Showing papers in "Russian Mathematical Surveys in 1971"

On matrices depending on parameters

Abstract: 1. Классические результаты (Аполлоний, Декарт, Ньютон, Харнак). Шестнадцатая проблема Гильберта.

Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces

Abstract: This paper is based on the papers, written mainly during the last decade, on the investigation and solution of boundary value problems in the theory of analytic functions on finite oriented Riemann surfaces. In the introduction we give a short survey of the fundamental work on this topic, beginning with the classical results of Riemann and right up to the research of contemporary authors. The main content of this paper consists of the material presented in §§ 2-6. Here we find explicit expressions for analogues to the Cauchy kernel, we construct the general solution, and give a complete sketch of the solubility of Riemann's boundary value problem for a single unknown piecewise meromorphic function in the case of composite contours on a closed oriented Riemann surface. In this context we give a new version for the solution of Jacobi's inversion problem. In §§ 7 and 8 we consider the case of Riemann surfaces of algebraic functions, we investigate the hyperelliptic case in detail, and we give applications. § 9 is devoted to boundary value problems on Riemann surfaces with boundary. We present the ideas of the method of passage to the double and the method of pasting. In § 10 we give a survey of results on the Hilbert boundary value problem for multiply-connected domains and we mention some new results of the author. In § 11 we give a survey of the literature on our topic that is not treated in the main part of the paper.

Homological methods in the theory of locally convex spaces

Abstract: This paper deals with the categories of direct and inverse countable spectra of locally convex spaces and with the functors of inductive and projective limits defined on these categories. We study the homological properties of such functors, introduce their satellites, and search for conditions for these satellites to vanish. We then apply the accumulated information about the functors of the limiting processes to certain problems in the theory of locally convex spaces: topological properties of a locally convex inductive limit, the homomorphism of the adjoint operator, the possibility of extending and lifting a map and the properties of the augmentation functor. We also consider examples of certain "pathologies".

Cyclotomic fields and modular curves

Abstract: The first chapter of this article contains an exposition of the work of Iwasawa and Mazur on the arithmetic of Abelian varieties over cyclotomic fields. The study of questions arising here leads us in the second chapter to the use of the zeta-function apparatus, and the conjectures of Weil and Birch - Swinnerton-Dyer; this permits us to obtain conditional formulae for the order of the Tate-Shafarevich group.

The stationary phase method and pseudodifferential operators

Abstract: In the paper asymptotic expansions are calculated for integrals of rapidly oscillating functions, in which , and are smooth functions, and is real-valued. The results obtained serve to develop a calculus of pseudodifferential operators and generalizations of them, the Fourier integral operators.

Basic concepts and theorems of the evolutionary genetics of free populations

Abstract: It is well known that the principles of biological inheritance, initiated by Mendel in 1865, allow of an exact mathematical formulation For this reason classical genetics can be regarded as a mathematical discipline This article is concerned with the direction in mathematical genetics that stems from the widely known papers of Hardy and Weinberg (1908) It scarcely touches upon purely probabilistic and statistical questions, but uses probabilities (mean values of frequencies) as state coordinates in an "infinitely large" population Change of state (evolution) occurs under the action of a certain quadratic operator The paper has two aspects: 1) the structure of free populations; 2) the behaviour of trajectories The fundamental investigations on these problems were carried out by S N Bernstein (1923-1924) and Reiersol (1962) Certain additional results directed towards completing the theory have been found recently by the author and are published here for the first time At the beginning of the paper we give a short sketch of the basic notions of classical genetics, in essence simply a minimal glossary The reader who is familiar with the elements of genetics to the extent, for example, of the popular tract of Auerbach [1] or the appropriate chapters of the textbook by Villee [2], could omit this sketch For a deeper study of the biological material the books of McKusick [3], Stern [4] and Mayr [5] are recommended The elementary mathematical questions of genetics are concerned with certain guiding principles in probability theory (see, for instance, [6]-[8]) The textbooks and monographs [9]-[15] are devoted to mathematical genetics The sources listed here apply but little to the problems of the present work The main results are concentrated in §§ 4, 5, 9, 11 The remaining sections play an auxiliary role

On discrete models of the kinetic boltzmann equation

Abstract: In this paper we discuss on simple models the connection between the kinetic equations and the equations of gas dynamics. We analyze the energy integrals of the gas dynamical model and elucidate their connections with Boltzmann's H-theorem.

Geometric theory of banach spaces. part ii. geometry of the unit sphere

Abstract: Interest in a geometrical approach to the study of Banach spaces is due to the following circumstance. Banach spaces have rich linear topological properties, which are extremely convenient in applications. However, the definition of a -space is inseparably linked with a norm, that is, with a fixed geometrical object - the unit ball , whereas the linear topological properties depend (by definition) only on the topology of the space, that is, on a class of bounded convex bodies. Thus, we are naturally led to the question: what can be said about the linear topological properties of a space in isometric terms, that is, whilst remaining within the framework of a given norm? The possibility of a productive investigation in this direction is essentially an infinite-dimensional situation, since in the finite-dimensional case the linear topology of a space is uniquely determined by the dimension. In view of the simplicity of the topological properties of n-dimensional spaces, the aim and fundamental object of investigation are geometrical (for example, the geometry of convex bodies). In the infinite-dimensional case topological questions give rise to enough concern. In this paper I follow tradition and give the main attention to results that lie in the topological channel, although it seems to me that an intrinsic study of the geometric object (an infinite-dimensional convex body) is no less interesting.

Linear Problems of Complex Analysis

Abstract: This article attempts to give a linearized form of the basic theorems of complex analysis (the Oka-Cartan theory). With this aim we study simultaneously: a) the isomorphism problem for spaces of holomorphic functions and , ; b) the existence of a linear separation of singularities for the space , where , and () are holomorphically convex domains in a complex manifold , and, in a more general setting, the splitting of the Cech complex of a coherent sheaf over a holomorphically convex domain ; c) the existence of a linear extension for holomorphic functions on a submanifold , and more generally, the splitting of a global resolution of a coherent sheaf. In several cases (for strictly pseudoconvex domains) these questions can be answered affirmatively. The proofs are based on the theory of Hilbert scales and bounds for solutions of the -problem in weighted -spaces. Counterexamples show that the same questions may also have negative answers.

Formal groups and their role in the apparatus of algebraic topology

Abstract: CONTENTSIntroduction § 1. Formal groups § 2. Cobordism and bordism theory § 3. The formal group of geometric cobordisms § 4. Two-valued formal groups and power systems § 5. Fixed points of periodic mappings in terms of formal groupsAppendix I. Steenrod powers in cobordisms and a new method of computing the bordism ring of quasicomplex manifoldsAppendix II. The Adams conjectureReferences

The initial and final behaviour of trajectories of markov processes

Abstract: The initial and final behaviour of the trajectories of Markov processes is studied within the theory of Martin boundaries. We propose a simpler approach, based on a direct investigation of the class of Markov processes with a given transition function and the class of Markov processes with a given cotransition function. In the class processes with the following property are distinguished: the probability of any event, determined by an arbitrarily small initial (final) section of a trajectory, is equal to 0 or 1. Every process of decomposes uniquely into such ergodic processes, and the corresponding measure completely describes the initial (final) behaviour of trajectories. The theory is invariant with respect to reversal of time. Based on the results of the present paper we shall study in a subsequent publication the excessive measures and excessive functions associated with a Markov process. A brief account of the main ideas of this work (for processes with non-random births and deaths) was given in the author's invited address at the International Congress of Mathematicians in Nice (1970).

Linear operators in hilbert spaces with g-metric

Abstract: In this article we present the basic facts of a general theory of linear operators in Hilbert spaces equipped with a Hermitian-bilinear metric. In particular, we investigate the spectra of operators of certain classes, most frequently encountered in applications.

On the theory of non-linear minimax problems

Abstract: This article is a survey of recent results on non-linear minimax problems. The following questions are considered: the directional differentiability of the maximum function; necessary conditions for a minimax and their geometrical interpretation; sufficient conditions for a local minimax; methods of successive approximation to find the stationary points of the maximum function; properties of the maximin function. These questions are set out first of all for the discrete case (Ch.?I) and then for the general case (Ch.?II); in the first chapter the accent is on methods of successive approximation, while in the second it is on the tie-up between the theory, as it has evolved, and certain classical results.

Crossed products of von neumann algebras

Abstract: The paper is concerned with von Neumann algebras with finite trace and their -automorphisms, and with crossed products. A detailed investigation is made of the problem of constructing hyperfinite factors of type II1 by means of crossed products. Some new results are obtained on subfactors of hyperfinite factors of type II1 and also some new information on the trajectory theory of measure-preserving transformations.

on the Solubility of Differential Equations with Simple Characteristics

Abstract: A survey is given of papers devoted to the problem of the existence of solutions to linear differential and pseudo-differential equations of principal type. The main results in this field are due to Lewy, Hormander, Nirenberg, Treves and the author. We also give a new theorem of maximal generality on local solubility of equations of principal type. By way of illustration to the exposition we mention as examples: the Lewy operator; the operator arising from the solution of the problem with directional derivatives for elliptic second order equations; non-singular operators.

The method of intrinsic boundary conditions in the theory of difference boundary value problems

Abstract: In this article we consider general systems of difference equations with constant coefficients in arbitrary many-dimensional network domains. We define the boundary of a network domain in a certain way and give a formula that expresses the values of a solution at each point of the network domain in terms of its values at points of the boundary. We use this formula to derive necessary and sufficient conditions, which we call 'intrinsic boundary conditions', for a vector function given on the boundary of a network domain to be extendable everywhere in this domain to a solution. This formula allows us to appreciate that it is natural to consider a general boundary value problem for the systems in question. The method we suggest for investigating and calculating solutions of difference boundary value problems consists in going from the original problem to the problem on the boundary that arises when we consider a combination of given and intrinsic boundary conditions. We present results obtained by the method of intrinsic boundary conditions. In the main they relate to non-stationary problems in simple and composite domains and have various degrees of effectiveness. We refer in the paper to other methods only to appreciate the position of the new method among those already available; it interacts with them and supplements them.

On local smoothness of generalized and hypoellipticity of second order differential equations

Abstract: The paper gives a survey of results on hypoellipticity of second order differential operators. In particular, our new theorems together with earlier results give a necessary and sufficient condition for hypoellipticity of a general second order differential operator with analytic coefficients, provided that it is not fully degenerate at any point of the domain. We dedicate this paper to I. G. Petrovskii on the occasion of his seventieth birthday.

Asymptotic behaviour of the spectral function of an elliptic equation

Abstract: In this paper we consider the asymptotic behaviour of the spectral function of an elliptic differential (pseudodifferential) equation or system of equations. For the case of differential operators this problem has been widely studied, and various methods have been developed for its solution (see [1] and [2] for a survey of these methods). We consider in this paper just one of these methods. It is based on a study of the structure of the fundamental solution of the Cauchy problem for a hyperbolic differential (pseudodifferential) equation. The method we use is called "the method of geometrical optics". For a system of first order differential equations it was originally developed in detail by Lax [8], and for pseudodifferential equations by Hormander [2] and independently by Eskin [17], [18] and Maslov [19]. In [2] Hormander also investigates the asymptotic behaviour of the spectral function for an elliptic pseudodifferential first order operator. Using some important results of Seeley [5] one can then derive the asymptotic behaviour of the spectral function of an elliptic differential operator of arbitrary order. Similar methods have previously been applied by the author in [3], [4] for second order elliptic differential operators. In this article we give a partial account of the results of [8] and [2]. We also present some new results due to the author, concerning both the structure of the fundamental solution of the Cauchy problem and the asymptotic behaviour of the spectral function.

The parametrix of elliptic operators with infinitely many independent variables

Abstract: Among the differential equations with infinitely many independent variables most attention has been paid to second-order parabolic equations. The paper begins with a brief review of the results obtained for the Cauchy problem for such equations. The parametrix is constructed for a certain class of elliptic differential operators with infinitely many variables. This parametrix is generated by a measure in the corresponding function space. A composition formula for an elliptic differential operator and its parametrix is proved. Applications are given to the theory of the solubility of elliptic equations with infinitely many variables.