2003 Approx. 200 p. 9 illus. Hardcover
0-387-95529-1
Pell's equation is part of a central area of algebraic number
theory that treats quadratic forms and the structure of the rings
of integers in algebraic number fields. It is an ideal topic to
lead college students, as well as some talented and motivated
high school students, to a better appreciation of the power of
mathematical technique. Even at the specific level of quadratic
diophantine equations, there are unsolved problems, and the
higher degree analogues of Pell's equation, particularly beyond
the third, do not appear to have been well studied. In this
focused exercise book, the topic is motivated and developed
through sections of exercises which will allow the readers to
recreate known theory and provide a focus for their algebraic
practice. There are several explorations that encourage the
reader to embark on their own research. A high school background
in mathematics is all that is needed to get into this book, and
teachers and others interested in mathematics who do not have (or
have forgotten) a background in advanced mathematics may find
that it is a suitable vehicle for keeping up an independent
interest in the subject.
Contents: The square root of 2.- Problems leading to Pell's
equation and preliminary investigations.- Quadratic surds.- The
fundamental solution.- Tracking down the fundamental solution.-
Pell's equation and pythagorean triples.- The cubic analogue of
Pell's equation.- Analogues of the fourth and higher degrees.- A
finite version of Pell's equation.- Answers and solutions.
Series: Problem Books in Mathematics.
2003 XIV, S. 423 Softcover
3-540-00191-3
Numerical Optimization has numerous applications in engineering
sciences, operations research, economics, finance, etc. Starting
with illustrations of this ubiquitous character, this book is
essentially devoted to numerical algorithms for optimization,
which are exposed in a tutorial way. It covers fundamental
algorithms as well as more specialized and advanced topics for
unconstrained and constrained problems. The theoretical bases of
the subject, such as optimality conditions, Lagrange multipliers
or duality, although recalled, are assumed known. Most of the
algorithms described in the book are explained in a detailed
manner, allowing straightforward implementation. This level of
detail is intended to familiarize the reader with some of the
crucial questions of numerical optimization: how algorithms
operate, why they converge, difficulties that may be encountered
and their possible remedies. Theoretical aspects of the
approaches chosen are also addressed with care, often using
minimal assumptions.
Keywords: Optimization algorithms, nonsmooth optimization,
sequential quadratic programming, interior-point methods
Series: Universitext.
2003 Approx. 1000 p. Hardcover
88-470-0180-3
This book can be an invaluable instrument for overviewing the
latest and newest issues in mathematical aspects of scientific
computing, discovering new applications and the most recent
developments in the old ones. Topics include applications like
fluid dynamics, electromagnetism, structural mechanics, kinetic
models, free boundary problems, and methodologies like a
posteriori estimates, adaptivity, discontinuous Galerkin methods,
domain decomposition techniques, and numerical linear algebra.
ENUMATH Conferences provide a forum for discussing recent aspects
of Numerical Mathematics, they convene leading experts and young
scientists with a special emphasis on contributions from Europe.
Readers will get an insight into the state of the art of
Numerical Mathematics and, more generally, into the field of
Advanced Applications.
Keywords: Applied Mathematics, Numerical, Numerical Mathematics
Contents: Flow problems.- Electromagnetic problems.- Elasticity
and structures.- Numerical problems in finance.- Numerical
methods for kinetic equations.- Singular free boundary problems.-
Other applications.- A posteriori estimates and adaptivity.-
Discontinuous Galerkin finite element methods.- Domain
decomposition methods.- ODE, IDE, and related problems.-
Numerical linear algebra.
2003 XII, 452 p. 27 illus. Hardcover
1-85233-667-6
Further Algebra and Applications is the second volume of a new
and revised edition of P.M. Cohn's classic three-volume text
"Algebra" which is widely regarded as one of the most
outstanding introductory algebra textbooks. For this edition, the
text has been reworked and updated into two self-contained,
companion volumes, covering advanced topics in algebra for second-
and third-year undergraduate and postgraduate research students.
The first volume, "Basic Algebra", covers the important
results of algebra; this companion volume focuses on the
applications and covers the more advanced parts of topics such as:
- groups and algebras
- homological algebra
- universal algebra
- general ring theory
- representations of finite groups
- coding theory
- languages and automata
The author gives a clear account, supported by worked examples,
with full proofs. There are numerous exercises with occasional
hints, and some historical remarks.
Keywords: Algebra, PI-rings, Skew fields, Universal algebra
Contents: Conventions on Terminology.- Preface.- Universal
Algebra.- Homological Algebra.- Further Group Theory.- Algebras.-
Central Simple Algebras.- Representation Theory of Finite Groups.-
Noetherian Rings and Polynomial Identities.- Rings without
Finiteness Assumptions.- Skew Fields.- Coding Theory.- Languages
and Automata.- Bibliography.- List of Notations.- Author Index.-
Subject Index.
2003 XXI, 469 p. Hardcover
0-387-95541-0
Point processes and random measures find wide applicability in
telecommunications, earthquakes, image analysis, spatial point
patterns, and stereology, to name but a few areas. The authors
have made a major reshaping of their work in their first edition
of 1988 and now present their Introduction to the Theory of Point
Processes in two volumes with sub-titles "Elementary Theory
and Models" and "General Theory and Structure".
Volume One contains the introductory chapters from the first
edition, together with an informal treatment of some of the later
material intended to make it more accessible to readers primarily
interested in models and applications. The main new material in
this volume relates to marked point processes and to processes
evolving in time, where the conditional intensity methodology
provides a basis for model building, inference, and prediction.
There are abundant examples whose purpose is both didactic and to
illustrate further applications of the ideas and models that are
the main substance of the text.
Volume Two returns to the general theory, with additional
material on marked and spatial processes. The necessary
mathematical background is reviewed in appendices located in
Volume One. Daryl Daley is a Senior Fellow in the Centre for
Mathematics and Applications at the Australian National
University, with research publications in a diverse range of
applied probability models and their analysis; he is co-author
with Joe Gani of an introductory text in epidemic modelling.
David Vere-Jones is an Emeritus Professor at Victoria University
of Wellington, widely known for his contributions to Markov
chains, point processes, applications in seismology.
Contents: Early History.- Basic Properties of the Poisson Process.-
Simple Results for Stationary Point Processes on the Line.-
Renewal Processes.- Finite Point Processes.- Models Constructed
via Conditioning: Cox, Cluster, and Marked Point Processes.-
Conditional Intensities and Likelihoods.- Second Order Properties
of Stationary Point Processes.
Series: Probability and its Applications.
2003 IX, 167 p. Softcover
3-540-00235-9
Symmetries and invariance principles play an important role in
various branches of mathematics. This book deals with measures
having weak symmetry properties. Even mild conditions ensure that
all invariant Borel measures on a second countable locally
compact space can be expressed as images of specific product
measures under a fixed mapping. The results derived in this book
are interesting for their own and, moreover, a number of
carefully investigated examples underline and illustrate their
usefulness and applicability for integration problems, stochastic
simulations and statistical applications.
Keywords: Measure and integration theory, sufficient statistics,
stochastic simulations, invariant measures
Series: Lecture Notes in Mathematics. Volume. 1808
2003 VI, 120 p. Softcover
3-540-00277-4
Domain decomposition methods are a well established tool for an
efficient numerical solution of partial differential equations,
in particular for the coupling of different model equations and
of different discretization methods. Based on the approximate
solution of local boundary value problems either by finite or
boundary element methods, the global problem is reduced to an
operator equation on the skeleton of the domain decomposition.
Different variational formulations then lead to hybrid domain
decomposition methods.
Keywords: Domain decomposition methods, coupling of finite and
boundary element methods, stability analysis of mixed
approximation schemes
Series: Lecture Notes in Mathematics. Volume. 1809