Table of contents

We give a decomposition of the Tate height into components possessing quadraticity with respect to a group law, and on the basis of this decomposition we obtain an estimate for the number of K-points with integral coordinates of certain algebraic curves.

Assuming Tate's finiteness conjecture, we prove some consequences of Tate's conjecture on homomorphisms of abelian varieties.

It is proved that for any finitely based variety of groups there exists a finitely based left factor such that their product is infinitely based (the left factor is a Burnside variety).

In this paper it is shown that the study of projective metabelian Lie algebras of finite rank reduces to a partial solution of Serre's problem on projective modules over polynomial rings. It is also observed that projective commutative-associative algebras of dimension 1 are isomorphic to the ring of polynomials in one variable over the ground field.

A description is given of the set of algebraic equations that connect the components of the solutions of a system of linear differential equations. The result thus obtained is applied to the proof of a theorem on the estimates of the orders of the zeros.

The following theorem is proved: If the Sylow 2-subgroups of a finite simple group are isomorphic to the Sylow 2-subgroups of , , then is isomorphic with .

It is proved that on any compact, connected, smooth manifold of dimension greater than two there exist smooth flows preserving a given measure with smooth positive density and ergodic with respect to it. (The smoothness is everywhere of infinite order.)

We study A- and B-integrability of n-dimensional conjugate functions and determine when the Fourier series of the conjugate function is its conjugate series.

We solve the problem of the best quadrature formula of the form for the following classes of differentiable periodic functions: , (where is a convex modulus of continuity and is odd), and .

Sufficient optimality conditions are proved in the form of a maximum principle for the time-optimal problem of transfer from a set into a set , where an object's behavior is described by the differential imbedding .

In this paper we develop a new asymptotic method for pseudodifferential operators in the case of characteristics of variable multiplicity; the th term of the asymptotics is expressed in terms of an -dimensional integral of a rapidly oscillating function of arguments, where is the dimension of the space .

For differential operators and ; with constant complex coefficients in the halfspace we present a precise description of the "space of traces" of elements in the completion of the space with respect to the metric is the norm in . We consider the case of the metric in detail. We establish necessary and sufficient conditions for validity of the inequality

for all is the form in .