• Bulletin of the Korean Mathematical Society
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• v.37 no.3
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• pp.429-435
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• 2000

Our main goal is to show that if there exist Jordan derivations D, E and G on a noncommutative 2-torsion free prime ring R such that$(G^2(x)+E(x))D(x)=0\ or\ D(x)(G^2(x)+E(x))=0\ for\ all\ x\inR$, then we have D=o or E=0, G=0.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.907-913
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• 2010

Let D be an integrally closed domain with quotient field K, * be a star operation on D, X, Y be indeterminates over D, $N_*\;=\;\{f\;{\in}\;D[X]|\;(c_D(f))^*\;=\;D\}$ and $R\;=\;D[X]_{N_*}$. Let b be the b-operation on R, and let $*_c$ be the star operation on D defined by $I^{*_c}\;=\;(ID[X]_{N_*})^b\;{\cap}\;K$. Finally, let Kr(R, b) (resp., Kr(D, $*_c$)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X, Y). In this paper, we show that Kr(R, b) $\subseteq$ Kr(D, $*_c$) and Kr(R, b) is a kfr with respect to K(Y) and X in the notion of [2]. We also prove that Kr(R, b) = Kr(D, $*_c$) if and only if D is a $P{\ast}MD$. As a corollary, we have that if D is not a $P{\ast}MD$, then Kr(R, b) is an example of a kfr with respect to K(Y) and X but not a Kronecker function ring with respect to K(Y) and X.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.885-898
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• 2012

Let m, n, r be nonzero fixed positive integers, R a 2-torsion free prime ring, Q its right Martindale quotient ring, and L a non-central Lie ideal of R. Let D : $R{\rightarrow}R$ be a skew derivation of R and $E(x)=D(x^{m+n+r})-D(x^m)x^{n+r}-x^mD(x^n)x^r-x^{m+n}D(x^r)$. We prove that if $E(x)=0$ for all $x{\in}L$, then D is a usual derivation of R or R satisfies $s_4(x_1,{\ldots},x_4)$, the standard identity of degree 4.

• The Pure and Applied Mathematics
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• v.15 no.3
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• pp.259-296
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• 2008

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation D : A $\rightarrow$ A such that $D(x)^2$[D(x),x] $\in$ rad(A) or [D(x),x]$D(x)^2$ $\in$ rad(A) for all x $\in$ A. In this case, we have D(A) $\subseteq$ rad(A).

• The Pure and Applied Mathematics
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• v.15 no.2
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• pp.179-201
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• 2008

Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D\;:\;A{\rightarrow}A$ such that $D(x)[D(x),x]^2\;{\in}\;rad(A)$ or $[D(x), x]^2 D(x)\;{\in}\;rad(A)$ for all $x\;{\in}\ A$. In this case, we have $D(A)\;{\subseteq}\;rad(A)$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.11 no.1
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• pp.41-56
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• 2007

A 3-partite 2-digraph is an orientation of a 3-partite multi-graph that is without loops and contains at most two edges between any pair of vertices from distinct parts. Let D(X, Y, Z) be a 3-partite 2-digraph with ${\mid}X{\mid}=l,\;{\mid}Y{\mid}=m,\;{\mid}Z{\mid}=n$. For any vertex v in D(X, Y, Z), let $d^+_{\nu}\;and\;d^-_{\nu}$ denote the outdegree and indegree respectively of v. Define $p_x=2(m+n)+d^+_x-d^-_x,\;q_y=2(l+n)+d^+_y-d^-_y\;and\;r_z=2(l+m)+d^+_z-d^-_z$ as the marks (or 2-scores) of x in X, y in Y and z in Z respectively. In this paper, we characterize the marks of 3-partite 2-digraphs and give a constructive and existence criterion for sequences of non-negative integers in non-decreasing order to be the mark sequences of some 3-partite 2-digraph.

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.401-416
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• 1994

We study the existence of an intermediate solution of nonlinear elliptic boundary value problems (BVP) of the form $$ (BVP) {\Delta u = f(x,u,\Delta u), in \Omega {Bu(x) = \phi(x), on \partial\Omega, $$ where $\Omega$ is a smooth bounded domain in $R^n, n \geq 1, and \partial\Omega \in C^{2,\alpha}, (0 < \alpha < 1), \Delta$ is the Laplacian operator, $\nabla u = (D_1u, D_2u, \cdots, D_nu)$ denotes the gradient of u and $$ Bu(x) = p(x)u(x) + q(x)\frac{d\nu}{du} (x), $$ where $\frac{d\nu}{du} denotes the outward normal derivative of u on $\partial\Omega$.

• 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.

• Kyungpook Mathematical Journal
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• v.53 no.4
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• pp.565-571
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• 2013

In this paper, we investigate the commutativity of a semiprime ring R admitting a generalized derivation F with associated derivation D satisfying any one of the properties: (i) $F(x){\circ}D(y)=[x,y]$, (ii) $D(x){\circ}F(y)=F[x,y]$, (iii) $D(x){\circ}F(y)=xy$, (iv) $F(x{\circ}y)=[F(x) y]+[D(y),x]$, and (v) $F[x,y]=F(x){\circ}y-D(y){\circ}x$ for all x, y in some appropriate subsets of R.

• Journal of the Korea Computer Graphics Society
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• v.16 no.3
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• pp.11-19
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• 2010

This paper describes how to define and implement the schema for 3D virtual objects with physical units so that the objects can be compared in virtual environments based on physical properties, such as length, according to the specified units. We define physical units for virtual objects using the International System of Units and based on the X3D (Extensible 3D) specification. The schema must be defined with validation so that it does not violate the original X3D data structure. In this paper, we have extended the original X3D schema with a physical unit specification, and demonstrate the difference between units-specified and non-units-specified 3D scenes using an X3D browser that we developed.