Boundary-value problem with symmetric boundary conditions and the integral condition to the potential is considered. The estimates of the minimal eighenvalue with different parameter α in the integral condition are odtained 

The classes of one-dimensional nonlinear evolution equations of the third order, which solutions are interconnected by generalized Miura and Backlund trunsformations, are obtained. The research is based on construction of Lax pairs by Calogero (J.Math.Phys. 32 (1991)(1) 72–74). It is proved that arbitrary one-dimensional nonlinear evolution equations, which can be presented in the form of local conservation law, admits a bunch (non-enumerable set) of Lax pairs by Calogero depended on several spectral parameters 

The dynamics of a point mass on a spring with damping and impacts with a delimiter is studied. It is supposed, that this dynamics is described on free flight segments by a linear equation. For the system considered the conditions of chaotic and periodic behavior of solutions are given 

The problem of acoustic echo attenuation has important applications because of existence of a resounance in a closed loop system loudspeaker–hall– microphone–loudspeaker. A connection of this problem with methods of Toeplitz matrix inversion and with a theory of the Szeg”o and Schur polynomials is described. A superfast Schur algorithm is presented for solution of the coefficients identification problem. Its complexity is calculated. The echo canceller system also contains a hybrid implementation of a linear convolution that admits a real time hardware implementation 

Статья посвящена 120-летию со дня рождения Сергея Натановича Бернштейна — одного из величайших математиков двадцатого века — и основана на личных воспоминаниях автора — одного из учеников С.Н.Бернштейна 

В статье рассматривается обобщение систем Ковалевской, Чаплыгина, Горячева–Чаплыгина и Богоявленского на пучке скобок, а также описывается метод введения разделяющих переменных и переменных “действие–угол” для этих систем. Для волчка Ковалевской найден альтернативный метод интегрирования на пучке путем преобразования координат, сводящих систему Ковалевской к системе Неймана, а также рассмотрена аналогия Колосова. В конце статьи получено обобщение системы Гаффе на пучок скобок 

The problem of a motion of a rigid body of movement on the perfectly rough surface is considered. The conditions of existence of Chaplygin-type integral is obtained. It is shown, that these conditions are valid in the case of motion of non homogeneous dynamically symmetric sphere (a top) on the perfectly rough plane or internal surface of the sphere 

Two wavelet bases related with discrete Haar-Chrestenson transform are considered in a space of discrete N-periodic signals with N=ns. Analogues of well-known theorems of shift, convolution and correlation for these bases are obtained 

Estimate from below for the number of limit cycles of Lienard's equation is obtained. A method of composing dynamical systems with definite number of limit cycles is suggested and applied to the investigation of cubic Lienard's equation 

Weak numerical method of the fourth order for the stochastic differential Ито equations with scalar non additive noise is proposed. The unified Taylor-Ito expansion is used for the construction of this method