Series: Cambridge Studies in Advanced Mathematics (No. 112)
Hardback (ISBN-13: 9780521886512)
Page extent: 224 pages
Size: 228 x 152 mm
This book deals with equations that have played a central role in the interplay between partial differential equations and probability theory. Most of this material has been treated elsewhere, but it is rarely presented in a manner that makes it easy to understand for people whose background is probability theory. Many results are given new proofs designed for readers with limited expertise in analysis. The author covers the theory of linear, second order partial differential equations of parabolic and elliptic type. Many of the techniques have antecedents in probability theory, although the book also covers a few purely analytic techniques. In particular, a chapter is devoted to the DeGiorgi-Moser-Nash estimates and the concluding chapter gives an introduction to the theory of pseudodifferential operators and their application to hypoellipticity, including the famous theorem of Lars Hormander.
* Minimal demands on reader for prior knowledge of partial differential
equations * Proofs are designed for readers with limited analytical background
* The selection of material is based on the author’s own experience during
his career
Contents
1. Kolmogorov’s forward, basic results; 2. Non-elliptic regularity results; 3. Preliminary elliptic regularity results; 4. Nash theory; 5. Localization; 6. On a manifold; 7. Subelliptic estimates and Hormander’s theorem.
Hardback (ISBN-13: 9780521849319)
Paperback (ISBN-13: 9780521614108)
126 line diagrams 30 half-tones 24 tables 203 exercises
Page extent: 512 pages
Size: 253 x 177 mm
This textbook, for second- or third-year students of computer science, presents insights, notations, and analogies to help them describe and think about algorithms like an expert, without grinding through lots of formal proof. Solutions to many problems are provided to let students check their progress, while class-tested PowerPoint slides are on the web for anyone running the course. By looking at both the big picture and easy step-by-step methods for developing algorithms, the author guides students around the common pitfalls. He stresses paradigms such as loop invariants and recursion to unify a huge range of algorithms into a few meta-algorithms. The book fosters a deeper understanding of how and why each algorithm works. These insights are presented in a careful and clear way, helping students to think abstractly and preparing them for creating their own innovative ways to solve problems.
* Includes lots of exercises, with selected solutions in the text; PowerPoint
slides for instructors available from the web * Presents the big picture
and step-by-step methods for developing algorithms, while avoiding common
pitfalls * Appendices give background on big-O notation, logic, asymptotics,
and other mathematical tools
Contents
Part I. Iterative Algorithms and Loop Invariants: 1. Measures of progress and loop invariants; 2. Examples using more of the input loop invariant; 3. Abstract data types; 4. Narrowing the search space: binary search; 5. Iterative sorting algorithms; 6. Euclid’s GCD algorithm; 7. The loop invariant for lower bounds; Part II. Recursion: 8. Abstractions, techniques, and theory; 9. Some simple examples of recursive algorithms; 10. Recursion on trees; 11. Recursive images; 12. Parsing with context-free grammars; Part III. Optimization Problems: 13. Definition of optimization problems; 14. Graph search algorithms; 15. Network flows and linear programming; 16. Greedy algorithms; 17. Recursive backtracking; 18. Dynamic programming algorithms; 19. Examples of dynamic programming; 20. Reductions and NP-completeness; 21. Randomized algorithms; Part IV. Appendix: 22. Existential and universal quantifiers; 23. Time complexity; 24. Logarithms and exponentials; 25. Asymptotic growth; 26. Adding made easy approximations; 27. Recurrence relations; 28. A formal proof of correctness; Part V. Exercise Solutions.
Hardback (ISBN-13: 9780521876933)
5 line diagrams 20 half-tones
Page extent: 210 pages
Size: 247 x 174 mm
Since its emergence in the early twentieth century, quantum theory has become the fundamental physical paradigm, and is essential to our understanding of the world. Providing a deeper understanding of the microscopic world through quantum theory, this supplementary text reviews a wider range of topics than conventional textbooks. Emphasis is given to modern entanglement, quantum teleportation, and Bose-Einstein condensation. Macroscopic quantum effects of practical relevance, for example superconductivity and the quantum Hall effect, are also described. Looking to the future, the author discusses the exciting prospects for quantum computing. Physical, rather than formal, explanations are given, and mathematical formalism is kept to a minimum so readers can understand the concepts more easily. Theoretical discussions are combined with a description of the corresponding experimental results. This book is ideal for undergraduate and graduate students in quantum theory and quantum optics.
* Provides a deeper understanding of the microscopic world through quantum
theory * Covers a wide spectrum of topics, including modern achievements
such as Bose-Einstein condensation * Physical explanations are favoured
over mathematical reasoning, making the book easy to read
Contents
Preface; 1. Unexpected findings; 2. Quantum states; 3. Measurement; 4. Correlations; 5. Philosophy; 6. Interaction; 7. Conservation laws; 8. Spin and statistics; 9. Macroscopic quantum effects; 10. Quantum computing; References; Index.
Series: Cambridge Studies in Advanced Mathematics (No. 113)
Hardback (ISBN-13: 9780521889698)
14 line diagrams 3 half-tones 3 tables 90 exercises 11 figures 2 colour figures 80 worked examples
Page extent: 300 pages
Size: 228 x 152 mm
With roots in the nineteenth century, Lie theory has since found many and varied applications in mathematics and mathematical physics, to the point where it is now regarded as a classical branch of mathematics in its own right. This graduate text focuses on the study of semisimple Lie algebras, developing the necessary theory along the way. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semisimple Lie algebras. Written in an informal style, this is a contemporary introduction to the subject which emphasizes the main concepts of the proofs and outlines the necessary technical details, allowing the material to be conveyed concisely. Based on a lecture course given by the author at the State University of New York at Stony Brook, the book includes numerous exercises and worked examples and is ideal for graduate courses on Lie groups and Lie algebras.
* Exposition emphasizes the main concepts rather than technical details
of the proofs, thus making it possible to cover a lot of material in relatively
concise work * Numerous exercises and worked examples, as well as a sample
syllabus, make this an ideal text for a graduate course on Lie Groups and
Lie Algebras * Focusses on semisimple Lie algebras and their representations;
contains material rarely included in standard textbooks such as BGG resolution
Contents
Preface; 1. Introduction; 2. Lie groups: Basic definitions; 3. Lie groups and Lie algebras; 4. Representations of Lie groups and Lie algebras; 5. Structure theory of Lie algebras; 6. Complex Semisimple Lie algebras; 7. Root systems; 8. Representations of Semisimple Lie Algebras; Overview of the literature; A. Root systems and simple Lie algebras; B. Sample syllabus; List of notation; Index; Bibliography.
Series: Encyclopedia of Mathematics and its Applications (No. 123)
Hardback (ISBN-13: 9780521878647)
119 line diagrams 1 table
Page extent: 352 pages
Size: 234 x 156 mm
Algorithmic Aspects of Graph Connectivity is the first comprehensive book on this central notion in graph and network theory, emphasizing its algorithmic aspects. Because of its wide applications in the fields of communication, transportation, and production, graph connectivity has made tremendous algorithmic progress under the influence of the theory of complexity and algorithms in modern computer science. The book contains various definitions of connectivity, including edge-connectivity and vertex-connectivity, and their ramifications, as well as related topics such as flows and cuts. The authors comprehensively discuss new concepts and algorithms that allow for quicker and more efficient computing, such as maximum adjacency ordering of vertices. Covering both basic definitions and advanced topics, this book can be used as a textbook in graduate courses in mathematical sciences, such as discrete mathematics, combinatorics, and operations research, and as a reference book for specialists in discrete mathematics and its applications.
* Starting with basic materials of graph theory, this book covers up-to-date
topics and algorithms in the area of graph connectivity
Contents
1. Introduction; 2. MA ordering and forest decompositions; 3. Minimum cuts; 4. Cut enumeration; 5. Cactus representations; 6. Extreme vertex sets; 7. Edge-splitting; 8. Connectivity augmentation; 9. Source location problems; 10. Submodular and posi-modular set functions.