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Abstract: Abstract A Kaczmarz method is presented for solving a class of fuzzy linear systems of equations with crisp coefficient matrix and fuzzy right-hand side. The iterative scheme is established and the convergence theorem is provided. Numerical examples show that the method is effective. PubDate: 2021-12-01 DOI: 10.3103/S1066369X21120033

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Abstract: Abstract In the paper, we consider a boundary value problem for a second order functional-differential equation with sufficiently general linear homogeneous boundary conditions. On the basis of the theory of semi-ordered spaces and with the help of special topological methods, we prove the existence of a unique positive solution to the problem. PubDate: 2021-12-01 DOI: 10.3103/S1066369X2112001X

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Abstract: Abstract We consider the properties of systems of functions \(\Phi_1\) orthogonal with respect to a discrete-continuous Sobolev inner product of the form \(\langle f,g \rangle_S = f(a)g(a)+f(b)g(b)+\int_a^b f'(t)g'(t)dt\) . In particular, we study the completeness of systems \(\Phi_1\) in the Sobolev space \(W^1_{L^2}\) . Additionally, we analyze the properties of Fourier series with respect to systems \(\Phi_1\) , and prove that these series converge uniformly to functions from \(W^1_{L^2}\) . PubDate: 2021-12-01 DOI: 10.3103/S1066369X21120057

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Abstract: Abstract For a differential-difference equation with a positive fundamental solution we obtain exponential stability conditions with exact estimates of the exponent and the coefficient of the exponential decay. These estimates are expressed in terms of the largest of two possible real roots of the characteristic function. We prove that one can obtain exact estimates for any solution by estimating the fundamental solution, taking into account the norm of the initial function. We establish two-sided estimates for the fundamental solution in the case, when equation parameters are given as intervals. PubDate: 2021-12-01 DOI: 10.3103/S1066369X21120069

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Abstract: Abstract We study five-dimensional pseudo-Riemannian h-spaces \(H_{221}\) of type \(\{221\}\) . Necessary and sufficient conditions are determined under which \(H_{221}\) is a space of constant (zero) curvature. Nonhomothetical projective motions in \(H_{221}\) of nonconstant curvature are found, homotheties and isometries of the indicated spaces are investigated. Dimensions, basis elements, and structure equations of maximal projective Lie algebras acting in \(H_{221}\) of nonconstant curvature are determined. As a result, the classification of h-spaces \(H_{221}\) of type \(\{221\}\) by (non-homothetical) Lie algebras of infinitesimal projective and affine transformations is obtained. PubDate: 2021-12-01 DOI: 10.3103/S1066369X21120021

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Abstract: Abstract For the Gellerstedt equation with a singular coefficient in some mixed domain, when the ellipticity boundary coincides with the segment of the Oy axis and the normal curve of the equation, a problem with the Bitsadze–Samarskii conditions on the elliptic boundary and on the degeneration line is studied. The well-posedness of the formulated problem is proved. PubDate: 2021-12-01 DOI: 10.3103/S1066369X21120070

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Abstract: Abstract The structures of partially ordered sets of degrees of negative and positive representability of linear orders are studied. The focus is on the negative representability of linear orders and orders with endomorphisms. In particular, for these structures, we established the existence of incomparable, maximal, and minimal degrees, infinite chains and anti-chains, and also considered the connection with the concepts of the reducibility of enumerartions, splittable degrees and positive representations. PubDate: 2021-12-01 DOI: 10.3103/S1066369X21120045

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Abstract: Abstract In this paper, a theorem on \({\mid{C},\alpha\mid}_k\) summability of an infinite series is generalized for the \(\varphi-{\mid{C},\alpha; \delta\mid}_k\) summability method. Also, a known result dealing with \({\mid{C},1\mid}_k\) summability is given. PubDate: 2021-11-01 DOI: 10.3103/S1066369X21110049

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Abstract: Abstract In this paper, a general theorem on absolute Cesàro summability of an infinite series is proved by using an almost increasing sequence instead of a positive non-decreasing sequence PubDate: 2021-11-01 DOI: 10.3103/S1066369X21110025

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Abstract: Abstract We establish some generalizations of the classical inequalities by Polya–Szego and Makai about torsional rigidity of convex domains. The main idea of the proof is in using an exact isoperimetric inequality for Euclidean moments of domains. This inequality has a wide class of extremal regions and is of independent interest. PubDate: 2021-11-01 DOI: 10.3103/S1066369X21110086

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Abstract: Abstract We examine a third kind integral equation in the class of generalized functions. We show that the considered equation has similar solvability properties as the Fredholm equation of the second kind. PubDate: 2021-11-01 DOI: 10.3103/S1066369X21110050

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Abstract: Abstract A linear boundary-value problem for a system of integro-differential equations with weakly singular kernels is considered. Questions of the unique solvability and the construction of algorithms for finding solution to the considered problem are studied. Conditions for the solvability of the boundary-value problem for a system of integro-differential equations with weakly singular kernels are established using the Dzhumabaev parametrization method based on splitting the interval and introducing additional parameters. Necessary and sufficient conditions for the solvability of the two-point problem for integro-differential equations with weakly singular kernels are obtained. PubDate: 2021-11-01 DOI: 10.3103/S1066369X21110013

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Abstract: Abstract Formulas are obtained for calculating the curvatures of an implicitly defined curve in n-dimensional Euclidean space. For these curves, Beltrami's theorem is generalized, which was proved by Beltrami in the case of three-dimensional space. PubDate: 2021-11-01 DOI: 10.3103/S1066369X21110062

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Abstract: Abstract In this paper, Lotka–Volterra recurrent neural networks with time-varying delays on time scales are considered. Using Banach's fixed-point principle, the theory of calculus on time scales and suitable Lyapunov functional, some sufficient conditions for the existence, uniqueness and Stepanov-exponential stability of positive weighted Stepanov-like pseudo almost periodic solution on time scales to the recurrent neural networks are established. Finally, an illustrative example and simulations are presented to demonstrate the effectiveness of the theoretical findings of the paper. The results of this paper are new and generalize some previously-reported results in the literature. PubDate: 2021-11-01 DOI: 10.3103/S1066369X21110074

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Abstract: Abstract In this paper, we study linear differential games described by a system of linear differential-difference equations of neutral type under geometric constraints on the controls of the players. Modified third method in the pursuit problem and modified method of directional pursuit are established for differential-difference equations of neutral type. We obtain new sufficient conditions on parameters of the process guaranteeing finishing the game in a definite finite time. PubDate: 2021-11-01 DOI: 10.3103/S1066369X21110037

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Abstract: Abstract An algorithm for the approximate calculation of the coefficients of the Dulac series (an asymptotic series of the monodromy transformation) in the space of vector fields with a Newton diagram containing more than one edge and a monodromic singular point is proposed. The conditions for the applicability of this algorithm are obtained. The algorithm is implemented in the MAPLE package. Examples are given for the case of a Newton diagram consisting of two edges. PubDate: 2021-10-01 DOI: 10.3103/S1066369X21100030

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Abstract: Abstract We study a nonlinear Bianchi equation which contains power nonlinearities in unknown function and its first derivatives. The method of separation of variables is used for investigation. An exponential solution is found in the case of an autonomous equation with linearly homogeneous right-hand side. It is shown that the equation has a solution in the form of a polylinear function in the case when the right-hand side of the equation contains the product of power functions of independent variables. Also, we have found solutions in the form of linear combination of exponents and in the form of generalized polynomials. The conditions on the parameters of the equation are given under which the above-mentioned solutions exist. Theorems determining the possibility of decreasing the dimension of the equation are proved. In particular, the initial equation is reduced to an ordinary differential equation for the solutions in the form of one- dimensional traveling waves, and it is reduced to a partial differential equation of lesser dimension for the solutions in the form of multi-dimensional traveling waves. In the latter case, a solution is found in the form of a generalized polynomial in linear combinations of independent variables. PubDate: 2021-10-01 DOI: 10.3103/S1066369X21100042

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Abstract: Abstract In this paper, we study problems with initial conditions for equation of oscillations of a rectangular plate subject to various boundary conditions. We establish an energy inequality, which implies the uniqueness of solution to three initial-boundary value problems. In the case, when the plate is hinged at its edges, we prove existence and stability theorems for the problem solution in classes of regular and generalized solutions. PubDate: 2021-10-01 DOI: 10.3103/S1066369X21100054

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Abstract: Abstract We prove that a set is bi-immune if and only if its images under computable permutations are not generically computable or effectively densely computable sets. An example of a coarsely computable bi-immune set is constructed. It is also proved that for any set there is a permutation from the same Turing degree such that its image under this permutation is an effectively densely computable set. Upper densities of weakly 1-generic sets are studied. PubDate: 2021-10-01 DOI: 10.3103/S1066369X21100017

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Abstract: Abstract This paper is devoted to exact (in the sense of the order of smallness) estimates of the best rational approximations of functions with derivative of generalized finite variation on a finite segment of a straight line in uniform and integral metrics. The obtained results were announced in the authors' paper in 2014. They are analogous to the results of the first author, where A. Khatamov establishes exact (in the sense of the order of smallness) estimates of the best spline approximations of functions with derivative of generalized finite variation on a finite segment of a straight line in uniform and integral metrics. Results announced by the authors in 2014 generalize those obtained by N.Sh. Zagirov in 1982, namely, exact (in the sense of the order of smallness) estimates of rational approximations of functions with generalized finite variation in the integral metric, to the best rational approximations of functions with derivative of generalized finite variation on a finite segment in uniform and integral metrics. Generally speaking, the calculation of exact (in the sense of the order of smallness) estimates for the best approximations for any class of functions in any metric is a difficult problem. PubDate: 2021-10-01 DOI: 10.3103/S1066369X21100066