• Journal of the Chungcheong Mathematical Society
• /
• v.32 no.4
• /
• pp.509-524
• /
• 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.

• Journal of the Chungcheong Mathematical Society
• /
• v.28 no.4
• /
• pp.647-651
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.25 no.2
• /
• pp.257-262
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.21 no.2
• /
• pp.355-361
• /
• 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.

• Journal of the Chungcheong Mathematical Society
• /
• v.35 no.2
• /
• pp.171-176
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.35 no.4
• /
• pp.1309-1318
• /
• 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
• /
• v.55 no.3
• /
• pp.1084-1096
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.34 no.3
• /
• pp.951-965
• /
• 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
• /
• v.27 no.4
• /
• pp.709-730
• /
• 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.

• Journal of the Society of Disaster Information
• /
• v.9 no.4
• /
• pp.439-448
• /
• 2013

The Amendment of Misdemeanor Punishment Act which has been enforced since March 22, 2013 contains Section 3. 1. 41 (titled "Continuous Harassment") which provides that stalking is one's act of "attempt to continuously approach other person or persons against the latter's express intent and thereby demand interviews or associations to them, or commit behaviors such as monitoring, following (shadowing) and ambush over and over again." In South Korea, any wrongdoer who does such act of stalking shall be fined 80,000 Korean won under the Enforcement Decree of Misdemeanor Punishment Act. However, in view of the seriousness of ever-diversified and skillful stalking crimes, we have a question of whether the application of current Misdemeanor Punishment Act would be effective in reality.In order to get over current limitations of Misdemeanor Punishment Act as a part of direct regulatory control over stalking crimes and more properly cope with stalking crimes, this study give the following suggestions.