# Mathematics

## 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.

## 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 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.

## 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.

## a-accessible Domains, a Nonsmooth Case

Petrozavodsk State University, Russia, 185910, Petrozavodsk, Lenin st., 33, amokira@rambler.ru, VstarV@list.ru This paper continues the study of a-accessible domains in Rn. They are starlike domains and satisfy cone condition which is important for applications. Conditions of ®-accessibility of domain, defined by the inequality F(x) < 0, is obtained for a continuous function F in Rn.

## Jordan–Dirichlet Theorem for Functional Differential Operator with Involution

In this paper the problem of decomposability of a function f(x) into Fourier series with respect to the system of eigenfunctions of a functional-differential operator with involution Ly = y′(1 − x) + ®y′(x) + p1(x)y(x) + p2(x)y(1−x), y(0) = °y(1) is investigated. Based on the study of the resolvent of the operator easier and using the method of contour integration of the resolvent, we obtain the sufficient conditions for the convergence of the Fourier series for a function f(x) (analogue of the Jordan–Dirichlet’s theorem).

## Cohomology of the Lie Algebra of Vector Fields on Some One-dimensional Orbifold

I. M. Gelfand and D. B. Fuchs have proved that the cohomology algebra of the Lie algebra of vector fields on the unit circle is isomorphic to the tensor product of the polynomial ring with one generator of degree two and the exterior algebra with one generator of degree three. In the present paper the cohomology of the Lie algebra of vector fields on the one-dimensional orbifold S1/Z2 are studied. S1/Z2 is the orbit space under the Z2 group action on the unit circle by reflection in the Ox axis.

## Dirac System with Undifferentiable Potential and Antiperiodic Boundary Conditions

The object of the paper is Dirac system with antiperiodic boundary conditions and complex-valued conditions potential. A new method is suggested for investigating spectral properties of this boundary problem. The method is based on the formulas of the transform operators type. It is rather elementary and simple. Using this method asymptotic behaviour of eigenvalues is specificated and it is proved that eigen and associated functions form Riesz basis with brackets in the space of quadratic summerable two-dimensional vector-functions since eigenvalues may be multiple.

## Algorithm Variable Order, Step and the Configuration Variables for Solving Stiff Problems

An inequality for stability control of a Ceschino’s scheme of second order of accuracy is constructed. A numerical formula of order one is developed that is based on the stages of the this method and its stability interval is extended to 32. On a base of L-stable (2,1)-scheme and a numerical Ceschino’s formula, an algorithm of alternating structure, in which an efficient numerical formula is chosen on an every step by a stability criterion, is constructed. The algorithm is intended for solving stiff and non-stiff problems.

## Hyperbolic Parallelograms of the Plane b H

Hyperbolic parallelograms on a Hyperbolic Plane b H of the positive curvature in the Cayley – Klein model are investigated. We conducted their classification, obtained the metric correlations between the measure of angles and the expressions of lengths of the edges through a measure of included angles.