Showing papers in "Russian Mathematical Surveys in 1969"
TL;DR: In this paper, the authors present a guide to the literature on monoidal transformations and their application to embeddings, including the Riemann-Roch theorem for embedding.
Abstract: CONTENTS Introduction Guide to the literature § 1. The Grothendieck groups and § 2. and cycles § 3. Self-intersection and exterior powers § 4. Projectivized bundles § 5. Computation of and the splitting principle § 6. Computation of (conclusion) § 7. as a covariant functor § 8. -filtration of the ring § 9. Filtration and dimension § 10. The connection between and § 11. Chern classes and the Adams operation § 12. The structure of monoidal transformations § 13. The behaviour of under a monoidal transformation § 14. The behaviour of under a monoidal transformation (continuation) § 15. The behaviour of under a monoidal transformation (conclusion) § 16. The Adams operations and the homomorphism of the direct image § 17. The sheaf of differentials § 18. The Riemann-Roch theorem for embeddings § 19. The Riemann-Roch theorem for projections References
140 citations
TL;DR: In this paper, the authors studied conditions under which stationary oscillations can be obtained from non-stationary ones in the limit of an elliptic self-adjoint second order operator acting in an infinite domain with a finite boundary.
Abstract: The paper deals with the asymptotic behaviour (as ) of the solutions of some non-stationary problems in mathematical physics. The main aim of the paper is to clarify conditions under which stationary oscillations can be obtained from non-stationary ones in the limit . We study the case of an elliptic self-adjoint second order operator acting in an infinite domain with a finite boundary. We also discuss some higher order operators, as well as the Laplace operator in a domain of special type with an infinite boundary.
139 citations
TL;DR: In this paper, a detailed account of the theory of Martin boundaries for Markov processes with a countable number of states and discrete time is given. And the generalization of this notion due to Hunt is discussed in the concluding section.
Abstract: 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.
93 citations
TL;DR: In this paper, a survey of the connections between non-oscillation and other properties of the solutions, and described known tests for nonoscillation, and established some new results of this kind.
Abstract: In this article we give a survey of the connections between non-oscillation and other properties of the solutions, and describe known tests for non-oscillation. We also establish some new results of this kind.
75 citations
TL;DR: In this article, the algebraic apparatus is described as a set of fundamental concepts, including divisors, differential forms, and intersection indices, for algebraic groups and differential forms.
Abstract: CONTENTS The algebraic apparatusChapter I. Fundamental concepts § 1. Plane algebraic curves § 2. Closed subsets of affine spaces § 3. Rational functions § 4. Quasiprojective varieties § 5. Products and mappings of quasiprojective varieties § 6. Dimension Chapter II. Local properties § 1. Simple and singular points § 2. Expansion in power series § 3. Properties of simple points § 4. Construction of birational isomorphisms § 5. Normal varieties Chapter III. Divisors and differential forms § 1. Divisors § 2. Divisors on curves § 3. Algebraic groups § 4. Differential forms § 5. Examples and applications of differential forms Chapter IV. Intersection indices § 1. Definitions and fundamental properties § 2. Applications of intersection indices § 3. Birational isomorphisms of surfaces
58 citations
56 citations
TL;DR: In this article, it was shown that a function whose coefficients in the series are all zero must itself be zero, and conditions under which the series, or a subsequence of its partial sums, converges to the corresponding function.
Abstract: This article is concerned with the representation of functions in domains of the complex plain by series in the systems , , .In § 1 we construct systems biorthogonal to the systems , , and find the asymptotic behaviour of functions of these systems.In § 2 we determine in a natural way the coefficients of the series in the systems in question by means of the biorthogonal systems. We also find the asymptotic behaviour of the coefficients for large indices. We obtain formulae for the remainder, that is, the difference between the function and a partial sum of the corresponding series.In § 3 we prove that a function whose coefficients in the series are all zero must itself be zero. This result makes it possible, in principle, to reconstruct a function from the coefficients of its series.In § 4 we give conditions under which the series, or a subsequence of its partial sums, converges to the corresponding function.
52 citations
TL;DR: The Borel-Moore theory as discussed by the authors is an extension of the Steenrod homology theory that is very close to the Borel -Moore theory, being isomorphic to it when the coefficient module is finitely generated.
Abstract: In this article we give a description of a certain new homology theory on fairly wide categories of topological spaces, as well as a survey of papers concerning this construction. In contrast to the Aleksandrov-Cech homology theory this theory satisfies all the Eilenberg-Steenrod axioms, including exactness. In the end it turns out to be equivalent to the Steenrod homology theory and very close to the Borel-Moore homology theory, being isomorphic to it when the coefficient module is finitely generated (without this condition the Borel-Moore theory is not well-defined). We show that many results of the Borel-Moore theory take their most definitive and natural form in the homology theory under discussion. The methods of sheaf theory, which are used in the Borel-Moore homology theory, can be applied just as effectively in our case.
50 citations
TL;DR: A survey of the literature on the theory of approximation in the Hausdorff metric, and certain related questions is given in this article, where a number of estimates are given of the best approximation of functions and curves in the plane relative to the Hhausdorff distance.
Abstract: The paper is a survey of the literature on the theory of approximation in the Hausdorff metric, and certain related questions.In the first chapter the definition of the Hausdorff distance is given, together with some of its properties. The relation between the Hausdorff and uniform distances is also discussed.The second chapter gives a survey of results relating to the calculation of e-entropy, e-capacity, and widths relative to the Hausdorff distance.A central position is occupied by the third chapter, where a number of estimates are given of the best approximation of functions and curves in the plane relative to the Hausdorff distance. A theorem is proved here on the existence of a universal estimate of the best approximation relative to the Hausdorff distance for all bounded functions. The question of the approximation of convex functions and curves by polygons, relative to the uniform and Hausdorff distances, is treated separately.Chapter 4 is devoted to linear approximations relative to the Hausdorff distance and the convergence of sequences of positive and convex linear operators.In a short final chapter a new problem in the theory of approximations is proposed.
36 citations
TL;DR: In this article, the authors give an account of the general theory, which includes as particular cases all the results listed in this paper, as well as a discussion of the relation between Martin's theory and the space of exits of Markov processes.
Abstract: Martin's theory makes it possible to describe the sets of all non-negative harmonic and superharmonic functions in an arbitrary domain of euclidean space. To each Markov process there corresponds the class of so-called excessive functions, analogous in their properties to the class of non-negative superharmonic functions. The study of this class is closely connected with the study of the space of exits of a Markov process. Corresponding results for discrete Markov chains were obtained by Doob, Hunt and Watanabe, and for certain types of processes with variable time by Kunita and Watanabe. The paper gives an account of the general theory, which includes as particular cases all the results listed.
31 citations
TL;DR: In this paper, the authors dealt with the classification of representations of the anticommutation relations in the space of occupation numbers, which 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 quasi-invariant.
Abstract: 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.
TL;DR: In this article, a survey of the literature on pseudodifferential partial differential equations of the first order is given, and the results of the energy inequalities and identity of the weak and the strong solutions of the strong solution are given.
Abstract: A large group of problems for systems of partial differential equations of the first order is studied by common methods; for these problems the theorem on energy inequalities is proved, under the assumption that the system (or rather, its characteristic matrix) is symmetric, and also the theorem on the identity of the weak and the strong solutions; these two theorems are used to prove existence and uniqueness of the strong solution. The methods are applicable to a number of problems for symmetric hyperbolic systems of the first order and for symmetric stationary systems that need not be elliptic.Recently new possibilities of developing and applying these methods by using pseudodifferential operators have been discovered, and these are far from being exhausted at the present time.In § 1 the problem is stated and a brief survey of the literature is given. In §§ 2-6 the three theorems mentioned above are set out with proofs suitable for systems of pseudodifferential operators of the first order in a bounded domain. § 7 deals with boundary value problems for symmetrizable systems, more general than the symmetric systems.
TL;DR: A general survey of the mathematical problems associated with representations of the commutation relations and with their physical interpretation can be found in this paper, where only sketches of proofs of the propositions are given; they can be easily completed by the reader.
Abstract: This paper is an appendix to that of V. Ya. Golodets "Classification of representations of the anticommutation relations." Our purpose is to give a general survey of the mathematical problems associated with representations of the commutation relations and with their physical interpretation. In the majority of cases only sketches of proofs of the propositions are given; they can, however, easily be completed by the reader.
TL;DR: In this paper, the statistical sum of a plane Ising model in the absence of an external magnetic field is computed in the presence of magnetic fields. But the model is not considered in this paper.
Abstract: In this paper the statistical sum of a plane Ising model in the absence of an external magnetic field is computed.
TL;DR: The main object of as discussed by the authors is to give an account of the results of Cartan on the real cohomology algebra of a homogeneous space associated with a compact group, which is based on the theory of Lie algebras and connected with them.
Abstract: The main object of this article is to give an account of the results of H. Cartan on the real cohomology algebra of a homogeneous space associated with a compact group. The exposition is based on the theory of Lie algebras and algebras connected with them and contains a number of incidental constructions and results of an algebraic character.
TL;DR: In this paper, it was shown that every identical relation in an Ω-algebra over a field of characteristic zero is equivalent to a system of polylinear identical relations, from which it follows that the study of ǫ-algebras with arbitrary identical relations reduces to that of à −algebra with polylinear à > 0 relations.
Abstract: Two (unconnected) propositions on Ω-algebras with identical relations are proved. The first of these (Theorem 1, in § 1) generalizes to Ω-algebras a known fact from the theory of associative linear algebras, which asserts that every finite-dimensional algebra is an algebra with identical relations (more exactly, every algebra A of dimension m over a field P satisfies a so-called standard identity of degree m + 1).In § 2 we prove that every identical relation in an Ω-algebra over a field of characteristic zero is equivalent to a system of polylinear identical relations (Theorem 2), from which it follows that the study of Ω-algebras with arbitrary identical relations reduces to that of Ω-algebras with polylinear identical relations. This theorem is proved in practically the same way as the corresponding proposition for ordinary algebras with identical relations, that is, algebras with a single binary multiplication (see for example, Mal'tsev [1]); it is clearly a generalization of it.
TL;DR: In this paper, the authors studied properties of free and near-free linear -algebras lying in a variety given by permutational identities, and derived a method of proof for varieties of the freeness theorem analogous to the Dehn-Magnus theorem for groups.
Abstract: In this paper we study properties of free and near-free linear -algebras (over a field) lying in a variety given by permutational identities. Examples of sub-identities in the case of ordinary linear algebras (with a single binary operation) are the commutative and anticommutative laws. The identities, studied by Polin in [4] are also special cases of identities of this kind.The auxiliary results derived in the first two sections yield a method of proof for varieties of the freeness theorem analogous to the Dehn-Magnus theorem for groups [7], Zhukov's theorem for non-associative algebras [2], and Shirshov's theorems for commutative and anticommutative algebras [5] and Lie algebras [6]. Zhukov' s theorem [2] and Shirshov's theorem [5] are special cases of our proposition. We note that although generally speaking a subalgebra of a free algebra in need not be free in the variety, the freeness theorem is always true for such varieties.It is known that for non-associative rings, in contrast to the case of linear algebras, the theorem on subrings of a free ring is false in the most general case. However, using the comparison of an -ring with a linear -algebra over the rational field, we obtain in ??3 a freeness theorem for -rings.The author expresses his indebtedness to A.?G.?Kurosh for valuable advice and remarks during the progress of the work, and for help in preparing the manuscript for the printer.
TL;DR: In this paper, a study of varieties of multioperator algebras given by identities of a special form is presented, and the main result of this paper comprises the freeness theorem mentioned above for subalgesbras of a free multioperator algebra, as well as parallel theorems in Shirshov's papers [2], and the methods of this last article are maintained without essential modifications.
Abstract: 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.
TL;DR: In this article, principal derived polylinear operators on an?-algebra A over an infinite field P are classified in terms of partial algebras, that is, of clones.
Abstract: 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.