# Mathematics

## Symmetry Axes of Planar Polynomial Differential Systems

The notion of *N*-type axis of symmetry is introduced. It is proved that the vector field defined by system of the differential equations with *n*order polynomials in a right hand, cannot have even number of axes of symmetry *N*-type at *n *=2*m,m *∈N. For *n *=2*,*3 full research of the given system on *N*-symmetry is carried out.

## Аbout Some Boundary Problems in the Semispace for a Class of Pseudo-Differential Equations with Degeneracy

Boundary problems in the halfspace for one class of the pseudodifferential equations are considered. The coercetive a priori estimations and theorems of the existence of solutions for these problems are established.

## On Multiple Completeness of the Root Functions for a Class of the Pencils of Differential Operators

A polinomial pencil of ordinary differential operators of n-th order generated by a homogeneous differential expression with constant coefficients and by two-point boundary conditions of a special structure with lcondition sinzero only (1 ≤ l ≤ n−1) isconsidered in the space L 2 [0,1]. The case is studied, when the roots of the characteristic equation lieonaray coming from theorigin. Asufficient condition of m-fold completeness of the system of root functions for m ≤ n−l inthe space L 2 [0,1] isfound. Anaccuracy of obtained result is shown.

## Integrability of a Partial Case of the Lowner Equation

We give a quadrature solution to the partial case of the Lowner¨ equation for the upper half-plane.

## Asymptotic Properties of Polynomials pˆα,βn (x), Orthogonal on Any Sets in the Сase of Integers α and β

Asymptotic properties of polynomials *p*ˆ* ^{α,β}_{n }*(

*x*)

*,*orthogonal with weight (1−

*x*)

_{j}*(1+*

^{α}*x*)

_{j}*∆*

^{β}*t*on any finite set of

_{j }*N*points from segment [−1

*,*1] are investigated. Namely an asymptotic formula is proved in which asymptotic behaviour of these polynomials as

*n*tends to infinity together with

*N*is closely related to asymptotic behaviour of the Jacobi polynomials.

## Cohomology Rings of Semicubecal Sets

The aim of this paper is to define the structure of the ring over the graded cohomology group of a semicubecal set with coefficients in a ring with unity.

## The M.A. Lavrentiev Inverse Problem on Mapping of Half-Plane Onto Polygon with Infinite Set of Vertices

The authors consider a generalization of the M.A.Lavrentiev inverse problem on a conformal mapping of half-plane onto interiority of a polygon for the case where the set of vertices of this polygon is infinite. We assume that the inner angles at unknown vertices and the image of the vertices under the conformal mapping on the real line are given. Under certain restrictions on values of the angles and on the sequence of points of the real line that are preimages of the vertices the formula for such a mapping is obtained.

## The Approached Calculation of Eigenvalues of the Discrete Operator by Means of Spectral Traces of Resolvent Degree

Letadiscreteself-adjointoperatorT actsinaseparableHilbertspace and have the kernel resolvent, and eigenvalues and eigenfunctions of the operator T be known. In the paper the method of calculation of eigenvalues of the perturbed operator T + P is considered. Resolvent of this operator is presented as convergent Neumann series on eigenfunctions of the operator T. The point of the method is that at ﬁrst is found a set of numbers which approximate traces of the resolvent degrees of the operator T + P.

## About the Congruences of Two-Generated Monoid

The congruences of two-generated monoid which generated by pair of words of length 2 are considered over two-letter alphabet. It is shown that number of equivalence classes for words of length n is equal to n + 1. The number of words in each class is found.

## The Problem of Convergence in Point Trigonometric Interpolation Process of Lagrange

An analogue of the characteristic of R. Salem is obtained for a trigonometric Lagrange interpolation process on the matrix of equally spaced nodes.