467 research outputs found

    Characterizations of minimal T3 1/2 L-topological spaces

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    AbstractFor L a closed sets lattice, we prove the existence and a characterization theorem of the minimal elements of T3 1/2 LX, where T3 1/2 LX is the set of all closed sets L-topologies on X ordered by inclusion

    Topology from enrichment: the curious case of partial metrics

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    For any small quantaloid \Q, there is a new quantaloid \D(\Q) of diagonals in \Q. If \Q is divisible then so is \D(\Q) (and vice versa), and then it is particularly interesting to compare categories enriched in \Q with categories enriched in \D(\Q). Taking Lawvere's quantale of extended positive real numbers as base quantale, \Q-categories are generalised metric spaces, and \D(\Q)-categories are generalised partial metric spaces, i.e.\ metric spaces in which self-distance need not be zero and with a suitably modified triangular inequality. We show how every small quantaloid-enriched category has a canonical closure operator on its set of objects: this makes for a functor from quantaloid-enriched categories to closure spaces. Under mild necessary-and-sufficient conditions on the base quantaloid, this functor lands in the category of topological spaces; and an involutive quantaloid is Cauchy-bilateral (a property discovered earlier in the context of distributive laws) if and only if the closure on any enriched category is identical to the closure on its symmetrisation. As this now applies to metric spaces and partial metric spaces alike, we demonstrate how these general categorical constructions produce the "correct" definitions of convergence and Cauchyness of sequences in generalised partial metric spaces. Finally we describe the Cauchy-completion, the Hausdorff contruction and exponentiability of a partial metric space, again by application of general quantaloid-enriched category theory.Comment: Apart from some minor corrections, this second version contains a revised section on Cauchy sequences in a partial metric spac

    Neutrosophic Crisp Set Theory

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    Since the world is full of indeterminacy, the Neutrosophics found their place into contemporary research. We now introduce for the first time the notions of Neutrosophic Crisp Sets and Neutrosophic Topology on Crisp Sets. We develop the 2012 notion of Neutrosophic Topological Spaces and give many practical examples. Neutrosophic Science means development and applications of Neutrosophic Logic, Set, Measure, Integral, Probability etc., and their applications in any field. It is possible to define the neutrosophic measure and consequently the neutrosophic integral and neutrosophic probability in many ways, because there are various types of indeterminacies, depending on the problem we need to solve. Indeterminacy is different from randomness. Indeterminacy can be caused by physical space, materials and type of construction, by items involved in the space, or by other factors. In 1965 [51], Zadeh generalized the concept of crisp set by introducing the concept of fuzzy set, corresponding to the situation in which there is no precisely defined set;there are increasing applications in various fields, including probability, artificial intelligence, control systems, biology and economics. Thus, developments in abstract mathematics using the idea of fuzzy sets possess sound footing. In accordance, fuzzy topological spaces were introduced by Chang [12] and Lowen [33]. After the development of fuzzy sets, much attention has been paid to the generalization of basic concepts of classical topology to fuzzy sets and accordingly developing a theory of fuzzy topology [1-58]. In 1983, the intuitionistic fuzzy set was introduced by K. Atanassov [55, 56, 57] as a generalization of the fuzzy set, beyond the degree of membership and the degree of non-membership of each element. In 1999 and 2002, Smarandache [71, 72, 73, 74] defined the notion of Neutrosophic Sets, which is a generalization of Zadeh’s fuzzy set and Atanassov\u27s intuitionistic fuzzy set

    On MV - topologies

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    En este trabajo estamos interesados en un tipo particular de topología fuzzy llamada MV-topología, la cual usa operaciones MV-algebraicas para generar abiertos fuzzy. Estos espacios topológicos fuzzy permiten generalizaciones naturales de definiciones y resultados importantes de la topología clásica. En este sentido, desarrollamos algunos conceptos y resultados centrales, con el proprósito de extender los correspondientes de la topología clásica, y al mismo tiempo siguiendo la ruta de la bien conocida teoría de espacios topológicos fuzzy. Mostramos que las MV-topologías son un tipo de topología fuzzy que goza de muy "buen comportamiento" matemático, en el sentido de que la mayoría de definiciones y resultados clásicos de topología general encuentran una extensión o adaptación natural en este marco. Entre otros resultados, también extendemos el concepto de haz para el caso en el que el espacio base es un espacio MV-topológico, y mostramos una representación por "MV-haces" para una clase de MV-álgebras.DoctoradoDOCTOR(A) EN CIENCIAS - MATEMÁTICA

    Initial Characterized L-spaces and Characterized L- topological Groups

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    In this research work, new topological notions are proposed and investigated. The notions are named initial characterized L-spaces and characterized L-topological groups. The properties of such notions are deeply studied. We show that the intitial characterized L-space for an characterized L-spaces exists. By this notion, the notions of characterized L-subspace and characterized product L-space are introduced and studied. More information can be found in the full paper

    Categorical Ontology of Complex Systems, Meta-Systems and Theory of Levels: The Emergence of Life, Human Consciousness and Society

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    Single cell interactomics in simpler organisms, as well as somatic cell interactomics in multicellular organisms, involve biomolecular interactions in complex signalling pathways that were recently represented in modular terms by quantum automata with ‘reversible behavior’ representing normal cell cycling and division. Other implications of such quantum automata, modular modeling of signaling pathways and cell differentiation during development are in the fields of neural plasticity and brain development leading to quantum-weave dynamic patterns and specific molecular processes underlying extensive memory, learning, anticipation mechanisms and the emergence of human consciousness during the early brain development in children. Cell interactomics is here represented for the first time as a mixture of ‘classical’ states that determine molecular dynamics subject to Boltzmann statistics and ‘steady-state’, metabolic (multi-stable) manifolds, together with ‘configuration’ spaces of metastable quantum states emerging from complex quantum dynamics of interacting networks of biomolecules, such as proteins and nucleic acids that are now collectively defined as quantum interactomics. On the other hand, the time dependent evolution over several generations of cancer cells --that are generally known to undergo frequent and extensive genetic mutations and, indeed, suffer genomic transformations at the chromosome level (such as extensive chromosomal aberrations found in many colon cancers)-- cannot be correctly represented in the ‘standard’ terms of quantum automaton modules, as the normal somatic cells can. This significant difference at the cancer cell genomic level is therefore reflected in major changes in cancer cell interactomics often from one cancer cell ‘cycle’ to the next, and thus it requires substantial changes in the modeling strategies, mathematical tools and experimental designs aimed at understanding cancer mechanisms. Novel solutions to this important problem in carcinogenesis are proposed and experimental validation procedures are suggested. From a medical research and clinical standpoint, this approach has important consequences for addressing and preventing the development of cancer resistance to medical therapy in ongoing clinical trials involving stage III cancer patients, as well as improving the designs of future clinical trials for cancer treatments.\ud \ud \ud KEYWORDS: Emergence of Life and Human Consciousness;\ud Proteomics; Artificial Intelligence; Complex Systems Dynamics; Quantum Automata models and Quantum Interactomics; quantum-weave dynamic patterns underlying human consciousness; specific molecular processes underlying extensive memory, learning, anticipation mechanisms and human consciousness; emergence of human consciousness during the early brain development in children; Cancer cell ‘cycling’; interacting networks of proteins and nucleic acids; genetic mutations and chromosomal aberrations in cancers, such as colon cancer; development of cancer resistance to therapy; ongoing clinical trials involving stage III cancer patients’ possible improvements of the designs for future clinical trials and cancer treatments. \ud \u

    On Initial and Final Characterized L- topological Groups

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    In this research work, new topological notions are proposed and investigated. The notions are named finalcharacterized L-spaces and initial and final characterized L-topological groups. The properties of such notionsare deeply studied. We show that all the final lefts and all the final characterized L-spaces are uniquely exist inthe category CRL-Sp and hence CRL-Sp is topological category over the category SET of all sets. By the notion offinal characterized L-space, the notions of characterized qoutien pre L-spaces and characterized sum L-spacesare introduced and studied. The characterized L-subspaces together with their related inclusion mappings andthe characterized quotient pre L-spaces together with their related canonical surjections are the equalizers andco-equalizers, respectively in CRL-Sp. Moreover, we show that the initial and final lefts and then the initial andfinal characterized L-topological groups uniquely exist in the category CRL-TopGrp. Hence, the category CRLTopGrpis topological category over the category Grp of all groups. By the notion of initial and finalcharacterized L-topological groups, the notions of characterized L-subgroups, characterized product Ltopologicalgroups and characterized L-topological quotient groups are introduced and studied., we show that thecategory CRL-TopGrp is concrete and co-concrete category of the category L-Top. More details can be found in the full paper

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