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Abstract: For \( 1<p,q<\infty \) , we find necessary and sufficient conditions for the validity of a discrete Hardy-type inequality $$ \left (\sum \limits _{n=1}^{\infty } (Af)_n ^q\right )^{\frac {1}{q}} \le C\left (\sum \limits _{k=1}^{\infty } f_k ^p\right )^{\frac {1}{p}}$$ for a class of matrix operators of the form \((Af)_n=\sum \limits _{k=1}^{n}a_{n,k}f_k \) , where \(n\ge 1 \) . PubDate: 2022-02-01

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Abstract: We consider mixed boundary value problems for one pseudohyperbolic equation in a quarter plane. We assume that the boundary value problems satisfy the Lopatinskiĭ condition. We prove theorems on unique solvability in anisotropic Sobolev spaces with exponential weight and establish some estimates for the solutions. PubDate: 2022-02-01

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Abstract: We study a homogeneous compound renewal process (c.r.p.) \(Z(t) \) . It is assumed that the elements of the sequence that rules the process satisfy Cramér’s moment condition \([{\bf C}_0] \) . We consider the family of processes $$ z_T(t):=\frac 1xZ(tT),\enspace \enspace 0\le t\le 1,$$ where \(x=x_T\sim T \) as \(T\to \infty \) . Conditions are proposed under which the extended large deviation principle holds for the trajectories \( z_T\) in the space \((\mathbb {V},\rho B) \) of functions with bounded variation, endowed with Borovkov’s metric. If the trajectories of the process \(Z(t) \) are monotone with probability 1 then, under the same condition, we prove the classical trajectory large deviation principle. PubDate: 2022-02-01

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Abstract: We deduce a number of new estimates for the proximity of a binomial distribution to the corresponding Poisson distribution in the uniform metric and propose a combined approach to estimate this uniform distance when, for small \(n\) and large \(p \) , the estimation is performed by computer calculating and the estimates obtained in the paper are used for the remaining values of \(n \) and \(p \) . PubDate: 2022-02-01

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Abstract: We study linear stability of steady states for a certain generalization (namely, nonisothermal flows under the influence of magnetic field) of the Pokrovskiĭ–Vinogradov basic rheological model which describes flows of solutions and melts of incompressible viscoelastic polymeric media. We prove that the linear problem describing magnetohydrodynamic (MHD) flow of polymers in an infinite plane channel has the following property: For a certain behavior of magnetic field outside of the channel, there exists a solution of the problem whose amplitude grows exponentially (in the class of functions that are periodic with respect to the variable changing along the side of the channel). PubDate: 2022-02-01

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Abstract: Let \(G \) be a finite group of Lie type \(E_7 \) or \(E_8\) over a field \(\mathbb {F}_q \) and let \(W \) be the Weyl group of \(G \) . In the present article, we find all maximal tori \(T \) of the group \(G \) that admit complements in the algebraic normalizer \(N(G,T) \) . For every group under consideration except for the simply connected group \(E_7(q)\) , we prove the following assertion: If \(w\in W\) and the corresponding torus \(T\) lacks the complement then there exists a lift of \(w\) in \(N(G,T) \) of order \( w \) . In the exceptional case, we find all elements \(w \) admitting a lift in \(N(G,T) \) of order \( w \) . PubDate: 2021-10-01 DOI: 10.1134/S1055134421040027

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Abstract: We study the degrees of decidable categoricity for almost prime models and their relationship with the degrees of the sets of complete formulas. We show that a result of Goncharov, Harizanov, and Miller for models of infinite signature is valid for models of the signature of graphs. PubDate: 2021-10-01 DOI: 10.1134/S1055134421040039

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Abstract: We prove the Wigner law for conjugacy matrices of generalized non-oriented weighted random graphs under some simple conditions on probabilities of the graph edges. PubDate: 2021-10-01 DOI: 10.1134/S1055134421040040

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Abstract: The article is devoted to the study of boundary value problems for a fractional-order convection-diffusion equation with memory effect. We construct two-layer monotone schemes with weights of the second order accuracy with respect to the time and space variables. We prove the uniqueness and stability for the solution with respect to the initial data and right-hand side and also the convergence of the solution of the difference scheme to the solution of the corresponding differential problem. PubDate: 2021-10-01 DOI: 10.1134/S1055134421040015

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Abstract: The paper is devoted to the study of the metric properties of regular and semiregular polyhedra in Euclidean spaces. In the first part, we prove that every regular polytope of dimension greater or equal than 4, and different from 120-cell in \(\mathbb {E}^4 \) is such that the set of its vertices is a Clifford–Wolf homogeneous finite metric space. The second part of the paper is devoted to the study of special properties of Archimedean solids. In particular, for each Archimedean solid, its description is given as the convex hull of the orbit of a suitable point of a regular tetrahedron, cube or dodecahedron under the action of the corresponding isometry group. PubDate: 2021-07-01 DOI: 10.1134/S1055134421030019

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Abstract: We investigate the Diophantine equation of the form \(x^2-dy^2=4t \) , with $$ d=k^2\big (-m+(k^2m-2)u\big )^2-4\big (m+(k^2m-1)u\big ), $$ where \(k\) and \(m \) are odd and \(u \) is even, and \(4t \) is a sufficiently small natural number. We obtain a complete description of the set of solutions to such an equation. PubDate: 2021-07-01 DOI: 10.1134/S1055134421030020

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Abstract: We study two types of multidimensional compound renewal processes (c.r.p.). We assume that the elements of the sequences that control the processes satisfy Cramér’s moment condition. Wide conditions are proposed under which the large deviation principle holds for finite-dimensional distributions of the processes. PubDate: 2021-07-01 DOI: 10.1134/S1055134421030032

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Abstract: We study the quotient spaces of the unit ball by some group of Möbius transformations. For mappings of such spaces, we obtain a bound for the distortion of a modulus of a family of spheres. As an application, we prove theorems on the local and boundary behavior of Orlicz–Sobolev classes on the quotient spaces. PubDate: 2021-07-01 DOI: 10.1134/S1055134421030044

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Abstract: We consider an important class of differential-difference equations of neutral type and study asymptotic properties of their solutions. We find necessary and sufficient conditions for exponential stability and represent them in geometric terms as a domain in the space of parameters. We analyze the behavior of the solutions on the boundary of the domain where stability is lost by various reasons. We consider asymptotic properties of the solutions together with the corresponding properties of their derivatives. PubDate: 2021-04-01 DOI: 10.1134/S1055134421020012

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Abstract: This is the final part of the paper on the dimensions of Jordan blocks in the images of regular unipotent elements from subsystem subgroups of type \(C_2 \) in \(p\) -restricted irreducible representations of groups of type \(C_n\) in characteristic \(p\geq 11 \) with locally small highest weights. Here the case where \(n>3 \) and the restriction of a representation considered to a canonical subgroup of type \(A_1\) containing such element has a weight not less than \(p\) , is investigated. PubDate: 2021-04-01 DOI: 10.1134/S1055134421020024

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Abstract: We consider a mathematical model describing the production of the components of some living system under the influence of positive and negative feedback. The model is presented in the form of the Cauchy problem for a nonlinear delay integro-differential equation. A theorem of the existence, uniqueness, and nonnegativity of the solutions to the model on the half-axis is proved for nonnegative initial data. The questions of the asymptotic behavior of the solutions and the stability of the equilibria of the model are investigated. Sufficient conditions for the asymptotic stability are obtained for nontrivial equilibria and the boundaries of their attraction domains are estimated. Examples illustrating the application of the obtained theoretical results are given. PubDate: 2021-04-01 DOI: 10.1134/S1055134421020036

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Abstract: We find new conditions for existence of strong solutions of ordinary differential equations with random right-hand side, stochastic differential equations with measurable random drift, and their trajectory analogs with symmetric integrals. We show that solutions of Itô equations satisfy a parabolic equation along trajectories of a Wiener process. PubDate: 2021-04-01 DOI: 10.1134/S1055134421020048

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Abstract: We construct a minimal cubature formula of degree \(2 \) for a torus in \({\mathbb R}^3 \) . PubDate: 2021-02-01 DOI: 10.1134/S1055134421010053

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Abstract: We find an exact estimate for the algorithmic complexity of the class of autostable ordered Abelian groups. PubDate: 2021-02-01 DOI: 10.1134/S1055134421010041

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Abstract: We find conditions for existence of two cycles for a five-dimensional piecewise-linear dynamical system that models functioning of a circular gene network. Conditions for existence of a cycle were obtained by the authors earlier. The phase portrait of a system is divided into subdomains (or blocks). With the use of such a discretization, we construct a combinatorial scheme for passages of trajectories between blocks. For the second cycle, we show that such a scheme depends on the parameters of a system. PubDate: 2021-02-01 DOI: 10.1134/S1055134421010016