Series: Carus Mathematical Monographs
University of Chicago
Paperback (ISBN 0883850397)
A classic advanced textbook, containing a cross-section of ideas,
techniques and results that give the reader an unparalleled
introductory overview of the subject. The author gives an
integrated presentation of overall theory and its applications
in, for example, the study of groups of matrices, group
representations, and in settling the problems of Burnside and
Kurosh. Readers are also informed of open questions. Definitions
are kept to a minimum and the statements of the theorems are
sharp and clear.
Reissue of classic account
Written by one of the key figures in the subject
Corrected and with some extra comments about recent developments
Contents
1. The Jacobson radical; 2. Semisimple rings; 3. Commutativity
theorems; 4. Simple algebras; 5. Representations of finite
groups; 6. Polynomial identities; 7. Goldie’s theorem; 8. The
Golob-Shafarevitch theorem.
Review
‘This beautiful book is the result of the author's wide and
deep knowledge of the subject matter combined with his gift for
exposition …The well selected material is offered in an
integrated presentation of the structure theory of noncommutative
(associative rings) and its applications. Besides seeing the
theory at work in the study of groups of matrices, group
representations, and in settling the problems of Burnside and
Kurosh, there are, or there are given the bases for the
construction of counterexamples, so the reader can see how a
theorem fails under weaker assumptions. Readers are also informed
of open questions and of the latest generalizations not presented
in the book … the style is lively and smooth. Definitions are
kept to a minimum and the statements of the theorems are sharp
and clear. This book will appeal to many a reader. It would be
wonderful as a textbook … those interested in studying or
reviewing its subject matter or looking for a rounded account of
it could do no better than choosing this book for this purpose.’
AMS Bulletin
Series: Encyclopedia of Mathematics and its Applications (No.
98)
Hardback (ISBN-10: 0521782015 | ISBN-13: 9780521782012)
This is first modern treatment of orthogonal polynomials from the
viewpoint of special functions. The coverage is encyclopaedic,
including classical topics such as Jacobi, Hermite, Laguerre,
Hahn, Charlier and Meixner polynomials as well as those (e.g.
Askey-Wilson and Al-Salam - Chihara polynomial systems)
discovered over the last 50 years: multiple orthogonal
polynomials are dicussed for the first time in book form. Many
modern applications of the subject are dealt with, including
birth- and death- processes, integrable systems, combinatorics,
and physical models. A chapter on open research problems and
conjectures is designed to stimulate further research on the
subject. Exercises of varying degrees of difficulty are included
to help the graduate student and the newcomer. A comprehensive
bibliography rounds off the work, which will be valued as an
authoritative reference and for graduate teaching, in which role
it has already been successfully class-tested.
A comprehensive coverage of all the orthogonal polynomials
discovered in the last fifty years, as well as classical work,
and a complete chapter devoted to open problems and conjectures
Contains a chapter on multiple orthogonal polynomials, a first in
book form
Contains applications of the subject to such areas as birth- and
death-processes, integrable systems, combinatorics, and physical
models
Contents
1. Preliminaries; 2. Orthogonal polynomials; 3. Differential
equations; 4. Jacobi polynomials; 5. Some inverse problems; 6.
Discrete orthogonal polynomials; 7. Zeros and inequalities; 8.
Polynomials orthogonal on the unit circle; 9. Linearization,
connections and integral representations; 10. The Sheffer
classification; 11. q-series preliminaries; 12. q-summation
theorems; 13. Some q-orthogonal polynomials; 14. Exponential and
q-Bessel functions; 15. The Askey-Wilson polynomials; 16. The
Askey-Wilson operators; 17. q-Hermite polynomials on the unit
circle; 18. Discrete q-orthogonal polynomials; 19. Fractional and
q-fractional calculus; 20. Polynomial solutions to functional
equations; 21. Some indeterminate moment problems; 22. The
Riemann?Hilbert problem; 23. Multiple orthogonal polynomials; 24.
Research problems; Bibliography; Index.
Hardback (ISBN-10: 0521857007 | ISBN-13: 9780521857000)
The quantitative analysis of biological sequence data is based on
methods from statistics coupled with efficient algorithms from
computer science. Algebra provides a framework for unifying many
of the seemingly disparate techniques used by computational
biologists. This book offers an introduction to this mathematical
framework and describes tools from computational algebra for
designing new algorithms for exact, accurate results. These
algorithms can be applied to biological problems such as aligning
genomes, finding genes and constructing phylogenies. As the first
book in the exciting and dynamic area, it will be welcomed as a
text for self-study or for advanced undergraduate and beginning
graduate courses.
Hardback (ISBN-10: 0521850150 | ISBN-13: 9780521850155)
Paperback (ISBN-10: 0521615259 | ISBN-13: 9780521615259)
Textbook
Lecturers can request inspection copies of this title.
Courses: As main text: - (Introduction to) Measure and
Integration - (Introduction to) Abstract measure theory - (Introduction
to) Lebesgue Integration - Measures and Integrals - Integrals and
Operators - Probability and Measure As supplementary reading: - (Introduction
to) Functional Analysis - (Introduction to) Real Analysis - (Introduction
to) Probability Theory - (Introduction to) Stochastic Processes -
Martingales
Levels: THIRD/FOURTH YEAR
This is a concise and elementary introduction to measure and
integration theory as it is nowadays needed in many parts of
analysis and probability theory. The basic theory - measures,
integrals, convergence theorems, Lp-spaces and multiple integrals
- is explored in the first part of the book. The second part then
uses the notion of martingales to develop the theory further,
covering topics such as Jacobi's generalized transformation
Theorem, the Radon-Nikodym theorem, differentiation of measures,
Hardy-Littlewood maximal functions or general Fourier series.
Undergraduate calculus and an introductory course on rigorous
analysis are the only essential prerequisites, making this text
suitable for both lecture courses and for self-study. Numerous
illustrations and exercises are included and these are not merely
drill problems but are there to consolidate what has already been
learnt and to discover variants, sideways and extensions to the
main material. Hints and solutions will be available on the
internet.
Introduction to a central mathematical topic accessible for
undergraduates
Easy to follow exposition with numerous illustrations and
exercises included. Hints and solutions are available on the
internet.
Text is suitable for classroom use as well as for self-study
Contents
Prelude; Dependence chart; Prologue; 1. The pleasures of
counting; 2. o-algebras; 3. Measures; 4. Uniqueness of measures;
5. Existance of measures; 6. Measurable mappings; 7. Measurable
functions; 8. Integration of positive functions; 9. Integrals of
measurable functions and null sets; 10. Convergence theroems and
their applications; 11. The function spaces; 12. Product measures
and Fubini’s theorem; 13. Integrals with respect to image
measures; 14. Integrals of images and Jacobi’s transformation
rule; 15. Uniform integrability and Vitali’s convergence
theorem; 16. Martingales; 17. Martingale convergence theorems; 18.
The Radon-Nikodym theorem and other applications of martingales;
19. Inner product spaces; 20. Hilbert space; 21. Conditional
expectations in ; 22. Conditional expectations in ; 23.
Orthonormal systems and their convergence behaviour; Appendix a:
Lim inf and lim supp; Appendix b: Some facts from point-set
topology; Appendix c: The volume of a parallelepiped; Appendix d:
Non-measurable sets; Appendix e: A summary of the Riemann
integral; Further reading; Bibliography; Notation index; Name and
subject index.