ASSOCIATIVE FUNCTIONS: TRIANGULAR NORMS AND COPULAS
by Claudi Alsina (Universitat Politécnica de Catalunya, Barcelona, Spain), Maurice J Frank (Illinois Institute of Technology, Chicago, USA) & Berthold Schweizer (University of Massachusetts, Amherst, USA)
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.
Contents:
- Representation Theorems for Associative Functions
- Functional Equations Involving t-Norms
- Inequalities Involving t-Norms
- Appendices:
- Examples and Counterexamples
- Open Problems
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Readership: Mathematicians, statisticians, economists, financial analysts, and
other scientists; advanced undergraduate and graduate students interested infunctional equations, copulas and their applications.
"... followed by a 'boom' of researches concerning associativity and its applications, among which the work of C Alsina is most remarkable. The present book, written with two collaborators, gives a proof for this statement ... This book concludes with two appendices and an extensive bibliography. This is a marvelous book!"
"This well-written and very readable book is probably quite unique: its main subject is a single functional equation, that of associativity (joined by some more or less related equations) and several of its equations."
