Table of contents

The paper is a continuation of work (MR 49 #801) in which in the case of a "noncontracting" unbounded domain there is distinguished a geometric characteristic of the domain that determines (under the fulfillment of a certain condition of "regularity" of the domain) the rate of stabilization for of the solution in of the following second boundary value problem for a parabolic equation:

in which the initial function decreases sufficiently rapidly as . It is proved in the present paper that the same characteristic also determines the rate of stabilization of the solution in a class of "contracting" domains . In this case, as in the case of a "noncontracting" domain, tends to zero as like : there exist estimates of the function from above and from below having such an order of decrease. Bibliography: 11 titles.

Suppose is a finite group of automorphisms of an associative algebra with an identity element over a field . Let . Assume that is a supernilpotent radical which is closed under the taking of subalgebras and satisfies the following condition: if and is a nonempty set, then the ring of matrices all but a finite number of whose columns are zero is radical. Theorem. If is a two-sided ideal of and , then implies . Examples of radicals satisfying the above conditions are Baer's lower radical, the locally nilpotent radical, the locally finite radical, and also the algebraic kernel and Köthe radical, if is uncountable. Bibliography: 6 titles.

In this paper a theorem on the rank of the group of rational points of an abelian variety with a sufficiently large ring of endomorphisms is proved. It is applied to construct nontrivial factors of the Jacobians of modular curves with finite groups of rational points and to prove finiteness theorems for modular curves. Bibliography: 20 titles.

The classical boundary theorem of F. and M. Riesz asserts that, if the radial limits of a bounded holomorphic function in the disk lie in a set of capacity zero for a set of positive measure on the circle , then . The main result of this paper is the proof of an analogous theorem for maps , where is a domain in . We take as uniqueness set on the boundary any set of positive Lebesgue measure on a generating submanifold. Bibliography: 12 titles.

Necessary and sufficient conditions are obtained for bisingular integral operators on piecewise smooth Ljapunov curves with discontinuous coefficients in the -spaces with a weight to be Noetherian. The Banach algebra generated by these operators is studied; a regularizer is constructed in the case of continuous coefficients. Bibliography: 40 titles.

On a compact manifold we examine the equation

(1)

We assume that is a second-order elliptic selfadjoint positive definite differential operator and that the coefficients of the operator and of the function are analytic on . It is well known that equation (1) has a unique global solution defined on the whole (as a consequence of the Cauchy-Kowalewski theorem there are many local solutions). In this paper we obtain an explicit expression for the value of at a point in terms of the Taylor coefficients of the right-hand side at , and of the coefficients of the operator. By the same token we obtain an expression for the solution of the global problem in terms of the local data of this problem. Bibliography: 7 titles.