Expected publication date is March 3, 2004
Description
In this book, award-winning author Goro Shimura
treats new areas
and presents relevant expository material
in a clear and readable
style. Topics include Witt's theorem and
the Hasse principle on
quadratic forms, algebraic theory of Clifford
algebras, spin
groups, and spin representations. He also
includes some basic
results not readily found elsewhere.
The two principle themes are:
(1) Quadratic Diophantine equations;
(2) Euler products and Eisenstein series
on orthogonal groups and
Clifford groups.
The starting point of the first theme is
the result of Gauss that
the number of primitive representations of
an integer as the sum
of three squares is essentially the class
number of primitive
binary quadratic forms. Presented are a generalization
of this
fact for arbitrary quadratic forms over algebraic
number fields
and various applications. For the second
theme, the author proves
the existence of the meromorphic continuation
of a Euler product
associated with a Hecke eigenform on a Clifford
or an orthogonal
group. The same is done for an Eisenstein
series on such a group.
Beyond familiarity with algebraic number
theory, the book is
mostly self-contained. Several standard facts
are stated with
references for detailed proofs.
Goro Shimura won the 1996 Steele Prize for
Lifetime Achievement
for "his important and extensive work
on arithmetical
geometry and automorphic forms".
Contents
Introduction
Algebraic theory of quadratic forms, Clifford
algebras, and spin
groups
Quadratic forms, Clifford groups, and spin
groups over a local or
global field
Quadratic diophantine equations
Groups and symmetric spaces over R
Euler products and Eisenstein series on orthogonal
groups
Euler products and Eisenstein series on Clifford
groups
Appendix
References
Frequently used symbols
Index
Details:
Series: Mathematical Surveys and Monographs,
Volume: 109
Publication Year: 2004
ISBN: 0-8218-3573-4
Paging: 275 pp.
Binding: Hardcover
Expected publication date is May 12, 2004
Description
"The Painleve equations themselves are
really a wonder. They
still continue to give us fresh mysteries
... One reason that I
wrote this book is to tell you how impressed
I am by the
mysteries of the Painleve equations."
--from the Preface
The six Painleve equations (nonlinear ordinary
differential
equations of the second order with nonmovable
singularities) have
attracted the attention of mathematicians
for more than 100 years.
These equations and their solutions, the
Painleve transcendents,
nowadays play an important role in many areas
of mathematics,
such as the theory of special functions,
the theory of integrable
systems, differential geometry, and mathematical
aspects of
quantum field theory.
The present book is devoted to the symmetry
of Painleve equations
(especially those of types II and IV). The
author studies
families of transformations for several types
of Painleve
equations--the so-called Backlund transformations--which
transform solutions of a given Painleve equation
to solutions of
the same equation with a different set of
parameters. It turns
out that these symmetries can be interpreted
in terms of root
systems associated to affine Weyl groups.
The author describes
the remarkable combinatorial structures of
these symmetries, and
it shows how they are related to the theory
of $\tau$-functions
associated to integrable systems.
Prerequisites include undergraduate calculus
and linear algebra
with some knowledge of group theory. The
book is suitable for
graduate students and research mathematicians
interested in
special functions and the theory of integrable
systems.
Contents
What is a Backlund transformation?
The symmetric form
$\tau$-functions
$\tau$-functions on the lattice
Jacobi-Trudi formula
Getting familiar with determinants
Guass decomposition and birational transformations
Lax formalism
Appendix
Bibliography
Details:
Series: Translations of Mathematical Monographs,Volume:
223
Publication Year: 2004
ISBN: 0-8218-3221-2
Paging: approximately 168 pp.
Binding: Hardcover
Series: AIP Conference Proceedings
2003, XI, 416 p., Hardcover
ISBN: 0-7354-0162-4
About this book
The conference reviewed the use of the Metropolis
Algorithm and
the Monte Carlo Method in the physical sciences,
highlighted
recent developments, and noted the spread
of the Monte Carlo
Method to other fields. It emphasized current
algorithms and
algorithmic issues, but also recorded valuable
historical
perspectives about the development of the
Metropolis Algorithm.
Written for:
Physicists and chemists who do computer simulation;
science
historians
Keywords:
Mathematical and computational physics
mathematical applications in physical chemistra
history of science
192 pages , 6 1/8 x 9 , 35 illus., paperback
ISBN: 3-7643-3258-1, published 2004
ABOUT THIS BOOK
"The Book is highly recommended as a
text for an
introductory course in nonlinear analysis
and bifurcation theory...
reading is fluid and very pleasant... style
is informal but far
from being imprecise." - Mathematical
Reviews
Here is a book that will be a joy to the
mathematician or
graduate student of mathematics - or even
the well-prepared
undergraduate - who would like, with a minimum
of background and
preparation, to understand some of the beautiful
results at the
heart of nonlinear analysis. Based on carefully-expounded
ideas
from several branches of topology, and illustrated
by a wealth of
figures that attest to the geometric nature
of the exposition,
the book will be of immense help in providing
its readers with an
understanding of the mathematics of the nonlinear
phenomena that
characterize our real world.
New to the second edition: New chapters will
supply additional
applications of the theory and techniques
presented in the book.
* Several new proofs, making the second edition
more self-contained
* New section devoted exclusively to bifurcation
theory.
This book is ideal for self-study for mathematicians
and students
interested in such areas of geometric and
algebraic topology,
functional analysis, differential equations,
and applied
mathematics. It is a sharply focused and
highly readable view of
nonlinear analysis by a practicing topologist
who has seen a
clear path to understanding.
TABLE OF CONTENTS
Part I: Fixed Point Existence Theory. The
Topological Point of
View. Ascoli-Arzela Theory. Brouwer Fixed
Point Theory. Schauder
Fixed Point Theory. The Forced Pendulum.
Equilibrium Heat
Distribution. Generalized Bernstein Theory.-
Part II: Degree
Theory. Some Topological Background. Brouwer
Degree. Leray-Schauder
Degree. Properties of the Leray-Schauder
Degree. Knobloch's
Theorem. More About the Pendulum.- Part III:
Bifurcation Theory.
A Separation Theorem. Compact Linear Operators.
The Degree
Calculation. The Krasnoselskii-Rabinowitz
Theorem. Nonlinear
Sturm-Liouville Theory. Euler Buckling.-
Appendix A: Singular
Homology.- Appendix B: Additivity and Product
Properties.-
References.- Index
Series: Progress in Nonlinear Differential
Equations and Their
Applications, Vol. 56
410 pages , 6 1/8 x 9 , hardcover
ISBN: 3-7643-4146-7, published 2004
ABOUT THIS BOOK
This book introduces a new, state-of-the-art
method for the study
of the asymptotic behavior of solutions to
evolution partial
differential equations; much of the text
is dedicated to the
application of this method to a wide class
of nonlinear diffusion
equations. The underlying theory hinges on
a new stability
result, formulated in the abstract setting
of infinite-dimensional
dynamical systems, which states that under
certain hypotheses,
the omega-limit set of a perturbed dynamical
system is stable
under arbitrary asymptotically small perturbations.
The Stability Theorem is examined in detail
in the first chapter,
followed by a review of basic results and
methods-many original
to the authors-for the solution of nonlinear
diffusion equations.
TABLE OF CONTENTS
Introduction * Classical Theory * Tools and
Transformations *
Special Solutions * Weak Theory * Propagation
Properties *
Propagation of High Values * Appendix I *
Appendix II *
Bibiliography * Index