• Journal of Korean Society of Industrial and Systems Engineering
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• v.35 no.4
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• pp.244-248
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• 2012

F-Measure is one of the external measures for evaluating the validity of clustering results. Though it has clear advantages over other widely used external measures such as Purity and Entropy, F-Measure has inherently been less sensitive than other validity measures. This insensitivity owes to the definition of F-Measure that counts only most influential portions. In this research, we present $F_n$-Measure, an external cluster evaluation measure based on F-Measure. $F_n$-Measure is so sensitive that it can detect their difference in the cases that F-Measure cannot detect the difference in clustering results. We compare $F_n$-Measure to F-Measure for a few clustering results and show which measure draws better result based upon homogeneity and completeness.

• Bulletin of the Korean Mathematical Society
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• v.57 no.3
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• pp.569-581
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• 2020

In this paper, we introduce the new general concept of usual expansiveness which is called "positively weak measure expansiveness" and study the basic properties of positively weak measure expansive C¹-differentiable maps on a compact smooth manifold M. And we prove that the following theorems. (1) Let 𝓟𝓦𝓔 be the set of all positively weak measure expansive differentiable maps of M. Denote by int(𝓟𝓦𝓔) is a C¹-interior of 𝓟𝓦𝓔. f ∈ int(𝓟𝓦𝓔) if and only if f is expanding. (2) For C¹-generic f ∈ C¹ (M), f is positively weak measure-expansive if and only if f is expanding.

• Honam Mathematical Journal
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• v.8 no.1
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• pp.51-74
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• 1986

The thery of measure is significant in that we extend from it to the theory of integration. AS specific metric outer measures we can take Hausdorff outer measure and Lebesgue-Stieltjes outer measure connecting measure with monotone functions.([12]) The purpose of this paper is to find some properties of Lebesgue-Stieltjes measure by extending it from $R^1$ to $R^n(n{\geq}1)$ $({\S}3)$ and differentiation of the integral defined by Borel measure $({\S}4)$. If in detail, as follows. We proved that if $_n{\lambda}_{f}^{\ast}$ is Lebesgue-Stieltjes outer measure defined on a finite monotone increasing function $f:R{\rightarrow}R$ with the right continuity, then $$_n{\lambda}_{f}^{\ast}(I)=\prod_{j=1}^{n}(f(b_j)-f(a_j))$$, where $I={(x_1,...,x_n){\mid}a_j$<$x_j{\leq}b_j,\;j=1,...,n}$. (Theorem 3.6). We've reached the conclusion of an extension of Lebesgue Differentiation Theorem in the course of proving that the class of continuous function on $R^n$ with compact support is dense in $L^p(d{\mu})$ ($1{\leq$}p<$\infty$) (Proposition 2.4). That is, if f is locally $\mu$-integrable on $R^n$, then $\lim_{h\to\0}\left(\frac{1}{{\mu}(Q_x(h))}\right)\int_{Qx(h)}f\;d{\mu}=f(x)\;a.e.(\mu)$.

• Fire Science and Engineering
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• v.10 no.3
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• pp.19-28
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• 1996

This paper present the way the fire detector determines whether a fire has broken out or not using the fuzzy measure. This method is based on Dempster's combination rule using the belief measure. The detector indicate a 'Fire'(F) or 'Nonfire'(N) when it determines whether a fire has broken out or not. To determine this, the fuzzy rule is applied in the setting value for the heat and smoke detector which is used. As a result, It is proved that the final decision can be determined more exactly whether a fire has broken out or not in proportion to the frequency of the fuzzy measure and the value of Bel (F).

• The Journal of the Korea Contents Association
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• v.2 no.2
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• pp.91-97
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• 2002

Let (X, F, g) be a fuzzy measure space. Then for any h$\in$ $L^{0}$ (X) , a$\in$[0 , 1] , and $A\in$F ∫$_{A}$aㆍh($\chi$)┬g＝aㆍ∫$_{A}$h($\chi$)┬g with the t-seminorm ┬(x, y)= xy. And we prove that the Seminormed fuzzy integral has some linearity properties only for ｛0,1｝-classes of fuzzy measure as follow, For any f, h$\in$ $L^{0}$ ($\chi$), any a, b$\in$R+: af＋bh$\in$ $L^{0}$ ($\chi$)⇒ ∫$_{A}$(af＋bh)┬g＝a∫$_{A}$f┬g＋b∫$_{A}$h┬g; if and only if g is a probability measure fulfilling g(A) $\in$｛0, 1｝ for all $A\in$F.n$F.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.935-955
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• 2020

In this paper we present a measurable version of the Smale's spectral decomposition theorem for homeomorphisms on compact metric spaces. More precisely, we prove that if a homeomorphism f on a compact metric space X is invariantly measure expanding on its chain recurrent set CR(f) and has the eventually shadowing property on CR(f), then f has the spectral decomposition. Moreover we show that f is invariantly measure expanding on X if and only if its restriction on CR(f) is invariantly measure expanding. Using this, we characterize the measure expanding diffeomorphisms on compact smooth manifolds via the notion of Ω-stability.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1131-1142
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• 2018

A notion of measure expansivity for homeomorphisms was introduced by Morales recently as a generalization of expansivity, and he obtained many interesting dynamic results of measure expansive homeomorphisms in [8]. In this paper, we introduce a concept of weak measure expansivity for homeomorphisms which is really weaker than that of measure expansivity, and show that a diffeomorphism f on a compact smooth manifold is $C^1$-stably weak measure expansive if and only if it is ${\Omega}$-stable. Moreover we show that $C^1$-generically, if f is weak measure expansive, then f satisfies both Axiom A and the no cycle condition.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.3
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• pp.579-586
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• 2009

Let $\varphi$ be a complete probability measure on $\mathbb{R}$, let $m_{\varphi}$ be the analogue of Wiener measure over paths on [0, T] and let f(t) be continuously differentiable on [0, T]. In this note, we give the analogue of Wiener measure $m_{\varphi}$ of {x in C[0, T]$\mid$x(0) < f(0) and $x(s_0){\geq}f(s_{0})$ for some $s_{0}$ in [0, T]} by use of integral equation techniques. This result is a generalization of Park and Paranjape's 1974 result[1].

• Journal of the Chungcheong Mathematical Society
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• v.36 no.3
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• pp.171-180
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• 2023

Investigating local dynamics requires precise control to effectively manage the subtle differences that distinguish it from global dynamics. This paper aims to study the localized perspective of the recently proposed continuum-wise expansive measures [13]. Let f : M → M be a diffeomorphism on a closed smooth manifold M and let p be a hyperbolic periodic point of f. We prove that if the homoclinic class H_f (p) of f associated to p is C¹-robustly measure continuum-wise expansive then it is hyperbolic.

• Journal of the Korean Institute of Intelligent Systems
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• v.12 no.6
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• pp.583-588
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• 2002

Recently, Szmidt and Kacprzyk[Fuzzy Sets and Systems 118(2001) 467-477] proposed a non-probabilistic-type entropy measure for intuitionistic fuzzy sets. Tt is a result of a geometric interpretation of intuitionistic fuzzy sets and uses a ratio of distances between them. They showed that the proposed measure can be defined in terms of the ratio of intuitionistic fuzzy cardinalities: of $F\bigcapF^c and F\bigcupF^c$, while applying the Hamming distances. In this note, while applying the Euclidean distances, it is also shown that the proposed measure can be defined in terms of the ratio of some function of intuitionistic fuzzy cardinalities: of $F\bigcapF^c and F\bigcupF^c$.