# Mathematics

## Polynomials, Orthogonal on Non-Uniform Grids

Asymptotic properties of polynomials pˆn(t), orthogonal with weight ∆tj on any finite set of 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 Lasiandra polynomials. Furthermore are investigated the approximating properties of the sums by Fourier on these polynomials..

## Generalization of Method A. A. Dorodnicyn Close Calculation of Eigenvalues and Eigenvectors of Symmetric Matrices on Case of Self-Conjugate Discrete Operators

Let the discrete self-conjugate operator A operates in separable Hilbert space H and has the kernel resolvent with simple spectrum. Self-conjugate and limited operator B operates also in H. Then it is possible to find such number ε > 0, that eigenvalues and eigenfunctions of the perturbation operatorA+εB will be calculated on a method of Dorodnicyn.

## tolerant space, space of tolerant loops, homotopy groups of tolerant space.

In the article is proved the theorem about isomorphism between homotopy groups of initial tolerant space and homotopy groups with descremented by one dimension of space of tolerant loops.

## The Classification of Complexes of Lines in Zeroth Order Frame in F ̄2 3 Space

Complexes of lines in hyperbolic type of biflag space introduced by the author are studied by the method of external Cartan forms. We prove that 5 non-special variants of complexes exist in mentioned space in zero order neighborhood. For every complex a first-order moving flag was drawn.

## Approximating Properties of Solutions of the Differential Equation with Integral Boundary Condition

With the use of the solution of the first-order differential equation the approximations to the continuous functions with integral boundary conditions are constructed.

## Hilbert Generalizations b-Bessel Systems

The notion of b-Bessel systems that generalizes the known classic notion of Bessel systems is introduced, the criteria of Bessel property of the systems are established. Some properties of the space of coefficients corresponding to the b-basis generalizing the classic notion of Schauder basis are studied.

## On One Special Mapping

The paper deals with the study of a specialmapping in connection with the structure of extremal functions in exact Kolmogorov inequalities on the half-line in the uniform metrics.

## On Weighted Analogs of Wiener’s and Levy’s Theorems for Fourier – Vilenkin Series

In this paper we find the general form of complex homomorphism for some subalgebras of absolutely convergent Fourier – Vilenkin series algebra. As a corollary, we obtain weighted analogs of Wiener’s and Levy’s theorems for Fourier – Vilenkin series.

## Variable Order and Step Algorithm Based on a Stages of Runge – Kutta Method of Third Order of Accuracy

An inequality for the stability control of 3-stage Runge – Kutta method of 3th order of accuracy is obtained. Method of first order with expanded stability domain is constructed. Algorithm of variable order is formulated. The results of stiff system computations are provided, which confirm an increase in efficiency for the variable order method as compared to a calculation with fixed scheme.

## Analysis of Heterogeneous Queueing Networks with Batch Movements of Customers

Closed exponential queueing network with different classes of customers and batch movements is considered. To model evolution of given network Markov chains are used. Two approaches to stationary distribution calculation for given type queueing networks are presented. Formulas for basic stationary characteristics are given.