Softarisons: theory and practice

[EN] This paper introduces the concept of softarison. Softarisons merge soft set theory with the theory of binary relations. Their purpose is the comparison of alternatives in a parameterized environment. We develop the basic theory and interpretations of softarisons. Then, the normative idea of ‘optimal’ alternatives is discussed in this context. We argue that the meaning of ‘optimality’ can be adjusted to fit in with the structure of each problem. A sufficient condition for the existence of an optimal alternative for unrestricted sets of alternatives is proven. This result means a counterpart of Weierstrass extreme value theorem for softarisons; thus, it links soft topology with the act of choice. We also provide a decision-making procedure—the minimax algorithm—when the alternatives are compared through a softarison. A case-study in the context of group interviews illustrates both the application of softarisons as an evaluation tool, and the computation of minimax solutions.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

Fuzzy Sets, Fuzzy Logic and Their Applications

The present book contains 20 articles collected from amongst the 53 total submitted manuscripts for the Special Issue “Fuzzy Sets, Fuzzy Loigic and Their Applications” of the MDPI journal Mathematics. The articles, which appear in the book in the series in which they were accepted, published in Volumes 7 (2019) and 8 (2020) of the journal, cover a wide range of topics connected to the theory and applications of fuzzy systems and their extensions and generalizations. This range includes, among others, management of the uncertainty in a fuzzy environment; fuzzy assessment methods of human-machine performance; fuzzy graphs; fuzzy topological and convergence spaces; bipolar fuzzy relations; type-2 fuzzy; and intuitionistic, interval-valued, complex, picture, and Pythagorean fuzzy sets, soft sets and algebras, etc. The applications presented are oriented to finance, fuzzy analytic hierarchy, green supply chain industries, smart health practice, and hotel selection. This wide range of topics makes the book interesting for all those working in the wider area of Fuzzy sets and systems and of fuzzy logic and for those who have the proper mathematical background who wish to become familiar with recent advances in fuzzy mathematics, which has entered to almost all sectors of human life and activity

On the relationship between plane and solid geometry

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned area

Ontology-based context representation and reasoning for object tracking and scene interpretation in video

Computer vision research has been traditionally focused on the development of quantitative techniques to calculate the properties and relations of the entities appearing in a video sequence. Most object tracking methods are based on statistical methods, which often result inadequate to process complex scenarios. Recently, new techniques based on the exploitation of contextual information have been proposed to overcome the problems that these classical approaches do not solve. The present paper is a contribution in this direction: we propose a Computer Vision framework aimed at the construction of a symbolic model of the scene by integrating tracking data and contextual information. The scene model, represented with formal ontologies, supports the execution of reasoning procedures in order to: (i) obtain a high-level interpretation of the scenario; (ii) provide feedback to the low-level tracking procedure to improve its accuracy and performance. The paper describes the layered architecture of the framework and the structure of the knowledge model, which have been designed in compliance with the JDL model for Information Fusion. We also explain how deductive and abductive reasoning is performed within the model to accomplish scene interpretation and tracking improvement. To show the advantages of our approach, we develop an example of the use of the framework in a video-surveillance application.This work was supported in part by Projects CICYT TIN2008- 06742-C02-02/TSI, CICYT TEC2008-06732-C02-02/TEC, SINPROB, CAM MADRINET S-0505/TIC/0255 and DPS2008–07029-C02–02.Publicad

Convex Soft Geometries

This paper introduces convex soft geometries. The import of this model is that it extends a successful field of research, namely, convex geometries, to account for alternatives characterized by a multiplicity of attributes. Related concepts include soft convex hulls and a notion of extreme elements. Some fundamental results are proven. Among them, we guarantee that an anti-exchange property holds true. Importantly, the existence of extreme elements is guaranteed hence proving their suitability for applied studies. Both results are natural (but non-trivial) extensions of comparable outputs for convex geometries. We present a detailed research program that might stimulate future investigations on this new research area