# Mathematics

## Cohomology of Lie algebra of vector fields on S1/Z2

In the present paper we calculate the diagonal cohomology of Lie algebra of vector fields on S1/Z2 with coefficients in the space of smooth functions and 1-forms, one-dimensional and two-dimensional cohomology with coefficients in R.

## Acts and partial acts over semilattices

We consider the acts and the partial acts over semilattices. We obtain a necessary and sufficient condition to be a partially ordered set X an act over a semilattice. The properties of partial acts are investigated and a sufficient condition is found for the expansion of partial act X over a semilattice S to a full S-act.

## Determination of the Boundary in the Local Charzynski–Tammi Conjecture for the Fifth Coefficient

## To a Solution of the Inhomogeneous Riemann–Hilbert Boundary Value Problem for Analytic Function in Multiconnected Circular Domain in a Special Case

The author offers a new approach to solution of the Riemann–Hilbert boundary value problem for analytic function in multiconnected

circular domain. This approach is based on construction of solution of corresponding homogeneous problem, when analytic in domain

function is being defined by known boundary values of its argument. The author considers a special case of a problem when the

index of a problem is more than zero and on unit of less order of connectivity of domain. Resolvability of a problem depends on

## A U-set for system of character of the zero-dimensional group under convergent over cubes

In this work we consider system of characters of the compact zero-dimensional group G and study uniqueness sets forN-fold multiple series for system of character a zero-dimensional group under convergent over cubes (in other words, U-sets). We proof that every finite set is a U-set and show that countable set with only one limit point is a U-set.

## The problem of Leont'ev on entire functions of completely regular growth

We consider an entire function of exponential type with all its zeros are simple and form a sequence with the index condensation zero. On the set of zeros a function of its derivative is growing quickly. Required to determine whether original function have complete regularity of growth. This problem, which arose in the theory of representation of analytic functions by exponential series was posed by A. F. Leontiev more than forty years ago and has not yet been solved.

## Structure of the inverse for the integral operator of special kind

Algebra (with identity) generated by integral operators on the spaces of continuous periodic functions is considered. This algebra is proved to be an inverse-closed subalgebra in the algebra of all bounded linear operators.

## On 2-fold completeness of the eigenfunctions for the strongly irregular quadratic pencil of differential operators of second order

A class of strongly irregular pencils of ordinary differential operators of second order with constant coefficients is considered. The roots of the characteristic equation of the pencils from this class are supposed to lie on a straight line coming through the origin and on the both side of the origin. Exact interval on which the system of eigenfunctions is 2-fold complete in the space of square summable functions is finded.

## Matrix representation of dilation operator on the product of zero-dimensional locally compact Abelian groups

In the real wavelet analysis dd-dimensional dilation operator may be written with the help of an integer-valued d×dmatrix. We find the matrix representation of the dilation operator on the product of zero-dimensional locally compact Abelian groups.

## On necessary conditions for a minimum of a quadratic functional with a Stieltjes integral and zero coefficient of the highest derivative on the part of the interval

In this paper we obtain a necessary condition for an extremum of a quadratic functional with a Stieltjes integral in the case where the coefficient of the highest derivative may vanish on a part of the interval. It is shown that the resulting mathematical model has the property of non-degeneracy. It is proved that a Variable boundary problem that arises as a necessary condition for an extremum is an “intermediate” position between the boundary value problems of fourth- and second-order – the solution space has dimension three.