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This paper contains several additional observations which serve to revise and further unify the proof of the approximation theorem.

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This paper contains a solution to the semisimplicity problem for modular and integral representation algebras of finite groups over an arbitrary field. (Some of the results of this paper were announced in Dokl. Akad. Nauk SSSR 198 (1971), 509-512 = Soviet Math. Dokl. 12 (1971), 817-821; MR 43 #7522.)

In this paper we formulate operators which constitute an essential generalization of the Riemann-Liouville fractional integro-differentiation operator. With the aid of these operators fundamentally new analogues of the classical formulas of Cauchy, Schwarz, and Poisson for the representation of analytic and harmonic functions in the interior of a circle are established. These formulas enable us to give a complete structural representation for the broad classes of harmonic and analytic functions associated with the generalized operators. Finally, in this paper we also establish sufficient conditions for the solvability of the Hausdorff and Stieltjes moment problems for some general families of sequences of positive numbers.

This paper, like the note on integral geometry in the last number of the "Uspekhi" , is an addendum to my paper [1]. The main idea of the paper is contained in § 2, where we pose the problem of describing linear elliptic equations and their boundary problems in topological terms. The most important of the properties in the large of the solutions of these equations and problems are preserved under small deformations of the problem and must therefore be, in some sense, homotopy invariants. The discovery and study of these invariants is the right way to sort out the whole multiplicity of boundary problems for elliptic equations and to classify these problems. § 1 is introductory, and the reader may skim through it lightly if he wishes. In this section we introduce the basic definitions and notations and, in preparation for the discussion of general boundary problems for elliptic equations in § 2, we advance some general considerations about the nature of these problems.In the preparation of this paper M. S. Agranovich and Z. Ya. Shapiro have given me considerable help, and I take this opportunity of expressing my thanks to them.

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.

In this paper we consider principal derived polylinear operators on an ?-algebra A over an infinite field P. We clarify them in terms of partial algebras, that is, of clones. The classification allows us also to classify the multioperator structures on a vector space A for various systems of multioperators. The idea of discussing clones comes from Cohn's book [1] and the papers of Whitlock [2], Khion [3] and Dicker [4]. We also use certain concepts of Higgins [5] relating to partial algebras. The author expresses his sincere thanks to A.?G.?Kurosh for his guidance on this work.

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.

The problem whether subalgebras of free algebras of various varieties are free plays an important role in general algebra. For some varieties of linear algebras over a field the problem was solved by Kurosh [1] and Shirshov [2], [3]. Kurosh [4] introduced the concept of multioperator algebra over a field and proved that every subalgebra of a free multioperator algebra is free. This paper is devoted to a study of varieties of multioperator algebras given by identities of a special form; particular cases are the commutative and anticommutative laws for classical linear algebras. The main result of the paper comprises the freeness theorem mentioned above for subalgebras of a free multioperator algebra, as well as parallel theorems in Shirshov's papers [2] on the freeness of subalgebras of a free commutative and a free anticommutative algebra; the methods of this last article are maintained without essential modifications.

The paper considers the properties of vector analogs of the Sobolev spaces, , which appear in the study of various models of continuous media. Korn's inequality, proved in the paper, makes it possible to reduce the problem of compactness of the imbedding operators in these spaces to the scalar case and, consequently, to apply the scalar imbedding theorem of S. L. Sobolev. Heretofore, Korn's inequality in the general form was known only for .Bibliography: 18 items.

In this paper a classification is given of actions of finite cyclic groups on quasicomplex manifolds in terms of the invariants of cobordism theory. Moreover, the methods of the paper allow one to understand the geometric nature of known results of a series of authors on actions of cyclic groups of prime order. Bibliography: 11 items.

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.

In this paper we answer a question posed by Ju. I. Manin: for which quasicharacters does there exist a generalized Neron pairing? It turns out that the quasicharacter must trivialize some explicitly described group of roots of unity. In addition, we establish in this paper a connection between generalized Neron pairings and the biextensions of Mumford.

This paper is devoted to problems of the representation and best approximation of the Cauchy kernel on the whole real axis - ∞ < x < + ∞. The method proposed here for the solution of such problems is based on the attraction of systems of rational functions with fixed poles which are orthogonal or biorthogonal on the whole axis. Bibliography: 9 items.

In this paper we study families of unipotent algebraic groups over integral rings. The main results relate to the geometry of such families. In particular, we prove that, under some hypotheses, the space of such a family is isomorphic to an affine space over the base. We give counterexamples showing that in the case of an arbitrary base ring the basic facts of the theory of unipotent algebraic groups over a field cease to be true. For a certain class of the group schemes that we consider we prove results on cohomology, extensions and deformations.

In this paper, the author proves the existence of a versal family of deformations for any projective algebraic variety. Bibliography: 6 items.

In this paper, complex Lie groups acting transitively and effectively on complex manifolds with solvable (nilpotent) fundamental groups are studied. It is shown that if is nilpotent, then locally , where is semisimple and is nilpotent. In the case when is solvable, the Levi decomposition of the group is direct if and only if the stationary subgroup contains a maximal unipotent subgroup of the semisimple part. The question of the existence of transitive semisimple groups on is considered.

This paper deals with three objects of the theory of formations: -stable groups of automorphisms, -coradicals and -abnormal maximal subgroups.Bibliography: 26 items.

In this paper we consider the trace identities satisfied in a full matrix algebra of order n. For the case of a field of characteristic zero we prove that all trace identities are consequences of one obtained from the Hamilton-Cayley theorem.

The paper outlines the way in which the constructive method obtained for symmetrizing nth Kronecker powers of induced representations (see ibid., vol. 6, no. 12, 1807 (1973)) may be applied to space group representations. Also a step by step procedure is given for finding the symmetrized cubes of space group representations, with zinc-blende as an example, which may be used independently of the general theory.

In this paper, one considers spectral and metric properties of flows (and cascades) which arise by restricting the action of a semisimple Lie group on a homogeneous space to a one-parameter subgroup.

Godel's incompleteness theorem states that there is no system of axioms and rules of inference such that the totality of all assertions deducible from the axioms is the same as the totality of all true assertions in arithmetic (indeed, for every consistent system one can construct effectively a true but unprovable assertion). The present article is devoted to a proof of this theorem, based on the concepts and methods of the theory of algorithms; the necessary information from the theory of algorithms is provided. The paper does not require specialized knowledge of any kind (in particular, none from mathematical logic), but assumes only a familiarity with elementary mathematical terminology and symbolism.

The paper deals with closed Hermitian operators and their selfadjoint extensions. New criteria are obtained for Hermitian operators to be regular in the sense of M. A. Krasnosel'skiĭ. For arbitrary closed Hermitian operators an expression is found for the extent to which an extended generalized resolvent fails to be an -function. A new class of operators is found and investigated for which is an -function for all admissible elements .Bibliography: 14 items.

This article presents results on Chebyshev sets and related questions. It is mainly concerned with proving the connectedness, "solarity" and convexity of Chebyshev sets and their generalizations under additional assumptions. Also considered are certain questions of approximative compactness, continuity of the metric projection, and various approximative-geometric properties of sets. We study the relationships between "suns" and conditions for the convexity of "suns". Papers on the above topics are surveyed.

In this paper we obtain algebraic criteria that isolate the class of Kummer surfaces of Jacobians of curves of genus 2 within the class of K3 surfaces. In addition, we prove analogues of Torelli's theorems for these surfaces and find a "standard" representation for them as a complete intersection of three quadrics in P5.

We prove that the zeta-functions can be continued analytically to the whole plane and obtain a functional equation for the zeta-functions with non-abelian characters of arbitrary simple central algebras over fields of algebraic numbers. By "non-abelian characters' we mean positive-definite zonal spherical functions on the group of ideles of the algebra in question that belong to automorphic functions on the group of ideles. The proofs are based on the Poisson formula for the additive group of ideles of the algebra, adapted for a specially constructed weight function (the "excision" weight function), which vanishes together with its Fourier transform on elements of norm zero (theta-formula). The paper consists of four chapters. Chapter 1 contains a survey of the necessary facts on simple algebras over local fields and their zeta-functions. Chapter 2 is fundamental and deals with the construction of the "excision" weight functions. Chapter 3 contains a survey of the basic facts on restricted direct products of locally compact groups. In Chapter 4 the technique of "excision" weight functions and Fourier analysis on the group of ideles is applied to the study of the zeta-functions of simple algebras over number fields.

Research on the construction and analysis of models of digital computing machines has been developed in recent times mainly in two directions. The basis of the first direction is the method proposed by Glushkov of processing discrete information from the point of view of the theory of automata. The second direction is connected with the work of Lyapunov, Ershov, Glebov and other mathematicians. In this work mathematical models of computing machines are regarded from the standpoint of general algebra. Bordering on this there are also several papers devoted to various ways of formalizing the concept of a computer programme (these questions are dealt with in detail in the survey by Ershov and Lyapunov [26]). The present survey (without claiming to be exhaustive) is concerned with ways of constructing and studying algebraic-automatic and algebraic models of computing machines. The main attention is given to methods of constructing corresponding models. In conclusion the authors describe one way of modifying an algebraic model, taking into account some elements of the structure of the machine and questions of the operation of a programme inserted in its memory. We assume that the reader is familiar with some of the basic concepts of general algebra (semigroup, isomorphism, partially ordered set, etc.). This information can be found, for example, in the book of Kurosh [35].

The paper proves the following theorem (a generalization of the corresponding Grauert-Fischer theorem): if there is a holomorphic deformation Xs of a compact complex space (in the sense of Serre) X, where the parameter s of the deformation runs through the complex space S, and if all the Xs are isomorphic to X, then the deformation is trivial. Bibliography: 7 references.

In this paper we consider the group of similitudes of a nondegenerate symmetric or antisymmetric bilinear form over a global field.For representations of this group, realized as functions on the factor-space of the adeles of this group modulo the principal adeles, we construct certain zeta-functions possessing an Euler product and a meromorphic continuation.Bibliography: 7 items.

The paper contains a detailed account of the theory of Martin boundaries for Markov processes with a countable number of states and discrete time. The probabilistic method of Hunt is used as a basis. This method is modified so as not to go outside the limits of the usual notion of a Markov process. The generalization of this notion due to Hunt is discussed in the concluding section.

The study of group properties and group structure can proceed from various initial principles. Usually, properties of a group are obtained from the properties of its elements. However, in a number of investigations essentially the converse problems are solved: to study the influence of the properties of the set of all subgroups of a group on the properties of its elements. This explains to what extent purely set-theoretical operations given on the set of all subgroups of a group G (this set forms a lattice S(G)) determine the properties of the group operation itself. The investigation of connections between the structure of a group and the structure of its lattice of subgroups has become one of the main general approaches in the study of groups. There are many interesting results in this field, but also many important unsolved problems. These problems seem to be difficult, and apparently new ideas and mathematical techniques are needed for their solution. The present survey (without claiming to exhaust the whole material) is devoted to the so-called lattice-theoretical questions of group theory and contains recent results. The main attention is given to infinite groups. The structures of various classes of finite groups, their lattice isomorphisms and homomorphisms, and many related questions are treated in Suzuki's comprehensive monograph [1]. General surveys of the results (up to 1961) for infinite soluble, nilpotent and radical groups are contained in the papers of Kontorovich, Pekelis, and Starostin [2] and Plotkin [3]. Finally, the basic notions (for group theory see [4], [5], for lattice theory [6]) and a brief survey of results in this field are contained in the new edition of Kurosh's monograph [5]. Under the circumstances there is naturally some overlap between this work and the above articles. This can be justified by the intention to make the report as complete as possible.

The paper deals with the classification of representations of the anticommutation relations in the "space of occupation numbers". The problem reduces to the determination of all the realizations of a certain denumerable discrete commutative group as a group of automorphisms of a measure space leaving the measure quasiinvariant, and also to the construction of such measures.

In this paper Wick and anti-Wick operator symbols are studied in connection with expansion into normal and antinormal series in terms of generation and annihilation operators. By the aid of the Wick and anti-Wick symbol of the operator a series of characteristic spectral properties are identified for . In particular, results are presented concerning necessary and sufficient conditions (separately) for to belong to the classes of bounded operators, completely continuous operators and nuclear operators, and also concerning bounds on the spectrum of , and the asymptotic behavior of the number of eigenvalues below ; and for positive selfadjoint operators a bound is obtained for the trace of the Green function: Bibliography: 14 titles.

The paper investigates the number of solutions of an algebraic congruence , where is a prime. Under certain conditions for the polynomial the asymptotic formula is obtained by elementary methods.

In this paper nonlinear equations A(u) = h are studied, where the operator does not necessarily satisfy the well-known condition of coerciveness. With the equation A(u) = h, which is in general not solvable for an arbitrary right side h, we associate a certain equation of the form B*A(u) = h, which is always solvable. Then the original equation A(u) = h is solvable up to Ker B*. This gives a description of the domain of values of the original operator A(u). Bibliography: 9 titles.

In this paper we give a classification of affine surfaces which are quasihomogeneous with respect to an algebraic group. In particular, we also obtain a classification of affine surfaces which are homogeneous with respect to an algebraic group.

In this article we introduce a new language to describe many problems of differential geometry: for example, problems connected with the theory of pseudogroups, Lie equations, foliations, characteristic classes, etc. This is the language of infinite-dimensional Lie algebras and their homogeneous spaces. It is closely connected with the general idea of formal differential geometry set forth by I. M. Gel'fand in his lecture at the International Congress of Mathematicians in Nice. In addition to a detailed account of the theory of homogeneous spaces of infinite-dimensional Lie algebras, this article contains applications of this theory to the characteristic classes of foliations. It also includes results on these questions from earlier papers by I. M. Gel'fand, D. A. Kazhdan, D. B. Fuks, and the authors.

In this paper we examine a general method for estimating double sums with real quadratic character and give some applications to the theory of quadratic forms and the theory of divisors of quadratic fields.

In this paper we describe the quiver representations (see Manuscripta Math. 6(1972), 71-103) which do not contain the problem of reducing a pair of matrices by similarity transformations.

A Magnus group is a group in which the intersection of the lower central series is trivial and its factors are torsion free. The main result of the paper is the following theorem.Theorem. If is the variety of all nilpotent groups of a certain class or the variety of all metabelian groups, or their intersection, and if free groups of and of are Magnus groups, then the -product of any Magnus -groups is a Magnus group.Bibliography: 18 items.

In this paper the author studies the question of conditions for the imbedding of an amalgam of groups, having a certain abstract property, in a group with the same property, where particular attention is paid to normal amalgams with one amalgamated subgroup. Bibliography: 11 titles.

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".

In the paper one investigates symplectic abelian groups that are the group-theoretical analog of symplectic linear spaces. Bibliography: 3 items.

One of the important aims of the modem constructive theory of functions is to establish relationships between the structural properties of functions and sequences of approximations to them. The foundations of work in this field were laid by Jackson, Bernstein, and de la Vall?e-Poussin. Subsequent developments were made by Zygmund, Kolmogorov, Nikol'skii, Pavard, and others. Jackson' s classical inequality and the fundamental converse theorem of Bernstein-de la Vall?e-Poussin, which were initially established for approximations to continuous functions by algebraic and trigonometric polynomials, have been generalized in various directions. Direct and converse theorems have been obtained for algebraic and trigonometric approximations in spaces other than?C, for spaces of almost periodic functions, for approximations by eigenfunctions of a Sturm-Liouville problem, and so on. The purpose of the present paper is to set forth the basic direct and converse theorems of the theory of approximations in Banach spaces. The main technique of the investigation is the use of strongly continuous semigroups of operators and the resolvents of operators generating these semigroups. Under certain conditions on the resolvent (see Ch.?II, ??1), general direct and converse theorems are established for approximations by eigen-subspaces of a generating operator. These general theorems include as special cases many of the previously known results in the constructive theory of functions.

In this paper we derive a formula for the l-adic Euler characteristic of a one-parameter family of algebraic varieties. We define an algebraic analog of the local monodromy of isolated singularities of algebraic hypersurfaces, defined in the complex case by Milnor. We discuss various conjectures connected with the definition of the conductor of a family of algebraic varieties. Bibliography: 26 items.

The paper studies connections between the Nielsen and Schreier properties in varieties of linear -algebras, and also between systems of identical relations such that the theorem about subalgebras lying in the variety is valid, or such that the theorem about finitely generated subalgebras analogous to that of Žukov is valid.Bibliography: 17 items.

In this paper polynilpotent verbal products of groups are regarded as polyverbal in the sense of O. N. Golovin. It is proved that among them only the nilpotent products have a finite basis of polyidentities.

The paper arose as a result of solution of a problem of finding necessary and sufficient conditions characterizing a pair of associative rings or algebras A, B, where A is a densely imbedded ideal in B. It turns out that the methods by which one succeeds in obtaining this solution permit treatment of the even more general situation of the so-called distributive Ω-semigroups, for which the corresponding Ω-algebras are commutative. This situation, with Ω empty, includes the case of semigroups also. Bibliography: 27 items.