O. N. Vasilenko, Moscow State University, Russia

Number-Theoretic Algorithms in Cryptography

Translations of Mathematical Monographs, Volume: 232
2006; approx. 248 pp; hardcover
ISBN-10: 0-8218-4090-8

Expected publication date is August 28, 2006. Algorithmic number theory is a rapidly developing branch of number theory, which, in addition to its mathematical importance, has substantial applications in computer science and cryptography. Among the algorithms used in cryptography, the following are especially important:

algorithms for primality testing;
factorization algorithms for integers and for polynomials in one variable;
applications of the theory of elliptic curves;
algorithms for computation of discrete logarithms;
algorithms for solving linear equations over finite fields;
algorithms for performing arithmetic operations on large integers.
The book describes the current state of these and some other algorithms. It also contains extensive bibliography. For this English translation, additional references were prepared and commented on by the author.

Readership

Graduate students and research mathematicians interested in algorithmic number theory and its applications.

Table of Contents

Primality testing and construction of large primes
Factorization of integers with exponential complexity
Factorization of integers with subexponential complexity
Application of elliptic curves to primality testing and factorization of integers
Algorithms for computing discrete logarithm
Factorization of polynomials over finite fields
Reduced lattice bases and their applications
Factorization of polynomials over the field of rational numbers with polynomial complexity
Discrete Fourier transform and its applications
High-precision integer arithmetic
Solving systems of linear equations over finite fields
Facts from number theory
Bibliography
Index

Jorg Bewersdorff

Galois Theory for Beginners: A Historical Perspective

Student Mathematical Library, Volume: 35
2006; 180 pp; softcover
ISBN-10: 0-8218-3817-2
Expected publication date is October 11, 2006.

Galois theory is the culmination of a centuries-long search for a solution to the classical problem of solving algebraic equations by radicals. In this book, Bewersdorff follows the historical development of the theory, emphasizing concrete examples along the way. As a result, many mathematical abstractions are now seen as the natural consequence of particular investigations.

Few prerequisites are needed beyond general college mathematics, since the necessary ideas and properties of groups and fields are provided as needed. Results in Galois theory are formulated first in a concrete, elementary way, then in the modern form. Each chapter begins with a simple question that gives the reader an idea of the nature and difficulty of what lies ahead. The applications of the theory to geometric constructions, including the ancient problems of squaring the circle, duplicating the cube, and trisecting an angle, and the construction of regular n-gons are also presented.

This book is suitable for undergraduates and beginning graduate students.

Readership

Undergraduates and graduate students interested in Galois Theory.

Table of Contents

Cubic equations
Casus irreducibilis: The birth of the complex numbers
Biquadratic equations
Equations of degree n and their properties
The search for additional solution formulas
Equations that can be reduced in degree
The construction of regular polygons
The solution of equations of the fifth degree
The Galois group of an equation
Algebraic structures and Galois theory
Epilogue
Index

Fan Chung, University of California at San Diego, La Jolla, CA,
and Linyuan Lu, University of South Carolina, Columbia, SC

Complex Graphs and Networks

CBMS Regional Conference Series in Mathematics, Number: 107
2006; 264 pp; softcover
ISBN-10: 0-8218-3657-9
Expected publication date is September 24, 2006.

Through examples of large complex graphs in realistic networks, research in graph theory has been forging ahead into exciting new directions. Graph theory has emerged as a primary tool for detecting numerous hidden structures in various information networks, including Internet graphs, social networks, biological networks, or, more generally, any graph representing relations in massive data sets.

How will we explain from first principles the universal and ubiquitous coherence in the structure of these realistic but complex networks? In order to analyze these large sparse graphs, we use combinatorial, probabilistic, and spectral methods, as well as new and improved tools to analyze these networks. The examples of these networks have led us to focus on new, general, and powerful ways to look at graph theory. The book, based on lectures given at the CBMS Workshop on the Combinatorics of Large Sparse Graphs, presents new perspectives in graph theory and helps to contribute to a sound scientific foundation for our understanding of discrete networks that permeate this information age.

Readership

Graduate students and research mathematicians interested in combinatorics (graph theory) and its applications to large networks

Table of Contents

Graph theory in the information age
Old and new concentration inequalities
A generative model--the preferential attachment scheme
Duplication models for biological networks
Random graphs with given expected degrees
The rise of the giant component
Average distance and the diameter
Eigenvalues of the adjacency matrix of G(\mathbf{w})
The semi-circle law for G(\mathbf{w})
Coupling on-line and off-line analyses of random graphs
The configuration model for power law graphs
The small world phenomenon in hybrid graphs
Bibliography
Index

Edited by: J. Marshall Ash and Roger L. Jones, DePaul University, Chicago, IL

Harmonic Analysis: Calderon-Zygmund and Beyond

Contemporary Mathematics,Volume: 411
2006; 147 pp; softcover
ISBN-10: 0-8218-3920-9

Starting in the early 1950's, Alberto Calderon, Antoni Zygmund, and their students developed a program in harmonic analysis with far-reaching consequences. The title of these proceedings reflects this broad reach. This book came out of a DePaul University conference honoring Stephen Vagi upon his retirement in 2002. Vagi was a student of Calderon in the 1960's, when Calderon and Zygmund were at their peak.

Two authors, Kenig and Gatto, were students of Calderon; one, Muckenhoupt, was a student of Zygmund;and one, Sadosky was a student of both. . Two others studied under Zygmund's student Elias Stein. The remaining authors all have close connections with the Calderon-Zygmund school of analysis.

This book should interest specialists in harmonic analysis and those curious to see it applied to partial differential equations and ergodic theory.

In the first article, Adam Koranyi summarizes Vagi's work. Four additional articles cover various recent developments in harmonic analysis: Eduardo Gatto studies spaces with doubling and non-doubling measures; Cora Sadosky, product spaces; Benjamin Muckenhoupt, Laguerre expansions; and Roger Jones, singular integrals. Charles Fefferman and Carlos Kenig present applications to partial differential equations and Stephen Wainger gives an application to ergodic theory. The final article records some interesting open questions from a problem session that concluded the conference.

Readership

Graduate students and research mathematicians interested in harmonic analysis.

Table of Contents

A. Koranyi -- The work of Stephen Vagi
A. E. Gatto -- On fractional calculus associated to doubling and non-doubling measures
C. Fefferman -- Fluids and singular integrals
C. Kenig -- The well-posedness of non-linear dispersive equations: Some recent developments
C. Sadosky -- The BMO extended family in product spaces
B. Muckenhoupt -- Mean convergence of Cesaro means of Laguerre expansions
R. L. Jones -- Variation inequalities for singular integrals and related operators
S. Wainger -- A maximal function on the discrete Heisenberg group with applications to ergodic theory
B. Muckenhoupt, R. L. Jones, P. Janakiraman, D. Ryabogin, and J. Xiao -- Problems

Qing Han, University of Notre Dame, IN, and Jia-Xing Hong, Fudan University, Shanghai, China

Isometric Embedding of Riemannian Manifolds in Euclidean Spaces

Mathematical Surveys and Monographs, Volume: 130
2006; 260 pp; hardcover
ISBN-10: 0-8218-4071-1

The question of the existence of isometric embeddings of Riemannian manifolds in Euclidean space is already more than a century old. This book presents, in a systematic way, results both local and global and in arbitrary dimension but with a focus on the isometric embedding of surfaces in {\mathbb R}^3. The emphasis is on those PDE techniques which are essential to the most important results of the last century. The classic results in this book include the Janet-Cartan Theorem, Nirenberg's solution of the Weyl problem, and Nash's Embedding Theorem, with a simplified proof by Gunther. The book also includes the main results from the past twenty years, both local and global, on the isometric embedding of surfaces in Euclidean 3-space.

The work will be indispensable to researchers in the area. Moreover, the authors integrate the results and techniques into a unified whole, providing a good entry point into the area for advanced graduate students or anyone interested in this subject. The authors avoid what is technically complicated. Background knowledge is kept to an essential minimum: a one-semester course in differential geometry and a one-year course in partial differential equations.

Readership

Graduate students and research mathematicians interested in geometric analysis, differential geometry, and PDEs.

Table of Contents

Isometric embedding of Riemannian manifolds
Fundamental theorems
Surfaces in low dimensional Euclidean spaces
Local isometric embedding of surfaces in \mathbb{R}^3
Basic equations
Nonzero Gauss curvature
Gauss curvature changing sign cleanly
Nonnegative Gauss curvature
Nonpositive Gauss curvature
Global isometric embedding of surfaces in \mathbb{R}^3
Deformation of surfaces
The Weyl problem
Complete negatively curved surfaces
Boundary value problems
Bibliography
Index

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