Table of contents

The Littlewood conjecture is proved: for any integers

where is an absolute constant.

Bibliography: 19 titles.

Exact values of linear widths (diameters), Kolmogorov widths and Gel'fand widths, in -spaces are found for some classes of functions, using a convex majorant of the modulus of continuity of a function or its derivative. Precise constants in the Jackson type theorems on approximation of continuous and differentiable functions are also found. These results are interpreted from the point of view of problems of optimal coding and optimal recovery of functions and their derivatives.

Bibliography: 23 titles.

The almost periodicity of infinite-zone potentials with lacunae which tend to zero sufficiently fast is proved. The proof is based on the study of the infinite problem of Jacobi inversion.

Bibliography: 14 titles.

In the article an asymptotic formula is determined for the number of representations of a natural number as the sum of the squares of four integers, two of which are prime.

Bibliography: 25 titles.

This paper contains a proof of the Tate conjecture on algebraic cycles on surfaces with strongly degenerate reduction over function fields and the Hodge conjecture on cycles on simple abelian varieties of the first and second types according to Albert's classification (under certain restrictions on the dimension of the variety and the center of its ring of endomorphisms).

Bibliography: 27 titles.

A Galois correspondence theorem is proved for any finite-dimensional Lie -algebra of outer derivations of a prime ring of positive characteristic. A theorem is proved on the existence of a locally finite ideal, in the sense of Širšov, over the ring of constants of such a Lie -algebra. Extension and rigidity theorems are also obtained.

Bibliography: 14 titles.