The first of a two volume set showcasing current research in model theory and its connections with number theory, algebraic geometry, real analytic geometry and differential algebra. Each volume contains a series of expository essays and research papers around the subject matter of a Newton Institute Semester on Model Theory and Applications to Algebra and Analysis. The articles convey outstanding new research on topics such as model theory and conjectures around Mordell-Lang; arithmetic of differential equations, and Galois theory of difference equations; model theory and complex analytic geometry; o-minimality; model theory and noncommutative geometry; definable groups of finite dimension; Hilbert's tenth problem; and Hrushovski constructions. With contributions from so many leaders in the field, this book will undoubtedly appeal to all mathematicians with an interest in model theory and its applications, from graduate students to senior researchers and from beginners to experts.

The second of a two volume set showcasing current research in model theory and its connections with number theory, algebraic geometry, real analytic geometry and differential algebra. Each volume contains a series of expository essays and research papers around the subject matter of a Newton Institute Semester on Model Theory and Applications to Algebra and Analysis. The articles convey outstanding new research on topics such as model theory and conjectures around Mordell-Lang; arithmetic of differential equations, and Galois theory of difference equations; model theory and complex analytic geometry; o-minimality; model theory and non-commutative geometry; definable groups of finite dimension; Hilbert's tenth problem; and Hrushovski constructions. With contributions from so many leaders in the field, this book will undoubtedly appeal to all mathematicians with an interest in model theory and its applications, from graduate students to senior researchers and from beginners to experts.

This book appeared about ten years ago in Gennan. It started as notes for a course which I gave intermittently at the ETH over a number of years. Following repeated suggestions, this English translation was commissioned by Springer; they were most fortunate in finding translators whose mathemati cal stature, grasp of the language and unselfish dedication to the essentially thankless task of rendering the text comprehensible in a second language, both impresses and shames me. Therefore, my thanks go to Dr. Roberto Minio, now Darmstadt and Professor Charles Thomas, Cambridge. The task of preparing a La'JEX-version of the text was extremely daunting, owing to the complexity and diversity of the symbolisms inherent in the various parts of the book. Here, my warm thanks go to Barbara Aquilino of the Mathematics Department of the ETH, who spent tedious but exacting hours in front of her Olivetti. The present book is not primarily intended to teach logic and axiomat ics as such, nor is it a complete survey of what was once called "elementary mathematics from a higher standpoint." Rather, its goal is to awaken a certain critical attitude in the student and to help give this attitude some solid foun dation. Our mathematics students, having been drilled for years in high-school and college, and having studied the immense edifice of analysis, regrettably come away convinced that they understand the concepts of real numbers, Euclidean space, and algorithm."

The modern discussion on the concept of truthlikeness was started in 1960. In his influential Word and Object, W. V. O. Quine argued that Charles Peirce's definition of truth as the limit of inquiry is faulty for the reason that the notion 'nearer than' is only "defined for numbers and not for theories." In his contribution to the 1960 International Congress for Logic, Methodology, and Philosophy of Science at Stan ford, Karl Popper defended the opposite view by defining a compara tive notion of verisimilitude for theories. was originally introduced by the The concept of verisimilitude Ancient sceptics to moderate their radical thesis of the inaccessibility of truth. But soon verisimilitudo, indicating likeness to the truth, was confused with probabilitas, which expresses an opiniotative attitude weaker than full certainty. The idea of truthlikeness fell in disrepute also as a result of the careless, often confused and metaphysically loaded way in which many philosophers used - and still use - such concepts as 'degree of truth', 'approximate truth', 'partial truth', and 'approach to the truth'. Popper's great achievement was his insight that the criticism against truthlikeness - by those who urge that it is meaningless to speak about 'closeness to truth' - is more based on prejudice than argument."

This monograph contains the results of our joint research over the last ten years on the logic of the fixed point operation. The intended au dience consists of graduate students and research scientists interested in mathematical treatments of semantics. We assume the reader has a good mathematical background, although we provide some prelimi nary facts in Chapter 1. Written both for graduate students and research scientists in theoret ical computer science and mathematics, the book provides a detailed investigation of the properties of the fixed point or iteration operation. Iteration plays a fundamental role in the theory of computation: for example, in the theory of automata, in formal language theory, in the study of formal power series, in the semantics of flowchart algorithms and programming languages, and in circular data type definitions. It is shown that in all structures that have been used as semantical models, the equational properties of the fixed point operation are cap tured by the axioms describing iteration theories. These structures include ordered algebras, partial functions, relations, finitary and in finitary regular languages, trees, synchronization trees, 2-categories, and others."

This is a motivated presentation of recent results on tree transducers, applied to studying the general properties of formal models and for providing semantics to context-free languages. The authors consider top-down tree transducers, macro tree transducers, attributed tree transducers, and macro attributed tree transducers. A unified terminology is used to define them, and their transformational capacities are compared. This handbook on tree transducers will serve as a base for further research.

In 1963, the first author introduced a course in set theory at the University of Illinois whose main objectives were to cover Godel's work on the con sistency of the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH), and Cohen's work on the independence of the AC and the GCH. Notes taken in 1963 by the second author were taught by him in 1966, revised extensively, and are presented here as an introduction to axiomatic set theory. Texts in set theory frequently develop the subject rapidly moving from key result to key result and suppressing many details. Advocates of the fast development claim at least two advantages. First, key results are high lighted, and second, the student who wishes to master the subject is com pelled to develop the detail on his own. However, an instructor using a "fast development" text must devote much class time to assisting his students in their efforts to bridge gaps in the text."

This collection of papers has its origin in a conference held at the Uni- versity of Toronto in June of 1988. The theme of the conference was Physicalism in Mathematics: Recent Work in the Philosophy of Math- ematics. At the conference, papers were read by Geoffrey Hellman (Minnesota), Yvon Gauthier (Montreal), Michael Hallett (McGill), Hartry Field (USC), Bob Hale (Lancaster & St Andrew's), Alasdair Urquhart (Toronto) and Penelope Maddy (Irvine). This volume supplements updated versions of six of those papers with contributions by Jim Brown (Toronto), John Bigelow (La Trobe), John Burgess (Princeton), Chandler Davis (Toronto), David Papineau (Cambridge), Michael Resnik (North Carolina at Chapel Hill), Peter Simons (Salzburg) and Crispin Wright (St Andrews & Michigan). Together they provide a vivid, expansive snapshot of the exciting work which is currently being carried out in philosophy of mathematics. Generous financial support for the original conference was provided by the Social Sciences & Humanities Research Council of Canada, the British Council, and the Department of Philosophy together with the Office of Internal Relations at the University of Toronto. Additional support for the production of this volume was gratefully received from the Social Sciences & Humanities Research Council of Canada.

One of the most remarkable recent occurrences in mathematics is the re-founding, on a rigorous basis, the idea of infinitesimal quantity, a notion which played an important role in the early development of the calculus and mathematical analysis. In this new and updated edition, basic calculus, together with some of its applications to simple physical problems, are presented through the use of a straightforward, rigorous, axiomatically formulated concept of 'zero-square', or 'nilpotent' infinitesimal - that is, a quantity so small that its square and all higher powers can be set, to zero. The systematic employment of these infinitesimals reduces the differential calculus to simple algebra and, at the same time, restores to use the "infinitesimal" methods figuring in traditional applications of the calculus to physical problems - a number of which are discussed in this book. This edition also contains an expanded historical and philosophical introduction.

Fuzzy technology has emerged as one of the most exciting new concepts available. Fuzzy Logic and its Applications... covers a wide range of the theory and applications of fuzzy logic and related systems, including industrial applications of fuzzy technology, implementing human intelligence in machines and systems. There are four main themes: intelligent systems, engineering, mathematical foundations, and information sciences. Both academics and the technical community will learn how and why fuzzy logic is appreciated in the conceptual, design and manufacturing stages of intelligent systems, gaining an improved understanding of the basic science and the foundations of human reasoning.

This book grew out of my interest in what is common to three disciplines: mathematics, philosophy, and history. The origins of Zermelo's Axiom of Choice, as well as the controversy that it engendered, certainly lie in that intersection. Since the time of Aristotle, mathematics has been concerned alternately with its assumptions and with the objects, such as number and space, about which those assumptions were made. In the historical context of Zermelo's Axiom, I have explored both the vagaries and the fertility of this alternating concern. Though Zermelo's research has provided the focus for this book, much of it is devoted to the problems from which his work originated and to the later developments which, directly or indirectly, he inspired. A few remarks about format are in order. In this book a publication is indicated by a date after a name; so Hilbert 1926, 178 refers to page 178 of an article written by Hilbert, published in 1926, and listed in the bibliography.

This volume constitutes the thoroughly refereed post-conference proceedings of the 7th International Conference on Curves and Surfaces, held in Avignon, in June 2010. The conference had the overall theme: "Representation and Approximation of Curves and Surfaces and Applications." The 39 revised full papers presented together with 9 invited talks were carefully reviewed and selected from 114 talks presented at the conference. The topics addressed by the papers range from mathematical foundations to practical implementation on modern graphics processing units and address a wide area of topics such as computer-aided geometric design, computer graphics and visualisation, computational geometry and topology, geometry processing, image and signal processing, interpolation and smoothing, scattered data processing and learning theory and subdivision, wavelets and multi-resolution methods.

In 1965 Juris Hartmanis and Richard E. Stearns published a paper "On the Computational Complexity of Algorithms." The field of complexity theory takes its name from this seminal paper and many of the major concepts and issues of complexity theory were introduced by Hartmanis in subsequent work. In honor of the contribution of Juris Hartmanis to the field of complexity theory, a special session of invited talks by Richard E. Stearns, Allan Borodin and Paul Young was held at the third annual meeting of the Structure in Complexity conference, and the first three chapters of this book are the final versions of these talks. They recall intellectual and professional trends in Hartmanis' contributions. All but one of the remainder of the chapters in this volume originated as a presentation at one of the recent meetings of the Structure in Complexity Theory Conference and appeared in preliminary form in the conference proceedings. In all, these expositions form an excellent description of much of contemporary complexity theory.

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.

The popular literature on mathematical logic is rather extensive and written for the most varied categories of readers. College students or adults who read it in their free time may find here a vast number of thought-provoking logical problems. The reader who wishes to enrich his mathematical background in the hope that this will help him in his everyday life can discover detailed descriptions of practical (and quite often -- not so practical ) applications of logic. The large number of popular books on logic has given rise to the hope that by applying mathematical logic, students will finally learn how to distinguish between necessary and sufficient conditions and other points of logic in the college course in mathematics. But the habit of teachers of mathematical analysis, for example, to stick to problems dealing with sequences without limit, uniformly continuous functions, etc. has, unfortunately, led to the writing of textbooks that present prescriptions for the mechanical construction of definitions of negative concepts which seem to obviate the need for any thinking on the reader's part. We are most certainly not able to enumerate everything the reader may draw out of existing books on mathematical logic, however.

Substructural logics are by now one of the most prominent branches of the research field usually labelled as "nonclassical logics" - and perhaps of logic tout court. Over the last few decades a vast amount of research papers and even some books have been devoted to this subject. The aim of the present book is to give a comprehensive account of the "state of the art" of substructural logics, focusing both on their proof theory (especially on sequent calculi and their generalizations) and on their semantics (both algebraic and relational).
Readership: This textbook is designed for a wide readership: graduate students in either philosophy, mathematics, theoretical computer science or theoretical linguistics with no previous knowledge of the subject (except for a working knowledge of elementary logic) will be gradually introduced into the field starting from its basic foundations; specialists and researchers in the area will find an up-to-date survey of the most important current research topics and problems.

This volume has a dual significance to the ESPRIT Basic Research efforts towards forging strong links between European academic and industrial teams carrying out research, often interdisciplinary, at the forefront of Information Technology. Firstly, it consists of the proceedings of the "Symposium on Computational Logic" - held on the occasion of the 7th ESPRIT Conference Week in November 1990 - whose organisation was inspired by the work of Basic Research Action 3012 (COMPULOG). This is a consortium which has attracted world-wide interest, with requests for collaboration throughout Europe, the US and Japan. The work of COMPULOG acts as a focal point in this symposium which is broadened to cover the work of other eminent researchers in the field, thus providing a review of the state of the art in computational logic, new and important contributions in the field, but also a vision of the future. Secondly, this volume is the first of an ESPRIT Basic Research Series of publications of research results. It is expected that the quality of content and broad distribution of this series will have a major impact in making the advances achieved accessible to the world of academic and industrial research alike. At this time, all ESPRIT Basic Research Actions have completed their first year and it is most encouraging and stimulating to see the flow of results such as the fine examples presented in this symposium.

This is the second volume of a two volume collection on Structural Complexity. This volume assumes as a prerequisite knowledge about the topics treated in Volume I, but the present volume itself is nearly self-contained. As in Volume I, each chapter of this book ends with a section entitled "Bibliographical Remarks", in which the relevant references for the chapter are briefly commented upon. These sections might also be of interest to those wanting an overview of the evolution of the field, as well as relevant related results which are not included in the text. Each chapter includes a section of exercises. The reader is encouraged to spend some time on them. Some results presented as exercises are occasionally used later in the text. A reference is provided for the most interesting and for the most useful exercises. Some exercises are marked with a * to indicate that, to the best knowledge of the authors, the solution has a certain degree of difficulty. Many topics from the field of Structural Complexity are not treated in depth, or not treated at all. The authors bear all responsibility for the choice of topics, which has been made based on the interest of the authors on each topic. Many friends and colleagues have made suggestions or corrections. In partic ular we would like to express our gratitude to Richard Beigel, Ron Book, Rafael Casas, Jozef Gruska, Uwe Schoning, Pekka Orponen, and Osamu Watanabe.

In the six years since the first edition of this book was published, the field of Structural Complexity has grown quite a bit. However, we are keeping this volume at the same basic level that it had in the first edition, and the only new result incorporated as an appendix is the closure under complementation of nondeterministic space classes, which in the previous edition was posed as an open problem. This result was already included in our Volume II, but we feel that due to the basic nature of the result, it belongs to this volume. There are of course other important results obtained during these last six years. However, as they belong to new areas opened in the field they are outside the scope of this fundamental volume. Other changes in this second edition are the update of some Bibliograph ical Remarks and references, correction of many mistakes and typos, and a renumbering of the definitions and results. Experience has shown us that this new numbering is a lot more friendly, and several readers have confirmed this opinion. For the sake of the reader of Volume II, where all references to Volume I follow the old numbering, we have included here a table indicating the new number corresponding to each of the old ones."

Descriptive set theory and definable proper forcing are two areas of set theory that developed quite independently of each other. This monograph unites them and explores the connections between them. Forcing is presented in terms of quotient algebras of various natural sigma-ideals on Polish spaces, and forcing properties in terms of Fubini-style properties or in terms of determined infinite games on Boolean algebras. Many examples of forcing notions appear, some newly isolated from measure theory, dynamical systems, and other fields. The descriptive set theoretic analysis of operations on forcings opens the door to applications of the theory: absoluteness theorems for certain classical forcing extensions, duality theorems, and preservation theorems for the countable support iteration. Containing original research, this text highlights the connections that forcing makes with other areas of mathematics, and is essential reading for academic researchers and graduate students in set theory, abstract analysis and measure theory.

Winner of the 1983National Book Award

..".a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor..." - The New Yorker (1983National Book Award edition)

Mathematics has been a human activity for thousands of years. Yet only a few people from the vast population of users are professional mathematicians, who create, teach, foster, and apply it in a variety of situations. The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it. They also believe that mathematics should be taught to non-mathematics majors in such a way as to instill an appreciation of the power and beauty of mathematics. Many people from around the world have told the authors that they have done precisely that with the first edition and they have encouraged publication of this revised edition complete with exercises for helping students to demonstrate their understanding. This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about. It will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications.

The text is ideal for 1) a GE course for Liberal Arts students 2) a Capstone course for perspective teachers 3) a writing course for mathematics teachers. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto ([email protected]) upon request.

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Now approaching its tenth year, this hugely successful book presents an unusual attempt to publicise the field of Complex Dynamics. The text was originally conceived as a supplemented catalogue to the exhibition "Frontiers of Chaos," seen in Europe and the United States, and describes the context and meaning of these fascinating images. A total of 184 illustrations - including 88 full-colour pictures of Julia sets - are suggestive of a coffee-table book.
However, the invited contributions which round off the book lend the text the required formality. Benoit Mandelbrot gives a very personal account, in his idiosyncratic self-centred style, of his discovery of the fractals named after him and Adrien Douady explains the solved and unsolved problems relating to this amusingly complex set.

The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite. . . - David Hilbert (1862-1943) Infinity is a fathomless gulf, There is a story attributed to David Hilbert, the preeminent mathe into which all things matician whose quotation appears above. A man walked into a vanish. hotel late one night and asked for a room. "Sorry, we don't have o Marcus Aurelius (121- 180), Roman Emperor any more vacancies," replied the owner, "but let's see, perhaps and philosopher I can find you a room after alL" Leaving his desk, the owner reluctantly awakened his guests and asked them to change their rooms: the occupant of room #1 would move to room #2, the occupant of room #2 would move to room #3, and so on until each occupant had moved one room over. To the utter astonish ment of our latecomer, room #1 suddenly became vacated, and he happily moved in and settled down for the night. But a numbing thought kept him from sleep: How could it be that by merely moving the occupants from one room to another, the first room had become vacated? (Remember, all of the rooms were occupied when he arrived.

This book corresponds to a mathematical course given in 1986/87 at the University Louis Pasteur, Strasbourg. This work is primarily intended for graduate students. The following are necessary prerequisites : a few standard definitions in set theory, the definition of rational integers, some elementary facts in Combinatorics (maybe only Newton's binomial formula), some theorems of Analysis at the level of high schools, and some elementary Algebra (basic results about groups, rings, fields and linear algebra). An important place is given to exercises. These exercises are only rarely direct applications of the course. More often, they constitute complements to the text. Mostly, hints or references are given so that the reader should be able to find solutions. Chapters one and two deal with elementary results of Number Theory, for example : the euclidean algorithm, the Chinese remainder theorem and Fermat's little theorem. These results are useful by themselves, but they also constitute a concrete introduction to some notions in abstract algebra (for example, euclidean rings, principal rings ... ). Algorithms are given for arithmetical operations with long integers. The rest of the book, chapters 3 through 7, deals with polynomials. We give general results on polynomials over arbitrary rings. Then polynomials with complex coefficients are studied in chapter 4, including many estimates on the complex roots of polynomials. Some of these estimates are very useful in the subsequent chapters.

The lecture courses in this work are derived from the SERC 'Logic for IT' Summer School and Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles; put together in this book they form an invaluable introduction to proof theory that is aimed at both mathematicians and computer scientists.