James J. Tattersall
Providence College, Rhode Island

Elementary Number Theory in Nine Chapters,2nd Edition

Hardback ISBN 0-521-85014-2
Paperback ISBN 0-521-61524-0

This textbook is intended to serve as a one-semester introductory course in number theory and in this second edition it has been revised throughout and many new exercises have been added. Historical perspective is included and emphasis is given to some of the subject's applied aspects; in particular the field of cryptography is highlighted. At the heart of the book are the major number theoretic accomplishments of Euclid, Fermat, Gauss, Legendre, and Euler, and to fully illustrate the properties of numbers and concepts developed in the text, a wealth of exercises have been included. It is assumed that the reader will have ‘pencil in hand’ and ready access to a calculator or computer. For students new to number theory, whatever their background, this is a stimulating and entertaining introduction to the subject.

Contents

1. The intriguing natural numbers; 2. Divisibility; 3. Prime numbers; 4. Perfect and amicable numbers; 5. Modular arithmetic; 6. Congruences of higher degree; 7. Cryptography; 8. Representations; 9. Partitions; Tables; Answers to selected exercises; Bibliography.

Roger Carter
University of Warwick

Lie Algebras of Finite and Affine Type

Series: Cambridge Studies in Advanced Mathematics

Hardback ISBN 0-521-85138-6

- available from September 2005 (

Lie algebras have many varied applications, both in mathematics and mathematical physics. This book provides a thorough but relaxed mathematical treatment of the subject, including both the Cartan-Killing-Weyl theory of finite dimensional simple algebras and the more modern theory of Kac-Moody algebras. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. The first half of the book deals with classification of the finite dimensional simple Lie algebras and of their finite dimensional irreducible representations. The second half introduces the theory of Kac-Moody algebras, concentrating particularly on those of affine type. A brief account of Borcherds algebras is also included. An Appendix gives a summary of the basic properties of each Lie algebra of finite and affine type.

Contents

1. Basic concepts; 2. Representations of soluble and nilpotent Lie algebras; 3. Cartan subalgebras; 4. The Cartan decomposition; 5. The root systems and the Weyl group; 6. The Cartan matrix and the Dynkin diagram; 7. The existence and uniqueness theorems; 8. The simple Lie algebras; 9. Some universal constructions; 10. Irreducible modules for semisimple Lie algebras; 11. Further properties of the universal enveloping algebra; 12. Character and dimension formulae; 13. Fundamental modules for simple Lie algebras; 14. Generalized Cartan matrices and Kac-Moody algebras; 15. The classification of generalised Cartan matrices; 16 The invariant form, root system and Weyl group; 17. Kac-Moody algebras of affine type; 18. Realisations of affine Kac-Moody algebras; 19. Some representations of symmetrisable Kac-Moody algebras; 20. Representations of affine Kac-Moody algebras; 21. Borcherds Lie algebras; Appendix.

G. A. Young / Imperial College of Science, Technology and Medicine, London
R. L. Smith / University of North Carolina, Chapel Hill

Essentials of Statistical Inference

Series: Cambridge Series in Statistical and Probabilistic Mathematics (No. 16)

Hardback ISBN 0-521-83971-8
- available from August 2005

This engaging textbook presents the concepts and results underlying the Bayesian, frequentist and Fisherian approaches to statistical inference, with particular emphasis on the contrasts between them. Aimed at advanced undergraduates and graduate students in mathematics and related disciplines, it covers in a concise treatment both basic mathematical theory and more advanced material, including such contemporary topics as Bayesian computation, higher-order likelihood theory, predictive inference, bootstrap methods and conditional inference. It contains numerous extended examples of the application of formal inference techniques to real data, as well as historical commentary on the development of the subject. Throughout, the text concentrates on concepts, rather than mathematical detail, while maintaining appropriate levels of formality. Each chapter ends with a set of accessible problems. Some prior knowledge of probability is assumed, while some previous knowledge of the objectives and main approaches to statistical inference would be helpful but is not essential.

Contents

Preface; Introduction; 1. Decision theory; 2. Bayesian methods; 3. Hypothesis testing; 4. Special models; 5. Sufficiency and completeness; 6. Two-sided tests and conditional inference; 7. Likelihood theory; 8. Higher-order theory; 9. Predictive inference; 10. Bootstrap methods; References; Index.

Geoff Der University of Glasgow, Scotland
Brian S Everitt Institute of Psychiatry, King's College, London, UK

Statistical Analysis of Medical Data Using SAS

ISBN: 1-58488-469-X
Publication Date: 8/15/2005
Number of Pages: 448

Offers the ability to interpret complex medical data for non-statisticians
Contains step-by-step procedures for analysis
Includes all relevant SAS codes
Features end-of-chapter exercises to build expertise with the software

Clinicians, not being trained experts in advanced statistical analysis, encounter increasing amounts of medical data, and require knowledge of sophisticated statistical methods to interpret this data and glean meaningful results. This text demonstrates how to use the SAS software package to analyze medical data, and translate it into a usable and meaningful format for practitioners. The authors detail particular analytical procedures, but focus on allowing users of SAS to implement the procedures and to correctly interpret the output. The book is useful to SAS users who are not primarily statisticians, but will also be of use to applied statisticians working with medical data.

Rex F Galbraith University College London, England, UK

Statistics for Fission Track Analysis

Series: Interdisciplinary Statistics

ISBN: 1-58488-533-5
Publication Date: 5/25/2005
Number of Pages: 224

Offers the first collection of scholastic literature on FT dating
Provides a basic reference for research works in fission track laboratories
Includes the key application area of FT dating - thermal history analysis

Statistical Foundations of Fission Track Analysis is the first book to address this important dating method, which is used extensively in geological and geoscience applications. Revered in the oil exploration industry, fission track (FT) dating is complementary to other methods based on radioactive decay and can be applied to older ages than radiocarbon dating: FT dating provides information dating back millions of years, rather than mere thousands. This book brings together important research literature in one volume. It includes useful applications as well as statistical foundations, key issues in thermal history analysis, and important numerical examples for aid in calculations.

Brady West /Kathleen B Welch CSAR, Ann Arbor, Michigan, USA
Brenda Gillespie University of Michigan, Ann Arbor, Michigan, USA

Linear Mixed Models: A Practical Guide using Statistical Software

ISBN: 1-58488-480-0
Publication Date: 11/15/2006
Number of Pages: 288

Offers a clear, introductory explanation of the theory and rationale behind mixed models
Presents real-world datasets that contain common problems that must be addressed by researchers using mixed models
Provides software code and the datasets discussed in the book on a supporting Web site

This book provides a clear explanation of mixed modeling techniques, introducing their application through the analyses of real datasets and presenting each example using the most commonly used software packages - R, SAS, SPSS, and STATA. The authors describe common research designs and corresponding data structures for which mixed models analysis is an appropriate statistical tool, and they include detailed descriptions on how to set up datasets for these types of analyses. The treatment also includes real-world datasets that contain common problems, such as missing data, that must be addressed using mixed models. A supporting Web site provides software code and the datasets discussed in the book.

Table of Contents

Dingyi Pei Graduate School of the Chinese Academy of Sciences, China

Authentification Codes and Combinatorial Designs

Series: Discrete Mathematics and Its Applications

ISBN: 1-58488-473-8
Publication Date: 12/15/2005
Number of Pages: 256

Covers combinatorial designs and their applications to authentication codes
Provides an in-depth analysis of unconditionally secure authentication schemes
Covers both symmetric and asymmetric codes

Authentication Codes and Combinatorial Designs addresses an important area in cryptography, namely the authentication of codes. It ties together the notion of authentication codes and combinatorial designs and demonstrates how ideas from combinatorics can be used for cryptographic applications. This is a comprehensive and self-contained treatment of the subject that covers optimal authentication codes, unconditionally secure authentication schemes, and both symmetric and asymmetric authentication codes. The authors prove the information-theoretic and combinatorial bounds and the conditions for achieving these bounds in simple and clean and comprehensive language of mathematics.