• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제11권2호
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• pp.9-14
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• 2007

A signed 3-partite graph is a 3-partite graph in which each edge is assigned a positive or a negative sign. Let G(U, V, W) be a signed 3-partite graph with $U\;=\;\{u_1,\;u_2,\;{\cdots},\;u_p\},\;V\;=\;\{v_1,\;v_2,\;{\cdots},\;v_q\}\;and\;W\;=\;\{w_1,\;w_2,\;{\cdots},\;w_r\}$. Then, signed degree of $u_i(v_j\;and\;w_k)$ is $sdeg(u_i)\;=\;d_i\;=\;d^+_i\;-\;d^-_i,\;1\;{\leq}\;i\;{\leq}\;p\;(sdeg(v_j)\;=\;e_j\;=\;e^+_j\;-\;e^-_j,\;1\;{\leq}\;j\;{\leq}q$ and $sdeg(w_k)\;=\;f_k\;=\;f^+_k\;-\;f^-_k,\;1\;{\leq}\;k\;{\leq}\;r)$ where $d^+_i(e^+_j\;and\;f^+_k)$ is the number of positive edges incident with $u_i(v_j\;and\;w_k)$ and $d^-_i(e^-_j\;and\;f^-_k)$ is the number of negative edges incident with $u_i(v_j\;and\;w_k)$. The sequences ${\alpha}\;=\;[d_1,\;d_2,\;{\cdots},\;d_p],\;{\beta}\;=\;[e_1,\;e_2,\;{\cdots},\;e_q]$ and ${\gamma}\;=\;[f_1,\;f_2,\;{\cdots},\;f_r]$ are called the signed degree sequences of G(U, V, W). In this paper, we characterize the signed degree sequences of signed 3-partite graphs.

• 대한수학회보
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• 제33권2호
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• pp.193-204
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• 1996

Let U and V be nonempty compact subsets of two Hausdorff topological vector spaces. Suppose that a function $J : U \times V \to R$ is such that for each $\upsilon \in V, J(\cdot, \upsilon)$ is lower semi-continuous and convex on U, and for each $ u \in U, J(u, \cdot)$ is upper semi-continuous and concave on V.

• Korean Journal of Mathematics
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• 제17권2호
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• pp.197-210
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• 2009

We prove the existence of multiple solutions (${\xi},{\eta}$) for perturbations of the elliptic system with Dirichlet boundary condition $$(0.1)\;\begin{array}{lcr}A{\xi}+g_1({\xi}+ 2{\eta})=s{\phi}_1+h\;in\;{\Omega},\\A{\xi}+g_2({\xi}+ 2{\eta})=s{\phi}_1+h\;in\;{\Omega},\end{array}$$ where $lim_{u{\rightarrow}{\infty}}\frac{gj(u)}{u}={\beta}_j$, $lim_{u{\rightarrow}-{\infty}}\frac{gj(u)}{u}={\alpha}_j$ are finite and the nonlinearity $g_1+2g_2$ crosses eigenvalues of A.

• 한국정밀공학회:학술대회논문집
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• 한국정밀공학회 2010년도 추계학술대회 논문집
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• pp.819-820
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• 2010

• 한국정밀공학회:학술대회논문집
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• 한국정밀공학회 2013년도 춘계학술대회 논문집
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• pp.1037-1038
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• 2013