December 2003, ISBN 1-4020-1832-0, Hardbound
Book Series: SEMANTICS STRUCTURES IN COMPUTATION
: Volume 3
Domains are mathematical structures for information
and
approximation; they combine order-theoretic,
logical, and
topological ideas and provide a natural framework
for modelling
and reasoning about computation. The theory
of domains has proved
to be a useful tool for programming languages
and other areas of
computer science, and for applications in
mathematics.
Included in this proceedings volume are selected
papers of
original research presented at the 2nd International
Symposium on
Domain Theory in Chengdu, China. With authors
from France,
Germany, Great Britain, Ireland, Mexico,
and China, the papers
cover the latest research in these sub-areas:
domains and
computation, topology and convergence, domains,
lattices, and
continuity, and representations of domains
as event and logical
structures.
Researchers and students in theoretical computer
science should
find this a valuable source of reference.
The survey papers at
the beginning should be of particular interest
to those who wish
to gain an understanding of some general
ideas and techniques in
this area.
October 2003, ISBN 1-4020-1714-6, Hardbound
Book Series: FUNDAMENTAL THEORIES OF PHYSICS
: Volume 134
This book is the first systematic treatment
of measures on
projection lattices of von Neumann algebras.
It presents
significant recent results in this field.
One part is inspired by
the Generalized Gleason Theorem on extending
measures on the
projection lattices of von Neumann algebras
to linear functionals.
Applications of this principle to various
problems in quantum
physics are considered (hidden variable problem,
Wigner type
theorems, decoherence functional, etc.).
Another part of the
monograph deals with a fascinating interplay
of algebraic
properties of the projection lattice with
the continuity of
measures (the analysis of Jauch-Piron states,
independence
conditions in quantum field theory, etc.).
These results have no
direct analogy in the standard measure and
probability theory. On
the theoretical physics side, they are instrumental
in recovering
technical assumptions of the axiomatics of
quantum theories only
by considering algebraic properties of finitely
additive measures
(states) on quantum propositions.
Audience: This volume is of interest to both
students and experts
interested in operator theory and functional
analysis, measure
and probability theory, mathematical foundations
of quantum
theory and their interpretations, quantum
probability, quantum
information theory, quantum field theory
and quantum logics.
November 2003, ISBN 1-4020-1766-9, Hardbound
Book Series: MATHEMATICS AND ITS APPLICATIONS
: Volume 564
This monograph presents a comprehensive treatment
of recent
results on algebraic geometry as they apply
to coding theory and
cryptography, with the goal the study of
algebraic curves and
varieties with many rational points. They
book surveys recent
developments on abelian varieties, in particular
the
classification of abelian surfaces, hyperelliptic
curves, modular
towers, Kloosterman curves and codes, Shimura
curves and modular
jacobian surfaces. Applications of abelian
varieties to
cryptography are presented including a discussion
of
hyperelliptic curve cryptosystems. The inter-relationship
of
codes and curves is developed building on
Goppa's results on
algebraic-geometry cods. The volume provides
a source book of
examples with relationships to advanced topics
regarding Sato-Tate
conjectures, Eichler-Selberg trace formula,
Katz-Sarnak
conjectures and Hecke operators.
The book will be of use to mathematicians,
physicists and
engineers interested in the mathematical
methods of algebraic
geometry as they apply to coding theory and
cryptography.
January 2004, ISBN 1-4020-1833-9, Hardbound
January 2004, ISBN 1-4020-1834-7, Paperback
Book Series: NATO SCIENCE SERIES: II: Mathematics,
Physics and
Chemistry : Volume 131
This book consists of a series of expository
articles on
axiomatic, enriched and motivic homotopy
theory arising out of a
NATO Advanced Study Institute of the same
name at the Isaac
Newton Institute for the Mathematical Sciences
in Cambridge, UK
in September 2002. There are introductions
to abstract, stable
and enriched homotopy theory, motivic homotopy
categories, and
stacks. Articles include applications to
stable homotopy theory,
higher commutativity, K-theory and descent.
The list of authors
brings together experts from both the homotopy
theoretic and
motivic sides, bridging the cultural divide
between them.
February 2004, ISBN 1-4020-1835-5, Hardbound
February 2004, ISBN 1-4020-1836-3, Paperback
Book Series: NATO SCIENCE SERIES: II: Mathematics,
Physics and
Chemistry : Volume 132
This book contains discussions of some of
the most exciting
subjects in the intimately related fields
of integrability and
partial solvability. It presents a wide variety
of advanced
topics, such as the symmetry approach to
integrability and
partial solvability, partially and exactly
solvable many-body
systems, the interplay between chaos and
integrability, the
inverse scattering method for initial-boundary
problems, and new
methods for dealing with reductions and deformations
of
integrable systems. A special effort is made
to discuss the
present frontiers of the concept of integrability.
The articles cover some of the most active
areas in integrability
and partial and exact solvability. More precisely,
the following
topics are discussed: nonlinear harmonic
oscillators, chaotic
dynamics, initial-boundary nonlinear problems,
reductions and
deformations of integrable systems, Darboux
transformations, Yang-Baxter
equations and matrix solitons, superintegrable
systems, exactly
and quasi-exactly solvable spin and many-body
models.
Every mathematician must make the transition
from the
calculations of high school to the structural
and theoretical
approaches of graduate school. Essentials
of Mathematics provides
the knowledge needed to move onto advanced
mathematical work, and
a glimpse of what being a mathematician might
be like. No other
book takes this particular holistic approach
to the task.
The content is of two types. There is material
for a “Transitions”
course at the sophomore level; introductions
to logic and set
theory, discussions of proof writing and
proof discovery, and
introductions to the number systems (natural,
rational, real, and
complex). The material is presented in a
fashion suitable for a
Moore Method course, although such an approach
is not necessary.
An accompanying Instructor’s Manual provides
support for all
flavors of teaching styles. In addition to
presenting the
important results for student proof, each
area provides warm-up
and follow-up exercises to help students
internalize the material.
The second type of content is an introduction
to the professional
culture of mathematics. There are many things
that mathematicians
know but weren’t exactly taught. To give
college students a
sense of the mathematical universe, the book
includes narratives
on this kind of information. There are sections
on pure and
applied mathematics, the philosophy of mathematics,
ethics in
mathematical work, professional (including
student)
organizations, famous theorems, famous unsolved
problems, famous
mathematicians, discussions of the nature
of mathematics
research, and more. The prerequisites for
a course based on this
book include the content of high school mathematics
and a certain
level of mathematical maturity. The student
must be willing to
think on an abstract level. Two semesters
of calculus indicates a
readiness for this material.
Contents: 0. Mathematics: What is Mathematics?;
Pure vs. Applied:
What Kind of People are Mathematicians? Mathematics
Subject
Classification. 1. Logic: Mathematical Systems:
Essentials of
Logic. The Axiomatic Method; The Results
of Godel. 2. Set Theory:
Proofs; Essentials of Set Theory; Paradoxes;
Axiomatic Set Theory:
3. The Natural Numbers: Mathematical Symbols
and Notation;
Essentials of the Natural Number System;
Cantor’s Infinite
Arithmetic. 4. The Positive Rational Numbers:
Philosophy of
Mathematics; Essentials of the Positive Rational
Numbers; Ethics.
5. The Real Numbers: Famous Mathematical
Objects; Essentials of
Positive Real Numbers; Essentials of the
Real Number System;
Important Properties of the Real Number Line.
6. The Complex
Numbers: Famous Mathematicians; Essentials
of the Complex Number
System; Important Properties of the Complex
Numbers; 7. And
Beyond: What is Mathematical Research?; Famous
Unsolved Problems;
Professional Organizations; Extracurricular
Activities. Catalog
Code:ELM
ISBN:0-88385-729-4
186 pp., Hardbound, 2003