Sidney Redner
A Guide to First Passage Processes
Description
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 modern 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.
Chapter Contents
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 non-fractal networks; 6. Systems with spherical symmetry; 7. Wedge domains; 8. Application to simple reactions.
ISBN: 0-521-65248-0
Binding: Hardback
Pages: 352
John G. Harris
Linear Elastic Waves
Cambridge Texts in Applied Mathematics series
Description
Wave propagation and scattering are among the most fundamental processes that we use to comprehend the world around us. While these processes are often very complex, one way to begin to understand them is to study wave propagation in the linear approximation. This is a book describing such propagation using, as a
context, the equations of elasticity. Two unifying themes are used. The first is that an understanding of plane wave interactions is fundamental to understanding more complex wave interactions. The second is that waves are best understood in an asymptotic approximation where they are free of the complications of their
excitation and are governed primarily by their propagation environments. The topics covered include reflection, refraction, the propagation of interfacial waves, integral representations, radiation and diffraction, and propagation in closed and open waveguides. Linear Elastic Waves is an advanced level textbook directed at applied mathematicians, seismologists, and engineers.
Chapter Contents
1. Simple wave solutions; 2. Kinematical descriptions of waves; 3. Reflection, refraction and interfacial waves; 4. Green's tensor and integral representations; 5. Radiation and diffraction; 6. Guided waves and dispersion.
ISBN: 0-521-64368-6
Binding: Hardback (Paperback)
Pages: 208
C. Foias, Oliver Manley, Ricardo M. S. Rosa, R. Temam
Navier-Stokes Equations and Turbulence
Encyclopedia of Mathematics and its Applications series
Description
This book aims to bridge the gap between practising mathematicians and the practitioners of turbulence theory. It presents the mathematical theory of turbulence to engineers and physicists, and the physical theory of turbulence to mathematicians. The book is the result of many years of research by the authors
to analyze turbulence using Sobolev spaces and functional analysis. In this way the authors have
recovered parts of the conventional theory of turbulence, deriving rigorously from the Navier-Stokes equations what had been arrived at earlier by phenomenological arguments. The mathematical technicalities are kept to a minimum within the book, enabling the language to be at a level understood by a broad audience.
Each chapter is accompanied by appendices giving full details of the mathematical proofs and subtleties. This unique presentation should ensure a volume of interest to mathematicians, engineers, and physicists.
Chapter Contents
1. Introduction and overview of turbulence; 2. Elements of the mathematical theory of the Navier-stokes equations; 3. Finite dimensionality of flows; 4. Stationary statistical solutions of the Navier-Stokes equations, time averages, and attractors; 5. Time-dependent statistical solutions of the Navier-Stokes equations and
fully-developed turbulence.
ISBN: 0-521-36032-3
Binding: Hardback
Pages: 432
Isaac Chavel
Isoperimetric Inequalities
Differential Geometric and Analytic Perspectives
Cambridge Tracts in Mathematics series
Description
This introduction treats the classical isoperimetric inequality in Euclidean space and contrasting rough inequalities in noncompact Riemannian manifolds. In Euclidean space the emphasis is on a most general form of the inequality sufficiently precise to characterize the case of equality, and in Riemannian manifolds the
emphasis is on those qualitative features of the inequality which provide insight into the coarse geometry at infinity of Riemannian manifolds. The treatment in Euclidean space features a number of proofs of the classical inequality in increasing generality, providing in the process a transition from the methods of classical
differential geometry to those of modern geometric measure theory; and the treatment in Riemannian manifolds features discretization techniques, and applications to upper bounds of large time heat diffusion in Riemannian manifolds. The result is an introduction to the rich tapestry of ideas and techniques of
isoperimetric inequalities, a subject that has its beginnings in classical antiquity and which continues to inspire fresh ideas in geometry and analysis to this very day - and beyond!
Chapter Contents
Part I. Introduction: 1. The isoperimetric problem; 2. The isoperimetric inequality in the plane; 3. Preliminaries; 4. Bibliographic notes; Part II. Differential Geometric Methods: 1. The C2 uniqueness theory; 2. The C1 isoperimetric inequality; 3. Bibliographic notes; Part III. Minkowski Area and Perimeter: 1. The Hausdorff
metric on compacta; 2. Minkowski area and Steiner symmetrization; 3. Application: the Faber-Krahn inequality; 4. Perimeter; 5. Bibliographic notes; Part IV. Hausdorff Measure and Perimeter: 1. Hausdorff measure; 2. The area formula for Lipschitz maps; 3. Bibliographic notes; Part V. Isoperimetric Constants: 1.
Riemannian geometric preliminaries; 2. Isoperimetric constants; 3. Discretizations and isoperimetric inequalities; 4. Bibliographic notes; Part VI. Analytic Isoperimetric Inequalities: 1. L2-Sobolev inequalities; 2. The compact case; 3. Faber-Kahn inequalities; 4. The Federer-Fleming theorem: the discrete case; 5. Sobolev
inequalities and discretizations; 6. Bibliographic notes; Part VII. Laplace and Heat Operators: 1. Self-adjoint operators and their semigroups; 2. The Laplacian; 3. The heat equation and its kernels; 4. The action of the heat semigroup; 5. Simplest examples; 6. Bibliographic notes; Part VIII. Large-Time Heat Diffusion: 1. The main
problem; 2. The Nash approach; 3. The Varopoulos approach; 4. Coulhon's modified Sobolev inequality; 5. The denoument: geometric applications; 6. Epilogue: the Faber-Kahn method; 7. Bibliographic notes;
Bibliography.
ISBN: 0-521-80267-9
Binding: Hardback
Burkard Polster, Gunter Steinke
Geometries on Surfaces
Encyclopedia of Mathematics and its Applications series
Description
The projective, Mvbius, Laguerre, and Minkowski planes over the real numbers are just a few examples of a host of fundamental classical topological geometries on surfaces. This book summarizes all known major results and open problems related to these classical point-line geometries and their close (nonclassical)
relatives. Topics covered include: classical geometries; methods for constructing nonclassical geometries; classifications and characterizations of geometries. This work is related to many other fields including
interpolation theory, convexity, the theory of pseudoline arrangements, topology, the theory of Lie groups, and many more. The authors detail these connections, some of which are well-known, but many much less so. Acting both as a reference for experts and as an accessible introduction for graduate students, this book will
interest anyone wishing to know more about point-line geometries and the way they interact.
Chapter Contents
1. Geometries for pedestrians; 2. Flat linear spaces; 3. Spherical circle planes; 4. Toroidal circle planes; 5. Cylindrical circle planes; 6. Generalized quadrangles; 7. Tubular circle planes; Appendices.
ISBN: 0-521-66058-0
Binding: Hardback
Pages: 500