Table of contents

In our paper On asymptotic "eigenfunctions" of the Cauchy problem for a nonlinear parabolic equation (Mat. Sb. 126 (1985), 435-472 = Math. USSR Sb. 54 (1986), 421-455) the exponent of in formula (2.3) must be . The same exponent is used in (2.12') and in the construction of the upper and lower solutions before this formula. The factor was omitted from the right side of (3.3), and the lower estimate (3.4) (not used in the sequel) should read

in . The assumption "" in the first paragraph in the proof of Theorem 8 should be omitted, as well as the quantity "" in the following display.

This paper is devoted to the study of uniform quasiasymptotics of the solution of the second mixed problem in , , and of the Cauchy problem () for the linear hyperbolic equation

with initial conditions

A criterion for the existence of quasiasymptotics of the solution of order is established under the assumption that the function has quasiasymptotics of order and with a certain condition of "isoperimetric type" on the class of domains considered. Bibliography: 13 titles.

For general nonhomogeneous linear equations with one regular singular point, the questions of solvability, the structure of the corresponding Wronskians, and the Cauchy-Green functions are studied on a finite interval. Formulas for the resolvents of the resulting integral equations are obtained in explicit form. An effective method of constructing differential equations for the analytic parts of special functions is indicated, and an analytical method is worked out for their best asymptotic approximation, which at the same time gives a high degree of accuracy in their practical computation on computers. Bibliography: 17 titles.

The following question is studied. Suppose one is given a 2n-dimensional compact complex manifold with holomorphic symplectic 2-form. Are there obstructions to the existence of n independent meromorphic first integrals in involution, and if so, what are they like? The answer to this question is given for K3 surfaces, Beauville manifolds, and complex tori; in these cases there are obstructions of an analytic character. Whether there are any topological obstructions is an unsolved problem. Bibliography: 18 titles.

This paper is concerned with the questions of solvability and smoothness of weak solutions of the first boundary value problem for a parabolic equation of the form

Bibliography: 18 titles.

Let be the Hardy class in the unit disc and a finite Borel measure in . Carleson's theorem describes conditions on under which the corresponding imbedding operator (the Carleson operator) is bounded. From this theorem follows a criterion for compactness of in terms of . This paper is devoted to further study of the Carleson operator. Almost sharp upper bounds on the singular numbers of are presented in terms of the intensity of . For measures whose support is a set of nonzero linear measure adjacent to the unit circle (and when certain other conditions), an asymptotic formula is obtained. A study is begun of measures whose support has just one point on the unit circle. A solution of a problem from the theory of rational approximation, posed by A. A. Gonchar, is also presented. Bibliography: 17 titles.

In this paper the duality of multiobjective problems is studied with the help of the apparatus of conjugate multivalued mappings introduced by the author. A duality theory, apparently of independent interest, is first developed for multivalued mappings. This theory is then used to get dual relations in multiobjective problems. Bibliography: 19 titles.

A functor algebraically dual to the operator -functor is constructed on the category of -algebras, and the author shows that it defines a homology theory on this category. The author also proves that it coincides with Kasparov's homology -functor on a large class of -algebras, including commutative -algebras. This functor is used to describe a class of homotopy invariant higher signatures. Bibliography: 10 titles.

A new method of asymptotic integration is developed - the method of regularization - in the case when the spectrum of the variable limit operator is zero at isolated points. To describe the singular dependence of a solution on the perturbation, additional independent variables are introduced; the space of resonance-free solutions is introduced, in which the coefficients of regularized kind (the solution of the extended problem) are defined. Asymptotic convergence of the series thus obtained to the exact solution of the original singularly perturbed problem is proved. Bibliography: 14 titles.

Explicit formulas are obtained which express the images of nonhomogeneous theta series of arbitrary integral nondegenerate quadratic forms in an even number of variables under the action of Hecke operators as linear combinations of theta series of the same type. Bibliography: 4 titles.

Conditions are obtained of the type in Tikhonov's theorem which, when satisfied, make possible passage to the limit on the small parameter . Estimates in Hölder spaces of functions are obtained for the solution of the problem. The author determines how the rate of convergence of the solution to the limit function as depends on the smoothness of the functions contained in the equations and the boundary and initial conditions. Cases of both finite and infinite time intervals are considered. Bibliography: 14 titles.

Let be a sequence of points in the complex plane, and a sequence of positive numbers. Problem: under what relations between and can any function in be approximated in the uniform norm by finite linear combinations of exponentials with the coefficient restriction ? Here depends only on . An exact solution of the problem is given under the assumption that . Bibliography: 26 titles.

This article deals with the solution of certain old problems in descriptive set theory and topology connected with the structural properties of the Borel sets in the space of irrational numbers. At the center of this circle of problems is the question of L. V. Keldysh on the universality of an element strictly of class . The author's solution is based on the principle of determinancy, which, as Martin proved, is satisfied for Borel sets. Closely connected with the question of Keldysh are certain problems considered by Lusin, Aleksandrov and Urysohn, and others. In this paper an answer is given, in particular, to a question posed by them on the number of irreducible Borel sets in each Borel class (a Borel set is said to be irreducible if any nonempty open-closed subset of it is indistinguishable from the ground set by some classification of the Borel sets). Bibliography: 23 titles.

The maps of Wall groups of finite abelian 2-groups obtained on passing to a double covering and the maps of Wall groups induced by the embedding of a subgroup of index 2 are computed. Bibliography: 7 titles.

The classical theorem of Poincaré on recursion relations is generalized. As the main application, a conjecture of Gonchar is proved for the case of the mth row of the multipoint Padé approximants of a function holomorphic in some neighborhood of a given continuum. Bibliography: 15 titles.

For entire functions given by Dirichlet series

absolutely convergent in some results are proved which give best possible, or close to best possible, conditions sufficient for the relation

as outside some set, where is the central index of the Dirichlet series. Bibliography: 4 titles.

For intuitionistic provability calculus obtained from the intuitionistic propositional calculus by adjoining to the postulates of the latter the axioms , , and , an algebraic proof is given of the separation property: if and only if there exists a derivation of formula whose terms contain only those connectives that occur in . The proof is achieved by constructing an (isomorphic) embedding of pseudo-Boolean algebras, and on this basis then constructing embeddings, into -pseudo-Boolean algebras, of algebras whose classes approximate corresponding fragments of the calculus . Bibliography: 14 titles.