216 pages | 200 line illustrations | 246x189mm
978-0-19-922867-6 | Hardback | 06 January 2011
978-0-19-922868-3 Paperback 06 January 2011
Concise introduction: only 200 pages.
Self-contained: provides tutorial background on the mathematics and physics usually assumed.
Friendly tutorial style: clearly written, logical and easy to read.
Well illustrated: nearly 200 diagrams.
Unified approach: all the theory is linked through a Fourier transform.
The opportunities for doing scattering experiments at synchrotron and neutron facilities have grown rapidly in recent years and are set to continue to do so into the foreseeable future. This text provides a basic understanding of how these techniques enable the structure and dynamics of materials to be studied at the atomic and molecular level. Although mathematics cannot be avoided in a theoretical discussion, the aim has been to write a book that most scientists will still find approachable. To this end, the first two chapters are devoted to providing a tutorial background in the mathematics and physics that are implicitly assumed in other texts. Thereafter, the philosophy has been one of keeping things as simple as possible.
Users of synchrotron and neutron facilities: mostly graduate students and research scientists; also structural biologists, inorganic and physical chemists, condensed matter physicists, senior undergraduates in physics.
Part I: Some preliminaries
1: Studying matter at the atomic and molecular level
2: Waves, complex numbers and Fourier transforms
Part II: Elastic scattering
3: The basics of X-ray and neutrons scattering
4: Surfaces, interfaces and reflectivity
5: Small-angle scattering and the big picture
6: Liquids and amorphous materials
7: Periodicity, symmetry and crystallography
Part III: Inelastic scattering
8: Energy exchange and dynamical information
9: Examples of inelastic scattering
256 pages | 38 line drawings | 246x171mm
978-0-19-923486-8 | Hardback | August 2011 (estimated)
The name "random walk" for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of "Nature". The same year, a similar problem was formulated by Albert Einstein in one of his Annus Mirabilis works. Even earlier such a problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays the theory of random walks has proved useful in physics and chemistry (diffusion, reactions, mixing in flows), economics, biology (from animal spread to motion of subcellular structures) and in many other disciplines. The random walk approach serves not only as a model of simple diffusion but of many complex sub- and super-diffusive transport processes as well. This book discusses the main variants of random walks and gives the most important mathematical tools for their theoretical description.
Undergraduate and graduate students in physics and chemistry and lecturers in the same disciplines; postgraduate students in the same disciplines and in biology; scientific workers entering a field where random walk models are used.
1: Characteristic Functions
2: Generating Functions and Applications
3: Continuous Time Random Walks
4: CTRW and Aging Phenomena
5: Master Equations
6: Fractional Diffusion and Fokker-Planck Equations for Subdiffusion
7: Levy Flights
8: Coupled CTRW and Levy Walks
9: Simple Reactions: A+B->B
10: Random Walks on Percolation Structures
320 pages | 77 b&w | 235x156mm
978-0-19-976711-3 | Hardback | October 2011 (estimated)
Expander families enjoy a wide range of applications in mathematics and computer science, and their study is a fascinating one in its own right. Expander Families and Cayley Graphs: A Beginner's Guide provides an introduction to the mathematical theory underlying these objects.
The central notion in the book is that of expansion, which roughly means the quality of a graph as a communications network. Cayley graphs are certain graphs constructed from groups; they play a prominent role in the study of expander families. The isoperimetric constant, the second largest eigenvalue, the diameter, and the Kazhdan constant are four measures of the expansion quality of a Cayley graph. The book carefully develops these concepts, discussing their relationships to one another and to subgroups and quotients as well as their best-case growth rates.
Topics include graph spectra (i.e., eigenvalues); a Cheeger-Buser-type inequality for regular graphs; group quotients and graph coverings; subgroups and Schreier generators; the Alon-Boppana theorem on the second largest eigenvalue of a regular graph; Ramanujan graphs; diameter estimates for Cayley graphs; the zig-zag product and its relation to semidirect products of groups; eigenvalues of Cayley graphs; Paley graphs; and Kazhdan constants.
The book was written with undergraduate math majors in mind; indeed, several dozen of them field-tested it. The prerequisites are minimal: one course in linear algebra, and one course in group theory. No background in graph theory or representation theory is assumed; the book develops from scatch the required facts from these fields. The authors include not only overviews and quick capsule summaries of key concepts, but also details of potentially confusing lines of reasoning.
The book contains ideas for student research projects (for capstone projects, REUs, etc.), exercises (both easy and hard), and extensive notes with references to the literature.
Readership: Undergraduate students, graduate students, and faculty in mathematics
Preface
Notations and conventions
Introduction
Part 1. Basics
Chapter 1. Graph eigenvalues and the isoperimetric constant
Chapter 2. Subgroups and quotients
Chapter 3. The Alon-Boppana theorem
Part 2. Combinatorial techniques
Chapter 4. Diameters of Cayley graphs and expander families
Chapter 5. Zig-zag products
Part 3. Representation-theoretic techniques
Chapter 6. Representations of Finite Groups
Chapter 7. Representation theory and eigenvalues of Cayley graphs
Chapter 8. Kazhdan constants
Appendix A. Linear algebra
Appendix B. Asymptotic analysis of functions
Bibliography
Index
424 pages | 235x156mm
978-0-19-979373-0 | Hardback | November 2011 (estimated)
Advanced Topics in Linear Algebra presents, in an engaging style, novel topics linked through the Weyr matrix canonical form, a largely unknown cousin of the Jordan canonical form discovered by Eduard Weyr in 1885. The book also develops much linear algebra unconnected to canonical forms, that has not previously appeared in book form. It presents common applications of Weyr form, including matrix commutativity problems, approximate simultaneous diagonalization, and algebraic geometry, with the latter two having topical connections to phylogenetic invariants in biomathematics and multivariate interpolation. The Weyr form clearly outperforms the Jordan form in many situations, particularly where two or more commuting matrices are involved, due to the block upper triangular form a Weyr matrix forces on any commuting matrix.
In this book, the authors develop the Weyr form from scratch, and include an algorithm for computing it. The Weyr form is also derived ring-theoretically in an entirely different way to the classical derivation of the Jordan form. A fascinating duality exists between the two forms that allows one to flip back and forth and exploit the combined powers of each. The book weaves together ideas from various mathematical disciplines, demonstrating dramatically the variety and unity of mathematics. Though the book's main focus is linear algebra, it also draws upon ideas from commutative and noncommutative ring theory, module theory, field theory, topology, and algebraic geometry.
Advanced Topics in Linear Algebra offers self-contained accounts of the non-trivial results used from outside linear algebra, and lots of worked examples, thereby making it accessible to graduate students. Indeed, the scope of the book makes it an appealing graduate text, either as a reference or for an appropriately designed one or two semester course. A number of the authors' previously unpublished results appear as well.
Readership: Mathematicians, upper-level graduate students in mathematics, and those who coduct research in linear algebra.
Preface
Chapter 1. Background Linear Algebra
Chapter 2. The Weyr Form
Chapter 3. Centralizers
Chapter 4. The Module Setting
Chapter 5. Gerstenhaber's Theorem
Chapter 6. Approximate Simultaneous Diagonalization
Chapter 7. Algebraic Varieties
Bibliography