Advances in Elliptic Curve Cryptography

Author: Ian F. Blake
Publisher: Cambridge University Press
ISBN: 1139441221
Release Date: 2005-04-25
Genre: Mathematics

Since the appearance of the authors' first volume on elliptic curve cryptography in 1999 there has been tremendous progress in the field. In some topics, particularly point counting, the progress has been spectacular. Other topics such as the Weil and Tate pairings have been applied in new and important ways to cryptographic protocols that hold great promise. Notions such as provable security, side channel analysis and the Weil descent technique have also grown in importance. This second volume addresses these advances and brings the reader up to date. Prominent contributors to the research literature in these areas have provided articles that reflect the current state of these important topics. They are divided into the areas of protocols, implementation techniques, mathematical foundations and pairing based cryptography. Each of the topics is presented in an accessible, coherent and consistent manner for a wide audience that will include mathematicians, computer scientists and engineers.

The Geometry and Topology of Coxeter Groups LMS 32

Author: Michael W. Davis
Publisher: Princeton University Press
ISBN: 9781400845941
Release Date: 2012-11-26
Genre: Mathematics

The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.

Quanta of Maths

Author: Alain Connes
Publisher: American Mathematical Soc.
ISBN: 9780821852033
Release Date: 2010
Genre: Mathematics

The work of Alain Connes has cut a wide swath across several areas of mathematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics. Specific themes covered by the articles are as follows: entropy in operator algebras, regular $C^*$-algebras of integral domains, properly infinite $C^*$-algebras, representations of free groups and 1-cohomology, Leibniz seminorms and quantum metric spaces; von Neumann algebras, fundamental Group of $\mathrm{II}_1$ factors, subfactors and planar algebras; Baum-Connes conjecture and property T, equivariant K-homology, Hermitian K-theory; cyclic cohomology, local index formula and twisted spectral triples, tangent groupoid and the index theorem; noncommutative geometry and space-time, spectral action principle, quantum gravity, noncommutative ADHM and instantons, non-compact spectral triples of finite volume, noncommutative coordinate algebras; Hopf algebras, Vinberg algebras, renormalization and combinatorics, motivic renormalization and singularities; cyclotomy and analytic geometry over $F_1$, quantum modular forms; differential K-theory, cyclic theory and S-cohomology.

Handbook of Tilting Theory

Author: Dieter Happel
Publisher: Cambridge University Press
ISBN: 052168045X
Release Date: 2007-01-04
Genre: Mathematics

A handbook of key articles providing both an introduction and reference for newcomers and experts alike.

Nonlinear Dispersive Equations

Author: Jaime Angulo Pava
Publisher: American Mathematical Soc.
ISBN: 9780821848975
Release Date: 2009
Genre: Mathematics

This book provides a self-contained presentation of classical and new methods for studying wave phenomena that are related to the existence and stability of solitary and periodic travelling wave solutions for nonlinear dispersive evolution equations. Simplicity, concrete examples, and applications are emphasized throughout in order to make the material easily accessible. The list of classical nonlinear dispersive equations studied includes Korteweg-de Vries, Benjamin-Ono, and Schrodinger equations. Many special Jacobian elliptic functions play a role in these examples. The author brings the reader to the forefront of knowledge about some aspects of the theory and motivates future developments in this fascinating and rapidly growing field. The book can be used as an instructive study guide as well as a reference by students and mature scientists interested in nonlinear wave phenomena.

Robust Chaos and Its Applications

Author: Elhadj Zeraoulia
Publisher: World Scientific
ISBN: 9789814374071
Release Date: 2012
Genre: Mathematics

Robust chaos is defined by the absence of periodic windows and coexisting attractors in some neighborhoods in the parameter space of a dynamical system. This unique book explores the definition, sources, and roles of robust chaos. The book is written in a reasonably self-contained manner and aims to provide students and researchers with the necessary understanding of the subject. Most of the known results, experiments, and conjectures about chaos in general and about robust chaos in particular are collected here in a pedagogical form. Many examples of dynamical systems, ranging from purely mathematical to natural and social processes displaying robust chaos, are discussed in detail. At the end of each chapter is a set of exercises and open problems (more than 260 in the whole book) intended to reinforce the ideas and provide additional experiences for both readers and researchers in nonlinear science in general, and chaos theory in particular.

Error Correcting Linear Codes

Author: Anton Betten
Publisher: Springer Science & Business Media
ISBN: 9783540317036
Release Date: 2006-09-21
Genre: Mathematics

This text offers an introduction to error-correcting linear codes for researchers and graduate students in mathematics, computer science and engineering. The book differs from other standard texts in its emphasis on the classification of codes by means of isometry classes. The relevant algebraic are developed rigorously. Cyclic codes are discussed in great detail. In the last four chapters these isometry classes are enumerated, and representatives are constructed algorithmically.

Fascinating Mathematical People

Author: Donald J. Albers
Publisher: Princeton University Press
ISBN: 1400839556
Release Date: 2011-09-06
Genre: Mathematics

Fascinating Mathematical People is a collection of informal interviews and memoirs of sixteen prominent members of the mathematical community of the twentieth century, many still active. The candid portraits collected here demonstrate that while these men and women vary widely in terms of their backgrounds, life stories, and worldviews, they all share a deep and abiding sense of wonder about mathematics. Featured here--in their own words--are major research mathematicians whose cutting-edge discoveries have advanced the frontiers of the field, such as Lars Ahlfors, Mary Cartwright, Dusa McDuff, and Atle Selberg. Others are leading mathematicians who have also been highly influential as teachers and mentors, like Tom Apostol and Jean Taylor. Fern Hunt describes what it was like to be among the first black women to earn a PhD in mathematics. Harold Bacon made trips to Alcatraz to help a prisoner learn calculus. Thomas Banchoff, who first became interested in the fourth dimension while reading a Captain Marvel comic, relates his fascinating friendship with Salvador Dalí and their shared passion for art, mathematics, and the profound connection between the two. Other mathematical people found here are Leon Bankoff, who was also a Beverly Hills dentist; Arthur Benjamin, a part-time professional magician; and Joseph Gallian, a legendary mentor of future mathematicians, but also a world-renowned expert on the Beatles. This beautifully illustrated collection includes many photographs never before published, concise introductions by the editors to each person, and a foreword by Philip J. Davis. Some images inside the book are unavailable due to digital copyright restrictions.

Surveys on Surgery Theory

Author: Sylvain E. Cappell
Publisher: Princeton University Press
ISBN: 0691049386
Release Date: 2000
Genre: Mathematics

Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. There have been some extraordinary accomplishments in that time, which have led to enormously varied interactions with algebra, analysis, and geometry. Workers in many of these areas have often lamented the lack of a single source that surveys surgery theory and its applications. Indeed, no one person could write such a survey. The sixtieth birthday of C. T. C. Wall, one of the leaders of the founding generation of surgery theory, provided an opportunity to rectify the situation and produce a comprehensive book on the subject. Experts have written state-of-the-art reports that will be of broad interest to all those interested in topology, not only graduate students and mathematicians, but mathematical physicists as well. Contributors include J. Milnor, S. Novikov, W. Browder, T. Lance, E. Brown, M. Kreck, J. Klein, M. Davis, J. Davis, I. Hambleton, L. Taylor, C. Stark, E. Pedersen, W. Mio, J. Levine, K. Orr, J. Roe, J. Milgram, and C. Thomas.

Geometry of Isotropic Convex Bodies

Author: Silouanos Brazitikos
Publisher: American Mathematical Soc.
ISBN: 9781470414566
Release Date: 2014-04-24
Genre: Mathematics

The study of high-dimensional convex bodies from a geometric and analytic point of view, with an emphasis on the dependence of various parameters on the dimension stands at the intersection of classical convex geometry and the local theory of Banach spaces. It is also closely linked to many other fields, such as probability theory, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics. It is now understood that the convexity assumption forces most of the volume of a high-dimensional convex body to be concentrated in some canonical way and the main question is whether, under some natural normalization, the answer to many fundamental questions should be independent of the dimension. The aim of this book is to introduce a number of well-known questions regarding the distribution of volume in high-dimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the Kannan-Lovász-Simonovits conjecture. This book provides a self-contained and up to date account of the progress that has been made in the last fifteen years.

Recurrence Sequences

Author: Graham Everest
Publisher: American Mathematical Soc.
ISBN: 9781470423155
Release Date: 2015-09-03
Genre:

Recurrence sequences are of great intrinsic interest and have been a central part of number theory for many years. Moreover, these sequences appear almost everywhere in mathematics and computer science. This book surveys the modern theory of linear recurrence sequences and their generalizations. Particular emphasis is placed on the dramatic impact that sophisticated methods from Diophantine analysis and transcendence theory have had on the subject. Related work on bilinear recurrences and an emerging connection between recurrences and graph theory are covered. Applications and links to other areas of mathematics are described, including combinatorics, dynamical systems and cryptography, and computer science. The book is suitable for researchers interested in number theory, combinatorics, and graph theory.

From Gestalt Theory to Image Analysis

Author: Agnès Desolneux
Publisher: Springer Science & Business Media
ISBN: 9780387726359
Release Date: 2007-12-18
Genre: Computers

This book introduces a new theory in Computer Vision yielding elementary techniques to analyze digital images. These techniques are a mathematical formalization of the Gestalt theory. From the mathematical viewpoint the closest field to it is stochastic geometry, involving basic probability and statistics, in the context of image analysis. The book is mathematically self-contained, needing only basic understanding of probability and calculus. The text includes more than 130 illustrations, and numerous examples based on specific images on which the theory is tested. Detailed exercises at the end of each chapter help the reader develop a firm understanding of the concepts imparted.

Combinatorics

Author: Peter J. Cameron
Publisher: Cambridge University Press
ISBN: 9781107393370
Release Date: 1994-10-06
Genre: Mathematics

Combinatorics is a subject of increasing importance, owing to its links with computer science, statistics and algebra. This is a textbook aimed at second-year undergraduates to beginning graduates. It stresses common techniques (such as generating functions and recursive construction) which underlie the great variety of subject matter and also stresses the fact that a constructive or algorithmic proof is more valuable than an existence proof. The book is divided into two parts, the second at a higher level and with a wider range than the first. Historical notes are included which give a wider perspective on the subject. More advanced topics are given as projects and there are a number of exercises, some with solutions given.