Series: Encyclopedia of Mathematics and its Applications (No. 114)
Hardback (ISBN-13: 9780521461009)
Traditional game theory has been successful at developing strategy in games of incomplete information: when one player knows something that the other does not. But it has little to say about games of complete information, for example tic-tac-toe, solitaire and hex. This is the subject of combinatorial game theory. Most board games are a challenge for mathematics: to analyze a position one has to examine the available options, and then the further options available after selecting any option, and so on. This leads to combinatorial chaos, where brute force study is impractical. In this comprehensive volume, Jozsef Beck shows readers how to escape from the combinatorial chaos via the fake probabilistic method, a game-theoretic adaptation of the probabilistic method in combinatorics. Using this, the author is able to determine exact results about infinite classes of many games, leading to the discovery of some striking new duality principles.
* Learn how to escape the combinatorial chaos using the fake probabilistic
method to analyze games such as tic-tac-toe, hex and solitaire * Unique
and comprehensive text by the master of combinatorial game theory: describes
striking results and new duality principles * With nearly 200 figures,
plus many exercises and worked examples, the concepts are made fully accessible
Contents
Preface; A summary of the book in a nutshell; Part A. Weak Win and Strong Draw: 1. Win vs. weak win; 2. The main result: exact solutions for infinite classes of games; Part B. Basic Potential Technique - Game-Theoretic First and Second Moments: 3. Simple applications; 4. Games and randomness; Part C. Advanced Weak Win - Game-theoretic Higher Moment: 5. Self-improving potentials; 6. What is the biased meta-conjecture, and why is it so difficult?; Part D. Advanced Strong Draw - Game-theoretic Independence: 7. BigGame-SmallGame decomposition; 8. Advanced decomposition; 9. Game-theoretic lattice-numbers; 10. Conclusion; Complete list of open problems; What kind of games? A dictionary; Dictionary of the phrases and concepts; Appendix A. Ramsey numbers; Appendix B. Hales-Jewett theorem: Shelah’s proof; Appendix C. A formal treatment of positional games; Appendix D. An informal introduction to game theory; References.
Piet de Jong / Macquarie University, Sydney
Gillian Z. Heller /Macquarie University, Sydney
Generalized Linear Models for Insurance Data
Series: International Series on Actuarial Science
Hardback (ISBN-13: 9780521879149)
This is the only book actuaries will need to understand generalized linear models (GLMs) for insurance applications. GLMs are used in the insurance industry to support critical decisions. Amazingly, no text introduces GLMs in this context or addresses the problems specific to insurance data. Until now. Using insurance data sets, this practical, rigorous book treats GLMs, covers all standard exponential family distributions, extends the methodology to correlated data structures,and discusses recent developments which go beyond the GLM. The focus is on issues which are specific to insurance data, such as model selection in the presence of large data sets and the handling of varying exposure times. Exercises and data-based practicals help readers to consolidate their skills, with solutions and data sets given on the companion website. Although the book is package-independent, SAS code and output examples feature in an appendix and on the website.
* Tailored to needs of actuaries * All techniques illustrated on real data
sets relevant to insurance * Exercises and data-based practicals consolidate
skills
Contents
Preface; 1. Insurance data; 2. Response distributions; 3. Exponential family responses and estimation; 4. Linear modeling; 5. Generalized linear models; 6. Models for count data; 7. Categorical responses; 8. Continuous responses; 9. Correlated data; 10. Extensions to the Generalized linear model; Appendix 1. Computer code and output; Bibliography; Index.
Peter D. T. A. Elliott / University of Colorado, Boulder
Duality in Analytic Number Theory
Series: Cambridge Tracts in Mathematics (No. 122)
Now in Paperback (ISBN-13: 9780521058087)
In this stimulating book, aimed at researchers both established and budding, Peter Elliott demonstrates a method and a motivating philosophy that combine to cohere a large part of analytic number theory, including the hitherto nebulous study of arithmetic functions. Besides its application, the book also illustrates a way of thinking mathematically: historical background is woven into the narrative, variant proofs illustrate obstructions, false steps and the development of insight, in a manner reminiscent of Euler. It is shown how to formulate theorems as well as how to construct their proofs. Elementary notions from functional analysis, Fourier analysis, functional equations and stability in mechanics are controlled by a geometric view and synthesized to provide an arithmetical analogue of classical harmonic analysis that is powerful enough to establish arithmetic propositions until now beyond reach. Connections with other branches of analysis are illustrated by over 250 exercises, structured in chains about individual topics.
* Motivates and studies the form as well as proving results * Links history
into the mathematical narrative * Much new material
Contents
Preface; Notation; Introduction; 0. Duality and Fourier analysis; 1. Background philosophy; 2. Operator norm inequalities; 3. Dual norm inequalities; 4. Exercises: including the large sieve; 5. The Method of the Stable Dual (1): deriving the approximate functional equations; 6. The Method of the Stable Dual (2): solving the approximate functional equations; 7. Exercises: almost linear, almost exponential; 8. Additive functions of class La: a first application of the method; 9. Multiplicative functions of the class La: first approach; 10. Multiplicative functions of the class La: second approach; 11. Multiplicative functions of the class La: third approach; 12. Exercises: why the form? 13. Theorems of Wirsing and Halasz; 14. Again Wirsing’s theorem; 15. Exercises: the Prime Number Theorem; 16. Finitely distributed additive functions; 17. Multiplicative functions of the class La: mean value zero; 18. Exercises: including logarithmic weights; 19. Encounters with Ramanujan's function t(n); 20. The operator T on L2; 21. The operator T on La and other spaces; 22. Exercises: the operator D and differentiation; the operator T and the convergence of measures; 23. Pause: towards the discrete derivative; 24. Exercises: multiplicative functions on arithmetic progressions; Wiener phenomenon; 25. Fractional power large sieves; operators involving primes; 26. Exercises: probability seen from number theory; 27. Additive functions on arithmetic progressions: small moduli; 28. Additive functions on arithmetic progressions: large moduli; 29. Exercises: maximal inequalities; 30. Shifted operators and orthogonal duals; 31. Differences of additive functions; local inequalities; 32. Linear forms of additive functions in La; 33. Exercises: stability; correlations of multiplicative functions; 34. Further readings; 35. Ruckblick (after the manner of Johannes Brahms); References; Author index; Subject index.
J. C. Huang / University of Houston
Path-Oriented Program Analysis
Hardback (ISBN-13: 9780521882866)
This book presents a unique method for decomposing a computer program along its execution paths, for simplifying the subprograms so produced, and for recomposing a program from its subprograms. This method enables us to divide and conquer the complexity involved in understanding the computation performed by a program by decomposing it into a set of subprograms and then simplifying them to the furthest extent possible. The resulting simplified subprograms are generally more understandable than the original program as a whole. The method may also be used to simplify a piece of source code by following the path-oriented method of decomposition, simplification, and recomposition. The analysis may be carried out in such a way that the derivation of the analysis result constitutes a correctness proof. The method can be applied to any source code (or portion thereof) that prescribes the computation to be performed in terms of assignment statements, conditional statements, and loop constructs, regardless of the language or paradigm used.
* Describes a unique method, for simplifying programs for analysis * Lets
us divide and conquer the complexity involved in understanding the computation
performed by a program * Can be applied to any source code regardless of
the language or paradigm used
Contents
1. Introduction; 2. State constraints; 3. Subprogram simplification; 4. Program set; 5. Pathwise decomposition; 6. Tautological constraints; 7. Program recomposition; 8. Discussion; 9. Automatic generation of symbolic traces; Appendix.
Harder, Gunter
Lectures on Algebraic Geometry I
Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces
Series: Aspects of Mathematics E 35
2008. viii, 300 pp. Hardc.
ISBN: 978-3-528-03136-7 -
Algebraic Geometry: From Abel and Riemann until today
This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homological algebra, theory of sheaves, and sheaf cohomology are developed. These methods are indispensable for modern algebraic geometry, but they are also fundamental for other branches of mathematics and of great interest in their own.
In the last chapter of volume I these concepts are applied to the theory of compact Riemann surfaces. In this chapter the author makes clear how influential the ideas of Abel, Riemann and Jacobi were and that many of the modern methods have been anticipated by them.
Traces in Number Theory, Geometry and Quantum Fields
Aspects of Mathematics E 38
2007. ix, 223 pp. With 4 Fig. Hardc.
ISBN: 978-3-8348-0371-9
Traces and determinants arise in various guises in many areas of mathematics and mathematical physics: in regularization procedures in quantum fields theory, in the definition of correlation functions and partition functions, in index theory for manifolds and for noncommutative spaces, and in the study of dynamical systems, through zeta functions and zeta determinants, as well as in number theory in the study of zeta and L-functions. This volumes shows, through a series of concrete example, specific results as well as broad overviews, how similar methods based on traces and determinants arise in different perspectives in the fields of number theory, dynamical systems, noncommutative geometry, differential geometry and quantum field theory.
contents:
Number theory, dynamical systems, noncommutative geometry, differential geometry and quantum field theory are five areas of mathematics represented in this volume, which presents an overview of different ongoing research directions around the main theme of traces and zeta functions. This collection of articles arises from an activity that took place at the Max-Planck Institute for Mathematics in Bonn.
for:
- Researchers in the Fields of Global Analysis, Noncommutative Geometry, Number Theory, Dynamical Systems, Mathematical Physics
- Advanced Graduate Students
Series: Aspects of Mathematics E 35