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.
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
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.
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.
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.
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.