Piermarco Cannarsa, Universita di Roma, Tor Vergata, Carlo Sinestrari

Semiconcave Functions and Optimal Control

Series: Progress in Nonlinear Differential Equations and Their Applications, Vol. 4000

350 pages , 6 1/8 x 9 , paperback
ISBN: 0-8176-4084-3, published 2004

ABOUT THIS BOOK

Semiconcavity is a natural generalization of concavity that retains most of the good properties known in convex analysis, but arises in a wider range of applications. This text is the first comprehensive exposition of the theory of semiconcave functions, and of the role they play in optimal control and Hamilton--Jacobi equations. The first part he general theory, encompassing all key results and illustrating them with significant examples. The latter part is devoted to applications concerning the Bolza problem in calculus of variations and optimal exit time problems for nonlinear control systems. The exposition is essentially self-contained since the book includes all prerequisites from convex analysis, nonsmooth analysis, and viscosity solutions. A central role in the present work is reserved for the study of singularities. Singularities are first investigated for general semiconcave functions, then sharply estimated for solutions of Hamilton--Jacobi equations, and finally analyzed in connection with optimal trajectories of control systems. Researchers in optimal control, the calculus of variations, and partial differential equations will find this book useful as a state-of-the-art reference for semiconcave functions. Graduate students will profit from this text as it provides a handy---yet rigorous---introduction to modern dynamic programming for nonlinear control systems.

Klaus Ecker, Freie Universitat Berlin, Berlin, Germany

Regularity Theory for Mean Curvature Flow

Series: Progress in Nonlinear Differential Equations and Their Applications, Vol. 57

144 pages , 6 1/8 x 9 , 16 illus., hardcover
ISBN: 0-8176-3243-3, published 2004

ABOUT THIS BOOK

Mean curvature flow and related geometric evolution equations are important tools in mathematics and mathematical physics. A major example is Hamilton's Ricci flow program, which has the aim of settling Thurston's geometrization conjecture, with recent major progress due to Perelman. Another important application of a curvature flow process is the resolution of the famous Penrose conjecture in general relativity by Huisken and Ilmanen.

Under mean curvature flow, surfaces usually develop singularities in finite time. This work presents techniques for the study of singularities of mean curvature flow and is largely based on the work of K. Brakke, although more recent developments are incorporated.

Key features and topics of this self-contained, systematic exposition:

* A detailed account of techniques (including some of Brakke's original ones) leading to a proof of Brakke's main regularity theorem
* Preliminary material begins with the concept of mean curvature flow, illustrated with important examples and special solutions including a detailed discussion of homethetic solutions
* Local pointwise estimates on geometric quantities for smooth solutions of mean curvature flow are derived in a streamlined presentation
* Rescaling methods, monotonicity formulas, and mean value inequalities are presented
* Two local regularity theorems and an estimate of the singular set are established
* Definitions and facts for hypersurfaces in Euclidean space, used throughout the text, are listed in an appendix, along with some background on geometric measure theory
* Relations to the regularity theory for minimal surfaces, as in Allard's and de Giorgi's work, are outlined
* Good bibliography and index

Graduate students and researchers in nonlinear PDEs, differential geometry, geometric measure theory and mathematical physics will benefit from this work.

TABLE OF CONTENTS

Introduction * Preliminaries, Examples and Global Behaviour * Local Estimates for Smooth Solutions * Integral Behaviour, Monotonicity and Consequences * Regularity Theory at the First Singular Time * Appendix * A. Geometry of Hypersurfaces * B. Evolution Equations for Mean Curvature Flow * C. Some Background on Geometric Measure Theory * References * Index

Jeffrey A. Hogan, University of Arkansas, Fayetteville, AR
Joseph D. Lakey, New Mexico State University, Las Cruces, NM

Time Frequency and Time-Scale Methods
Adaptive Decompositions, Uncertainty Principles, and Sampling

Series: Applied and Numerical Harmonic Analysis

400 pages , 6 1/8 x 9 , 40 illus., hardcover
ISBN: 0-8176-4276-5, published 2004

ABOUT THIS BOOK

The goal of the book is to develop a deeper understanding of the roles of time-frequency or Fourier and Gabor analysis, and time-scale or wavelet analysis, when the various tools are properly assembled in a larger context. While researchers at the forefront of developments in time-frequency scale (TFS) analysis are well aware of the benefits of such a unified approach, there remains a gap in the larger community of practitioners concerning precisely the strengths and limitations of Gabor analysis versus wavelets. The book fills this gap by presenting the interface between time-frequency and time-scale methods as a rich area of work. Topics and Features- *Self-contained core material is presented in the first part of the book *Second part addresses new TFS methods and algorithms *Presents an integrated approach to using Fourier/Gabor methods and wavelet methods *Real world applications of Gabor theory, wavelets to PDEs and wavelets to compression *Develops the ideas of phase space and the uncertainty principle focusing on the role of the metaplectic group and the flexibility in constructing analysis tools *Explanations at the end of each chapter address historical context, major developments and new directions. Electrical engineers and applied mathematicians in signal and image processing analysis and communication theory will find an authoritative, systematic presentation of this active field of modern analysis and applications. Researchers and professionals in wavelets and mathematical signal analysis will find this an up-to-date resource.

TABLE OF CONTENTS

1. Preliminaries
2. Discrete and Continuous Fourier Analysis
3. Continuous and Discrete Wavelet Analysis
4. Time-frequency Versus Time-scale
5. Fourier and Wavelets: the Best of Both Worlds
6. Phase Space and the Metaplectic Group
7. All About Uncertainty
8. Function Spaces and Operator Theory
9. Regular and Irregular Sampling in Fourier and Wavelet Analysis
10. Further Applications

Archimedes, Edited and translated by Reviel Netz

The Works of Archimedes
Translation and Commentary
Volume 1, The Two Books On the Sphere and the Cylinder

February 2004 | Hardback | 386 pages 108 line diagrams | ISBN: 0-521-66160-9

Archimedes was the greatest scientist of antiquity and one of the greatest of all time. This book is Volume 1 of the first authoritative translation of his works into English. It is also the first publication of a major ancient Greek mathematician to include a critical edition of the diagrams, and the first translation into English of Eutocius’ ancient commentary on Archimedes. Furthermore, it is the first work to offer recent evidence based on the Archimedes Palimpsest, the major source for Archimedes, lost between 1915 and 1998. A commentary on the translated text studies the cognitive practice assumed in writing and reading the work, and it is Reviel Netz’s aim to recover the original function of the text as an act of communication. Particular attention is paid to the aesthetic dimension of Archimedes’ writings. Taken as a whole, the commentary offers a groundbreaking approach to the study of mathematical texts.

Contents

Introduction; On Sphere and Cylinder Book I; On Sphere and Cylinder Book II; Eutocius’ Commentary to On Sphere and Cylinder Book I; Eutocius’ Commentary to On Sphere and Cylinder Book II.

Desmond Higham

An Introduction to Financial Option Valuation
Mathematics, Stochastics and Computation

March 2004 | Paperback | 276 pages 120 exercises 95 figures 80 worked examples | ISBN: 0-521-54757-1
March 2004 | Hardback | 276 pages 120 exercises 95 figures 80 worked examples | ISBN: 0-521-83884-3

This is a lively textbook providing a solid introduction to financial option valuation for undergraduate students armed with a working knowledge of a first year calculus. Written in a series of short chapters, its self-contained treatment gives equal weight to applied mathematics, stochastics and computational algorithms. No prior background in probability, statistics or numerical analysis is required. Detailed derivations of both the basic asset price model and the Black?Scholes equation are provided along with a presentation of appropriate computational techniques including binomial, finite differences and in particular, variance reduction techniques for the Monte Carlo method. Each chapter comes complete with accompanying stand-alone MATLAB code listing to illustrate a key idea. Furthermore, the author has made heavy use of figures and examples, and has included computations based on real stock market data. Solutions to exercises are available from solutions@cambridge.org.

Contents

1. Introduction; 2. Option valuation preliminaries; 3. Random variables; 4. Computer simulation; 5. Asset price movement; 6. Asset price model: part I; 7. Asset price model: part II; 8. Black?Scholes PDE and formulas; 9. More on hedging; 10. The Greeks; 11. More on the Black?Scholes formulas; 12. Risk neutrality; 13. Solving a nonlinear equation; 14. Implied volitility; 15. The Monte Carlo method; 16. The binomial method; 17. Cash-or-nothing options; 18. American options; 19. Exotic options; 20. Historical volatility; 21. Monte Carlo part II: variance reduction by antithetic variates; 22. Monte Carlo part III: variance reduction by control variates; 23. Finite difference methods; 24. Finite difference methods for the Black?Scholes