Table of contents

The two-parameter family of bifurcation diagrams Σ of the moment map is investigated in the integrable Kovalevskaya-Yehia case for the motion of a rigid body. A method is developed which is useful for calculating the bifurcation set Θ in the parameter space which corresponds to bifurcations of diagrams in Σ and for classifying these bifurcations. The properties of the sets Σ and Θ are thoroughly investigated, and details of the modifications the bifurcation diagrams undergo as the value of the parameter crosses Θ are described. Illustrations which explain the structure of the different types of diagram and their interrelations are given.

Bibliography: 22 titles.

For a connected simply connected semisimple algebraic group G we prove the existence of invariant tensors in certain tensor powers of rational G-modules and establish relations between the existence of such invariant tensors and stability of diagonal actions of G on affine algebraic varieties.

Bibliography: 12 titles.

It is proved that the Cesàro space , , contains a complemented subspace isomorphic to if and only if either or . A class of subspaces of this space that contain complemented copies of the space is distinguished.

Bibliography: 16 titles.

Suppose that is a free -generated associative ring with the identity . In 1993 Zelmanov put the following question: is it true that the nilpotency degree of has exponential growth?

We give the definitive answer to Zelmanov's question by showing that the nilpotency class of an -generated associative algebra with the identity is smaller than , where

This result is a consequence of the following fact based on combinatorics of words. Let , and be positive integers. Then all words over an alphabet of cardinality whose length is not less than are either -divisible or contain ; a word is -divisible if it can be represented in the form so that are placed in lexicographically decreasing order. Our proof uses Dilworth's theorem (according to V.N. Latyshev's idea). We show that the set of not -divisible words over an alphabet of cardinality has height over the set of words of degree , where

Bibliography: 40 titles.

We prove that the bound from the theorem on 'economic' maps is best possible. Namely, for we construct a map from an -dimensional simplex to an -dimensional Euclidean space for which (and for any close map) there exists a -dimensional plane whose preimage has cardinality not less than the upper bound from the theorem on 'economic' maps.

Bibliography: 16 titles.

This paper is concerned with the problem of best recovery for a fractional power of the Laplacian of a smooth function on  from an exact or approximate Fourier transform for it, which is known on some convex subset of . A series of optimal recovery methods is constructed. Information about the Fourier transform outside some ball centred at the origin proves redundant--it is not used by the optimal methods. These optimal methods differ in the way they 'process' key information.

Bibliography: 12 titles.

The qualitative properties of solutions to the Cauchy problem for a degenerate parabolic equation containing a nonlinear operator of Baouendi-Grushin type and with gradient absorption whose density depends on time, as well as the space variables, are investigated. Bounds for the diameter of the support of the solution which are sharp with respect to time are obtained, together with its maximum. A condition which determines whether or not the phenomenon of decay to zero of the total mass of the solution occurs is discovered.

Bibliography: 35 titles.