Table of contents

In this paper we study the average value of the function τ_k(n), the number of representations of n as a product of k natural factors, n≤x, with a remainder term which is uniform in x and k.

In this paper we prove the uniform boundedness of the torsion of elliptic curves over algebraic number fields of fixed degree under the condition that their invariants belong to fields over which the rank of the curve V² = U⁴

– 1

is bounded.

In this paper we answer a question posed by Ju. I. Manin: for which quasicharacters does there exist a generalized Néron pairing? It turns out that the quasicharacter must trivialize some explicitly described group of roots of unity. In addition, we establish in this paper a connection between generalized Néron pairings and the biextensions of Mumford.

In this paper we study the connection between free projective metabelian Lie algebras of finite rank and Serre's problem. We prove that projective metabelian Lie algebras of rank two are free.

We establish a sufficient condition for stability of Riemannian manifolds, i.e. a property according to which every equimorphism of this manifold can be topologically extended to its absolute.

In this paper the classical Cauchy-Bochner integral representation is generalized to a more extensive class of functions holomorphic in tubular domains T^C over a cone C.

Any bounded holomorphic function defined on an analytic closed submanifold in general position in a strictly pseudoconvex domain can be continued to a bounded holomorphic function in the entire domain.

Conditions are advanced in the paper which, imposed on a ring of holomorphic functions of several variables and a finite collection of functions in that ring, are sufficient for the whole ring to be generated as the closed ideal spanned by the functions in the collection.

An equiconvergence theorem for nonharmonic Fourier series of the form and ordinary Fourier series is proved for functions in , , when the exponents are the roots of a member of a certain class of entire functions.

A study is made of the asymptotic behavior of the eigenvalues and of expansions in the root vectors of the class of integral operators specified in the title. If some natural conditions, ensuring "regularity" of the asymptotic behavior of the spectrum, are imposed on the kernel, the root vectors form a basis in L_p(0,T) (1 < p < ∞) and a Riesz basis in L₂(0,T).

In this paper the regularization of a singularity with respect to a parameter is derived by means of an extension of the original operator and subsequent application of perturbation theory in an unbounded space, and used to solve an "extended" problem asymptotically. It is proved that this asymptotic solution is unique. An appropriate restriction of the asymptotic solution thus obtained will be an asymptotic solution of the original problem; this restriction is also unique. The theory of this method is illustrated by an example of an ordinary linear system of general form.

In this work we establish necessary and sufficient optimality conditions, and a theorem on the uniqueness of optimizing functions for control systems of general type.