# Mathematics

## 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.

## 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.

## 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

## 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. It has been proved that

## 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).

## 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