# Mathematics

## On Almost Nilpotent Varieties with Integer PI-Exponent

We study almost nilpotent varieties of algebras over a field of zero characteristic. Earlier in the class of algebras with identical relation x(yz) ≡ 0 and in the class of all commutative metabelian algebras countable sets of varieties with integer PI-exponent were defined. Only the existence of almost nilpotent subvariety in each defined variety was proved. In the paper by means of combinatorial methods and methods of the representation theory of symmetric groups we prove that earlier defined varieties are almost nilpotent.

## Equiconvergence Theorem for Integral Operator with Involution

In the paper, the integral operator with kernel having discontinuities of the first kind at the lines t = x and t = 1 − x is studied. The equiconvergence of Fourier expansions for arbitrary integrable function f(x) in eigenfunctions and associated functions of the considered operator and expansions of linear combination of functions f(x) and f(1 − x) in trigonometric system is proved. The equiconvergence is studied using the method based on integration of the resolvent using spectral value. Methods, developed by A. P.

## Generalized Absolute Convergence of Series with Respect to Multiplicative Systems of Functions of Generalized Bounded Variation

A. Zygmund proved that a 2π-periodic function with bounded variation and from any Lipschitz class Lip(α) has absolutely convergent Fourier series. This result was extended to many classes of functions of generalized bounded variation (for example, functions of bounded Jordan-Wiener p-variation, functions of bounded Λ-variation introduced by D. Waterman et al) and to different spaces defined with the help of moduli of continuity.

## On the Geometric Structure of the Continuos Mappings Preserving the Orientation of Simplexes

It is easy to show that if a continuous open map preserves the orientation of allsimplexes, the nit is affine. The class of continuous open maps f : D ⊂ R m → R n that preserve the orientation of simplexes from a given subset of a set of simplexes with vertices in the domain D ⊂ R m is considered. In this paper, questions of the geometric structure of linear inverse images of such mappings are studied.

## Linear Difference Equation of Second Order in a Banach Space and Operators Splitting

In differential and difference equations classical textbooks, the n-th order differential and difference equations reducing by standard substitution to first-order differential and difference equations system is described. Each of the cohering equations can be written in the operator form. Naturally there is a question of coincidence of a number of properties of differential and difference equations (operators) of the second order and the corresponding functional equations (operators) of first order.

## On Recovering Integro-Differential Operators from the Weyl Function

We study inverse problems of spectral analysis for second order integro-differential operators, which are a perturbation of the Sturm–Liouville operator by the integral Volterra operator. We pay the main attention to the nonlinear inverse problem of recovering the potential from the given Weyl function provided that the kernel of the integral operator is known a priori. We obtain properties of the spectral characteristics and the Weyl function, provide an algorithm for constructing the solution of the inverse problem and establish the uniqueness of the solution.

## On an Inner Estimate of a Convex Body by the Lebesgue Set of Convex Differentiable Function

A finite-dimentional problem of embedding the largest by the inclusion of lower Lebesgue set of given convex function f(x) in a given convex body D ⊂ R p is considered. This problem is the generalization of the problem of inscribed ball (function f(x) is some norm, and the Lebesgue sets are the corresponding balls). The function f(x) must be differentiable on R p possibly expending the point 0 p and 0 p is the uniqueness point of minimum. Mathematical formalization of this problem is proposed in the form of finding maximin of a function of the difference of arguments.

## Embeddings of Generalized Bounded Variation Function Spaces into Spaces of Functions with Given Majorant of Average Modulus of Continuity

In the present paper we study embeddings of some spaces of functions of generalized bounded variation into classes of functions with given majorant of average modulus of continuity introduced by B. Sendov and V. Popov. We consider the spaces ΛBV (p) of functions of bounded (Λ − p)-variation suggested by D. Waterman (for p = 1) and M. Shiba (for p > 1) and spaces V (v(n)) of functions with given majorant of its modulus of variation. The last quantity was introduced by Z. A. Chanturia. The necessary and sufficient conditions of such embeddings are proved.

## Well-posedness of the Dirichlet Problem for One Class of Degenerate Multi-dimensional Hyperbolic-parabolic Equations

It has been shown by Hadamard that one of the fundamental problems of mathematical physics, the analysis of the behavior of oscillating string is an ill-posed problem when the boundary-value conditions are imposed on the entire boudary of the domain. As noted by A. V. Bitsadze and A. M. Nakhushev, the Dirichlet problem is ill-posed not only for the wave equation but for hyperbolic PDEs in general.

## On Multiple Completeness of the Root Functions of the Pencils of Differential Operators with Constant Coefficients

A class of the pencils of ordinary differential operators of n-th order with constant coefficients is considered. The roots of the characteristic equation of the pencils from this class are supposed to lie on a straight line containing the origin, provided that one of the roots lies on one part from the origin, the rest lie on another part. The cases when the system of root functions is m-fold (3 ≤ m ≤ n − 1) complete in the space of square summable functions on main interval are described.