We study families of unitary operators on a Hilbert space $H$, commuting to within a constant. It is shown that if such a family generates in the norm topology the algebra $B(H)$ of all bounded linear operators, then the dimension of $H$ is finite and there exist some unitary operators $v_1, v_2,\ldots, v_m \in S$ which generate $B(H)$ such that $\left( \lambda_jv_j\right)^n=I\quad$ ($j=1,\ldots,m$), where $n=\dim H$ and $\lambda_j$ - some constants which belong to the unit circle. 2000 Mathematics Subject Classification: 42A70. Top of the page.

The estimation of function is found which satisfies to a task of the Goursat with the data on characteristics. 2000 Mathematics Subject Classification: 35L70. Top of the page.

Nonstationarity character for a solution of heterogeneous linear differential equation in Hilbert space are studied . The right side of equation $L_{t}y=x(t)\ $is defined by semigroup of operators with bounded infinitesimal quasi-unitary operator. To investigate nonstationarity properties, an infinitesimal function has been introduced. The canonical representations for infinitesimal function have been obtained with the help of triangular models of quasiunitary operators. {\it 2000 Mathematics Subject Classification: 47A45, 60G12. Top of the page.

A task decides for the guided robust system synthesis of the limited management satisfying to set to limitations and translating any point phase space in beginning of coordinates for eventual time. The decision is based on a method functions of controllability.2000 Mathematics Subject Classification: 47A45. Top of the page.

In the paper several conditions are presented that relate the stabilizability properties of linear stationary discrete systems $x(k+1)=Ax(k) + Bu(k)$ } it 2000 Mathematics Subject Classification: 93B05. Top of the page.

In present work, we introduce the Weyl intervals associated with the truncated completely indeterminate Nevanlinna-Pick interpolation problem in the class R[a,b]. It is shown that when adding interpolation condition, the Weyl matrix intervals are embedded. The filling of the Weyl intervals with values of solutions of the Nevanlinna-Pick interpolation problem in the corresponding point is proved. We introduce the limiting Weyl intervals that are completely filled with values of solutions of the problem. We also obtained a criteria for the completely indetermination of the infinite Nevanlinna-Pick problem. it 2000 Mathematics Subject Classification: 47A57, 42A82. Top of the page.

In this paper we consider the regularization of the Zakharov system. This regularization is based on the singular perturbation of Laplas operator by biharmonic operator with a small coefficient. The main result is the convergence of attractors of singular perturbed systems to the attractor of two dimensional Zakharov system. it 2000 Mathematics Subject Classification: 35Q55, 35B40 . Top of the page.

The discrete model of thin elastic layer is suggested in the paper. On the basis of proposed model the problem of longitudinal harmonic wave propagation have been considered. The results received with the help of suggested discrete model were compared with the results of continuous theories on the basis of dispersion curves. It is shown that for medium and long waves discrete model yields satisfactory results not conceding alternative continuous models of plate theories and approaches by its characteristics to the model based on the equations of elasticity theory. it 2000 Mathematics Subject Classification: 74J05. Top of the page.

For operators close to unitary operator analog of Branges transform has been constructed. In this case Hilbert's space of entire functions is generated by two-dimensional vector-functions. The view of reproducing kernel in this space has been found and Parseval's identity for constructed generalization of Branges transform has been obtained. it 2000 Mathematics Subject Classification: 47A45. Top of the page.

We introduce an analogue of Jessen function of the subharmonic almost periodic function $u(z)$ and investigate its relation with asymptotic properties of the $u(z)$. it 2000 Mathematics Subject Classification: 42A75. Top of the page.

Numerical analytical method for the spectrum analysis of electromagnetic field in flat resonator is suggested in the paper. The method provides rapid convergence and high precision in case of smooth resonator boundaries. The question of potential method by-effects leading to appearance of parasitic frequencies in spectrum is touched upon. The results of numerical experiments illustrating possibilities of the method are analyzed. it 2000 Mathematics Subject Classification: 45F99. Top of the page.

The classification of complex submanifolds $F^l\subset {\bf C}^{l+p}$ with nondegenerate totally geodesic Grassmann image is obtained. it 2000 Mathematics Subject Classification: 53B25. Top of the page.

In the paper some previous results of author, obtained for the Livsic-Brodskii $J$-nodes with strongly regular $J$-inner characteristic matrix functions are extended onto the case of operator valued characteristic functions. This is done after investigation more general problem for the class of strongly $H_2$-regular pairs of operators $A\in L(X)$ and $C\in L(X,Y)$ that give condition on such a pair, when the semigroup of operators $T(t)=e^{iAt}, t\geq 0$, is bi-stable. it 2000 Mathematics Subject Classification: 47A57, 42A82. Top of the page.

The aim of this paper is to thoroughly study the radii of the Weyl matrix balls associated with the matricial Hamburger moment problem. In particular, we study the multiplicative structure of the radii. Let $m$ denote the size of matrix moments. As is known the ranks of the limit radii $m_+$ and $m_-$ satisfy inequations $ 0\leq m_+ , m_- \leq m $, and reach extremum $m$ simultaneously. Let $m_+$ and $m_-$ be any integer numbers, which satisfy inequations $ 0\leq m_+ , m_- \leq m - 1$. It is proved in the paper, that there exists such the Hamburger moment problem that $m_+$ and $m_-$ are the ranks of the limit radii. it 2000 Mathematics Subject Classification: 47A57, 42A82. Top of the page.

The method of asymptotic integration of a singular perturbed nonlinear system of differential equations with neterovs boundary-value conditional and a degenerate matrix at the derivatives is offered. Coefficients solution system of differential equations be find with algebraic systems equations. Examine it is possible cases deed boundary-value linear functional on general solution system. The method be find unknown value with boundary-value condition is show every case. it 2000 Mathematics Subject Classification: 34E05. Top of the page.