79 research outputs found

    Weakly fuzzy topological entropy

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    summary:In 2005, İ. Tok fuzzified the notion of the topological entropy R. A. Adler et al. (1965) using the notion of fuzzy compactness of C. L. Chang (1968). In the present paper, we have proposed a new definition of the fuzzy topological entropy of fuzzy continuous mapping, namely weakly fuzzy topological entropy based on the notion of weak fuzzy compactness due to R. Lowen (1976) along with its several properties. We have shown that the topological entropy R. A. Adler et al. (1965) of continuous mapping ψ ⁣:(X,τ)(X,τ)\psi \colon (X,\tau )\rightarrow (X,\tau ), where (X,τ)(X,\tau ) is compact, is equal to the weakly fuzzy topological entropy of ψ ⁣:(X,ω(τ))(X,ω(τ))\psi \colon (X,\omega (\tau ))\rightarrow (X,\omega (\tau )). We have also established an example that shows that the fuzzy topological entropy of İ. Tok (2005) cannot give such a bridge result to the topological entropy of Adler et al. (1965). Moreover, our definition of the weakly fuzzy topological entropy can be applied to find the topological entropy (namely weakly fuzzy topological entropy hw(ψ)h_w(\psi )) of the mapping ψ ⁣:XX\psi \colon X\rightarrow X (where XX is either compact or weakly fuzzy compact), whereas the topological entropy ha(ψ)h_a(\psi ) of Adler does not exist for the mapping ψ ⁣:XX\psi \colon X\rightarrow X (where XX is non-compact weakly fuzzy compact). Finally, a product theorem for the weakly fuzzy topological entropy has been established

    Transitivity in Fuzzy Hyperspaces

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    Given a metric space (X, d), we deal with a classical problem in the theory of hyperspaces: how some important dynamical properties (namely, weakly mixing, transitivity and point-transitivity) between a discrete dynamical system f : (X, d) → (X, d) and its natural extension to the hyperspace are related. In this context, we consider the Zadeh’s extension fbof f to F(X), the family of all normal fuzzy sets on X, i.e., the hyperspace F(X) of all upper semicontinuous fuzzy sets on X with compact supports and non-empty levels and we endow F(X) with different metrics: the supremum metric d∞, the Skorokhod metric d0, the sendograph metric dS and the endograph metric dE. Among other things, the following results are presented: (1) If (X, d) is a metric space, then the following conditions are equivalent: (a) (X, f) is weakly mixing, (b) ((F(X), d∞), fb) is transitive, (c) ((F(X), d0), fb) is transitive and (d) ((F(X), dS)), fb) is transitive, (2) if f : (X, d) → (X, d) is a continuous function, then the following hold: (a) if ((F(X), dS), fb) is transitive, then ((F(X), dE), fb) is transitive, (b) if ((F(X), dS), fb) is transitive, then (X, f) is transitive; and (3) if (X, d) be a complete metric space, then the following conditions are equivalent: (a) (X × X, f × f) is point-transitive and (b) ((F(X), d0) is point-transitive

    One-class classifiers based on entropic spanning graphs

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    One-class classifiers offer valuable tools to assess the presence of outliers in data. In this paper, we propose a design methodology for one-class classifiers based on entropic spanning graphs. Our approach takes into account the possibility to process also non-numeric data by means of an embedding procedure. The spanning graph is learned on the embedded input data and the outcoming partition of vertices defines the classifier. The final partition is derived by exploiting a criterion based on mutual information minimization. Here, we compute the mutual information by using a convenient formulation provided in terms of the α\alpha-Jensen difference. Once training is completed, in order to associate a confidence level with the classifier decision, a graph-based fuzzy model is constructed. The fuzzification process is based only on topological information of the vertices of the entropic spanning graph. As such, the proposed one-class classifier is suitable also for data characterized by complex geometric structures. We provide experiments on well-known benchmarks containing both feature vectors and labeled graphs. In addition, we apply the method to the protein solubility recognition problem by considering several representations for the input samples. Experimental results demonstrate the effectiveness and versatility of the proposed method with respect to other state-of-the-art approaches.Comment: Extended and revised version of the paper "One-Class Classification Through Mutual Information Minimization" presented at the 2016 IEEE IJCNN, Vancouver, Canad

    Orbit Tracing Properties on Hyperspaces and Fuzzy Dynamical Systems

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    [EN] Let X be a compact metric space and a continuous map f:X-->X which defines a discrete dynamical system (X,f). The map f induces two natural maps, namely \bar{f}:K(X)-->K(X) on the hyperspace K(X) of non-empty compact subspaces of X and the Zadeh¿s extension \hat{f}:F(X)-->F(X) on the space F(X) of normal fuzzy set. In this work, we analyze the interaction of some orbit tracing dynamical properties, namely the specification and shadowing properties of the discrete dynamical system (X,f) and its induced discrete dynamical systems (K(X),\bar{f}) and (F(X),\hat{f}). Adding an algebraic structure yields stronger conclusions, and we obtain a full characterization of the specification property in the hyperspace, in the fuzzy space, and in the phase space X if we assume that the later is a convex compact subset of a (metrizable and complete) locally convex space and f is a linear operator.This work was supported by MCIN/AEI/10.13039/501100011033, Project PID2019-105011GB-I00, and by Generalitat Valenciana, Project PROMETEU/2021/070.Bartoll Arnau, S.; Martínez Jiménez, F.; Peris Manguillot, A.; Ródenas Escribá, FDA. (2022). Orbit Tracing Properties on Hyperspaces and Fuzzy Dynamical Systems. Axioms. 11(12):1-11. https://doi.org/10.3390/axioms11120733111111

    Toward a multilevel representation of protein molecules: comparative approaches to the aggregation/folding propensity problem

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    This paper builds upon the fundamental work of Niwa et al. [34], which provides the unique possibility to analyze the relative aggregation/folding propensity of the elements of the entire Escherichia coli (E. coli) proteome in a cell-free standardized microenvironment. The hardness of the problem comes from the superposition between the driving forces of intra- and inter-molecule interactions and it is mirrored by the evidences of shift from folding to aggregation phenotypes by single-point mutations [10]. Here we apply several state-of-the-art classification methods coming from the field of structural pattern recognition, with the aim to compare different representations of the same proteins gathered from the Niwa et al. data base; such representations include sequences and labeled (contact) graphs enriched with chemico-physical attributes. By this comparison, we are able to identify also some interesting general properties of proteins. Notably, (i) we suggest a threshold around 250 residues discriminating "easily foldable" from "hardly foldable" molecules consistent with other independent experiments, and (ii) we highlight the relevance of contact graph spectra for folding behavior discrimination and characterization of the E. coli solubility data. The soundness of the experimental results presented in this paper is proved by the statistically relevant relationships discovered among the chemico-physical description of proteins and the developed cost matrix of substitution used in the various discrimination systems.Comment: 17 pages, 3 figures, 46 reference

    Nonlinear dynamics and modeling of heart and brain signals

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