2003 | Hardback | 368 pages 43 line diagrams 131 exercises |
ISBN: 0-521-62197-6
The central theme of this book is an exposition of the geometric
technique of calculating syzygies. It is written from a point of
view of commutative algebra, and without assuming any knowledge
of representation theory the calculation of syzygies of
determinantal varieties is explained. The starting point is a
definition of Schur functors, and these are discussed from both
an algebraic and geometric point of view. Then a chapter on
various versions of Bott痴 Theorem leads on to a careful
explanation of the technique itself, based on a description of
the direct image of a Koszul complex. Applications to
determinantal varieties follow, plus there are also chapters on
applications of the technique to rank varieties for symmetric and
skew symmetric tensors of arbitrary degree, closures of conjugacy
classes of nilpotent matrices, discriminants and resultants.
Numerous exercises are included to give the reader insight into
how to apply this important method.
Contents
1. Introduction; 2. Schur functions and Schur complexes; 3.
Grassmannians and flag varieties; 4. Bott's theorem; 5. The
geometric technique; 6. The determinantal varieties; 7. Higher
rank varieties; 8. The nilpotent orbit closures; 9. Resultants
and discriminants.
2003 | Hardback | 368 pages 106 line diagrams | ISBN: 0-521-77296-6
Numerical simulation of compressible, inviscid time-dependent
flow is a major branch of computational fluid dynamics. Its
primary goal is to obtain accurate representation of the time
evolution of complex flow patterns, involving interactions of
shocks, interfaces, and rarefaction waves. The Generalized
Riemann Problem (GRP) algorithm, developed by the authors for
this purpose, provides a unifying 壮hell・which comprises some
of the most commonly used numerical schemes of this process. This
monograph gives a systematic presentation of the GRP methodology,
starting from the underlying mathematical principles, through
basic scheme analysis and scheme extensions (such as reacting
flow or two-dimensional flows involving moving or stationary
boundaries). An array of instructive examples illustrates the
range of applications, extending from (simple) scalar equations
to computational fluid dynamics. Background material from
mathematical analysis and fluid dynamics is provided, making the
book accessible to both researchers and graduate students of
applied mathematics, science and engineering.
Contents
Preface; List of figures; 1. Introduction; Part I. Basic Theory:
2. Scalar conservation laws; Appendix A - entropy conditions for
scalar conservation laws; 3. The GRP method for scalar
conservation laws; Appendix B - convergence of the Godunov
scheme; 4. Systems of conservation laws; Appendix C - Riemann
solver for a y-law gas; 5. The generalized Riemann problem (GRP)
for compressible fluid dynamics; Appendix D - the MUSCL scheme; 6.
Analytical and numerical treatment of fluid dynamical problems;
Part II. Numerical Implementation: 7. From the GRP algorithm to
scientific computing; 8. Geometric extensions; 9. A physical
extension: reacting flow; 10. Wave interaction in a duct - a
comparative study; Bibliography; Glossary; Index.
2003 | Hardback | 496 pages | ISBN: 0-521-75307-4
The focus of this book is on algebro-geometric
solutions of
completely integrable nonlinear partial differential
equations in
(1+1)-dimensions, also known as soliton equations.
Explicitly
treated integrable models include the KdV,
AKNS, sine-Gordon, and
Camassa-Holm hierarchies as well as the classical
massive
Thirring system. An extensive treatment of
the class of algebro-geometric
solutions in the stationary as well as time-dependent
contexts is
provided. The formalism presented includes
trace formulas,
Dubrovin-type initial value problems, Baker-Akhiezer
functions,
and theta function representations of all
relevant quantities
involved. The book uses techniques from the
theory of
differential equations, spectral analysis,
and elements of
algebraic geometry (most notably, the theory
of compact Riemann
surfaces). The presentation is rigorous,
detailed, and self-contained,
with ample background material provided in
various appendices.
Detailed notes for each chapter together
with an exhaustive
bibliography enhance the presentation offered
in the main text.
Contents
Introduction; 1. The KdV hierarchy; 2. The
sGmKdV hierarchy; 3.
The AKNS hierarchy; 4. The classical massive
Thirring system; 5.
The Camassa-Holm hierarchy; Appendix A. Algebraic
curves and
their theta functions; Appendix B. KdV-type
curves; Appendix C.
AKNS-type curves; Appendix D. Asymptotic
spectral parameter
expansions; Appendix E. Lagrange interpolation;
Appendix F.
Symmetric functions; Appendix G. KdV and
AKNS Darboux-type
transformations; Appendix H. Elliptic functions;
Appendix I:
Herglotz functions; Appendix J. Weyl-Titchmarsh
theory; List of
symbols; Bibliography; Index.
2003 | Hardback | 264 pages 1 line diagram | ISBN: 0-521-81411-1
2003 | Paperback | 264 pages 1 line diagram | ISBN: 0-521-89110-8
At first glance the prime numbers appear to be distributed in a
very irregular way amongst the integers, but it is possible to
produce a simple formula that tells us (in an approximate but
well defined sense) how many primes we can expect to find that
are less than any integer we might choose. The prime number
theorem tells us what this formula is and it is indisputably one
of the great classical theorems of mathematics. This textbook
gives an introduction to the prime number theorem suitable for
advanced undergraduates and beginning graduate students. The
author's aim is to show the reader how the tools of analysis can
be used in number theory to attack a "real"problem, and
it is based on his own experiences of teaching this material.
Contents
Preface; 1. Foundations; 2. Some important Dirichlet series and
arithmetic functions; 3. The basic theorems; 4. Prime numbers in
residue classes: Dirichlet's theorem; 5. Error estimates and the
Riemann hypothesis; 6. An 粗lementary・proof of the prime
number theorem; Appendices; Bibliography; Index.
2003 | Hardback | 566 pages 65 line diagrams 23 tables | ISBN:
0-521-81496-0
Abstract regular polytopes stand at the end of more than two
millennia of geometrical research, which began with regular
polygons and polyhedra. They are highly symmetric combinatorial
structures with distinctive geometric, algebraic or topological
properties; in many ways more fascinating than traditional
regular polytopes and tessellations. The rapid development of the
subject in the past 20 years has resulted in a rich new theory,
featuring an attractive interplay of mathematical areas,
including geometry, combinatorics, group theory and topology.
Abstract regular polytopes and their groups provide an appealing
new approach to understanding geometric and combinatorial
symmetry. This is the first comprehensive up-to-date account of
the subject and its ramifications, and meets a critical need for
such a text, because no book has been published in this area of
classical and modern discrete geometry since Coxeter's Regular
Polytopes (1948) and Regular Complex Polytopes (1974). The book
should be of interest to researchers and graduate students in
discrete geometry, combinatorics and group theory.
Contents
Foreword; 1. Classical regular polytopes; 2. Regular polytopes; 3.
Coxeter groups; 4. Amalgamation; 5. Realizations; 6. Regular
polytopes on space-forms; 7. Mixing; 8. Twisting; 9. Unitary
groups and hermitian forms; 10. Locally toroidal 4-polytopes: I;
11. Locally toroidal 4-polytopes: II; 12. Higher toroidal
polytopes; 13. Regular polytopes related to linear groups; 14.
Miscellaneous classes of regular polytopes; Bibliography; Indices.