Series: Encyclopedia of Mathematics and its Applications (No. 38)
Paperback (ISBN-13: 9780521111560)
This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are based on deep facts from analysis and function theory, such as duality theory and comparison theorems; these are presented in chapters 1 and 3. In keeping with the author's intention to make the book as self-contained as possible, chapter 2 contains an introduction to polynomial and spline approximation. Chapters 4 to 7 apply the theory to specific classes of functions. The last chapter deals with n-widths and generalises some of the ideas of the earlier chapters. Each chapter concludes with commentary, exercises and extensions of results. A substantial bibliography is included. Many of the results collected here have not been gathered together in book form before, so it will be essential reading for approximation theorists.
Preface; 1. Best approximation and duality in extremal problems; 2. Polynomials and spline-functions as approximating tools; 3. Comparison theorems and inequalities for the norms of functions and their derivatives; 4. Polynomial approximation of classes of functions with bounded r-th derivative in Lp; 5. Spline approximation of classes of functions with bounded r-th derivative; 6. Exact constants in Jackson inequalities; 7. Approximation of classes of functions determined by modulus of continuity; 8. N-widths of functional classes and closely related extremal problems; Appendixes; References.
Series: London Mathematical Society Lecture Note Series (No. 361)
Paperback (ISBN-13: 9780521747660)
After a forty-year lull, the study of word-values in groups has sprung back into life with some spectacular new results in finite group theory. These are largely motivated by applications to profinite groups, including the solution of an old problem of Serre. This book presents a comprehensive account of the known results, both old and new. The more elementary methods are developed from scratch, leading to self-contained proofs and improvements of some classic results about infinite soluble groups. This is followed by a detailed introduction to more advanced topics in finite group theory, and a full account of the applications to profinite groups. The author presents proofs of some very recent results and discusses open questions for further research. This self-contained account is accessible to research students, but will interest all research workers in group theory.
* Self-contained account of word-values in groups, from first principles
* Contains complete proofs of old and new results * A user-friendly introduction
to new results in finite and pro-finite groups
Preface; 1. Generalities; 2. Verbally elliptic classes; 3. Words of infinite width; 4. Words and profinite groups; 5. Algebraic and analytic groups; Appendix; Bibliography; Index.
Hardback (ISBN-13: 9780521845632)
A student-friendly style, over 100 illustrations, and numerous exercises are brought together in this textbook for advanced undergraduate and beginning graduate students in physics and mathematics. Lewis Ryder develops the theory of general relativity in detail. Covering the core topics of black holes, gravitational radiation, and cosmology, he provides an overview of general relativity and its modern ramifications. The book contains chapters on gravitational radiation, cosmology, and connections between general relativity and the fundamental physics of the microworld. It explains the geometry of curved spaces and contains key solutions of Einstein's equations - the Schwarzschild and Kerr solutions. Mathematical calculations are worked out in detail, so students can develop an intuitive understanding of the subject, as well as learn how to perform calculations. The book also includes topics concerned with the relation between general relativity and other areas of fundamental physics. Password protected solutions for instructors are available at www.cambridge.org/9780521845632.
* Student-friendly style, over 100 illustrations, and several exercises
with solutions available online * Mathematical calculations are worked
out in detail to help develop an intuitive understanding of the subject
* Includes topics that overlap with other areas of fundamental physics,
to help the reader assess how GR relates to other physics subjects
Preface; Notation; Important formulae and physical constants; 1. Introduction; 2. Special relativity, non-inertial effects and electromagnetism; 3. Differential geometry I: vectors, forms and absolute differentiation; 4. Differential geometry II: geodesics and curvature; 5. Einstein field equations, the Schwarzschild solution and experimental test of general relativity; 6. Gravitomagnetic effects: gyroscopes and clocks; 7. Gravitational collapse and black holes; 8. Action principles, conservation laws and the Cauchy problem; 9. Gravitational radiation; 10. Cosmology; 11. Gravitation and field theory; References; Index.
Series: Cambridge Monographs on Mathematical Physics
Hardback (ISBN-13: 9780521877879)
Quantum field theory in curved spacetime has been remarkably fruitful. It can be used to explain how the large-scale structure of the universe and the anisotropies of the cosmic background radiation that we observe today first arose. Similarly, it provides a deep connection between general relativity, thermodynamics, and quantum field theory. This book develops quantum field theory in curved spacetime in a pedagogical style, suitable for graduate students. The authors present detailed, physically motivated, derivations of cosmological and black hole processes in which curved spacetime plays a key role. They explain how such processes in the rapidly expanding early universe leave observable consequences today, and how in the context of evaporating black holes, these processes uncover deep connections between gravitation and elementary particles. The authors also lucidly describe many other aspects of free and interacting quantized fields in curved spacetime.
* Emphasises physically motivated derivations to gradually introduce the
material to the reader * Takes the reader from introductory to research
level, by including comprehensive discussions of applications to cosmology
and black holes * Describes many other aspects of free and interacting
quantized fields in curved spacetime to provide a deeper understanding
of the material
Preface; Conventions and notation; 1. Quantum fields in Minkowski spacetime; 2. Basics of quantum fields in curved spacetimes; 3. Expectation values quadratic in fields; 4. Particle creation by black holes; 5. The one-loop effective action; 6. The effective action: non-gauge theories; 7. The effective action: gauge theories; Appendixes; References; Index.