• The Pure and Applied Mathematics
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• v.17 no.4
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• pp.363-372
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• 2010

In this paper we obtain the Hyers-Ulam stability of functional equations $f(x+y)=f(x)+f(y)+In\;{\alpha}^{2xy-1}$ and $f(x+y)=f(x)+f(y)+In\;{\beta(x,y)^{-1}$ which is related to the exponential and beta functions.

• The Pure and Applied Mathematics
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• v.15 no.2
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• pp.103-110
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• 2008

In this paper, we obtain the general solution and the stability of the cubic functional equation f(2x + y, 2z + w) + f(2x - y, 2z - w) = 2f(x + y, z + w) + 2f(x - y, z - w) + 12f(x, z). The cubic form $f(x,\;y)\;=\;ax^3\;+\;bx^2y\;+\;cxy^2\;+\;dy^3$ is a solution of the above functional equation.

• Honam Mathematical Journal
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• v.38 no.2
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• pp.231-241
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• 2016

In this paper we will define the tight integral closure of a finite set of ideals of a ring relative to a module and we will study some related results.

• Honam Mathematical Journal
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• v.30 no.2
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• pp.363-368
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• 2008

In this paper, we find some properties of multiplication modules and give an alternative proof of a Theorem of P.F. Smith.

• Honam Mathematical Journal
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• v.39 no.4
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• pp.475-483
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• 2017

In this paper we consider the tight closure of an ideal relative to a module whose its zero submodule has a primary decomposition.

• The Pure and Applied Mathematics
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• v.17 no.3
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• pp.231-247
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• 2010

In this paper, we investigate the generalized Hyers-Ulam stability of a bi-Jensen functional equation $4f(\frac{x\;+\;y}{2},\;\frac{z\;+\;w}{2})$ = f(x, z) + f(x, w) + f(y, z) + f(y, w). Also, we establish improved results for the stability of a bi-Jensen equation on the punctured domain.

• The Pure and Applied Mathematics
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• v.6 no.2
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• pp.103-111
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• 1999

We find some conditions under which G(f)-sequence of a CW-pair (X, A) is exact. And we also introduce a G(f)-sequence for a CW-triple (X, A, B) and examine when the sequence is exact.

• Journal of the Korean Mathematical Society
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• v.54 no.1
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• pp.59-68
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• 2017

Let $F=R^{(n)}$ be a free R-module of finite rank $n{\geq}2$. In this paper, we characterize the prime submodules of F with at most n generators when R is a $Pr{\ddot{u}}fer$ domain. We also introduce the notion of prime matrix and we show that when R is a valuation domain, every finitely generated prime submodule of F with at least n generators is the row space of a prime matrix.

• Journal of the Korean Society of Fisheries and Ocean Technology
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• v.32 no.2
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• pp.157-163
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• 1996

To ceramic film, $SiO_2$ and $Al_2O_3$, coated on pure Fe and stainless steel(SUS41O) by RF magnetron sputtering, the adherence between mm and substarte was studied. The adherence index (${\chi}$) was determined by the measure of micro hardness test. Also, the corrosion resistance on oxide coatings was studied using electrochemical measurement. The main results obtained are as the following: 1) In the micro-hardness test, with $1{\mu}m$ thickness mm, it has only one the value of ${\chi}$. Above $2{\mu}$thickness fIlm, however, get another value of ${\chi}$as the cracks in fIlm. 2) The oxide fIlm adhere well on the mild materials such as pure steel than high intensity materials like stainless. 3) Alumina($Al_2O_3$) coated materials have better corrosion resistance than silica($SiO_2$)coated materials

• The Pure and Applied Mathematics
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• v.23 no.3
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• pp.247-263
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• 2016

Let $M_{1}f(x,y):=\frac{3}{4}f(x+y)-\frac{1}{4}f(-x-y)+\frac{1}{4}f(x-y)+\frac{1}{4}f(y-x)-f(x)-f(y)$, $M_{2}f(x,y):=2f(\frac{x+y}{2})+f(\frac{x-y}{2})+f(\frac{y-x}{2})-f(x)-f(y)$. Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) $N(M_{1}f(x,y),t){\geq}N({\rho}M_{2}f(x,y),t)$ where ρ is a fixed real number with |ρ| < 1, and (0.2) $N(M_{2}f(x,y),t){\geq}N({\rho}M_{1}f(x,y),t)$ where ρ is a fixed real number with |ρ| < $\frac{1}{2}$.