August 2003 | Hardback | 440 pages 140 line
diagrams | ISBN: 0-521-58304-7
August 2003 | Paperback | 440 pages 140 line
diagrams | ISBN: 0-521-58750-6
The theory of dynamical systems is a major
mathematical
discipline closely intertwined with all main
areas of mathematics.
It has greatly stimulated research in many
sciences and given
rise to the vast new area variously called
applied dynamics,
nonlinear science, or chaos theory. This
introduction for senior
undergraduate and beginning graduate students
of mathematics,
physics, and engineering combines mathematical
rigor with copious
examples of important applications. It covers
the central
topological and probabilistic notions in
dynamics ranging from
Newtonian mechanics to coding theory. Readers
need not be
familiar with manifolds or measure theory;
the only prerequisite
is a basic undergraduate analysis course.
The authors begin by
describing the wide array of scientific and
mathematical
questions that dynamics can address. They
then use a progression
of examples to present the concepts and tools
for describing
asymptotic behavior in dynamical systems,
gradually increasing
the level of complexity. The final chapters
introduce modern
developments and applications of dynamics.
Subjects include
contractions, logistic maps, equidistribution,
symbolic dynamics,
mechanics, hyperbolic dynamics, strange attractors,
twist maps,
and KAM-theory.
Contents
1. What is a dynamical system?; Part I. Simple
Behavior in
Dynamical Systems: 2. Systems with stable
asymptotic behavior; 3.
Linear maps and linear differential equations;
Part II.
Complicated Behavior in Dynamical Systems:
4. Quasiperiodicity
and uniform distribution on the circle; 5.
Quasiperiodicity and
uniform distribution in higher dimension;
6. Conservative
systems; 7. Simple systems with complicated
orbit structure; 8.
Entropy and chaos; 9. Simple dynamics as
a tool; Part III.
Panorama of Dynamical Systems: 10. Hyperbolic
dynamics; 11.
Quadratic maps; 12. Homoclinic tangles; 13.
Strange attractors;
14. Diophantine approximation and applications
of dynamics to
number theory; 15. Variational methods, twist
maps, and closed
geodesics; Appendix; Solutions.
August 2003 | Paperback | 426 pages 36 line
diagrams | ISBN: 0-521-53569-7
(London Mathematical Society Lecture Note
Series, vol. 300)
This book introduces the reader to the geometry
of surfaces and
submanifolds in the conformal n-sphere. Various
models for Mobius
geometry are presented: the classical projective
model, the
quaternionic approach, and an approach that
uses the Clifford
algebra of the space of homogeneous coordinates
of the classical
model; the use of 2-by-2 matrices in this
context is elaborated.
For each model in turn applications are discussed.
Topics
comprise conformally flat hypersurfaces,
isothermic surfaces and
their transformation theory, Willmore surfaces,
orthogonal
systems and the Ribaucour transformation,
as well as analogous
discrete theories for isothermic surfaces
and orthogonal systems.
Certain relations with curved flats, a particular
type of
integrable system, are revealed. Thus this
book will serve both
as an introduction to newcomers (with background
in Riemannian
geometry and elementary differential geometry)
and as a reference
work for researchers.
Contents
Introduction; 0. Preliminaries: the Riemannian
point of view; 1.
The projective model; 2. Application: conformally
flat
hypersurfaces; 3. Application: isothermic
and Willmore surfaces;
4. A quaternionic model; 5. Application:
smooth and discrete
isothermic surfaces; 6. A Clifford algebra
model; 7. A Clifford
algebra model; Vahlen matrices; 8. Applications:
orthogonal
systems, isothermic surfaces; Conclusion.
August 2003 | Hardback | 328 pages | ISBN:
0-521-81593-2
(Cambridge Tracts in Mathematics, vol. 158)
A ring is called quasi-Frobenius if it is
right or left
selfinjective, and right or left artinian
(all four combinations
are equivalent). The study of these rings
grew out of the theory
of representations of a finite group as a
group of matrices over
a field, and the subject is intimately related
to duality, the
duality from right to left modules induced
by the hom functor and
the duality related to annihilators. The
present extent of the
theory is vast, and this book makes no attempt
to be
encyclopedic; instead it provides an elementary,
self-contained
account of the basic facts about these rings
at a level allowing
researchers and graduate students to gain
entry to the field.
Contents
1. Background; 2. Mininjective rings; 3.
Semiperfect mininjective
rings; 4. Min-CS rings; 5. Principally injective
and FP-rings; 6.
Simple-injective and dual rings; 7. FGF rings;
8. Johns rings; 9.
A generic example; Appendix A. Morita equivalence;
Appendix B.
Semiperfect and semiregular rings; Appendix
C. The Camps-Dicks
theorem.
August 2003 | Paperback (Hardback) | 536
pages | ISBN: 0-521-53428-3
Now available in a fully revised and updated
new edition, this
well established textbook provides a straightforward
introduction
to the theory of probability. The presentation
is entertaining
without any sacrifice of rigour; important
notions are covered
with the clarity that the subject demands.
Topics covered include
conditional probability, independence, discrete
and continuous
random variables, basic combinatorics, generating
functions and
limit theorems, and an introduction to Markov
chains. The text is
accessible to undergraduate students and
provides numerous worked
examples and exercises to help build the
important skills
necessary for problem solving.
Contents
1. Probability; 2. Conditional probability
and independence; 3.
Counting; 4. Random variables: distribution
and expectation; 5.
Random vectors: independence and dependence;
6. Generating
functions and their applications; 7. Continuous
random variables;
8. Jointly continuous random variables; 9.
Markov chains;
Appendix.
August 2003 | Hardback (Paperback) | 400
pages | ISBN: 0-521-81309-3
August 2003 | Paperback (Hardback) | 400
pages | ISBN: 0-521-01253-8
This undergraduate introduction to analytic
number theory
develops analytic skills in the course of
studying ancient
questions on polygonal numbers, perfect numbers
and amicable
pairs. The question of how the primes are
distributed amongst all
the integers is central in analytic number
theory. This
distribution is determined by the Riemann
zeta function, and
Riemann’s work shows how it is connected
to the zeroes of his
function, and the significance of the Riemann
Hypothesis.
Starting from a traditional calculus course
and assuming no
complex analysis, the author develops the
basic ideas of
elementary number theory. The text is supplemented
by series of
exercises to further develop the concepts,
and includes brief
sketches of more advanced ideas, to present
contemporary research
problems at a level suitable for undergraduates.
In addition to
proofs, both rigorous and heuristic, the
book includes extensive
graphics and tables to make analytic concepts
as concrete as
possible.
Contents1. Sums and differences; 2. Products
and divisibility; 3.
Order and magnitude; 4. Counterexamples;
5. Averages; 6. Prime
number theorems; 7. Series; 8. The Basel
problem; 9. Euler’s
product; 10. The Riemann zeta function; 11.
Pell’s equation; 12.
Elliptic curves; 13. Symmetry; 14. Explicit
formula