Table of contents

The last sentence of part 1 of the proof of Proposition 2 in §5 of the author's paper On indices of exponential separation (Math. USSR Sb. 52 (1985), no. 2, 439-470) should read: "Since this has been proved for every , lower semicontinuity of the function is proved." In the fourth line from the bottom of p. 453 the subscript should be replaced by . In the tenth line of p. 455 the superscript should be replaced by . In formula (2) on p. 456 should be replaced by and in the ninth line from the bottom of p. 456 the subscript should be replaced by . In the eighth line of p. 460 should be replaced by .

Let be the best approximation to the function in the Hardy space by rational functions of degree at most . It is shown that, for example, satisfies the condition (, ) if and only if belongs to the Hardy-Besov space . Rational approximation is also considered in () and . Some applications of the results are given. Bibliography: 29 titles.

Formulas are proved for the spectral asymptotics of a multidimensional Schrödinger operator with potential which behaves nonregularly at infinity. Bibliography: 17 titles.

The author describes the soluble minimal irreducible subgroups of , where and are prime numbers, , , and is an arbitrary subfield of the field of real numbers. He proves that up to conjugacy, there exist exactly 4 soluble minimal irreducible subgroups in , , , and , where each is a Sylow 2-subgroup of and , , and are minimal transitive groups of permutation matrices of degree and are metabelian groups, each of which is generated by two matrices, and and are soluble groups of class 3 with three generators:

where is the order of the number 2 modulo , is the order of modulo , and is the order of modulo . The properties of subspaces generated by the rows of circulants over a prime finite field are investigated. The connection between these properties and the problem of describing certain classes of minimal irreducible linear groups is indicated. Bibliography: 18 titles.

In the following three cases criteria are found for complements of divisors in compact complex manifolds to be hyperbolically embedded in the sense of Kobayashi: for divisors with normal crossings, for arbitrary divisors in complex surfaces, and for unions of hyperplanes in projective space. A criterion is given for two-dimensional polynomial polyhedra to be hyperbolically embedded, and Iitaka's conjecture about conditions for hyperbolicity of the complement of a set of projective lines is confirmed. Upper semicontinuity is proved for the Kobayashi-Royden pseudometrics and Kobayashi-Eisenman pseudovolumes of a family of complex manifolds containing degenerate fibers, and conditions are given under which the hyperbolic length (volume) on the smooth part of a degenerate fiber is the limit of the hyperbolic length (volume) on the nonsingular fibers. Bibliography: 28 titles.

An explicit classification is obtained for the maximal arithmetic subgroups of indefinite orthogonal -groups of type , where is a global field (). As a preliminary, a local analogue of the maximality problem is considered, and conditions are determined under which, for groups of the type indicated, local maximality implies global. Bibliography: 14 titles.

Complete proofs are given of criteria for analytic (or -closed) continuation of CR-functions (or CR-forms), which were formulated in part I of the article. Bibliography: 16 titles.

Fredholm equations of the first and second kinds are considered in an abstract Hilbert space. A unified solvability criterion is given for them in terms of nonnegativity of bilinear forms. For Fredholm equations of the second kind it is possible to pass from an arbitrary equation to an equation with nonnegative operator, some solution of which can be used to construct a solution of the original equation that is minimal in norm. Questions connected with restrictions on the norms of solutions are answered for these equations. The results are then carried over to Fredholm integral equations of both kinds and to linear algebraic systems. Bibliography: 4 titles.

Conditions are given which can be imposed on the coefficients of an orthogonal series of Jacobi polynomials and which, when fulfilled, guarantee that the series is a Fourier-Jacobi series; and the question of its convergence in mean is solved. The results are analogues of known theorems for cosine-series due to Kolmogorov, Szidon, Telyakovskiĭ, and the author. Bibliography: 9 titles.

The author considers rational elliptic modules over an algebraically closed complete extension and proves that is transcendental over for modules of rank . Transcendence of the periods of the lattices corresponding to rational modules of the form is also proved. Bibliography: 3 titles.

This paper presents a new theory of self-correction in circuits, one which differs from earlier methods by being completely based upon properties of Boolean functions, thus guaranteeing the universality of the methods which have already been developed. The principal result is the construction of a self-correcting, information-nonredundant Boolean function expansion, which admits an analogue in any nontrivial control system and permits the construction of asymptotically nonredundant self-correcting circuits. The Boolean functions include a broad spectrum of special classes.

This article investigates analytic solutions of a convolution equation and of systems of two convolution equations with a single unknown function. The characteristic functions of all the convolution operators studied here are entire functions of exponential type. A general representation is determined for solutions of homogeneous and inhomogeneous equations and of systems of such equations in the form of absolutely convergent series in entire functions (as a rule, exponentials forming an absolutely representing system). A criterion is established for solvability of a system of two inhomogeneous convolution equations with a single unknown function. The main results are obtained with the help of nontrivial expansions of zero in convex domains with respect to functions forming an absolutely representing system. Bibliography: 19 titles.

Imbedding theorems are established for classes of functions whose derivatives belong to various symmetric spaces. The functions studied can fail to vanish on the boundary of the domain. A class of domains is singled out for which the imbedding theorems have the same form as for function spaces satisfying a zero boundary condition. Bibliography: 20 titles.

Asymptotic expansions are studied for integrals of holomorphic forms over cycles in the Milnor fibration close to a critical level surface of a holomorphic function. Some regular features are found for the set of exponents in such expansions, and computational results are given. Asymptotic expansions are also considered for real integrals connected with analytic functions. A connection is established between expansions of integrals in a complex region and a real region. Bibliography: 30 titles.

Properties of cyclic -subgroups of order 4 in finite groups are studied. A consequence of the results is the Corollary.Suppose that the 2-group is a -subgroup of a finite group , and that is a simple group of characteristic 2 type. Then either is elementary, or , , , , , , , , , , or . Bibliography: 13 titles.

This paper investigates the arithmetic properties of the values of certain hypergeometric functions at points in an imaginary quadratic field I. Under certain conditions it is proved that these values are linearly independent over I. An effective lower bound is given for a linear form, which contains the maximum modulus of its coefficients. An estimate is also given which depends on the bounds within which each of the coefficients varies. Bibliography: 11 titles.

A smooth -dimensional manifold equipped with the structure of a tangent pseudovector bundle over a certain smooth -dimensional base manifold is studied in this paper from a local point of view. Under the assumption that a special affine connection is given in , a general path space is obtained. The method used here is based on the isotropy groups of the first and second kind and the choice of special systems of coordinates. Bibliography: 22 titles.

Let be the set of compact (i.e., contained in some compact subgroup) elements of a topological group , and let be its closure. The following assertions are proved: Theorem 1.A compact connected semisimple Lie group has a free dense subgroup each of whose nonidentity elements is a generator of a maximal torus in .Theorem 2.Suppose that a connected Lie group has no nontrivial compact elements in its center and coincides with the closure of its commutator group, and let be its Lie algebra. The following conditions are equivalent: (i) . (ii) has a dense subgroup of compact elements. (iii) , where is a nilpotent ideal and is a semisimple compact algebra whose adjoint action on does not have a zero weight. (iv) , where is a nilpotent connected simply connected normal subgroup and is a semisimple compact connected subgroup whose center acts (by conjugations) regularly on .Corollary.A locally compact connected group that coincides with the closure of its commutator group has a dense subgroup of compact elements if and only if . Bibliography: 16 titles.