• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.471-479
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• 2008

Let $R\;{\in}\;C^{m{\times}m}$ and $S\;{\in}\;C^{n{\times}n}$ be nontrivial unitary involutions, i.e., $R^*\;=\;R\;=\;R^{-1}\;{\neq}\;I_m$ and $S^*\;=\;S\;=\;S^{-1}\;{\neq}\;I_m$. We say that $G\;{\in}\;C^{m{\times}n}$ is a generalized reflexive matrix if RGS = G. The set of all m ${\times}$ n generalized reflexive matrices is denoted by $GRC^{m{\times}n}$. In this paper, an efficient method for the least squares solution $X\;{\in}\;GRC^{m{\times}n}$ of the matrix equation AXB = D with arbitrary coefficient matrices $A\;{\in}\;C^{p{\times}m}$, $B\;{\in}\;C^{n{\times}q}$and the right-hand side $D\;{\in}\;C^{p{\times}q}$ is developed based on the canonical correlation decomposition(CCD) and, an explicit formula for the general solution is presented.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.473-485
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• 2010

We consider the differential equation $$(x^2\;-\;1)u_{xx}\;+\;2xyu_{xy}\;+\;(y^2\;-\;1)u_{yy}\;+\;gxu_x\;+\;gyu_y\;=\;\lambda_nu,\;(*)$$ where $\lambda_n\;=\;n(n\;+\;9\;-\;1)$. We show that the differential equation (*) has a polynomial set as solutions if $g\;{\neq}\;-1$, -3, -5, $\cdots$. Also, we construct an orthogonalizing distributional weight for g < 1 and $g\;{\neq}\;1$, 0, -1, $\cdots$ by regularizing a one-dimensional integral with a singularity on the endpoint of the interval.

• Bulletin of the Korean Mathematical Society
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• v.31 no.1
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• pp.93-97
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• 1994

Previously T.Porter [3] has given a relative Chinese Remainder Theorem under the hypothesis that given ring R has at least one .tau.-closed maximal ideal (by his notation Ma $x_{\tau}$(R).neq..phi.). In this short paper we drop his overall hypothesis that Ma $x_{\tau}$(R).neq..phi. and give the proof and some related results with this Theorem. In this paper R will always denote a commutative ring with identity element and all modules will be unitary left R-modules unless otherwise specified. Let .tau. be a given hereditarty torsion theory for left R-module category R-Mod. The class of all .tau.-torsion left R-modules, dented by J is closed under homomorphic images, submodules, direct sums and extensions. And the class of all .tau.-torsionfree left R-modules, denoted by F, is closed under taking submodules, injective hulls, direct products, and isomorphic copies ([2], Proposition 1.7 and 1.10).

• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.815-823
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• 2010

Let ${\tau}\;{\neq}\;\delta_0$ be either a power bounded radial measure with compact support on the unit disc D with $\tau(D)\;=\;1$ such that there is a $\delta$ > 0 so that ${\mid}\hat{\tau}(s){\mid}\;{\neq}\;1$ for every $s\;{\in}\;\Sigma(\delta)$ \ {0,1}, or just a radial probability measure on D. Here, we provide a decomposition of the set X = {$h\;{\in}\;L^{\infty}(D)\;{\mid}\;lim_{n{\rightarrow}{\infty}}\;h\;*\;\tau^n$ exists}. Let $\tau_1$, ..., $\tau_n$ be measures on D with above mentioned properties. Here, we prove that if $f\;{in}\;L^{\infty}(D^n)$ satisfies an invariant volume mean value property with respect to $\tau_1$, ..., $\tau_n$, then f is n-harmonic.

• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.431-447
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• 2011

We investigate the asymptotic behavior of positive solutions to the elliptic equation (0.1) ${\Delta}u+|x|^{l_1}u^p+|x|^{l_2}u^q=0$ in $\mathbb{R}^n$. We obtain a conclusion that, for n $\geq$ 3, -2 < $l_2$ < $l_1$ $\leq$ 0 and q > p > 1, any positive radial solution to (0.1) has the following properties: $lim_{r{\rightarrow}{\infty}}r^{\frac{2+l_1}{p-1}}\;u$ and $lim_{r{\rightarrow}0}r^{\frac{2+l_2}{q-1}}\;u$ always exist if $\frac{n+1_1}{n-2}$ < p < q, $p\;{\neq}\;\frac{n+2+2l_1}{n-2}$, $q\;{\neq}\;\frac{n+2+2l_2}{n-2}$. In addition, we prove that the singular positive solution of (0.1) is unique under some conditions.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.343-348
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• 1995

Let X and Y be two normed spaces and D a nonempty convex subset of X. Let $T : X \ to L(X,Y)$ be a mapping, where L(X,Y) is the space of all continuous linear mappings from X into Y. And let $C : D \to 2^Y$ be a set-valued map such that for each $x \in D$, C(x) is a convex cone in Y such that Int $C(x) \neq 0 and C(x) \neq Y$, where Int denotes the interior.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1245-1252
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• 2011

Consider the $L^2$-adjoint $s_g^{'*}$ of the linearization of the scalar curvature $s_g$. If ker $s_g^{'*}{\neq}0$ on an n-dimensional compact manifold, it is well known that the scalar curvature $s_g$ is a non-negative constant. In this paper, we study the structure of the level set ${\varphi}^{-1}$(0) and find the behavior of Ricci tensor when ker $s_g^{'*}{\neq}0$ with $s_g$ > 0. Also for a nontrivial solution (g, f) of $z=s_g^{'*}(f)$ on an n-dimensional compact manifold, we analyze the structure of the regular level set $f^{-1}$(-1). These results give a good understanding of the given manifolds.

• Journal of the Korean Mathematical Society
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• v.52 no.1
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• pp.97-111
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• 2015

Let R be a commutative ring with $1{\neq}0$. In this paper, we introduce the concept of weakly 2-absorbing primary ideal which is a generalization of weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever a, b, $c{\in}R$ and $0{\neq}abc{\in}I$, then $ab{\in}I$ or $ac{\in}\sqrt{I}$ or $bc{\in}\sqrt{I}$. A number of results concerning weakly 2-absorbing primary ideals and examples of weakly 2-absorbing primary ideals are given.

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.217-232
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• 2016

We make a study of two generalizations of regular rings, concentrating our attention on the structure of idempotents. A ring R is said to be right attaching-idempotent if for $a{\in}R$ there exists $0{\neq}b{\in}R$ such that ab is an idempotent. Next R is said to be generalized regular if for $0{\neq}a{\in}R$ there exist nonzero $b{\in}R$ such that ab is a nonzero idempotent. It is first checked that generalized regular is left-right symmetric but right attaching-idempotent is not. The generalized regularity is shown to be a Morita invariant property. More structural properties of these two concepts are also investigated.

• Kyungpook Mathematical Journal
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• v.54 no.1
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• pp.65-72
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• 2014

A ring R is called weakly semicommutative ring if for any a, $b{\in}R^*$ = R\{0} with ab = 0, there exists $n{\geq}1$ such that either an $a^n{\neq}0$ and $a^nRb=0$ or $b^n{\neq}0$ and $aRb^n=0$. In this paper, many properties of weakly semicommutative rings are introduced, some known results are extended. Especially, we show that a ring R is a strongly regular ring if and only if R is a left SF-ring and weakly semicommutative ring.