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Author: Ying-ming Liu
Publisher: World Scientific
ISBN: 9814518204
Category : Mathematics
Languages : en
Pages : 365
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Book Description
Fuzzy set theory provides us with a framework which is wider than that of classical set theory. Various mathematical structures, whose features emphasize the effects of ordered structure, can be developed on the theory. Fuzzy topology is one such branch, combining ordered structure with topological structure. This branch of mathematics, emerged from the background — processing fuzziness, and locale theory, proposed from the angle of pure mathematics by the great French mathematician Ehresmann, comprise the two most active aspects of topology on lattice, which affect each other.This book is the first monograph to systematically reflect the up-to-date state of fuzzy topology. It emphasizes the so-called “pointed approach” and the effects of stratification structure appearing in fuzzy sets.The monograph can serve as a reference book for mathematicians, researchers, and graduate students working in this branch of mathematics. After an appropriate rearrangements of the chapters and sections, it can also be used as a text for undergraduates.
Author: Ying-ming Liu
Publisher: World Scientific
ISBN: 9814518204
Category : Mathematics
Languages : en
Pages : 365
Get Book Here
Book Description
Fuzzy set theory provides us with a framework which is wider than that of classical set theory. Various mathematical structures, whose features emphasize the effects of ordered structure, can be developed on the theory. Fuzzy topology is one such branch, combining ordered structure with topological structure. This branch of mathematics, emerged from the background — processing fuzziness, and locale theory, proposed from the angle of pure mathematics by the great French mathematician Ehresmann, comprise the two most active aspects of topology on lattice, which affect each other.This book is the first monograph to systematically reflect the up-to-date state of fuzzy topology. It emphasizes the so-called “pointed approach” and the effects of stratification structure appearing in fuzzy sets.The monograph can serve as a reference book for mathematicians, researchers, and graduate students working in this branch of mathematics. After an appropriate rearrangements of the chapters and sections, it can also be used as a text for undergraduates.
Author: N. Palaniappan
Publisher: Chapman & Hall/CRC
ISBN: 9780849324161
Category : Mathematics
Languages : en
Pages : 0
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Book Description
In recent years, many concepts in mathematics, engineering, computer science, and many other disciplines have been in a sense redefined to incorporate the notion of fuzziness. Designed for graduate students and research scholars, Fuzzy Topology imparts the concepts and recent developments related to the various properties of fuzzy topology. The author first addresses fundamental problems, such as the idea of a fuzzy point and its neighborhood structure and the theory of convergence. He then studies the connection between fuzzy topological spaces and topological spaces and introduces fuzzy continuity and product induced spaces. Chapter Three examines fuzzy nets, fuzzy upper and lower limits, and fuzzy convergence and is followed by a study of fuzzy metric spaces. The treatment then introduces the concept of fuzzy compactness before moving to initial and final topologies and the fuzzy Tychnoff theorem. The final sections of the book cover connectedness, complements, separation axioms, and uniform spaces.
Author: Ulrich Höhle
Publisher: Springer Science & Business Media
ISBN: 9780792383888
Category : Business & Economics
Languages : en
Pages : 732
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Book Description
Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory is a major attempt to provide much-needed coherence for the mathematics of fuzzy sets. Much of this book is new material required to standardize this mathematics, making this volume a reference tool with broad appeal as well as a platform for future research. Fourteen chapters are organized into three parts: mathematical logic and foundations (Chapters 1-2), general topology (Chapters 3-10), and measure and probability theory (Chapters 11-14). Chapter 1 deals with non-classical logics and their syntactic and semantic foundations. Chapter 2 details the lattice-theoretic foundations of image and preimage powerset operators. Chapters 3 and 4 lay down the axiomatic and categorical foundations of general topology using lattice-valued mappings as a fundamental tool. Chapter 3 focuses on the fixed-basis case, including a convergence theory demonstrating the utility of the underlying axioms. Chapter 4 focuses on the more general variable-basis case, providing a categorical unification of locales, fixed-basis topological spaces, and variable-basis compactifications. Chapter 5 relates lattice-valued topologies to probabilistic topological spaces and fuzzy neighborhood spaces. Chapter 6 investigates the important role of separation axioms in lattice-valued topology from the perspective of space embedding and mapping extension problems, while Chapter 7 examines separation axioms from the perspective of Stone-Cech-compactification and Stone-representation theorems. Chapters 8 and 9 introduce the most important concepts and properties of uniformities, including the covering and entourage approaches and the basic theory of precompact or complete [0,1]-valued uniform spaces. Chapter 10 sets out the algebraic, topological, and uniform structures of the fundamentally important fuzzy real line and fuzzy unit interval. Chapter 11 lays the foundations of generalized measure theory and representation by Markov kernels. Chapter 12 develops the important theory of conditioning operators with applications to measure-free conditioning. Chapter 13 presents elements of pseudo-analysis with applications to the Hamilton–Jacobi equation and optimization problems. Chapter 14 surveys briefly the fundamentals of fuzzy random variables which are [0,1]-valued interpretations of random sets.
Author: Xuzhu Wang
Publisher: Springer Science & Business Media
ISBN: 3540783105
Category : Mathematics
Languages : en
Pages : 227
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Book Description
Mathematics of Fuzziness – Basic Issues introduces a basic notion of ‘fuzziness’ and provides a conceptual mathematical framework to characterize such fuzzy phenomena in Studies in Fuzziness and Soft Computing. The book systematically presents a self-contained introduction to the essentials of mathematics of fuzziness ranging from fuzzy sets, fuzzy relations, fuzzy numbers, fuzzy algebra, fuzzy measures, fuzzy integrals, and fuzzy topology to fuzzy control in a strictly mathematical manner. It contains most of the authors’ research results in the field of fuzzy set theory and has evolved from the authors’ lecture notes to both undergraduate and graduate students over the last three decades. A lot of exercises in each chapter of the book are particularly suitable as a textbook for any undergraduate and graduate student in mathematics, computer science and engineering. The reading of the book will surely lay a solid foundation for further research on fuzzy set theory and its applications.
Author: S.E. Rodabaugh
Publisher: Springer Science & Business Media
ISBN: 9781402015151
Category : Mathematics
Languages : en
Pages : 488
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Book Description
Topological and Algebraic Structures in Fuzzy Sets has these unique features: -strategically located at the juncture of fuzzy sets, topology, algebra, lattices, foundations of mathematics; -major studies in uniformities and convergence structures, fundamental examples in lattice-valued topology, modifications and extensions of sobriety, categorical aspects of lattice-valued subsets, logic and foundations of mathematics, t-norms and associated algebraic and ordered structures; -internationally recognized authorities clarify deep mathematical aspects of fuzzy sets, particularly those topological or algebraic in nature; -comprehensive bibliographies and tutorial nature of longer chapters take readers to the frontier of each topic; -extensively referenced introduction unifies volume and guides readers to chapters closest to their interests; -annotated open questions direct future research in the mathematics of fuzzy sets; -suitable as a text for advanced graduate students.
Author: David Weisburd
Publisher: Springer Science & Business Media
ISBN: 0387096876
Category : Social Science
Languages : en
Pages : 258
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Book Description
Putting Crime in its Place: Units of Analysis in Geographic Criminology focuses on the units of analysis used in geographic criminology. While crime and place studies have been a part of criminology from the early 19th century, growing interest in crime places over the last two decades demands critical reflection on the units of analysis that should form the focus of geographic analysis of crime. Should the focus be on very small units such as street addresses or street segments, or on larger aggregates such as census tracts or communities? Academic researchers, as well as practical crime analysts, are confronted routinely with the dilemma of deciding what the unit of analysis should be when reporting on trends in crime, when identifying crime hot spots or when mapping crime in cities. In place-based crime prevention, the choice of the level of aggregation plays a particularly critical role. This peer reviewed collection of essays aims to contribute to crime and place studies by making explicit the problems involved in choosing units of analysis in geographic criminology. Written by renowned experts in the field, the chapters in this book address basic academic questions, and also provide real-life examples and applications of how they are resolved in cutting-edge research. Crime analysts in police and law enforcement agencies as well as academic researchers studying the spatial distributions of crime and victimization will learn from the discussions and tools presented.
Author: Radim Bělohlávek
Publisher: Oxford University Press
ISBN: 0190200014
Category : Mathematics
Languages : en
Pages : 545
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Book Description
The main part of the book is a comprehensive overview of the development of fuzzy logic and its applications in various areas of human affair since its genesis in the mid 1960s. This overview is then employed for assessing the significance of fuzzy logic and mathematics based on fuzzy logic.
Author: Phil Diamond
Publisher: World Scientific
ISBN: 9789810217310
Category : Mathematics
Languages : en
Pages : 192
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Book Description
The primary aim of the book is to provide a systematic development of the theory of metric spaces of normal, upper semicontinuous fuzzy convex fuzzy sets with compact support sets, mainly on the base space ?n. An additional aim is to sketch selected applications in which these metric space results and methods are essential for a thorough mathematical analysis.This book is distinctly mathematical in its orientation and style, in contrast with many of the other books now available on fuzzy sets, which, although all making use of mathematical formalism to some extent, are essentially motivated by and oriented towards more immediate applications and related practical issues. The reader is assumed to have some previous undergraduate level acquaintance with metric spaces and elementary functional analysis.
Author: Ying-ming Liu
Publisher: World Scientific
ISBN: 9789810228620
Category : Mathematics
Languages : en
Pages : 372
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Book Description
Fuzzy set theory provides us with a framework which is wider than that of classical set theory. Various mathematical structures, whose features emphasize the effects of ordered structure, can be developed on the theory. Fuzzy topology is one such branch, combining ordered structure with topological structure. This branch of mathematics, emerged from the background ? processing fuzziness, and locale theory, proposed from the angle of pure mathematics by the great French mathematician Ehresmann, comprise the two most active aspects of topology on lattice, which affect each other.This book is the first monograph to systematically reflect the up-to-date state of fuzzy topology. It emphasizes the so-called ?pointed approach? and the effects of stratification structure appearing in fuzzy sets.The monograph can serve as a reference book for mathematicians, researchers, and graduate students working in this branch of mathematics. After an appropriate rearrangements of the chapters and sections, it can also be used as a text for undergraduates.
Author: Ibtesam Alshammari
Publisher: Infinite Study
ISBN:
Category : Mathematics
Languages : en
Pages : 12
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Book Description
Picture fuzzy nano topological spaces is an extension of intuitionistic fuzzy nano topological spaces. Every decision in life ends with an answer such as yes or no, or true or false, but we have an another component called abstain, which we have not yet considered. This work is a gateway to study such a problem. This paper motivates an enquiry of the third component – abstain - in practical problems. The aim of this paper is to investigate the contemporary notion of picture fuzzy nano topological spaces and explore some of its properties. The stated properties are quantified with numerical data. Furthermore, an algorithm for Multiple Attribute Decision-Making (MADM) with an application regarding the file selection of building material under uncertainty by using picture fuzzy nano topological spaces is developed. As a practical problem, a comparison table is presented to show the difference between the novel concept and the existing methods.
