ISBN-13: 978-0-9715766-1-2
ISBN-10: 0-9715766-1-0
640 pages, Hardcover
Student-friendly but rigorous book aimed primarily at third or
fourth year undergraduates. The pace is slow but thorough, with
an abundance of motivations, examples, and counterexamples.
Arguments used within proofs are explicitly cited, with
references to where they were first proved. Many examples have
solutions that make use of several results or concepts, so that
students can see how various techniques can blend together into
one.
Includes solutions to all odd-numbered exercises
This book is designed as an introduction to basic functional
analysis at the senior/graduate level. It has been written in
such a way that a well-motivated undergraduate student can follow
and appreciate the material without undue difficulties while an
advanced graduate student can also find topics of interest:
topological vector spaces, Kolmogorov's normability criterion,
Tychonov's classification of finite-dimensional Hausdorff
topological vector spaces, and the theorems of Korovkin and
Muntz, to mention a few.
Textbooks in functional analysis (or more generally, in
mathematics) are often unnecessarily demanding -- written in a
concise manner with few examples and motivations. Proofs in such
textbooks are sometimes so terse that much time and energy are
required of students just to verify their logical correctness,
let alone understand the ideas behind them. This is counter-productive:
students are given the impression that mathematical proofs are
mysterious; the proofs fail to convince readers of the validity
of the theorems; and students are deprived of an opportunity to
learn useful techniques and principles in problem solving.
In contrast, the pace of this book is deliberately slow but
thorough. I chose to write a textbook that I would like to have
studied from as a student - one that is mathematically rigorous
but leisurely, with lots of motivations and examples. This is a
student-friendly book that can be read and enjoyed by a
reasonably well-motivated undergrad. Proofs are developed in
detail, with all steps justified. Almost all previously proven
results used within proofs and solutions to examples are
explicitly cited, and referred to by number. This eliminates
unnecessary time and frustration spent figuring out exactly which
results were used and where they were first proved.
table of contents
with contributions by Adrien Douady, William Dunbar, and
Roland Roeder, as well as Sylvain Bonnot, David Brown, Allen
Hatcher, Chris Hruska, and Sudeb Mitra
with forewords by Clifford Earle and William Thurston
ISBN-13: 978-0-9715766-2-9; ISBN-10: 0-9715766-2-9
459 pages, Hardcover
From the foreword by William Thurston:
I have long held a great admiration and appreciation for John
Hamal Hubbard and his passionate engagement with mathematics....This
book develops a rich and interesting, interconnected body of
mathematics that is also connected to many outside subjects. I
commend it to you....I only wish that I had had access to a
source of this caliber much earlier
in my career.
Between 1970 and 1980, William Thurston astonished the
mathematical world by announcing the four theorems discussed in
this book:
Not only are the theorems of extraordinary beauty in themselves,
but the methods of proof Thurston introduced were so novel and
displayed such amazing geometric insight that to this day they
have barely entered the accepted methods of mathematicians in the
field.
....
The book is divided into two volumes. The first sets up the
Teichmuller theory necessary for discussing Thurston's theorems;
the second proves Thurston's theorems, providing more background
material where
necessary, in particular for the two hyperbolization theorems.
....
I have tried very hard to make this book accessible to a second-year
graduate student: I am assuming the results of a pretty solid
first year of graduate studies, but very little beyond, and I
have included appendices with proofs of anything not ordinarily
in such courses. I never refer to the
literature for some difficult but important result. Such
references are the bane of readers, who often find the sight
differences of assumptions and incompatible notationsan
insurmountable obstacle.
table of contents
Description
The material collected in this volume reflects the active present
of this area of mathematics, ranging from the abstract theory of
gradient flows to stochastic representations of non-linear
parabolic PDE's. Articles will highlight the present as well as
expected future directions of development of the field with
particular emphasis on applications. The article by Ambrosio and
Savare discusses the most recent development in the theory of
gradient flow of probability measures. After an introduction
reviewing the properties of the Wasserstein space and
corresponding subdifferential calculus, applications are given to
evolutionary partial differential equations. The contribution of
Herrero provides a description of some mathematical approaches
developed to account for quantitative as well as qualitative
aspects of chemotaxis. Particular attention is paid to the limits
of cell's capability to measure external cues on the one hand,
and to provide an overall description of aggregation models for
the slim mold Dictyostelium discoideum on the other. The chapter
written by Masmoudi deals with a rather different topic -
examples of singular limits in hydrodynamics. This is nowadays a
well-studied issue given the amount of new results based on the
development of the existence theory for rather general systems of
equations in hydrodynamics. The paper by DeLellis addreses the
most recent results for the transport equations with regard to
possible applications in the theory of hyperbolic systems of
conservation laws. Emphasis is put on the development of the
theory in the case when the governing field is only a BV function.
The chapter by Rein represents a comprehensive survey of results
on the Poisson-Vlasov system in astrophysics. The question of
global stability of steady states is addressed in detail. The
contribution of Soner is devoted to different representations of
non-linear parabolic equations in terms of Markov processes.
After a brief introduction on the linear theory, a class of non-linear
equations is investigated, with applications to stochastic
control and differential games. The chapter written by Zuazua
presents some of the recent progresses done on the problem of
controllabilty of partial differential equations. The
applications include the linear wave and heat equations,parabolic
equations with coefficients of low regularity, and some fluid-structure
interaction models.
Contents
Preface
Contributors
1.L. Ambriosio, G. Savare: Gradient flows of probability measures
2.M.A. Herrero: The mathematics of chemotaxis
3.N. Masmoudi: Examples of singular limits in hydrodynamics
4. C. DeLellis: Notes on hyperbolic systems of conservation laws
and transport equations
5. G. Rein: Collisionless kinetic equations from astrophysics -
the Vlasov-Poisson system
6. H.M. Stochastic representations for non-linear parabolic PDE's
7. E. Zuazua Controllability and observability of partial
differential equations: Some results and open problems
Index
ISBN-10: 0-19-921390-9
ISBN-13: 978-0-19-921390-0
Estimated publication date: January 2007
656 pages, 117 halftones & line illus., 234x156 mm
Reviews
'Review from previous edition ...a carefully-researched, thorough
and well-presented text.' - Contemporary Physics
'...very clearly written and essentially self-contained... not
only a very good and thorough introduction to the subject, but
also a precious reference for researchers.' - Foundations of
Physics
''This book covers a large set of topics, normally not covered in
standard physics curricula ... I recommend this book to
physicists interested in widening their horizons in the
directions covered by the book ... I do not know of any other
source providing such a systematic and well written introduction
into this area of research.'' - Mathematical Reviews
Description
Self-contained introduction.
Combines fundamental questions and specific applications.
Develops new mathematical techniques.
Explains computer simulation techniques.
Numerous specific examples.
This book treats the central physical concepts and mathematical
techniques used to investigate the dynamics of open quantum
systems. To provide a self-contained presentation the text begins
with a survey of classical probability theory and with an
introduction into the foundations of quantum mechanics with
particular emphasis on its statistical interpretation. The
fundamentals of density matrix theory, quantum Markov processes
and dynamical semigroups are developed. The most important master
equations used in quantum optics and in the theory of quantum
Brownian motion are applied to the study of many examples.
Special attention is paid to the theory of environment induced
decoherence, its role in the dynamical description of the
measurement process and to the experimental observation of
decohering Schrodinger cat states.
The book includes the modern formulation of open quantum systems
in terms of stochastic processes in Hilbert space. Stochastic
wave function methods and Monte Carlo algorithms are designed and
applied to important examples from quantum optics and atomic
physics, such as Levy statistics in the laser cooling of atoms,
and the damped Jaynes-Cummings model. The basic features of the
non-Markovian quantum behaviour of open systems are examined on
the basis of projection operator techniques. In addition, the
book expounds the relativistic theory of quantum measurements and
discusses several examples from a unified perspective, e.g. non-local
measurements and quantum teleportation. Influence functional and
super-operator techniques are employed to study the density
matrix theory in quantum electrodynamics and applications to the
destruction of quantum coherence are presented.
The text addresses graduate students and lecturers in physics and
applied mathematics, as well as researchers with interests in
fundamental questions in quantum mechanics and its applications.
Many analytical methods and computer simulation techniques are
developed and illustrated with the help of numerous specific
examples. Only a basic understanding of quantum mechanics and of
elementary concepts of probability theory is assumed.
Readership: Undergraduate and graduate students in physics,
theoretical physics, and applied mathematics, as well as
researchers and lecturers in the field of quantum mechanics.