Description
The purpose of thermodynamics and statistical
physics is to understand the equilibrium
of a gas or the different states of matter.
To understand the strange fractal sets
appearing when one iterates a quadratic polynomial
is one of the goals of the theory of holomorphic
dynamical systems. These two theories are
strongly linked: The laws
of thermodynamics happen to be an extremely
powerful tool for understanding the objects
of holomorphic dynamical systems. A "thermodynamic
formalism" has been
developed, bringing together notions that
are a priori unrelated. While the deep reasons
of this parallelism remain unknown, the goal
of this book is to describe this
formalism both from the physical and mathematical
point of view in order to understand how
it works and how useful it can be.
This translation is a slightly revised version
of the original French edition. The main
changes are in Chapters 5 and 6 and consist
of clarification of some proofs and a new
presentation of the basics in iteration of
polynomials.
Contents
Introduction
The ergodic hypothesis
The concept of entropy
Entropy in ergodic theory
The Perron-Frobenius-Ruelle theorem
Conformal repellers
Iteration of quadratic polynomials
Phase transitions
Hausdorff measures and dimension
Bibliography
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: SMF/AMS Texts and Monographs Volume:
2
Publication Year: 2000
ISBN: 0-8218-1948-8
Paging: 82 pp. Binding: Softcover
--------------------------------------------------------------------------------
Edited by: Jeffrey Adams, University of Maryland,
College Park, MD,
and David Vogan, Massachusetts Institute
of Technology, Cambridge, MA
Description
This book contains written versions of the
lectures given at the PCMI Graduate Summer
School on the representation theory of Lie
groups. The volume begins with lectures by A. Knapp and P. Trapa outlining
the state of the subject around the year
1975, specifically, the fundamental results
of Harish-Chandra on the general
structure of infinite-dimensional representations
and the Langlands classification.
Additional contributions outline developments
in four of the most active areas of research
over the past 20 years. The clearly written
articles present results to date, as
follows: R. Zierau and L. Barchini discuss
the construction of representations on Dolbeault
cohomology spaces. D. Vogan describes the
status of the Kirillov-Kostant
"philosophy of coadjoint orbits"
for unitary representations. K. Vilonen presents
recent advances in the theory of "localization"
introduced by Beilinson and Bernstein.
And Jian-Shu Li covers Howe's theory of "dual
reductive pairs".
Each contributor to the volume presents the
topics in a unique, comprehensive, and accessible
manner geared toward advanced graduate students
and researchers.
Students should have completed the standard
introductory graduate courses for full comprehension
of the work. The book would also serve well
as a supplementary text
for a course on introductory infinite-dimensional
representation theory.
Contents
A. W. Knapp and P. E. Trapa, Representations
of semisimple Lie groups
Introduction
Some representations of $SL(n,\Bbb{R})$
Semsimple groups and structure theory
Introduction to representation theory
Cartan subalgebras and highest weights
Action by the Lie algebra
Cartan subgroups and global characters
Discrete series and asymptotics
Langlands classification
Bibliography
R. Zierau, Representations in Dolbeault cohomology
Introduction
Complex flag varieties and orbits under a
real form
Open $G_0$-orbits
Examples, homogeneous bundles
Dolbeault cohomology, Bott-Borel-Weil theorem
Indefinite harmonic theory
Intertwining operators I
Intertwining operators II
The linear cycle space
Bibliography
L. Barchini, Unitary representations attached
to elliptic orbits. A geometric approach
Introduction
Globalizations
Dolbeault cohomology and maximal globalization
$L^2$-cohomology and discrete series representations
Indefinite quantization
Bibliography
D. A. Vogan, Jr., The method of adjoint orbits
for real reductive groups
Introduction
Some ideas from mathematical physics
The Jordan decomposition and three kinds
of quantization
Complex polarizations
The Kostant-Sekiguchi correspondence
Quantizing the action of $K$
Associated graded modules
A good basis for associated graded modules
Proving unitarity
Exercises
Bibliography
K. Vilonen, Geometric methods in representation
theory
Introduction
Overview
Derived categories of constructible sheaves
Equivariant derived categories
Functors to representations
Matsuki correspondence for sheaves
Characteristic cyles
The character formula
Microlocalization of Matsuki = Sekiguchi
Homological algebra (appendix by M. Hunziker)
Bibliography
Jian-Shu Li, Minimal representations and
reductive dual pairs
Introduction
The oscillator representation
Models
Duality
Classification
Unitarity
Minimal representations of classical groups
Dual pairs in simple groups
Bibliography
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: IAS/Park City Mathematics Series,
Volume: 8
Publication Year: 1999
ISBN: 0-8218-1941-0
Paging: approximately 340 pp.
Binding: Hardcover
@
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Salma Kuhlmann, University of Saskatchewan, Saskatoon, SK, Canada
Description
Model theoretic algebra has witnessed remarkable
progress in the last few years. It has found
profound applications in other areas of mathematics,
notably in algebraic
geometry and in singularity theory.
Since Wilkie's results on the o-minimality
of the expansion of the reals by the exponential
function, and most recently even by all Pfaffian
functions, the study of
o-minimal expansions of the reals has become
a fascinating topic. The quest for analogies
between the semi-algebraic case and the o-minimal
case has set a direction to
this research.
Through the Artin-Schreier Theory of real
closed fields, the structure of the non-archimedean
models in the semi-algebraic case is well
understood. For the o-minimal
case, so far there has been no systematic
study of the non-archimedean models. The
goal of this monograph is to serve this purpose.
The author presents a detailed description
of the non-archimedean models of the elementary
theory of certain o-minimal expansions of
the reals in which the exponential
function is definable. The example of exponential
Hardy fields is worked out with particular
emphasis. The basic tool is valuation theory,
and a sufficient amount of
background material on orderings and valuations
is presented for the convenience of the reader.
Contents
Preliminaries on valued and ordered modules
Non-archimedean exponential fields
Valuation theoretic interpretation of the
growth and Taylor axioms
The exponential rank
Construction of exponential fields
Models for the elementary theory of the reals
with restricted analytic functions and exponentiation
Exponential Hardy fields
The model theory of contraction groups
Bibliography
Index
List of notation
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Fields Institute Monographs,
Volume: 12
Publication Year: 2000
ISBN: 0-8218-0943-1
Paging: 166 pp.
Binding: Hardcover
--------------------------------------------------------------------------------
Joe Albree, Auburn University at Montgomery,
AL, and David C. Arney
and V. Frederick Rickey, United States Military
Academy, West Point, NY
Description
This book reveals the rich collection of
mathematical works located at the nation's
first military school, the U.S. Military
Academy at West Point. It outlines the relevant history of the Academy, discusses the mathematics
department and curriculum, and describes
the development of the library during the
nineteenth century. A major part
of this book is an annotated catalog of the
more than 1300 works published between 1496
and 1915 found in the West Point library.
Mathematics and its instruction greatly influenced
the development of the Academy, the technological
growth of America's army, and the standards
of the military
profession. These events, in turn, were crucial
to the overall development of mathematics,
mechanics, and engineering during the nineteenth
century in the United States.
Three individuals played a prominent role
in this chronicle: Sylvanus Thayer, Charles
Davies, and Albert Church.
Listed are rare and historically valuable
works in a broad range of mathematical subjects.
The collection clearly shows the strong European
influence on the early
Academy. Also listed are numerous textbooks
by West Point faculty and graduates; significant
contributions were made by these writers
to algebra, geometry, calculus,
descriptive geometry, mechanics, surveying,
and mathematics education.
This book provides an important resource
for the general audience as well as for those
in pursuit of more scholarly information.
It contains many interesting photographs
and valuable details about the West Point
collection. It is a must-have for anyone
interested in mathematical books and collections.
Contents
Introduction
Catalog of the West Point collection
Catalog of 1803 Inventory of books, maps
and charts, belonging to the Military Academy
at West Point
Photographs
Portraits in the collection
Frontispieces in the collection
Details:
Publisher: American Mathematical Society,
London Mathematical Society
Distributor: American Mathematical Society
Series: History of Mathematics,
Volume: 18
Publication Year: 2000
ISBN: 0-8218-2059-1
Paging: 272 pp.
Binding: Hardcover
--------------------------------------------------------------------------------
R. A. Minlos, Institute for Problems of Information
Transmission, Moscow, Russia
Description
This book presents a mathematically rigorous
approach to the main ideas and phenomena
of statistical physics. The introduction
addresses the physical motivation,
focussing on the basic concept of modern
statistical physics, that is the notion of
Gibbsian random fields.
Properties of Gibbsian fields are analyzed
in two ranges of physical parameters: "regular"
(corresponding to high-temperature and low-density
regimes) where no phase
transition is exhibited, and "singular"
(low temperature regimes) where such transitions
occur.
Next, a detailed approach to the analysis
of the phenomena of phase transitions of
the first kind, the Pirogov-Sinai theory,
is presented. The author discusses this theory
in a general way and illustrates it with
the example of a lattice gas with three types
of particles. The conclusion gives a brief
review of recent developments arising from
this theory.
The volume is written for the beginner, yet
advanced students will benefit from it as
well. The book will serve nicely as a supplementary
textbook for course study. The
prerequisites are an elementary knowledge
of mechanics, probability theory and functional
analysis.
Contents
The subject and the main notions of equilibrium
statistical physics
Typical systems of statistical physics (Phase
space, dynamics, microcanonical measure)
Statistical ensembles (Microcanonical and
canonical ensembles, equivalence of ensembles)
Statistical ensembles-Continuation (the system
of indistinguishable particles and the grand
canonical ensemble)
The thermodynamic limit and the limit Gibbs
distribution
The existence and some ergodic properties
of limiting Gibbs distributions for nonsingular
values of parameters
The correlation functions and the correlation
equations
Existence of the limit correlation function
(for large positive $\mu$ or small $\beta$)
Decrease of correlations for the limit Gibbs
distribution and some corollaries (Representativity
of mean values, distribution of fluctuations,
ergodicity)
Thermodynamic functions
Phase transitions
Gibbs distributions with boundary configurations
An example of nonuniqueness of Gibbs distributions
Phase transitions in more complicated models
The ensemble of contours (Pirogov-Sinai theory)
Deviation: The ensemble of geometric configurations
of contours
The Pirogov-Sinai equations (Completion of
the proof of the main theorem)
Epilogue. What is next?
Bibliography
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: University Lecture Series,
Volume: 19
Publication Year: 2000
ISBN: 0-8218-1337-4
Paging: 103 pp.
Binding: Softcover
--------------------------------------------------------------------------------
Daniel W. Stroock, Massachusetts Institute of Technology, Cambridge, MA
Description
This book aims to bridge the gap between
probability and differential geometry. It
gives two constructions of Brownian motion
on a Riemannian manifold: an extrinsic one
where the manifold is realized as an embedded
submanifold of Euclidean space and an intrinsic
one based on the "rolling" map.
It is then shown how geometric quantities
(such as curvature) are reflected by the
behavior of Brownian paths and how that behavior
can be used to extract information about
geometric quantities. Readers
should have a strong background in analysis
with basic knowledge in stochastic calculus
and differential geometry.
Professor Stroock is a highly-respected expert
in probability and analysis. The clarity
and style of his exposition further enhance
the quality of this volume. Readers will
find an inviting introduction to the study
of paths and Brownian motion on Riemannian
manifolds.
Contents
Brownian motion in Euclidean space
Diffusions in Euclidean space
Some addenda, extensions, and refinements
Doing it on a manifold, an extrinsic approach
More about extrinsic Riemannian geometry
Bochner's identity
Some intrinsic Riemannian geometry
The bundle of orthonormal frames
Local analysis of Brownian motion
Perturbing Brownian paths
References
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Mathematical Surveys and Monographs,
Volume: 74
Publication Year: 2000
ISBN: 0-8218-2020-6
Paging: 269 pp.
Binding: Hardcover
--------------------------------------------------------------------------------
Greg Hjorth, University of California, Los Angeles, CA
Description
Actions of Polish groups are ubiquitous in
mathematics. In certain branches of ergodic
theory and functional analysis, one finds
a systematic study of the group of
measure-preserving transformations and the
unitary group. In logic, the analysis of
countable models intertwines with results
concerning the actions of the infinite
symmetric group.
This text develops the theory of Polish group
actions entirely from scratch, ultimately
presenting a coherent theory of the resulting
orbit equivalence classes that may allow complete classification by invariants
of an indicated form. The book concludes
with a criterion for an orbit equivalence
relation classifiable by countable structures
considered up to isomorphism.
This self-contained volume offers a complete
treatment of this active area of current
research and develops a difficult general
theory classifying a class of mathematical
objects up to some relevant notion of isomorphism
or equivalence.
Contents
An outline
Definitions and technicalities
Turbulence
Classifying homeomorphisms
Infinite dimensional group representations
A generalized Scott analysis
GE groups
The dark side
Beyond Borel
Looking ahead
Ordinals
Notation
Bibliography
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Mathematical Surveys and Monographs,
Volume: 75
Publication Year: 2000
ISBN: 0-8218-2002-8
Paging: 195 pp.
Binding: Hardcover
--------------------------------------------------------------------------------
Frederick P. Gardiner, Brooklyn College, CUNY, NY, and Nikola Lakic, Lehman College, CUNY, Bronx, NY
Description
The Teichm?ller space $T(X)$ is the space
of marked conformal structures on a given
quasiconformal surface $X$. This volume uses
quasiconformal mapping to give a
unified and up-to-date treatment of $T(X)$.
Emphasis is placed on parts of the theory
applicable to noncompact surfaces and to
surfaces possibly of infinite analytic
type.
The book provides a treatment of deformations
of complex structures on infinite Riemann
surfaces and gives background for further
research in many areas. These
include applications to fractal geometry,
to three-dimensional manifolds through its
relationship to Kleinian groups, and to one-dimensional
dynamics through its
relationship to quasisymmetric mappings.
Many research problems in the application
of function theory to geometry and dynamics
are suggested.
Contents
Quasiconformal mapping
Riemann surfaces
Quadratic differentials, Part I
Quadratic differentials, Part II
Teichm?ller equivalence
The Bers embedding
Kobayashi's metric on Teichmuller space
Isomorphisms and automorphisms
Teichmuller uniqueness
The mapping class group
Jenkins-Strebel differentials
Measured foliations
Obstacle problems
Asymptotic Teichm?ller space
Asymptotically extremal maps
Universal Teichmuller space
Substantial boundary points
Earthquake mappings
Bibliography
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Mathematical Surveys and Monographs,
Volume: 76
Publication Year: 2000
ISBN: 0-8218-1983-6
Paging: 372 pp.
Binding: Hardcover
@--------------------------------------------------------------------------------
A. V. Fursikov, Moscow State University, Russia
Description
This volume presents the analysis of optimal
control problems for systems described by
partial differential equations. The book
offers simple and clear exposition of main
results in this area. The methods proposed
by the author cover cases where the controlled
system corresponds to well-posed or ill-posed
boundary value problems,
which can be linear or nonlinear. The uniqueness
problem for the solution of nonlinear optimal
control problems is analyzed in various settings.
Solutions of several
previously unsolved problems are given. In
addition, general methods are applied to
the study of two problems connected with
optimal control of fluid flows described
by
the Navier-Stokes equations.
Contents
The existence of solutions to optimal control
problems
Optimality system for optimal control problems
The solvability of boundary value problems
for a dense set of data
The problem of work minimization in accelerating
still fluid to a prescribed velocity
Optimal boundary control for nonstationary
problems of fluid flow and nonhomogeneous
boundary value problems for the Navier-Stokes
equations
The Cauchy problem for elliptic equations
in a conditionally well-posed formulation
The local exact controllability of the flow
of incompressible viscous fluid
Bibliography
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Translations of Mathematical Monographs,
Volume: 187
Publication Year: 2000
ISBN: 0-8218-1382-X
Paging: 305 pp.
Binding: Hardcover
--------------------------------------------------------------------------------
Edited by: E. L. Grinberg, S. Berhanu, M.
Knopp, and G. Mendoza, Temple University,
Philadelphia, PA,
and E. T. Quinto, Tufts University, Medford,
MA
Description
This book presents the proceedings from the
conference honoring the work of Leon Ehrenpreis.
Professor Ehrenpreis worked in many different
areas of mathematics and
found connections among all of them. For
example, one can find his analytic ideas
in the context of number theory, geometric
thinking within analysis, transcendental
number theory applied to partial differential
equations, and more. The conference brought
together the communities of mathematicians
working in the areas of interest to
Professor Ehrenpreis and allowed them to
share the research inspired by his work.
The collection of articles here presents
current research on PDEs, several complex
variables, analytic number theory, integral
geometry, and tomography. The work of
Professor Ehrenpreis has contributed to basic
definitions in these areas and has motivated
a wealth of research results. This volume
offers a survey of the fundamental
principles that unified the conference and
influenced the mathematics of Leon Ehrenpreis.
Contents
M. Agranovsky -- On a problem of injectivity
for the Radon transform on a paraboloid
J. C. ?lvarez -- Anti-self-dual symplectic
forms and integral geometry
T. T. Banh -- Holomorphic extendibility of
functions via nonlinear Fourier transforms
C. A. Berenstein and A. Yger -- Division-interpolation
methods and Nullstellens?tze
J. Boman -- Uniqueness and non-uniqueness
for microanalytic continuation of ultradistributions
G. Boros and V. H. Moll -- A rational Landen
transformation. The case of degree six
R. W. Braun, R. Meise, and B. A. Taylor --
A perturbation result for linear differential
operators admitting a global right inverse
on $\Cal{D}^\prime$
S. Catto, J. Huntley, N. Moh, and D. Tepper
-- Weyl's law with error estimate
F. Colombo, I. Sabadini, and D. C. Struppa
-- Dirac equation in the octonionic algebra
W. Culp-Ressler and W. d. A. Pribitkin --
A note on Siegel's proof of Hamburger's theorem
A. D'Agnolo -- Sheaves and $\Cal{D}$-modules
in integral geometry
J. P. D'Angelo -- Positivity conditions and
squared norms of holomorphic polynomial mappings
L. Ehrenpreis, P. Kuchment, and A. Panchenko
-- The exponential x-ray transform and Fritz
John's equation. I. Range description
C. L. Epstein and G. M. Henkin -- Two lemmas
in local analytic geometry
H. M. Farkas and I. Kra -- Partitions and
theta constant identities
J. Gasqui and H. Goldschmidt -- The Radon
transform and spectral rigidity of the Grassmannians
D. Geller -- Complex powers of convolution
operators on the Heisenberg group
A. Greenleaf, A. Seeger, and S. Wainger --
Estimates for generalized Radon transforms
in three and four dimensions
E. Grinberg and I. Pesenson -- Irregular
sampling and the Radon transform
P. Guerzhoy -- Involution of $\Lambda$-adic
analytic spaces and the $U_p$-operator for
half-integral weight modular forms
R. C. Gunning -- Maximal sequences of compact
Riemann surfaces
J. Hounie and J. Tavares -- On BMO singularities
of solutions of analytic complex vector fields
A. A. Himonas and G. Misiolek -- The initial
value problem for a fifth order shallow water
equation on the real line
X. Huang, J. Merker, and F. Meylan -- Mappings
between degenerate real analytic hypersurfaces
in $C^n$
A. Katsevich -- Analysis of artifacts in
local tomography with nonsmooth attenuation
M. I. Knopp and W. d. A. Pribitkin -- The
Hecke convergence factor and modular forms
of weight zero
J. J. Kohn -- Hypoellipticity at points of
infinite type
M. Derridj and D. S. Tartakoff -- Semi-global
analytic regularity for ${{\overline{\partial}}_b}$
on CR submanifolds of $\Bbb{C}^2$
L. Lanzani -- Cauchy transform and Hardy
spaces for rough planar domains
A. Meziani -- On real analytic planar vector
fields near the character set
A. M. Odlyzko and E. M. Rains -- On longest
increasing subsequences in random permutations
P. C. Pasles -- Convergence of Poincar? series
with two complex coweights
W. d. A. Pribitkin -- Eisenstein series and
Eichler integrals
N. K. Stanton -- Real hypersurfaces with
no infinitesimal CR automorphisms
F. Treves -- Exension of cohomology classes
Y. Zhou and E. T. Quinto -- Two-radius support
theorems for spherical Radon transforms on
manifolds
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Contemporary Mathematics,
Volume: 251
Publication Year: 2000
ISBN: 0-8218-1148-7
Paging: approximately 463 pp.
Binding: Softcover