Series: CMS Books in Mathematics
2004, Approx. 320 p., Hardcover
ISBN: 0-387-40562-3
About this book
The main purpose of this book is to show
how ideas from
combinatorial group theory have spread to
two other areas of
mathematics: the theory of Lie algebras and
affine algebraic
geometry. Some of these ideas, in turn, came
to combinatorial
group theory from low-dimensional topology
at the beginning of
the 20th Century. This book is divided into
three fairly
independent parts. Part I provides a brief
exposition of several
classical techniques in combinatorial group
theory, namely,
methods of Nielsen, Whitehead, and Tietze.
Part II contains the
main focus of the book. Here the authors
show how the
aforementioned techniques of combinatorial
group theory found
their way into affine algebraic geometry,
a fascinating area of
mathematics that studies polynomials and
polynomial mappings.
Part III illustrates how ideas from combinatorial
group theory
contributed to the theory of free algebras.
The focus here is on
Schreier varieties of algebras (a variety
of algebras is said to
be Schreier if any subalgebra of a free algebra
of this variety
is free in the same variety of algebras).
Table of contents
Preface.- Introduction.- I. Groups: Introduction.
Classical
Techniques. Test Elements. Other Special
Elements. Automorphic
Orbits.- II. Polynomial Algebras: Introduction.
The Jacobian
Conjecture. The Cancellation Conjecture.
Nagata's Problem. The
Embedding Problem. Coordinate Polynomials.
Test Polynomials.- III.
Free Nielsen-Schreier Algebras: Introduction.
Schreier Varieties
of Algebras. Rank Theorems and Primitive
Elements. Generalized
Primitive Elements. Free Leibniz Algebras.-
References.-
Notations.- Author Index.- Subject Index.
Series: Applied Mathematical Sciences , Vol.
157
2004, Approx. 470 pp., Hardcover
ISBN: 0-387-40853-3
About this textbook
Homology is a powerful tool used by mathematicians
to study the
properties of spaces and maps that are insensitive
to small
perturbations. This book uses a computer
to develop a
combinatorial computational approach to the
subject. The core of
the book deals with homology theory and its
computation.
Following this is a section containing extensions
to further
developments in algebraic topology, applications
to computational
dynamics, and applications to image processing.
Included are
exercises and software that can be used to
compute homology
groups and maps. The book will appeal to
researchers and graduate
students in mathematics, computer science,
engineering, and
nonlinear dynamics.
Table of contents
Preview.- Cubical Homology.- Computing Homology
Groups.- Chain
Maps and Reduction Algorithms.- Preview of
Maps.- Homology of
Maps.- Computing Homology of Maps.- Prospects
in Digital Image
Processing.- Homological Algebra.- Nonlinear
Dynamics.- Homology
of Topological Polyhedra.- Topology.- Algebra.-
Syntax of
Algorithms.
Series: Lecture Notes in Mathematics , Vol.
1824
2003, XIII, 188 p., Softcover
ISBN: 3-540-20072-X
About this book
The volume develops the foundations of differential
geometry so
as to include finite-dimensional spaces with
singularities and
nilpotent functions, at the same level as
is standard in the
elementary theory of schemes and analytic
spaces. The theory of
differentiable spaces is developed to the
point of providing a
handy tool including arbitrary base changes
(hence fibred
products, intersections and fibres of morphisms),
infinitesimal
neighbourhoods, sheaves of relative differentials,
quotients by
actions of compact Lie groups and a theory
of sheaves of Frechet
modules paralleling the useful theory of
quasi-coherent sheaves
on schemes. These notes fit naturally in
the theory of
C^\infinity-rings and C^\infinity-schemes,
as well as in the
framework of Spallek?s C^\infinity-standard
differentiable
spaces, and they require a certain familiarity
with commutative
algebra, sheaf theory, rings of differentiable
functions and
Frechet spaces.
Table of contents
Introduction.- 1. Differentiable Manifolds.-
2. Differentiable
Algebras.- 3. Differentiable Spaces.- 4.
Topology of
Differentiable Spaces.- 5. Embeddings.- 6.
Topological Tensor
Products.- 7. Fibred Products.- 8. Topological
Localization.- 9.
Finite Morphisms.- 10. Smooth Morphisms.-
11. Quotients by
Compact Lie Groups.- A. Sheaves of Frechet
Modules.- B. Space of
Jets.- References.- Index.
Series: Lecture Notes in Mathematics
Series: Fondazione C.I.M.E., Firenze , Vol.
1825
2003, XIII, 163 p., Softcover
ISBN: 3-540-20099-1
About this book
This volume aims to disseminate a number
of new ideas that have
emerged in the last few years in the field
of numerical
simulation, all bearing the common denominator
of the "multiscale"
or "multilevel" paradigm. This
covers the presence of
multiple relevant "scales" in a
physical phenomenon;
the detection and representation of "structures",
localized in space or in frequency, in the
solution of a
mathematical model; the decomposition of
a function into "details"
that can be organized and accessed in decreasing
order of
importance; and the iterative solution of
systems of linear
algebraic equations using "multilevel"
decompositions
of finite dimensional spaces.
Table of contents
Preface.- A. Cohen: Theoretical Applied and
Computational Aspects
of Nonlinear Approximation.- W. Dahmen: Multiscale
and Wavelet
Methods for Operator Equations.- J. H. Bramble:
Multilevel
Methods in Finite Elements
2003, Approx. 500 p., Hardcover
ISBN: 0-387-40349-3
About this book
This volume contains a selection of the most
important papers in
the theory of chaotic attractors over the
past 40 years. It is
dedicated to James Yorke - a pioneer in the
field and a recipient
of the 2003 Japan prize - on the occasion
of his 60th birthday.
The volume includes an introduction to Yorke's
work and an
overview of key developments in the theory
of chaotic attractors.
Table of contents
From the contents: Introduction (I. Who is
James Yorke? II. Chaos
and SLYRB measures: The development of the
theory of chaotic
attractors).- E.N. Lorenz, Deterministic
nonperiodic flow (1963).-
K. Krzyzewski and W. Szlenk, On invariant
measures for expanding
differentiable mappings (1969).- A. Lasota
and J.A. Yorke, On the
existence of invariant measures for piecewise
monotonic
transformations (1973).- R. Bowen and D.
Ruelle, The ergodic
theory of Axiom A flows (1975).- T.-Y. Li
and J.A. Yorke, Period
three implies chaos (1975).- R.M. May, Simple
mathematical models
with very complicated dynamics (1976).- M.
Henon, A two-
dimensional mapping with a strange attractor
(1976).- E. Ott,
Strange attractors and chaotic motions of
dynamical systems (1981).-
F. Hofbauer and G. Keller, Ergodic properties
of invariant
measures for piecewise monotonic transformations
(1982).- D. J.
Farmer, E. Ott and J.A. Yorke, The dimension
of chaotic
attractors (1983).- P. Grassberger and I.
Procaccia, Measuring
the strangeness of strange attractors (1983).-
M. Rychlik,
Invariant measures and variational principle
for Lozi
applications (1983).- P. Collet and Y. Levy,
Ergodic properties
of the Lozi mappings (1984). ...
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Series: Texts in Applied Mathematics , Vol.
13
2004, Approx. 440 p. 21 illus., Hardcover
ISBN: 0-387-00444-0
About this textbook
Partial differential equations are fundamental
to the modeling of
natural phenomena, arising in every field
of science.
Consequently, the desire to understand the
solutions of these
equations has always had a prominent place
in the efforts of
mathematicians; it has inspired such diverse
fields as complex
function theory, functional analysis and
algebraic topology. Like
algebra, topology, and rational mechanics,
partial differential
equations are a core area of mathematics.
This book aims to
provide the background necessary to initiate
work on a Ph.D.
thesis in PDEs for beginning graduate students.
Prerequisites
include a truly advanced calculus course
and basic complex
variables. Lebesgue integration is needed
only in Chapter 10, and
the necessary tools from functional analysis
are developed within
the course. The book can be used to teach
a variety of different
courses. This new edition features new problems
throughout and
the problems have been rearranged in each
section from simplest
to most difficult. New examples have also
been added. The
material on Sobolev spaces has been rearranged
and expanded. A
new section on nonlinear variational problems
with "Young-measure"
solutions appears. The reference section
has also been expanded.
Table of contents
Introduction* Characteristics* Conservation
Laws and Shocks*
Maximum Principles* Distributions* Function
Spaces* Sobolev
Spaces * Operator Theory * Linear Elliptic
Equations * Nonlinear
Elliptic Equations * Energy Methods for Evolution
Problems *
Semigroup Methods * References * Index
Series: Graduate Texts in Mathematics , Vol.
111
2004, Approx. 400 pp. 50 illus., Hardcover
ISBN: 0-387-95490-2
About this textbook
This book is an introduction to the theory
of elliptic curves,
ranging from its most elementary aspects
to current research. The
first part, which grew out of Tate's Haverford
lectures, covers
the elementary arithmetic theory of elliptic
curves over the
rationals. The next two chapters recast the
arguments used in the
proof of the Mordell theorem into the context
of Galois
cohomology and descent theory. This is followed
by three chapters
on the analytic theory of elliptic curves,
including such topics
as elliptic functions, theta functions, and
modular functions.
Next, the theory of endomorphisms and elliptic
curves over
infinite and local fields are discussed.
The book then continues
by providing a survey of results in the arithmetic
theory,
especially those related to the conjecture
of the Birch and
Swinnerton-Dyer. This new edition contains
three new chapters
which explore recent directions and extensions
of the theory of
elliptic curves and the addition of two new
appendices. The first
appendix, written by Stefan Theisan, examines
the role of Calabi-Yau
manifolds in string theory, while the second,
by Otto Forster,
discusses the use of elliptic curves in computing
theory and
coding theory. Dale Husemoller is a member
of the faculty at the
Max Planck Institute of Mathematics in Bonn.
Table of contents
Introduction to Rational Points on Plane
Curves * Elementary
Properties of the Chord-Tangent Group Law
on a Cubic Curve *
Plane Algebraic Curves * Factorial Rings
and Elimination Theory *
Elliptic Curves and Their Isomorphism * Families
of Elliptic
Curves and Geometric Properties of Torsion
Points * Reduction mod
p and Torsion Points * Proof of Mordell's
Finite Generation
Theorem * Galois Cohomology and Isomorphism
Classification of
Elliptic Curves over Arbitrary Fields * Descent
and Galois
Cohomology * Elliptic and Hypergeometric
Functions * Theta
Functions * Modular Functions * Endomorphisms
of Elliptic Curves
* Elliptic Curves over Finite Fields * Elliptic
Curves over Local
Fields * Elliptic Curves over Global Fields
and l-adic
Representations * L-Functions of an Elliptic
Curve and Its
Analytic Continuation * Remarks on the Birch
and Swinnerton-Dyer
Conjecture * Remarks on the Modular Curves
Conjecture and
Fermat's Last Theorem * Higher Dimensional
Analogs of Elliptic
Curves: Calabi-Yau Varieties * Families of
Elliptic Curves *
Appendix I: Calabi-Yau Manifolds and String
Theory * Appendix II:
Elliptic Curves in Algorithmic Number Theory
* Appendix III:
Guide to the Exercises * Bibliography * Index