Hardback (ISBN-13: 9780521765398)
23 b/w illus. 532 exercises
Page extent: 392 pages
Size: 253 x 215 mm
This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.
* More than 250 good examples and 500 exercises * Comprehensive treatment in only 400 pages * Concentrates on results useful for application
1. Measure theory; 2. Laws of large numbers; 3. Central limit theorems; 4. Random walks; 5. Martingales; 6. Markov chains; 7. Ergodic theorems; 8. Brownian motion; Appendix A. Measure theory details.
Paperback (ISBN-13: 9780521168281)
Page extent: 84 pages
Size: 216 x 140 mm
This book was first published in 1952. It is largely devoted to the object of proving the Vinogradov-Goldbach theorem: that every sufficiently large odd number is the sum of three primes. In the course of proving this, T. Estermann, formerly Professor of Mathematics at the University of London, supplies numerous theories and results on characters and primes in arithmetic progressions. The author also ensures that the proofs presented to the reader are both clear and remarkably concise. The volume at hand addresses the Riemann zeta function, primes in arithmetical progression, and the ways in which odd numbers can be represented as the sum of three primes. At the end of the book is an index and a seven-page section of theorems and formulae for reference. This volume is both interesting and accessible, and will appeal to all with an enthusiasm for mathematics and problem solving.
Preface; Preface to the second impression; Remarks on notation; 1. The Riemann zeta function and a refinement of the prime number theorem; 2. The number of primes in an arithmetical progression; 3. The representations of an odd number as a sum of three primes; Theorems and formulae for reference.
Hardback (ISBN-13: 9780521766999)
150 b/w illus.
Page extent: 475 pages
Size: 247 x 174 mm
If you work with data and need easy access to clear, reliable definitions and explanations of modern statistical and statistics-related concepts, then look no further than this dictionary. Nearly 4000 terms are defined, covering medical, survey, theoretical, and applied statistics, including computational and graphical aspects. Entries are provided for standard and specialized statistical software. In addition, short biographies of over 100 important statisticians are given. Definitions provide enough mathematical detail to clarify concepts and give standard formulae when these are helpful. The majority of definitions then give a reference to a book or article where the user can seek further or more specialized information, and many are accompanied by graphical material to aid understanding.
* 4th edition includes new entries reflecting changes in the field, particularly a growing interest in Bayesian statistics and machine learning * Trusted author in touch with contemporary statistical practice * Carefully chosen pointers to the literature give users access to information beyond the basic definitions
Preface; Acknowledgements; Notes on use; Sources; The Dictionary.
Series: London Mathematical Society Lecture Note Series (No. 377)
Paperback (ISBN-13: 9780521738668)
65 exercises
Page extent: 330 pages
Size: 228 x 152 mm
This is the first elementary introduction to Galois cohomology and its applications. The first part is self-contained and provides the basic results of the theory, including a detailed construction of the Galois cohomology functor, as well as an exposition of the general theory of Galois descent. The author illustrates the theory using the example of the descent problem of conjugacy classes of matrices. The second part of the book gives an insight into how Galois cohomology may be used to solve algebraic problems in several active research topics, such as inverse Galois theory, rationality questions or the essential dimension of algebraic groups. Assuming only a minimal background in algebra, the main purpose of this book is to prepare graduate students and researchers for more advanced study.
* Presents the basic theory using detailed proofs * Provides a wide range of applications of Galois cohomology * Only prerequisites are Galois theory, tensor products of vector spaces and algebras
Foreword Jean-Pierre Tignol; Introduction; Part I. An Introduction to Galois Cohomology: 1. Infinite Galois theory; 2. Cohomology of profinite groups; 3. Galois cohomology; 4. Galois cohomology of quadratic forms; 5. Etale and Galois algebras; 6. Groups extensions and Galois embedding problems; Part II. Applications: 7. Galois embedding problems and the trace form; 8. Galois cohomology of central simple algebras; 9. Digression: a geometric interpretation of H1 (-, G); 10. Galois cohomology and Noether's problem; 11. The rationality problem for adjoint algebraic groups; 12. Essential dimension of functors; References; Index.
Series: Cambridge Studies in Advanced Mathematics (No. 128)
Hardback (ISBN-13: 9780521113670)
65 b/w illus. 11 colour illus. 95 exercises
Page extent: 200 pages
Size: 228 x 152 mm
Graph theory meets number theory in this stimulating book. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann/Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta are produced, showing you cannot hear the shape of a graph. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum chaos and random matrix theory, plus expander/Ramanujan graphs of interest in computer science. Created for beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and exercises, both theoretical and computer-based, are included throughout.
* Makes connections with quantum chaos and random matrix theory, plus Ramanujan graphs, which are of interest to computer scientists * Explains key ideas using lots of well-chosen illustrations, alongside theoretical and computer-based exercises * Perfect for beginning graduate students, or established researchers who want a stimulating introduction to the topic
List of illustrations; Preface; Part I. A Quick Look at Various Zeta Functions: 1. Riemann's zeta function and other zetas from number theory; 2. Ihara's zeta function; 3. Selberg's zeta function; 4. Ruelle's zeta function; 5. Chaos; Part II. Ihara's Zeta Function and the Graph Theory Prime Number Theorem: 6. Ihara zeta function of a weighted graph; 7. Regular graphs, location of poles of zeta, functional equations; 8. Irregular graphs: what is the RH*; 9. Discussion of regular Ramanujan graphs; 10. The graph theory prime number theorem; Part III. Edge and Path Zeta Functions: 11. The edge zeta function; 12. Path zeta functions; Part IV. Finite Unramified Galois Coverings of Connected Graphs: 13. Finite unramified coverings and Galois groups; 14. Fundamental theorem of Galois theory; 15. Behavior of primes in coverings; 16. Frobenius automorphisms; 17. How to construct intermediate coverings using the Frobenius automorphism; 18. Artin L-functions; 19. Edge Artin L-functions; 20. Path Artin L-functions; 21. Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function; 22. The Chebotarev Density Theorem; 23. Siegel poles; Part V. Last Look at the Garden: 24. An application to error-correcting codes; 25. Explicit formulas; 26. Again chaos; 27. Final research problems; References; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 137)
Hardback (ISBN-13: 9780521190220)
3 b/w illus. 170 exercises
Page extent: 250 pages
Size: 234 x 156 mm
The algebraic theory of automata was created by Schutzenberger and Chomsky over 50 years ago and there has since been a great deal of development. Classical work on the theory to noncommutative power series has been augmented more recently to areas such as representation theory, combinatorial mathematics and theoretical computer science. This book presents to an audience of graduate students and researchers a modern account of the subject and its applications. The algebraic approach allows the theory to be developed in a general form of wide applicability. For example, number-theoretic results can now be more fully explored, in addition to applications in automata theory, codes and non-commutative algebra. Much material, for example, Schutzenberger's theorem on polynomially bounded rational series, appears here for the first time in book form. This is an excellent resource and reference for all those working in algebra, theoretical computer science and their areas of overlap.
* A comprehensive exposition of the theory * Includes numerous number-theoretic applications and a new proof of Soittola’s theorem * An excellent resource for graduate students and researchers
Preface; Part I. Rational Series: 1. Rational series; 2. Minimization; 3. Series and languages; 4. Rational expressions; Part II. Arithmetic: 5. Automatic sequences and algebraic series; 6. Rational series in one variable; 7. Changing the semiring; 8. Positive series in one variable; Part III. Applications: 9. Matrix semigroups and applications; 10. Noncommutative polynomials; 11. Codes and formal series; 12. Semisimple syntactic algebras; Open problems and conjectures; References; Index of notation; Index.