Table of contents

The authors obtain results in the theory of multiple trigonometric sums that in essence take account of the domain of variation of the principal parameters.

Bibliography: 6 titles.

In this paper it is shown that the image of the Galois group under an -adic representation in the Tate module of an abelian variety has an algebraic Lie algebra which contains the scalar matrices as a subalgebra (Serre's conjecture). This paper also proves the finiteness of the intersection of a subgroup of an abelian variety all of whose elements have order equal to a power of a fixed number with a wide class of subvarieties.

Bibliography: 13 titles.

A method for constructing group-invariant solutions of differential equations is described. At the foundation of the method lies a reduction of the dimension of the base of a bundle of -jets of functions by means of a passage to the manifolds of orbits of the contact action of the Lie group of partial symmetries of the differential equation. Only the orbits of a certain submanifold of are considered, an extension of an involutive system of first-order differential equations associated with the action of the group.

Bibliography: 7 titles.

The paper gives a negative answer to the following question of M. Wriedt: Is it true that in every projective limit of reflexive Banach spaces there exists a normlike metric for which all closed hyperplanes are proximinal?

In particular, it is shown that if is a nuclear Fréchet space nonisomorphic to the space of all sequences , then for an arbitrary normlike metric on  inducing the topology , there exist nonproximinal closed hyperplanes.

Bibliography: 14 titles.

Let be a manifold with volume element . For any neighborhood , let be the group of diffeomorphisms of  that are concentrated in , and in this group let be the component of the identity. We compute the inductive limit of the family with respect to the natural inclusions for .

Bibliography: 15 titles.

This paper studies estimates and the asymptotic behavior as for the probabilities , and , , where , the being independent identically distributed random variables with mean zero, and and are nonrandom functions. Under certain restrictions on the boundaries and logarithmic asymptotes of these probabilities are found in the case when the satisfy (respectively) a two-sided or a one-sided Cramér condition. The method is based on an absolutely continuous substitution for the original probability measure.

Bibliography: 18 titles.

In this paper the author obtains a formula representing the free term of the Laurent expansion of the zeta functions of absolute and ray classes of a real quadratic field at the point 1, as well as the value of Hecke's L-function corresponding to the signature character. The formula contains Dedekind sums and sums analogous to them in which functions depending on the same arguments as in ordinary Dedekind sums appear.

Bibliography: 4 titles.

This paper studies birational automorphisms of algebraic threefolds representable in the form of conic bundles. It is shown that, under certain conditions on the degeneration curve of a conic bundle, the automorphisms preserve the conic bundle structure.

Bibliography: 18 titles.

In this paper an analogue of Jackson's inequality is established for the Hardy spaces : if , then

; , and .

Bibliography: 15 titles.

This paper gives a biregular classification of smooth Fano 3-folds (i.e. varieties with ample anticanonical sheaf) for which there exists a morphism onto a smooth rational surface whose fibers are smooth rational curves.

Bibliography: 9 titles.