Mathematics

2001, No. 4

Y.A. Rodichev, A.N. Stepanov

TO THE MEMORY OF А.A. MARTYNOV

L.M. Berkovich

ALEXANDER ANDRONOV AND NONLINEAR SCIENCE

The international Conference "Progress in Nonlinear Science" (Nizhny Novgorod, July, 2-6, 2001) was dedicated to the centenary of Alexander A. Andronov (1901-1952), the outstanding Russian scientist, one of the most prominent persons in the theory of dynamical systems and one of the pioneers of nonlinear science.

E.G. Abramochkin

HERMITE-LAGUERRE-GAUSSIAN FUNCTIONS

A unity of two-dimensional Hermite-Gaussian and Laguerre-Gaussian functions is proposed by introducing an additional parameter. Continuous changing of the introduced parameter allows to transform Hermite-Gaussian functions into Laguerre-Gaussian functions keeping some important properties of both function families, for example, the orthogonality in the space L₂(R²). The functions (called Hermite-Laguerre-Gaussian functions) are investigated similar to classic orthogonal polynomials. Some properties of this function family (including algebraic and integral representations, the Rodrigues formula, and symmetry properties with respect to the introduced parameter) are obtained. It is shown that the functions are invariant with respect to some integral transformations of Fourier type.

E.M. Knutova

ASYMPTOTIC PROPERTIES OF STUDENT'S CONDITIONAL DISTRIBUTIONS IN HILBERT SPACE

The paper is devoted to the study of asymptotic properties of conditional distributions wich are generated by finite-dimensional projections of Student's measure in real Hilbert space. Almost sure convergence of conditional distributions to normality is proved. The proof is based on Schoenberg representation of conditional distributions in the form of Gaussian mixture, it is also based on the Laplace method of the integrals asymptotic finding. The strong law of large numbers for the scheme of series of conditional distributions families is stated. Some properties of logarithmic derivatives and logarithmic gradients of Student's measure and their connection with limit conditional distributions are considered.

I.V. Konopleva, B.V. Loginov

GENERALIZED JORDAN STRUCTURE AND SYMMETRY OF RESOLVING SYSTEMS IN BRANCHING THEORY

Nonlinear differential equations in the Banach spaces with degenerate Fredholm operator at the derivative are considered. The aim of this paper is reduction of the original problem to finite-dimensional bifurcation equation in the root-subspace (BEqR) and resolving systems (RS). The application of intertwining operators properties (non-group symmetry) is given as well as group symmetry for the nonlinear equations. RS reduction possibilities both by the unknowns number and equations one are disscused. Variants of Grobman-Hartman theorem are proved. Under the group symmetry conditions of the original nonlinear equation the connections between the reduction (dimension lowering) possibilities of the corresponding branching equation and its potentiality properties are studied. Applications to capillary-gravity surface wave theory are considered.

S.Y. Popov

STANDARD INTEGRAL MODEL OF ALGEBRAIC TORI

Integral models play a significant role in the theory of linear algebraic groups over a field of the arithmetical type. The choice of such models is not unique. Thus, it is an important problem to define the integral model which is not trivial. In this paper, in the first place, the properties of the standard integral model of algebraic tori are studied. This model was proposed by Professor V.E.Voskresenskii. It is determined by the inner parameters of a torus and has some extreme properties. Secondly, the paper is devoted to the reduction of the standard integral model modulo prime. The main result is the structural theorem which establishes the correspondence between the type of the reduction and the structure of the minimal splitting field of algebraic tori.

S.Y. Shatskih

SOME PROPERTIES OF LOGARITHMIC DERIVATIVES OF STABLE ELLIPTICALLY CONTOURED MEASURES

The paper is devoted to the study of asymptotic properties of conditional distributions which are generated by finite-dimensional projections of stable elliptically contoured measures in real Hilbert space. The connection between conditional distributions of stable elliptically contoured measures and their logarithmic derivatives is stated. The statistic independence and the strong law of large numbers for the sequence of logarithmic derivatives multiplied by the fixed random variable are proved.