Series: Perspectives in Logic
Hardback (ISBN-13: 9780521884396)
Foundations of mathematics is the study of the most basic concepts and
logical structure of mathematics, with an eye to the unity of human knowledge.
Almost all of the problems studied in this book are motivated by an overriding
foundational question: What are the appropriate axioms for mathematics*
Through a series of case studies, these axioms are examined to prove particular
theorems in core mathematical areas such as algebra, analysis, and topology,
focusing on the language of second-order arithmetic, the weakest language
rich enough to express and develop the bulk of mathematics. In many cases,
if a mathematical theorem is proved from appropriately weak set existence
axioms, then the axioms will be logically equivalent to the theorem. Furthermore,
only a few specific set existence axioms arise repeatedly in this context,
which in turn correspond to classical foundational programs. This is the
theme of reverse mathematics, which dominates the first half of the book.
The second part focuses on models of these and other subsystems of second-order
arithmetic. Additional results are presented in an appendix.
* Includes revised and newly added material from the original ASL edition
1. Introduction; Part I. Development of Mathematics within Subsystems of Z2: 2. Recursive comprehension; 3. Arithmetical comprehension; 4. Weak Konig’s lemma; 5. Arithmetical transfinite recursion; 6. Pi 11 comprehension; Part II. Models of Subsystems of Z2: 7. Beta-models; 8. Omega-models; 9. Non-omega-models; 10. Additional results.
Series: Encyclopedia of Mathematics and its Applications
Hardback (ISBN-13: 9780521897303)
Sub-Riemannian manifolds are manifolds with the Heisenberg principle built in. This comprehensive text and reference begins by introducing the theory of sub-Riemannian manifolds using a variational approach in which all properties are obtained from minimum principles, a robust method that is novel in this context. The authors then present examples and applications, showing how Heisenberg manifolds (step 2 sub-Riemannian manifolds) might in the future play a role in quantum mechanics similar to the role played by the Riemannian manifolds in classical mechanics. Sub-Riemannian Geometry: General Theory and Examples is the perfect resource for graduate students and researchers in pure and applied mathematics, theoretical physics, control theory, and thermodynamics interested in the most recent developments in sub-Riemannian geometry.
* Useful to both pure and applied mathematicians as well as theoretical
physicists * Discusses the most recent development in sub-Riemannian geometry
* Provides both the theoretical tools to understand material with plenty
of examples
Part I. General Theory: 1. Introductory chapter; 2. Basic properties; 3. Horizontal connectivity; 4. Hamilton-Jacobi theory; 5. Hamiltonian formalism; 6. Lagrangian formalism; 7. Connections on sub-Riemannian manifolds; 8. Gauss’ theory of sub-Riemannian manifolds; Part II. Examples and Applications: 9. Heisenberg manifolds; 10. Examples of Heisenberg manifolds; 11. Grushin manifolds; 12. Hormander manifolds. Appendix A: local non-solvability; Appendix B: fibre bundles.
Hardback (ISBN-13: 9780521815987)
One-dimensional quantum systems show fascinating properties beyond the scope of the mean-field approximation. However, the complicated mathematics involved is a high barrier to non-specialists. Written for graduate students and researchers new to the field, this book is a self-contained account of how to derive the exotic quasi-particle picture from the exact solution of models with inverse-square interparticle interactions. The book provides readers with an intuitive understanding of exact dynamical properties in terms of exotic quasi-particles which are neither bosons nor fermions. Powerful concepts, such as the Yangian symmetry in the Sutherland model and its lattice versions, are explained. A self-contained account of non-symmetric and symmetric Jack polynomials is also given. Derivations of dynamics are made easier, and are more concise than in the original papers, so readers can learn the physics of one-dimensional quantum systems through the simplest model.
* Written for graduate students and researchers new to the field, providing
an intuitive understanding of the subject * Provides an insight of Yangian
theory and Jack polynomials in the context of theoretical physics * Derivations
of dynamics are easier and more concise than in the original papers, so
readers can learn the physics through the simplest model
Preface; 1. Introduction; Part I. Physical Properties: 2. Single-component Sutherland model; 3. Multi-component Sutherland model; 4. Spin chain with 1/r2 interactions; 5. SU(K) spin chain; 6. Supersymmetric t-J model with 1/r2 interaction; Part II. Mathematics Related to 1/r2 Systems: 7. Jack polynomials; 8. Yang-Baxter relations and orthogonal eigenbasis; 9. SU(K) and supersymmetric Yangians; 10. Uglov’s theory; Epilogue; References; List of symbols; Index.
Heritage of European Mathematics
ISBN 978-3-03719-059-3
December 2008, 144 pages, hardcover, 17 x 24 cm.
Thomas Harriot (c.*1560*1621) was a mathematician and astronomer, known
not only for his work in algebra and geometry, but also for his wide-ranging
interests in ballistics, navigation, and optics (he discovered the sine
law of refraction now known as Snell’s law).
By about 1614, Harriot had developed finite difference interpolation methods for navigational tables. In 1618 (or slightly later) he composed a treatise entitled ‘De numeris triangularibus et inde de progressionibus arithmeticis, Magisteria magna’, in which he derived symbolic interpolation formulae and showed how to use them. This treatise was never published and is here reproduced for the first time. Commentary has been added to help the reader to follow Harriot’s beautiful but almost completely nonverbal presentation. The introductory essay preceding the treatise gives an overview of the contents of the ‘Magisteria’ and describes its influence on Harriot’s contemporaries and successors over the next sixty years. Harriot’s method was not superseded until Newton, apparently independently, made a similar discovery in the 1660s. The ideas in the ‘Magisteria’ were spread primarily through personal communication and unpublished manuscripts, and so, quite apart from their intrinsic mathematical interest, their survival in England during the seventeenth century provides an important case study in the dissemination of mathematics through informal networks of friends and acquaintances.
EMS Monographs in Mathematics
ISBN 978-3-03719-068-5
January 2009, 600 pages, hardcover, 16.5 x 23.5 cm.
In 1965 Penrose introduced the fundamental concept of a trapped surface, on the basis of which he proved a theorem which asserts that a spacetime containing such a surface must come to an end. The presence of a trapped surface implies, moreover, that there is a region of spacetime, the black hole, which is inaccessible to observation from infinity.
A major challenge since that time has been to find out how trapped surfaces actually form, by analyzing the dynamics of gravitational collapse. The present monograph achieves this aim by establishing the formation of trapped surfaces in pure general relativity through the focusing of gravitational waves.
The theorems proved in the present monograph constitute the first foray
into the long-time dynamics of general relativity in the large, that is,
when the initial data are no longer confined to a suitable neighborhood
of trivial data. The main new method, the short pulse method, applies to
general systems of Euler*Lagrange equations of hyperbolic type, and provides
the means to tackle problems which have hitherto seemed unapproachable.
This monograph will be of interest to people working in general relativity, geometric analysis, and partial differential equations.