# Mathematics

## On Transformation Operator for the System of Dirac Equations with Summable Potentials

The paper proves the existence of the transformation operator for summable functions and finds the analogy of the formula for the potential with the transformation operator in the given case.

## Asymptotics Around the Degeneration Spot of Heat Equation Solution with Strong Degeneration

The paper deals with heat equation with strong degeneration. It is known that for such problems initial conditions are not stated at t = 0 as there exists the only smooth solution of such equation. The paper investigates a class of uniqueness of the solution and studies solvability of the problem in spaces of continuous functions. An asymptotic representation of solution around the degeneration spot is built, i.e. the main part of the solution is defined at t → +0 and residuals are estimated.

## An Inverse Problem for Quasilinear Elliptic Equations

The article examines incorrect return problems in the defining unknown factors in the quasilinear elliptic equation. Theorems of existence, uniqueness and stability have been proved. The consecutive approach method is used for the construction of the regulating algorithm for defining several factors.

## Recovering Differential Operators on a Graph with a Cycle and with Generalized Matching Conditions

The solution of the inverse spectral problem is obtained for second-order differential operators on a graph with a cycle and with generalized matching conditions in the internal vertex.

## On Convergence of Fourier – Vilenkin Series in L p [0, 1), 0 < p ≤ 1

In this paper we study convergence a.e. and L p -convergence (0 < p ≤ 1) of Fourier –Vilenkin series under some tauberian conditions on Fourier coefficients of a function. In the case of Fourier – Walsh series these results are obtained by F. Moricz.

## Approximative Properties of Mixed Series by Lagerre’s Polynomials on Classes of Smooth Functions

Approximative properties of mixed series by Lagerre’s polynomials on classes of smooth functions that given on axle [0, ∞) are viewed. Inequality that corresponds to Lebesgue inequality for trigonometric Fourier sums was found for evaluation of deflection of smooth function from it’s partial sums of mixed series by Lagerre’s polynomials. Evaluations for corresponding Lebesgue function of partial sums of mixed series by Lagerre’s polynomials were found.

## 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.

## About Approximation Multinominals, Orthogonal on Any Grids

In this work are investigated approximation properties of multinominals pˆn(x), orthogonal with weight ∆tj on the any grids consisting of final number of points of a piece [−1, 1]. Namely the approximation formula, in which is established at increase n together with N, approximation behaviour of these multinominals close to approximation behaviour of multinominals Lasiandra.

## O РАЗРЕШИМОСТИ ОБРАТНОЙ ЗАДАЧИ ШТУРМА – ЛИУВИЛЛЯ В СИММЕТРИЧНОМ СЛУЧАЕ

Necessary and sufficient conditions are provided for the solvability of the inverse problem of recovering Sturm – Liouville operator from its spectrum in the central symmetry case.

## On Laguerre Expansions Summability by the Linear Methods

This paper studies a problem of Laguerre expansions summability via methods defined by triangular matrices. The conditions on a matrix and an expandable function are obtained to guarantee the convergence of corresponding linear means at the Lebesgue point t = 0.