Raquel Garcia Catalán

Features of the extension of a statistical measure of complexity to continuous systems.

We discuss some aspects of the extension to continuous systems of a statistical measure of complexity introduced by López-Ruiz, Mancini, and Calbet [Phys. Lett. A 209, 321 (1995)]. In general, the extension of a magnitude from the discrete to the continuous case is not a trivial process and requires some kind of choice. In the present study, several possibilities appear available. One of them is examined in detail. Some interesting properties desirable for any magnitude of complexity are discovered on this particular extension.

AGGREGATION OF PREFERENCES IN CRISP AND FUZZY SETTINGS: FUNCTIONAL EQUATIONS LEADING TO POSSIBILITY RESULTS

We analyze various models introduced in social choice to aggregate individual preferences. We show that on the basis of most of these models there is a system of functional equations such that, in many cases, the origin of impossibility results in a social choice model is the non-existence of a solution for the corresponding system. Among the functional equations considered, we pay a particular attention to general means and associativity, proving that the existence of an associative bivariate mean is equivalent to the existence of a semilatticial partial order. This key result allows us to explain how the knowledge of associative bivariate means can be used to solve social choice paradoxes. In our analysis we deal both with crisp and fuzzy settings.

Oversampling and preservation of tightness in affine frames

The problem of how an oversampling of translations affects the bounds of an affine frame has been proposed by Chui and Shi. In particular, they proved that tightness is preserved if the oversampling factor is coprime with the dilation factor. In this paper we study, in the dyadic dilation case, oversampling of translation by factors which do not satisfy the above condition, and prove that tightness is preserved only in the case of affine frames generated by wavelets having frequency support with very particular properties.

Numerical representability of fuzzy total preorders

Abstract We introduce the concept of a fuzzy total preorder. Then we analyze its numerical representability through a real-valued order-preserving function defined for each α-cut.

Reinterpreting a fuzzy subset by means of a Sincov's functional equation

Throughout the paper it is shown that the classical definition of a fuzzy subset carries additional structures. The concept of a fuzzy subset is regarded from an alternative point of view: namely, the characteristic function of a fuzzy subset may be reinterpreted in terms of a Sincov’s functional equation in two variables. Since the solutions of a Sincov’s functional equation are also closely related to the existence of representable total preorders, an special attention is paid to the relationship between fuzzy subsets and total preorders defined on a universe. Some possible applications of this approach are pointed out in the final section

Functional equations related to fuzzy sets and representable orderings1

The original definition of a fuzzy set hides several additional structures related to functional equations. In the present paper, we analyze the separability functional equation and some of its variants, proving that the solutions of those equations can be described in terms of pairs of linked fuzzy sets on the same universe. In addition, we show that representable total preorders, interval orders and semiorders on a universe can also be analyzed throughout separability functional equations and, consequently, by means of fuzzy sets. Other functional equations related to fuzzy sets, as well as some relationship between this setting and other approaches as the construction of quasi-metrics on a universe are also pointed out as a by-product.

On the Structure of Acyclic Binary Relations

We investigate the structure of acyclic binary relations from different points of view. On the one hand, given a nonempty set we study real-valued bivariate maps that satisfy suitable functional equations, in a way that their associated binary relation is acyclic. On the other hand, we consider acyclic directed graphs as well as their representation by means of incidence matrices. Acyclic binary relations can be extended to the asymmetric part of a linear order, so that, in particular, any directed acyclic graph has a topological sorting.

Geometrical aggregation of finite fuzzy sets

This work has been partially supported by the research projects TIN2013-47605-P, ECO2015-65031-R, MTM2015-63608-P (MINECO/AEI-FEDER, UE), TIN2016-77356-P (MINECO/AEI-FEDER, UE) and TIN2016-81731-REDT (LO-DISCO II and MINECO/AEI-FEDER, UE).

An axiomatic approach to finite means

Abstract In this paper we analyze the notion of a finite mean from an axiomatic point of view. We discuss several axiomatic alternatives, with the aim of establishing a universal definition reconciling all of them and exploring theoretical links to some branches of Mathematics as well as to multidisciplinary applications.

Binary relations coming from solutions of functional equations: orderings and fuzzy subsets

Electronic version of an article published as International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems Vol. 25, Suppl. 1 (December 2017) 19-42 DOI: 10.1142/S0218488517400025