Showing papers in "Russian Mathematical Surveys in 1972"

Gibbs measures in ergodic theory

Abstract: In this paper we introduce the concept of a Gibbs measure, which generalizes the concept of an equilibrium Gibbs distribution in statistical physics. The new concept is important in the study of Anosov dynamical systems. By means of this concept we construct a wide class of invariant measures for dynamical systems of this kind and investigate the problem of the existence of an invariant measure consistent with a smooth structure on the manifold; we also study the behaviour under small random excitations as . The cases of discrete time and continuous time are treated separately.

Limit-compact and condensing operators

Abstract: The paper contains a survey of investigations concerned with three new concepts: limit-compact operators, measures of non-compactness, and condensing operators. A measure of non-compactness is a function of a set that is invariant under the transition to the closed convex hull of the set. If a certain measure of non-compactness is defined in a space, a condensing operator is defined, roughly speaking, as an operator that decreases the measure of non-compactness of any set whose closure is not compact. The more general concept of a limit-compact operator is defined by means of a property common to all condensing operators; it can be formulated in terms not related to measures of non-compactness. The theory of limit-compact operators can be regarded as a simultaneous generalization of the theory of completely continuous and contracting operators. For non-linear operators the main result is the construction of the theory of the rotation of limit-compact vector fields and, in particular, the proof of a number of new fixed-point principles (Chapter 3 of the present paper). In the theory of linear operators a number of results are obtained that are related to the concept of a Fredholm operator and the Fredholm spectrum of an operator (Chapter 2). The theory of measures of non-compactness and condensing operators has found different applications in general topology, in the theory of ordinary differential equations, functional-differential equations, partial differential equations, the theory of extrema of functionals, etc. The paper contains several examples concerning differential equations in a Banach space and functional-differential equations of neutral type. These examples do not have a special significance but are chosen merely to illustrate the methods. They are therefore investigated with neither maximal generality nor completeness.

Lectures on bifurcations in versal families

Abstract: Речь идет о приложениях идей и методов теории особенностей гладких отображений к анализу бифуркаций в локальных задачах теории обыкновенных дифференциальных уравнений. Преимущество общей точки зрения теории особенностей состоит, главным образом, в том, что она указывает nравильную постановку вопросов; к их решению сделаны лишь nервые шаги, о которых и рассказывалось в лекциях.

Non-linear potential theory

Abstract: CONTENTSIntroduction Part I § 1. The space § 2. -potentials of generalized functions with finite -energy § 3. The maximum principle. A generalization of a theorem of Evans-Vasilesko. Lemmas on sequences of potentials § 4. The -capacity of a compact set. The capacity potential § 5. The capacity potential of an analytic set. The measurability of an analytic set relative to -capacity and -complete functions § 6. An estimate of the potential in terms of the modulus of continuity of the measure § 7. Metric properties of sets of zero -capacity § 8. Use of Bessel potentials (the case ). The capacity § 9. Guide to the literature Part II § 1. Auxiliary information on the spaces § 2. Generalized functions with finite -energy and -potential § 3. The maximum principle. A generalization of a theorem of Evans-Vasilesko § 4. The -capacity of a compact set. The capacity potential § 5. The capacity potential of an analytic set. Measurability of analytic sets relative to -capacity and -complete functions § 6. An estimate of the potential in terms of the modulus of continuity of the measure § 7. Metric properties of sets of zero -capacity § 8. Use of Bessel potentials (the case ). The capacity References

Minkowski duality and its applications

Abstract: This article is an account of problems grouped around the concept of Minkowski duality - one of the central constructions in convex analysis. The article consists of an introduction, four sections, and a commentary.In § 1 we set out the main facts about H-convex elements and introduce the Minkowski-Fenchel and the Minkowski-Moreau schemes; we consider the space of H-convex sets. Here we collect together the main examples, namely the convex and sublinear functions, and the stable, normal, and convex sets in the sense of Fan, amongst others. § 2 is concerned mainly with representations of positive functionals over continuous H-convex functions and sets. Here we also establish the links between such constructions and the Choquet theory.In § 3 we introduce various characterizations of H-convexity in the form of theorems on supremal generators. In particular, we consider in detail theorems on the definability of convergence of sequences of operators in terms of their convergence on a cone. Other applications of supremal generators are also given.In § 4 problems of isoperimetric type (with an arbitrary number of constraints) in the geometry of convex surfaces are analyzed as problems of programming in a space of convex sets. We examine as particular examples exterior and interior isoperimetric problems, the Uryson problem, and others.

Five Lectures on Three-Dimensional Varieties

Abstract: This article is a reproduction of a course of five lectures delivered in the seminar on algebraic geometry at the Moscow State University. The lectures are devoted to an account of the immense progress that has recently been made in the theory of three-dimensional varieties. Unfortunately these results have been accessible to Soviet mathematicians only in the form of announcements. While these lectures contain a solution of Luroth's problem (see Lecture 3), their main object is to make extensive use of the Abel-Jacobi mapping for families of curves on three-dimensional varieties.

Differential operators on a cubic cone

Abstract: Consider in the space C with the coordinates x{ , x2, x3 the surface X defined by the equation x\ + x\ + x\ = 0. We prove the following theorem: T H E O R E M 1. Let D{X) be the ring of regular differential operators on X, and Da the ring of germs at the point 0 of analytic operators on X. Then 1°. the rings D(X) and Da are not Noetherian; 2°. for any natural number k the rings D{X) and Da are not generated by the subspaces Dk (Dak, respectively) of operators of order not exceeding k. In particular, the rings D{X) and Da are not finitely generated. Theorem 1 answers questions raised in Malgrange's survey article [ 1 ] . The ring D(X) has an interesting structure (see Proposition 1). We denote by E(X) the ring of regular functions on X{E(X) = C[x,, x2, x3]/[x\ + x\ + x 3 3]) and by D(X) the ring of regular differential operators on X. By Dk we denote the space of operators of order not exceeding k. Setting ax{f){x) = /(λχ) and b\(3))(f) ~ aK(DaX-x(f)) for λ e C* we define an action of the group C* oo in the spaces E{X) and D(X). It is clear that E{X) = © E(X), where E{X) is the finite-dimensional space of homogenous functions of degree i on X. We call an operator 3 e D(X) homogenous of degree i (i e Z) if b%{3) = V-Sb for all λ e C* (equivalent definition: 2b {E {X)) C E(X) for all n). We denote by D' the space of all such operators and set D\ = D' η Dk.

The cohomology of abelian varieties over a number field

Abstract: This article is a survey of results on the arithmetic of Abelian varieties that have been obtained by cohomological methods. It consists of an Introduction and six sections. In the Introduction the main facts to be proved in the article are stated. They are concentrated around two arithmetical problems: the determination of the rank of an Abelian variety over a number field and the related problem of the structure of the group of locally trivial principal homogeneous spaces (the Tate-Shafarevich group); also the investigation of the behaviour of points of finite order on an Abelian variety and the related problem of divisibility of principal homogeneous spaces.The first section recalls the proofs of the necessary facts from the Galois cohomology of finite modules. The basic results relating to the first of the problems mentioned are proved in §§ 3-4. The fifth and sixth sections are devoted to the problem of the divisibility of points and of principal homogeneous spaces; a certain cohomological finiteness theorem is also proved here.

REPRESENTATION OF FUNCTIONS BY SERIES AND CLASSES ϕ(L)

Abstract: In the paper we give a survey of results concerning the problems of representing functions by series in various systems, and of the uniqueness of such a representation. The account covers older results as well as those of recent years due to several authors. Much attention is also paid to questions of the validity of the Weierstrass theorem and of representing functions in the classes (L).

Convexity of values of vector integrals, theorems on measurable choice and variational problems

Abstract: We give an account of applications of measurable many-valued mappings and theorems on convexity of finite-dimensional vector integrals to several variational problems. Theorems on convexity are carried over to vector integrals with values in function spaces, and with the help of these we obtain a maximum principle as a necessary and sufficient extremum condition and an existence theorem for a non-linear variational problem with operator constraints of integral equality type, similar to Monge's problem on mass displacement.

Partial differential inequalities

Abstract: ContentsIntroductionChapter 1. Variational elliptic inequalities ??1.1. First example ??1.2. Variational inequalities ??1.3. Variational inequalities and many-valued operators ??1.4. Examples ??1.5. The method of penalization ??1.6. The method of regularization ??1.7. Interpretation of variational inequalities and multipliers ??1.8. Regularity ??1.9. Comparison and convexity properties of solutions Chapter 2. Parabolic evolution inequalities (I) ??2.1. First examples ??2.2. The variational formulation. The strong solution ??2.3. The variational formulation. The weak solution ??2.4. Examples ??2.5. The method of elliptic regularization ??2.6. The method of penalization ??2.7. Some generalizations ??2.8. Quasi-stationary problems ??2.9. Regularity ??2.10. Remarks Chapter 3. Parabolic evolution inequalities (II). The flow of a non-Newtonian fluid ??3.1. Statement of the problem ??3.2. The main results ??3.3. The principle of the proof Chapter 4. Inequalities of the second order with respect to?t ??4.1. First example ??4.2. The general variational problem ??4.3. Examples ??4.4. Methods for existence proofs Chapter 5. Variational inequalities for Maxwell operators ??5.1. Statement of the problem ??5.2. Results Chapter 6. Other problems ??6.1. Coupled phenomena ??6.2. A problem with a decay ??6.3. Behaviour at infinity. Almost periodic solution Chapter 7. General remarks on the numerical approximation of solutions of inequalities ??7.1. Discrete schemes for evolution inequalities ??7.2. Analysis of stability ??7.3. The solution of stationary problems Chapter 8. Singular layer phenomena ??8.1. Generalities ??8.2. An example Chapter 9. Open problems References

Integral representation of excessive measures and excessive functions

Abstract: One of the central results of classical potential theory is the theorem on the representation of an arbitrary non-negative superharmonic function in the form of a sum of a Green's potential and a Poisson integral. We obtain similar integral representations for the excessive measures and functions connected with an arbitrary Markov transition function. Many authors have studied the homogeneous excessive measures connected with a homogeneous transition function. We begin with the inhomogeneous case and then reduce the homogeneous case to it. The method proposed gives a considerable gain in generality. The investigation is carried out in the language of convex measurable spaces and in contrast to previous papers no topological arguments are used. Our basis are the results obtained in [3] (also without topology) on the integral representation of Markov processes with a given transition function. For the reduction of the homogeneous case to the inhomogeneous we use a theorem from the theory of dynamical systems due to Yu. I. Kifer and S. A. Pirogov (see the Appendix at the end of this paper).

The convergence of distributions of functionals on stochastic processes

Abstract: The first part of the paper contains a survey of the present state of research in the general problems of convergence of stochastic processes. An important place is given to theorems on weak convergence of distributions on metric spaces. Then a different approach is proposed to the study of convergence of distributions of functionals on stochastic processes, which is connected with the approximation of the trajectories of a process by some family of functions. We think of approximation in terms of the nearness of the functionals in question. Using this approach we obtain all the main results on convergence in specific function spaces that are known at present. These results are obtained in their most general form without the requirement that the limiting processes should belong to the space under discussion. New limit theorems are also obtained and among them theorems for processes with discontinuities of the second kind and others.

Absolute convergence of fourier series with respect to complete orthonormal systems

Abstract: This article gives a survey of results on absolute convergence of Fourier series with respect to the trigonometric system, the Haar system and arbitrary complete orthonormal systems. It clarifies which of the propositions on absolute convergence of trigonometric Fourier series and Fourier-Haar series remain valid for arbitrary complete orthonormal systems.

The cauchy problem and other related problems for convolution equations

Abstract: Spaces of generalized functions with exponential asymptotic behaviour are considered. Convolutors in these spaces are completely described. It is shown that a convolution equation is uniquely soluble if and only if there exists a fundamental solution that is a convolutor. The explicit description of convolutors renders this condition effective. In particular, Petrovskii's correctness condition is obtained in the case of differential equations. A calculus of pseudodifferential operators with inhomogeneous symbols of constant strength is constructed; the solubility of the Cauchy problem can be proved by means of this calculus for a certain class of differential equations with variable coefficients.

Notes on mathematical economics

Abstract: These notes are based on individual lectures of a course on mathematical economics given by the author in the autumn of 1971 in the Faculty of Mathematics and Mechanics of Moscow State University. § 1 describes the properties of neoclassical production functions and types of technological progress, Ramsey's model (distribution of income between consumption and accumulation), and on the basis of a simple example it is shown how Pontryagin's maximum principle is used to find an optimum plan. The exposition is based essentially on the material of [40]. In § 2, following Debreu, the author considers models of pure exchange (without production) and explains the structure of sets of equilibrium states in them; in this analysis, considerable use is made of Sard's lemma on regular values of smooth mappings, and simple considerations on the indices of vector fields. These notes pursue limited methodical aims; they are intended for mathematicians and economists who show a reserved optimism about the use of mathematical methods in economics.

the Development of Higher Education in the U.S.S.R. during the First Three Five-Year Plans

Abstract: In this article we consider questions connected with the reorganization of higher schools (in particular, of technical higher schools) from the July 1928 and November 1929 Plenary Sessions of the All-Union Communist Party (Bolsheviks) until the Second World War. It was in this period that our high school system took on the form that still exists today. In the subsequent periods there was an increase in the number of students and of colleges, both in the evening class and in the correspondence sections, the teaching syllabuses were brought into line with the new achievements in science and technology. But the teaching methods have remained unchanged.

Some lattice-theoretical properties of groups and semi-groups

Abstract: During the period following the publication of the survey [5] a number of new papers appeared in which connections between the structure of an algebraic system (a group, a semigroup or a topological group) and the lattice of its subsystems (subgroups, subsemigroups, closed subgroups) are studied.In a sense the present article is a continuation of [5], although its style differs somewhat in that it includes fragments of proofs of the most interesting facts.It also considers other lattices similar to the subgroup lattice of a discrete group. Accordingly it contains five sections studying the subgroup lattice of infinite groups (§ 1), the subsemigroup lattice of these groups (§ 2), the subsemigroup lattice of a semigroup (§ 3), the subgroup lattice in groups with various finiteness conditions (§ 4), and finally the lattice of closed subgroups of a topological group (§ 5). All the definitions necessary for an understanding of the new results are given here. Definitions of other concepts that are already known well-enough can be found in [5] or in Kurosh's book [4].The authors have tried to examine all the available relevant literature; this is listed at the end of the article. Titles cited in [5] are repeated here only when they are directly referred to in the text in connection with new results not mentioned in [5].