# Mathematics

## Optimality Solutions in Games with Preference Relations

For n person games with preference relations some types of optimality solutions are introduced. Elementary properties of their solutions are considered. One sufficient condition for nonempty Cα-core is found.

## Parametrization of Bivariate Nonseparable Haar Wavelets

A parametrization of all orthogonal wavelet bases for Haar multiresolution analysis is derived. The bases generated by three piecewise constant wavelets {ηi(x, y)}, i = 1, 2, 3, supported on [0, 1] × [0, 1], with values aij ∈ R, i = 1, 2, 3, j = 1, 2, 3, 4 are considered.

## The Characteristic of Stability of the Solution in the Problem of Convex Compact Set Asphericity

We consider the problem of stability of the solution in the problem of asphericity of a convex set with respect to the error of defining the compact set. It is shown that the optimal value of the criterion function (an asphericity indicator) is stable. Properties of the setvalued mapping, that puts to a convex compact compact set the centers of its asphericity are also investigated. It is proved that this mapping is semicontinious from above everywhere in the space of convex compact sets.

## Λ-Summability and Multiplicators of Holder Classes of Fourierseries with Respect ̈ to Character Systems

Let G be a Vilenkin group of bounded type. We obtain nessesary and sufficient conditions of uniform Λ-summability for all Fourier series of f ∈ C(G) and one of Λ-summability in L 1 (G) for all Fourier series of f ∈ L 1 (G). Also we extend some T. Quek and L. Yap results to the case of general modulus of continuity.

## 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.