Table of contents

The concept of local measurability is introduced for a selfadjoint operator with respect to a JW-algebra, and a theorem is proved on realization of ordered Jordan algebras of operators.

Bibliography: 33 titles.

In this paper it is shown that over an algebraically closed field there exist only finitely many mutually nonisomorphic toroidal Fano varieties. We give a complete description of toroidal Fano varieties with a centrally symmetric fan. We prove the rationality of a special class of toroidal Fano varieties and consider applications to problems of rationality of algebraic groups. Bibliography: 17 titles.

This article studies representations of semisimple Lie algebras arising naturally in the -adic cohomology of algebraic varieties defined over global fields. A conjecture is formulated about the restrictions the index of the cohomology space and the Hodge numbers of a variety impose on the weights of a represention. The conjecture is proved for ordinary varieties over function fields. An analog of the conjecture is valid for the rational cohomology of varieties defined over the field of complex numbers.

Bibliography: 43 titles.

The author proves an analogue of Men'shov's theorem (on correction up to a function with uniformly convergent Fourier series) for an arbitrary locally compact abelian group of finite topological dimension. The spectrum of the corrected function can be placed in a prescribed "sparse" set.

Bibliography: 11 titles.

The Yang-Mills field is studied in the case of the SU(2) algebra. An explicit three-instanton solution is constructed. This solution is a rational function of free real parameters which vary in .

Bibliography: 13 titles.

Complete asymptotic expansions are found for the first eigenvalues and eigenfunctions of classical problems for the Laplace operator in two- and three-dimensional domains with small holes.

Bibliography: 17 titles.

The leading term of the asymptotics of a solution of the nonlinear hyperbolic system

for large times is constructed and justified. A version of the method of the inverse problem reducing to the solution of a matrix problem of linear conjugation on the complex plane of the spectral parameter is used to solve this system. The coefficients of the asymptotics of are expressed explicitly in terms of the elements of the Riemann matrix realizing the linear conjugation. A theorem is proved on the approximation of the exact solution by the asymptotics constructed.

Bibliography: 14 titles.

A new proof is offered of a differential criterion of regularity for the adjoint representation of a semisimple connected group that does not use the existence of a section in the set of regular elements. Using the ideas of this proof, similar results are obtained for certain linear actions with a Cartan subspace, and, conversely, the existence of a section in the set of regular elements is proved.

Bibliography: 7 titles.

An extension of a class of variational problems and evolution variational inequalities in nonreflexive spaces is constructed.

The concept of a generalized solution for problems of this class is introduced, and relations that they satisfy are obtained. Extensions of a variational problem for nonparametric surfaces of prescribed mean curvature and two problems from the theory of ideal plasticity are considered as examples.

Bibliography: 14 titles.