• Honam Mathematical Journal
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• v.1 no.1
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• pp.21-25
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• 1979

Abelian군(群)의 presheaf에 관한 직적(直積)과 직화(直和)를 Category 입장에서 정의(定義)하고 presheaf $F_{\lambda}\;({\lambda}{\epsilon}{\Lambda})$들의 두 직적(直積)(또는 直和)은 서로 동형적(同型的) 관계(關係)에 있으며, 특히 ${\phi}:X{\rightarrow}Y$가 homeomorphism이라 하고 ${\phi}_*F$를 X상(上)의 presheaf F의 direct image이라 하면 (1) $({\phi}_*F, \;{\phi}_*(f_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$가 $({\phi}_*F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}$의 직적(直積)일 때 오직 그때 한하여 $(F,\;(f_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$는 $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$의 직적(直積)이다. (2) $({\phi}_*F,\;{\phi}_*(l_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$가 $({\phi}_*F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}$의 직화(直和)일 때 오직 그때 한하여 $(F,\;(l_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$는 $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$의 직화(直和)이다. Let $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$ be an indexed set of presheaves of abelian group on topological space X. We can define the cartesian product $$\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda}$$ of $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$ by $$(\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda})(U)=\prod_{{\lambda}{\epsilon}{\Lambda}}(F_{\lambda}(U))$$ for U open in X $${\rho}_v^u:\;(\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda})(U){\rightarrow}(\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda})(V)((s_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}{\rightarrow}(_{\lambda}{\rho}_v^u(s_{\lambda}))_{{\lambda}{\epsilon}{\Lambda}})$$ for $V{\subseteq}U$ open in X where $_{\lambda}{\rho}^U_V$ is a restriction of $F_{\lambda}$, And we have natural presheaf morphisms ${\pi}_{\lambda}$ and ${\iota}_{\lambda}$ such that ${\pi}_{\lambda}(U):\;({\prod}_\;F_{\lambda})(U){\rightarrow}F_{\lambda}(U)((s_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}{\rightarrow}s_{\lambda})$ ${\iota}_{\lambda}(U):\;F_{\lambda}(U){\rightarrow}({\prod}\;F_{\lambda})(U)(s_{\lambda}{\rightarrow}(o,o,{\cdots}\;{\cdots}o,s_{\lambda},o,{\cdots}\;{\cdots}o)$ for $(s_{\lambda}){\epsilon}{\prod}_{\lambda}\;F_{\lambda}(U)$ and $(s_{\lambda}){\epsilon}F_{\lambda}(U)$.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.461-468
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• 2008

We define an $\epsilon$-fuzzy congruence, which is a weakened fuzzy congruence, find the $\epsilon$-fuzzy congruence generated by the union of two $\epsilon$-fuzzy congruences on a semigroup, and characterize the $\epsilon$-fuzzy congruences generated by fuzzy relations on semigroups. We also show that the collection of all $\epsilon$-fuzzy congruences on a semigroup is a complete lattice and that the collection of $\epsilon$-fuzzy congruences under some conditions is a modular lattice.

• Korean Journal of Mathematics
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• v.14 no.1
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• pp.71-77
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• 2006

We find the ${\epsilon}$-fuzzy equivalence relation generated by the union of two ${\epsilon}$-fuzzy equivalence relations on a set, find the ${\epsilon}$-fuzzy equivalence relation generated by a fuzzy relation on a set, and find sufficient conditions for the composition ${\mu}{\circ}{\nu}$ of two ${\epsilon}$-fuzzy equivalence relations ${\mu}$ and ${\nu}$ to be the ${\epsilon}$-fuzzy equivalence relation generated by ${\mu}{\cup}{\nu}$. Also we study fuzzy partitions of ${\epsilon}$-fuzzy equivalence relations.

• Proceedings of the Korean Information Science Society Conference
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• 2005.07a
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• pp.901-903
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• 2005

본 논문에서는 평면 상에 두 볼록집합 P와 Q가 주어졌을 때, P를 강체운동 하에서 수평 이동 및 회전이동하여 Q와 겹치는 영역이 근사적으로 최대가 되는 알고리즘을 제시한다. 임의의 양의 상수 $\epsilon$이 주어졌을 때, 본 알고리즘은 가장 많이 겹치는 넓이의 $1-\epsilon$ 배를 보장하는 P의 강체운동을 $O((1/\epsilon)$ 번의 기하 질의 와 $O((1/{\epsilon}^2)log(1/$\epsilon)) 시간 내에 구할 수 있다. 특히 P와 Q가 볼록다각형 일 때, $O((1/\epsilon)log\;n+(1/{\epsilon}^2)log(1/\epsilon))$ 시간에 구한다. 만약 수평 이동만 사용할 경우는 $O((1/\epsilon)log\;n+(1/\epsilon)log(1/\epsilon))$ 시간에 구할 수 있다.

• Korean Journal of Mathematics
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• v.26 no.4
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• pp.757-775
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• 2018

For any non-negative real number ${\epsilon}_0$, we shall introduce a concept of the ${\epsilon}_0$-dense subset of $R^m$. Applying this concept, for any sequence {${\epsilon}_n$} of positive real numbers, we also introduce the concept of the {${\epsilon}_n$}-attainable sequence and of the points of {${\epsilon}_n$}-attainable ace in the open subset of $R^m$. We also study the characteristics of those sequences and of the points of {${\epsilon}_n$}-dense ace. And we research the conditions that an {${\epsilon}_n$}-attainable sequence has no {${\epsilon}_n$}-attainable ace. We hope to reconsider the social consideration on the ace in social life by referring to these concepts about the aces.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.1
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• pp.69-86
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• 2019

In this paper, we introduce a concept of the ${\epsilon}_0$-limits of vector and multiple valued sequences in $R^m$. Using this concept, we study about the concept of the ${\epsilon}_0$-dense subset and of the points of ${\epsilon}_0$-dense ace in the open subset of $R^m$. We also investigate the properties and the characteristics of the ${\epsilon}_0$-dense subsets and of the points of ${\epsilon}_0$-dense ace.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.835-845
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• 2017

The $GCF_{\epsilon}$ expansion is a new class of continued fractions induced by the transformation $T_{\epsilon}:(0, 1]{\rightarrow}(0, 1]$: $T_{\epsilon}(x)={\frac{-1+(k+1)x}{1+k-k{\epsilon}x}}$ for $x{\in}(1/(k+1),1/k]$. Under the algorithm $T_{\epsilon}$, every $x{\in}(0,1]$ corresponds to an increasing digits sequences $\{k_n,n{\geq}1\}$. Their basic properties, including the ergodic properties, law of large number and central limit theorem have been discussed in [4], [5] and [7]. In this paper, we study the large deviation for the $GCF_{\epsilon}$ expansion and show that: $\{{\frac{1}{n}}{\log}k_n,n{\geq}1\}$ satisfies the different large deviation principles when the parameter ${\epsilon}$ changes in [-1, 1], which generalizes a result of L. J. Zhu [9] who considered a case when ${\epsilon}(k){\equiv}0$ (i.e., Engel series).

• Journal of the Korean Mathematical Society
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• v.52 no.3
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• pp.637-647
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• 2015

For generalized continued fraction (GCF) with parameter ${\epsilon}(k)$, we consider the size of the set whose partial quotients increase rapidly, namely the set $$E_{\epsilon}({\alpha}):=\{x{\in}(0,1]:k_{n+1}(x){\geq}k_n(x)^{\alpha}\;for\;all\;n{\geq}1\}$$, where ${\alpha}$ > 1. We in [6] have obtained the Hausdorff dimension of $E_{\epsilon}({\alpha})$ when ${\epsilon}(k)$ is constant or ${\epsilon}(k){\sim}k^{\beta}$ for any ${\beta}{\geq}1$. As its supplement, now we show that: $$dim_H\;E_{\epsilon}({\alpha})=\{\frac{1}{\alpha},\;when\;-k^{\delta}{\leq}{\epsilon}(k){\leq}k\;with\;0{\leq}{\delta}<1;\\\;\frac{1}{{\alpha}+1},\;when\;-k-{\rho}<{\epsilon}(k){\leq}-k\;with\;0<{\rho}<1;\\\;\frac{1}{{\alpha}+2},\;when\;{\epsilon}(k)=-k-1+\frac{1}{k}$$. So the bigger the parameter function ${\epsilon}(k_n)$ is, the larger the size of $E_{\epsilon}({\alpha})$ becomes.

• Korean Journal of Mathematics
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• v.27 no.3
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• pp.707-722
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• 2019

For any non-negative real number ${\epsilon}_0$, we shall introduce a concept of the ${\epsilon}_0$-Cauchy sequence in a normed linear space V and also introduce a concept of the ${\epsilon}_0$-completeness in those spaces. Finally we introduce a concept of the generalized Banach spaces with these concepts.

• Management Science and Financial Engineering
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• v.10 no.2
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• pp.103-118
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• 2004

$\epsilon$-sensitivity analysis is a kind of methods for performing sensitivity analysis for linear programming. Its main advantage is that it can be directly applied for interior-point methods with a little computation. Although $\epsilon$-sensitivity analysis was proposed several years ago, there have been no studies on its relationship with other sensitivity analysis methods. In this paper, we discuss the relationship between $\epsilon$-sensitivity analysis and sensitivity analysis using an optimal basis. First. we present a property of $\epsilon$-sensitivity analysis, from which we derive a simplified formula for finding the characteristic region of $\epsilon$-sensitivity analysis. Next, using the simplified formula, we examine the relationship between $\epsilon$-sensitivity analysis and sensitivity analysis using optimal basis when an $\epsilon$-optimal solution is sufficiently close to an optimal extreme solution. We show that under primal nondegeneracy or dual non degeneracy of an optimal extreme solution, the characteristic region of $\epsilon$-sensitivity analysis converges to that of sensitivity analysis using an optimal basis. However, for the case of both primal and dual degeneracy, we present an example in which the characteristic region of $\epsilon$-sensitivity analysis is different from that of sensitivity analysis using an optimal basis.