# Mathematics

## On the class of exponentially growing sequences that are not uniformly distributed modulo one

The paper presents a family of exponentially growing but not uniformly distributed sequences modulo one.

## Integrals of the Loewner equation with exponential driving function

We consider the qualitative local behavior of trajectories for the ordinary Loewner differential equation with a driving function which is inverse to the exponential function of an integer power. All the singular points and the corresponding singular solutions are described. It is shown that this driving function generates solutions to the Loewner equation which map conformally a half-plane slit along a smooth curve onto the upper half-plane. The asymptotical correspondence between harmonic measures of two slit sides is derived.

## About generating set of the invariant subalgebra of free restricted Lie algebra

Suppose that L=L(X) is the free Lie p-algebra of finite rank k with free generating set X={x1,…,xk} on a field of positive characteristic. Let G is nontrivial finite group of homogeneous automorphisms L(X). Our main purpose to prove that LG subalgebra of invariants is is infinitely generated. We have more strongly result. Let Y=∪∞n=1Yn be homogeneous free generating set for the algebra of invariants LG, elements Yn are of degree n relatively X, n≥1. Consider the corresponding generating function H(Y,t)=∑∞n=1|Yn|tn.

## Parameters Recovering Algorithm for One Class of Irrationalities

In this article we study one class of irrationalities whichmay be defined as covergent series with rational coefficients. This class contain a lot of well known constants such as ln 2, ¼, e.t.c. We consider the problem of determination parameters of rational coefficients by rational approximation of irrationality. We deduced the lower and upper bounds and present an algorithm for determination of unknown parameters. Also, we present some results of practical calculations.

## On the A. V. Mikhalev’s Problem for Lie Algebras

Weakened A. V. Mikhalev’ sproblem about the prime radical of artinian Lie algebras is solved.

## Well-posedness of the Dirichlet problem in a cylindrical domain for multidimensional elliptic-parabolic equation

A unique solvability of classic solutions to Dirichlet's problem in the cylindrical domain for the model multidimensional elliptic-parabolic equation is shown in the article

## Mixed problem for simplest hyperbolic first order equations with involution

In this paper investigates the mixed problem for the first order differential equation with involution at the potential and with periodic boundary conditions. Using the received refined asymptotic formulas for eigenvalues and eigenfunctions of the corresponding spectral problem, the application of the Fourier method is substantiated. We used techniques, which allow to avoid investigation of the uniform convergence of the series, obtained by term by term differentiation of formal solution on method of Fourier.

## Parabolic parallelograms of the plane Ĥ

Parabolic parallelograms on a Hyperbolic Plane Hˆ with 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.

## Projective and injective descriptions in the complex domain. Duality

Research of a invariant subspaces of a differential operators infinite order in a complex domain generated many issues, related with transition to dual problems. This work devoted overcome these difficulties

## Asymptotic properties and weighted estimation of polynomials, orthogonal on the nonuniform grids with Jacobi weight

Current work is devoted to investigation of properties of polynomials, orthogonal with Jacobi weight on nonuniform grid where. In case of integer for such discrete orthonormal polynomials asymptotic formula with was obtained, where classical Jacobi polynomial, remainder term. As corollary of asymptotic formula it was deduced weighted estimation polynomials on segment [−1,1].