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Books > Science & Mathematics > Mathematics > Mathematical foundations
Introduction to Fuzzy Systems provides students with a self-contained introduction that requires no preliminary knowledge of fuzzy mathematics and fuzzy control systems theory. Simplified and readily accessible, it encourages both classroom and self-directed learners to build a solid foundation in fuzzy systems. After introducing the subject, the authors move directly into presenting real-world applications of fuzzy logic, revealing its practical flavor. This practicality is then followed by basic fuzzy systems theory. The book also offers a tutorial on fuzzy control theory, based mainly on the well-known classical Proportional-Integral-Derivative (PID) controllers theory and design methods. In particular, the text discusses fuzzy PID controllers in detail, including a description of the new notion of generalized verb-based fuzzy-logic control theory. Introduction to Fuzzy Systems is primarily designed to provide training for systems and control majors, both senior undergraduate and first year graduate students, to acquaint them with the fundamental mathematical theory and design methodology required to understand and utilize fuzzy control systems.
This book presents a mathematically-based introduction into the fascinating topic of Fuzzy Sets and Fuzzy Logic and might be used as textbook at both undergraduate and graduate levels and also as reference guide for mathematician, scientists or engineers who would like to get an insight into Fuzzy Logic. Fuzzy Sets have been introduced by Lotfi Zadeh in 1965 and since then, they have been used in many applications. As a consequence, there is a vast literature on the practical applications of fuzzy sets, while theory has a more modest coverage. The main purpose of the present book is to reduce this gap by providing a theoretical introduction into Fuzzy Sets based on Mathematical Analysis and Approximation Theory. Well-known applications, as for example fuzzy control, are also discussed in this book and placed on new ground, a theoretical foundation. Moreover, a few advanced chapters and several new results are included. These comprise, among others, a new systematic and constructive approach for fuzzy inference systems of Mamdani and Takagi-Sugeno types, that investigates their approximation capability by providing new error estimates. "
This ambitious exposition by Malik and Mordeson on the fuzzification of discrete structures not only supplies a solid basic text on this key topic, but also serves as a viable tool for learning basic fuzzy set concepts "from the ground up" due to its unusual lucidity of exposition. While the entire presentation of this book is in a completely traditional setting, with all propositions and theorems provided totally rigorous proofs, the readability of the presentation is not compromised in any way; in fact, the many ex cellently chosen examples illustrate the often tricky concepts the authors address. The book's specific topics - including fuzzy versions of decision trees, networks, graphs, automata, etc. - are so well presented, that it is clear that even those researchers not primarily interested in these topics will, after a cursory reading, choose to return to a more in-depth viewing of its pages. Naturally, when I come across such a well-written book, I not only think of how much better I could have written my co-authored monographs, but naturally, how this work, as distant as it seems to be from my own area of interest, could nevertheless connect with such. Before presenting the briefest of some ideas in this direction, let me state that my interest in fuzzy set theory (FST) has been, since about 1975, in connecting aspects of FST directly with corresponding probability concepts. One chief vehicle in carrying this out involves the concept of random sets."
This is the first book to collect essays from philosophers, mathematicians and computer scientists working at the exciting interface of algorithmic learning theory and the epistemology of science and inductive inference. Readable, introductory essays provide engaging surveys of different, complementary, and mutually inspiring approaches to the topic, both from a philosophical and a mathematical viewpoint. Building upon this base, subsequent papers present novel extensions of algorithmic learning theory as well as bold, new applications to traditional issues in epistemology and the philosophy of science. The volume is vital reading for students and researchers seeking a fresh, truth-directed approach to the philosophy of science and induction, epistemology, logic, and statistics.
The present anthology has its origin in two international conferences that were arranged at Uppsala University in August 2004: "Logicism, Intuitionism and F- malism: What has become of them?" followed by "Symposium on Constructive Mathematics." The rst conference concerned the three major programmes in the foundations of mathematics during the classical period from Frege's Begrif- schrift in 1879 to the publication of Godel' ] s two incompleteness theorems in 1931: The logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. The main purpose of the conf- ence was to assess the relevance of these foundational programmes to contemporary philosophy of mathematics. The second conference was announced as a satellite event to the rst, and was speci cally concerned with constructive mathematics-an activebranchofmathematicswheremathematicalstatements-existencestatements in particular-are interpreted in terms of what can be effectively constructed. C- structive mathematics may also be characterized as mathematics based on intuiti- isticlogicand, thus, beviewedasadirectdescendant ofBrouwer'sintuitionism. The two conferences were successful in bringing together a number of internationally renowned mathematicians and philosophers around common concerns. Once again it was con rmed that philosophers and mathematicians can work together and that real progress in the philosophy and foundations of mathematics is possible only if they do. Most of the papers in this collection originate from the two conferences, but a few additional papers of relevance to the issues discussed at the Uppsala c- ferences have been solicited especially for this volume."
This book offers an up-to-date, comprehensive account of determinantal rings and varieties, presenting a multitude of methods used in their study, with tools from combinatorics, algebra, representation theory and geometry. After a concise introduction to Groebner and Sagbi bases, determinantal ideals are studied via the standard monomial theory and the straightening law. This opens the door for representation theoretic methods, such as the Robinson-Schensted-Knuth correspondence, which provide a description of the Groebner bases of determinantal ideals, yielding homological and enumerative theorems on determinantal rings. Sagbi bases then lead to the introduction of toric methods. In positive characteristic, the Frobenius functor is used to study properties of singularities, such as F-regularity and F-rationality. Castelnuovo-Mumford regularity, an important complexity measure in commutative algebra and algebraic geometry, is introduced in the general setting of a Noetherian base ring and then applied to powers and products of ideals. The remainder of the book focuses on algebraic geometry, where general vanishing results for the cohomology of line bundles on flag varieties are presented and used to obtain asymptotic values of the regularity of symbolic powers of determinantal ideals. In characteristic zero, the Borel-Weil-Bott theorem provides sharper results for GL-invariant ideals. The book concludes with a computation of cohomology with support in determinantal ideals and a survey of their free resolutions. Determinants, Groebner Bases and Cohomology provides a unique reference for the theory of determinantal ideals and varieties, as well as an introduction to the beautiful mathematics developed in their study. Accessible to graduate students with basic grounding in commutative algebra and algebraic geometry, it can be used alongside general texts to illustrate the theory with a particularly interesting and important class of varieties.
A compilation of papers presented at the 2001 European Summer Meeting of the Association for Symbolic Logic, Logic Colloquium '01 includes surveys and research articles from some of the world's preeminent logicians. Two long articles are based on tutorials given at the meeting and present accessible expositions of research in two active areas of logic, geometric model theory and descriptive set theory of group actions. The remaining articles cover seperate research topics in many areas of mathematical logic, including applications in Computer Science, Proof Theory, Set Theory, Model Theory, Computability Theory, and aspects of Philosophy. This collection will be of interest not only to specialists in mathematical logic, but also to philosophical logicians, historians of logic, computer scientists, formal linguists and mathematicians in the areas of algebra, abstract analysis and topology. A number of the articles are aimed at non-specialists and serve as good introductions for graduate students.
In this book, Yurii L. Ershov posits the view that computability-in the broadest sense-can be regarded as the Sigma-definability in the suitable sets. He presents a new approach to providing the Godel incompleteness theorem based on systematic use of the formulas with the restricted quantifiers. The volume also includes a novel exposition on the foundations of the theory of admissible sets with urelements, using the Gandy theorem throughout the theory's development. Other topics discussed are forcing, Sigma-definability, dynamic logic, and Sigma-predicates of finite types."
This monograph presents a new theory for analysis, comparison and design of nonlinear smoothers, linking to established practices. Although a part of mathematical morphology, the special properties yield many simple, powerful and illuminating results leading to a novel nonlinear multiresolution analysis with pulses that may be as natural to vision as wavelet analysis is to acoustics. Similar to median transforms, they have the advantages of a supporting theory, computational simplicity, remarkable consistency, full trend preservation, and a Parceval-type identity. Although the perspective is new and unfamiliar to most, the
reader can verify all the ideas and results with simple simulations
on a computer at each stage. The framework developed turns out to
be a part of mathematical morphology, but the additional specific
structures and properties yield a heuristic understanding that is
easy to absorb for practitioners in the fields like signal- and
image processing.
Fuzzy logic techniques have had extraordinary growth in various engineering systems. The developments in engineering sciences have caused apprehension in modern years due to high-tech industrial processes with ever-increasing levels of complexity. Advanced Fuzzy Logic Approaches in Engineering Science provides innovative insights into a comprehensive range of soft fuzzy logic techniques applied in various fields of engineering problems like fuzzy sets theory, adaptive neuro fuzzy inference system, and hybrid fuzzy logic genetic algorithms belief networks in industrial and engineering settings. The content within this publication represents the work of particle swarms, fuzzy computing, and rough sets. It is a vital reference source for engineers, research scientists, academicians, and graduate-level students seeking coverage on topics centered on the applications of fuzzy logic in high-tech industrial processes.
"The Art of Proof" is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.
This is a monograph that details the use of Siegel's method and the classical results of homotopy groups of spheres and Lie groups to determine some Gottlieb groups of projective spaces or to give the lower bounds of their orders. Making use of the properties of Whitehead products, the authors also determine some Whitehead center groups of projective spaces that are relevant and new within this monograph.
Game theory has already proved its tremendous potential for con?ict resolution problems in the ?elds of Decision Theory and Economics. In the recent past, there have been attempts to extend the results of crisp game theory to those con?ict resolution problems which are fuzzy in nature e.g. Nishizaki and Sakawa [61] and references cited there in. These developments have lead to the emergence of a new area in the literature called fuzzy games. Another area in the fuzzy decision theory, which has been growing very fast is the area of fuzzy mathematical programming and its applications to various branches of sciences, Engineering and Management. In the crisp scenario, there exists a beautiful relationship between two person zero sum matrix game theory and duality in linear p- gramming. It is therefore natural to ask if something similar holds in the fuzzy scenario as well. This discussion essentially constitutes the core of our presentation. The objective of this book is to present a systematic and focussed study of the application of fuzzy sets to two very basic areas of decision theory, namely Mathematical Programming and Matrix Game Theory.
This book is devoted primarily to topics in interpolation for scalar, matrix and operator valued functions. About half the papers are based on lectures which were delivered at a conference held at Leipzig University in August 1994 to commemorate the 80th anniversary of the birth of Vladimir Petrovich Potapov. The volume also contains the English translation of several important papers relatively unknown in the West, two expository papers written especially for this volume, and historical material based on reminiscences of former colleagues, students and associates of V.P. Potapov. Numerous examples of interpolation problems of the Nevanlinna-Pick and CarathA(c)odory-FejA(c)r type are included as well as moment problems and problems of integral representation in assorted settings. The major themes cover applications of the Potapov method of fundamental matrix inequalities, multiplicative decompositions of J-inner matrix valued functions, the abstract interpolation problem, canonical systems of differential equations and interpolation in spaces with an indefinite metric. This book should appeal to a wide range of readers: mathematicians specializing in pure and applied mathematics and engineers who work in systems theory and control. The book will be of use to graduate students and mathematicians interested in functional analysis.
This monograph shows that, through a recourse to the concepts and methods of abstract algebraic logic, the algebraic theory of regular varieties and the concept of analyticity in formal logic can profitably interact. By extending the technique of Plonka sums from algebras to logical matrices, the authors investigate the different classes of models for logics of variable inclusion and they shed new light into their formal properties. The book opens with the historical origins of logics of variable inclusion and on their philosophical motivations. It includes the basics of the algebraic theory of regular varieties and the construction of Plonka sums over semilattice direct systems of algebra. The core of the book is devoted to an abstract definition of logics of left and right variable inclusion, respectively, and the authors study their semantics using the construction of Plonka sums of matrix models. The authors also cover Paraconsistent Weak Kleene logic and survey its abstract algebraic logical properties. This book is of interest to scholars of formal logic.
Introduction to Fuzzy Reliability treats fuzzy methodology in hardware reliability and software reliability in a relatively systematic manner. The contents of this book are organized as follows. Chapter 1 places reliability engineering in the scope of a broader area, i.e. system failure engineering. Readers will find that although this book is confined to hardware and software reliability, it may be useful for other aspects of system failure engineering, like maintenance and quality control. Chapter 2 contains the elementary knowledge of fuzzy sets and possibility spaces which are required reading for the rest of this book. This chapter is included for the overall completeness of the book, but a few points (e.g. definition of conditional possibility and existence theorem of possibility space) may be new. Chapter 3 discusses how to calculate probist system reliability when the component reliabilities are represented by fuzzy numbers, and how to analyze fault trees when probabilities of basic events are fuzzy. Chapter 4 presents the basic theory of profust reliability, whereas Chapter 5 analyzes the profust reliability behavior of a number of engineering systems. Chapters 6 and 7 are devoted to probist reliability theory from two different perspectives. Chapter 8 discusses how to model software reliability behavior by using fuzzy methodology. Chapter 9 includes a number of mathematical problems which are raised by applications of fuzzy methodology in hardware and software reliability, but may be important for fuzzy set and possibility theories.
Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and, of course, mathematicians.
Boolean valued analysis is a technique for studying properties of an arbitrary mathematical object by comparing its representations in two different set-theoretic models whose construction utilises principally distinct Boolean algebras. The use of two models for studying a single object is a characteristic of the so-called non-standard methods of analysis. Application of Boolean valued models to problems of analysis rests ultimately on the procedures of ascending and descending, the two natural functors acting between a new Boolean valued universe and the von Neumann universe. This book demonstrates the main advantages of Boolean valued analysis which provides the tools for transforming, for example, function spaces to subsets of the reals, operators to functionals, and vector-functions to numerical mappings. Boolean valued representations of algebraic systems, Banach spaces, and involutive algebras are examined thoroughly. Audience: This volume is intended for classical analysts seeking new tools, and for model theorists in search of challenging applications of nonstandard models.
Many years of practical experience in teaching discrete mathematics form the basis of this text book. Part I contains problems on such topics as Boolean algebra, k-valued logics, graphs and networks, elements of coding theory, automata theory, algorithms theory, combinatorics, Boolean minimization and logical design. The exercises are preceded by ample theoretical background material. For further study the reader is referred to the extensive bibliography. Part II follows the same structure as Part I, and gives helpful hints and solutions. Audience: This book will be of great value to undergraduate students of discrete mathematics, whereas the more difficult exercises, which comprise about one-third of the material, will also appeal to postgraduates and researchers.
This second edition of "A Beginner's Guide to Finite Mathematics" takes a distinctly applied approach to finite mathematics at the freshman and sophomore level. Topics are presented sequentially: the book opens with a brief review of sets and numbers, followed by an introduction to data sets, histograms, means and medians. Counting techniques and the Binomial Theorem are covered, which provides the foundation for elementary probability theory; this, in turn, leads to basic statistics. This new edition includes chapters on game theory and financial mathematics. Requiring little mathematical background beyond high school algebra, the text will be especially useful for business and liberal arts majors.
This volume collects chapters that examine representation theory as connected with affine Lie algebras and their quantum analogues, in celebration of the impact Vyjayanthi Chari has had on this area. The opening chapters are based on mini-courses given at the conference "Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification", held on the occasion of Chari's 60th birthday at the Catholic University of America in Washington D.C., June 2018. The chapters that follow present a broad view of the area, featuring surveys, original research, and an overview of Vyjayanthi Chari's significant contributions. Written by distinguished experts in representation theory, a range of topics are covered, including: String diagrams and categorification Quantum affine algebras and cluster algebras Steinberg groups for Jordan pairs Dynamical quantum determinants and Pfaffians Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification will be an ideal resource for researchers in the fields of representation theory and mathematical physics.
This book contains the proceedings of the International Symposium on Mathematical Morphology and its Applications to Image and Signal Processing IV, held June 3-5, 1998, in Amsterdam, The Netherlands. The purpose of the work is to provide the image analysis community with a sampling of recent developments in theoretical and practical aspects of mathematical morphology and its applications to image and signal processing. Among the areas covered are: digitization and connectivity, skeletonization, multivariate morphology, morphological segmentation, color image processing, filter design, gray-scale morphology, fuzzy morphology, decomposition of morphological operators, random sets and statistical inference, differential morphology and scale-space, morphological algorithms and applications. Audience: This volume will be of interest to research mathematicians and computer scientists whose work involves mathematical morphology, image and signal processing.
Fundamentals of Fuzzy Sets covers the basic elements of fuzzy set theory. Its four-part organization provides easy referencing of recent as well as older results in the field. The first part discusses the historical emergence of fuzzy sets, and delves into fuzzy set connectives, and the representation and measurement of membership functions. The second part covers fuzzy relations, including orderings, similarity, and relational equations. The third part, devoted to uncertainty modelling, introduces possibility theory, contrasting and relating it with probabilities, and reviews information measures of specificity and fuzziness. The last part concerns fuzzy sets on the real line - computation with fuzzy intervals, metric topology of fuzzy numbers, and the calculus of fuzzy-valued functions. Each chapter is written by one or more recognized specialists and offers a tutorial introduction to the topics, together with an extensive bibliography.
This book discusses major theories and applications of fuzzy soft multisets and their generalization which help researchers get all the related information at one place. The primary objective of this book is to help bridge the gap to provide a textbook on the theories in fuzzy soft multisets and their applications in real life. It is targeted to researchers and students working in the field of fuzzy set theory, multiset theory, soft set theory and their applications. Uncertainty, vagueness and the representation of imperfect knowledge have been a problem in many fields of research, including artificial intelligence, network and communication, signal processing, machine learning, computer science, information technology, as well as medical science, economics, environments and engineering. There are many mathematical tools for dealing with uncertainties. They include fuzzy set theory, multiset theory, soft set theory and soft multiset theory. |
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