• 충청수학회지
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• 제32권4호
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• pp.509-524
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• 2019

Let Act(G, X) be the set of all continuous actions of a finitely generated group G on a compact metric space X. In this paper, we study the concepts of topologically stable points and continuous shadowable points of a group action T ∈ Act(G, X). We show that if T is expansive then the set of continuous shadowable points is contained in the set of topologically stable points.

• 대한수학회보
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• 제61권4호
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• pp.1137-1159
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• 2024

In the paper, we prove that if a continuous map of a compact uniform space is equicontinuous and pointwise topologically stable, then it is persistent. We also show that if a sequence of uniformly expansive continuous maps of a compact uniform space has a uniform limit and the uniform shadowing property, then the limit is topologically stable. In addition, we introduce the concepts of shadowable points and topologically stable points for a continuous map of a compact topological space and obtain that every shadowable point of an expansive continuous map of a compact topological space is topologically stable.

• 충청수학회지
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• 제28권4호
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• pp.647-651
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• 2015

Let (X, d) be a compact metric space, and let $f:X{\rightarrow}X$ be a continuous map. We consider that if a positively continuum-wise expansiveness continuous map f has the positively shadowing property in the nonwandering set, then the topological entropy is positive.

• 대한수학회논문집
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• 제25권2호
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• pp.257-262
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• 2010

Let f be a $C^2$-diffeomorphism on a closed surface which satisfies the Axiom A. Then f is in the $C^2$-interior of the set of all diffeomorphisms having the inverse shadowing property with respect to the class of the continuous methods if and only if f satisfies the strong transversality condition.

• 대한수학회논문집
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• 제21권2호
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• pp.355-361
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• 2006

We prove that if a continuous surjective map f on a compact metric space X has the average shadowing property, then every point x is chain recurrent. We also show that if a homeomorphism f has more than two fixed points on $S^1$, then f does not satisfy the average shadowing property. Moreover, we construct a homeomorphism on a circle which satisfies the shadowing property but not the average shadowing property. This shows that the converse of the theorem 1.1 in [6] is not true.

• 충청수학회지
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• 제35권2호
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• pp.171-176
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• 2022

In this article, we investigate the transitivity and chain transitivity on multi-valued dynamical systems. For compact-valued continuous dynamics, we prove that the notion of transitivity is expressed by the notions of the shadowing property and chain transitivity under locally maximal condition.

• 대한수학회논문집
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• 제35권4호
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• pp.1309-1318
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• 2020

In this paper we extend the concept of topological stability from continuous maps to the corresponding induced maps and prove that a continuous map is topologically stable if and only if its induced map also is topologically stable.

• Nuclear Engineering and Technology
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• 제55권3호
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• pp.1084-1096
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• 2023

On-the-fly(OTF) generation of Spectral Superhomogenization(SSPH) factors is analyzed in the multigroup(MG) whole core calculation employing pin-wise continuous energy(CE) slowing-down solutions. The motivation for the work is to avoid the huge computing time required for the generation of a parametrized SSPH factor library(PSSL) which is used to resolve the angular dependency of MG resonance cross sections, and also to exploit the advantage of flexible choice of a MG structure by using CE slowing-down solutions. Two pin-wise CE slowing-down methods, the equivalent Dancoff cell method and the shadowing effect correction method, are evaluated with the OTF SSPH method. The effectiveness of the OTF SSPH method is examined for various simplified and realistic core problems with various MG structures. It is demonstrated that the computing time overhead of this method is negligible whereas the solution accuracy is considerably enhanced.

• 대한수학회논문집
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• 제34권3호
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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 $n{\in}{\mathbb{N}}$.

• Nonlinear Functional Analysis and Applications
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• 제27권4호
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• pp.709-730
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• 2022

We consider the nonlinear matrix equation (NMEs) of the form 𝓤 = 𝓠 + Σ^k_i=1 𝓐^*_iℏ(𝓤)𝓐_i, where 𝓠 is n × n Hermitian positive definite matrices (HPDS), 𝓐₁, 𝓐₂, . . . , 𝓐_m are n × n matrices, and ~ is a nonlinear self-mappings of the set of all Hermitian matrices which are continuous in the trace norm. We discuss a sufficient condition ensuring the existence of a unique positive definite solution of a given NME and demonstrate this sufficient condition for a NME 𝓤 = 𝓠 + 𝓐^*₁(𝓤²/900)𝓐₁ + 𝓐^*₂(𝓤²/900)𝓐₂ + 𝓐^*₃(𝓤²/900)𝓐₃. In order to do this, we define 𝓕𝓖_w-contractive conditions and derive fixed points results based on aforesaid contractive condition for a mapping in extended Branciari b-metric distance followed by two suitable examples. In addition, we introduce weak well-posed property, weak limit shadowing property and generalized Ulam-Hyers stability in the underlying space and related results.