Mathematics

On One Exceptional Case of the First Basic Three-Element Carleman-Type Boundary Value Problem for Bianalytic Functions in a Circle

Authors: Perelman N. R.

This article considers a non-degenerate (nonreducible to two-element) three-element problem of Carleman type for bianalytic functions in an exceptional case, that is, when one of the coefficients of the boundary condition vanishes at a finite number of contour points. The unit circle is taken as the contour. For this case, an algorithm for solving the problem is constructed, which consists in reducing the boundary conditions of this problem to a system of four Fredholm type equations of the second kind.

On the Geometry of Three-dimensional Pseudo-Riemannian Homogeneous Spaces. II

Authors: Mozhey N. P.

The problem of establishing links between the curvature and the topological structure of a manifold is one of the important problems of the geometry. In general, the purpose of the research of manifolds of various types is rather complicated. Therefore, it is natural to consider this problem in a narrower class of pseudo-Riemannian manifolds, for example, in the class of homogeneous pseudo-Riemannian manifolds. This paper is a continuation of the part I.

On the Positive Solutions of a Model System of Nonlinear Ordinary Differential Equations

Authors: Kobilzoda M. M., Naimov A. N.

This article investigates the properties of positive solutions of a model system of two nonlinear ordinary differential equations with variable coefficients. We found the new conditions on coefficients for which an arbitrary solution (x(t), y(t)) with positive initial values x(0) and y(0) is positive, nonlocally continued and bounded at t > 0. For this conditions we investigated the question of global stability of positive solutions via method of constructing the guiding function and the method of limit equations.

Symmetrization in Clean and Nil-Clean Rings

Authors: Danchev P. V.

We introduce and investigate D-clean and D-nil-clean rings as well as some other closely related symmetric versions of cleanness and nil-cleanness. A comprehensive structural characterization is given for these symmetrically clean and symmetrically nil-clean rings in terms of Jacobson radical and its quotient. It is proved that strongly clean (resp., strongly nil-clean) rings are always D-clean (resp., D-nil-clean).Our results corroborate our recent findings published in Bull. Irkutsk State Univ., Math. (2019) and Turk. J. Math. (2019).

The External Estimate of the Compact Set by Lebesgue Set of the Convex Function

Authors: Abramova V. V., Dudov S. I., Osipcev M. A.

The finite-dimensional problem of embedding a given compact D ⊂ R p into the lower Lebesgue set G(α) = {y ∈ R p : f(y) 6 α} of the convex function f(·) with the smallest value of α due to the offset of D is considered. Its mathematical formalization leads to the problem of minimizing the function φ(x) = max y∈D f(y − x) on R p . The properties of the function φ(x) are researched, necessary and sufficient conditions and conditions for the uniqueness of the problem solution are obtained.