Richard M. Weiss

Quadrangular Algebras.

Paper | 2005 | ISBN: 0-691-12460-4
144 pp. | 6 x 9 | 1 line illus.

This book introduces a new class of non-associative algebras related to certain exceptional algebraic groups and their associated buildings. Richard Weiss develops a theory of these "quadrangular algebras" that opens the first purely algebraic approach to the exceptional Moufang quadrangles. These quadrangles include both those that arise as the spherical buildings associated to groups of type E6, E7, and E8 as well as the exotic quadrangles "of type F4" discovered earlier by Weiss. Based on their relationship to exceptional algebraic groups, quadrangular algebras belong in a series together with alternative and Jordan division algebras. Formally, the notion of a quadrangular algebra is derived from the notion of a pseudo-quadratic space (introduced by Jacques Tits in the study of classical groups) over a quaternion division ring. This book contains the complete classification of quadrangular algebras starting from first principles. It also shows how this classification can be made to yield the classification of exceptional Moufang quadrangles as a consequence. The book closes with a chapter on isotopes and the structure group of a quadrangular algebra.

Quadrangular Algebras is intended for graduate students of mathematics as well as specialists in buildings, exceptional algebraic groups, and related algebraic structures including Jordan algebras and the algebraic theory of quadratic forms.

Richard M. Weiss is William Walker Professor of Mathematics at Tufts University. He is the author of The Structure of Spherical Buildings (Princeton) and the coauthor (with Jacques Tits) of Moufang Polygons. He received a Humboldt Research Prize in 2004.

Table of Contents:

Preface vii
Chapter 1. Basic Definitions 1
Chapter 2. Quadratic Forms 11
Chapter 3. Quadrangular Algebras 21
Chapter 4. Proper Quadrangular Algebras 29
Chapter 5. Special Quadrangular Algebras 37
Chapter 6. Regular Quadrangular Algebras 45
Chapter 7. Defective Quadrangular Algebras 59
Chapter 8. Isotopes 77
Chapter 9. Improp er Quadrangu lar Algebras 83
Chapter 10. Existence 95
Chapter 11. Moufang Quadrangles 109
Chapter 12. The Structure Group 125
Bibliography 133
Index 134

Scheidemann, Volker

Introduction to Complex Analysis in Several Variables

2005, VIII, 171 p., Softcover
ISBN: 3-7643-7490-X

About this book

The book gives a comprehensive introduction to complex analysis in several variables. One major focus of the book is extension phenomena alien to the one-dimensional theory (Hartog's Kugelsatz, theorem of Cartan-Thullen, Bochner's theorem). The book primarily aims at students starting to work in the field of complex analysis in several variables and teachers who want to prepare a university lecture. Therefore, the book contains many examples and supporting exercises.

Written for:
Graduate students and teachers in mathematics and engineering, physicists, engineers

Keywords:

Biholomorphic maps
Cartan-Thullen theory
Complex variables
Holomorphic functions
Kugelsatz

Table of Contents

William Shaw
University of Oxford

Complex Analysis with MATHEMATICA

Hardback (ISBN-10: 0521836263 | ISBN-13: 9780521836265)

Complex Analysis with Mathematica offers a new way of learning and teaching a subject that lies at the heart of many areas of pure and applied mathematics, physics, engineering and even art. This book offers teachers and students an opportunity to learn about complex numbers in a state-of-the-art computational environment. The innovative approach also offers insights into many areas too often neglected in a student treatment, including complex chaos and mathematical art. Thus readers can also use the book for self-study and for enrichment. The use of Mathematica enables the author to cover several topics that are often absent from a traditional treatment. Students are also led, optionally, into cubic or quartic equations, investigations of symmetric chaos and advanced conformal mapping. A CD is included which contains a live version of the book: in particular all the Mathematica code enables the user to run computer experiments.

* Integration of a course on complex variables with the symbolic computation system Mathematica

* An introduction to twistor methods and the use of complex variables for 3 and 4 dimensional problems.

* Integration of a course on complex variables with Mathematica software, supported by the accompanying CD

Contents

Preface; 1. Why you need complex numbers; 2. Complex algebra and geometry; 3. Cubics, quartics and visualization of complex roots; 4. Newton-Raphson iteration and complex fractals; 5. A complex view of the real logistic map; 6. The Mandelbrot set; 7. Symmetric chaos in the complex plane; 8. Complex functions; 9. Sequences, series and power series; 10. Complex differentiation; 11. Paths and complex integration; 12. Cauchy’s theorem; 13. Cauchy’s integral formula and its remarkable consequences; 14. Laurent series, zeroes, singularities and residues; 15. Residue calculus: integration, summation and the augment principle; 16. Conformal mapping I: simple mappings and Mobius transforms; 17. Fourier transforms; 18. Laplace transforms; 19. Elementary applications to two-dimensional physics; 20. Numerical transform techniques; 21. Conformal mapping II: the Schwarz-Christoffel transformation; 22. Tiling the Euclidean and hyperbolic planes; 23. Physics in three and four dimensions I; 24. Physics in three and four dimensions II; Index.

Ed. by Cogdell, James W. / Jiang, Dihua / Kudla, Stephen S. / Soudry, David / Stanton, Robert J.

Automorphic Representations, L-Functions and Applications: Progress and Prospects
Proceedings of a conference honoring Steve Rallis on the occasion of his 60th birthday, The Ohio State University, March 27-30, 2003

October 2005. 24 x 17 cm. Approx. 440 pages. Cloth.
ISBN 3-11-017939-3

Series: Ohio State University Mathematical Research Institute Publications 11
Subjects: Mathematics / Algebra, Number theory
Language: English

This volume is the proceedings of the conference on Automorphic Representations, L-functions and Applications: Progress and Prospects, held at the Department of Mathematics of The Ohio State University, March 27?30, 2003, in honor of the 60th birthday of Steve Rallis.


The theory of automorphic representations, automorphic L-functions and their applications to arithmetic continues to be an area of vigorous and fruitful research. The contributed papers in this volume represent many of the most recent developments and directions, including

Rankin-Selberg L-functions (Bump, Ginzburg-Jiang-Rallis, Lapid-Rallis)

the relative trace formula (Jacquet, Mao-Rallis)

automorphic representations (Gan-Gurevich, Ginzburg-Rallis-Soudry)

representation theory of p-adic groups (Baruch, Kudla-Rallis, Moeglin, Cogdell-Piatetski-Shapiro-Shahidi)

p-adic methods (Harris-Li-Skinner, Vigneras), and

arithmetic applications (Chinta-Friedberg-Hoffstein).

The survey articles by Bump, on the Rankin?Selberg method, and by Jacquet, on the relative trace formula, should be particularly useful as an introduction to the key ideas about these important topics.

This volume should be of interest both to researchers and students in the area of automorphic representations, as well as to mathematicians in other areas interested in having an overview of current developments in this important field.

Ed. by Ciliberto, Ciro / Geramita, Antony V. / Harbourne, Brian / Miro-Roig, Rosa Maria / Ranestad, Kristian

Projective Varieties with Unexpected Properties
A Volume in Memory of Giuseppe Veronese. Proceedings of the international conference ‘Varieties with Unexpected Properties’, Siena, Italy, June 8?13, 2004

24 x 17 cm. VIII, 392 pages. Cloth.
ISBN 3-11-018160-6
Series: [de Gruyter Proceedings in Mathematics]
Subjects: Mathematics / Algebra, Number theory
Language: English
to be published November 2005

This volume contains refereed papers related to the lectures and talks given at a conference held in Siena (Italy) in June 2004. Also included are research papers that grew out of discussions among the participants and their collaborators. All the papers are research papers, but some of them also contain expository sections which aim to update the state of the art on the classical subject of special projective varieties and their applications and new trends like phylogenetic algebraic geometry.

The topic of secant varieties and the classification of defective varieties is central and ubiquitous in this volume. Besides the intrinsic interest of the subject, it turns out that it is also relevant in other fields of mathematics like expressions of polynomials as sums of powers, polynomial interpolation, rank tensor computations, Bayesian networks, algebraic statistics and number theory.

Emmanouil, Ioannis

Idempotent Matrices over Complex Group Algebras

Series: Universitext
2006, XIII, 280 p., Softcover
ISBN: 3-540-27990-3

About this textbook

The theory of idempotent matrices with entries in complex group algebras has recently experienced a revival, in view of its close relationship with deep geometric problems and conjectures. The relevant questions studied in this book for general groups are motivated by specific examples. A variety of techniques is employed from commutative algebra, homological algebra and functional analysis.

The book can serve as an introduction to this lively research area. The pace is suitable for independent study and the level of the presentation not very demanding. The exercises at the end of each chapter form an essential part of the book.

Table of contents

Introduction.- Motivating Examples.- Reduction to Positive Characteristic.- A Homological Approach.- Completions of CG.- Appendices: Tools from Commutative Algebra.- Discrete Ring-valued Integrals.- Frobenius' Density Theorem.- Homological Techniques.- Comparison of Projections.

Viatcheslav Mukhanov

Physical Foundations of Cosmology

Hardback (ISBN-10: 0521563984 | ISBN-13: 9780521563987)

Publication is planned for November 2005 | 448 pages | 247 x 174 mm

Courses: Cosmology, Astrophysics As a supplement to: Statistical Physics, Particle Physics, The Standard Model and Beyond, Quantum Field Theory, Statistical Mechanics, Nuclear Physics, Field Theory and Astrophysics courses
Levels: ADVANCED UNDERGRADUATE AND GRADUATE
Inflationary cosmology has been developed over the last twenty years to remedy serious shortcomings in the standard hot big bang model of the universe. Taking an original approach, this textbook explains the basis of modern cosmology and shows where the theoretical results come from. The book is divided into two parts; the first deals with the homogeneous and isotropic model of the Universe, the second part discusses how inhomogeneities can explain its structure. Established material such as the inflation and quantum cosmological perturbation are presented in great detail, however the reader is brought to the frontiers of current cosmological research by the discussion of more speculative ideas. An ideal textbook for both advanced students of physics and astrophysics, all of the necessary background material is included in every chapter and no prior knowledge of general relativity and quantum field theory is assumed.

* Presents detailed derivations of all basic results needed in cosmology, including robust predictions of inflation

* Contains an analytical treatment of nucleosynthesis, recombination and CMB fluctuations

* Provides elementary introductions to more advanced topics

Contents

Part I. Homogeneous Isotropic Universe: 1. Kinematics and dynamics of an expanding universe; 2. Propagation of light and horizons; 3. The hot universe; 4. The very early universe; 5. Inflation I: homogeneous limit; Part II. Inhomogeneous Universe: 6. Gravitational instability in Newtonian theory; 7. Gravitational instability in general relativity; 8. Inflation II: origin of the primordial inhomogeneities; 9. Cosmic microwave background anisotropies; 10. Bibliography.