Claudi Alsina
Polytechnic University of Catalonia
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Featured researches published by Claudi Alsina.
Archive | 2006
Claudi Alsina; Maurice J Frank; Berthold Schweizer
The functional equation of associativity is the topic of Abels first contribution to Crelles Journal. Seventy years later, it was featured as the second part of Hilberts 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.
Journal of Mathematical Analysis and Applications | 1983
Claudi Alsina; E Trillas; L Valverde
INTRODUCTION It was proved by Bellman and Giertz [2] that, under reasonable hypotheses (especially distributivity), the only truth-functional logical connectives for fuzzy sets are the usual min and max. The following easy argument proves that distributivity, monotonicity and boundary conditions are essential assumptions: x = F(x, 1) = F(x, G(l, 1)) = G(F(x, l), F(x, 1)) = G(x, x), max(x, Y) = G(max(x, Y), max(x, Y>> > G(x, Y> > max(G(x, 01, G(O, Y)) = ma+, Y), i.e., G(x, y) = max(x, y). Here F and G are, respectively, functions from (0, l]
Aequationes Mathematicae | 1992
Claudi Alsina; B. Schweizer; A. Sklar
SummaryIn this paper we give a new definition of a probabilistic normed space. This definition, which is based on a characterization of normed spaces by means of a betweenness relation, includes the earlier definition of A. N. Šerstnev as a special case and leads naturally to the definition of the principal class of probabilistic normed spaces, the Menger spaces.
Statistics & Probability Letters | 1993
Claudi Alsina; Roger B. Nelsen; Berthold Schweizer
We characterize the class of binary operations \/o on distribution functions which are both induced pointwise, in the sense that the value of \/o(F, G) at g is a function of F(t) and G(t) (e.g. mixtures), and derivable from functions on random variables (e.g. convolution).
Fuzzy Sets and Systems | 1985
Claudi Alsina
Abstract In this paper we find the general solution of the functional equation S ∗ (T(x, y), T(x, N(y))) = x , where S ∗ is a t-conorm, T is a t-norm and N is a strong negation on the unit interval. In particular the result yields a family of connectives for fuzzy sets.
IEEE Transactions on Fuzzy Systems | 2002
Enric Trillas; Claudi Alsina
This paper deals with the logical equivalence of the classical propositional calculus [p/spl and/q/spl rarr/r]=[(p/spl rarr/r)V(q/spl rarr/r)]. This equality seems to play a central role in a recent discussion around a paper of Combs and Andrews (1998). After reconsidering the equivalence in lattices, its validity in the standard theories of fuzzy sets endowed with an implication operator is studied.
soft computing | 1999
Enric Trillas; Claudi Alsina; Joan Jacas
Abstract We clarify which space of functions in [0, 1]E would be reasonable in fuzzy logic in order to avoid self-contradiction.
Archive | 1985
Claudi Alsina
In this paper we solve for a given e > 0 the inequality
Archive | 2009
Claudi Alsina; Justyna Sikorska; M Santos Tomás
International Journal of General Systems | 2002
Enric Trillas; Claudi Alsina; Ana Pradera
{{\text{d}}_{\text{L}}}\left( {\tau \left( {F\left( {j/a} \right),F\left( {j/b} \right.} \right)} \right),F\left( {j/a + \left. b \right)} \right) \leqslant \varepsilon ,