Adler, R.J., Taylor, Jonathan

Random Fields and Geometry

Series: Springer Monographs in Mathematics
2007, Approx. 480 p., 30 illus., Hardcover
ISBN-10: 0-387-48112-5
ISBN-13: 978-0-387-48112-8

About this book

This self-contained monograph focuses on recent important developments in the study of random fields, stochastic processes defined over high dimensional parameter spaces. While it replaces Adler's 1981 classic The Geometry of Random Fields, this is not an update, but a completely new work with a completely new way of handling both the Geometry and the Probability that are its central themes.

There are three quite distinct parts to the monograph. Part I provides a comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorising measures, Borell and Slepian inequalities. The treatment is didactic and user-friendly. Part II is about Geometry, both integral and Riemannian, and the material included here is what is needed for the over-riding probabilistic theme of the book. It contains a quick review of both these geometric settings, followed by carefully presented introductions to topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory and tube formulae. This is the only place in which all these topics, necessary for the study of random fields can be found in a concise, self-contained, treatment. The most important part of the book is in Part III, which is about the geometry of excursion sets of random fields and the related eEuler characteristic approachf to extremal probabilities. This part contains path-breaking material of both theoretical and practical importance and is unique in the way in which it intertwines probabilistic and geometric problems.

Applications of this theory, which are significant and cover areas as widespread as brain imaging, physical oceanography and astrophysics, will be treated in a separate volume with Keith Worsley.

This monograph will be of interest to probabilists and statisticians, both applied and theoretical, along with mathematicians interested in learning about new relationships between geometry and probability. It is also a basic reference text for those who will eventually be interested mainly in the companion volume of applications. Given the clear and pedagogical style of the book and the current importance of research in random fields this comprehensive and definitive work will serve as an indispensable reference work, while at the same time being an excellent text for self study and graduate courses in probability, statistics, analysis and geometry.

Table of contents

Preface.- Part I. Gaussian Processes. Gaussian Fields. Gaussian Inequalities. Orthogonal Expansions. Excursion Probabilities. Stationary Fields.- Parat II. Geometry. Integral Geometry. Differential Geometry. Piecewise Smooth Manifolds. Critical Point Theory. Volume of Tubes.- Part III. The Geometry of Random Fields. Random Fields on Euclidean Spaces. Random Fields on Manifolds. Mean Intrinsic Volumes. Excursion Probabilities for Smooth Fields. Non-Gaussian Geometry.- References.- Index.

Gikhman, Iosif I., Skorokhod, Anatoli V.

The Theory of Stochastic Processes III

Series: Classics in Mathematics
Originally published as Vol. 232 in the series: Grundlehren der mathematischen Wissenschaften
Reprint of the 1st ed. Berlin Heidelberg New York 1979, 2007, VII, 387 p., Hardcover
ISBN-10: 3-540-49940-7
ISBN-13: 978-3-540-49940-4

From the Reviews:

"Gihman and Skorohod have done an excellent job of presenting the theory in its present state of rich imperfection."
D.W. Stroock in Bulletin of the American Mathematical Society, 1980

"To call this work encyclopedic would not give an accurate picture of its content and style. Some parts read like a textbook, but others are more technical and contain relatively new results. ... The exposition is robust and explicit, as one has come to expect of the Russian tradition of mathematical writing. The set when completed will be an invaluable source of information and reference in this ever-expanding field."
K.L. Chung in American Scientist, 1977

"The dominant impression is of the authors' mastery of their material, and of their confident insight into its underlying structure."
J.F.C. Kingman in Bulletin of the London Mathematical Society, 1977

Table of contents

Martingales and Stochastic Integrals.- Stochastic Differential Equations.- Stochastic Differential Equations for Continuous Processes and Continuous Markov Processes in Rm.- Remarks.- Bibliography.- Appendix: Corrections to Volumes I and II.- Subject Index

Nicolaescu, Liviu

An Invitation to Morse Theory

Series: Universitext
2007, Approx. 245 p., 32 illus., Softcover
ISBN-10: 0-387-49509-6
ISBN-13: 978-0-387-49509-5

About this textbook

As the the title suggests, the goal of this little book is to give the reader a taste of the "unreasonable effectiveness" of Morse theory. The main idea behind this technique can be easily visualized. Suppose M is a smooth, compact manifold which for simplicity we assume is embedded in an Euclidean space E. We would like to understand basic topological invariants of M such as its homology and we attempt a "slicing" technique. We fix a unit vector u in E and we start slicing M with the family of hyperplanes perpendicular to u. Such a hyperplane will in general intersect M along a submanifold (slice). The manifold can be recovered by continuously stacking the slices on top of each other in the same order as they were cut out of M.

Think of the collection of slices as a deck of cards of various shapes. If we let these slices continuously pile up in the order they were produced we notice and increasing stack of slices. As this stack grows we notice that there are moments of time when its shape suffers a qualitative change. Morse theory is about extracting quantifiable information by studying the evolution of the shape of this growing stack of slices.

This self-contained treatment of Morse Theory focuses on applications and is intended for a graduate course on differential or algebraic topology. This is the first textbook to include topics such as Morse-Smale flows, min-max theory, moment maps and equivariant cohomology, and complex Morse theory. The exposition is enhanced with examples, problems, and illustrations.

Table of contents

Preface.- Notations and conventions.- Morse functions.- The topology of Morse functions.- Applications.- Basics of complex Morse theory.- Exercises and solutions.- References.- Index.

Sanders, Jan A., Verhulst, Ferdinand, Murdock, James

Averaging Methods in Nonlinear Dynamical Systems, 2nd ed.

Series: Applied Mathematical Sciences , Vol. 59
2007, Approx. 430 p., Hardcover
ISBN-10: 0-387-48916-9
ISBN-13: 978-0-387-48916-2

About this book

Perturbation theory and in particular normal form theory has shown strong growth in recent decades. This book is a drastic revision of the first edition of the averaging book. The updated chapters represent new insights in averaging, in particular its relation with dynamical systems and the theory of normal forms. Also new are survey appendices on invariant manifolds. One of the most striking features of the book is the collection of examples, which range from the very simple to some that are elaborate, realistic, and of considerable practical importance. Most of them are presented in careful detail and are illustrated with illuminating diagrams.

Table of contents

Basic Material and Asymptotics.- Averaging: the Periodic Case.- Methodology of Averaging.- Averaging: General Case.- Attraction.- Periodic Averaging and Hyperbolicity.- Averaging Over Angles.- Passage Through Resonance.- From Averaging to Normal Forms.- Hamiltonian Normal Form Theory.- Classical (First Level) Normal Form Theory.- Nilpotent (Classical) Normal Form.- Higher Level Normal Form Theory.- A. The The History of the Theory of Averaging.- B. A 4-dimensional Example of Hopf Bifurcation.- C. Invariant Manifolds by Averaging.- D. Celestial Mechanics.- E. On Averaging Methods for Partial Differential Equations.- References.- Index of Definitions & Descriptions.

Cohen, Henri

Number Theory
Volume I: Elementary and Algebraic Methods for Diophantine Equations

Series: Graduate Texts in Mathematics , Vol. 239
2007, XII, 550 p., Hardcover
ISBN-10: 0-387-49922-9
ISBN-13: 978-0-387-49922-2

About this textbook

The central theme is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects.

The first is the local aspect: one can do analysis in p-adic fields, and here the author starts by looking at solutions in finite fields, then proceeds to lift these solutions to local solutions using Hensel lifting. The second aspect is the global aspect: the use of number fields, and in particular of class groups and unit groups. The third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject, and embodies in a beautiful way the local and global aspects of Diophantine problems. In fact, these functions are defined through the local aspects of the problems, but their analytic behavior is intimately linked to the global aspects.

Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included 5 appendices on these techniques. These appendices were written by Henri Cohen, Yann Bugeaud, Maurice Mignotte, Sylvain Duquesne, and Samir Siksek, and contain material on the use of Galois representations, the superfermat equation, Mihailescufs proof of Catalanfs Conjecture, and applications of linear forms in logarithms.

Table of contents

Preface.- Introduction to Diophantine Equations.- Abelian Groups, Lattices, and Finite Fields.-Basic Algebraic Number Theory.- p-adic Fields.-Quadratic Forms and Local-Global Principles.- Some Diophantine Equations.- Eilliptic Curves.- Diophantine Aspects of Elliptic Curves.- Bibliography.- Index.

Cohen, Henri

Number Theory
Volume II: Analytic and Modern Tools

Series: Graduate Texts in Mathematics , Vol. 240
2007, XII, 500 p., Hardcover
ISBN-10: 0-387-49893-1
ISBN-13: 978-0-387-49893-5

About this textbook

This book deals with several aspects of what is now called "explicit number theory." The central theme is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects.

The first is the local aspect: one can do analysis in p-adic fields, and here the author starts by looking at solutions in finite fields, then proceeds to lift these solutions to local solutions using Hensel lifting. The second aspect is the global aspect: the use of number fields, and in particular of class groups and unit groups. The third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject, and embodies in a beautiful way the local and global aspects of Diophantine problems. In fact, these functions are defined through the local aspects of the problems, but their analytic behavior is intimately linked to the global aspects.

Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included 5 appendices on these techniques. These appendices were written by Henri Cohen, Yann Bugeaud, Maurice Mignotte, Sylvain Duquesne, and Samir Siksek, and contain material on the use of Galois representations, the superfermat equation, Mihailescufs proof of Catalanfs Conjecture, and applications of linear forms in logarithms.

Table of contents

Bernoulli Polynomials and the Gamma Functions.- Dirichlet Series.- Applications of Linear Forms in Logarithms.- Rational Points on Higher Genus Curves.- The Modular Approach to Diophantine Equations.- The Superfermat Equation.- Catalanfs Equation.- Bibliography.- Index.