Series: Historical Perspectives on Modern Economics
Hardback (ISBN-13: 9780521562669)
36 b/w illus.
Page extent: 416 pages
Drawing on a wealth of new archival material, including personal correspondence and diaries, Robert Leonard tells the fascinating story of the creation of game theory by Hungarian Jewish mathematician John von Neumann and Austrian economist Oskar Morgenstern. Game theory first emerged amid discussions of the psychology and mathematics of chess in Germany and fin-de-siecle Austro-Hungary. In the 1930s, on the cusp of anti-Semitism and political upheaval, it was developed by von Neumann into an ambitious theory of social organization. It was shaped still further by its use in combat analysis in World War II and during the Cold War. Interweaving accounts of the periodfs economics, science, and mathematics, and drawing sensitively on the private lives of von Neumann and Morgenstern, Robert Leonard provides a detailed reconstruction of a complex historical drama.
* Shows how game theory emerged out of discussions about psychology, behavior, and mathematics of chess in the early 20th century * Shows the crucial importance of social upheaval and the rise of anti-Semitism to the further development of the theory * Shows how von Neumann intended game theory to be less a theory of individual strategic behavior than a new analysis of the social order
Introduction; Part I. Struggle and Equilibrium: From Lasker to von Neumann: 1. 'The strangest states of mind': chess, psychology and Emanuel Lasker's Kampf; 2. 'Deeply rooted yet alien': Hungarian Jews and mathematicians; 3. From Budapest to Gottingen: an apprenticeship in modern mathematics; 4. 'The futile search for the perfect formula': von Neumann's minimax theorem; Part II. Oskar Morgenstern and Interwar Vienna: 5. Equilibrium on trial: the young Morgenstern and the Austrian school; 6. Wrestling with complexity: Wirtschaftsprognose and beyond; 7. Ethics and the excluded middle: Karl Menger and social science; 8. From Austroliberalism to Anschluss: the Viennese economists in the 1930s; Part III. From War to Cold War: 9. Mathematics and the social order: von Neumann's return to game theory; 10. Ars combinatoria: writing the theory of games; 11. Morgenstern's catharsis; 12. Von Neumann's war; 13. Social science and the 'present danger': game theory and psychology at the RAND Corporation, 1946*1960; Conclusion.
Hardback (ISBN-13: 9780521896719)
Paperback (ISBN-13: 9780521721493)
147 b/w illus. 4 colour illus. 125 exercises
Page extent: 345 pages
The link between the physical world and its visualization is geometry. This easy-to-read, generously illustrated textbook presents an elementary introduction to differential geometry with emphasis on geometric results. Avoiding formalism as much as possible, the author harnesses basic mathematical skills in analysis and linear algebra to solve interesting geometric problems, which prepare students for more advanced study in mathematics and other scientific fields such as physics and computer science. The wide range of topics includes curve theory, a detailed study of surfaces, curvature, variation of area and minimal surfaces, geodesics, spherical and hyperbolic geometry, the divergence theorem, triangulations, and the Gauss*Bonnet theorem. The section on cartography demonstrates the concrete importance of elementary differential geometry in applications. Clearly developed arguments and proofs, colour illustrations, and over 100 exercises and solutions make this book ideal for courses and self-study. The only prerequisites are one year of undergraduate calculus and linear algebra.
* Assumes only one year of undergraduate calculus and linear algebra * Equips the reader for further study in mathematics as well as other fields such as physics and computer science * Over 100 exercises and solutions
Preface; Notation; 1. Euclidean geometry; 2. Curve theory; 3. Classical surface theory; 4. The inner geometry of surfaces; 5. Geometry and analysis; 6. Geometry and topology; 7. Hints for solutions to (most) exercises; Formulary; List of symbols; References; Index.
Series: Encyclopedia of Mathematics and its Applications
Hardback (ISBN-13: 9780521847520)
91 b/w illus. 7 tables
Page extent: 748 pages
This collection of papers presents a series of in-depth examinations of a variety of advanced topics related to Boolean functions and expressions. The chapters are written by some of the most prominent experts in their respective fields and cover topics ranging from algebra and propositional logic to learning theory, cryptography, computational complexity, electrical engineering, and reliability theory. Beyond the diversity of the questions raised and investigated in different chapters, a remarkable feature of the collection is the common thread created by the fundamental language, concepts, models, and tools provided by Boolean theory. Many readers will be surprised to discover the countless links between seemingly remote topics discussed in various chapters of the book. This text will help them draw on such connections to further their understanding of their own scientific discipline and to explore new avenues for research.
* Emphasis on algorithms and applications * Authors are the leading experts in this field
Part I. Algebraic Structures: 1. Compositions and clones of Boolean functions Reinhard Poschel and Ivo Rosenberg; 2. Decomposition of Boolean functions Jan C. Bioch; Part II. Logic: 3. Proof theory Alasdair Urquhart; 4. Probabilistic analysis of satisfiability algorithms John Franco; 5. Optimization methods in logic John Hooker; Part III. Learning Theory and Cryptography: 6. Probabilistic learning and Boolean functions Martin Anthony; 7. Learning Boolean functions with queries Robert H. Sloan, Balazs Szorenyi and Gyorgy Turan; 8. Cryptography and error-correcting codes Claude Carlet; 9. Vectorial Boolean functions for cryptography Claude Carlet; Part IV. Graph Representations and Efficient Computation Models: 10. Binary decision diagrams Beate Bollig, Martin Sauerhoff, Detlef Sieling and Ingo Wegener; 11. Circuit complexity Matthias Krause and Ingo Wegener; 12. Fourier transforms and threshold circuit complexity Jehoshua Bruck; 13. Neural networks and Boolean functions Martin Anthony; 14. Decision lists and related classes of Boolean functions Martin Anthony; Part V. Applications in Engineering: 15. Hardware equivalence and property verification J. H. Roland Jiang and Tiziano Villa; 16. Synthesis of multi-level Boolean networks Tiziano Villa, Robert K. Brayton and Alberto L. Sangiovanni-Vincentelli; 17. Boolean aspects of network reliability Charles J. Colbourn.
Hardback (ISBN-13: 9780521764827)
34 b/w illus.
Page extent: 450 pages
Aimed at graduate students in physics and mathematics, this book provides an introduction to recent developments in several active topics at the interface between algebra, geometry, topology and quantum field theory. The first part of the book begins with an account of important results in geometric topology. It investigates the differential equation aspects of quantum cohomology, before moving on to noncommutative geometry. This is followed by a further exploration of quantum field theory and gauge theory, describing AdS/CFT correspondence, and the functional renormalization group approach to quantum gravity. The second part covers a wide spectrum of topics on the borderline of mathematics and physics, ranging from orbifolds to quantum indistinguishability and involving a manifold of mathematical tools borrowed from geometry, algebra and analysis. Each chapter presents introductory material before moving on to more advanced results. The chapters are self-contained and can be read independently of the rest.
* Examines current active research topics in mathematics and physics * Chapters are self-contained and present introductory material before moving on to more advanced results * Contains both mathematics and physics presentations, with a focus on interdisciplinary aspects
Introduction; 1. The impact of QFT on low-dimensional topology Paul Kirk; 2. Differential equations aspects of quantum cohomology Martin A. Guest; 3. Index theory and groupoids Claire Debord and Jean-Marie Lescure; 4. Renormalization Hopf algebras and combinatorial groups Alessandra Frabetti; 5. BRS invariance for massive boson fields Jose M. Gracia-Bondia; 6. Large N field theories and geometry David Berenstein; 7. Functional renormalization group equations, asymptotic safety, and quantum Einstein gravity Martin Reuter and Frank Saueressig; 8. When is a differentiable manifold the boundary of an orbifold* Andres Angel; 9. Canonical group quantization, rotation generators and quantum indistinguishability Carlos Benavides and Andres Reyes-Lega; 10. Conserved currents in Kahler manifolds Jaime R. Camacaro and Juan Carlos Moreno; 11. A symmetrized canonical determinant on odd-class pseudodifferential operators Marie-Francoise Ouedraogo; 12. Some remarks about cosymplectic metrics on maximal flag manifolds Marlio Paredes and Sofia Pinzon; 13. Heisenberg modules over real multiplication noncommutative tori and related algebraic structures Jorge Plazas; Index.
Hardback (ISBN-13: 9780521896030)
65 b/w illus.
Page extent: 340 pages
Group theory has long been an important computational tool for physicists, but, with the advent of the Standard Model, it has become a powerful conceptual tool as well. This book introduces physicists to many of the fascinating mathematical aspects of group theory, and mathematicians to its physics applications. Designed for advanced undergraduate and graduate students, this book gives a comprehensive overview of the main aspects of both finite and continuous group theory, with an emphasis on applications to fundamental physics. Finite groups are extensively discussed, highlighting their irreducible representations and invariants. Lie algebras, and to a lesser extent Kac-Moody algebras, are treated in detail, including Dynkin diagrams. Special emphasis is given to their representations and embeddings. The group theory underlying the Standard Model is discussed, along with its importance in model building. Applications of group theory to the classification of elementary particles are treated in detail.
* A comprehensive overview of the main aspects of both finite and continuous group theory, with an emphasis on applications to fundamental physics * Finite groups and Lie algebras are extensively discussed * Applications of group theory to the classification of elementary particles are treated in detail
1. Preface: the pursuit of symmetries; 2. Finite groups: an introduction; 3. Finite groups: representations; 4. Hilbert Spaces; 5. SU(2); 6. SU(3); 7. Classification of compact simple Lie algebras; 8. Lie algebras: representation theory; 9. Finite groups: the road to simplicity; 10. Beyond Lie algebras; 11. The groups of the Standard Model; 12. Exceptional structures; Appendices; References; Bibliography; Index.