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Author: S.E. Rodabaugh Publisher: Springer Science & Business Media ISBN: 9401702314 Category : Mathematics Languages : en Pages : 468
Book Description
This volume summarizes recent developments in the topological and algebraic structures in fuzzy sets and may be rightly viewed as a continuation of the stan dardization of the mathematics of fuzzy sets established in the "Handbook", namely the Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Volume 3 of The Handbooks of Fuzzy Sets Series (Kluwer Academic Publish ers, 1999). Many of the topological chapters of the present work are not only based upon the foundations and notation for topology laid down in the Hand book, but also upon Handbook developments in convergence, uniform spaces, compactness, separation axioms, and canonical examples; and thus this work is, with respect to topology, a continuation of the standardization of the Hand book. At the same time, this work significantly complements the Handbook in regard to algebraic structures. Thus the present volume is an extension of the content and role of the Handbook as a reference work. On the other hand, this volume, even as the Handbook, is a culmination of mathematical developments motivated by the renowned International Sem inar on Fuzzy Set Theory, also known as the Linz Seminar, held annually in Linz, Austria. Much of the material of this volume is related to the Twenti eth Seminar held in February 1999, material for which the Seminar played a crucial and stimulating role, especially in providing feedback, connections, and the necessary screening of ideas.
Author: S.E. Rodabaugh Publisher: Springer Science & Business Media ISBN: 9401702314 Category : Mathematics Languages : en Pages : 468
Book Description
This volume summarizes recent developments in the topological and algebraic structures in fuzzy sets and may be rightly viewed as a continuation of the stan dardization of the mathematics of fuzzy sets established in the "Handbook", namely the Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, Volume 3 of The Handbooks of Fuzzy Sets Series (Kluwer Academic Publish ers, 1999). Many of the topological chapters of the present work are not only based upon the foundations and notation for topology laid down in the Hand book, but also upon Handbook developments in convergence, uniform spaces, compactness, separation axioms, and canonical examples; and thus this work is, with respect to topology, a continuation of the standardization of the Hand book. At the same time, this work significantly complements the Handbook in regard to algebraic structures. Thus the present volume is an extension of the content and role of the Handbook as a reference work. On the other hand, this volume, even as the Handbook, is a culmination of mathematical developments motivated by the renowned International Sem inar on Fuzzy Set Theory, also known as the Linz Seminar, held annually in Linz, Austria. Much of the material of this volume is related to the Twenti eth Seminar held in February 1999, material for which the Seminar played a crucial and stimulating role, especially in providing feedback, connections, and the necessary screening of ideas.
Author: Jana, Chiranjibe Publisher: IGI Global ISBN: 1799801926 Category : Mathematics Languages : en Pages : 439
Book Description
In the world of mathematics, the study of fuzzy relations and its theories are well-documented and a staple in the area of calculative methods. What many researchers and scientists overlook is how fuzzy theory can be applied to industries outside of arithmetic. The framework of fuzzy logic is much broader than professionals realize. There is a lack of research on the full potential this theoretical model can reach. The Handbook of Research on Emerging Applications of Fuzzy Algebraic Structures provides emerging research exploring the theoretical and practical aspects of fuzzy set theory and its real-life applications within the fields of engineering and science. Featuring coverage on a broad range of topics such as complex systems, topological spaces, and linear transformations, this book is ideally designed for academicians, professionals, and students seeking current research on innovations in fuzzy logic in algebra and other matrices.
Author: Ulrich Höhle Publisher: Springer Science & Business Media ISBN: 1461550793 Category : Mathematics Languages : en Pages : 722
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: Liu Ying-Ming Publisher: World Scientific ISBN: 9814518204 Category : Computers Languages : en Pages : 364
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. Contents:Fuzzy Topological SpacesOperations on Fuzzy Topological SpacesL-Valued Stratification SpacesConvergence TheoryConnectednessSome Properties Related to CardinalsSeparation (I)Separation (II)CompactnessCompactificationParacompactnessUniformity and ProximityMetric SpacesRelations Between Fuzzy Topological Spaces and Locales Readership: Senior undergraduates, graduate students, and researchers in mathematics and computer science. keywords:Fuzzy;Topology;Fuzzy Lattice;Lattice-valued Topology;Multiple Choice Principle;Coincident Neighborhood Structure;Level Structure;Pointlike Structure;Ordered Structure;Locale “This will be a very useful reference book for everyone working in this field.” Mathematical Reviews
Author: Wojciech P. Ziarko Publisher: Springer Science & Business Media ISBN: 1447132386 Category : Computers Languages : en Pages : 486
Book Description
The objective of this book is two-fold. Firstly, it is aimed at bringing to gether key research articles concerned with methodologies for knowledge discovery in databases and their applications. Secondly, it also contains articles discussing fundamentals of rough sets and their relationship to fuzzy sets, machine learning, management of uncertainty and systems of logic for formal reasoning about knowledge. Applications of rough sets in different areas such as medicine, logic design, image processing and expert systems are also represented. The articles included in the book are based on selected papers presented at the International Workshop on Rough Sets and Knowledge Discovery held in Banff, Canada in 1993. The primary methodological approach emphasized in the book is the mathematical theory of rough sets, a relatively new branch of mathematics concerned with the modeling and analysis of classification problems with imprecise, uncertain, or incomplete information. The methods of the theory of rough sets have applications in many sub-areas of artificial intelligence including knowledge discovery, machine learning, formal reasoning in the presence of uncertainty, knowledge acquisition, and others. This spectrum of applications is reflected in this book where articles, although centered around knowledge discovery problems, touch a number of related issues. The book is intended to provide an important reference material for students, researchers, and developers working in the areas of knowledge discovery, machine learning, reasoning with uncertainty, adaptive expert systems, and pattern classification.
Author: Ulrich Höhle Publisher: Springer ISBN: 9781461550808 Category : Mathematics Languages : en Pages : 716
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: Sunil Jacob John Publisher: Springer Nature ISBN: 303057654X Category : Technology & Engineering Languages : en Pages : 251
Book Description
This book offers a self-contained guide to the theory and main applications of soft sets. It introduces readers to the basic concepts, the algebraic and topological structures, as well as hybrid structures, such as fuzzy soft sets and intuitionistic fuzzy sets. The last part of the book explores a range of interesting applications in the fields of decision-making, pattern recognition, and data science. All in all, the book provides graduate students and researchers in mathematics and various applied science fields with a comprehensive and timely reference guide to soft sets.
Author: S.E. Rodabaugh
Author: S.E. Rodabaugh
Author: S. E. Rodabaugh
Author: Jana, Chiranjibe
Author: Ulrich Höhle
Author: Liu Ying-Ming
Author: Wojciech P. Ziarko
Author: Florentin Smarandache
Author: Ulrich Höhle
Author: Etienne E. Kerre
Author: Sunil Jacob John