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Abstract: We describe the divisors of complex valued homogeneous harmonic polynomials which are products of linear forms on \(\mathbb R^{3}\) , and characterize homogeneous polynomials \(p\) that admit a couple of linear forms \(\ell _{1}\) and \(\ell _{2} \) such that \(\ell _{1}^{m}p \) and \(\ell _{2}^{m}p \) are harmonic for some \(m\in \mathbb N \) . The latter gives an example of a pair of spherical harmonics whose set of common zeros has a length compatible with the sharp upper bound of this quantity for a single harmonic. PubDate: 2022-05-01

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Abstract: We apply the inverse scattering method to integrating the loaded Korteweg–de Vries equation with a self-consistent source of integral type in the class of rapidly decreasing complex-valued functions. PubDate: 2022-05-01

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Abstract: An exponential upper bound is obtained for the tail probability of the centered and normalized number of cycles in an Erdös-Rényi random graph where every edge occurs with the same probability, independently of the others. PubDate: 2022-05-01

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Abstract: We obtain the necessary and sufficient condition for existence of a universal \(\Sigma \) -function in the hereditarily finite superstructure over a structure. We apply this condition to various well-known classes of structures. PubDate: 2022-05-01

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Abstract: We consider the global Morrey-type spaces \({GM}_{p(\cdot ),\theta (\cdot ),w(\cdot )}(\Omega )\) with variable exponents \(p(x) \) , \(\theta (x)\) , and \(w(x,r) \) defining these spaces. In the case of unbounded sets \(\Omega \subset {\mathbb {R}}^{n}\) , we prove the boundedness of the Hardy–Littlewood maximal operator and potential-type operator in these spaces. We prove Spanne-type results on the boundedness of the Riesz potential \({I}^{\alpha } \) in global Morrey-type spaces with variable exponents \( {GM}_{p(\cdot ),\theta (\cdot ),w(\cdot )}(\Omega ) \) . PubDate: 2022-05-01

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Abstract: We establish a connection between physical structures and Lie groups and prove that each physical structure of rank \((n+1,2)\) , \(n\in \mathbb {N} \) , on a smooth manifold is isotopic to an almost \(n \) -transitive Lie group of transformations. We also prove that each almost \(n\) -transitive Lie group of transformations is isotopic to a physical structure of rank \((n+1,2) \) . PubDate: 2022-05-01

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Abstract: We consider an operator represented by the sum of a series in the values of the resolvent of a densely defined closed operator in a complex Banach space. We describe the left inverse for this operator, apply this result to regularization of equations of the first kind, and consider several examples. PubDate: 2022-05-01

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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 DOI: 10.1134/S1055134422010035

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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 DOI: 10.1134/S1055134422010023

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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 DOI: 10.1134/S1055134422010047

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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 DOI: 10.1134/S1055134422010059

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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 DOI: 10.1134/S1055134422010011

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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