Mathematics

1999, No. 2

BIRATIONAL GEOMETRY AND ARITHMETIC OF LINEAR ALGEBRAIC GROUPS, IV

The previous chapters have been published in the issues of Vestnik SamGU No. 2, No. 4, 1997 and No. 2 1998. In this part of the work the theory of R-equivalence on linear algebraic groups is stated. The class group of R-equivalence is also a birational invariant in the category of linear k-groups and this category is the most natural area of the application of R-equivalence. The concept of R-equivalence was introduced in the arithmetic of varieties by Manin. This section contains the results of Colliot-Thelene and Sansuc on the computation of R-equivalence of algebraic tori and the author`s result on investigation of R-equivalence on simply connected semisimple groups and on the connections between the class group of R-equivalence with Whitehead group SK₁(A) which has been studiced by Platonov. The R-equivalence on the groups of adjoint type is described including recent results of Merkurjev. In conclusion the class numbers of algebraic tori of two most often encountered types are calculated. It leads to interesting indexformulas in the arithmetic of number fields.

A.K. Kairakbaev

CONSTRUCTION OF THE AVERAGE DIFFERENTIAL INCLUSION FOR SOME MODEL OF GYROSCOPE

The mathematical model of the noncontactly suspended gyroscope on a mobile platform is investigated when the platform`s acceleration being function of time and some variable, the latter has a velocity of the change which is involved in the assign segment. By the averaging schemes it is shown that the exact upper approximate differential inclusion exists in the upper approximation problem. It is determined that the right-hand side of the constructed differential inclusion belongs to a two-dimensional plane of the slow variables`s space. The slow variables of the issue system have been evaluated on the asimptotic great segment.

V.A. Kushmantseva

ДОПОЛНЯЕМЫЕ ГИПЕРПОДПРОСТРАНСТВА БАНАХОВЫХ РЕШЕТОК

It is proved that if X is separable æ-complete real Banach lattice withoute bands isometric to Hilbert space, then in X there are no 1-complemented hypersubspaces (i.e. subspaces with codimention one).

A.V. Linkov

SUBSTANTIATION OF A METHOD THE FOURIER FOR BOUNDARY VALUE PROBLEMS WITH AN INVOLUTE DEVIATION

The problems of substantiation of a method the Fourier for boundary value problems of mathematical physics, in which the differential operator contains an involute deviation in a partial derivative, are considered in this paper. The series of outcomes guaranteeing a correctness of applicability of a method the Fourier is obtained.

N.N. Evdokimova, L.S. Pulkina

A NONLOCAL PROBLEM FOR A DEGENERATE HYPERBOLIC EQUATION

In this paper the existance, uniqeness and stability of generalized solution for a degenerate hyperbolic equation are prooved.