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Register a new userAdmissible orders on fuzzy numbers
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Publication date
2020
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English
Authors
Nicolas Zumelzu
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From the more than two hundred partial orders for fuzzy numbers proposed in the literature, only a few are total. In this paper, we introduce the notion of admissible order for fuzzy numbers equipped with a partial order, i.e. a total order which refines the partial order. In particular, it is given special attention to the partial order proposed by Klir and Yuan in 1995. Moreover, we propose a method to construct admissible orders on fuzzy numbers in terms of linear orders defined for intervals considering a strictly increasing upper dense sequence, proving that this order is admissible for a given partial order. Finally, we use admissible orders to ranking the path costs in fuzzy weighted graphs.
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Ranking of intuitionsitic fuzzy number plays a vital role in decision making and other intuitionistic fuzzy applications. In this paper, we propose a new ranking method of intuitionistic fuzzy number based on distance measure. We first define a distance measure for interval numbers based on Lp metric and further generalize the idea for intuitionistic fuzzy number by forming interval with their respective value and ambiguity indices. Finally, some comparative results are given in tabular form.
The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. In this paper, the notion of fuzzy gyronorms on gyrogroups is introduced. The relations of fuzzy metrics (in the sense of George and Veeramani), fuzzy gyronorms and gyronorms on gyrogroups are studied. Also, the fuzzy metric structures on fuzzy normed gyrogroups are discussed. In the last, the fuzzy metric completion of a gyrogroup with an invariant metric are studied. We mainly show that let $d$ be an invariant metric on a gyrogroup $G$ and $(widehat{G},widehat{d})$ is the metric completion of the metric space $(G,d)$; then for any continuous $t$-norm $ast$, the standard fuzzy metric space $(widehat{G},M_{widehat{d}},ast)$ of $(widehat{G},widehat{d})$ is the (up to isometry) unique fuzzy metric completion of the standard fuzzy metric space $(G,M_d,ast)$ of $(G,d)$; furthermore, $(widehat{G},M_{widehat{d}},ast)$ is a fuzzy metric gyrogroup containing $(G,M_d,ast)$ as a dense fuzzy metric subgyrogroup and $M_{widehat{d}}$ is invariant on $widehat{G}$. Applying this result, we obtain that every gyrogroup $G$ with an invariant metric $d$ admits an (up to isometric) unique complete metric space $(widehat{G},widehat{d})$ of $(G,d)$ such that $widehat{G}$ with the topology introduced by $widehat{d}$ is a topology gyrogroup containing $G$ as a dense subgyrogroup and $widehat{d}$ is invariant on $widehat{G}$.
In this paper we prove that Neutrosophic Set (NS) is an extension of Intuitionistic Fuzzy Set (IFS) no matter if the sum of single-valued neutrosophic components is < 1, or > 1, or = 1. For the case when the sum of components is 1 (as in IFS), after applying the neutrosophic aggregation operators one gets a different result from that of applying the intuitionistic fuzzy operators, since the intuitionistic fuzzy operators ignore the indeterminacy, while the neutrosophic aggregation operators take into consideration the indeterminacy at the same level as truth-membership and falsehood-nonmembership are taken. NS is also more flexible and effective because it handles, besides independent components, also partially independent and partially dependent components, while IFS cannot deal with these. Since there are many types of indeterminacies in our world, we can construct different approaches to various neutrosophic concepts. Also, Regret Theory, Grey System Theory, and Three-Ways Decision are particular cases of Neutrosophication and of Neutrosophic Probability. We extended for the first time the Three-Ways Decision to n-Ways Decision, and the Spherical Fuzzy Set to n-HyperSpherical Fuzzy Set and to n-HyperSpherical Neutrosophic Set.
Prime Numbers clearly accumulate on defined spiral graphs,which run through the Square Root Spiral. These spiral graphs can be assigned to different spiral-systems, in which all spiral-graphs have the same direction of rotation and the same -- second difference -- between the numbers, which lie on these spiral-graphs. A mathematical analysis shows, that these spiral graphs are caused exclusively by quadratic polynomials. For example the well known Euler Polynomial x2+x+41 appears on the Square Root Spiral in the form of three spiral-graphs, which are defined by three different quadratic polynomials. All natural numbers,divisible by a certain prime factor, also lie on defined spiral graphs on the Square Root Spiral (or Spiral of Theodorus, or Wurzelspirale). And the Square Numbers 4, 9, 16, 25, 36 even form a highly three-symmetrical system of three spiral graphs, which divides the square root spiral into three equal areas. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. With the help of the Number-Spiral, described by Mr. Robert Sachs, a comparison can be drawn between the Square Root Spiral and the Ulam Spiral. The shown sections of his study of the number spiral contain diagrams, which are related to my analysis results, especially in regards to the distribution of prime numbers.
We prove some theorems which give sufficient conditions for the existence of prime numbers among the terms of a sequence which has pairwise relatively prime terms.
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