# Mathematics

## On Determination of Functional-Differential Pencils on Closed Sets from the Weyl-Type Function

Second-order functional-differential pencils on closed sets are considered with nonlinear dependence on the spectral parameter. Properties of their spectral characteristics are obtained and the inverse problem is studied, which consists in recovering coefficients of the pencil from the given Weyl-type function. The statement and the study of inverse spectral problems essentially depend on the structure of the closed set. We consider an important subclass of closed sets when the set is a unification of a finite number of closed intervals and isolated points.

## Smooth Approximations in C[0, 1]

The first orthonormal basis in the space of continuous functions was constructed by Haar in 1909. In 1910, Faber integrated the Haar system and obtained the first basis of continuous functions in the space of continuous functions. Schauder rediscovered this system in 1927. All functions of Faber – Shauder are piecewise linear, and partial sums are inscribed polygons. There was many attempts to build smooth analogues of the Faber – Schauder basis. In 1965, K. M. Shaidukov succeeded. The functions he constructed were smooth, but consisted of parabolic arcs.

## On the Approximate Solution of a Class of Weakly Singular Integral Equations

The work is devoted to the study of the solution of one class of weakly singular surface integral equations of the second kind. First, a Lyapunov surface is partitioned into “regular” elementary parts, and then a cubature formula for one class of weakly singular surface integrals is constructed at the control points. Using the constructed cubature formula, the considered integral equation is replaced by a system of algebraic equations.

## 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.

## 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.

## 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).