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On Fuzzifications of Discrete Dynamical Systems
, 2008
"... Let X denote a locally compact metric space and ϕ: X → X be a continuous map. In the 1970s L. Zadeh presented an extension principle, helping us to fuzzify the dynamical system (X,ϕ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, ..."
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Let X denote a locally compact metric space and ϕ: X → X be a continuous map. In the 1970s L. Zadeh presented an extension principle, helping us to fuzzify the dynamical system (X,ϕ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets and Systems 61 (1994)] and we generalize Zadeh’s original extension principle. In this paper we study basic properties, such as the continuity of socalled gfuzzifications. We also show that, for any gfuzzification: (i) a uniformly convergent sequence of uniformly convergent maps on X induces a uniformly convegent sequence of continuous maps on the space of fuzzy sets, and (ii) a conjugacy (a semiconjugacy, resp.) between two discrete dynamical systems can be extended to a conjugacy (a semiconjugacy, resp.) between fuzzified dynamical systems. Moreover, at the end of this paper we show that there are connections between gfuzzifications and crisp dynamical systems via setvalued dynamical systems and skewproduct (triangular) maps. Throughout this paper we consider different topological structures in the space of fuzzy sets; namely, the sendograph, endograph and levelwise topologies.
The Concepts of Tightness for Fuzzy Set Valued Random Variables
"... In this paper, we introduce several concepts of tightness for a sequence of random variables taking values in the space of normal and uppersemicontinuous fuzzy sets with compact support in Rp and give some characterizations of their concepts. Also, counterexamples for the relationships between the ..."
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In this paper, we introduce several concepts of tightness for a sequence of random variables taking values in the space of normal and uppersemicontinuous fuzzy sets with compact support in Rp and give some characterizations of their concepts. Also, counterexamples for the relationships between the concepts of tightness are given.