# Mathematics

## Polynomials Orthogonal with Respect to Sobolev Type Inner Product Generated by Charlier Polynomials

The problem of constructing of the Sobol ev orthogonal

## The Solution of the Inverse Boundary Value Problem for a Wing Proﬁle, Located Close to Rectilinear Screen, in a New Setting

The paper shows the inve r se boundar y value problem for the airfoil located near the solid rectilinear boundary and streamlined by a potential ﬂow of the incom pressible inviscid ﬂuid with speed parallel to the boundary,

## Inverse Problem for Sturm – Liouville Operators in the Com plex Plane

The inverse problem for the standard Sturm – Liouville equation with a spectral parameter ρ and a potential function, piecewise-entire on a rectiﬁable curve γ ⊂ C, on which only the starting point is given, is studied for the ﬁrst time. A function Q that is bounded on a curve γ is piecewise-entire on it if γ can be splitted by a ﬁnite number of points into parts on which Q coincides with entire functions, different in neighboring parts. The split points, the initial and ﬁnal points of the curve are called critical points. The problem is to ﬁnd all the critical

## To Chang Theorem. III

Various multilinear polynomials of Capelli type belonging to a free associative algebra F {X ∪ Y } over an arbitrary ﬁeld F generated by a countable set X ∪ Y are considered. The formulas expressing coefﬁcients of polynomial Chang R(¯x, ¯y|¯w) are found. It is proved that if the characteristic of ﬁeld F is not equal two
then polynomial R(¯x, ¯y| ¯w) may be represented by different ways in the form of sum of two consequencesof standard polynomial S

## On Application of Elliptic Curves in Some Electronic Voting Protocols

Electronic voting protocols allow us to carry out voting procedure in which ballots exist only electronically. These protocols provide the secret nature of vote. The main property of electronic voting protocols is the universal checkability, i.e. provision of an opportunity to any person interested, including detached onlookers to check correctness of counting of votes at any moment. In operation cryptography protocols of electronic vote of Shauma - Pederson and Kramera - Franklin - Shoyenmeykersa - Yunga are considered.

## On the Representation of Functions by Absolutely Convergent Series by H -system

The paper deals with the representation of absolutely convergent series of functions in spaces of homogeneous type. The definition of a system of Haar type (H-system) associated to a dyadic family on a space of homogeneous type X is given in the Introduction. It is proved that for almost everywhere (a.e.) finite and measurable on a set  X  function f there exists an absolutely convergent series by the system H, which converges to  f  a.e. on  X .

## Asymptotic Formulae for Weight Numbers of the Sturm – Liouville Boundary Problem on a Star-shaped Graph [ Асимптотические формулы для весовых чисел краевой задачи Штурма – Лиувилля на графе-звезде]

В статье исследована краевая задача Штурма–Лиувилля на графе Γ определенного вида. Граф Γ имеет m ребер, смежных с одной внутренней вершиной, а остальные m вершин являются верши- нами степени 1. Краевая задача на данном графе задается дифференциальными выражениями Штурма–Лиувилля с вещественными потенциалами, краевыми условиями Дирихле и стандартными условиями склейки. Определенная таким образом краевая задача имеет счетное множество собственных значений. Мы рассмотрим вычеты диагональных элементов матрицы Вейля в собственных значениях, которые назовем весовыми числами.

## Stability of Periodic Billiard Trajectories in Triangle

The problem of stability of periodic billiard trajectories in triangles is considered. The notion of stability means the preservation of a period and qualitative structure of a trajectory (its combinatorial type) for sufficiently small variations of a triangle. The geometric, algebraic and fan unfoldings are introduced for stable trajectories description. The new method of fan coding, using these unfoldings, is proposed. This method permits to simplify the stability analysis.