Table of contents

Let be a Siegel modular form of degree , weight and character for the congruence subgroup of the Siegel modular group . Suppose that is an eigenfunction for all Hecke operators with index relatively prime to . It is proven that for each fixed, symmetric, semi-integral, positive definite matrix of order and for each Dirichlet character , equal to zero on all prime divisors of , the Dirichlet series

where are the Fourier coefficients of and is the set of integral matrices of order with positive determinant, has an expansion as an Euler product which can be explicitly calculated. Bibliography: 13 titles.

Suppose that is the error of the best method of integration in the class with nodes  of multiplicities , i.e. . It is then shown that for and for every system of multiplicities  with for , the lower bound

is attained for some nodes  with exactly the multiplicities . Moreover, and  . Bibliography: 20 titles.

The work is concerned with an optimal control problem on an infinite time interval. A simple method of interpretating the Euler equation is given, which does not require the use of approximation theory. Necessary conditions for a weak extremum in the form of a local maximum principle are obtained. Bibliography: 9 titles.

In the category of right unitary modules over the associative ring  with 1, one can define weak  purity, where is the set of right ideals of satisfying certain conditions. This is a generalization of the concept of neatness in Abelian group theory. Using the properties of weak -purity, several classes of rings can be characterized. Moreover, an affirmative answer can be given to question 18 (question 14 in the English translation) of A.P. Mishina and L.A. Skornjakov's book Abelian groups and modules, which deals with properties of -high purity. Groups of weakly -pure and -high extensions are studied. Bibliography: 15 titles.

Some canonical forms for pointwise controllable systems with retarded argument are considered. Bibliography: 4 titles.

We establish analogs of the theorem of Montessus de Ballore for rational approximations (of the type of the Padé approximants) to series of orthogonal polynomials and of Faber polynomials. Bibliography: 11 titles.

We introduce the concept of an elliptic family on the manifold M in a trace algebra. We define the Chern character of an elliptic family. We also introduce the algebra of formal symbols on Rⁿ with coefficients in a trace algebra. We establish a connection between the Chern characters of an elliptic family on M in the algebra of formal symbols on Rⁿ and of the elliptic family on M × R²ⁿ formed by the leading terms of the symbols. Bibliography: 8 titles.