Paperback (ISBN-13: 9780898716931)
Page extent: 300 pages
Size: 247 x 174 mm
Weight: 0.484 kg
Originally published in 1981, The Geometry of Random Fields remains an important text for its coverage and exposition of the theory of both smooth and non-smooth random fields; closed form expressions for various geometric characteristics of the excursion sets of smooth, stationary, Gaussian random fields over N-dimensional rectangles; descriptions of the local behavior of random fields in the neighborhoods of high maxima; and a treatment of the Markov property for Gaussian fields. The Geometry of Random Fields is essential reading for researchers in probability and statistics, with no prior knowledge of geometry required. Since the book was originally published it has become a standard reference in areas of physical oceanography, cosmology, and neuroimaging. It is written at a level accessible to nonspecialists, including advanced undergraduates and early graduate students.
* Self-contained classic text on the theory of smooth and non-smooth random fields * Accessible to readers who are specialists in neither probability nor geometry * Covers both smooth and non-smooth fields, whereas other books on the subject tend to treat only one of the two scenarios
Preface to the Classics edition; Preface; Corrections and comments; 1. Random fields and excursion sets; 2. Homogeneous fields and their spectra; 3. Sample function regularity; 4. Geometry and excursion characteristics; 5. Some expectations; 6. Local maxima and high-level excursions; 7. Some non-Gaussian fields; 8. Sample function erraticism and Hausdorff dimension; Appendix. The Markov property for Gaussian fields; References; Author index; Subject index.
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Paperback (ISBN-13: 9780521143554)
Page extent: 229 pages
Size: 244 x 170 mm
Disordered magnetic systems enjoy non-trivial properties which are different and richer than those observed in their pure, non-disordered counterparts. These properties dramatically affect the thermodynamic behaviour and require specific theoretical treatment. This 2006 book deals with the theory of magnetic systems in the presence of frozen disorder, in particular paradigmatic and well-known spin models such as the Random Field Ising Model and the Ising Spin Glass. This is a unified presentation using a field theory language which covers mean field theory, dynamics and perturbation expansion within the same theoretical framework. Particular emphasis is given to the connections between different approaches such as statics vs. dynamics, microscopic vs. phenomenological models. The book introduces some useful and little-known techniques in statistical mechanics and field theory. This book will be of great interest to graduate students and researchers in statistical physics and basic field theory.
* Includes end of chapter summaries * Each chapter is organised in the clearest way possible * Many computations are explained, so readers can reproduce the results and understand the techniques
1. A brief introduction; 2. The Random Field Ising model; 3. The dynamical approach; 4. The p=2 spherical model; 5. Mean field spin glasses: one-step RSB; 6. The Sherrington*Kirkpatrick model; 7. Mean field via TAP equations; 8. Spin glass above D=6; 9. Propagators, mostly replicon; 10. Ward*Takahashi identities and Goldstone modes; 11. Alternative approaches and conclusions; Appendices; Index.
Hardback (ISBN-13: 9780521764100)
Paperback (ISBN-13: 9780521128216)
175 b/w illus.
Page extent: 320 pages
Size: 247 x 174 mm
This easy-to-read book demonstrates how a simple geometric idea reveals fascinating connections and results in number theory, the mathematics of polyhedra, combinatorial geometry, and group theory. Using a systematic paper-folding procedure it is possible to construct a regular polygon with any number of sides. This remarkable algorithm has led to interesting proofs of certain results in number theory, has been used to answer combinatorial questions involving partitions of space, and has enabled the authors to obtain the formula for the volume of a regular tetrahedron in around three steps, using nothing more complicated than basic arithmetic and the most elementary plane geometry. All of these ideas, and more, reveal the beauty of mathematics and the interconnectedness of its various branches. Detailed instructions, including clear illustrations, enable the reader to gain hands-on experience constructing these models and to discover for themselves the patterns and relationships they unearth.
* Accessible to anyone interested in symmetry, numbers and patterns * Assumes no mathematical knowledge beyond school level * Open questions encourage the reader to pursue the subject further on their own
Preface; 1. Flexagons - a beginning thread; 2. Another thread - 1-period paper folding; 3. More paper folding threads - 2-period paper-folding; 4. A number-theory thread - folding numbers, a number trick, and some titbits; 5. The polyhedron thread - building some polyhedra and defining a regular polyhedron; 6. Constructing dipyramids and rotating rings from straight strips of triangles; 7. Continuing the paper-folding and number theory threads; 8. A geometry and algebra thread - constructing, and using, Jennifer's puzzle; 9. A polyhedral geometry thread - constructing braided platonic solids and other woven polyhedra; 10. Combinatorial and symmetry threads; 11. Some golden threads - constructing more dodecahedra; 12. More combinatorial threads - collapsoids; 13. Group theory - the faces of the tri-hexaflexagon; 14. Combinatorial and group theory threads - extended face planes of the platonic solids; 15. A historical thread - involving the Euler characteristic, Descartes' total angular defect, and Polya's dream; 16. Tying some loose ends together - symmetry, group theory, homologues, and the Polya enumeration theorem; 17. Returning to the number theory thread - generalized quasi-order and coach theorems; References; Index.
Series: Mathematical Association of America Textbooks
Hardback (ISBN-13: 9780883857625)
51 b/w illus.
Page extent: 410 pages
Size: 254 x 178 mm
Weight: 0.87 kg
Combinatorics is mathematics of enumeration, existence, construction, and optimization questions concerning finite sets. This text focuses on the first three types of questions and covers basic counting and existence principles, distributions, generating functions, recurrence relations, Polya theory, combinatorial designs, error correcting codes, partially ordered sets, and selected applications to graph theory including the enumeration of trees, the chromatic polynomial, and introductory Ramsey theory. The only prerequisites are single-variable calculus and familiarity with sets and basic proof techniques. It is flexible enough to be used for undergraduate courses in combinatorics, second courses in discrete mathematics, introductory graduate courses in applied mathematics programs, as well as for independent study or reading courses. It also features approximately 350 reading questions spread throughout its eight chapters. These questions provide checkpoints for learning and prepare the reader for the end-of-section exercises of which there are over 470.
* Flexible enough to be used for undergraduate courses in combinatorics, second courses in discrete mathematics, introductory graduate courses in applied mathematics programs, as well as for independent study or reading courses * 350 reading questions are spread through the chapters, providing checkpoints for learning to prepare the reader for the end-of-section exercises * Travel notes enrich the material of each section with anecdotes, open problems, suggestions for further reading and biographical information about mathematicians involved in the discoveries
Preface; Before you go; Notation; Part I. Principles of Combinatorics: 1. Typical counting questions, the product principle; 2. Counting, overcounting, the sum principle; 3. Functions and the bijection principle; 4. Relations and the equivalence principle; 5. Existence and the pigeonhole principle; Part II. Distributions and Combinatorial Proofs: 6. Counting functions; 7. Counting subsets and multisets; 8. Counting set partitions; 9. Counting integer partitions; Part III. Algebraic Tools: 10. Inclusion-exclusion; 11. Mathematical induction; 12. Using generating functions, part I; 13. Using generating functions, part II; 14. techniques for solving recurrence relations; 15. Solving linear recurrence relations; Part IV. Famous Number Families: 16. Binomial and multinomial coefficients; 17. Fibonacci and Lucas numbers; 18. Stirling numbers; 19. Integer partition numbers; Part V. Counting Under Equivalence: 20. Two examples; 21. Permutation groups; 22. Orbits and fixed point sets; 23. Using the CFB theorem; 24. Proving the CFB theorem; 25. The cycle index and Polya's theorem; Part VI. Combinatorics on Graphs: 26. Basic graph theory; 27. Counting trees; 28. Colouring and the chromatic polynomial; 29. Ramsey theory; Part VII. Designs and Codes: 30. Construction methods for designs; 31. The incidence matrix, symmetric designs; 32. Fisher's inequality, Steiner systems; 33. Perfect binary codes; 34. Codes from designs, designs from codes; Part VIII. Partially Ordered Sets: 35. Poset examples and vocabulary; 36. Isomorphism and Sperner's theorem; 37. Dilworth's theorem; 38. Dimension; 39. Mobius inversion, part I; 40. Mobius inversion, part II; Bibliography; Hints and answers to selected exercises.