Series: Encyclopaedia of Mathematical Sciences, Vol. 134
2005, Approx. 240 p., 5 illus., Hardcover
ISBN: 3-540-24241-4
About this book
The theory of linear algebraic monoids culminates in a coherent
blend of algebraic groups, convex geometry, and semigroup theory.
The book discusses all the key topics in detail, including
classification, orbit structure, representations, universal
constructions, and abstract analogues. An explicit cell
decomposition is constructed for the wonderful compactification,
as is a universal deformation for any semisimple group. A final
chapter summarizes important connections with other areas of
algebra and geometry. The book will serve as a solid basis for
further research. Open problems are discussed as they arise and
many useful exercises are included.
Table of contents
Introduction.- Background.- Algebraic Monoids.- Regularity
Conditions.- Classification of Reductive Monoids.- Universal
Constructions.- Orbit Structure of Reductive Monoids.- The Monoid
Analogue of the Bruhat Decomposition.- Representations and Blocks
of Algebraic Monoids.- Monoids of Lie Type.- Cell Decomposition
of Algebraic Monoids.- Conjugacy Classes.- The Centralizer of a
Semisimple Element.- Combinatorics Related to Algebraic Monoids.-
Related Developments.- References.- Index.
Series: Universitext,
2005, XII, 308 p., Hardcover
ISBN: 1-85233-905-5
About this textbook
This book provides a lucid and concise introduction to the basic
results concerning the notion of an order. Although it is mainly
intended for beginning postgraduates, the prerequisities are
minimal and selected parts can profitably be used to broaden the
horizon of the advanced undergraduate.
The treatment is modern, with a slant towards recent developments
in the theory of residuated lattices and ordered regular
semigroups.
Topics covered include:
- residuated mappings
- Galois connections
- modular, distributive, and complemented lattices
- Boolean algebras
- pseudocomplemented lattices
- Stone algebras
- Heyting algebras
- ordered groups
- lattice-ordered groups
- representable groups
- Archimedean ordered structures
- ordered semigroups
- naturally ordered regular and inverse Dubreil-Jacotin
semigroups
Featuring material that has been hitherto available only in
research articles, and an account of the range of applications of
the theory, there are also many illustrative examples and
numerous exercises throughout, making it ideal for use as a
course text, or as a basic introduction to the field for
researchers in mathematics, logic and computer science.
Table of contents
Ordered sets. residuated mappings.- Lattices. lattice morphisms.-
Regular equivalences.- Modular lattices.- Distributive lattices.-
Complementation. Boolean algebras.- Pseudocomplementation. Stone
algebras. Heyting algebras.- Congruences. subdirectly irreducible
algebras.- Ordered groups.- Archimedean ordered structures.-
Ordered semigroups. residuated semigroups.- Epimorphic group
images. Dubreil-Jacotin semigroups.- Ordered regular semigroups.-
Structure theorems.- References.- Index
Series: Graduate Texts in Mathematics, Vol. 135
2005, Approx. 500 p., Hardcover
ISBN: 0-387-24766-1
About this textbook
This is a graduate textbook covering an especially broad range of
topics. The first part of the book contains a careful but rapid
discussion of the basics of linear algebra, including vector
spaces, linear transformations, quotient spaces, and isomorphism
theorems. The author then proceeds to modules, emphasizing a
comparison with vector spaces. A thorough discussion of inner
product spaces, eigenvalues, eigenvectors, and finite dimensional
spectral theory follows, culminating in the finite dimensional
spectral theorem for normal operators. The second part of the
book is a collection of topics, including metric vector spaces,
metric spaces, Hilbert spaces, tensor products, and affine
geometry. The last chapter discusses the umbral calculus, an area
of modern algebra with important applications.
The new edition has been thoroughly revised and contains a new
chapter on convexity and separation, as well as new material on
positive linear functionals, a topic that is useful in finance
and optimization.
Table of contents
* Vector Spaces * Linear Transformations * The Isomorphism
Theorems * Modules I: Basic Properties * Modules II: Free and
Noetherian Modules * Modules over a Principal Ideal Domain * The
Structure of a Linear Operator * Eigenvalues and Eigenvectors *
Real and Complex Inner Product Spaces * Structure Theory for
Normal Operators * Metric Vector Spaces: The Theory of Bilinear
Forms * Metric Spaces * Hilbert Spaces * Tensor Products *
Positive Solutions to Linear Systems: Convexity and Separation *
Affine Geometry * Operator Factorizations: QR and Singular Value
* The Umbral Calculus * References * Index
Series: Undergraduate Texts in Mathematics,
2005, Approx. 400 p., Hardcover
ISBN: 0-387-23233-8
About this textbook
Difference equations are models of the world around us. From
clocks to computers to chromosomes, processing discrete objects
in discrete steps is a common theme. Difference equations arise
naturally from such discrete descriptions and allow us to pose
and answer such questions as: How much? How many? How long?
Difference equations are a necessary part of the mathematical
repertoire of all modern scientists and engineers.
In this new text, designed for sophomores studying mathematics
and computer science, the authors cover the basics of difference
equations and some of their applications in computing and in
population biology. Each chapter leads to techniques that can be
applied by hand to small examples or programmed for larger
problems. Along the way, the reader will use linear algebra and
graph theory, develop formal power series, solve combinatorial
problems, visit Perron?Frobenius theory, discuss pseudorandom
number generation and integer factorization, and apply the Fast
Fourier Transform to multiply polynomials quickly.
The book contains many worked examples and over 250 exercises.
While these exercises are accessible to students and have been
class-tested, they also suggest further problems and possible
research topics.
Table of contents
Preface * Fibonacci Numbers * Homogeneous Linear Recurrence
Relations * Finite Difference Equations * Generating Functions *
Nonnegative Difference Equations * Leslie's Population Matrix
Model * Matrix Difference Equations * Modular Recurrences *
Computational Complexity * Some Nonlinear Recurrences * Appendix
A: Worked Examples * Appendix B: Complex Numbers * Appendix C:
Highlights of Linear Algebra * Appendix D: Roots in the Unit
Circle * References * Index
2005, Approx. 630 p., Hardcover
ISBN: 0-387-22925-6
About this book
A renowned mathematician who considers himself both applied and
theoretical in his approach, Peter Lax has spent most of his
professional career at NYU, making significant contributions to
both mathematics and computing. He has written several important
published works and has received numerous honors including the
National Medal of Science, the Lester R. Ford Award, the
Chauvenet Prize, the Semmelweis Medal, the Wiener Prize, and the
Wolf Prize. Several students he has mentored have become leaders
in their fields.
These two volumes span the years from 1952 up until 1999, and
cover many varying topics, from functional analysis, partial
differential equations, and numerical methods to conservation
laws, integrable systems and scattering theory. After each paper,
or collection of papers, is a commentary placing the paper in
context and where relevant discussing more recent developments.
Many of the papers in these volumes have become classics and
should be ready by any serious student of these topics. In terms
of insight, depth, and breadth, Las has few equals. The reader of
this selecta will quickly appreciate his brilliance as well as
his masterful touch. Having this collection of papers in one
place allows one to follow the evolution of his ideas and
mathematical interests and to appreciate how many of these papers
initiated topics that developed lives of their own.
Table of contents
* Preface * Table of Contents * List of Publications * Partial
Differential Equations * Difference Approximations to PDE *
Hyperbolic Systems of Conservation Laws * Integrable Systems *
Integrable Systems
Series: Differential and Integral Equations and Their
Applications
ISBN: 1-58488-520-3
Publication Date: 8/15/2005
Number of Pages: 356
Provides review of the main techniques of the theory of
elliptical equations
Offers comparative analysis of various approaches to differential
equations on manifolds with singularities
Includes numerous illustrations and exercises to aid
comprehension
This much-needed book on elliptic theory presents a systematic
exposition of both analytical and topological aspects of the
theory of elliptical differential equations, a relatively new
field that is often not addressed in detail. It compares various
approaches to differential equations on manifolds with
singularities. It includes reviews of the main techniques of
elliptical theory and analytic aspects of differential operators
and examines the Index and Lefschetz theories, applications of
vibrations of thin shells, scattering theory, and Schroedinger
and Sobolev problems. Numerous examples, exercises, and
illustrations enhance understanding of the concepts of elliptic
theory.
2005, XVIII, 598 p., Hardcover
ISBN: 0-387-22926-4
About this book
A renowned mathematician who considers himself both applied and theoretical in his approach, Peter Lax has spent most of his professional career at NYU, making significant contributions to both mathematics and computing. He has written several important published works and has received numerous honors including the National Medal of Science, the Lester R. Ford Award, the Chauvenet Prize, the Semmelweis Medal, the Wiener Prize, and the Wolf Prize. Several students he has mentored have become leaders in their fields.
These two volumes span the years from 1952 up until 1999, and cover many varying topics, from functional analysis, partial differential equations, and numerical methods to conservation laws, integrable systems and scattering theory. After each paper, or collection of papers, is a commentary placing the paper in context and where relevant discussing more recent developments. Many of the papers in these volumes have become classics and should be ready by any serious student of these topics. In terms of insight, depth, and breadth, Las has few equals. The reader of this selecta will quickly appreciate his brilliance as well as his masterful touch. Having this collection of papers in one place allows one to follow the evolution of his ideas and mathematical interests and to appreciate how many of these papers initiated topics that developed lives of their own.
Table of contents
List of Publications.- Acknowledgment.- Scattering Theory in Euclidean Space.- Scattering Theory for Automorphic Functions.- Functional Analysis.- Analysis.- Algebra.