Joseph H Silverman, Brown University

Friendly Introduction to Number Theory, A, 2/e

Copyright 2001, 400 pp.
Cloth format
ISBN 0-13-030954-0

Summary

For courses in Elementary Number Theory for non-math majors, for mathematics education students, and for Computer Science students.

This is an introductory undergraduate text designed to entice non-math majors into learning some mathematics, while teaching them to think mathematically at the same time. Starting with nothing more than basic high school algebra, the reader is gradually led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing style is informal and includes many numerical examples, which are analyzed for patterns and used to make conjectures. The emphasis is on the methods used for proving theorems rather than on specific results.

Features

・NEW-8 new chapters-1) Number Theory and Imaginary Numbers; 2) Gaussian Integers and Unique Factorization; 3) Irrational Numbers and Transcendental Numbers; 4) Fibonacci's Rabbits and Linear Recurrence Sequences; 5) Generating Functions; 6) Sums of Powers; 7) Binomial Coefficients and Pascal's Triangle; and 8) Primality Testing and Carmichael Numbers.
・NEW-Many new problems and computer exercises-Over 250 new problems. Problems relocated to the end of each chapter.
　　ﾟThis provides students with more logical flow of information.
・Low key introduction to Number Theory.
　　ﾟAllows students to explore an area of math different from standard calculus sequences.
・Five basic steps emphasized-Experimentation, pattern recognition, hypothesis formation, hypothesis testing, and formal proof.
　　ﾟEncourages students to make mathematical discovers on their own through use of open-ended problems.
・RSA cryptosystem, elliptic curves, and Fermat's Last Theorem are featured.
　　ﾟEnables students to see real-life applications of mathematics.
・Proof of Fermat's Last theorem by Andrew Wiles-Overview provided.
　　ﾟGives students an introduction to one of the most significant mathematical achievements of the 20th century.

Yuri L. Ershov
Novosibirsk State University and Institute of Discrete Mathematics and Informatics, Russia

Multi-Valued Fields

SIBERIAN SCHOOL OF ALGEBRA AND LOGIC
For more than 30 years, the author has studied the model-theoretic aspects of the theory of valued fields and multi-valued fields. Many of the key results included in this book were obtained by the author whilst preparing the manuscript. Thus the unique overview of the theory, as developed in the book, has been previously unavailable.

The book deals with the theory of valued fields and mutli-valued fields. The theory of Prufer rings is discussed from the `geometric' point of view. The author shows that by introducing the Zariski topology on families of valuation rings, it is possible to distinguish two important subfamilies of Prufer rings that correspond to Boolean and near Boolean families of valuation rings. Also, algebraic and model-theoretic properties of multi-valued fields with near Boolean families of valuation rings satisfying the local-global principle are studied. It is important that this principle is elementary, i.e., it can be expressed in the language of predicate calculus. The most important results obtained in the book include a criterion for the elementarity of an embedding of a multi-valued field and a criterion for the elementary equivalence for multi-valued fields from the class defined by the additional natural elementary conditions (absolute unramification, maximality and almost continuity of local elementary properties). The book concludes with a brief chapter discussing the bibliographic references available on the material presented, and a short history of the major developments within the field.

Contents

1. Valuation Rings. 2. Multi-Valued Fields. 3. Local-Global Properties of Near Boolean Families. 4. Model-Theoretic Properties of Multi-Valued Fields. Bibliographical and Historical Remarks. Index.

Hardbound, ISBN 0-306-11068-7
June 2001, 280 pp.

Peyret, R., Universite de Nice-Sophia, Nice, France

Spectral Methods with Application to Incompressible Viscous Flow

2002. Approx. 260 pp. Hardcover
0-387-95221-7

This well-written book explains the theory of spectral methods and their aplication to incompressible fluid flow in clear and elementary terms. It begins with an introduction to the fundamentals of spectral methods and then moves on to cover, in particular, the Fourier and Chebyshev methods. Examples and exercises are included. Chapters 4 and 5 handle streamfunction-vorticity and velocity-pressure for Navier -Stokes equations. Chapter 6 addresses special topics such as self- adaptive coordinate transform, domain decomposition, treatment of singularities, and free-surface flow. The work will be useful to those teaching in the field at the graduate level, as well as to researchers working in the area.

Contents: Introduction * Chapter 1. Fundamentals of Spectral Methods * Chapter 2
Fourier Method * Chapter 3 Chebyshev Method * Chapter 4 Navier-Stokes
equations: Streamfunction-vorticity * Chapter 5 Navier-Stokes
equations: Velocity-pressure * Chapter 6 Special Topics * Self-
adaptive coordinate transform * Domain decomposition * Treatment of
singularities * Free-surface flow * Conclusion * References

Series: Texts in Applied Mathematics. VOL. 100

Reichl, L.E., University of Texas, Austin, TX, USA

The Transition to Chaos in Conservative Classical Systems
Quantum Manifestations

2nd ed. 2002. Approx. 580 pp. 180 figs. Hardcover
0-387-98788-6

Based on courses given at the universities of Texas and California, this book treats an active field of research that touches upon the foundations of physics and chemistry. It presents, in as simple a manner as possible, the basic mechanisms that determine the dynamical evolution of both classical and quantum systems in sufficient generality to include quantum phenomena. The book begins with a discussion of Noether theorem, integrability, KAM theory, and a definition of chaotic behavior; continues with a detailed discussion of area-preserving maps, integrable quantum systems, spectral properties, path integrals, and periodically driven systems; and concludes by showing how to apply the ideas to stochastic systems. The presentation is complete and self-contained; appendices provide much of the needed mathematical background, and there are extensive references to the current literature; while problems at the ends of chapters help students clarify their understanding. This new edition has an updated presentation throughout, and a new chapter on open quantum systems.

Series: Institute for Nonlinear Science.