Showing papers in "Russian Mathematical Surveys in 1970"

Dynamical systems with elastic reflections

Abstract: In this paper we consider dynamical systems resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second quadratic form is negative-definite at each point of the boundary, where the boundary is taken to be equipped with the field of inward normals. We prove that such systems are ergodic and are K-systems. The basic method of investigation is the construction of transversal foliations for such systems and the study of their properties.

The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms

Abstract: In 1964 Kolmogorov introduced the concept of the complexity of a finite object (for instance, the words in a certain alphabet). He defined complexity as the minimum number of binary signs containing all the information about a given object that are sufficient for its recovery (decoding). This definition depends essentially on the method of decoding. However, by means of the general theory of algorithms, Kolmogorov was able to give an invariant (universal) definition of complexity. Related concepts were investigated by Solomonoff (U.S.A.) and Markov. Using the concept of complexity, Kolmogorov gave definitions of the quantity of information in finite objects and of the concept of a random sequence (which was then defined more precisely by Martin-Lof). Afterwards, this circle of questions developed rapidly. In particular, an interesting development took place of the ideas of Markov on the application of the concept of complexity to the study of quantitative questions in the theory of algorithms. The present article is a survey of the fundamental results connected with the brief remarks above.

On small random perturbations of dynamical systems

Abstract: In this paper we study the effect on a dynamical system of small random perturbations of the type of white noise: where is the -dimensional Wiener process and as . We are mainly concerned with the effect of these perturbations on long time-intervals that increase with the decreasing . We discuss two problems: the first is the behaviour of the invariant measure of the process as , and the second is the distribution of the position of a trajectory at the first time of its exit from a compact domain. An important role is played in these problems by an estimate of the probability for a trajectory of not to deviate from a smooth function by more than during the time . It turns out that the main term of this probability for small and has the form , where is a certain non-negative functional of . A function , the minimum of over the set of all functions connecting and , is involved in the answers to both the problems. By means of we introduce an independent of perturbations relation of equivalence in the phase-space. We show, under certain assumption, at what point of the phase-space the invariant measure concentrates in the limit. In both the problems we approximate the process in question by a certain Markov chain; the answers depend on the behaviour of on graphs that are associated with this chain. Let us remark that the second problem is closely related to the behaviour of the solution of a Dirichlet problem with a small parameter at the highest derivatives.

Embeddings and immersions in Riemannian geometry

Abstract: This article is a significantly expanded version of a paper read by one of the authors to the Moscow Mathematical Society [18]. It consists of two chapters and eleven appendices. The first chapter contains a survey of known results, as a rule with precise statements and references, but without full proofs. In the second chapter, the fundamental embedding theorems are set out in detail, among them some new results concerning a bound for the smallest dimension of a euclidean space in which any compact riemannian manifold of given dimension can be embedded, and also to the corresponding local problem. In Appendices 1, 3, 4, 5, 7, 8 and 9, proofs of miscellaneous propositions in the survey are given. Appendices 2 and 6 are needed for other appendices, but are interesting in their own right. In Appendix 10 a more general embedding problem is considered. Appendix 11 has a bearing on Chapter 2 and is of an analytic nature. We wish to thank I. Ya. Bakel'man, A. L. Verner, Yu. A. Volkov, S. P. Geisberg, V. L. Eidlin and Ya. M. Eliashberg for their help in the preparation of the article. We are especially grateful to Yu. D. Burago who at our request modified one of his inequalities to suit our requirements (Appendix 2).

On polyanalytic functions

Abstract: Polyanalytic functions of a complex variable (a class which is one of the most important natural generalizations of that of analytic functions) have been the subject of a considerable number of papers in recent years. The present paper is a fairly complete survey of results obtained so far in this branch of complex analysis.

RESONANCE THEOREMS and SUPERLINEAR OPERATORS

Abstract: This article is concerned with problems of the metrical theory of functions. We establish criteria for systems of measurable functions to be systems of convergence in measure, systems of convergence almost everywhere, etc. The exposition is based on a unified method, using superlinear operators.

The homotopy structure of the linear group of a Banach space

Abstract: The homotopy type of the linear group of an infinite-dimensional Banach space is as important in the theory of Banach manifolds and bundles as is the (stable) homotopy structure of the orthogonal and unitary groups in the theory of finite-dimensional vector bundles and in -theory (for more details see [4]). Kuiper has proved [20] the contractibility of the linear group of a Hilbert space , and Neubauer has given a positive answer [34] to the question of the contractibility of , , and . At the same time there are examples (the first of which was given by Douady [11]) of Banach spaces with a non-contractible and disconnected linear group. In [30] the author drew attention to the fact that the constructions of Kuiper and Neubauer could be formalized to provide a general procedure for proving (or analysing) the contractibility of the linear group . This enables us to settle the question of the homotopy structure of the linear groups of many specific Banach spaces. The present paper reviews the results that have been obtained up till now on the contractibility of the linear group of Banach spaces. In § 1 examples are given of Banach spaces with homotopically non-trivial linear groups. The general procedure for analysing the contractibility of , Theorem 1, is set out in § 2, and the problem of obtaining explicit analytic conditions necessary for the applicability of this procedure is solved in § 3. In §§ 4-6 examples are given of many specific Banach spaces (of smooth and of measurable functions), and the contractibility of their linear groups is proved. § 7 contains remarks on the general procedure and unsolved questions.

Programming a computer to play chess

Abstract: This paper presents an account of the mathematical foundations of the construction of game playing programs and methods of putting them on computers. We give a description of the principles used in organizing and processing information in the chess programs devised by the authors during the years 1961-6 for the electronic computer M-20. We also give some games played according to these programs.

Geometric theory of Banach spaces. Part I. The theory of basis and minimal systems

Abstract: In recent years substantial success has been achieved in the study of geometric and linear topological properties of Banach spaces (B-spaces). Our aim is to present some of the results that have been obtained since the appearance of the well-known survey of the geometric theory of B-spaces, the monograph of M. M. Day "Normed Linear Spaces". The term "geometric theory" which we use is to a large extent conventional. At present the principal method of investigating B-spaces is to study special sequences of elements of a space; this is more reminiscent of the methods of analysis than of geometry. In the first part of the survey we give an account of the apparatus of the theory of sequences and demonstrate its potential in investigating topological properties of Banach spaces. At the same time a general look at the whole host of facts, which make the current theory so rich, becomes possible in the study of the geometric structure of the unit sphere, that is, of the isometric properties of a space. This approach to the investigation of Banach spaces will be developed in a second part. These two parts do not exhaust the contemporary theory of normed linear spaces, which consists of at least two other large branches: the finite-dimensional Banach spaces or Minkowski spaces and the investigation of isomorphisms and embeddings. Each of these domains has recently received a fundamental stimulus to its development. It is sufficient for the reader of the present article to be acquainted with the elements of functional analysis as given in Chapters 1-5 of [72] or in Chapters 1-4 of [45]. We shall omit the proofs of statements that are given in sufficient detail in the Russian literature, or that can be obtained by methods illustrated by other examples. In addition, we shall not mention proofs that would lead us away from the exposition of the method.

The development and applications of the asymptotic method of lyusternik and vishik

Abstract: The asymptotic method of Lyusternik and Vishik has now been greatly expanded and has found important applications in many fields of mechanics The principal virtues of the method lie in its conceptual simplicity and in the fact that it can be applied to broad classes of partial differential equations with a small parameter occurring in the highest derivatives The present paper has two chapters In the first the method is illustrated by the example of the degeneration into elliptical equations of elliptical equations of higher orders, and the basic papers in which the method is developed and extended are surveyed The cases when the boundary layer functions are defined by partial differential equations (parabolic, elliptic and hyperbolic boundary layers), and also the phenomenon of the interior boundary layer, are illustrated in greater detail Recommendations are made for the application of the Newton diagram method in boundary layer problems when there is a small parameter entering arbitrarily Among the most interesting applications of the method we note the problem of the passage past a muffled body of the ultrasonic flow of a viscous gas (Markov and Chudov); number of problems in the non-linear theory of thin flexible plates and shells (Srubshchik and Yudovich); and problems in the motion of a rigid body with cavities containing a viscous fluid (Moiseev, Chernous'ko, and others) The second chapter consists mainly of the author's results on differential equations in a Banach space containing a small parameter in the leading derivatives Here the Lyusternik-Vishik method has been very little applied so far despite its undoubted advantages in this type of problem Of special interest are the cases when the boundary problem is situated on the spectrum The results of the chapter can be applied, in particular, to systems of ordinary differential equations, to integral-differential equations, and also to parabolic equations

Metric properties of measure preserving homeomorphisms

Abstract: We study "typical" metric (ergodic) properties of measure preserving homeomorphisms of regularly connected cellular polyhedra and of some other spaces. In 1941 Oxtoby and Ulam proved (for a narrower class of spaces) that ergodicity is such a property. Using a modification of their construction and the method of approximating metric automorphisms by periodic ones, we prove in this paper that almost all properties that are "typical" for the metric automorphisms of the Lebesgue spaces are also "typical" for the situation under discussion.

the General Theory of Relaxation Processes for Convex Functionals

Abstract: This article sets out a theory of the convergence of minimization processes convex functionals that reduce the value of the functional at each step. A geometrical language, independent of the algorithmic structure, is used to describe the processes: the language of relaxation angles and factors. Convergence conditions are derived and the rate of convergence and stability of the process are studied in this terminology. Translation from the language of concrete algorithms to the geometrical terminology is not difficult, and thanks to this the theory has a wide area of applications: gradient and operator-gradient processes, processes of Newtonian type, coordinate relaxation, Jacobi processes and relaxation for the Rayleigh functional.

Geometric problems in quasilinear elliptic equations

Abstract: In their survey reports A. D. Aleksandrov and A. V. Pogorelov [1] and N. V. Efimov [2] give a detailed account of the deep relationships between the theory of surfaces and the theory of partial differential equations; they also highlight the main results and research problems on the boundary of geometry and analysis connected with Gaussian curvature of surfaces and its various generalizations. More recently problems on the boundary of geometry and the theory of quasilinear equations have been intensively investigated in various directions. In this article we shall be concerned with just two closely related problems from among the diverse questions that have been studied in this field: the geometric methods of estimating the solution to the Dirichlet problem for a quasilinear elliptic equation, and the construction of a non-parametric hypersurface with given mean curvature in Riemannian space. Even for the case of zero mean curvature (minimal surfaces) these questions present considerable difficulties. We shall not pay special attention to the minimal surface situation, but we draw attention to the very full presentation of the research in this field in the surveys by R. Osserman [3], [40] and J. C. C. Nitsche [4]. This article is a considerably enlarged account of the lectures given by the author in the Second and Third All-Union symposia on geometry in the large in 1967 and 1969 [15], [35].

Duality in mathematical programming and some problems of convex analysis

Abstract: In the paper we explain a general construction method for a problem that is dual in a certain sense to a given extremal problem and we illustrate the method by examples of various classes of problems of mathematical programming. In addition, we consider some problems in the theory of convex sets and convex functions, investigations that were stimulated chiefly by the needs of mathematical programming.

The topology of functional manifolds and the calculus of variations in the large

Abstract: In this article we consider problems of the calculus of variations in the large on Riemannian manifolds. We give a survey of results on one-dimensional and many-dimensional problems, and we investigate the problem of estimating the number of simple closed geodesics.

Superlinear point-set maps and models of economic dynamics

Abstract: CONTENTSIntroduction ??1. Superlinear maps and their duals ??2. Spectral properties of superlinear maps ??3. Optimal trajectories and their characteristics ??4. The asymptotic behaviour of optimal trajectories ??5. A class of extremal problems ??6. Historical and bibliographical notesReferences

The present state of the theory of games

Abstract: A great amount of information about the theory of games of very varied mathematical content has been accumulated in recent years. The account by Karlin in his monograph [64] of only some of the most highly developed branches of the theory of antagonistic games took up about 500 pages. A common approach to the theory of games as a whole has not yet been worked out. This article attempts to survey systematically the basic branches and directions of the theory of games in its present state. The general definition of a game as a formalized representation of a conflict is taken as a basis. All the "forms" of games considered earlier can be obtained from this definition as particular cases. A systematic look at the theory of games, as in the case of normative theory, enables us to place many of the results of the theory of games in fairly natural groupings. Without being able to give either an exhaustive or even a fairly full description of these results the author has restricted himself to an account of the most typical of them. The amount of detail is not uniform and is inversely proportional to the accessibility of the original material. Facts that can be found in Russian publications are merely noted or just mentioned. In particular, questions considered in the author's survey article [32] are only very briefly touched upon. Specific assertions given in this paper are mainly illustrative in character and some could be replaced by others without detriment. We consider practically all the significant branches of the theory of games with the exception of differential games. Although the theory of differential games has a well-defined place in a number of directions of the theory of games, its methods and problems are becoming increasingly independent. Detailed surveys of the theory of differential games are given in [59], [97]. The author does not touch upon the history of the theory of games and refers the interested reader to the addendum to the Russian edition of von Neumann and Morgenstern's basic monograph "The Theory of Games and Economic Behaviour" [82]. The present article is similar in concept to a paper of the same name given by the author at the First All-Union Conference on the Theory of Games at Erevan in November 1968 (see [123] and [37]), but there are essential differences in the selection of material and the presentation.

Degenerating elliptic differential and pseudo-differential operators

Abstract: The present paper is a survey of some results concerning higher-order elliptic differential operators which degenerate on the boundary of a domain. The principal aspect in the study of such operators is that of investigating the corresponding ordinary equations with parameters which degenerate at a single point. The parametrix for such operators is a degenerating pseudo-differential operator. We investigate the properties of the kernels of these pseudo-differential operators.

On the structure of ii1-factors

Abstract: ContentsIntroduction § 1. Central sequences § 2. Residual arrays § 3. The construction of non-isomorphic II1-factorsReferences

Lenin, science and education

Abstract: In this article we present material relating to Lenin's activity as Head of the Soviet Government in the fields of soviet science, culture and education, without of course making any claim to completeness. In spite of the enormous difficulties which the Soviet country had lived through in the years of civil war and of outside intervention and, in particular, had experienced at the beginning of the period of reconstruction, in those very years the network of colleges grew, the network of scientific research institutes and other scientific establishments was started, and the necessary foundation for the further development of Soviet science was laid. Here we tell of the perspicacity of the founder of the Soviet state and of his faith in the future. The greater part of the documents quoted was written by Lenin; but numerous documents from the period in question signed by other statesmen, have a direct connection with Lenin, since any largescale measure in the fields of science, culture, and education was carried out with his approval. We present the material divided into sections, but chronological order is observed in each section. Every document we quote is given with its full title.

Tikhonov semifields and certain problems in general topology

Abstract: In this article we give a detailed presentation of a number of results connected with the problem of the measurability of cardinals and their topological equivalents (stated in the language of Cech compactifications and Hewitt spaces). The presentation is directed from a single view point by a systematic use of Tikhonov semifields (that is, the Tikhonov products of a certain number of copies of the real line). At the end we examine the "geometrical" properties of Tikhonov semifields based on Gleason's results on the representation of functions on an uncountable direct product in the form of a composition. In particular, we give the result of Gel'fand and Fuks that on the "sphere" of a Tikhonov semifield any continuous real function is constant.

Problems of value distribution in dimensions higher than unity

Abstract: Consider two -dimensional complex manifolds and , where is assumed to be compact. Suppose that on a form is give, which defines an element of volume, and on a function with isolated critical points and such that the domain is relatively compact for all . For each point we construct on a form of bidegree with certain special properties which allow us to use a more or less standard techniques to prove the following "first main theorem": if a holomorphic map is non-degenerate for at least one point, then where denotes the integral , and the integral ; here is the number of points (including multiplicities) such that . Under various conditions on the exhaustion and the mapping we obtain various theorems which assert that when these conditions hold, then the quantity grows for almost all (over some subsequence of numbers ) at the same rate as . We also consider the case of real manifolds and smooth maps. Here we obtain analogous results, though by different methods.