Series: Encyclopedia of Mathematics and its Applications (No. 140)
Hardback (ISBN-13: 9781107002029)
42 b/w illus. 1000 exercises
Page extent: 530 pages
Size: 234 x 156 mm
Real Analysis Through Modern Infinitesimals provides a course on mathematical analysis based on Internal Set Theory (IST) introduced by Edward Nelson in 1977. After motivating IST through an ultrapower construction, the book provides a careful development of this theory representing each external class as a proper class. This foundational discussion, which is presented in the first two chapters, includes an account of the basic internal and external properties of the real number system as an entity within IST. In its remaining fourteen chapters, the book explores the consequences of the perspective offered by IST as a wide range of real analysis topics are surveyed. The topics thus developed begin with those usually discussed in an advanced undergraduate analysis course and gradually move to topics that are suitable for more advanced readers. This book may be used for reference, self-study, and as a source for advanced undergraduate or graduate courses.
* Will appeal to readers with no background in mathematical logic * Emphasis on applications is ideal for readers seeking experience in applying modern infinitesimals * Flexible teaching resource - instructors can apply the material to a wide variety of courses
Preface; Introduction; Part I. Elements of Real Analysis: 1. Internal set theory; 2. The real number system; 3. Sequences and series; 4. The topology of R; 5. Limits and continuity; 6. Differentiation; 7. Integration; 8. Sequences and series of functions; 9. Infinite series; Part II. Elements of Abstract Analysis: 10. Point set topology; 11. Metric spaces; 12. Complete metric spaces; 13. Some applications of completeness; 14. Linear operators; 15. Differential calculus on Rn; 16. Function space topologies; A. Vector spaces; B. The b-adic representation of numbers; C. Finite, denumerable, and uncountable sets; D. The syntax of mathematical languages; References; Index.
IRMA Lectures in Mathematics and Theoretical Physics Vol. 16
ISBN 978-3-03719-079-1
June 2010, 964 pages, hardcover, 17 x 24 cm.
The purpose of this handbook is to give an overview of some recent developments in differential geometry related to supersymmetric field theories. The main themes covered are:
special geometry and supersymmetry
generalized geometry
geometries with torsion
para-geometries
holonomy theory
symmetric spaces and spaces of constant curvature
conformal geometry
wave equations on Lorentzian manifolds
D-branes and K-theory
The intended audience consists of advanced students and researchers working in differential geometry, string theory and related areas. The emphasis is on geometrical structures occurring on target spaces of supersymmetric field theories. Some of these structures can be fully described in the classical framework of pseudo-Riemannian geometry. Others lead to new concepts relating various fields of research, such as special Kahler geometry or generalized geometry.
EMS Tracts in Mathematics Vol. 12
ISBN 978-3-03719-084-5
DOI 10.4171/084
July 2010, 675 pages, hardcover, 17 x 24 cm.
This three-volume set is a comprehensive study of the tractability of multivariate problems. The present second volume deals with algorithms using standard information consisting of function values for the approximation of linear and selected nonlinear functionals. An important example is numerical multivariate integration.
The proof techniques used in volumes I and II are quite different. It is especially hard to establish meaningful lower error bounds for the approximation of functionals by using finitely many function values. Here, the concept of decomposable reproducing kernels is helpful, allowing it to find matching lower and upper error bounds for some linear functionals. It is then possible to conclude tractability results from such error bounds.
Tractability results even for linear functionals are very rich in variety. There are infinite-dimensional Hilbert spaces for which the approximation with an arbitrarily small error of all linear functionals requires only one function value. There are Hilbert spaces for which all nontrivial linear functionals suffer from the curse of dimensionality. This holds for unweighted spaces, where the role of all variables and groups of variables is the same. For weighted spaces one can monitor the role of all variables and groups of variables. Necessary and sufficient conditions on the decay of the weights are given to obtain various notions of tractability.
The text contains extensive chapters on discrepancy and integration, decomposable kernels and lower bounds, the Smolyak/sparse grid algorithms, lattice rules and the CBC (component-by-component) algorithms. This is done in various settings. Path integration and quantum computation are also discussed.
The book is of interest for researchers working in computational mathematics, especially in approximation of high-dimensional problems. It is also well suited for graduate courses and seminars. 61 open problems are listed to stimulate future research in tractability.
ISBN: 978-0-470-62451-7
Hardcover
384 pages
October 2010
Written by an avid user of topological groups, this book provides a concrete introduction to metric and topological groups. Important metric spaces are first introduced with the construction of a concrete metric, and a topological characterization of their open sets is immediately established. Many proofs are presented in a purely topological fashion. It is this use of topologies that affords the opportunity to discuss the limitations of some of the metric group results as well as the topological machinery needed to generalize them. The book focuses on metric cases, thus keeping many proofs in familiar territory for readers new to the topic. Readers learn how to replace the sequences with nets to obtain a general proof, and the result is an increased emphasis on topological groups and less on general topology. The exercises have been designed to help hold the interest of advanced and beginning readers, and a variety of calculations, remarks, and necessary related facts have been placed within the exercises and are used throughout the text. When used in proofs, the calculations, remarks, and facts are carefully referred to so that advanced readers will know what is needed in the proof and will be able to move forward at a faster pace. Novice readers will have to reference the exercise and workout a solution to become more engaged with the material.
380 pages | 168 b/w line and halftone illustrations | 246x189mm
978-0-19-959457-3 | Hardback | November 2010 (estimated)
978-0-19-959458-0 | Paperback
Elucidates the paradox of chaotic motion
Discusses chaotic systems without the use of calculus
Companion CD, with point-and-click animated experiments
Uniquely comprehensive discussion at this level
Includes historical perspective on chaos theory
Based on only elementary mathematics, this engaging account of chaos theory bridges the gap between introductions for the layman and college-level texts. It develops the science of dynamics in terms of small time steps, describes the phenomenon of chaos through simple examples, and concludes with a close look at a homoclinic tangle, the mathematical monster at the heart of chaos. The presentation is enhanced by many figures, animations of chaotic motion (available on a companion CD), and biographical sketches of the pioneers of dynamics and chaos theory. To ensure accessibility to motivated high school students, care has been taken to explain advanced mathematical concepts simply, including exponentials and logarithms, probability, correlation, frequency analysis, fractals, and transfinite numbers. These tools help to resolve the intriguing paradox of motion that is predictable and yet random, while the final chapter explores the various ways chaos theory has been put to practical use.
Readership: General readers, high-school and undergraduate students, and practising scientists who want a quick and easily accessible introduction to chaos theory.
1: Chaos Everywhere
2: Galileo Galilei --- Birth of a New Science
3: Isaac Newton --- Dynamics Perfected
4: Celestial Mechanics --- Clockwork Universe
5: Pendulum --- Linear and Nonlinear
6: Josephson Effect --- Synchronization
7: Chaos Forgets the Past
8: Chaos Takes a Random Walk
9: Chaos Makes Noise
10: Edward Lorenz --- Butterfly Effect
11: Chaos Comes of Age
12: Tilt-A-Whirl --- Chaos at the Amusement Park
13: Billiard-Ball Chaos --- Atomic Disorder
14: Iterated Maps --- Chaos Made Simple
15: State Space --- Going with the Flow
16: Strange Attractor
17: Fractal Geometry
18: Stephen Smale --- Horseshoe Map
19: Henri Poincare --- Topological Tangle
20: Chaos Goes to Work