Hardback (ISBN-13: 9780521194587)
29 b/w illus. 1 table
Page extent: 344 pages
Size: 247 x 174 mm
Analyzing the behavior of complex networks is an important element in the design of new man-made structures such as communication systems and biologically engineered molecules. Because any complex network can be represented by a graph, and therefore in turn by a matrix, graph theory has become a powerful tool in the investigation of network performance. This self-contained book provides a concise introduction to the theory of graph spectra and its applications to the study of complex networks. Covering a range of types of graphs and topics important to the analysis of complex systems, this guide provides the mathematical foundation needed to understand and apply spectral insight to real-world systems. In particular, the general properties of both the adjacency and Laplacian spectrum of graphs are derived and applied to complex networks. An ideal resource for researchers and students in communications networking as well as in physics and mathematics.
* General properties of both the adjacency and Laplacian spectrum of graphs are derived and applied to complex networks * Proofs are written in a deductive and comprehensive manner, presenting all the derivations required in one place * Practical examples illustrate how mathematical and statistical tools can be applied to real-world networks
Preface; Acknowledgements; 1. Introduction; Part I. Spectra of Graphs: 2. Algebraic graph theory; 3. Eigenvalues of the adjacency matrix; 4. Eigenvalues of the Laplacian Q; 5. Spectra of special types of graphs; 6. Density function of the eigenvalues; 7. Spectra of complex networks; Part II. Eigensystem and Polynomials: 8. Eigensystem of a matrix; 9. Polynomials with real coefficients; 10. Orthogonal polynomials; List of symbols; Bibliography; Index.
Hardback (ISBN-13: 9780521195751)
40 b/w illus. 69 exercises
Page extent: 304 pages
Size: 247 x 174 mm
Canonical methods are a powerful mathematical tool within the field of gravitational research, both theoretical and experimental, and have contributed to a number of recent developments in physics. Providing mathematical foundations as well as physical applications, this is the first systematic explanation of canonical methods in gravity. The book discusses the mathematical and geometrical notions underlying canonical tools, highlighting their applications in all aspects of gravitational research from advanced mathematical foundations to modern applications in cosmology and black hole physics. The main canonical formulations, including the Arnowitt-Deser-Misner (ADM) formalism and Ashtekar variables, are derived and discussed. Ideal for both graduate students and researchers, this book provides a link between standard introductions to general relativity and advanced expositions of black hole physics, theoretical cosmology or quantum gravity.
* Gives a thorough account of gravity theory, from advanced mathematical foundations to modern applications in cosmology and black hole physics * Provides mathematical foundations as well as physical applications to give a systematic explanation of canonical methods in gravity * Touches on large areas of theoretical gravitational research: cosmology, black holes, quantum gravity
1. Introduction; 2. Isotropic cosmology: a prelude; 3. Hamiltonian formulation of general relativity; 4. Model systems and perturbations; 5. Global and asymptotic properties; 6. Quantum gravity; Appendix; References; Index.
Series: Classroom Resource Materials
Hardback (ISBN-13: 9780883857687)
Page extent: 316 pages
Size: 254 x 178 mm
Weight: 0.71 kg
te students to advanced problem solving and undergraduate research in two ways. Firstly, it provides a colourful tour of classical analysis which places a wide variety of problems in their historical context. Secondly, it helps students gain an understanding of mathematical discovery and proof. In demonstrating a variety of possible solutions to the same sample exercise, the reader will come to see how the connections between apparently inapplicable areas of mathematics can be exploited in problem-solving. This book will serve as excellent preparation for participation in mathematics competitions, as a valuable resource for undergraduate mathematics reading courses and seminars and as a supplement text in a course on analysis. It can also be used in independent study, since the chapters are free-standing.
* Understanding is facilitated by placing mathematical problems in their historical context * The interplay between continuous and discrete mathematics is emphasized by applying induction, recursion and combinatorics to traditional problems in classical analysis * Prepares the reader for participation in mathematics competitions, and includes many problems from previous competitions
Preface; 1. Two classical inequalities; 2. A new approach for proving inequalities; 3. Means generated by an integral; 4. The L'Hopital monotone rule; 5. Trigonometric identities via complex numbers; 6. Special numbers; 7. On a sum of cosecants; 8. The gamma products in simple closed forms; 9. On the telescoping sums; 10. Summation of subseries in closed form; 11. Generating functions for powers of Fibonacci numbers; 12. Identities for the Fibonacci powers; 13. Bernoulli numbers via determinants; 14. On some finite trigonometric power sums; 15. Power series; 16. Six ways to sum Ā(2); 17. Evaluations of some variant Euler sums; 18. Interesting series involving binomial coefficients; 19. Parametric differentiation and integration; 20. Four ways to evaluate the Poisson integral; 21. Some irresistible integrals; Solutions to selected problems.
Hardback (ISBN-13: 9781107003873)
11 b/w illus. 2 tables
Page extent: 328 pages
Size: 253 x 215 mm
This interdisciplinary study of infinity explores the concept through the prism of mathematics and then offers more expansive investigations in areas beyond mathematical boundaries to reflect the broader, deeper implications of infinity for human intellectual thought. More than a dozen world-renowned researchers in the fields of mathematics, physics, cosmology, philosophy and theology offer a rich intellectual exchange among various current viewpoints, rather than displaying a static picture of accepted views on infinity. The book starts with a historical examination of the transformation of infinity from a philosophical and theological study to one dominated by mathematics. It then offers technical discussions on the understanding of mathematical infinity. Following this, the book considers the perspectives of physics and cosmology: can infinity be found in the real universe* Finally, the book returns to questions of philosophical and theological aspects of infinity.
* Gathers disparate disciplines to explore the concept and meaning of infinity * Accessible to readers without advanced mathematics and scientific backgrounds who are deeply interested in a topic * Book shows a 'dispute in action', polemics between different viewpoints and opinions rather than a static panorama of accepted views
Introduction Rudy Rucker; Part I. Perspectives on Infinity from History: 1. Infinity as a transformative concept in science and theology Wolfgang Achtner; Part II. Perspectives on Infinity from Mathematics: 2. The mathematical infinity Enrico Bombieri; 3. Warning signs of a possible collapse of contemporary mathematics Edward Nelson; Part III. Technical Perspectives on Infinity from Advanced Mathematics: 4. The realm of the infinite W. Hugh Woodin; 5. A potential subtlety concerning the distinction between determinism and nondeterminism W. Hugh Woodin; 6. Concept calculus: much better than Harvey M. Friedman; Part IV. Perspectives on Infinity from Physics and Cosmology: 7. Some considerations on infinity in physics Carlo Rovelli; 8. Cosmological intimations of infinity Anthony Aguirre; 9. Infinity and the nostalgia of the stars Marco Bersanelli; 10. Infinities in cosmology Michael Heller; Part V. Perspectives on Infinity from Philosophy and Theology: 11. God and infinity: directions for future research Graham Oppy; 12. Notes on the concept of the infinite in the history of Western metaphysics David Bentley Hart; 13. God and infinity: theological insights from Cantor's mathematics Robert J. Russell; 14. A partially skeptical response to Hart and Russell Denys A. Turner.
Hardback (ISBN-13: 9780521761581)
Paperback (ISBN-13: 9780521132503)
768 exercises
Page extent: 536 pages
Size: 253 x 215 mm
This second edition of Daniel W. Stroock's text is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis. It is intended to provide readers with an introduction to probability theory and the analytic ideas and tools on which the modern theory relies. It includes more than 750 exercises. Much of the content has undergone significant revision. In particular, the treatment of Levy processes has been rewritten, and a detailed account of Gaussian measures on a Banach space is given.
* Presents a novel selection and treatment of probability theory * The reader will see how probability theory can be used in other branches of mathematics
1. Sums of independent random variables; 2. The central limit theorem; 3. Infinitely divisible laws; 4. Levy processes; 5. Conditioning and martingales; 6. Some extensions and applications of martingale theory; 7. Continuous parameter martingales; 8. Gaussian measures on a Banach space; 9. Convergence of measures on a Polish space; 10. Wiener measure and partial differential equations; 11. Some classical potential theory.