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This volume presents answers to some natural questions of a general analytic character that arise in the theory of Banach spaces. I believe that altogether too many of the results presented herein are unknown to the active abstract analysts, and this is not as it should be. Banach space theory has much to offer the prac- titioners of analysis; unfortunately, some of the general principles that motivate the theory and make accessible many of its stunning achievements are couched in the technical jargon of the area, thereby making it unapproachable to one unwilling to spend considerable time and effort in deciphering the jargon. With this in mind, I have concentrated on presenting what I believe are basic phenomena in Banach spaces that any analyst can appreciate, enjoy, and perhaps even use. The topics covered have at least one serious omission: the beautiful and powerful theory of type and cotype. To be quite frank, I could not say what I wanted to say about this subject without increasing the length of the text by at least 75 percent. Even then, the words would not have done as much good as the advice to seek out the rich Seminaire Maurey-Schwartz lecture notes, wherein the theory's development can be traced from its conception. Again, the treasured volumes of Lindenstrauss and Tzafriri also present much of the theory of type and cotype and are must reading for those really interested in Banach space theory.

Due to a steady flow of requests over several years, Springer-Verlag now provides a corrected reprint of this text. It is designed to serve as a text for a first semester graduate level course for students in digital communication systems. As a pre- requisite, it is presumed that the reader has an understanding of basic probability and stochastic processes. The treatment of digital communications in this book is intended to serve as an introduction to the subject. Part one is a development of the elements of statistical communication theory and radar detection. The text begins with a general model of a communication system which is extensively developed and the performance analyses of various conventional systems. The first part also serves as introductory material for the second part of the text which is a comprehensive study of the theory of transmitter optimization for coherent and noncoherent digital commu- nication systems, that is, the theory of signal design.

Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century. The book is divided into four parts. In the first, Lang presents the general analytic theory starting from scratch. Most of this can be read by a student with a basic knowledge of complex analysis. The next part treats complex multiplication, including a discussion of Deuring's theory of l-adic and p-adic representations, and elliptic curves with singular invariants. Part three covers curves with non-integral invariants, and applies the Tate parametrization to give Serre's results on division points. The last part covers theta functions and the Kronecker Limit Formula. Also included is an appendix by Tate on algebraic formulas in arbitrary charactistic.

This book was written to myself at about the time I began graduate studies in anthropology-the sort of thing a Samuel Beckett character might do. It is about the conduct of research. In a very real sense the purpose is partially to compensate for the inadequacies of my professors. Perhaps this is what education is about. The effort has not been an unqualified success, but it has been extremely gratifying. I was trained in anthropology. After completing the Ph. D. I went to Stanford on a post-doctoral fellowship. At the time, this was a novelty and the depart- ment was not prepared for such a thing. To stay occupied I began attending lectures, seminars, and discussion groups in mathematics and statistics. This was about the luckiest choice I ever made. The excitement was easily as intense as that which I experienced upon encountering anthropology. On one oc- casion I innocently and independently proved a theorem that had first been done 2000 years earlier. It is currently used as an exercise in high school mathematics so it is neither difficult nor arcane. Learning all this did not tarnish my sense of discovery. (On reflection I am puzzled by my failure to have seen all this "e;beauty"e; when I was exposed to it as an undergraduate. The unparalleled excellence of the Stanford program was undoubtedly responsible for my belated conversion.

A serendipitous discovery in nuclear physics has led to a useful tool in materials science. In the late 1950s, scientists at General Electric (among them the author) discovered that when mica is exposed to energetic charged particles (such as are emitted in radioactive decay or occur in cosmic rays), the particles leave latent tracks in the material. When such a material is chemically etched, the tracks are revealed as narrow, deep pits, whose size and shape is determined both by the particle that made the track and by the technique used in etching. It soon turned out that glass, plastics, or certain other materials can be similarly treated. This discovery paved the way not only for a new and useful method of measuring radioactivity, it has also found widespread applications in other fields, ranging from geology and materials science to archaeology and art history. Thus, for example, naturally produced tracks can be used to estimate the age of a mineral deposit or an archaeological material; and deliberately produced tracks can be used to make extremely fine filters. Fleischer presents the history of these developments and discusses the applications of the technique in a way that will be interesting to anyone with a minimal knowledge of physics.

This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised.

In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "e;undergraduate topology"e; proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does nr~ understand the simplest topological facts, such as the rcason why knots exist. In my opinion, a well-balanced introduction to topology should stress its intuitive geometric aspect, while admitting the legitimate interest that analysts and algebraists have in the subject. At any rate, this is the aim of the present book. In support of this view, I have followed the historical development where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recreations like the seven bridges; rather, it resulted from the l'isualization of problems from other parts of mathematics-complex analysis (Riemann), mechanics (Poincare), and group theory (Dehn). It is these connec- tions to other parts of mathematics which make topology an important as well as a beautiful subject.

Algorithmic Algebra studies some of the main algorithmic tools of computer algebra, covering such topics as Grobner bases, characteristic sets, resultants and semialgebraic sets. The main purpose of the book is to acquaint advanced undergraduate and graduate students in computer science, engineering and mathematics with the algorithmic ideas in computer algebra so that they could do research in computational algebra or understand the algorithms underlying many popular symbolic computational systems: Mathematica, Maple or Axiom, for instance. Also, researchers in robotics, solid modeling, computational geometry and automated theorem proving community may find it useful as symbolic algebraic techniques have begun to play an important role in these areas. The book, while being self-contained, is written at an advanced level and deals with the subject at an appropriate depth. The book is accessible to computer science students with no previous algebraic training. Some mathematical readers, on the other hand, may find it interesting to see how algorithmic constructions have been used to provide fresh proofs for some classical theorems. The book also contains a large number of exercises with solutions to selected exercises, thus making it ideal as a textbook or for self-study.

Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory. This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.

Intended for first- or second-year undergraduates, this introduction to discrete mathematics covers the usual topics of such a course, but applies constructivist principles that promote - indeed, require - active participation by the student. Working with the programming language ISETL, whose syntax is close to that of standard mathematical language, the student constructs the concepts in her or his mind as a result of constructing them on the computer in the syntax of ISETL. This dramatically different approach allows students to attempt to discover concepts in a "e;Socratic"e; dialog with the computer. The discussion avoids the formal "e;definition-theorem"e; approach and promotes active involvement by the reader by its questioning style. An instructor using this text can expect a lively class whose students develop a deep conceptual understanding rather than simply manipulative skills. Topics covered in this book include: the propositional calculus, operations on sets, basic counting methods, predicate calculus, relations, graphs, functions, and mathematical induction.

The aim of this book is to provide an introduction for students and nonspecialists to a fascinating relation between combinatorial geometry and algebraic geometry, as it has developed during the last two decades. This relation is known as the theory of toric varieties or sometimes as torus embeddings. Chapters I-IV provide a self-contained introduction to the theory of convex poly- topes and polyhedral sets and can be used independently of any applications to algebraic geometry. Chapter V forms a link between the first and second part of the book. Though its material belongs to combinatorial convexity, its definitions and theorems are motivated by toric varieties. Often they simply translate algebraic geometric facts into combinatorial language. Chapters VI-VIII introduce toric va- rieties in an elementary way, but one which may not, for specialists, be the most elegant. In considering toric varieties, many of the general notions of algebraic geometry occur and they can be dealt with in a concrete way. Therefore, Part 2 of the book may also serve as an introduction to algebraic geometry and preparation for farther reaching texts about this field. The prerequisites for both parts of the book are standard facts in linear algebra (including some facts on rings and fields) and calculus. Assuming those, all proofs in Chapters I-VII are complete with one exception (IV, Theorem 5.1). In Chapter VIII we use a few additional prerequisites with references from appropriate texts.

This book provides a unified introduction to a variety of computational algorithms for Bayesian and likelihood inference. In this third edition, I have attempted to expand the treatment of many of the techniques discussed. I have added some new examples, as well as included recent results. Exercises have been added at the end of each chapter. Prerequisites for this book include an understanding of mathematical statistics at the level of Bickel and Doksum (1977), some understanding of the Bayesian approach as in Box and Tiao (1973), some exposure to statistical models as found in McCullagh and NeIder (1989), and for Section 6. 6 some experience with condi- tional inference at the level of Cox and Snell (1989). I have chosen not to present proofs of convergence or rates of convergence for the Metropolis algorithm or the Gibbs sampler since these may require substantial background in Markov chain theory that is beyond the scope of this book. However, references to these proofs are given. There has been an explosion of papers in the area of Markov chain Monte Carlo in the past ten years. I have attempted to identify key references-though due to the volatility of the field some work may have been missed.

The Nature of the Book This book is a text for a first course in statistical concepts and methods. It introduces the analysis of data and statistical inference and explains various methods in enough detail so that the student can apply them. Little mathematical background is required; only high school algebra is used. No mathematical proof is given in the body of the text, although algebraic demonstrations are given in appendices at the ends of some chapters. The exposition is based on logic, verbal explanations, figures, and numerical examples. The verbal and conceptual levels are higher than the mathematical leveL The concepts and methods of statistical analysis are illustrated by more than 100 interesting real-life data sets. Some examples are taken from daily life; many deal with the behavioral sciences, some with business, the health sciences, the physical sciences, and engineering. The exercises are of varying degrees of difficulty. This book is suitable for undergraduates in many majors and for grad­ uate students in the health sciences, behavioral sciences, and education. It has grown out of our experience over many years of teaching such courses. An earlier text by T. W. Anderson and S. L. Sclove, The Statistical Analysis of Data, had similar objectives and was of a similar nature.

Biophysical Chemistry: Molecules to Membranes is a one-semester textbook for graduate and senior undergraduate students. Developed over several years of teaching, the approach differs from that of other texts by emphasizing thermodynamics of aqueous solutions, by rigorously treating electrostatics and irreversible phenomena, and by applying these principles to topics of biochemistry and biophysics. The main sections are: (1) Basic principles of equilibrium thermodynamics. (2) Structure and behavior of solutions of ions and molecules. The discussions range from properties of bulk water to the solvent structure of solutions of small molecules and macromolecules. (3) Physical principles are extended for the non-homogenous and non-equilibrium nature of biological processes. Areas included are lipid/water systems, transport phenomena, membranes, and bio-electrochemistry. This new textbook will provide an essential foundation for research in cellular physiology, biochemistry, membrane biology, as well as the derived areas bioengineering, pharmacology, nephrology, and many others.

This book is a revision of Random Point Processes written by D. L. Snyder and published by John Wiley and Sons in 1975. More emphasis is given to point processes on multidimensional spaces, especially to pro- cesses in two dimensions. This reflects the tremendous increase that has taken place in the use of point-process models for the description of data from which images of objects of interest are formed in a wide variety of scientific and engineering disciplines. A new chapter, Translated Poisson Processes, has been added, and several of the chapters of the fIrst edition have been modifIed to accommodate this new material. Some parts of the fIrst edition have been deleted to make room. Chapter 7 of the fIrst edition, which was about general marked point-processes, has been eliminated, but much of the material appears elsewhere in the new text. With some re- luctance, we concluded it necessary to eliminate the topic of hypothesis testing for point-process models. Much of the material of the fIrst edition was motivated by the use of point-process models in applications at the Biomedical Computer Labo- ratory of Washington University, as is evident from the following excerpt from the Preface to the first edition. "e;It was Jerome R. Cox, Jr. , founder and [1974] director of Washington University's Biomedical Computer Laboratory, who ftrst interested me [D. L. S.