There seems to be no doubt that geometry originates from such practical activ ities as weather observation and terrain survey. But there are different manners, methods, and ways to raise the various experiences to the level of theory so that they finally constitute a science. F. Engels said, "The objective of mathematics is the study of space forms and quantitative relations of the real world. " Dur ing the time of the ancient Greeks, there were two different methods dealing with geometry: one, represented by the Euclid's "Elements," purely pursued the logical relations among geometric entities, excluding completely the quantita tive relations, as to establish the axiom system of geometry. This method has become a model of deduction methods in mathematics. The other, represented by the relevant work of Archimedes, focused on the study of quantitative re lations of geometric objects as well as their measures such as the ratio of the circumference of a circle to its diameter and the area of a spherical surface and of a parabolic sector. Though these approaches vary in style, have their own features, and reflect different viewpoints in the development of geometry, both have made great contributions to the development of mathematics. The development of geometry in China was all along concerned with quanti tative relations."

Die Weltwirtschaft steht an der Schwelle zu einem Zeitalter intelligenter Systeme. Viele Untemehmen werden sich kiinftig im zunehmend globalen Wettbewerb nur dann behaupten konnen, wenn es ihnen gelingt, ihre Produkte und Verfahren mit einem hohen MaB an Intelligenz auszustatten (, Machine Intelligence Quotient'). Gerade die Bundesrepublik Deutschland ist als sogenanntes Hochlohnland und mit ihren sehr begrenzten namrlichen Ressourcen zur Sicherung ihres wirtschaftlichen Wohlstandes auf eine schnelle Umsetzung fortschrittlicher Methoden und Technologien angewiesen. Eine zentrale Rolle bei der Entwicklung intelligenter Produkte und Verfahren spielen neue methodische Ansatze der Informatik. Die gesellschaftlichen Krafte in der Bundesrepublik haben dies erkannt, bis hin zu GewerkschaftsfUhrem, die intelligente Produkte fiir den Weltrnarkt fordem. Der erste Absatz dieses V orwortes starnmt iibrigens von Giinther Einert, dem Minister fUr Wirtschaft, Mittelstand und Technologie des Landes NRW. Neben der Mikro-Elektronik und der Mikro-Systemtechnik inklusive cler Mikro Sensorik als Basistechnologien werden Fuzzy-Logik, Kiinstliche Neuronale Netze und Evolutionare Algorithmen sowie Kombinationen davon als Problemlosungstechniken bei der Entwicklung der geforderten intelligenten Produkte eine wesentliche Rolle spielen. In einem irrt der Minister allerdings: neu sind die methodischen Ansatze, auf die er sich bezieht, durchaus nicht. Neu ist nur ihre erfolgreiche Umsetzung. Wir haben die Thematik der 4. Dortmunder Fuzzy-Tage erweitert und dies mit dem Untertitel "Fachtagung iiber Computational Intelligence" zum Ausdruck gebracht. Wir sind nicht die ersten, die eine Fachtagung tiber Computational Intelligence ankiindigen (das war IEEE), aber wir sind nach meiner Kenntnis die ersten, die eine so1che Fachtagung durchfUhren."

The book concerns limit theorems and laws of large numbers for scaled unionsof independent identically distributed random sets. These results generalizewell-known facts from the theory of extreme values. Limiting distributions (called union-stable) are characterized and found explicitly for many examples of random closed sets. The speed of convergence in the limit theorems for unions is estimated by means of the probability metrics method.It includes the evaluation of distances between distributions of random sets constructed similarly to the well-known distances between distributions of random variables. The techniques include regularly varying functions, topological properties of the space of closed sets, Choquet capacities, convex analysis and multivalued functions. Moreover, the concept of regular variation is elaborated for multivalued (set-valued) functions. Applications of the limit theorems to simulation of random sets, statistical tests, polygonal approximations of compacts, limit theorems for pointwise maxima of random functions are considered. Several open problems are mentioned. Addressed primarily to researchers in the theory of random sets, stochastic geometry and extreme value theory, the book will also be of interest to applied mathematicians working on applications of extremal processes and their spatial counterparts. The book is self-contained, and no familiarity with the theory of random sets is assumed.

Following the success of Logic for Mathematicians, Dr Hamilton has written a text for mathematicians and students of mathematics that contains a description and discussion of the fundamental conceptual and formal apparatus upon which modern pure mathematics relies. The author's intention is to remove some of the mystery that surrounds the foundations of mathematics. He emphasises the intuitive basis of mathematics; the basic notions are numbers and sets and they are considered both informally and formally. The role of axiom systems is part of the discussion but their limitations are pointed out. Formal set theory has its place in the book but Dr Hamilton recognises that this is a part of mathematics and not the basis on which it rests. Throughout, the abstract ideas are liberally illustrated by examples so this account should be well-suited, both specifically as a course text and, more broadly, as background reading. The reader is presumed to have some mathematical experience but no knowledge of mathematical logic is required.

In the last 25 years, the fuzzy set theory has been applied in many disciplines such as operations research, management science, control theory, artificial intelligence/expert system, etc. In this volume, methods and applications of fuzzy mathematical programming and possibilistic mathematical programming are first systematically and thoroughly reviewed and classified. This state-of-the-art survey provides readers with a capsule look into the existing methods, and their characteristics and applicability to analysis of fuzzy and possibilistic programming problems. To realize practical fuzzy modelling, we present solutions for real-world problems including production/manufacturing, transportation, assignment, game, environmental management, resource allocation, project investment, banking/finance, and agricultural economics. To improve flexibility and robustness of fuzzy mathematical programming techniques, we also present our expert decision-making support system IFLP which considers and solves all possibilities of a specific domain of (fuzzy) linear programming problems. Basic fuzzy set theories, membership functions, fuzzy decisions, operators and fuzzy arithmetic are introduced with simple numerical examples in aneasy-to-read and easy-to-follow manner. An updated bibliographical listing of 60 books, monographs or conference proceedings, and about 300 selected papers, reports or theses is presented in the end of this study.

During the Fall Semester of 1987, Stevo Todorcevic gave a series of lectures at the University of Colorado. These notes of the course, taken by the author, give a novel and fast exposition of four chapters of Set Theory. The first two chapters are about the connection between large cardinals and Lebesque measure. The third is on forcing axioms such as Martin's axiom or the Proper Forcing Axiom. The fourth chapter looks at the method of minimal walks and p-functions and their applications. The book is addressed to researchers and graduate students interested in Set Theory, Set-Theoretic Topology and Measure Theory.

The Set Theory and Applications meeting at York University, Ontario, featured both contributed talks and a series of invited lectures on topics central to set theory and to general topology. These proceedings contain a selection of the resulting papers, mostly announcing new unpublished results.

These proceedings include the papers presented at the logic meeting held at the Research Institute for Mathematical Sciences, Kyoto University, in the summer of 1987. The meeting mainly covered the current research in various areas of mathematical logic and its applications in Japan. Several lectures were also presented by logicians from other countries, who visited Japan in the summer of 1987.

The papers collected in this volume are most of the material presented at the Advanced School on Mathematical Models for the Semantics of Parallelism, held in Rome, September 24- October 1, 1986. The need for a comprehensive and clear presentation of the several semantical approaches to parallelism motivated the stress on mathematical models, by means of which comparisons among different approaches can also be performed in a perspicuous way.

Ordered structures have been increasingly recognized in recent years due to an explosion of interest in theoretical computer science and all areas of discrete mathematics. This book covers areas such as ordered sets and lattices. A key feature of ordered sets, one which is emphasized in the text, is that they can be represented pictorially. Lattices are also considered as algebraic structures and hence a purely algebraic study is used to reinforce the ideas of homomorphisms and of ideals encountered in group theory and ring theory. Exposure to elementary abstract algebra and the rotation of set theory are the only prerequisites for this text. For the new edition, much has been rewritten or expanded and new exercises have been added.

The global biodiversity crisis is one of humanity's most urgent problems, but even quantifying biological diversity is a difficult mathematical and conceptual challenge. This book brings new mathematical rigour to the ongoing debate. It was born of research in category theory, is given strength by information theory, and is fed by the ancient field of functional equations. It applies the power of the axiomatic method to a biological problem of pressing concern, but it also presents new theorems that stand up as mathematics in their own right, independently of any application. The question 'what is diversity?' has surprising mathematical depth, and this book covers a wide breadth of mathematics, from functional equations to geometric measure theory, from probability theory to number theory. Despite this range, the mathematical prerequisites are few: the main narrative thread of this book requires no more than an undergraduate course in analysis.

A comprehensive work in finite-value systems that covers the latest achievements using the semi-tensor product method, on various kinds of finite-value systems. These results occupy the highest position in the analysis and control of this field. It not only covers all aspects of research in finite-value systems, but also presents the mathematical derivation for each conclusion in depth. The book contains examples to provide a better understanding of the practical applications of finite-value systems. It will serve as a textbook for graduate students of Cybernetics, Mathematical, and Biology, and a reference for readers interested in the theory of finite-value systems.

Continues to focuses specifically on probability rather than probability and statistics Offers a conversational presentation rather than theorem/proof and includes examples based on engineering applications as it highlights Excel computations Presents a review of set theory and updates all descriptions so they are more understandable such as events versus outcomes Additional new material includes distributions such as beta and lognormal, a section on counting principles for defining probabilities, a section on mixture distributions, and a pair of distribution summary tables A solutions manual is available for qualified textbook adoptions

The breathtakingly rapid pace of change in computing makes it easy to overlook the pioneers who began it all. The Universal Computer: The Road from Leibniz to Turing explores the fascinating lives, ideas, and discoveries of seven remarkable mathematicians. It tells the stories of the unsung heroes of the computer age - the logicians.

Model theory is one of the central branches of mathematical logic. The field has evolved rapidly in the last few decades. This book is an introduction to current trends in model theory, and contains a collection of articles authored by top researchers in the field. It is intended as a reference for students as well as senior researchers.

Alex Oliver and Timothy Smiley provide a natural point of entry to what for most readers will be a new subject. Plural logic deals with plural terms ('Whitehead and Russell', 'Henry VIII's wives', 'the real numbers', 'the square root of -1', 'they'), plural predicates ('surrounded the fort', 'are prime', 'are consistent', 'imply'), and plural quantification ('some things', 'any things'). Current logic is singularist: its terms stand for at most one thing. By contrast, the foundational thesis of this book is that a particular term may legitimately stand for several things at once; in other words, there is such a thing as genuinely plural denotation. The authors argue that plural phenomena need to be taken seriously and that the only viable response is to adopt a plural logic, a logic based on plural denotation. They expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists. A formal system of plural logic is presented in three stages, before being applied to Cantorian set theory as an illustration. Technicalities have been kept to a minimum, and anyone who is familiar with the classical predicate calculus should be able to follow it. The authors' approach is an attractive blend of no-nonsense argumentative directness and open-minded liberalism, and they convey the exciting and unexpected richness of their subject. Mathematicians and linguists, as well as logicians and philosophers, will find surprises in this book. This second edition includes a greatly expanded treatment of the paradigm empty term zilch, a much strengthened treatment of Cantorian set theory, and a new chapter on higher-level plural logic.

The Art of Proving Binomial Identities accomplishes two goals: (1) It provides a unified treatment of the binomial coefficients, and (2) Brings together much of the undergraduate mathematics curriculum via one theme (the binomial coefficients). The binomial coefficients arise in a variety of areas of mathematics: combinatorics, of course, but also basic algebra (binomial theorem), infinite series (Newton's binomial series), differentiation (Leibniz's generalized product rule), special functions (the beta and gamma functions), probability, statistics, number theory, finite difference calculus, algorithm analysis, and even statistical mechanics. The book is very suitable for advanced undergraduates or beginning graduate students and includes various exercises asking them to prove identities. Students will find that the text and notes at the end of the chapters encourages them to look at binomial coefficients from different angles. With this learning experience, students will be able to understand binomial coefficients in a new way. Features: Provides a unified treatment of many of the techniques for proving binomial coefficient identities. Ties together several of the courses in the undergraduate mathematics curriculum via a single theme. A textbook for a capstone or senior seminar course in mathematics. Contains several results by the author on proof techniques for binomial coefficients that are not well-known. Ideal for self-study, it contains a large number of exercises at the end of each chapter, with hints or solutions for every exercise at the end of the book.

The Art of Proving Binomial Identities accomplishes two goals: (1) It provides a unified treatment of the binomial coefficients, and (2) Brings together much of the undergraduate mathematics curriculum via one theme (the binomial coefficients). The binomial coefficients arise in a variety of areas of mathematics: combinatorics, of course, but also basic algebra (binomial theorem), infinite series (Newton's binomial series), differentiation (Leibniz's generalized product rule), special functions (the beta and gamma functions), probability, statistics, number theory, finite difference calculus, algorithm analysis, and even statistical mechanics. The book is very suitable for advanced undergraduates or beginning graduate students and includes various exercises asking them to prove identities. Students will find that the text and notes at the end of the chapters encourages them to look at binomial coefficients from different angles. With this learning experience, students will be able to understand binomial coefficients in a new way. Features: Provides a unified treatment of many of the techniques for proving binomial coefficient identities. Ties together several of the courses in the undergraduate mathematics curriculum via a single theme. A textbook for a capstone or senior seminar course in mathematics. Contains several results by the author on proof techniques for binomial coefficients that are not well-known. Ideal for self-study, it contains a large number of exercises at the end of each chapter, with hints or solutions for every exercise at the end of the book.

Model theory begins with an audacious idea: to consider statements about mathematical structures as mathematical objects of study in their own right. While inherently important as a tool of mathematical logic, it also enjoys connections to and applications in diverse branches of mathematics, including algebra, number theory and analysis. Despite this, traditional introductions to model theory assume a graduate-level background of the reader. In this innovative textbook, Jonathan Kirby brings model theory to an undergraduate audience. The highlights of basic model theory are illustrated through examples from specific structures familiar from undergraduate mathematics, paying particular attention to definable sets throughout. With numerous exercises of varying difficulty, this is an accessible introduction to model theory and its place in mathematics.

Set theory is a rich and beautiful subject whose fundamental concepts permeate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since nearly all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In this text, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, ordinal numbers, and cardinal numbers, are developed within the framework of axiomatic set theory. The reader will need to be comfortable reading and writing mathematical proofs. The proofs in this textbook are rigorous, clear, and complete, while remaining accessible to undergraduates who are new to upper-level mathematics. Exercises are included at the end of each section in a chapter, with useful suggestions for the more challenging exercises.

At the heart of this short introduction to category theory is the idea of a universal property, important throughout mathematics. After an introductory chapter giving the basic definitions, separate chapters explain three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties all three together. The book is suitable for use in courses or for independent study. Assuming relatively little mathematical background, it is ideal for beginning graduate students or advanced undergraduates learning category theory for the first time. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations. Copious exercises are included.