• 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 $n{\in}{\mathbb{N}}$.

• Asian Pacific Journal of Cancer Prevention
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• v.15 no.6
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• pp.2559-2563
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• 2014

Purpose: We investigated the expression of epithelial $Ca^{2+}$ channel transient receptor potential vanilloid (TRPV) 5 and 6 in non-small-cell lung cancer (NSCLC) and assessed their prognostic role in patients after surgical resection. Materials and Methods: From January 2008 to January 2009, 145 patients who had undergone surgical resection of NSCLCs were enrolled in the study. Patient clinical characteristics were retrospectively reviewed. Fresh tumor samples as well as peritumor tissues were analyzed for TRPV5/6 expression using immune-histochemistry (IHC) and quantitative reverse transcriptase-polymerase chain reaction (RT-PCR). Patients were grouped based on their TRPV5 and TRPV6 levels in the tumor tissues, followed up after surgery, and statistically analyzed to examine the prognostic roles of TRPV5 and TRPV6 on patients' survival after surgical resection of NSCLCs. Results: Using IHC, among the 145 patients who had undergone surgical resection of NSCLCs, strong protein expression (grade${\geq}2$) of TRPV5 and TRPV6 was observed in a lower percentage of primary tumor tissues than in non-tumor tissues of same patients. Similar findigns were obtained with the RT-PCR test for mRNA levels. Decreased overall mRNA levels of TRPV5 and TRPV6 were associated with a worse overall survival rate (p=0.004 and p=0.003 respectively) and shorter recurrence-free survival (p<0.001 and p<0.001 respectively). The combining effect of TRPV5 and TRPV6 on survival was further investigated using multivariate analysis. The results showed that a combination of low expression of TRPV5 and TRPV6 could be an independent predictor of poor recurrence-free survival (p=0.002). Conclusions: Decreased expression of TRPV5/6 in tumor tissues was observed in NSCLC patients and was associated with shorter median survival time after surgical resection. Combined expression of TRPV5 and TRPV6 in tumor tissues demonstrated promising prognostic value in NSCLC patients.