• 대한수학회보
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• 제52권2호
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• pp.505-511
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• 2015

In this paper, we investigate relationships among property ($k-{\beta}$), weak property (${\beta}_k$), k-nearly uniformly convexity and property ($A_k$).

• Korean Journal of Mathematics
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• 제20권4호
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• pp.415-422
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• 2012

In this paper, we define the weak property (${\beta}_{\kappa}$) and get the following strict implications. $$(UC){\Rightarrow}w-({\beta}_1){\Rightarrow}w-({\beta}_2){\Rightarrow}\;{\cdots}\;{\Rightarrow}w-({\beta}_{\infty}){\Rightarrow}(BS)$$.

• Korean Journal of Mathematics
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• 제14권2호
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• pp.161-168
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• 2006

In this paper, we study property (${\beta}$) and B-convexity in reflexive Banach spaces. It is shown that k-uniform convexity implies B-convexity and property (${\beta}$). We also show that there is a Banach space with property (${\beta}$) which cannot be equivalently renormed to be B-convex.

• 충청수학회지
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• 제28권3호
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• pp.419-429
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• 2015

In this paper, we study properties SVEP, Bishop's property (${\beta}$), property (${\delta}$), decomposability, property $({\beta})_{\epsilon}$, subscalarity, Kato spectrum, generalized Kato decomposition, finite ascent for bounded linear operators in Banach spaces.

• Journal of applied mathematics & informatics
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• 제29권1_2호
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• pp.459-468
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• 2011

Let T ${\in}$ L(X), S ${\in}$ L(Y), A ${\in}$ L(X, Y) and B ${\in}$ L(Y, X) such that SA = AT, TB = BS, AB = S and BA = T. Then S and T shares the same local spectral properties SVEP, Bishop's property (${\beta}$), property $({\beta})_{\epsilon}$, property (${\delta}$) and and subscalarity. Moreover, the operators ${\lambda}I$ - T and ${\lambda}I$ - S have many basic operator properties in common.

• 폴리머
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• 제25권5호
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• pp.635-641
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• 2001

포접 기능을 가져 악취의 분리제거에 사용될 수 있을 것으로 보이는 $\beta$-시클로덱스트린 및 $\beta$-시클로덱스트린에 벤조산을 포접시킨 물질($\beta$-시클로덱스트린/벤조산 포접체)에 시아누르산 염화물을 반응시켜 클로로트리아진 유도체를 제조하고, 이를 면섬유와 반응시켜 새로운 소취 기능성 섬유를 제조하였다. 포접체의 형성은 적외선 분광분석 및 가시-자외선 분광분석기를 사용하여 확인하였으며, 반응기의 도입은 EDX로 확인하였다. 소취성능은 제조한 소취기능성 섬유를 컬럼에 채우고, 암모니아 기체를 흘려 보냈을 때 빠져나온 암모니아를 물에 용해시킨 용액의 농도를 0.1N 염산으로 적정하여 평가하였다. 섬유에 부착된 $\beta$-시클로덱스트린 단위가 증가할수록 소취성이 증가하였으며, $\beta$-시클로덱스트린에 벤조산을 포접한 경우에 소취성이 보다 증가하였다. 이는 포접체에 존재하는 벤조산과 암모니아의 결합때문이라고 생각되었다.

• Korean Journal of Mathematics
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• 제22권3호
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• pp.463-469
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• 2014

In this paper, we investigate relationship between superreflexivity and weak property (${\beta}_k$). Indeed, we get the following diagram.

• 호남수학학술지
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• 제31권2호
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• pp.137-145
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• 2009

Let T ${\in}$ $\mathcal{L}$(X), S ${\in}$ $\mathcal{L}$(Y ), A ${\in}$ $\mathcal{L}$(X, Y ) and B ${\in}$ $\mathcal{L}$(Y,X) such that SA = AT, TB = BS, AB = S and BA = T. Then S and T shares that same local spectral properties SVEP, property (${\beta}$), property $({\beta})_{\epsilon}$, property (${\delta}$) and decomposability. From these common local spectral properties, we give some results related with Aluthge transforms and subscalar operators.

• Journal of applied mathematics & informatics
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• 제24권1_2호
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• pp.437-448
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• 2007

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let $T_1,\;T_2\;and\;T_3\;:\;K{\rightarrow}E$ be nonexpansive mappings with nonempty common fixed points set. Let $\{\alpha_n\},\;\{\beta_n\},\;\{\gamma_n\},\;\{\alpha'_n\},\;\{\beta'_n\},\;\{\gamma'_n\},\;\{\alpha'_n\},\;\{\beta'_n\}\;and\;\{\gamma'_n\}$ be real sequences in [0, 1] such that ${\alpha}_n+{\beta}_n+{\gamma}_n={\alpha}'_n+{\beta'_n+\gamma}'_n={\alpha}'_n+{\beta}'_n+{\gamma}'_n=1$, starting from arbitrary $x_1{\in}K$, define the sequence $\{x_n\}$ by $$\{zn=P({\alpha}'_nT_1x_n+{\beta}'_nx_n+{\gamma}'_nw_n)\;yn=P({\alpha}'_nT_2z_n+{\beta}'_nx_n+{\gamma}'_nv_n)\;x_{n+1}=P({\alpha}_nT_3y_n+{\beta}_nx_n+{\gamma}_nu_n)$$ with the restrictions $\sum^\infty_{n=1}{\gamma}_n<\infty,\;\sum^\infty_{n=1}{\gamma}'_n<\infty,\; \sum^\infty_{n=1}{\gamma}'_n<\infty$. (i) If the dual $E^*$ of E has the Kadec-Klee property, then weak convergence of a $\{x_n\}$ to some $x^*{\in}F(T_1){\cap}{F}(T_2){\cap}(T_3)$ is proved; (ii) If $T_1,\;T_2\;and\;T_3$ satisfy condition(A'), then strong convergence of $\{x_n\}$ to some $x^*{\in}F(T_1){\cap}{F}(T_2){\cap}(T_3)$ is obtained.

• 대한수학회보
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• 제50권6호
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• pp.1981-1988
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• 2013

We say a positive integer n satisfies the Lehmer property if ${\phi}(n)$ divides n - 1, where ${\phi}(n)$ is the Euler's totient function. Clearly, every prime satisfies the Lehmer property. No composite integer satisfying the Lehmer property is known. In this article, we show that every composite integer of the form $D_{p,n}=np^n+1$, for a prime p and a positive integer n, or of the form ${\alpha}2^{\beta}+1$ for ${\alpha}{\leq}{\beta}$ does not satisfy the Lehmer property.