• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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• pp.735-743
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• 2011

Let f be a diffeomorphism of a closed n-dimensional smooth manifold. In this paper, we introduce the notion of $C^1$-stably periodic shadowing property for a closed f-invariant set, and prove that for a transitive set ${\Lambda}$, if f has the $C^1$-stably periodic shadowing property on ${\Lambda}$, then ${\Lambda}$ admits a dominated splitting.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.2
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• pp.297-307
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• 2014

The present paper gives the notion of a derivation on a CI-algebra X and investigates related properties. We define a set $Fix_d(X)$ by $Fix_d(X)=\{x{\in}X{\mid}d(x)=x\}$, where d is a derivation on a CI-algebra X. We show that $Fix_d(X)$ is a subalgebra of X. Also, we prove some one-to-one and onto derivation theorems. Moreover, we study a regular derivation on a CI-algebra and an isotone derivation on a transitive CI-algebra.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.967-979
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• 2019

Chaotic dynamical systems, preferably on a Cantor-like space with some arithmetic operations are considered as good pseudo-random number generators. There are many definitions of chaos, of which Devaney-chaos and pos itive topological entropy seem to be the strongest. Let $A=\{0,1,{\dots},p-1\}$. We define a continuous map on $A^{\mathbb{Z}}$ using addition with a carry, in combination with the shift map. We show that this map gives rise to a dynamical system with positive entropy, which is also Devaney-chaotic: i.e., it is transitive, sensitive and has a dense set of periodic points.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.247-262
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• 2024

In this paper we introduce and study the notions of topological sensitivity and its stronger forms on semiflows and on product semiflows. We give a relationship between multi-topological sensitivity and thick topological sensitivity on semiflows. We prove that for a Urysohn space X, a syndetically transitive semiflow (T, X, 𝜋) having a point of proper compact orbit is syndetic topologically sensitive. Moreover, it is proved that for a T₃ space X, a transitive, nonminimal semiflow (T, X, 𝜋) having a dense set of almost periodic points is syndetic topologically sensitive. Also, wherever necessary examples/counterexamples are given.

• Communications of the Korean Mathematical Society
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• v.9 no.3
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• pp.687-700
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• 1994

Throughout this paper we will let H denote the complete Heyting algebra ($H, \vee, \wedge, *$) with order reversing involution *. 0 and 1 denote the supermum and the infimum of $\emptyset$, respectively. Given any set X, any element of $H^X$ is called H-fuzzy set (or, simply f.set) in X and will be denoted by small Greek letters, such as $\mu, \nu, \rho, \sigma$. $H^X$ inherits a structure of H with order reversing involution in natural way, by definding $\vee, \wedge, *$ pointwise (sam notations of H are usual). If $f$ is a map from a set X to a set Y and $\mu \in H^Y$, then $f^{-1}(\mu)$ is the f.set in X defined by f^{-1}(\mu)(x) = \mu(f(x))$. Also for $\sigma \in H^X, f(\sigma)$ is the f.set in Y defined by $f(\sigma)(y) = sup{\sigma(x) : f(x) = y}$ ([4]). A preorder R on a set X is reflexive and transitive relation on X, the pair (X,R) is called preordered set. A map $f$ from a preordered set (X, R) to another one (Y,T) is said to be preorder preserving (inverting) if for $x,y \in X, xRy$ implies $f(x)T f(y) (resp. f(y)Tf(x))$. For the terminology and notation, we refer to [10, 11, 13] for category theory and [7] for H-fuzzy semitopogenous spaces.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.387-400
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• 2014

We prove that the set of k-type nonwandering points of a Z2-action T can be decomposed into a disjoint union of closed and T-invariant sets $B_1,{\ldots},B_l$ such that $T|B_i$ is topologically k-type transitive for each $i=1,2,{\ldots},l$, if T is expansive and has the shadowing property.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.951-965
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• 2019

Let $f:X{\rightarrow}X$ be a continuous map on a compact metric space (X, d) and for an arbitrary $x{\in}X$, $${\mathcal{SC}}_d(x,f):=\{y{\mid}x{\text{ can be strong }}d-{\text{chain to }}y\}$$. We give an example to show that ${\mathcal{SC}}_d(x,f)$ is dependent on the metric d on X but it is a closed and f-invariant set. We prove that if ${\mathcal{SC}}_d(x,f){\supseteq}{\Omega}(f)$ or f has the asymptotic-average shadowing property, then ${\mathcal{SC}}_d(x,f)=X$. Also, we show that if f has the shadowing property, then ${\lim}\;{\sup}_{n{\in}{\mathbb{N}}}\{f^n\}={\mathcal{SC}}_d(f)$ where ${\mathcal{SC}}_d(f)=\{(x,y){\mid}y{\in}{\mathcal{SC}}_d(x,f)\}$. For each $n{\in}{\mathbb{N}}$, we give an example in which ${\mathcal{SCR}}_d(f^n){\neq}{\mathcal{SCR}}_d(f)$. In spite of it, we prove that if $f^{-1}:(X,d){\rightarrow}(X,d)$ is an equicontinuous map, then ${\mathcal{SCR}}_d(f^n)={\mathcal{SCR}}_d(f)$ for all $n{\in}{\mathbb{N}}$.

• Bulletin of the Korean Mathematical Society
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• v.29 no.1
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• pp.117-123
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• 1992

The dimension problem has been one of central themes in the theory of ordered sets. In this paper we focus on amalgamated ordered sets. Although some results can be obviously applied to infinite cases, we assume throughout that all ordered set are finite. If A and B are ordered sets whose orders agree on A.cap.B, then the amalgam of A and B is defined to the the set A.cup.B in which the order is the transitive closure of the union of the two orders, i.e., the smallest order containing the two orders, and is denoted by A .or. B .leq. dim A + dim B for any ordered sets A and B. But it is quite surprising that the dimension of the amalgam of certain 2-dimensional ordered sets can be arbitrarily large.

• Communications of the Korean Mathematical Society
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• v.19 no.2
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• pp.355-373
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• 2004

An extended directed triple system of order v, denoted by EDTS(v), is a pair (V, (equation omitted)) where V is a v-set and (equation omitted) is a set of transitive triples of elements of V such that every ordered pair of elements of V is contained in exactly one member of (equation omitted). We obtain a necessary and sufficient condition for the existence of cyclic EDTS(v)s, and when k=1 or 2, we also obtain a necessary and sufficient condition for the existence of k-rotational EDTS(v)s.

• Journal of the Korean Institute of Intelligent Systems
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• v.11 no.2
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• pp.141-147
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• 2001

소프트웨어의 측정값에 근거하여 소프트웨어 품질에 관한 의사결정을 할 때, 동치관계의 요구조건인 추이적(transitive) 특성이 항상 만족되는 것은 아니다. 순환수(cyclomatic number)가 거의 비슷한 프로그램에서, 하나의 \"구조적인\" 프로그램 범주에 속하고 또 다른 하나는 \"비구조적인\" 프로그램 범주에 속한다고 명확히 분류할 수 있는가하는 점이다. 따라서, 본 연구에서는 동치관계보다는 허용적 관계를 만족하는 허용적 러프집합에 근거한 소프트웨어 분류기준을 제시하고자 한다. 분류기준을 생성하기 위한 실험 데이터 집합을 수집하고, 집합 내의 각 원소에 관한 허용적 클래스들을 생성한 후, 각 허용적 클래스들의 중심값을 클러스터링하여 분류기준을 생성한다. 생성된 분류기준을 또 다른 실험 집합에 적용하여 비교 분석한 결과 생성된 분류기준이 타당함을 보여준다.생성된 분류기준이 타당함을 보여준다.