# Mathematics

## On Explicit and Exact Solutions of the Markushevich Boundary Problem for Circle

In the article the Markushevich boundary problem on the circle is considered for the case when the first coefficient of the problem is an arbitrary function from the Holder class and the second coefficient is the boundary value of a function that is meromorphic in the unit disk. An explicit method of solution of the given problem is proposed, the number of linearly independent solutions of the homogeneous problem and the number of solvability conditions are calculated, the general solution of the problem is found.

## Finite Closed 5-Loops of Extended Hyperbolic Plane

There are four types of finite closed 5-loops which are invariant by the fundamental group G and singled out on the extended hyperbolic plane H2. It is proved that convex 5-loops belong to two types. The interior of the first type 5-loop coincides with the plane H2 . The 5-loop of the second type allows the partition into two simple loops of three and four dimension. Its interior coincides with the interior of the component of the simple 4-loop. The topological 5-loop properties are researched.

## Solvability of Boundary Value Problems for the Schrodinger Equation with Purely Imaginary Coefficient

The paper examines regional problems for nonlinear Schrodinger equation when factor of the equation is the square-summable function that has a square-summable derivative. In this process, theorems of existence and uniqueness of the solution of the boundary value problems under consideration have been proved.

## Leray – Serra Spectral Sequence for Tolerant Quasifibering of Tolerant Ways

The article constructs Leray – Serra homological spectral sequence for tolerant quasifibering of tolerant ways and computes the two first members of this sequence.

## 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.