R. V. Gamkrelidze – författare

Offering the insights of L.S. Pontryagin, one of the foremost thinkers in modern mathematics, the second volume in this four-volume set examines the nature and processes that make up topological groups. Already hailed as the leading work in this subject for its abundance of examples and its thorough explanations, the text is arranged so that readers can follow the material either sequentially or schematically. Stand-alone chapters cover such topics as topological division rings, linear representations of compact topological groups, and the concept of a lie group.

Offering the insights of L.S. Pontryagin, one of the foremost thinkers in modern mathematics, the second volume in this four-volume set examines the nature and processes that make up topological groups. Already hailed as the leading work in this subject for its abundance of examples and its thorough explanations, the text is arranged so that readers can follow the material either sequentially or schematically. Stand-alone chapters cover such topics as topological division rings, linear representations of compact topological groups, and the concept of a lie group.

This volume contains three articles: "Asymptotic methods in the theory of ordinary differential equations" b''y V. F. Butuzov, A. B. Vasil''eva, and M. V. Fedoryuk, "The theory of best ap­ proximation in Dormed linear spaces" by A. L. Garkavi, and "Dy­ namical systems with invariant measure" by A. ''VI. Vershik and S. A. Yuzvinskii. The first article surveys the literature on linear and non­ linear singular asymptotic problems, in particular, differential equations with a small parameter. The period covered by the survey is primarily 1962-1967. The second article is devoted to the problem of existence, characterization, and uniqueness of best approximations in Banach spaces. One of the chapters also deals with the problem of the convergence of positive operators, inasmuch as the ideas and methods of this theory are close to those of the theory of best ap­ proximation. The survey covers the literature of the decade 1958-1967. The third article is devoted to a comparatively new and rapid­ ly growing branch of mathematics which is closely related to many classical and modern mathematical disciplines. A survey is given of results in entropy theory, classical dynamic systems, ergodic theorems, etc. The results surveyed were primarily published during the period 1956-1967.

This volume contains five review articles, two in the Algebra part and three in the Geometry part, surveying the fields of cate­ gories and class field theory, in the Algebra part, and of Finsler spaces, structures on differentiable manifolds, and packing, cover­ ing, etc., in the Geometry part. The literature covered is primar­ Hy that published in 1964-1967. Contents ALGEBRA CATEGORIES ............... . 3 M. S. Tsalenko and E. G. Shul''geifer § 1. Introduction........... 3 § 2. Foundations of the Theory of Categories . . . . . 4 § 3. Fundamentals of the Theory of Categories . . . . . 6 § 4. Embeddings of Categories ... . . . . . . . . . . . . 14 § 5. Representations of Categories . . . . . . . . . . . . . 16 § 6. Axiomatic Characteristics of Algebraic Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 18 § 7. Reflective Subcategories; Varieties. . . 20 § 8. Radicals in Categories . . . . . . . 24 § 9. Categories with Involution. . . . . . 29 § 10. Universal Algebras in Categories . 30 § 11. Categories with Multiplication . . . 34 § 12. Duality of Functors. .. ....... 37 § 13. Homotopy Theory . . . . .. ........... 39 § 14. Homological Algebra in Categories. . . . . . 41 § 15. Concrete Categories . . . . .. ......... 44 § 16. Generalizations.. . . . . . . 45 Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 CLASS FIELD THEORY. FIELD EXTENSIONS. . . . . . . . 59 S. P. Demushkin 66 Literature Cited vii CONTENTS viii GEOMETRY 75 FINSLER SPACES AND THEIR GENERALIZATIONS ..

This volume contains five review articles, two in the Algebra part and three in the Geometry part, surveying the fields of cate­ gories and class field theory, in the Algebra part, and of Finsler spaces, structures on differentiable manifolds, and packing, cover­ ing, etc., in the Geometry part. The literature covered is primar­ Hy that published in 1964-1967. Contents ALGEBRA CATEGORIES ............... . 3 M. S. Tsalenko and E. G. Shul'geifer § 1. Introduction........... 3 § 2. Foundations of the Theory of Categories . . . . . 4 § 3. Fundamentals of the Theory of Categories . . . . . 6 § 4. Embeddings of Categories ... . . . . . . . . . . . . 14 § 5. Representations of Categories . . . . . . . . . . . . . 16 § 6. Axiomatic Characteristics of Algebraic Categories . . . . . . . . . . . . . . . . . . . . . . . . . . 18 § 7. Reflective Subcategories; Varieties. . . 20 § 8. Radicals in Categories . . . . . . . 24 § 9. Categories with Involution. . . . . . 29 § 10. Universal Algebras in Categories . 30 § 11. Categories with Multiplication . . . 34 § 12. Duality of Functors. .. ....... 37 § 13. Homotopy Theory . . . . .. ........... 39 § 14. Homological Algebra in Categories. . . . . . 41 § 15. Concrete Categories . . . . .. ......... 44 § 16. Generalizations.. . . . . . . 45 Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 CLASS FIELD THEORY. FIELD EXTENSIONS. . . . . . . . 59 S. P. Demushkin 66 Literature Cited vii CONTENTS viii GEOMETRY 75 FINSLER SPACES AND THEIR GENERALIZATIONS ..

This volume contains two review articles: "Stochastic Pro­ gramming" by Vo V. Kolbin, and "Application of Queueing-Theoretic Methods in Operations Research, " by N. Po Buslenko and A. P. Cherenkovo The first article covers almost all aspects of stochastic programming. Many of the results presented in it have not pre­ viously been surveyed in the Soviet literature and are of interest to both mathematicians and economists. The second article com­ prises an exhaustive treatise on the present state of the art of the statistical methods of queueing theory and the statistical modeling of queueing systems as applied to the analysis of complex systems. Contents STOCHASTIC PROGRAMMING V. V. Kolbin Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 § 1. The Geometry of Stochastic Linear Programming Problems. . . . . . . . . . . . . . . . . . . . 5 § 2. Chance-Constrained Problems . . . . . . . . . 8 § 3. Rigorous Statement of stochastic Linear Programming Problems . . . . . . . . . . 16 § 4. Game-Theoretic Statement of Stochastic Linear Programming Problems. . . . . . . . 18 § 5. Nonrigorous Statement of SLP Problems . . . 19 § 6. Existence of Domains of Stability of the Solutions of SLP Problems . . . . . . . . . 29 § 7. Stability of a Solution in the Mean. . . . . . . . . . . . 30 § 8. Dual Stochastic Linear Programming Problems. . . 37 § 9. Some Algorithms for the Solution of Stochastic Linear Programming Problems . . . . . . . . . . 40 § 10. Stochastic Nonlinear Programming: Some First Results . . . . . . . . . . . . . . . . . . . . . . 42 § 11. The Two-Stage SNLP Problem. . . . . . . . . . . . 47 § 12. Optimality and Existence of a Plan in Stochastic Nonlinear Programming Problems. 58 Literature Cited . . . . . . . . . . . . . . . . .. . . . . . . . . .

This volume contains two review articles: "Stochastic Pro­ gramming" by Vo V. Kolbin, and "Application of Queueing-Theoretic Methods in Operations Research, " by N. Po Buslenko and A. P. Cherenkovo The first article covers almost all aspects of stochastic programming. Many of the results presented in it have not pre­ viously been surveyed in the Soviet literature and are of interest to both mathematicians and economists. The second article com­ prises an exhaustive treatise on the present state of the art of the statistical methods of queueing theory and the statistical modeling of queueing systems as applied to the analysis of complex systems. Contents STOCHASTIC PROGRAMMING V. V. Kolbin Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 § 1. The Geometry of Stochastic Linear Programming Problems. . . . . . . . . . . . . . . . . . . . 5 § 2. Chance-Constrained Problems . . . . . . . . . 8 § 3. Rigorous Statement of stochastic Linear Programming Problems . . . . . . . . . . 16 § 4. Game-Theoretic Statement of Stochastic Linear Programming Problems. . . . . . . . 18 § 5. Nonrigorous Statement of SLP Problems . . . 19 § 6. Existence of Domains of Stability of the Solutions of SLP Problems . . . . . . . . . 29 § 7. Stability of a Solution in the Mean. . . . . . . . . . . . 30 § 8. Dual Stochastic Linear Programming Problems. . . 37 § 9. Some Algorithms for the Solution of Stochastic Linear Programming Problems . . . . . . . . . . 40 § 10. Stochastic Nonlinear Programming: Some First Results . . . . . . . . . . . . . . . . . . . . . . 42 § 11. The Two-Stage SNLP Problem. . . . . . . . . . . . 47 § 12. Optimality and Existence of a Plan in Stochastic Nonlinear Programming Problems. 58 Literature Cited . . . . . . . . . . . . . . . . .. . . . . . . . . .

This work is a continuation of earlier volumes under the heading "Probability Theory, Mathematical Statistics, and Theo- retical Cybernetics," published as part of the "Itogi Nauki" series. The present volume comprises a single review article, en- titled "Reliability of Discrete Systems," covering material pub- lished mainly in the last six to eight years and abstracted in "Referativnyi Zhurnal-Matematika" (Soviet Abstract Journal in Mathematics). The bibliography encompasses 313 items. The editors welcome inquiries regarding the present volume or the format and content of future volumes of the series; corre- spondence should be sent to the following address: Otdel Matemat- ika (Mathematics Section), Baltiiskaya ul., 14, Moscow, A-219. v Contents RELIABILITY OF DISCRETE SYSTEMS M. A. Gavrilov, V. M. Ostianu, and A. I. Potekhin Introduction ...* ...* ...1 ...CHAPTER 1. Assurance of Infallibility in Discrete Systems...5 1. State of the Art .*...*...5 2. Basic Definitions, Concepts, and Problem Formulations. 6 3. Redundancy Models...*...10 . * . . 4. Composition Methods ...17 5. Majority Methods ...*...27 ...6. Methods Using the Interweaving Model...35 7.Methods Using Effective-Coding Models...38 CHAPTER II. Assurance of Stability in Discrete System s...63 ...1. Basic Concepts and Definitions ...63 . . 2. Elimination of Inadmissible I-Races...69 . 3. Elimination of Inadmissible E-Races ...70 . 4. Elimination of Inadmissible M-Races ...73 . 5. Elimination of Inadmissible L-Races ...86 .

This volume contains five review articles, three in the Al­ gebra part and two in the Geometry part, surveying the fields of ring theory, modules, and lattice theory in the former, and those of integral geometry and differential-geometric methods in the calculus of variations in the latter. The literature covered is primarily that published in 1965-1968. v CONTENTS ALGEBRA RING THEORY L. A. Bokut'', K. A. Zhevlakov, and E. N. Kuz''min § 1. Associative Rings. . . . . . . . . . . . . . . . . . . . 3 § 2. Lie Algebras and Their Generalizations. . . . . . . 13 ~ 3. Alternative and Jordan Rings. . . . . . . . . . . . . . . . 18 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 MODULES A. V. Mikhalev and L. A. Skornyakov § 1. Radicals. . . . . . . . . . . . . . . . . . . 59 § 2. Projection, Injection, etc. . . . . . . . . . . . . . . . . . . 62 § 3. Homological Classification of Rings. . . . . . . . . . . . 66 § 4. Quasi-Frobenius Rings and Their Generalizations. . 71 § 5. Some Aspects of Homological Algebra . . . . . . . . . . 75 § 6. Endomorphism Rings . . . . . . . . . . . . . . . . . . . . . 83 § 7. Other Aspects. . . . . . . . . . . . . . . . . . . 87 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 91 LATTICE THEORY M. M. Glukhov, 1. V. Stelletskii, and T. S. Fofanova § 1. Boolean Algebras . . . . . . . . . . . . . . . . . . . . . " 111 § 2. Identity and Defining Relations in Lattices . . . . . . 120 § 3. Distributive Lattices. . . . . . . . . . . . . . . . . . . . . 122 vii viii CONTENTS § 4. Geometrical Aspects and the Related Investigations. . . . . . . . . . . . • . . • . . . . . . . . . • 125 § 5. Homological Aspects. . . . . . . . . . . . . . . . . . . . . . 129 § 6. Lattices ofCongruences and of Ideals of a Lattice . . 133 § 7. Lattices of Subsets, of Subalgebras, etc. . . . . . . . . 134 § 8. Closure Operators . . . . . . . . . . . . . . . . . . . . . . . 136 § 9. Topological Aspects. . . . . . . . . . . . . . . . . . . . . . 137 § 10. Partially-Ordered Sets. . . . . . . . . . . . . . . . . . . . 141 § 11. Other Questions. . . . . . . . . . . . . . . . . . . . . . . . . 146 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 GEOMETRY INTEGRAL GEOMETRY G. 1. Drinfel''d Preface . . . . . . . . .

This volume contains five review articles, three in the Al­ gebra part and two in the Geometry part, surveying the fields of ring theory, modules, and lattice theory in the former, and those of integral geometry and differential-geometric methods in the calculus of variations in the latter. The literature covered is primarily that published in 1965-1968. v CONTENTS ALGEBRA RING THEORY L. A. Bokut', K. A. Zhevlakov, and E. N. Kuz'min § 1. Associative Rings. . . . . . . . . . . . . . . . . . . . 3 § 2. Lie Algebras and Their Generalizations. . . . . . . 13 ~ 3. Alternative and Jordan Rings. . . . . . . . . . . . . . . . 18 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 MODULES A. V. Mikhalev and L. A. Skornyakov § 1. Radicals. . . . . . . . . . . . . . . . . . . 59 § 2. Projection, Injection, etc. . . . . . . . . . . . . . . . . . . 62 § 3. Homological Classification of Rings. . . . . . . . . . . . 66 § 4. Quasi-Frobenius Rings and Their Generalizations. . 71 § 5. Some Aspects of Homological Algebra . . . . . . . . . . 75 § 6. Endomorphism Rings . . . . . . . . . . . . . . . . . . . . . 83 § 7. Other Aspects. . . . . . . . . . . . . . . . . . . 87 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 91 LATTICE THEORY M. M. Glukhov, 1. V. Stelletskii, and T. S. Fofanova § 1. Boolean Algebras . . . . . . . . . . . . . . . . . . . . . " 111 § 2. Identity and Defining Relations in Lattices . . . . . . 120 § 3. Distributive Lattices. . . . . . . . . . . . . . . . . . . . . 122 vii viii CONTENTS § 4. Geometrical Aspects and the Related Investigations. . . . . . . . . . . . • . . • . . . . . . . . . • 125 § 5. Homological Aspects. . . . . . . . . . . . . . . . . . . . . . 129 § 6. Lattices ofCongruences and of Ideals of a Lattice . . 133 § 7. Lattices of Subsets, of Subalgebras, etc. . . . . . . . . 134 § 8. Closure Operators . . . . . . . . . . . . . . . . . . . . . . . 136 § 9. Topological Aspects. . . . . . . . . . . . . . . . . . . . . . 137 § 10. Partially-Ordered Sets. . . . . . . . . . . . . . . . . . . . 141 § 11. Other Questions. . . . . . . . . . . . . . . . . . . . . . . . . 146 Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 GEOMETRY INTEGRAL GEOMETRY G. 1. Drinfel'd Preface . . . . . . . . .

The present book contains three articles: "Systems of Linear Differential Equations," by V. P. Palamodov; "Fredholm Operators and Their Generalizations," by S. N. Krachkovskii and A. S. Di­ kanskii; and "Representations of Groups and Algebras in Spaces with an Indefinite Metric" by M. A. Naimark and R. S. Ismagilov. In the fi.rst article the accent is on those characteristics of systems of differential equations which distinguish the systems from the scalar case. Considerable space is devoted in particular to "nonquadratic systems," a topic that has very recently stimulated interest. The second article is devoted to the algebraic aspects of the theory of operators (determinant theory in particular) in Banach and linear topological spaces. The third article reflects the present state of the art in the given area of the theory of representations, which has been re­ ceiving considerable attention in connection with its applications in physics (particularly in quantum field theory) and in the theory of differential equations.

The present book contains three articles: "Systems of Linear Differential Equations," by V. P. Palamodov; "Fredholm Operators and Their Generalizations," by S. N. Krachkovskii and A. S. Di­ kanskii; and "Representations of Groups and Algebras in Spaces with an Indefinite Metric" by M. A. Naimark and R. S. Ismagilov. In the fi.rst article the accent is on those characteristics of systems of differential equations which distinguish the systems from the scalar case. Considerable space is devoted in particular to "nonquadratic systems," a topic that has very recently stimulated interest. The second article is devoted to the algebraic aspects of the theory of operators (determinant theory in particular) in Banach and linear topological spaces. The third article reflects the present state of the art in the given area of the theory of representations, which has been re­ ceiving considerable attention in connection with its applications in physics (particularly in quantum field theory) and in the theory of differential equations.

Offering the insights of L.S. Pontryagin, one of the foremost thinkers in modern mathematics, the second volume in this four-volume set examines the nature and processes that make up topological groups. Already hailed as the leading work in this subject for its abundance of examples and its thorough explanations, the text is arranged so that readers can follow the material either sequentially or schematically. Stand-alone chapters cover such topics as topological division rings, linear representations of compact topological groups, and the concept of a lie group.