Paperback (ISBN-13: 9780521036917)
First-passage properties underlie a wide range of stochastic processes, such as diffusion-limited growth, neuron firing and the triggering of stock options. This book provides a unified presentation of first-passage processes, which highlights its interrelations with electrostatics and the resulting powerful consequences. The author begins with a presentation of fundamental theory including the connection between the occupation and first-passage probabilities of a random walk, and the connection to electrostatics and current flows in resistor networks. The consequences of this theory are then developed for simple, illustrative geometries including the finite and semi-infinite intervals, fractal networks, spherical geometries and the wedge. Various applications are presented including neuron dynamics, self-organized criticality, diffusion-limited aggregation, the dynamics of spin systems and the kinetics of diffusion-controlled reactions. First-passage processes provide an appealing way for graduate students and researchers in physics, chemistry, theoretical biology, electrical engineering, chemical engineering, operations research and finance to understand all of these systems.
? Highlights first-passage processes; most books on probability theory and stochastic processes treat it as a subsidiary
? The emphasis is on physical intuition and how to solve problems rather than on theory
? A range of applications are presented as being part of first-passage processes
Contents
Preface; Errata; 1. First-passage fundamentals; 2. First passage in an interval; 3. Semi-infinite system; 4. Illustrations of first passage in simple geometries; 5. Fractal and nonfractal networks; 6. Systems with spherical symmetry; 7. Wedge domains; 8. Applications to simple reactions; References; Index.
Series: Acta Numerica (No. 16)
Hardback (ISBN-13: 9780521877435)
Acta Numerica is a high-impact factor, prestigious, annual publication containing invited surveys by leading researchers in numerical mathematics and scientific computing. The surveys present overviews of recent developments in their area and provide ‘state-of-the-art’ techniques and analyses. It is essential reading for all practitioners and researchers.
? High impact factor survey volume
? Contributors are leading researchers
? Covers topics of current interest and presents state-of-the-art overviews of them
Contents
1. Numerical weather prediction M. Cullen; 2. Molecular dynamics B. Leimkuhler; 3. Hyperbolic conservation laws K.W. Morton and T. Sonar; 4. Gibbs phenomenon and its avoidance in computing PDEs E. Tadmor; 5. Computational special functions N. Temme; 6. R functions in geometric approximation V. Shapiro.
Contributors
M. Cullen, B. Leimkuhler, K. W. Morton, T. Sonar, E. Tadmor, N. Temme, V. Shapiro
Hardback (ISBN-13: 9780521883405)
Paperback (ISBN-13: 9780521709798)
This unique book provides a meaningful resource for applied mathematics through Fourier analysis. It develops a unified theory of discrete and continuous (univariate) Fourier analysis, the fast Fourier transform, and a powerful elementary theory of generalized functions and shows how these mathematical ideas can be used to study sampling theory, PDEs, probability, diffraction, musical tones, and wavelets. The book contains an unusually complete presentation of the Fourier transform calculus. It uses concepts from calculus to present an elementary theory of generalized functions. FT calculus and generalized functions are then used to study the wave equation, diffusion equation, and diffraction equation. Real-world applications of Fourier analysis are described in the chapter on musical tones. A valuable reference on Fourier analysis for a variety of students and scientific professionals, including mathematicians, physicists, chemists, geologists, electrical engineers, mechanical engineers, and others.
? A modern introduction to Fourier analysis and its applications: shows how Fourier analysis can be used to study sampling theory, PDEs, probability, diffraction and real-world applications
? Accessible to students and researchers in mathematics, physics, chemistry, geology and engineering
? Over 540 exercises illustrate and extend mathematical ideas from the text, stimulating the development of problem-solving skills
Contents
1. Fourier's representation for functions on R, Tp, Z, and PN; 2. Convolution of functions on R, Tp, Z and PN; 3. The calculus for finding Fourier transforms of functions of R; 4. The calculus for finding Fourier transforms of functions of Tp, Z, and PN; 5. Operator identities associated with Fourier analysis; 6. The fast Fourier transform; 7. Generalized functions on R; 8. Sampling; 9. Partial differential equations; 10. Wavelets; 11. Musical tones; 12. Probability; Appendix 0. The impact of Fourier analysis; Appendix 1. Functions and their Fourier transforms; Appendix 2. The Fourier transform calculus; Appendix 3. Operators and their Fourier transforms; Appendix 4. The Whittaker-Robinson flow chart for harmonic analysis; Appendix 5. FORTRAN code for a Radix 2 FFT; Appendix 6. The standard normal probability distribution; Appendix 7. Frequencies of the piano keyboard; Index.
Series: Cambridge Tracts in Mathematics (No. 174)
Hardback (ISBN-13: 9780521874267)
Descriptive set theory and definable proper forcing are two areas of set theory that developed quite independently of each other. This monograph unites them and explores the connections between them. Forcing is presented in terms of quotient algebras of various natural sigma-ideals on Polish spaces, and forcing properties in terms of Fubini-style properties or in terms of determined infinite games on Boolean algebras. Many examples of forcing notions appear, some newly isolated from measure theory, dynamical systems, and other fields. The descriptive set theoretic analysis of operations on forcings opens the door to applications of the theory: absoluteness theorems for certain classical forcing extensions, duality theorems, and preservation theorems for the countable support iteration. Containing original research, this text highlights the connections that forcing makes with other areas of mathematics, and is essential reading for academic researchers and graduate students in set theory, abstract analysis and measure theory.
? Highlights the links between descriptive set theory and forcing; fully explores the connections that forcing makes with other areas of mathematics
? Contains several dozen determined infinite games
? Contains new research on this topic: essential reading for academic researchers and graduate students in the areas of set theory, abstract analysis and measure theory
Contents
1. Introduction; 2. Basics; 3. Properties; 4. Examples; 5. Operations; 6. Applications; 7. Questions; Bibliography; Index.
Series: Cambridge Monographs on Mathematical Physics
Paperback (ISBN-13: 9780521037037)
This book provides a rigorous introduction to the theory of complex angular momenta, based on the methods of field theory. It comprises an English translation of the series of lectures given by V. N. Gribov in 1969, when the physics of high-energy hadron interactions was being created. Besides their historical significance, these lectures contain material which is highly relevant to research today. The basic physical results and the approaches Gribov developed are now being rediscovered in an alternative context: in the microscopic theory of hadrons provided by quantum chromodynamics. The ideas and calculation techniques presented in this book are useful for analysing high-energy hadron scattering phenomena, deep inelastic lepton-hadron scattering, the physics of heavy ion collisions, kinetic phenomena in phase transitions, and will be instrumental in the analysis of electroweak processes at the next-generation particle accelerators, such as LHC and TESLA.
? Simple and intuitive introduction to the topic
? Based on lectures by V. N. Gribov, one of the most distinguished physicists in particle physics
? Unique in its approach
Contents
Foreword Yuri Dokshitzer; Introduction Yuri Dokshitzer and Leonid Frankfurt; 1. High energy hadron scattering; 2. Physics of the t-channel and complex angular momenta; 3. Singularities of partial waves and unitarity; 4. Properties of Regge poles; 5. Regge poles in high energy scattering; 6. Scattering of particles with spin; 7. Fermion Regge poles; 8. Regge poles in perturbation theory; 9. Reggeization of an electron; 10. Vector field theory; 11. Inconsistency of the Regge pole picture; 12. Two-reggeon exchange and branch point singularities in the l plane; 13. Properties of Mandelstam branch singularities; 14. Reggeon diagrams; 15. Interacting reggeons; 16. Reggeon field theory; 17. The structure of weak and strong coupling solutions; Appendix A: space-time description of the hadron interactions at high energies; Appendix B: character of inclusive spectra and fluctuations produced in inelastic processes by multi-pomeron exchange; Appendix C: theory of the heavy pomeron; Index.