# Mathematics

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## 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

In this article the Sturm-Liouville boundary value problem on the graph Γof a special structure is considered. The graph Γhas m edges, joined at one common vertex, and m vertices of degree 1. The boundary value problem is set by the Sturm-Liouville differential expression with real-valued potentials, the Dirichlet boundary conditions, and the standard matching conditions. This problem has a countable set of eigenvalues. We consider the so-called weight numbers, being the residues of the diagonal elements of the Weyl matrix in the eigenvalues.

## 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.

## Recurrence Relations for Polynomials Orthonormal on Sobolev, Generated by Laguerre Polynomials

In this paper we consider the system of polynomials (l_r,n)^a (r — natural number, n = 0,1,...), orthonormal with respect to the Sobolev inner product (Sobolev orthonormal polynomials) of the following type<f, g> = (sum _(v=0))^(r−1) f^(ν)(0)g ^(ν)(0) + (f _0)^∞ f^(r) (x)g^(r)(x)ρ^(x)dx  and generated by the classical orthonormal Laguerre polynomials.Recurrence relations are obtained for the system of Sobolev orthonormal polynomials, which can be used for studying various properties of these polynomials and calculate their values for any x and n.

## Approximation Properties of Dicrete Fourier Sums for Some Piecewise Linear Functions

Let N be a natural number greater than 1. We select N uniformly distributed points t_k = 2πk/N (0 < k < N − 1) on [0,2\pi]. Denote by  L_ n,N (f) = L _n,N (f,x)1 < n < ⌊N/2⌋  the trigonometric polynomial of order n

## Almost Periodic at Infinity Functions Relative to the Subspace of Functions Integrally Decrease at Infinity

In the paper we introduce and study a new class of almost periodic at infinity functions, which is defined by means of a subspace of  integrally decreasing at infinity functions. It
is wider than the class of almost periodic at infinity functions introduced in the papers of A.G.Baskakov (with respect to the subspace of functions vanishing at infinity). It suffices to turn
to the approximation theory for a new class of functions, where the Fourier coefficients are slowly varying at infinity functions with respect to the subspace of functions that decrease integrally

## Adjustment of Functions and Lagrange Interpolation Based on the Nodes Close to the Legendre Nodes

It is well known that the Lagrange interpolation of a continuous function based on the Chebyshev nodes may be divergent everywhere (for arbitrary nodes, almost everywhere) like the Fourier series of a summable function. On the other hand any measurable almost everywhere finite function can be “adjusted” in a set of arbitrarily small measure such that its Fourier series will be uniformly convergent. The question arises: does the class of continuous functions have a similar property with respect to any interpolation process?

## Connections of Nonzero Curvature on Three-dimensional Non-reductive Spaces

When a homogeneous space admits an invariant affine connection? If there exists at least one invariant
connection then the space is isotropy-faithful, but the isotropy-faithfulness is not sufficient for the space in
order to have invariantconnections. If a homogeneousspace is reductive, then the space admits an invariant