• Honam Mathematical Journal
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• v.1 no.1
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• pp.21-25
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• 1979

Abelian군(群)의 presheaf에 관한 직적(直積)과 직화(直和)를 Category 입장에서 정의(定義)하고 presheaf $F_{\lambda}\;({\lambda}{\epsilon}{\Lambda})$들의 두 직적(直積)(또는 直和)은 서로 동형적(同型的) 관계(關係)에 있으며, 특히 ${\phi}:X{\rightarrow}Y$가 homeomorphism이라 하고 ${\phi}_*F$를 X상(上)의 presheaf F의 direct image이라 하면 (1) $({\phi}_*F, \;{\phi}_*(f_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$가 $({\phi}_*F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}$의 직적(直積)일 때 오직 그때 한하여 $(F,\;(f_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$는 $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$의 직적(直積)이다. (2) $({\phi}_*F,\;{\phi}_*(l_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$가 $({\phi}_*F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}$의 직화(直和)일 때 오직 그때 한하여 $(F,\;(l_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$는 $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$의 직화(直和)이다. Let $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$ be an indexed set of presheaves of abelian group on topological space X. We can define the cartesian product $$\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda}$$ of $(F_{\lambda})_{{\lambda}{\epsilon}{\Lambda}})$ by $$(\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda})(U)=\prod_{{\lambda}{\epsilon}{\Lambda}}(F_{\lambda}(U))$$ for U open in X $${\rho}_v^u:\;(\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda})(U){\rightarrow}(\prod_{{\lambda}{\epsilon}{\Lambda}}\;F_{\lambda})(V)((s_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}{\rightarrow}(_{\lambda}{\rho}_v^u(s_{\lambda}))_{{\lambda}{\epsilon}{\Lambda}})$$ for $V{\subseteq}U$ open in X where $_{\lambda}{\rho}^U_V$ is a restriction of $F_{\lambda}$, And we have natural presheaf morphisms ${\pi}_{\lambda}$ and ${\iota}_{\lambda}$ such that ${\pi}_{\lambda}(U):\;({\prod}_\;F_{\lambda})(U){\rightarrow}F_{\lambda}(U)((s_{\lambda})_{{\lambda}{\epsilon}{\Lambda}}{\rightarrow}s_{\lambda})$ ${\iota}_{\lambda}(U):\;F_{\lambda}(U){\rightarrow}({\prod}\;F_{\lambda})(U)(s_{\lambda}{\rightarrow}(o,o,{\cdots}\;{\cdots}o,s_{\lambda},o,{\cdots}\;{\cdots}o)$ for $(s_{\lambda}){\epsilon}{\prod}_{\lambda}\;F_{\lambda}(U)$ and $(s_{\lambda}){\epsilon}F_{\lambda}(U)$.

• Proceedings of the Korean Society of Precision Engineering Conference
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• 1997.04a
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• pp.389-394
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• 1997

Consider a tetrhedron is composed of six dihedral angles .phi.(i=1,2..., 6), and a vertex of a tetrahedron is also three dihedral angles. It will assume that a vertex A, for an example, is composed of there angles definded such as .alpha..betha. and .gamma. !. then there is a corresponding angle can be given as .phi1.,.phi2.,.phi3.. Here, in order to differentiate between a conventional triangle and dihedral angle, if a dihedral angle degined in this paper is symbolized as .phi..LAMBDA.,the value of cos.theta.of .phi./sab a/, in a trigonometric function rule,can be defined to tecos.phi..LAMBD/sab A/., and it is defined as a tetradedral cosine .phi. or simply called a tecos.phi.. Moreover, in a simillar method, the dihedral angle of tetrahedron .phi..LAMBDA. is given as : value of sin .theta. can defind a tetra-sin.phi..LAMBDA., and value of tan .theta. of .phi..LAMBDA. is a tetra-tan .phi..LAMBDA. By induction it can derive that a tetrahedral geometry on the basis of suggesting a geometric tetrahedron

• Biomedical Science Letters
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• v.6 no.2
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• pp.131-139
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• 2000

RAPD-PCR was applied to identify the phylogenetic relationship between isolated Korean-type coliphages ($\phi$C1, $\phi$C2, $\phi$C3 and $\phi$C4) and well-known American coliphages ($\phi$T2, $\phi$T4, $\phi$T5, $\phi$T7 and ${\phi}{\lambda}$). Subsequently, a computer analysis was carried out with the results of RAPD-PCR. As a result, 9 individuals were divided into five groups. The Korean-type coliphages formed a single cluster which showed very high genetic similarity but the American-type coliphages revealed very low genetic similarity among them. In particular, the $\phi$T2와 $\phi$T4 (T$_{even}$ phages) made one sub-cluster among American coliphages, and they were very distant from $\phi$T5, $\phi$T7 and ${\phi}{\lambda}$. However, ${\phi}{\lambda}$ made a cluster with the Korean-type coliphages that we isolated. The genome size of Korean-type coliphages was ranged from 25,000 bp to 35,000 bp. Among them, the genome of $\phi$C2 was the smallest and that of $\phi$C1 was the biggest, while others were in the middle of the size.

• Korean Journal of Mathematics
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• v.14 no.2
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• pp.259-271
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• 2006

We consider a semilinear elliptic boundary value problem with Dirichlet boundary condition $Au+bu^+-au^-=t_{1{\phi}1}+t_{2{\phi}2}$ in ${\Omega}$ and ${\phi}_n$ is the eigenfuction corresponding to ${\lambda}_n(n=1,2,{\cdots})$. We have a concern with the multiplicity of solutions of the equation when ${\lambda}_1$ < a < ${\lambda}_2$ < b < ${\lambda}_3$.

• Publications of The Korean Astronomical Society
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• v.9 no.1
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• pp.69-84
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• 1994

The spectral types of stars can be classified by using Balmer discontinuity($D_B$) and wavelength(${\lambda}_B$) expressed in terms of effective temperatures appeared in Balmer discontinuity. In this research, in order to classify stars, we used the well established observational data of high dispersion spectrophotometry for the spectral types and luminosity classes of stars in the Breger(1976) catalogue. Balmer discontinuity by effective temperatures of stars was accurately measured, and the ${\lambda}_B$ was replaced to luminosity classes of MK system, because of the close relationship between the As and luminosity classes. We measured the energy gradients(${\phi}_R$) of stars which were expressed as a function of spectral types in the interval of ${\lambda}{\lambda}4,000{\sim}4600{\AA}$, and then obtained a new physical parameter(${\phi}$) from the $D_B$ and ${\phi}_B$. The new parameter, ${\phi}$ can be used instead of HD classifications of stars and can be used widely for spectrophotometrical classifications of stars.

• Communications of the Korean Mathematical Society
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• v.19 no.1
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• pp.119-127
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• 2004

Suppose that $\psi{\;}\in{\;}L^2(\mathbb{R})$ generates a wavelet frame (resp. Riesz basis) with bounds A and B. If $\phi{\;}\in{\;}L^2(\mathbb{R})$ satisfies $$\mid$\^{\psi}(\xi)\;\^{\phi}(\xi)$\mid${\;}<{\;}{\lambda}\frac{$\mid$\xi$\mid$^{\alpha}}{(1+$\mid$\xi$\mid$)^{\gamma}}$ for some positive constants $\alpha,{\;}\gamma,{\;}\lambda$ such that $1{\;}<1{\;}+{\;}\alpha{\;}<{\;}\gamma{\;}and{\;}{\lambda}^2M{\;}<{\;}A$, then $\phi$ also generates a wavelet frame (resp. Riesz basis) with bounds $A(1{\;}-{\;}{\lambda}\sqrt{M/A})^2{\;}and{\;}B(1{\;}+{\;}{\lambda}\sqrt{M/A})^2$, where M is a constant depending only on $\alpha,{\;}\gamma$ the dilation step a, and the translation step b.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.97-103
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• 2009

Suppose F is a field of characteristic not 2 and $F\;{\neq}\;Z_3$. Let $M_n(F)$ be the linear space of all $n{\times}n$ matrices over F, and let ${\Gamma}_n(F)$ be the subset of $M_n(F)$ consisting of all $n{\times}n$ involutory matrices. We denote by ${\Phi}_n(F)$ the set of all maps from $M_n(F)$ to itself satisfying A - ${\lambda}B{\in}{\Gamma}_n(F)$ if and only if ${\phi}(A)$ - ${\lambda}{\phi}(B){\in}{\Gamma}_n(F)$ for every A, $B{\in}M_n(F)$ and ${\lambda}{\in}F$. It was showed that ${\phi}{\in}{\Phi}_n(F)$ if and only if there exist an invertible matrix $P{\in}M_n(F)$ and an involutory element ${\varepsilon}$ such that either ${\phi}(A)={\varepsilon}PAP^{-1}$ for every $A{\in}M_n(F)$ or ${\phi}(A)={\varepsilon}PA^{T}P^{-1}$ for every $A{\in}M_n(F)$. As an application, the maps preserving inverses of matrces also are characterized.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.603-617
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• 1997

Consider a Sobolev inner product on the space of polynomials such as $$ \phi(p,q) = \lambda p(c)q(c) + <\tau,p'(x)q'(x)> $$ where $\tau$ is a moment functional and c and $\lambda$ are real constants. We investigate properties of orthogonal polynomials relative to $\phi(\cdot,\cdot)$ and give necessary and sufficient conditions under which such Sobolev orthogonal polynomials satisfy a spectral type differential equation with polynomial coefficients.

• Bulletin of the Korean Mathematical Society
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• v.33 no.1
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• pp.75-86
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• 1996

Let G be a finite simple connected graph with vertex set V(G) and edge set E(G). Let A(G) be the adjacency matrix of G. The characteristic polynomial of G is the characteristic polynomial $\Phi(G;\lambda) = det(\lambda I - A(G))$ of A(G). A zero of $\Phi(G;\lambda)$ is called an eigenvalue of G.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.3
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• pp.261-267
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• 2007

We show the existence of the positive solution for the system of the following nonlinear wave equations with Dirichlet boundary conditions $$u_{tt}-u_{xx}+av^+=s{\phi}_{00}+f$$, $$v_{tt}-v_{xx}+bu^+=t{\phi}_{00}+g$$, $$u({\pm}\frac{\pi}{2},t)=v({\pm}\frac{\pi}{2},t)=0$$, where $u_+=max\{u,0\}$, s, $t{\in}R$, ${\phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${\lambda}_{00}=1$ of the eigenvalue problem $u_{tt}-u_{xx}={\lambda}_{mn}u$ with $u({\pm}\frac{\pi}{2},t)=0$, $u(x,t+{\pi})=u(x,t)=u(-x,t)=u(x,-t)$ and f, g are ${\pi}$-periodic, even in x and t and bounded functions in $[-\frac{\pi}{2},\frac{\pi}{2}]{\times}[-\frac{\pi}{2},\frac{\pi}{2}]$ with $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}f{\phi}_{00}=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}g{\phi}_{00}=0$.