Krizek, M., Academy of Sciences, Prague, Czech Republic
Luca, F., UNAM, Morelia, Mexico
Somer, L., Catholic University of America, Washington, DC, USA

17 Lectures on Fermat Numbers
From Number Theory to Geometry

2002. Approx. 250 pp. 67 figs. Hardcover
0-387-95332-9

French mathematician Pierre de Fermat became most well known for his pioneering work in the area of number theory. His work with numbers has been attracting the attention of amateur and professional mathematicians for over 350 years. This book was written in honor of the 400th anniversary of his birth and is based on a series of lectures given by the authors. The purpose of this book is to provide readers with an overview of the many properties of Fermat numbers and to demonstrate their numerous appearances and applications in areas such as number theory, probability theory, geometry, and signal processing. This book introduces a general mathematical audience to basic mathematical ideas and algebraic methods connected with the Fermat numbers and will provide invaluable reading for the amateur and professional alike.
Michal Krizek is a senior researcher at the Mathematical Institute of the Academy of Sciences of the Czech Republic and Associate Professor in the Department of Mathematics and Physics at Charles University in Prague. Florian Luca is a researcher at the Mathematical Institute of the UNAM in Morelia, Mexico. Lawrence Somer is a Professor of Mathematics at The Catholic University of America in Washington, D. C.

Contents:
Foreword by Alena Solcova.- Table of Contents.- Preface.- Glossary of Symbols.- Introduction.- Fundamentals of Number Theory.- Basic Properties of Fermat Numbers.- The Most Beautiful Theorems on Fermat Numbers.- Primality of Fermat Numbers.- Divisibility of Fermat Numbers.- Factors of Fermat Numbers.- Connection With the Pascal Triangle.- Miscellaneous Results.- The Irrationality of the Sum of Some Reciprocals.- Fermat Primes and a Diophantine Equation.- Fermat's Little Theorem, Pseudoprimes, and Super-Pseudoprimes.- Generalizations of Fermat Numbers.- Open Problems.- Fermat Number Transform and Other Applications.- The Proof of Gauss's Theorem.- Euclidean Constructions of the Regular Heptadecagon.- Appendix A, B, C.- References.- Web Site Sources.- Name Index.- Subject Index.

Series: CMS Books in Mathematics. VOL. 10

Buff, R., New York, USA

Uncertain Volatility Models - Theory and Application

2002. Approx. 230 pp. With CD-ROM. Softcover
3-540-42657-4

This book describes Uncertain Volatility Models in mathematical finance and their computer implementation for portfolios of vanilla, barrier and American options in equity and FX markets. Uncertain Volatility Models place subjective constraints on the volatility of the stochastic process of the underlying asset and evaluate option portfolios under worst- and best-case scenarios. This book is for graduate students, researchers and practitioners who wish to study advanced aspects of volatility risk in portfolios of vanilla and exotic options. The reader is assumed to be familiar with arbitrage pricing theory.

Contents:
Computational Finance: Theory: Notation and Basic Definitions. Continuous Time Finance. Scenario-Based Evaluation and Uncertainty.- Algorithms for Uncertain Volatility Models: A Lattice Framework. Algorithms for Vanilla Options. Algorithms for Barrier Options. Algorithms for American Options. Exotic Volatility Scenarios. Algorithms for Minimum-entropy Calibration.- Object-Oriented Implementation: The Architecture of MtgLib. Towards Web-based Applications.

Series: Springer Finance.

Rosenbaum, P.R., University of Pennsylvania, Philadelphia, PA, USA (Ed.)

Observational Studies

2nd ed. 2002. Approx. 250 pp. 13 figs. Hardcover
0-387-98967-6

A sound statistical account of the principles and methods for the design and analysis of observational studies. Readers are assumed to have a working knowledge of basic probability and statistics, but otherwise the account is reasonably self- contained. Throughout there are extended discussions of actual observational studies to illustrate the ideas discussed, drawn from topics as diverse as smoking and lung cancer, lead in children, nuclear weapons testing, and placement programs for students. As a result, many researchers will find this an invaluable companion in their work.

Contents:
Observational Studies.- Randomized Experiments.- Overt Bias in Observational Studies.- Sensitivity to Hidden Bias.- Known Effects.- Multiple Reference Groups in Case-Referent Studies.- Multiple Control Groups.- Coherence and Focused Hypotheses.- Constructing Matched Sets and Strata.- Some Strategic Issues.

Series: Springer Series in Statistics.

Iris Anshel (Arithmetica Inc., USA), Michael Anshel (City University of New York, USA) & Dorian Goldfeld (Columbia University, USA)

CONTRIBUTIONS TO CONTEMPORARY CRYPTOGRAPHY

This monograph deals with contributions to certain areas of contemporary cryptography based on recent developments in mathematics and computer science, including: (1) public-key cryptography based on combinatorial group theory, with an introduction to the exciting new area of braid group cryptography; (2) construction of one-way functions and pseudorandom number generators from a very general class of zeta functions, namely the feasible Selberg class. A focused survey of the underlying methods is presented together with careful computational constructions, allowing the reader to pursue independent investigations.
The book should be helpful to the beginning graduate student in computer science or mathematics who is looking for research directions, and the expert in cryptography who is searching for new approaches to the critical problems of computational security.

Contents:

Sources of Contemporary Cryptographic Research: Classical Cryptography
Difficult Mathematical Problems
Complexity Issues
Group Theory and Public-Key Cryptography: An Algebraic Method for Constructing Key Agreement Protocols
Group-Theoretic Decision Problems and Cryptography
Group-Theoretic Key Agreement Protocols
Zeta Functions and Stream Cipher Cryptography: One-Way Functions and Pseudorandom Generators
Zeta Functions and Modular Forms
Constructing Cryptographic Primitives from Zeta Functions
Implementations: MUSE: Multistream Encryption System for Secure Communication
BRAID: A Key Agreement Protocol Based on Braid Groups
Appendices: Complexity
Group Theory
Number Theory

Readership: Graduate students and researchers in algebra and theoretical computer science.

200pp (approx.) Pub. date: Scheduled Spring 2002
ISBN 981-02-4829-6

David A. Thomas, Ball State University

Modern Geometry

ISBN: 0-534-36550-7
c 2002
320 pages. Casebound. 7 3/8 x 9

MODERN GEOMETRY is focused primarily on the needs of mathematics education majors, though it would serve well as the basis for an undergraduate mathematics elective at the junior or senior level. It will also be useful in a variety of professional development contexts for practicing mathematics teachers, including graduate mathematics education courses.

MODERN GEOMETRY was written to provide undergraduate and graduate level mathematics education students with an introduction to both Euclidean and non-Euclidean geometries, appropriate to their needs as future junior and senior high school mathematics teachers.

MODERN GEOMETRY provides a systematic survey of Euclidean, hyperbolic, transformation, fractal, and projective geometries. This approach is consistent with the recommendations of the National Council of Teachers of Mathematics (NCTM), the International Society for Technology in Education (ISTE), and other professional organizations active in the preparation and continuing professional development of K-12 mathematics teachers.

Table of Contents
1. Geometry Through the Ages.
Greek Geometry Before Euclid. Euclid and the Elements. Neutral Geometry. Famous Open Problems in Geometry.
2. Topics in Euclidean Geometry.
Elementary Constructions. Exploring Relationships Between Objects. Formal Geometric Proof.
3. Other Geometries.
The Concept of Parallelism. Points, Lines, and Curves in Poincare's Disc Model. Polygons in Hyperbolic Space. Congruence in Hyperbolic Space.
4. Transformation Geometry.
An Analytic Model of the Euclidean Plane. Representing Linear Transformations in 2-space with Matrices. The Direct Isometries: Translations and Rotations. Indirect Isometries: Reflections. Composition and Analysis of Transformations. Other Linear Transformations.
5. Fractal Geometry.
Introduction to Self-similarity. Fractal Dimension. Iterated Function Systems. From Order to Chaos. The Mandelbrot Set.
6. Projective Geometry.
Elements of Perspective Drawing. Introduction to Projective Geometry. The Cross Ratio. Applications of the Cross Ratio. Matrix Methods for 3-point Perspective Transformations. Applications of Geometry in Remote Sensing. Applications of Geometry in Terrain Rendering.
Index.