This is a classic introduction to set theory, suitable for students with no previous knowledge of the subject. Providing complete, up-to-date coverage, the book is based in large part on courses given over many years by Professor Hajnal. The first part introduces all the standard notions of the subject; the second part concentrates on combinatorial set theory. Exercises are included throughout and a new section of hints has been added to assist the reader.

This text presents methods of modern set theory as tools that can be usefully applied to other areas of mathematics. The author describes numerous applications in abstract geometry and real analysis and, in some cases, in topology and algebra. The book begins with a tour of the basics of set theory, culminating in a proof of Zorn's Lemma and a discussion of some of its applications. The author then develops the notions of transfinite induction and descriptive set theory, with applications to the theory of real functions. The final part of the book presents the tools of "modern" set theory: Martin's Axiom, the Diamond Principle, and elements of forcing. Written primarily as a text for beginning graduate or advanced level undergraduate students, this book should also interest researchers wanting to learn more about set theoretical techniques applicable to their fields.

The starting point of this book is Sperner's theorem, which answers the question: What is the maximum possible size of a family of pairwise (with respect to inclusion) subsets of a finite set? This theorem stimulated the development of a fast growing theory dealing with external problems on finite sets and, more generally, on finite partially ordered sets. This book presents Sperner theory from a unified point of view, bringing combinatorial techniques together with methods from programming, linear algebra, Lie-algebra representations and eigenvalue methods, probability theory, and enumerative combinatorics. Researchers and graduate students in discrete mathematics, optimisation, algebra, probability theory, number theory, and geometry will find many powerful new methods arising from Sperner theory.

In this book the authors present their research into the foundations of the theory of Polish groups and the associated orbit equivalence relations. The particular case of locally compact groups has long been studied in many areas of mathematics. Non-locally compact Polish groups occur naturally as groups of symmetries in such areas as logic (especially model theory), ergodic theory, group representations, and operator algebras. Some of the topics covered here are: topological realizations of Borel measurable actions; universal actions; applications to invariant measures; actions of the infinite symmetric group in connection with model theory (logic actions); dichotomies for orbit spaces (including Silver, Glimm-Effros type dichotomies and the topological Vaught conjecture); descriptive complexity of orbit equivalence relations; definable cardinality of orbit spaces.

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

This book provides an approach to the study of perturbation and discretization effects on the long-time behavior of dynamical and control systems. It analyzes the impact of time and space discretizations on asymptotically stable attracting sets, attractors, asumptotically controllable sets and their respective domains of attractions and reachable sets. Combining robust stability concepts from nonlinear control theory, techniques from optimal control and differential games and methods from nonsmooth analysis, both qualitative and quantitative results are obtained and new algorithms are developed, analyzed and illustrated by examples.

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.

This book 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 these algebras using the theory of bisimulations. Their approach is completely constructive, and contains both intuitionistic set theory and topos theory. In particular it provides a uniform description of various constructions of the cumulative hierarchy of sets in forcing models, sheaf models and realizability models. Graduate students and researchers in mathematical logic, category theory and computer science should find this book of great interest, and it should be accessible to anyone with a background in categorical logic.

This book is an excellent starting point for any curriculum in fuzzy systems fields such as computer science, mathematics, business/economics and engineering. It covers the basics leading to: fuzzy clustering, fuzzy pattern recognition, fuzzy database, fuzzy image processing, soft computing, fuzzy applications in operations research, fuzzy decision making, fuzzy rule based systems, fuzzy systems modeling, fuzzy mathematics. It is not a book designed for researchers - it is where you really learn the "basics" needed for any of the above-mentioned applications. It includes many figures and problem sets at the end of sections.

Strange Functions in Real Analysis, Third Edition differs from the previous editions in that it includes five new chapters as well as two appendices. More importantly, the entire text has been revised and contains more detailed explanations of the presented material. In doing so, the book explores a number of important examples and constructions of pathological functions. After introducing basic concepts, the author begins with Cantor and Peano-type functions, then moves effortlessly to functions whose constructions require what is essentially non-effective methods. These include functions without the Baire property, functions associated with a Hamel basis of the real line and Sierpinski-Zygmund functions that are discontinuous on each subset of the real line having the cardinality continuum. Finally, the author considers examples of functions whose existence cannot be established without the help of additional set-theoretical axioms. On the whole, the book is devoted to strange functions (and point sets) in real analysis and their applications.

From Simon & Schuster, Fuzzy Logic is about the revolutionary computer technology that is changing our world. Fuzzy logic is a way to program computers so that they can mimic the imprecise way that humans make decisions. This technology allows for many innovative applications, including cars that virtually drive themselves, washing machines that pick the right wash cycles and water temperature automatically and air conditioning and heaters that adjust to the number of people in the room.

The book systematically develops the nonlinear potential theory connected with the weighted Sobolev spaces, where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces occur as solutions spaces for degenerate elliptic partial differential equations. The Sobolev space theory covers results concerning approximation, extension, and interpolation, Sobolev and Poincaré inequalities, Maz'ya type embedding theorems, and isoperimetric inequalities. In the chapter devoted to potential theory, several weighted capacities are investigated. Moreover, "Kellogg lemmas" are established for various concepts of thinness. Applications of potential theory to weighted Sobolev spaces include quasi continuity of Sobolev functions, Poincaré inequalities, and spectral synthesis theorems.

Introduction Model theorists have often joked in recent years that the part of mathemat- ical logic known as "pure model theory" (or stability theory), as opposed to the older and more traditional "model theory applied to algebra" , turns out to have more and more to do with other subjects ofmathematics and to yield gen- uine applications to combinatorial geometry, differential algebra and algebraic geometry. We illustrate this by presenting the very striking application to diophantine geometry due to Ehud Hrushovski: using model theory, he has given the first proof valid in all characteristics of the "Mordell-Lang conjecture for function fields" (The Mordell-Lang conjecture for function fields, Journal AMS 9 (1996), 667-690). More recently he has also given a new (model theoretic) proof of the Manin-Mumford conjecture for semi-abelian varieties over a number field. His proofyields the first effective bound for the cardinality ofthe finite sets involved (The Manin-Mumford conjecture, preprint). There have been previous instances of applications of model theory to alge- bra or number theory, but these appl~cations had in common the feature that their proofs used a lot of algebra (or number theory) but only very basic tools and results from the model theory side: compactness, first-order definability, elementary equivalence...

This book comprises revised full versions of lectures given during the 9th European Summer School in Logic, Languages, and Information, ESSLLI'97, held in Aix-en-Provence, France, in August 1997. The six lectures presented introduce the reader to the state of the art in the area of generalized quantifiers and computation. Besides an introductory survey by the volume editor various aspects of generalized quantifiers are studied in depth.

This volume compiles geostatistical and spatial autoregressive data analyses involving georeferenced socioeconomic, natural resources, agricultural, pollution, and epidemiological variables. Benchmark analyses are followed by analyses of readily available data sets, emphasizing parallels between geostatistical and spatial autoregressive findings. Both SAS and SPSS code are presented for implementation purposes. This informative casebook will serve geographers, regional scientists, applied spatial statisticians, and spatial scientists from across disciplines.

The explanation of the formal duality of Kerdock and Preparata codes is one of the outstanding results in the field of applied algebra in the last few years. This result is related to the discovery of large sets of quad riphase sequences over Z4 whose correlation properties are better than those of the best binary sequences. Moreover, the correlation properties of sequences are closely related to difference properties of certain sets in (cyclic) groups. It is the purpose of this book to illustrate the connection between these three topics. Most articles grew out of lectures given at the NATO Ad vanced Study Institute on "Difference sets, sequences and their correlation properties." This workshop took place in Bad Windsheim (Germany) in August 1998. The editors thank the NATO Scientific Affairs Division for the generous support of this workshop. Without this support, the present collection of articles would not have been realized."

This book is dedicated to Andrzej Ehrenfeucht on the occasion of his 65th birthday. On personal invitation by the volume editors, 22 internationally well-known scientists from mathematical logics and theoretical computer science participated in this project honoring an excellent scientist with excellent papers centered around his scientific work.
The 22 invited papers are presented in topical sections on model theory, games and logic, graphs and algorithms, pattern matching and learning, combinatorics of words, algebra of languages, formal language theory, and computational molecular biology.

The Philosophy of Mathematics Today gives a panorama of the best current work in this lively field, through twenty essays specially written for this collection by leading figures. The topics include indeterminacy, logical consequence, mathematical methodology, abstraction, and both Hilbert's and Frege's foundational programmes. The collection will be an important source for research in the philosophy of mathematics for years to come.

Contributors Paul Benacerraf, George Boolos, John P. Burgess, Charles S. Chihara, Michael Detlefsen, Michael Dummett, Hartry Field, Kit Fine, Bob Hale, Richard G. Heck, Jnr., Geoffrey Hellman, Penelope Maddy, Karl-Georg Niebergall, Charles D. Parsons, Michael D. Resnik, Matthias Schirn, Stewart Shapiro, Peter Simons, W.W. Tait, Crispin Wright.

The purpose of this monograph is to develop a very general approach to the algebra ization of sententiallogics, to show its results on a number of particular logics, and to relate it to other existing approaches, namely to those based on logical matrices and the equational consequence developed by Blok, Czelakowski, Pigozzi and others. The main distinctive feature of our approachlies in the mathematical objects used as models of a sententiallogic: We use abstract logics, while the dassical approaches use logical matrices. Using models with more structure allows us to reflect in them the metalogical properties of the sentential logic. Since an abstract logic can be viewed as a "bundle" or family of matrices, one might think that the new models are essentially equivalent to the old ones; but we believe, after an overall appreciation of the work done in this area, that it is precisely the treatment of an abstract logic as a single object that gives rise to a useful -and beautiful- mathematical theory, able to explain the connections, not only at the logical Ievel but at the metalogical Ievel, between a sentential logic and the particular dass of models we associate with it, namely the dass of its full models. Traditionally logical matrices have been regarded as the most suitable notion of model in the algebraic studies of sentential logics; and indeed this notion gives sev eral completeness theorems and has generated an interesting mathematical theory."

The authors present some surprising connections that sets of uniqueness for trigonometic series have with descriptive set theory. They present many new results concerning the structure of sets of uniqueness and include solutions to some of the classical problems in this area. Topics covered include symmetric perfect sets and the solution to the Borel Basis Problem for U, the class of sets of uniqueness. To make the material accessible to both logicians, set theorists and analysts, the authors have covered in some detail large parts of the classical and modern theory of sets of uniqueness as well as the relevant parts of descriptive set theory. Because the book is essentially selfcontained and requires the minimum prerequisites, it will serve as an excellent introduction to the subject for graduate students and research workers in set theory and analysis.

Fuzzy logic has become an important tool for a number of different applications ranging from the control of engineering systems to artificial intelligence. In this concise introduction, the author presents a succinct guide to the basic ideas of fuzzy logic, fuzzy sets, fuzzy relations, and fuzzy reasoning, and shows how they may be applied. The book culminates in a chapter which describes fuzzy logic control: the design of intelligent control systems using fuzzy if-then rules which make use of human knowledge and experience to behave in a manner similar to a human controller. Throughout, the level of mathematical knowledge required is kept basic and the concepts are illustrated with numerous diagrams to aid in comprehension. As a result, all those curious to know more about fuzzy concepts and their real-world application will find this a good place to start.

This text provides a wide-ranging introduction to convex sets and functions, suitable for final-year undergraduates and also graduate students. Demanding only a modest knowledge of analysis and linear algebra, it discusses such diverse topics as number theory, classical extremum problems, combinatorial geometry, linear programming, game theory, polytopes, bodies of constant width, the gamma function, minimax approximation, and the theory of linear, classical, and matrix inequalities.

An advanced graduate course. Some knowledge of forcing is assumed, and some elementary Mathematical Logic, e.g. the Lowenheim-Skolem Theorem. A student with one semester of mathematical logic and 1 of set theory should be prepared to read these notes. The first half deals with the general area of Borel hierarchies. What are the possible lengths of a Borel hierarchy in a separable metric space? Lebesgue showed that in an uncountable complete separable metric space the Borel hierarchy has uncountably many distinct levels, but for incomplete spaces the answer is independent. The second half includes Harrington's Theorem - it is consistent to have sets on the second level of the projective hierarchy of arbitrary size less than the continuum and a proof and appl- ications of Louveau's Theorem on hyperprojective parameters.

In recent years, applied mathematics has been used in all novel disciplines of scientific development. Advances in Applied Mathematical Problems summarizes interdisciplinary work within the field of applied mathematics. The topics discussed in the book include: * Similarity Solutions of Spherical Shock Waves in a Self-Gravitating Ideal Gas * Dual Solutions for Finite Element Analysis of Unsteady Hydromagnetic Stagnation Point Flow of Water Nanofluid Generated by Stretching Sheet * Multiparametric modeling of carbon cycle in temperate wetlands for regional climate change analysis using satellite data * An Intelligent Neuro Fuzzy System for Pattern Classification * Fuzzy inventory model with demand, deterioration and inflation: a comparative study through NGTFN and CNTFN * Summability and its application for the stability of the system * Design Of Manufacturing, Control, And Automation Systems * SEIR - Application for Crop through Water and Soil Texture * Advances in radial basis functions * Modeling for Time Period of Natural Frequency for Non-Homogeneous Square Plate With Variable Thickness and Temperature Effect * A Study On Metric Fixed Point Theorems Satisfying Integral Type Contractions * Objective Function - In Radiometric Studies - Application to Agrs Surveys Associated With Radon * Modelling Kernel Function in Black body Radiation Inversion

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."