Table of contents

In this paper the construction is begun of a theory of Dirichlet series with Euler expansion which correspond to analytic automorphic forms for congruence subgroups of the integral symplectic group of genus 2. Namely, for an arbitrary positive integer a connection is revealed between the eigenvalues of an eigenfunction of all the Hecke operators , where is the principal congruence subgroup of degree of the group , and its Fourier coefficients. This connection can be written in the language of Dirichlet series in the form of identities; here an infinite sequence of identities arises, indexed by classes of positive definite integral primitive binary quadratic forms equivalent modulo the principal congruence subgroup of degree of . Bibliography: 15 titles.

An elliptic equation is considered in a domain that is obtained from a two-dimensional domain by removing a neighborhood of a segment. It is assumed that this neighborhood contracts to the segment as . An asymptotic expansion of the solution of the first boundary value problem in is constructed and justified. Bibliography: 5 titles.

In this paper it is proved that a function of n complex variables that is meromorphic in each variable separately on a special set X is globally meromorphic in the neighborhood of X. This result is an analog of a theorem of J. Siciak on separate holomorphicity. Bibliography: 8 titles.

Variational inequalities with a small parameter are considered, together with closely related nonlinear operator equations. The penalty method as applied to variational inequalities is studied, and Galerkin's method relative to nonlinear equations. Under conditions of monotonicity, coerciveness and hemicontinuity on the operators, uniform convergence (with respect to the small parameter) of the approximate solutions thus obtained to the exact solution is demonstrated. Bibliography: 12 titles.

A description is given of two-sided Noetherian rings over which all finitely generated modules are semichained. Bibliography: 31 titles.

A nonnegative solution of the equation is considered in an arbitrary domain with smooth boundary . The surface can contain a characteristic manifold . Conditions on are obtained ensuring that is always finite. These conditions turn out to be best possible. Bibliography: 4 titles.

General symmetric hypoelliptic systems of differential operators in with discrete spectrum are considered. Two-sided estimates, as , are found for , the number of eigenvalues in the interval . Under a regularity assumption on the behavior of the spectrum of the Weyl matrix symbol of the system, these estimates reduce to the asymptotics of with an estimate of the remainder term. In part the results are also new for the scalar case. Bibliography: 9 titles.

A connection is established between domains of meromorphy of a function of several complex variables and the degree of approximations by rational functions of special form. Bibliography: 5 titles.

In this paper results concerning the notion of dimension of an embedding, earlier obtained for embeddings of compacta in Euclidean space E_n for n≥6, are extended to the case n=5. For n=4 a reduction of the main problem to the so-called open four-dimensional Poincaré conjecture is given, and some sufficient conditions are given for n=3. Bibliography: 14 titles.

For an arbitrary simply connected domain D, an estimate is obtained for the norm of the holomorphic component of a meromorphic function having a given number of poles in D. The estimate is uniform with respect to D. Bibliography: 4 titles.