![]() The paper "A weak comparison principle for reaction-diffusion systems" deals with certain differential equations for which uniqueness of the Cauchy problem fails. This special issue addresses work at the interface between function spaces, hyperspaces, fuzzy structures, and asymmetric topology. Many of the impactful and highly cited papers in general topology and its applications have been published in the realm of this relationship. The interplay between fuzzy structures and the theory of hyperspaces and the asymmetric topology is a topic of broad appeal, encouraging new research on such topics and in many cases defining emerging research directions and laying the foundation for future innovations. Hyperspaces, asymmetric topology and fuzzy structures are interdisciplinary topics of increasing interest in the development of the general topology, and its applications in other areas of Mathematics and computer science, such as convex analysis and optimization, fractals and dynamical systems, mathematical economics, and image computing. Zadeh proposed in the sixties his theory of fuzzy sets, this subject became a productive field of research: fuzzy structures not only generalize most of the familiar structures but also they allow obtaining interesting applications in engineering. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Salvador Romaguera,1 Gerald Beer,2 and Manuel Sanchis3ġ Instituto Universitario de Matemática Pura y Aplicada, Universitat Politécnica de Valencia, 46022 Valencia, SpainĢĝepartment of Mathematics, California State University, Los Angeles, CA 90032, USAģ Institut de Matematiques i Aplicacions de Castellá (IMAC), Universitat Jaume I, Campus del Riu Sec S/N, 12071 Castellá, Spain Correspondence should be addressed to Salvador Romaguera 2 April 2013 Accepted 2 April 2013Ĭopyright © 2013 Salvador Romaguera et al. ![]() Furthermore, in the continuous complete case, the d_H-Scott topology coincides with the lower Vietoris topology, and the d_Q-Scott topology coincides with the upper Vietoris topology.Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 619707, 2 pages įunction Spaces, Hyperspaces, and Asymmetric and Fuzzy Structures Then we show that the Hoare and Smyth powerdomains of an algebraic complete quasi-metric space are again algebraic complete, with those quasi-metrics, and similarly that the corresponding powerdomains of continuous complete quasi-metric spaces are continuous complete. Through these isomorphisms again, the two powerdomains inherit quasi-metrics d_H and d_Q, respectively, that are reminiscent of the well-known Hausdorff metric. Turning to the corresponding hyperspaces, namely the same powerdomains, but equipped with the lower Vietoris and upper Vietoris topologies instead, this turns into homeomorphisms with the corresponding space of previsions, equipped with the so-called weak topology. ![]() There are natural isomorphisms between the Hoare and Smyth powerdomains, as used in denotational semantics, and spaces of discrete sublinear previsions, and of discrete normalized superlinear previsions, respectively. We show that the Kantorovich-Rubinstein quasi-metrics d_KR and d^a_KR of Part I extend naturally to various spaces of previsions, and in particular not just the linear previsions (roughly, measures) of Part I.
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