# Mathematics

## New Method for Investigating the Hilbert Boundary Value Problem with an Infinite Logarithmic Order Index

We consider the problem of identification of the analytical in the complex upper half plane by boundary condition on the entire real axis, according to which, the real part of the product, by the given on the real axis complex function and the boundary values of the desired analytical function equal zero everywhere on the real axis.

## On Customary Spaces of Leibniz –Poisson Algebras

Let K be a base field of characteristic zero. It is well known that in this case all information about varieties of linear algebras V contains in its polylinear components P_{n}(V), n ∈ N, where P_{n}(V) is a linear span of polylinear words of n different letters in a free algebra K(X,V). D. Farkas defined customary polynomials and proved that every Poisson PI-algebra satisfies some customary identity. Poisson algebras are special case of Leibniz –Poisson algebras.

## On Semigroups of Relations with the Operation of Left and Right Rectangular Products

A set of binary relations closed with respect to some collection of operations on relations forms an algebra called an algebra of relations. The class of all algebras (partially ordered algebras) isomorphic to algebras (partially ordered by set-theoretic inclusion ⊆ algebras) of relations with operations from is denoted by R{Ω} (R{Ω, ⊆}). An operation on relations is called primitive-positive if it can be defined by a formula of the first-order predicate calculus containing only existential quantifiers and conjunctions in its prenex normal form.

## The problem of optimal control for singularly perturbed system with delay with integral quadratic con-straints

The control problem for the singularly perturbed system with delay with indeterminate initial conditions and integral quadratic constraints on the control resources according to the minimax criterion is considered. Procedure is proposed for construction initial approximation of control response for minimax problem of control.

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

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.

## Symmetrization in Clean and Nil-Clean Rings

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).

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

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.

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

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

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 Inverse Nodal and Spectral Problems for Boundary Value Problems with Discontinuity Conditions Inside the Interval

The solution of inverse nodal and inverse spectral problems is presented for second-order differential operators on a finite interval with discontinuity conditions inside the interval. Connections between these two classes of inverse problems are established.