bound for the number of chromatic numbers of the graphs under examination is obtained.
Abstract: Graphs which are analogs of Kneser graphs are studied. The problem of determining the chromatic numbers of these graphs is considered. It is shown that their structure is similar to that of Kneser graphs. Upper and lower bounds for the chromatic numbers of the graphs under examination are obtained. For certain parameter values, an order-sharp estimate of the chromatic numbers is found, and in some cases, the exact value of the quantity in question is determined.
12 citations
TL;DR: In this article, the Dirichlet problem in the half-space for elliptic differential-difference equations with operators that are compositions of differential and difference operators is considered and an integral representation of the found solution in terms of a Poisson-type formula is constructed, and its convergence to zero as the time-like independent variable tends to infinity is proved.
Abstract: The Dirichlet problem in the half-space for elliptic differential-difference equations with operators that are compositions of differential and difference operators is considered. For this problem, classical solvability or solvability almost everywhere (depending on the constraints imposed on the boundary data) is proved, an integral representation of the found solution in terms of a Poisson-type formula is constructed, and its convergence to zero as the time-like independent variable tends to infinity is proved.
11 citations
TL;DR: In this article, the authors developed an approach to write efficient short-wave asymptotics based on the representation of the Maslov canonical operator in a neighborhood of generic caustics in the form of special functions of a composite argument.
Abstract: We develop an approach to writing efficient short-wave asymptotics based on the representation of the Maslov canonical operator in a neighborhood of generic caustics in the form of special functions of a composite argument. A constructive method is proposed that allows expressing the canonical operator near a caustic cusp corresponding to the Lagrangian singularity of type $$A_3$$
(standard cusp) in terms of the Pearcey function and its first derivatives. It is shown that, conversely, the representation of a Pearcey type integral via the canonical operator turns out to be a very simple way to obtain its asymptotics for large real values of the arguments in terms of Airy functions and WKB-type functions.
9 citations
TL;DR: In this paper, the authors give sufficient conditions on a matrix A ensuring the existence of a partition of this matrix into two submatrices with extremely small norm of the image of any vector.
Abstract: We give sufficient conditions on a matrix A ensuring the existence of a partition of this matrix into two submatrices with extremely small norm of the image of any vector. Under some weak conditions on a matrix A we obtain a partition of A with the extremely small $(1,q)$-norm of submatrices.
9 citations
TL;DR: In this paper, the problem of whether a formation β contains products of F(G)-subnormal β-subgroups of finite solvable groups is studied, and the supersolvability of any group G having three supersolvable subgroups whose indices in G are pairwise coprime is proved.
Abstract: A subgroup H of a finite group G is said to be F(G)-subnormal if it is subnormal in HF(G), where F(G) is the Fitting subgroup of G. In the paper, the problem of whether or not a formation β contains products of F(G)-subnormal β-subgroups of finite solvable groups is studied. In particular, solvable saturated formations β with this property are described. Formation properties of groups having three solvable F(G)-subnormal subgroups with pairwise coprime indices are studied. The supersolvability of any group G having three supersolvable F(G)-subnormal subgroups whose indices in G are pairwise coprime is proved.
TL;DR: In this paper, the existence of a nonlocal relaxation periodic solution of a logistic equation with sufficiently large parameter values is proved by using the large parameter method, and asymptotic estimates of the main characteristics of this solution are also constructed.
Abstract: A logistic equation with state- and parameter-dependent delay is considered. The existence of a nonlocal relaxation periodic solution of this equation is proved for sufficiently large parameter values. The proof is carried out by using the large parameter method. For large parameter values, asymptotic estimates of the main characteristics of this solution are also constructed.
TL;DR: An explicit representation of the Green function of the Dirichlet problem for the triharmonic equation in the unit ball of space of dimension greater than 2 is given in this article, where the authors also consider the problem of the tri-harmonic problem in the case of a fixed number of vertices.
Abstract: An explicit representation of the Green function of the Dirichlet problem for the triharmonic equation in the unit ball of space of dimension greater than 2 is given.
TL;DR: In this article, it was shown that the gradient projection method for minimizing a weakly convex function on a set converges with linear rate on a proximally smooth (nonconvex) set of special form (for example, on a smooth manifold).
Abstract: Let a weakly convex function (in the general case, nonconvex and nonsmooth) satisfy the quadratic growth condition. It is proved that the gradient projection method for minimizing such a function on a set converges with linear rate on a proximally smooth (nonconvex) set of special form (for example, on a smooth manifold), provided that the constant of weak convexity of the function is less than the constant in the quadratic growth condition and the constant of proximal smoothness for the set is sufficiently large. The connection between the quadratic growth condition on the function and other conditions is discussed.
TL;DR: In this paper, it was shown that a sortable ideal I is Freiman if and only if its sorted graph is chordal, and this characterization was used to give a complete classification of Freiman principal Borel ideals and of freiman Veronese-type ideals with constant bound.
Abstract: In this paper, it is shown that a sortable ideal I is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese-type ideals with constant bound.
TL;DR: In this paper, a new method for deriving estimates of the rate of convergence of optimal methods for solving problems of smooth (strongly) convex stochastic optimization is described.
Abstract: A new method for deriving estimates of the rate of convergence of optimal methods for solving problems of smooth (strongly) convex stochastic optimization is described. The method is based on the results of stochastic optimization derived from results on the convergence of optimal methods under the conditions of inexact gradients with small noises of nonrandom nature. In contrast to earlier results, all estimates in the present paper are obtained in model generality.
TL;DR: In this article, the authors prove existence and uniqueness theorems for the solution of the inverse problem of simultaneous determination of the t-dependent coefficients of u and ux in a nondivergent parabolic equation with two independent variables from integral observation of x.
Abstract: We prove existence and uniqueness theorems for the solution of the inverse problem of simultaneous determination of the t-dependent coefficients of u and ux in a nondivergent parabolic equation with two independent variables from integral observation of x. Estimates of the maxima of the moduli of these coefficients with constants explicitly expressed in terms of the input data of the problem are given. An example of an inverse problem to which the proved theorems apply is presented.
TL;DR: In this paper, an analogue of the Bombieri-Vinogradov theorem for the set of primes satisfying the condition that the ratio of α > 0 is 1/2.
Abstract: Let $$\alpha > 0$$
be any fixed noninteger. In the paper, we prove an analogue of the Bombieri–Vinogradov theorem for the set of primes $$p$$
satisfying the condition $$\{ p^{\alpha} \} < 1/2$$
. This generalizes the previous result of Gritsenko and Zinchenko.
TL;DR: In this article, mixed boundary value problems for strongly elliptic differential-difference equations and non-local mixed problems for ellipses in a cylinder are considered, and a connection between these problems and their unique solvability is established.
Abstract: Mixed boundary-value problems for strongly elliptic differential-difference equations and nonlocal mixed problems for elliptic differential equations in a cylinder are considered. A connection between these problems and their unique solvability is established.
TL;DR: In this paper, the authors studied the asymptotic behavior of the survival probability for multi-type branching processes in a random environment, where all the particles are of one type and the class of processes under consideration corresponds to intermediately subcritical processes.
Abstract: The asymptotic behavior of the survival probability for multi-type branching processes in a random environment is studied. In the case where all particles are of one type, the class of processes under consideration corresponds to intermediately subcritical processes. Under fairly general assumptions on the form of the generating functions of the laws of reproduction of particles, it is proved that the survival probability at a remote instant n of time for a process that started at the zero instant of time from one particle of any type is of the order of λnn−1/2, where λ ∈ (0, 1) is a constant defined in terms of the Lyapunov exponent for products of the mean-value matrices of the laws of reproduction of particles.
TL;DR: In this paper, the authors established the first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations.
Abstract: This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations.
TL;DR: For noncyclic one-relator groups, the residual nilpotence condition holds if and only if the residual p-finiteness condition holds for some prime number p.
Abstract: All groups in the family of Baumslag-Solitar groups (i.e., groups of the form G(m,n) = 〈a, b; a−1bma = bn〉, where m and n are nonzero integers) for which the residual nilpotence condition holds if and only if the residual p-finiteness condition holds for some prime number p are described. It has turned out, in particular, that the group G(pr, −pr), where p is an odd prime and r ≥ 1, is residually nilpotent, but it is residually q-finite for no prime q. Thus, an answer to the existence problem for noncyclic one-relator groups possessing such a property (formulated by McCarron in his 1996 paper) is obtained. A simple proof of the statement that an arbitrary residually nilpotent noncyclic one-relator group which has elements of finite order is residual p-finite for some prime p, which was announced in the same paper of McCarron, is also given.
TL;DR: In this paper, the authors studied the family of rearrangement invariant spaces E containing subspaces on which the E-norm is equivalent to the L1-norm and proved that any space with nonseparable second Kothe dual belongs to this family.
Abstract: We study the family of rearrangement invariant spaces E containing subspaces on which the E-norm is equivalent to the L1-norm and a certain geometric characteristic related to the Kadec–Pelczinski alternative is extremal. We prove that, after passing to an equivalent norm, any space with nonseparable second Kothe dual belongs to this family. In the course of the proof, we show that every nonseparable rearrangement invariant space E can be equipped with an equivalent norm with respect to which E contains a nonzero function orthogonal to the separable part of E.
TL;DR: In this paper, the Ramanujan-type congruences modulo 2 for q-series and theta-function identities were shown to be congruent tommodulo 2moverlined.
Abstract: For any given positive integersmand n, let pm(n) denote the number of overpartitions of n with no parts divisible by 4mand only the parts congruent tommodulo 2moverlined. In this paper, we prove Ramanujan-type congruences modulo 2 for pm(n) by applying q-series and Ramanujan’s theta-function identities.