Mathematical Surveys and Monographs, Volume: 127
2006; 316 pp; hardcover
ISBN-10: 0-8218-4096-7
ISBN-13: 978-0-8218-4096-2
Expected publication date is August 13, 2006.
This book covers one of the most exciting but most difficult
topics in the modern theory of dynamical systems: chaotic
billiards. In physics, billiard models describe various
mechanical processes, molecular dynamics, and optical phenomena.
The theory of chaotic billiards has made remarkable progress in
the past thirty-five years, but it remains notoriously difficult
for the beginner, with main results scattered in hardly
accessible research articles. This is the first and so far only
book that covers all the fundamental facts about chaotic
billiards in a complete and systematic manner. The book contains
all the necessary definitions, full proofs of all the main
theorems, and many examples and illustrations that help the
reader to understand the material. Hundreds of carefully designed
exercises allow the reader not only to become familiar with
chaotic billiards but to master the subject.
The book addresses graduate students and young researchers in
physics and mathematics. Prerequisites include standard graduate
courses in measure theory, probability, Riemannian geometry,
topology, and complex analysis. Some of this material is
summarized in the appendices to the book.
Readership
Graduate students and research mathematicians interested in
mathematical physics, statistical mechanics, dynamical systems,
and ergodic theory.
Table of Contents
Simple examples
Basic constructions
Lyapunov exponents and hyperbolicity
Dispersing billiards
Dynamics of unstable manifolds
Ergodic properties
Statistical properties
Bunimovich billiards
General focusing chaotic billiards
Afterword
Measure theory
Probability theory
Ergodic theory
Bibliography
Index
Mathematical Surveys and Monographs, Volume: 128
2006; approx. 560 pp; hardcover
ISBN-10: 0-8218-3651-X
ISBN-13: 978-0-8218-3651-4
Expected publication date is August 11, 2006.
At the turn of the twentieth century, the French mathematician
Paul Painleve and his students classified second order nonlinear
ordinary differential equations with the property that the
location of possible branch points and essential singularities of
their solutions does not depend on initial conditions. It turned
out that there are only six such equations (up to natural
equivalence), which later became known as Painleve I-VI.
Although these equations were initially obtained answering a
strictly mathematical question, they appeared later in an
astonishing (and growing) range of applications, including, e.g.,
statistical physics, fluid mechanics, random matrices, and
orthogonal polynomials. Actually, it is now becoming clear that
the Painleve transcendents (i.e., the solutions of the Painleve
equations) play the same role in nonlinear mathematical physics
that the classical special functions, such as Airy and Bessel
functions, play in linear physics.
The explicit formulas relating the asymptotic behaviour of the
classical special functions at different critical points, play a
crucial role in the applications of these functions. It is shown
in this book, that even though the six Painleve equations are
nonlinear, it is still possible, using a new technique called the
Riemann-Hilbert formalism, to obtain analogous explicit formulas
for the Painleve transcendents. This striking fact, apparently
unknown to Painleve and his contemporaries, is the key ingredient
for the remarkable applicability of these "nonlinear special
functions".
The book describes in detail the Riemann-Hilbert method and
emphasizes its close connection to classical monodromy theory of
linear equations as well as to modern theory of integrable
systems. In addition, the book contains an ample collection of
material concerning the asymptotics of the Painleve functions and
their various applications, which makes it a good reference
source for everyone working in the theory and applications of
Painleve equations and related areas.
Readership
Graduate students and research mathematicians interested in
special functions, in particular, Painleve transcendents.
Table of Contents
Introduction. Painleve transcendents as nonlinear special
functions
Part 1. Riemannian-Hilbert problem, isomonodromy method and
special functions
Systems of linear ordinary differential equations with rational
coefficients. Elements of the general theory
Monodromy theory and special functions
Inverse monodromy problem and Riemann-Hilbert factorization
Isomonodromy deformations. The Painleve equations
The isomonodromy method
Backlund transformations
Part 2. Asymptotics of the Painleve II transcendent. A case study
Asymptotic solutions of the second Painleve equation in the
complex plane. Direct monodromy problem approach
Asymptotic solutions of the second Painleve equation in the
complex plane. Inverse monodromy problem approach
PII asymptotics on the canonical six-rays. The purely imaginary
case
PII asymptotics on the canonical six-rays. Real-valued case
PII quasi-linear Stokes phenomenon
Part 3. Asymptotics of the third Painleve transcendent
PIII equation, an overview
Sine-Gordon reduction of PIII
Canonical four-rays. Real-valued solutions of SG-PIII
Canonical four-rays. Singular solutions of the SG-PIII
Asymptotics in the complex plane of the SG-PIII transcendent
Proof of Theorem 3.4
The Birkhoff-Grothendieck theorem with a parameter
Bibliography
Subject index
Contemporary Mathematics, Volume: 405
2006; 155 pp; softcover
ISBN-10: 0-8218-3755-9
ISBN-13: 978-0-8218-3755-9
Expected publication date is August 2, 2006.
This volume consists of a collection of papers that brings
together fundamental research in Radon transforms, integral
geometry, and tomography. It grew out of the Special Session at a
Sectional Meeting of the American Mathematical Society in 2004.
The book contains very recent work of some of the top researchers
in the field.
The articles in the book deal with the determination of
properties of functions on a manifold by integral theoretic
methods, or by determining the geometric structure of subsets of
a manifold by analytic methods. Of particular concern are ways of
reconstructing an unknown function from some of its projections.
Radon transforms were developed at the beginning of the twentieth
century by researchers who were motivated by problems in
differential geometry, mathematical physics, and partial
differential equations. Later, medical applications of these
transforms produced breakthroughs in imaging technology that
resulted in the 1979 Nobel Prize in Physiology and Medicine for
the development of computerized tomography. Today the subject
boasts substantial cross-disciplinary interactions, both in pure
and applied mathematics as well as medicine, engineering,
biology, physics, geosciences, and industrial testing. Therefore,
this volume should be of interest to a wide spectrum of
researchers both in mathematics and in other fields.
Readership
Graduate students and research mathematicians interested in real
analysis, integral geometry, and tomography.
Table of Contents
M. L. Agranovsky and E. T. Quinto -- Remarks on stationary sets
for the wave equation
C. Berenstein, F. Gavilanez, and J. Baras -- Network tomography
J. Boman -- On stable inversion of the attenuated Radon transform
with half data
M. Dobrescu and G. Olafsson -- Wavelet sets without groups
L. Ehrenpreis -- The Radon transform for functions defined on
planes
F. B. Gonzalez and J. Zhang -- The modified wave equation on the
sphere
A. Katsevich and A. Zamyatin -- Analysis of a family of exact
inversion formulas for cone beam computer tomography
A. Markoe -- The $k$-plane transform and Riesz potentials
E. Ournycheva and B. Rubin -- The composite cosine transform on
the Stiefel manifold and generalized zeta integrals
I. Pesenson -- Frames for spaces of Paley-Wiener functions on
Riemannian manifolds
J. Rennie -- Properties of the stationary sets for the wave
equation
Contemporary Mathematics, Volume: 406
2006; 270 pp; softcover
ISBN-10: 0-8218-3818-0
ISBN-13: 978-0-8218-3818-1
Expected publication date is August 4, 2006.
This book is based on lectures given during a Workshop on
Representations of Algebras and Related Topics. Some additional
articles are included in order to complete a panoramic view of
the main trends of the subject. The volume contains original
presentations by leading algebraists addressed to specialists as
well as to a broader mathematical audience. The articles include
new proofs, examples, and detailed arguments. Topics under
discussion include moduli spaces associated to quivers, canonical
basis of quantum algebras, categorifications and derived
categories, $A$-infinity algebras and functor categories, cluster
algebras, support varieties for modules and complexes, the
Gabriel-Roiter measure for modules, and selfinjective algebras.
Readership
Graduate students and research mathematicians interested in
representation theory.
Table of Contents
A. Bakke Buan and R. Marsh -- Cluster-tilting theory
C. Geis, L. Le Bruyn, and M. Reineke -- Introduction to moduli
spaces associated to quivers (With an appendix by Lieven Le Bruyn
and Markus Reineke)
A. Hubery -- From triangulated categories to Lie algebras: A
theorem of Peng and Xiao
B. Keller -- A-infinity algebras, modules and functor categories
H. Krause and D. Kussin -- Rouquier's theorem on representation
dimension
C. M. Ringel -- Foundation of the representation theory of Artin
algebras, using the Gabriel-Roiter measure.
R. Rouquier -- Categorification of $\mathfrak{sl}_2$ and braid
groups
A. Skowronski -- Selfinjective algebras: Finite and tame type
O. Solberg -- Support varieties for modules and complexes