• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.529-572
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• 2012

A basis of a subspace of $S_4({\Gamma}_0(79))$ is given and the formulas for the number of representations of positive integers by some direct sums of the quadratic forms $x^2_1+x_1x_2+20x^2_2$, $4x^2_1{\pm}x_1x_2+5x^2_2$, $2x^2_1{\pm}x_1x_2+10x^2_2$ are determined.

• Korean Journal of Mathematics
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• v.14 no.2
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• pp.227-232
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• 2006

We study some dynamical properties in the context of bitransformation groups, and show that if (H,X,T) is a bitransformation group such that (H,X) is almost periodic and (X/H,T) is pointwise almost periodic $T_2$ and $x{\in}X$, then $E_x=\{q{\in}E(H,X){\mid}qx{\in}{\overline{xT}\}$ is a compact $T_2$ topological group and $E_{qx}=E_x(q{\in}E(H,X))$ when H is abelian, where E(H,X) is the enveloping semigroup of the transformation group (H,X).

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.125-134
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• 1993

Let X be a compact Hausdorff space, and let C(X) (resp. $C_{R}$(X)) be the complex (resp. real) Banach algebra of all continuous complex-valued(resp. real-valued) functions on X with the pointwise operations and the supremum norm x. A Banach function algebra on X is a Banach algebra lying in C(X) which separates the points of X and contains the constants. A Banach function algebra on X equipped with the supremum norm is called a uniform algebra on X, that is, a uniformly closed subalgebra of C(X) which separates the points of X and contains the constants.s.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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• pp.427-433
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• 2013

Observing that every Tychonoff space X has an extension $kX$ which is a weakly Lindel$\ddot{o}$f space and the minimal quasi-F cover $QF(kX)$ of $kX$ is a weakly Lindel$\ddot{o}$f, we show that ${\Phi}_{kX}:QF(kX){\rightarrow}kX$ is a $z^{\sharp}$-irreducible map and that $QF({\beta}X)=QF(kX)$. Using these, we prove that $QF(kX)=kQF(X)$ if and only if ${\Phi}^k_X:kQF(X){\rightarrow}kX$ is an onto map and ${\beta}QF(X)=(QF{\beta}X)$.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.2
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• pp.139-146
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• 2007

In this paper, we present characterizations of the power function distribution by the independence of record values. We establish that $X{\in}$ POW(1, ${\nu}$) for ${\nu}$ > 0, if and only if $\frac{X_{L(n)}}{X_{L(n)}-X_{L(n+1)}}$ and $X_{L(n)}$ are independent for $n{\geq}1$. And we prove that $X{\in}$ POW(1, ${\nu}$) for ${\nu}$ > 0; if and only if $\frac{X_{L(n+1)}}{X_{L(n)}-X_{L(n+1)}}$ and $X_{L(n)}$ are independent for $n{\geq}1$. Also we characterize that $X{\in}$ POW(1, ${\nu}$) for ${\nu}$ > 0, if and only if $\frac{X_{L(n)}+X_{L(n+1)}}{X_{L(n)}-X_{L(n+1)}}$ and $X_{L(n)}$ are independent for $n{\geq}1$.

• Honam Mathematical Journal
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• v.34 no.1
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• pp.45-54
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• 2012

In this paper, we prove stabilities of multiplicative functional equations in three variables such as $r(\frac{x+y+z}{3})-r(x+y+z)$=$\frac{2r(\frac{x+y}{2})r(\frac{y+z}{2})r(\frac{z+x}{2})}{r(\frac{x+y}{2})r(\frac{y+z}{2})+r(\frac{y+z}{2})r(\frac{z+x}{2})+r(\frac{z+x}{2})r(\frac{x+y}{2})}$ and $r(\frac{x+y+z}{3})+r(x+y+z)$=$\frac{4r(\frac{x+y}{2})r(\frac{y+z}{2})r(\frac{z+x}{2})}{r(\frac{x+y}{2})r(\frac{y+z}{2})+r(\frac{y+z}{2})r(\frac{z+x}{2})+r(\frac{z+x}{2})r(\frac{x+y}{2})}$.

• Communications of the Korean Mathematical Society
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• v.10 no.4
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• pp.943-947
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• 1995

Assume that X is locally compact and Hausdorff. Then, we show that $\alpha X = sup {X \cup_f S(f)$\mid$f \in S^{\alpha}}$ for any compactification $\alpha X$ of X if and only if for any 2-point compatification $\gamma X$ of X with $\gamma X - X = {-\infty, +\infty}$, there exists a clopen subset A of \gamma X$ such that $-\infty \in A$ and $+\infty \notin A$. As a corollary, we obtain that if X is connected and locally connected, then $\alpha X = sup {X \cup_f S(f)$\mid$f \in S^{\alpha}}$ for any compactification $\alpha X$ of X if and only if X is 1-complemented.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1573-1590
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• 2020

In this paper, we classify all solutions bounded from below to uniformly elliptic equations of second order in the form of Lu(x) = a_ij(x)D_iju(x) + b_i(x)D_iu(x) + c(x)u(x) = f(x) or Lu(x) = D_i(a_ij(x)D_ju(x)) + b_i(x)D_iu(x) + c(x)u(x) = f(x) in unbounded cylinders. After establishing that the Aleksandrov maximum principle and boundary Harnack inequality hold for bounded solutions, we show that all solutions bounded from below are linear comb_inations of solutions, which are sums of two special solutions that exponential growth at one end and exponential decay at the another end, and a bounded solution that corresponds to the inhomogeneous term f of the equation.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.1-12
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• 2018

Let R be a 3!-torsion free noncommutative semiprime ring, and suppose there exists a Jordan derivation $D:R{\rightarrow}R$ such that $D^2(x)[D(x),x]=0$ or $[D(x),x]D^2(x)=0$ for all $x{\in}R$. In this case we have $f(x)^5=0$ for all $x{\in}R$. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A{\rightarrow}A$ such that $D^2(x)[D(x),x]{\in}rad(A)$ or $[D(x),x]D^2(x){\in}rad(A)$ for all $x{\in}A$. In this case, we show that $D(A){\subseteq}rad(A)$.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.2
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• pp.279-285
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• 2008

This paper presents some characterizations of the Weibull distribution by the independence of record values. We prove that $X{\sim}Weibull(1,{\alpha})$, ${\alpha}>0$ if and only if $\frac{X_{U(n+1)}}{X_{U(n+1)}-X_{U(n)}}$ and $X_{U(n+1)}$ for $n{\geq}1$ are independent. We show that $X{\sim}Weibull(1,{\alpha})$, ${\alpha}>0$ if and only if $\frac{X_{U(n+1)}}{X_{U(n+1)}-X_{U(n)}}$ and $X_{U(n+1)}$ for $n{\geq}1$ are independent. And we establish that $X{\sim}Weibull(1,{\alpha})$, ${\alpha}>0$ if and only if $\frac{X_{U(n+1)}+X_{U(n)}}{X_{U(n+1)}-X_{U(n)}}$ and $X_{U(n+1)}$ for $n{\geq}1$ are independent.