2003 | Paperback | 288 pages | ISBN: 0-521-64861-0
This volume comprises articles from four outstanding researchers
who work at the cusp of analysis and logic. The emphasis is on
active research topics; many results are presented that have not
been published before and open problems are formulated.
Considerable effort has been made by the authors to integrate
their articles and make them accessible to mathematicians new to
the area.
Contents
Introduction; Part I. Ultraproducts in Analysis: 1. Introduction;
2. Normed space structures; 3. Signatures; 4. Ultrapowers of
normed space structures; 5. Positive bounded formulas; 6. Basic
model theory; 7. Quantifier-free formulas; 8. Ultraproducts of
normed space structures; 9. Basic model theory II; 10. Isomorphic
ultrapowers; 11. Alternative formulations of the theory; 12.
Homogeneous structures; 13. More model theory; 14. Types;
References; Index; Part II. Actions of Polish Groups and
Classification Problems: 1. Introduction; 2. The general Glimm-Effros
dichotomy; 3. Actions of polish groups; 4. Actions of countable
groups; 5. Actions of locally compact groups; 6. Actions of the
infinite symmetric group; 7. Turbulence I: overview; 8.
Turbulence II: basic facts; 9. Turbulence III: induced actions;
10. Turbulence IV: some examples; 11. Turbulence V: calmness; 12.
Turbulence VI: the first main theorem; 13. Turbulence VII: the
second main theorem; References; Index; Part III. On Subspaces,
Asymptotic Structure, and Distortion of Banach Spaces;
Connections with Logic: 1. Introduction; 2. Background material:
the 60’s and 70’s; 3. The unconditional basic sequence
problem and connections with distortion; 4. Gowers’ dichotomy:
a block Ramsey theorem; 5. Distortion; 6. Aymptotic structure; 7.
Ordinal indices; 8. The homogeneous Banach space problem; 9.
Concluding remarks; References; Index.
2003 | Hardback | 276 pages 30 line diagrams | ISBN: 0-521-66103-X
Permutation group algorithms are one of the workhorses of
symbolic algebra systems computing with groups. They played an
indispensable role in the proof of many deep results, including
the construction and study of sporadic finite simple groups. This
book describes the theory behind permutation group algorithms, up
to the most recent developments based on the classification of
finite simple groups. Rigorous complexity estimates,
implementation hints, and advanced exercises are included
throughout. The central theme is the description of nearly linear
time algorithms, which are extremely fast both in terms of
asymptotic analysis and of practical running time. A significant
part of the permutation group library of the computational group
algebra system GAP is based on nearly linear time algorithms. The
book fills a significant gap in the symbolic computation
literature. It is recommended for everyone interested in using
computers in group theory, and is suitable for advanced graduate
courses.
Contents
1. Introduction; 2. Black-box groups; 3. Permutation groups: a
complexity overview; 4. Bases and strong generating sets; 5.
Further low-level algorithms; 6. A library of nearly linear time
algorithms; 7. Solvable permutation groups; 8. Strong generating
tests; 9. Backtrack methods; 10. Large-base groups.
2003 | Paperback | 225 pages | ISBN: 0-521-52548-9
The primary goal of this book is to give a brief introduction to
the main ideas of algebraic and geometric invariant theory. It
assumes only a minimal background in algebraic geometry, algebra
and representation theory. Topics covered include the symbolic
method for computation of invariants on the space of homogeneous
forms, the problem of finite-generatedness of the algebra of
invariants, the theory of covariants and constructions of
categorical and geometric quotients. Throughout, the emphasis is
on concrete examples which originate in classical algebraic
geometry. Based on lectures given at University of Michigan,
Harvard University and Seoul National University, the book is
written in an accessible style and contains many examples and
exercises. A novel feature of the book is a discussion of
possible linearizations of actions and the variation of quotients
under the change of linearization. Also includes the construction
of toric varieties as torus quotients of affine spaces.
Contents
1. The symbolic method; 2. The first fundamental theorem; 3.
Reductive algebraic groups; 4. Hilbert’s fourteenth problem; 5.
Algebra of covariants; 6. Quotients; 7. Linearization of actions;
8. Stability; 9. Numerical criterion of stability; 10. Projective
hypersurfaces; 11. Configurations of linear subspaces; 12. Toric
varieties.
2003 | Hardback | 256 pages 7 line diagrams 1 table 88
exercises | ISBN: 0-521-81998-9
This book describes a constructive approach to the Inverse Galois
problem: Given a finite group G and a field K, determine whether
there exists a Galois extension of K whose Galois group is
isomorphic to G. Further, if there is such a Galois extension,
find an explicit polynomial over K whose Galois group is the
prescribed group G. The main theme of the book is an exposition
of a family of ‘generic’ polynomials for certain finite
groups, which give all Galois extensions having the required
group as their Galois group. The existence of such generic
polynomials is discussed, and where they do exist, a detailed
treatment of their construction is given. The book also
introduces the notion of ‘generic dimension’ to address the
problem of the smallest number of parameters required by a
generic polynomial.
Contents
Introduction; 1. Preliminaries; 2. Groups of small degree; 3.
Hilbertian fields; 4. Galois theory of commutative rings; 5.
Generic extensions and generic polynomials; 6. Solvable groups I:
p-groups; 7. Solvable groups II: Frobenius groups; 8. The number
of parameters; Appendix A. Technical results; Appendix B.
Invariant theory; Bibliography; Index.
November 2002 | Paperback (Hardback) | 644 pages 21 colour
plates 3 tables 203 figures | ISBN: 0-521-53169-1
Now available in paperback, this wide-ranging text on modern
fluid mechanics research includes sections on modelling the
environment, physiology and magnetohydrodynamics. At the same
time, the book discusses basic physical phenomena such as
turbulence that still present fundamental challenges.
Conventional textbooks cannot hope to give graduate students more
than an inkling of what topics are currently being researched, or
how to make a choice between them. This book aims to rectify
matters, at least in part. It consists of eleven chapters that
each introduces a different branch of the subject. Though not
exhaustive, the coverage is broad: thin-film flows, Saffman-Taylor
fingering, flows in arteries and veins, convective and absolute
instabilities, turbulence, natural convection,
magnetohydrodynamics, solidification, geological fluid mechanics,
oceanography and atmospheric dynamics are all introduced and
reviewed by established authorities. Thus the book will not only
be suitable for graduate-level courses but also for specialists
seeking introductions to other areas.
Reviews
‘… a wide-ranging collection of articles … I recommend this
book without reservation, and it certainly belongs in any
technical library.’ John Miles, University of California, San
Diego
‘Each author was charged with being didactic rather than
providing a comprehensive survey of the literature surrounding
their subject. The eleven distinguished authors, including two of
the editors, succeeded marvellously carrying out their charge …
all chapters are exquisitely written and do provide a superb
introduction and significant understanding to their respective
topics … The book is a joy to read.’ Applied Mechanics
Reviews
‘… a very good contribution indeed … The authors are to be
congratulated … I very much enjoyed reading this book … I
hope it will find a wide readership among both new and
experienced fluid dynamists.’ Professor Frank T. Smith,
Mathematics Today
‘The book belongs on the shelf of any serious student of fluid
mechanics and many researchers whose work touches on the subject
of fluid mechanics in a substantial way.’ H. Aref, Journal of
Fluid Mechanics
Contributors
Stephen Davis, Yves Couder, Tim Pedley, Patrick Huerre, Javier
Jimenez, Paul Linden, Keith Moffatt, Grae Worster, Herbert
Huppert, Chris Garrett, Michael McIntyre
Contents
Preface; 1. Interfacial fluid dynamics Stephen Davis; 2. Viscous
fingering as an archetype for growth patterns Yves Couder; 3.
Blood flow in arteries and veins Tim Pedley; 4. Open shear flow
instabilities Patrick Huerre; 5. Turbulence Javier Jimenez; 6.
Convection in the environment Paul Linden; 7. Reflections on
magnetohydrodynamics Keith Moffatt; 8. Solidification of fluids
Grae Worster; 9. Geological fluid mechanics Herbert Huppert; 10.
The dynamic ocean Chris Garrett; 11. On global-scale atmospheric
and oceanic circulations Michael McIntyre; Index.