Modelling liquid–liquid equilibria for island type ternary systems

Closed miscibility gaps in ternary liquid mixtures, at constant temperature and pressure, are obtained if phase separations occur only in the ternary region, whilst all binary mixtures involved in the system are completely miscible. This type of behaviour, although not very frequent, has been observed for a certain number of systems. Nevertheless, we have found no information about the applicability of the common activity coefficient models, as NRTL and UNIQUAC, for these types of ternary systems. Moreover, any of the island type systems published in the most common liquid–liquid equilibrium data collections, are correlated with any model. In this paper, the applicability of the NRTL equation to model the LLE of island type systems is assessed using topological concepts related to the Gibbs stability test. A first attempt to correlate experimental LLE data for two island type ternary systems is also presented.

Voronoi cells via linear inequality systems

The theory and methods of linear algebra are a useful alternative to those of convex geometry in the framework of Voronoi cells and diagrams, which constitute basic tools of computational geometry. As shown by Voigt and Weis in 2010, the Voronoi cells of a given set of sites T, which provide a tesselation of the space called Voronoi diagram when T is finite, are solution sets of linear inequality systems indexed by T. This paper exploits systematically this fact in order to obtain geometrical information on Voronoi cells from sets associated with T (convex and conical hulls, tangent cones and the characteristic cones of their linear representations). The particular cases of T being a curve, a closed convex set and a discrete set are analyzed in detail. We also include conclusions on Voronoi diagrams of arbitrary sets.

Topological structures of complex belief systems

The concepts of substantive beliefs and derived beliefs are defined, a set of substantive beliefs S like open set and the neighborhood of an element substantive belief. A semantic operation of conjunction is defined with a structure of an Abelian group. Mathematical structures exist such as poset beliefs and join-semilattttice beliefs. A metric space of beliefs and the distance of belief depending on the believer are defined. The concepts of closed and opened ball are defined. S′ is defined as subgroup of the metric space of beliefs Σ and S′ is a totally limited set. The term s is defined (substantive belief) in terms of closing of S′. It is deduced that Σ is paracompact due to Stone's Theorem. The pseudometric space of beliefs is defined to show how the metric of the nonbelieving subject has a topological space like a nonmaterial abstract ideal space formed in the mind of the believing subject, fulfilling the conditions of Kuratowski axioms of closure. To establish patterns of materialization of beliefs we are going to consider that these have defined mathematical structures. This will allow us to understand better cultural processes of text, architecture, norms, and education that are forms or the materialization of an ideology. This materialization is the conversion by means of certain mathematical correspondences, of an abstract set whose elements are beliefs or ideas, in an impure set whose elements are material or energetic. Text is a materialization of ideology.

Coverage and invariability in fuzzy systems

This paper presents a new complex system systemic. Here, we are working in a fuzzy environment, so we have to adapt all the previous concepts and results that were obtained in a non-fuzzy environment, for this fuzzy case. The direct and indirect influences between variables will provide the basis for obtaining fuzzy and/or non-fuzzy relationships, so that the concepts of coverage and invariability between sets of variables will appear naturally. These two concepts and their interconnections will be analyzed from the viewpoint of algebraic properties of inclusion, union and intersection (fuzzy and non-fuzzy), and also for the loop concept, which, as we shall see, will be of special importance.