We consider two-dimensional bounded remainder sets, which are constructed by a hexagonal developmant of the torus. Also, we find optimization of boundaries for these sets.

 

Let v ∈ N. Let the function Φ(T) arbitrarily slow tend to +∞ with T → +∞. The asymptotical formulas for fractional moments of the Riemann zetafunction 2 R T T |ζ(σ + it)| 2/vdt for 1/2 + Φ(T)/ln T 6 σ < 1 and for fractional moments of the arithmetical Dirichlet series of second degree from Selberg’s class 2 R T T |L(σ + it)| 2/vdt for 1/2 + Φ(T)/ √ ln T 6 σ < 1, are obtained.

In this work we considered the matrix decomposition of the cubic irrational α satisfying the equation x 3 − 4x 2 − 5x − 1 = 0. For decomposition of the matrix α 1 = Y∞ k=0 310941 · k + 155427 156744 · k + 78333 61578 · k + 30882 31041 · k + 15564 an algorithm of transition to regular continued fraction is constructed.

In this paper, we prove: Theorem. Each volume A > A′ there are more A 5 ln A of polynomials of second degree with integer coefficients, senior coefficients are equal to two, each of which contains more A 5 ln1+ε A simple. (ε > 0 — constant)

 

This article solved the problem of conjugacy of subgroups of cyclic groups in a free product with amalgamation of cyclic subgroup and the HNN-extension of cyclic groups with the Union.

Density theorems about zeros of dirichlet polynomials approximating Dirichlet L-fuctions in the critical strip are obtained.

The problem of representing integers as sums of terms of s certain type is actual in number theory and its applications. We are interested in the average length of these expansions and the required number of auxiliary calculations. The paper deals with DBNS, chains and the polyadic (factorial) expansions of positive integers.

With V. Ch. Salihov’s estimates of linear forms in ln 2 and ln 3 we improve the asymptotic estimate of number of terms in DBNS.

We investigate the motion of a satellite in the gravitational field of a massive deformable planet. Planet is modeled as body that consists of a solid core and a viscoelastic shell of a Kelvin-Voigt material. The satellite is modeled as a point mass. The system of integro-differential equations for a motion of a mechanical system is got out from the variational principle of the d’Alembert-Lagrange according to the linear theory of elasticity. Approximate equations of motion in vector are constructed with asymptotic method of motions separation. This system of equations describes the dynamics of the "planet-satellite"with regard to the perturbations caused by elasticity and dissipation. To describe the evolution of the orbital parameters of a satellite, averaged differential equations were derived. Phase trajectories were constructed for particular cases, their stationary solutions were found and investigated on stability. In the case of the existence of two stationary orbits stationary solution that corresponding to the motion along the orbit of larger radius is asymptotically stable, and the orbit of smaller radius is unstable. Some of the planets in the solar system and their satellites are considered as examples. This problem is a model for the study of the tidal theory of planetary motion

In the paper a new multidimensional generalization of fraction part function is introduced. We obtain a formula which express the number of points from the orbit of irrational shift on multidimensional torus, lying in a given domain, in the terms of sums of multidimensional fraction parts. Also we give some application of this formula to various number-theoretic problems.

In this paper we try to give a generalization of the usual notion of extremum of real functions to the vector-valued functions of several real variables. Our aim is that in this generalization remain valid the usual properties and relations for extremum of real functions. A considered generalization is also characterized by an equivalent generalization. Our definitions and related results are illustrated by numerous examples.