The paper is dedicated to the 150-th anniversary of the famous scientist Aleksandr Mikhailovich Lyapunov who was an Academician of the Saint-Petersburg Academy of Sciences and a professor of the Kharkov Imperial University. We discuss the main results of A.M.Lyapunov in the field of the movement stability which form a basis of the contemporary stability theory. We retrace the ways of development of this theory - partial stability, Lyapunov vector functions, stability in Banach spaces. We discuss the development of the Lyapunov functions method to the control systems, namely, the controllability function method. On the basis of this method, the admissible synthesis problems for various classes of systems are solved. The results are illustrated by examples. 2000 Mathematics Subject Classification: 34D20, 93D30, 49J15, 93B50, 93B52. Top of the page.

In the paper the description of life and scientific activity of A. M. Lyapunov was given. The influence of Lyapunov's ideas in the proof of the Poincare conjecture and the geometric Thurston conjecture is shown. 2000 Mathematics Subject Classification: Top of the page.

A survey of the experimental and theoretical results on absolute and convective instability of the flows of a viscous liquid past deformable surfaces, in the pipes and channels with compliant walls is presented. The flow instability results in the tube collapse, decrease of the volumetric flow rate, wall oscillations, sound generation, and, finally, destruction of the system. Results of investigation of the flow stability in the multilayered isotropic and anisotropic viscoelastic tubes with different boundary conditions at the outer surface of the tube are presented. It is shown, one or several absolute unstable fluid based modes can be found at different model parameters. By a proper choice of the viscous and elastic coefficients and thickness of separate layers the most unstable mode can be stabilized and the absolute instability can be eliminated. As applied to the tubes of technical devices it enables avoiding significant energy loss, tube collapse and undesirable noise generation. In application to the blood vessels the developed models enable biomechanical interpretation of the parameters of both noise and wall oscillation curves measured in arteries and veins. 2000 Mathematics Subject Classification: 76E15,74F10, 76D05,76Z05. Top of the page.

Using the Lindstedt-Poincare method the spectra of the anharmonic oscillators with quartic, sextic and octic power of nonlinearity are obtained. The method and the computer software for solving the problem is developed. It is shown that the obtained formula for the spectrum of quartic oscillator fully coincides with one derived by the quantum normal form method. 2000 Mathematics Subject Classification: 42A70. Top of the page.

We investigate a new class of nonlinear control systems of O.D.E. which are not feedback linearizable in general. Our class is a generalization of the well-known feedback linearizable systems (the Jakubczyk-Respondek-Hunt-Su-Meyer criterion), and moreover it is a generalization of the triangular (or pure-feedback) forms studied before. We describe our "generalized triangular form" in coordinate-free terms of certain nested integrable distributions. Furthermore, our conditions are the global analog of the Jakubczyk-Respondek-Hunt-Su-Meyer conditions. Therefore, the problem of the feedback equivalence of a system to our generalized triangular form is solved globally in the whole state space by the definition of our class. We apply a specific procedure, which can be called "backstepping" (following the terminology of the standard adaptive control theory for strict-feedback forms), and solve the problem of global controllability for our class. Our "backstepping algorithm", in turn, is based on the construction of a certain discontinuous feedback law. We propose to treat our class as a new canonical form which is a nonlinear global analog of the Brunovsky canonical form on the one hand, and which is a global and coordinate-free generalization of the triangular form on the other hand. 2000 Mathematics Subject Classification: 93C10, 93B10, 93B11, 93B05, 93B52. Top of the page.

The singular part of the operator of the electrostatic problem for a spherical segment and sectional rounding of a cone was separated and inverted by the method of regularization. The method is based on the technics of contour integration and Abel integral transformation. The Fredholm integral equation of the second kind with a compact operator in Hilbert space $L_2(0,\gamma)$ was obtained. Some generalizations of problem are considered. 2000 Mathematics Subject Classification: 65N12, 35A25,78A45.   [ Full-text available (PDF) ] .     Top of the page.

In work it is shown, that in everyone not Dedekind primary FC-group there is a subgroup, which normalizer a maximum subgroup. 2000 Mathematics Subject Classification: 20F50. Top of the page.

In this paper we obtain explicit formulas for Blashke-Potapov product and study the multiplicative structure of the resolvent matrix of the completely indeterminate moment problem on finate interval. 2000 Mathematics Subject Classification: 47A57, 42A82. Top of the page.

It is discussed, when a two-dimensional surface with degenerated ellipse of normal curvature in four-dimensional Euclidean space does not belong to any three-dimensional sphere. 2000 Mathematics Subject Classification: 53A05. Top of the page.

Conformal mappings of the profiles of cracked and uncracked cylindrical shells on unit circle is built. Uniformity of the both districts mapping achieves whith the help of the Christoffel-Schwarz integral using. The result of the article is building of the functions, which leads to receiving of the solutions of the corresponding elasticity theory problems under comparable conditions. 2000 Mathematics Subject Classification: 30C20. Top of the page.

In this paper we classify those finite metric spaces $K$ which provide the Banach space ${\rm Lip}(K)$ of all real functions on $K$ with the alternative Daugavet property or, equivalently, with the property that the numerical index ${\rm n}({\rm Lip}(K))$ equals 1. Also, we characterize those metric spaces $K$ for which ${\rm Lip}(K)$ is isometric to $l_{\infty}^n$. 2000 Mathematics Subject Classification: 46B20. Top of the page.

Numerical models for the spectrum analysis of the finite open and closed cylindrical resonators with complicated cross-section are suggested. The comparative analysis results for calculated solutions are given. 2000 Mathematics Subject Classification: 45F99. Top of the page.

In the article, the determination of generalized canonical product of points of complex plane is entered. It is proved for a case, when points are the zeros of entire function of finite order, that it coincides with canonical product of Weyershtrass. 2000 Mathematics Subject Classification: 30D15, 30D20. Top of the page.