V. Armitage / W. F. Eberlein

Elliptic Functions

Series: London Mathematical Society Student Texts (No. 67)
Paperback (ISBN-13: 9780521785631 | ISBN-10: 0521785634)
Hardback (ISBN-13: 9780521780780 | ISBN-10: 0521780780)

In its first six chapters this text seeks to present the basic ideas and properties of the Jacobi elliptic functions as an historical essay, an attempt to answer the fascinating question: 'what would the treatment of elliptic functions have been like if Abel had developed the ideas, rather than Jacobi?' Accordingly, it is based on the idea of inverting integrals which arise in the theory of differential equations and, in particular, the differential equation that describes the motion of a simple pendulum. The later chapters present a more conventional approach to the Weierstrass functions and to elliptic integrals, and then the reader is introduced to the richly varied applications of the elliptic and related functions. Applications spanning arithmetic (solution of the general quintic, the functional equation of the Riemann zeta function), dynamics (orbits, Euler's equations, Green's functions), and also probability and statistics, are discussed.

* The emphasis on applications makes the book suitable for a lecture course in which students are invited to develop ideas as a basis for essays or projects ｿ Applications are referred to throughout and the book concludes with significant applications in algebra, arithmetic, classical geometry and applied mathematics ｿ A new look at the Jacobi functions based on a rigorous treatment of the differential equation for the simple pendulum and the inversion of integrals

Contents

1. The 'simple' pendulum; 2. Jacobian elliptic functions of a complex variable; 3. General properties of elliptic functions; 4. Theta functions; 5. The Jacobian elliptic functions for complex k; 6. Introduction to transformation theory; 7. The Weierstrass elliptic functions; 8. Elliptic integrals; 9. Applications of elliptic functions in geometry; 10. An application of elliptic functions in algebra solution of the general quintic equation; 11. An arithmetic application of elliptic functions; 12. Applications in mechanics and statistics and other topics; Appendix; Bibliography.

B. Bollobas / Trinity College, Cambridge
O. Riordan / Trinity College, Cambridge

Percolation

Hardback (ISBN-13: 9780521872324 | ISBN-10: 0521872324)

Percolation theory was initiated some fifty years ago as a mathematical framework for the study of random physical processes such as flow through a disordered porous medium. It has proved to be a remarkably rich theory, with applications beyond natural phenomena to topics such as network modelling. The aims of this book are twofold. First to present classical results in a way that is accessible to non-specialists. Second, to describe, for the first time in a book, recent results of Smirnov in conformal invariance, and outline the proof that the critical probability for random Voronoi percolation in the plane is 1/2. Throughout, the presentation is streamlined, with elegant and straightforward proofs requiring minimal background in probability and graph theory. Numerous examples illustrate the important concepts and enrich the arguments. All-in-all, it will be an essential purchase for mathematicians, physicists, electrical engineers and computer scientists working in this exciting area.

* Only book that systematically presents probabilistic and graph theory tools for percolation, and that describes Smirnov's results in conformal invariance ? Streamlined presentation requiring only minimal background in probability and graph theory * Numerous examples illustrate the important concepts and enrich the arguments

Contents

Preface; 1. Basic concepts; 2. Probabilistic tools; 3. Percolation on Z2 - the Harris-Kesten Theorem; 4. Exponential decay and critical probabilities - theorems of Menshikov and Aizenman & Barsky; 5. Uniqueness of the infinite open cluster and critical probabilities; 6. Estimating critical probabilities; 7. Conformal invariance - Smirnov's Theorem; 8. Continuum percolation; Bibliography; Index; List of notation.

Edited by Jesus M. F. Castillo / Universidad de Extremadura
William B. Johnson / Texas A & M University

Methods in Banach Space Theory

Series: London Mathematical Society Lecture Note Series (No. 337)
Paperback (ISBN-13: 9780521685689 | ISBN-10: 0521685680)

This book presents an overview of modern Banach space theory. It contains sixteen papers that reflect the wide expanse of the subject. Articles are gathered into five sections according to methodology rather than the topics considered. The sections are: geometrical methods; homological methods; topological methods; operator theoretic methods; and also function space methods. Each section contains survey and research papers describing the state-of-the-art in the topic considered as well as some of the latest most important results. Researchers working in Banach space theory, functional analysis or operator theory will find much of interest here.

* Presents overview of modern Banach space theory * Contains most up-to-date and important results in this topic * Ideal for all researchers working in Banach space theory, functional analysis or operator theory

Contents

Acknowledgements; Foreword; Part I. Geometrical Methods: 1. Saturated extensions, the attractors method and hereditarily James tree spaces Spiros A. Agyros, Alexander D. Arvanitakis and Andreas G. Tolias; 2. The Daugavet property for Lindenstrauss spaces J. Becerra and M. Martin; 3. Weakly null sequences in the Banach space I. Gasparis, E. Odell and B. Wahl; Part II. Homological Methods: 4. Yet another proof of Sobczyk's theorem Felix Cabello Sanchez; 5. The category of exact sequences of Banach spaces Jesus M. F. Castillo and Yolanda Moreno; 6. Extension problems for C(K)spaces and twisted sums N. J. Kalton; 7. Palamodov's questions from homological methods in the theory of locally convex spaces Jochen Wengenroth; Part III. Topological Methods: 8. Ordinal representability in Banach spaces M. J. Campion, J. C. Candeal, A. S. Granero and E. Indurain; 9. Overclasses of the class of Radon-Nikodym compact spaces Marian Fabian; 10. Convexity, compactness and distances A. S. Granero and Marcos Sanchez; Part IV. Operator Theory Methods: 11. Weyl's and Browder's theorems through the quasi-nilpotent part of an operator Pietro Aiena and Maria Teresa Biondi; 12. Multiplications and elementary operators in the Banach space setting Eero Saksman and Hans-Olav Tylli; 13. Interpolation of Banach spaces by the ?-method Jesus Suarez and Lutz Weis; Part V. Function Space Methods: 14. Solvability of an integral equation in BC(R+) J. Caballero, B. Lopez and K. Sadarangani; 15. Harold Bohr meets Stefan Banach Andreas Defant and Christopher Prengel; 16. Selected problems on the structure of complemented subspaces of Banach spaces Aleksander Pelczynski; List of participants.

Contributors

Spiros A. Agyros, Alexander D. Arvanitakis, Andreas G. Tolias, J. Becerra, M. Martin, I. Gasparis, E. Odell, B. Wahl, Felix Cabello Sanchez, Jesus M. F. Castillo, Yolanda Moreno, N. J. Kalton, Jochen Wengenroth, M. J. Campion, J. C. Candeal, A. S. Granero, E. Indurain, Marian Fabian, Marcos Sanchez, Pietro Aiena, Maria Teresa Biondi, Eero Saksman, Hans-Olav Tylli, Jesus Suarez, Lutz Weis, J. Caballero, B. Lopez, K. Sadarangani, Andreas Defant, Christopher Prengel, Aleksander Pelczynski

Richard J. Gardner / Western Washington University

Geometric Tomography, 2nd Edition

Series: Encyclopedia of Mathematics and its Applications
Hardback (ISBN-13: 9780521866804 | ISBN-10: 0521866804)
Paperback (ISBN-13: 9780521684934 | ISBN-10: 0521684935)

Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes. It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. The subject overlaps with convex geometry and employs many tools from that area, including some formulas from integral geometry. It also has connections to discrete tomography, geometric probing in robotics and to stereology. This comprehensive study provides a rigorous treatment of the subject. Although primarily meant for researchers and graduate students in geometry and tomography, brief introductions, suitable for advanced undergraduates, are provided to the basic concepts. More than 70 illustrations are used to clarify the text. The book also presents 66 unsolved problems. Each chapter ends with extensive notes, historical remarks, and some biographies. This new edition includes numerous updates and improvements, with some 300 new references bringing the total to over 800.

* Over 70 illustrations * Comprehensive bibliography has over 800 references * Well-written and clear; the opening chapters are accessible to advanced undergraduates

Contents

0. Background material; 1. Parallel X-rays of planar convex bodies; 2. Parallel X-rays in n dimensions; 3. Projections and projection functions; 4. Projection bodies and volume inequalities; 5. Point X-rays; 6. Chord functions and equichordal problems; 7. Sections, section functions, and point X-rays; 8. Intersection bodies and volume inequalities; 9. Estimates from projection and section functions; A. Mixed volumes and dual mixed volumes; B. Inequalities; C. Integral transforms.

Alexandra Shlapentokh / East Carolina University

Hilbert's Tenth Problem
Diophantine Classes and Extensions to Global Fields

Series: New Mathematical Monographs (No. 7)
Hardback (ISBN-13: 9780521833608 | ISBN-10: 0521833604)

In the late sixties Matiyasevich, building on the work of Davis, Putnam and Robinson, showed that there was no algorithm to determine whether a polynomial equation in several variables and with integer coefficients has integer solutions. Hilbert gave finding such an algorithm as problem number ten on a list he presented at an international congress of mathematicians in 1900. Thus the problem, which has become known as Hilbert's Tenth Problem, was shown to be unsolvable. This book presents an account of results extending Hilbert's Tenth Problem to integrally closed subrings of global fields including, in the function field case, the fields themselves. While written from the point of view of Algebraic Number Theory, the book includes chapters on Mazur's conjectures on topology of rational points and Poonen's elliptic curve method for constructing a Diophatine model of rational integers over a 'very large' subring of the field of rational numbers.

* Looks at the subject from the point of view of Algebraic Number Theory * Also includes information on Mazur's Conjectures and Poonen's elliptic curve method * Suitable for graduate students

Contents

1. Introduction; 2. Diophantine classes: definition and basic facts; 3. Diophantine equivalence and diophantine decidability; 4. Integrality at finitely many primes and divisibility of order at infinitely many primes; 5. Bound equations for number fields and their consequences; 6. Units of rings of W-integers of norm 1; 7. Diophantine classes over number fields; 8. Diophantine undecidability of function fields; 9. Bounds for function fields; 10. Diophantine classes over function fields; 11. Mazur's conjectures and their consequences; 12. Results of Poonen; 13. Beyond global fields; A. Recursion theory; B. Number theory; Bibliography; Index.