Associative functions : triangular norms and copulas / / Claudi Alsina, Maurice J. Frank, Berthold Schweizer

• Autore: Alsina Claudi
• Pubblicazione: Singapore ; ; Hackensack, NJ, : World Scientific, c2006
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (253 p.)
• Disciplina: 515/.7
• Soggetto topico: Functional equations; Associative law (Mathematics); Mathematical analysis; Functional equations - Study and teaching; Associative law (Mathematics) - Study and teaching; Mathematical analysis - Study and teaching
• Altri autori: SchweizerB (Berthold) FrankMaurice J
• Note generali: Bibliographic Level Mode of Issuance: Monograph
• Nota di bibliografia: Includes bibliographical references (p. 223-234) and index.
• Nota di contenuto: Preface -- Special symbols -- 1. Introduction. 1.1. Historical notes. 1.2. Preliminaries. 1.3. t-norms and s-norms. 1.4. Copulas -- 2. Representation theorems for associative functions. 2.1. Continuous, Archimedean t-norms. 2.2. Additive and multiplicative generators. 2.3. Extension to arbitrary closed intervals. 2.4. Continuous, non-Archimedean t-norms. 2.5. Non-continuous t-norms. 2.6. Families of t-norms. 2.7. Other representation theorems. 2.8. Related functional equations -- 3. Functional equations involving t-norms. 3.1. Simultaneous associativity. 3.2. n-duality. 3.3. Simple characterizations of Min. 3.4. Homogeneity. 3.5. Distributivity. 3.6. Conical t-norms. 3.7. Rational Archimedean t-norms. 3.8. Extension and sets of uniqueness -- 4. Inequalities involving t-norms. 4.1. Notions of concavity and convexity. 4.2. The dominance relation. 4.3. Uniformly close associative functions. 4.4. Serial iterates and n-copulas. 4.5. Positivity.
• Sommario/riassunto: 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.
• Titolo autorizzato: Associative functions
• ISBN: 9786611919344; 9781281919342; 1281919349; 9789812774200; 9812774203
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910953793103321
• Lo trovi qui: Univ. Federico II