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Author: Erich Peter Klement
Publisher: Springer Science & Business Media
ISBN: 9401595402
Category : Philosophy
Languages : en
Pages : 391
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Book Description
This book discusses the theory of triangular norms and surveys several applied fields in which triangular norms play a significant part: probabilistic metric spaces, aggregation operators, many-valued logics, fuzzy logics, sets and control, and non-additive measures together with their corresponding integrals. It includes many graphical illustrations and gives a well-balanced picture of theory and applications. It is for mathematicians, computer scientists, applied computer scientists and engineers.
Author: Erich Peter Klement
Publisher: Springer Science & Business Media
ISBN: 9401595402
Category : Philosophy
Languages : en
Pages : 391
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Book Description
This book discusses the theory of triangular norms and surveys several applied fields in which triangular norms play a significant part: probabilistic metric spaces, aggregation operators, many-valued logics, fuzzy logics, sets and control, and non-additive measures together with their corresponding integrals. It includes many graphical illustrations and gives a well-balanced picture of theory and applications. It is for mathematicians, computer scientists, applied computer scientists and engineers.
Author: Claudi Alsina
Publisher: World Scientific
ISBN: 9814478520
Category : Mathematics
Languages : en
Pages : 253
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Book Description
The functional equation of associativity is the topic of Abel's first contribution to Crelle's Journal. Seventy years later, it was featured as the second part of Hilbert's Fifth Problem, and it was solved under successively weaker hypotheses by Brouwer (1909), Cartan (1930) and Aczel (1949). In 1958, B Schweizer and A Sklar showed that the “triangular norms” introduced by Menger in his definition of a probabilistic metric space should be associative; and in their book Probabilistic Metric Spaces, they presented the basic properties of such triangular norms and the closely related copulas. Since then, the study of these two classes of functions has been evolving at an ever-increasing pace and the results have been applied in fields such as statistics, information theory, fuzzy set theory, multi-valued and quantum logic, hydrology, and economics, in particular, risk analysis.This book presents the foundations of the subject of associative functions on real intervals. It brings together results that have been widely scattered in the literature and adds much new material. In the process, virtually all the standard techniques for solving functional equations in one and several variables come into play. Thus, the book can serve as an advanced undergraduate or graduate text on functional equations.
Author: Erich Peter Klement
Publisher: Elsevier
ISBN: 0080459536
Category : Mathematics
Languages : en
Pages : 491
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Book Description
This volume gives a state of the art of triangular norms which can be used for the generalization of several mathematical concepts, such as conjunction, metric, measure, etc. 16 chapters written by leading experts provide a state of the art overview of theory and applications of triangular norms and related operators in fuzzy logic, measure theory, probability theory, and probabilistic metric spaces.Key Features:- Complete state of the art of the importance of triangular norms in various mathematical fields- 16 self-contained chapters with extensive bibliographies cover both the theoretical background and many applications- Chapter authors are leading authorities in their fields- Triangular norms on different domains (including discrete, partially ordered) are described- Not only triangular norms but also related operators (aggregation operators, copulas) are covered- Book contains many enlightening illustrations· Complete state of the art of the importance of triangular norms in various mathematical fields· 16 self-contained chapters with extensive bibliographies cover both the theoretical background and many applications· Chapter authors are leading authorities in their fields· Triangular norms on different domains (including discrete, partially ordered) are described· Not only triangular norms but also related operators (aggregation operators, copulas) are covered· Book contains many enlightening illustrations
Author: D. Butnariu
Publisher: Springer Science & Business Media
ISBN: 9401736022
Category : Business & Economics
Languages : en
Pages : 207
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Book Description
This book aims to present, in a unified approach, a series of mathematical results con cerning triangular norm-based measures and a class of cooperative games with Juzzy coalitions. Our approach intends to emphasize that triangular norm-based measures are powerful tools in exploring the coalitional behaviour in 'such games. They not and simplify some technical aspects of the already classical axiomatic the only unify ory of Aumann-Shapley values, but also provide new perspectives and insights into these results. Moreover, this machinery allows us to obtain, in the game theoretical context, new and heuristically meaningful information, which has a significant impact on balancedness and equilibria analysis in a cooperative environment. From a formal point of view, triangular norm-based measures are valuations on subsets of a unit cube [0, 1]X which preserve dual binary operations induced by trian gular norms on the unit interval [0, 1]. Triangular norms (and their dual conorms) are algebraic operations on [0,1] which were suggested by MENGER [1942] and which proved to be useful in the theory of probabilistic metric spaces (see also [WALD 1943]). The idea of a triangular norm-based measure was implicitly used under various names: vector integrals [DVORETZKY, WALD & WOLFOWITZ 1951], prob abilities oj Juzzy events [ZADEH 1968], and measures on ideal sets [AUMANN & SHAPLEY 1974, p. 152].
Author: Qingqing Hu
Publisher: Infinite Study
ISBN:
Category : Mathematics
Languages : en
Pages : 22
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Book Description
Neutrosophic triangular norms (t-norms) and their residuated lattices are not only the main research object of neutrosophic set theory, but also the core content of neutrosophic logic. Neutrosophic implications are important operators of neutrosophic logic. Neutrosophic residual implications based on neutrosophic t-norms can be applied to the fields of neutrosophic inference and neutrosophic control. In this paper, neutrosophic t-norms, neutrosophic residual implications, and the residuated lattices derived from neutrosophic t-norms are investigated deeply. First of all, the lattice and its corresponding system are proved to be a complete lattice and a De Morgan algebra, respectively. Second, the notions of neutrosophic t-norms are introduced on the complete lattice discussed earlier. The basic concepts and typical examples of representable and non-representable neutrosophic t-norms are obtained. Naturally, De Morgan neutrosophic triples are defined for the duality of neutrosophic t-norms and neutrosophic t-conorms with respect to neutrosophic negators. Third, neutrosophic residual implications generated from neutrosophic t-norms and their basic properties are investigated. Furthermore, residual neutrosophic t-norms are proved to be infinitely -distributive, and then some important properties possessed by neutrosophic residual implications are given. Finally, a method for producing neutrosophic t-norms from neutrosophic implications is presented, and the residuated lattices are constructed on the basis of neutrosophic t-norms and neutrosophic residual implications.
Author: Claudi Alsina
Publisher: World Scientific
ISBN: 9812566716
Category : Science
Languages : en
Pages : 253
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Book Description
The functional equation of associativity is the topic of Abel's first contribution to Crelle's Journal. Seventy years later, it was featured as the second part of Hilbert's Fifth Problem, and it was solved under successively weaker hypotheses by Brouwer (1909), Cartan (1930) and Aczel (1949). In 1958, B Schweizer and A Sklar showed that the ?triangular norms? introduced by Menger in his definition of a probabilistic metric space should be associative; and in their book Probabilistic Metric Spaces, they presented the basic properties of such triangular norms and the closely related copulas. Since then, the study of these two classes of functions has been evolving at an ever-increasing pace and the results have been applied in fields such as statistics, information theory, fuzzy set theory, multi-valued and quantum logic, hydrology, and economics, in particular, risk analysis.This book presents the foundations of the subject of associative functions on real intervals. It brings together results that have been widely scattered in the literature and adds much new material. In the process, virtually all the standard techniques for solving functional equations in one and several variables come into play. Thus, the book can serve as an advanced undergraduate or graduate text on functional equations.
Author: Witold Pedrycz
Publisher: John Wiley & Sons
ISBN: 0470724153
Category : Technology & Engineering
Languages : en
Pages : 1148
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Book Description
Although the notion is a relatively recent one, the notions and principles of Granular Computing (GrC) have appeared in a different guise in many related fields including granularity in Artificial Intelligence, interval computing, cluster analysis, quotient space theory and many others. Recent years have witnessed a renewed and expanding interest in the topic as it begins to play a key role in bioinformatics, e-commerce, machine learning, security, data mining and wireless mobile computing when it comes to the issues of effectiveness, robustness and uncertainty. The Handbook of Granular Computing offers a comprehensive reference source for the granular computing community, edited by and with contributions from leading experts in the field. Includes chapters covering the foundations of granular computing, interval analysis and fuzzy set theory; hybrid methods and models of granular computing; and applications and case studies. Divided into 5 sections: Preliminaries, Fundamentals, Methodology and Algorithms, Development of Hybrid Models and Applications and Case Studies. Presents the flow of ideas in a systematic, well-organized manner, starting with the concepts and motivation and proceeding to detailed design that materializes in specific algorithms, applications and case studies. Provides the reader with a self-contained reference that includes all pre-requisite knowledge, augmented with step-by-step explanations of more advanced concepts. The Handbook of Granular Computing represents a significant and valuable contribution to the literature and will appeal to a broad audience including researchers, students and practitioners in the fields of Computational Intelligence, pattern recognition, fuzzy sets and neural networks, system modelling, operations research and bioinformatics.
Author: Adam E Gawęda
Publisher: Springer
ISBN: 3319679465
Category : Technology & Engineering
Languages : en
Pages : 417
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Book Description
This book is a tribute to Professor Jacek Żurada, who is best known for his contributions to computational intelligence and knowledge-based neurocomputing. It is dedicated to Professor Jacek Żurada, Full Professor at the Computational Intelligence Laboratory, Department of Electrical and Computer Engineering, J.B. Speed School of Engineering, University of Louisville, Kentucky, USA, as a token of appreciation for his scientific and scholarly achievements, and for his longstanding service to many communities, notably the computational intelligence community, in particular neural networks, machine learning, data analyses and data mining, but also the fuzzy logic and evolutionary computation communities, to name but a few. At the same time, the book recognizes and honors Professor Żurada’s dedication and service to many scientific, scholarly and professional societies, especially the IEEE (Institute of Electrical and Electronics Engineers), the world’s largest professional technical professional organization dedicated to advancing science and technology in a broad spectrum of areas and fields. The volume is divided into five major parts, the first of which addresses theoretic, algorithmic and implementation problems related to the intelligent use of data in the sense of how to derive practically useful information and knowledge from data. In turn, Part 2 is devoted to various aspects of neural networks and connectionist systems. Part 3 deals with essential tools and techniques for intelligent technologies in systems modeling and Part 4 focuses on intelligent technologies in decision-making, optimization and control, while Part 5 explores the applications of intelligent technologies.
Author: Lotfi A Zadeh
Publisher: World Scientific Publishing
ISBN: 9813238194
Category : Computers
Languages : en
Pages : 610
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Book Description
Nowadays, voluminous textbooks and monographs in fuzzy logic are devoted only to separate or some combination of separate facets of fuzzy logic. There is a lack of a single book that presents a comprehensive and self-contained theory of fuzzy logic and its applications.Written by world renowned authors, Lofti Zadeh, also known as the Father of Fuzzy Logic, and Rafik Aliev, who are pioneers in fuzzy logic and fuzzy sets, this unique compendium includes all the principal facets of fuzzy logic such as logical, fuzzy-set-theoretic, epistemic and relational. Theoretical problems are prominently illustrated and illuminated by numerous carefully worked-out and thought-through examples.This invaluable volume will be a useful reference guide for academics, practitioners, graduates and undergraduates in fuzzy logic and its applications.
Author: Sergei O. Kuznetsov
Publisher: Springer Science & Business Media
ISBN: 3642218806
Category : Computers
Languages : en
Pages : 382
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Book Description
This book constitutes the refereed proceedings of the 13th International Conference on Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, RSFDGrC 2011, held in Moscow, Russia in June 2011. The 49 revised full papers presented together with 5 invited and 2 tutorial papers were carefully reviewed and selected from a total of 83 submissions. The papers are organized in topical sections on rough sets and approximations, coverings and granules, fuzzy set models, fuzzy set applications, compound values, feature seletion and reduction, clusters and concepts, rules and trees, image processing, and interactions and visualization.
