New trends on the numerical representability of semiordered structures

[EN] We introduce a survey, including the historical back-ground, on different techniques that have recently been issued in the search for a characterization of the representability of semiordered structures, in the sense of Scott and Suppes, by means of a real-valued function and a strictly positive threshold of discrimination.This work has been supported by the research projects MTM2007-62499, ECO2008-01297, MTM2009-12872-C02-02 and MTM2010-17844 (Spain)Abrísqueta, F.; Campión, M.; Catalán, R.; De Miguel, J.; Estevan, A.; Induráin, E.; Zudaire, M.... (2012). New trends on the numerical representability of semiordered structures. Mathware & Soft Computing Magazine. 19(1):25-37. http://hdl.handle.net/10251/57632S253719

Open questions in utility theory

Throughout this paper, our main idea is to explore different classical questions arising in Utility Theory, with a particular attention to those that lean on numerical representations of preference orderings. We intend to present a survey of open questions in that discipline, also showing the state-of-art of the corresponding literature.This work is partially supported by the research projects ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), and TIN2016-77356-P (MINECO/ AEI-FEDER, UE)

Two geometric constant for operators acting on a separable Banach space

The main result of this paper is the following: a separate Banach space X is reflexive if and only if the infimum of the Gelfand numbers of any bounded linear operator defined on X can be computed by means of just one sequence of nested, closed, finite condimensional subspaces with null intersections

Order embeddings with irrational codomain: Debreu properties of real subsets

The objective of this paper is to investigate the role of the set of irrational numbers as the codomain of order-preserving functions defined on topological totally preordered sets. We will show that although the set of irrational numbers does not satisfy the Debreu property it is still nonetheless true that any lower (respectively, upper) semicontinuous total preorder representable by a real-valued strictly isotone function (semicontinuous or not) also admits a representation by means of a lower (respectively, upper) semicontinuous strictly isotone function that takes values in the set of irrational numbers. These results are obtained by means of a direct construction. Moreover, they can be related to Cantor's characterization of the real line to obtain much more general results on the semicontinuous Debreu properties of a wide family of subsets of the real line