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This volume contains the invited lectures given at the Sixteenth British Combinatorial Conference, held in July 1997 at Queen Mary and Westfield College. It provides a valuable survey of the present state of knowledge in combinatorics.

The purpose of this book is to illustrate the use of linear logic in the application of proof theory to computer science. It contains tutorials introducing the application of linear logic, as well as advanced contributions on recent trends. It is an excellent introduction to research topics in the area.

This is an essentially self-contained monograph in an intriguing field of fundamental importance for Representation Theory, Harmonic Analysis, Mathematical Physics, and Combinatorics. It is a major source of general information about the double affine Hecke algebra, also called Cherednik's algebra, and its impressive applications.

This book presents thirteen papers by leading researchers in computational mathematics. A wide range of topics are covered, illustrating the diversity of the field and its applications. This book will be of interest to researchers and graduate students in all areas of mathematics involving numerical and symbolic computations.

This volume comprises articles from four outstanding researchers who work at the cusp of analysis and logic. The emphasis is on active research topics; many results are presented that have not been published before and open problems are formulated. Considerable effort has been made by the authors to integrate their articles and make them accessible to mathematicians new to the area.

This book provides a general introduction to Tits buildings, their geometries and the related classical linear groups. It also surveys recent developments in model theory, in particular explaining amalgamation methods which have produced examples and counterexamples in group theory and geometry. Suitable for graduate students and researchers.

This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems. It will be invaluable to mathematicians wanting to see how pure mathematics can be applied, and also engineers and computer scientists wishing to implement such systems.

The British Combinatorial Conference is one of the most well known meetings for combinatorialists. This volume collects the invited talks from the 1999 conference held at the University of Kent, spanning a broad range of combinatorial topics. All researchers into combinatorics will find this volume an outstanding and up-to-date resource.

This two-volume book contains selected papers from the international conference 'Groups St Andrews 1997 in Bath'. The articles cover a wide spectrum of modern group theory. Proceedings of earlier 'Groups St Andrews' conferences have had a major impact on the development of group theory and these volumes should be equally important.

The theory of von Neumann algebras has undergone rapid development since the work of Tonita, Takesaki and Conner. These notes, based on lectures given at the University of Newcastle upon Tyne, provide an introduction to the subject and demonstrate the important role of the theory of crossed products. Part I deals with general continuous crossed products and proves the commutation theorem and the duality theorem. Part II discusses the structure of Type III von Neumann algebras and considers crossed products with modular actions. Restricting the treatment to the case of o-finite von Neumann algebras enables the author to work with faithful normal states.

These lecture notes are devoted to an area of current research interest that bridges functional analysis and function theory. The unifying theme is the notion of subharmonicity with respect to a uniform algebra. The topics covered include the rudiments of Choquet theory, various classes of representing measures, the duality between abstract sub-harmonic functions and Jensen measures, applications to problems of approximation of plurisubharmonic functions of several complex variables, and Cole's theory of estimates for conjugate functions. Many of the results are published here for the first time in monograph form.

This book is concerned with the relations between graphs, error-correcting codes and designs, in particular how techniques of graph theory and coding theory can give information about designs. A major revision and expansion of a previous volume in this series, this account includes many examples and new results as well as improved treatments of older material. So that non-specialists will find the treatment accessible the authors have included short introductions to the three main topics. This book will be welcomed by graduate students and research mathematicians and be valuable for advanced courses in finite combinatorics.

Recursion theory - now a well-established branch of pure mathematics, having grown rapidly over the last 35 years - deals with the general (abstract) theory of those operations which we conceive as being `computable' by idealized machines.

Diophantine equations over number fields have formed one of the most important and fruitful areas of mathematics throughout civilisation. In recent years increasing interest has been aroused in the analogous area of equations over function fields. However, although considerable progress has been made by previous authors, none has attempted the central problem of providing methods for the actual solution of such equations. The latter is the purpose and achievement of this volume: algorithms are provided for the complete resolution of various families of equations, such as those of Thue, hyperelliptic and genus one type. The results are achieved by means of an original fundamental inequality, first announced by the author in 1982. Several specific examples are included as illustrations of the general method and as a testimony to its efficiency. Furthermore, bounds are obtained on the solutions which improve on those obtained previously by other means. Extending the equality to a different setting, namely that of positive characteristic, enables the various families of equations to be resolved in that circumstance. Finally, by applying the inequality in a different manner, simple bounds are determined on their solutions in rational functions of the general superelliptic equation. This book represents a self-contained account of a new approach to the subject, and one which plainly has not reached the full extent of its application. It also provides a more direct on the problems than any previous book. Little expert knowledge is required to follow the theory presented, and it will appeal to professional mathematicians, research students and the enthusiastic undergraduate.

The central theme of this book is the theorem of Ambrose and Singer, which gives for a connected, complete and simply connected Riemannian manifold a necessary and sufficient condition for it to be homogeneous. This is a local condition which has to be satisfied at all points, and in this way it is a generalization of E. Cartan's method for symmetric spaces. The main aim of the authors is to use this theorem and representation theory to give a classification of homogeneous Riemannian structures on a manifold. There are eight classes, and some of these are discussed in detail. Using the constructive proof of Ambrose and Singer many examples are discussed with special attention to the natural correspondence between the homogeneous structure and the groups acting transitively and effectively as isometrics on the manifold.

This book describes a fresh approach to the classification of of convex plane polygons and of convex polyhedra according to their symmetry properties, based on ideas of topology and transformation group theory. Although there is considerable agreement with traditional treatments, a number of new concepts emerge that present classical ideas in a quite new way.

This self-contained and relatively elementary introduction to functions of several complex variables and complex (especially compact) manifolds is intended to be a synthesis of those topics and a broad introduction to the field. Part I is suitable for advanced undergraduates and beginning postgraduates whilst Part II is written more for the graduate student. The work as a whole will be useful to professional mathematicians or mathematical physicists who wish to acquire a working knowledge of this area of mathematics. Many exercises have been included and indeed they form an integral part of the text. The prerequisites for understanding Part I would be met by any mathematics student with a first degree and together the two parts provide an introduction to the more advanced works in the subject.

This is a systematic mathematical study of differential (and more general self-adjoint) operators, with particular emphasis on spectral and scattering problems. Suitable for researchers in analysis or mathematical physics, this book could also be used as a text for an advanced course on the applications of analysis.

This is an introduction to, and survey of, the constructive approaches to pure mathematics. The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism. Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate. Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists. This account for non-specialists in these and other disciplines.

This book will be a valuable addition to the growing literature in the area and essential reading for all researchers in the field of soliton theory.

Employing geometry to classical mechanics can prove to be a fruitful exercise, and such methods have proved to have wide applications to physics and engineering. The main points that are covered are the stability of relative equilibria, geometric phases, mechanical integrators, stabilization and control, bifurcation of relative equilibria and chaos in mechanical systems.

For students and experts in commutative algebra, algebraic geometry, homological algebra and computational algebra, this book is a reference in the theory of Rees algebras and related topics. It features a discussion of advanced computational methods in algebra using Groebner basis theory.

The papers printed here explore many of the rapidly developing connections between ergodic theory and other branches of mathematics, giving the background of each area, the most outstanding results and the promising lines of research. They should form perfect starting points for beginning researchers.

These notes are based on a series of lectures given at the Advanced Research Institute of Discrete Applied Mathematics held at Rutgers University. Their aim is to link together algorithmic problems arising in knot theory, statistical physics and classical combinatorics.

Offers a new, algebraic, approach to set theory. The authors introduce a particular kind of algebra, the Zermelo-Fraenkel algebras, which arise from the familiar axioms of Zermelo-Fraenkel set theory. Furthermore the authors explicitly construct such algebras using the theory of bisimulations. Their approach is completely constructive, and contains both intuitionistic set theory and topos theory.

These proceedings contain surveys articles, research articles, and visionary articles that explain new approaches to important problems on the interface of pure mathematics and mathematical physics. Ideal for students and researchers, it gives a snapshot of the state of the field as well as defining directions for future research.

This is a short and easy-to-read account of the theory of Groebner bases and its applications. It is in two parts, the first consisting of tutorial lectures written by leading experts. The second part contains seventeen original papers on Groebner bases. In an appendix are English translations of the original German papers of Bruno Buchberger in which Groebner bases were introduced.

Seventeen articles from the most outstanding contemporary topics in algebraic geometry. Includes a beautiful exposition of the new simple approach to the resolution of singularities, a detailed essay on the A,D,E classification, a discussion of the new special Lagrangian approach to mirror symmetry, and two surveys of Gromow-Witten invariants.

This volume has grown out of lectures given by Professor Pfister over many years. The emphasis here is placed on results about quadratic forms that give rise to interconnections between number theory, algebra, algebraic geometry, and topology. This is a gem of a book bringing together 30 years' worth of results that are certain to interest anyone whose research touches on quadratic forms.

The purpose of these notes is to explain in detail some topics on the intersection of commutative algebra, representation theory and singularity theory. They are based on lectures given in Tokyo, but also contain new research. It is the first cohesive account of the area and will provide a useful synthesis of recent research for algebraists.