This is the first exposition of the theory of quasi-symmetric designs, that is, combinatorial designs with at most two block intersection numbers. The authors aim to bring out the interaction among designs, finite geometries, and strongly regular graphs. The book starts with basic, classical material on designs and strongly regular graphs and continues with a discussion of some important results on quasi-symmetric designs. The later chapters include a combinatorial construction of the Witt designs from the projective plane of order four, recent results dealing with a structural study of...

The present book is devoted to one of the newest branches of variety theory: varieties of group representations. In addition to its intrinsic value, it has numerous connections with varieties of groups, rings and Lie algebras, polynomial identities, group rings, etc., and provides results, methods and ideas that are of interest to a broad algebraic audience. The book presents a clear and detailed exposition of several central topics in the field, leading from initial definitions and problems to the most current advances and developments. Among the topics treated are stable and unipotent...

The study of special cases of elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centers of research in number theory. This book, addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Wei finite basis theorem, points of finite order (Nagell-Lutz), etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to...

This work is a self-contained treatise on the research conducted on squares by Pfister, Hilbert, Hurwitz, and others. Many classical and modern results and quadratic forms are brought together in this book, and the treatment requires only a basic knowledge of rings, fields, polynomials, and matrices. The author deals with many different approaches to the study of squares, from the classical works of the late nineteenth century, to areas of current research.

The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated...

The articles in these two volumes arose from papers given at the 1991 International Symposium on Geometric Group Theory, and they represent some of the latest thinking in this area. Many of the world's leading figures in this field attended the conference, and their contributions cover a wide diversity of topics. This second volume contains solely a ground breaking paper by Gromov, which provides a fascinating look at finitely generated groups. For anyone whose interest lies in the interplay between groups and geometry, these books will be an essential addition to their library.

The theory of blowup algebras--Rees algebras, associated graded rings, Hilbert functions, and birational morphisms--is undergoing a period of rapid development. One of the aims of this book is to provide an introduction to these developments. The emphasis is on deriving properties of rings from their specifications in terms of generators and relations. While this places limitations on the generality of many results, it opens the way for the application of computational methods. A highlight of the book is the chapter on advanced computational methods in algebra built on current understanding...

The subject of this book lies on the boundary between probability theory and the theory of function spaces. Here Professor Braverman investigates independent random variables in rearrangement invariant (r.i.) spaces. The significant feature of r.i. spaces is that the norm of an element depends on its distribution only, and this property allows the results and methods associated with r.i. spaces to be applied to problems in probability theory. On the other hand, probabilistic methods can also prove useful in the study of r.i. spaces. In this book new techniques are used and a number of...

In 1963 Walter Feit and John G. Thompson published a proof of a 1911 conjecture by Burnside that every finite group of odd order is solvable. This proof, which ran for 255 pages, was a tour-de-force of mathematics and inspired intense effort to classify finite simple groups. This book presents a revision and expansion of the first half of the proof of the Feit-Thompson theorem. Simpler, more detailed proofs are provided for some intermediate theorems. Recent results are used to shorten other proofs.

This volume discusses results about quadratic forms that give rise to interconnections among number theory, algebra, algebraic geometry, and topology. The author deals with various topics including Hilbert's 17th problem, the Tsen-Lang theory of quasi-algebraically closed fields, the level of topological spaces, and systems of quadratic forms over arbitrary fields. Whenever possible, proofs are short and elegant, and the author has made this book as self-contained as possible. This book brings together thirty years' worth of results certain to interest anyone whose research touches on...