LEADER 04775nam 22007934a 450 001 9910953793103321 005 20251116220544.0 010 $a9786611919344 010 $a9781281919342 010 $a1281919349 010 $a9789812774200 010 $a9812774203 024 3 $a9789812566713 035 $a(CKB)1000000000480153 035 $a(OCoLC)614464270 035 $a(CaPaEBR)ebrary10201442 035 $a(SSID)ssj0000106892 035 $a(PQKBManifestationID)11140633 035 $a(PQKBTitleCode)TC0000106892 035 $a(PQKBWorkID)10014195 035 $a(PQKB)10691981 035 $a(MiAaPQ)EBC1681474 035 $a(WSP)00006036 035 $a(Au-PeEL)EBL1681474 035 $a(CaPaEBR)ebr10201442 035 $a(CaONFJC)MIL191934 035 $a(OCoLC)879025280 035 $a(Perlego)847879 035 $a(BIP)13435284 035 $a(EXLCZ)991000000000480153 100 $a20061024d2006 uy 0 101 0 $aeng 135 $aurcn||||||||| 181 $ctxt 182 $cc 183 $acr 200 10$aAssociative functions $etriangular norms and copulas /$fClaudi Alsina, Maurice J. Frank, Berthold Schweizer 205 $a1st ed. 210 $aSingapore ;$aHackensack, NJ $cWorld Scientific$dc2006 215 $a1 online resource (253 p.) 300 $aBibliographic Level Mode of Issuance: Monograph 311 08$a9789812566713 311 08$a9812566716 320 $aIncludes bibliographical references (p. 223-234) and index. 327 $aPreface -- 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. 330 $aThe 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. 606 $aFunctional equations 606 $aAssociative law (Mathematics) 606 $aMathematical analysis 606 $aFunctional equations$xStudy and teaching$vTextbooks 606 $aAssociative law (Mathematics)$xStudy and teaching$vTextbooks 606 $aMathematical analysis$xStudy and teaching$vTextbooks 615 0$aFunctional equations. 615 0$aAssociative law (Mathematics) 615 0$aMathematical analysis. 615 0$aFunctional equations$xStudy and teaching 615 0$aAssociative law (Mathematics)$xStudy and teaching 615 0$aMathematical analysis$xStudy and teaching 676 $a515/.7 700 $aAlsina$b Claudi$0309455 701 $aSchweizer$b B$g(Berthold)$0368525 701 $aFrank$b Maurice J$01865802 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910953793103321 996 $aAssociative functions$94473020 997 $aUNINA