Bücher veröffentlicht von Springer New York

The theory of algebraic groups results from the interaction of various basic techniques from field theory, multilinear algebra, commutative ring theory, algebraic geometry and general algebraic representation theory of groups and Lie algebras. It is thus an ideally suitable framework for exhibiting basic algebra in action. To do that is the principal concern of this text. Accordingly, its emphasis is on developing the major general mathematical tools used for gaining control over algebraic groups, rather than on securing the final definitive results, such as the classification of the simple groups and their irreducible representations. In the same spirit, this exposition has been made entirely self-contained; no detailed knowledge beyond the usual standard material of the first one or two years of graduate study in algebra is pre- supposed. The chapter headings should be sufficient indication of the content and organisation of this book. Each chapter begins with a brief announcement of its results and ends with a few notes ranging from supplementary results, amplifications of proofs, examples and counter-examples through exercises to references. The references are intended to be merely suggestions for supplementary reading or indications of original sources, especially in cases where these might not be the expected ones. Algebraic group theory has reached a state of maturity and perfection where it may no longer be necessary to re-iterate an account of its genesis. Of the material to be presented here, including much of the basic support, the major portion is due to Claude Chevalley.

From the reviews: "e;Bela Bollobas introductory course on graph theory deserves to be considered as a watershed in the development of this theory as a serious academic subject. ... The book has chapters on electrical networks, flows, connectivity and matchings, extremal problems, colouring, Ramsey theory, random graphs, and graphs and groups. Each chapter starts at a measured and gentle pace. Classical results are proved and new insight is provided, with the examples at the end of each chapter fully supplementing the text... Even so this allows an introduction not only to some of the deeper results but, more vitally, provides outlines of, and firm insights into, their proofs. Thus in an elementary text book, we gain an overall understanding of well-known standard results, and yet at the same time constant hints of, and guidelines into, the higher levels of the subject. It is this aspect of the book which should guarantee it a permanent place in the literature."e; #Bulletin of the London Mathematical Society#1

Kummer's work on cyclotomic fields paved the way for the development of algebraic number theory in general by Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. However, the success of this general theory has tended to obscure special facts proved by Kummer about cyclotomic fields which lie deeper than the general theory. For a long period in the 20th century this aspect of Kummer's work seems to have been largely forgotten, except for a few papers, among which are those by Pollaczek [Po], Artin-Hasse [A-H] and Vandiver [Va]. In the mid 1950's, the theory of cyclotomic fields was taken up again by Iwasawa and Leopoldt. Iwasawa viewed cyclotomic fields as being analogues for number fields of the constant field extensions of algebraic geometry, and wrote a great sequence of papers investigating towers of cyclotomic fields, and more generally, Galois extensions of number fields whose Galois group is isomorphic to the additive group of p-adic integers. Leopoldt concentrated on a fixed cyclotomic field, and established various p-adic analogues of the classical complex analytic class number formulas. In particular, this led him to introduce, with Kubota, p-adic analogues of the complex L-functions attached to cyclotomic extensions of the rationals. Finally, in the late 1960's, Iwasawa [Iw 1 I] . made the fundamental discovery that there was a close connection between his work on towers of cyclotomic fields and these p-adic L-functions of Leopoldt-Kubota.

Knot theory is a kind of geometry, and one whose appeal is very direct because the objects studied are perceivable and tangible in everyday physical space. It is a meeting ground of such diverse branches of mathematics as group theory, matrix theory, number theory, algebraic geometry, and differential geometry, to name some of the more prominent ones. It had its origins in the mathematical theory of electricity and in primitive atomic physics, and there are hints today of new applications in certain branches of chemistryJ The outlines of the modern topological theory were worked out by Dehn, Alexander, Reidemeister, and Seifert almost thirty years ago. As a subfield of topology, knot theory forms the core of a wide range of problems dealing with the position of one manifold imbedded within another. This book, which is an elaboration of a series of lectures given by Fox at Haverford College while a Philips Visitor there in the spring of 1956, is an attempt to make the subject accessible to everyone. Primarily it is a text- book for a course at the junior-senior level, but we believe that it can be used with profit also by graduate students. Because the algebra required is not the familiar commutative algebra, a disproportionate amount of the book is given over to necessary algebraic preliminaries.

Combinatorics and graph theory have mushroomed in recent years. Many overlapping or equivalent results have been produced. Some of these are special cases of unformulated or unrecognized general theorems. The body of knowledge has now reached a stage where approaches toward unification are overdue. To paraphrase Professor Gian-Carlo Rota (Toronto, 1967), "e;Combinatorics needs fewer theorems and more theory. "e; In this book we are doing two things at the same time: A. We are presenting a unified treatment of much of combinatorics and graph theory. We have constructed a concise algebraically- based, but otherwise self-contained theory, which at one time embraces the basic theorems that one normally wishes to prove while giving a common terminology and framework for the develop- ment of further more specialized results. B. We are writing a textbook whereby a student of mathematics or a mathematician with another specialty can learn combinatorics and graph theory. We want this learning to be done in a much more unified way than has generally been possible from the existing literature. Our most difficult problem in the course of writing this book has been to keep A and B in balance. On the one hand, this book would be useless as a textbook if certain intuitively appealing, classical combinatorial results were either overlooked or were treated only at a level of abstraction rendering them beyond all recognition.

The present book grew out of introductory lectures on the theory offunctions of several variables. Its intent is to make the reader familiar, by the discussion of examples and special cases, with the most important branches and methods of this theory, among them, e.g., the problems of holomorphic continuation, the algebraic treatment of power series, sheaf and cohomology theory, and the real methods which stem from elliptic partial differential equations. In the first chapter we begin with the definition of holomorphic functions of several variables, their representation by the Cauchy integral, and their power series expansion on Reinhardt domains. It turns out that, in l:ontrast ~ 2 there exist domains G, G c en to the theory of a single variable, for n with G c G and G "e;# G such that each function holomorphic in G has a continuation on G. Domains G for which such a G does not exist are called domains of holomorphy. In Chapter 2 we give several characterizations of these domains of holomorphy (theorem of Cartan-Thullen, Levi's problem). We finally construct the holomorphic hull H(G} for each domain G, that is the largest (not necessarily schlicht) domain over en into which each function holomorphic on G can be continued.

This is a substantial expansion of the first edition. The last chapter on stochastic differential equations is entirely new, as is the longish section 9.4 on the Cameron-Martin-Girsanov formula. Illustrative examples in Chapter 10 include the warhorses attached to the names of L. S. Ornstein, Uhlenbeck and Bessel, but also a novelty named after Black and Scholes. The Feynman-Kac-Schrooinger development (6.4) and the material on re- flected Brownian motions (8.5) have been updated. Needless to say, there are scattered over the text minor improvements and corrections to the first edition. A Russian translation of the latter, without changes, appeared in 1987. Stochastic integration has grown in both theoretical and applicable importance in the last decade, to the extent that this new tool is now sometimes employed without heed to its rigorous requirements. This is no more surprising than the way mathematical analysis was used historically. We hope this modest introduction to the theory and application of this new field may serve as a text at the beginning graduate level, much as certain standard texts in analysis do for the deterministic counterpart. No monograph is worthy of the name of a true textbook without exercises. We have compiled a collection of these, culled from our experiences in teaching such a course at Stanford University and the University of California at San Diego, respectively. We should like to hear from readers who can supply VI PREFACE more and better exercises.

This book develops a theory of formal power series in noncommuting variables, the main emphasis being on results applicable to automata and formal language theory. This theory was initiated around 196O-apart from some scattered work done earlier in connection with free groups-by M. P. Schutzenberger to whom also belong some of the main results. So far there is no book in existence concerning this theory. This lack has had the unfortunate effect that formal power series have not been known and used by theoretical computer scientists to the extent they in our estimation should have been. As with most mathematical formalisms, the formalism of power series is capable of unifying and generalizing known results. However, it is also capable of establishing specific results which are difficult if not impossible to establish by other means. This is a point we hope to be able to make in this book. That formal power series constitute a powerful tool in automata and language theory depends on the fact that they in a sense lead to the arithmetization of automata and language theory. We invite the reader to prove, for instance, Theorem IV. 5. 3 or Corollaries III. 7. 8 and III. 7.- all specific results in language theory-by some other means. Although this book is mostly self-contained, the reader is assumed to have some background in algebra and analysis, as well as in automata and formal language theory.

Mathematics for infinite dimensional objects is becoming more and more important today both in theory and application. Rings of operators, renamed von Neumann algebras by J. Dixmier, were first introduced by J. von Neumann fifty years ago, 1929, in [254] with his grand aim of giving a sound founda- tion to mathematical sciences of infinite nature. J. von Neumann and his collaborator F. J. Murray laid down the foundation for this new field of mathematics, operator algebras, in a series of papers, [240], [241], [242], [257] and [259], during the period of the 1930s and early in the 1940s. In the introduction to this series of investigations, they stated Their solution 1 {to the problems of understanding rings of operators) seems to be essential for the further advance of abstract operator theory in Hilbert space under several aspects. First, the formal calculus with operator-rings leads to them. Second, our attempts to generalize the theory of unitary group-representations essentially beyond their classical frame have always been blocked by the unsolved questions connected with these problems. Third, various aspects of the quantum mechanical formalism suggest strongly the elucidation of this subject. Fourth, the knowledge obtained in these investigations gives an approach to a class of abstract algebras without a finite basis, which seems to differ essentially from all types hitherto investigated. Since then there has appeared a large volume of literature, and a great deal of progress has been achieved by many mathematicians.

In the past decade there has been a significant change in the freshman/ sophomore mathematics curriculum as taught at many, if not most, of our colleges. This has been brought about by the introduction of linear algebra into the curriculum at the sophomore level. The advantages of using linear algebra both in the teaching of differential equations and in the teaching of multivariate calculus are by now widely recognized. Several textbooks adopting this point of view are now available and have been widely adopted. Students completing the sophomore year now have a fair preliminary under- standing of spaces of many dimensions. It should be apparent that courses on the junior level should draw upon and reinforce the concepts and skills learned during the previous year. Unfortunately, in differential geometry at least, this is usually not the case. Textbooks directed to students at this level generally restrict attention to 2-dimensional surfaces in 3-space rather than to surfaces of arbitrary dimension. Although most of the recent books do use linear algebra, it is only the algebra of ~3. The student's preliminary understanding of higher dimensions is not cultivated.

The objective of this book is to make analytical methods available to students of ecology. The text deals with concepts of energy exchange, gas exchange, and chemical kinetics involving the interactions of plants and animals with their environments. The first four chapters are designed to show the applications of biophysical ecology in a preliminary, sim- plified manner. Chapters 5-10, treating the topics of radiation, convec- tion, conduction, and evaporation, are concerned with the physical environment. The spectral properties of radiation and matter are thoroughly described, as well as the geometrical, instantaneous, daily, and annual amounts of both shortwave and longwave radiation. Later chapters give the more elaborate analytical methods necessary for the study of photosynthesis in plants and energy budgets in animals. The final chapter describes the temperature responses of plants and animals. The discipline of biophysical ecology is rapidly growing, and some important topics and references are not included due to limitations of space, cost, and time. The methodology of some aspects of ecology is illustrated by the subject matter of this book. It is hoped that future students of the subject will carry it far beyond its present status. Ideas for advancing the subject matter of biophysical ecology exceed individual capacities for effort, and even today, many investigators in ecology are studying subjects for which they are inadequately prepared. The potential of modern science, in the minds and hands of skilled investigators, to of the interactions of organisms with their advance our understanding environment is enormous.

This book is the outgrowth of both a research program and a graduate course at the University of California, Irvine (UCI) since 1966, as well as a graduate course at the California State Polytechnic University, Pomona (Cal Poly Pomona). The research program, part of the UCI Pattern Recogni- tion Project, was concerned with the design of trainable classifiers; the graduate courses were broader in scope, including subjects such as feature selection, cluster analysis, choice of data set, and estimates of probability densities. In the interest of minimizing overlap with other books on pattern recogni- tion or classifier theory, we have selected a few topics of special interest for this book, and treated them in some depth. Some of this material has not been previously published. The book is intended for use as a guide to the designer of pattern classifiers, or as a text in a graduate course in an engi- neering or computer science curriculum. Although this book is directed primarily to engineers and computer scientists, it may also be of interest to psychologists, biologists, medical scientists, and social scientists.

The present essay stems from a history of polyhedra from 1750 to 1866 written several years ago (as part of a more general work, not published). So many contradictory statements regarding a Descartes manuscript and Euler, by various mathematicians and historians of mathematics, were encountered that it was decided to write a separate study of the relevant part of the Descartes manuscript on polyhedra. The contemplated short paper grew in size, as only a detailed treatment could be of any value. After it was completed it became evident that the entire manuscript should be treated and the work grew some more. The result presented here is, I hope, a complete, accurate, and fair treatment of the entire manuscript. While some views and conclusions are expressed, this is only done with the facts before the reader, who may draw his or her own conclusions. I would like to express my appreciation to Professors H. S. M. Coxeter, Branko Griinbaum, Morris Kline, and Dr. Heinz-Jiirgen Hess for reading the manuscript and for their encouragement and suggestions. I am especially indebted to Dr. Hess, of the Leibniz-Archiv, for his assistance in connection with the manuscript. I have been greatly helped in preparing the translation ofthe manuscript by the collaboration of a Latin scholar, Mr. Alfredo DeBarbieri. The aid of librarians is indispensable, and I am indebted to a number of them, in this country and abroad, for locating material and supplying copies.