• 대한수학회지
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• 제56권6호
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• pp.1441-1461
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• 2019

In this paper we consider the monotonicity of the lowest constant ${\lambda}_a^b(g)$ under the Ricci-Bourguignon flow and the normalized Ricci-Bourguignon flow such that the equation $$-{\Delta}u+au\;{\log}\;u+bRu={\lambda}_a^b(g)u$$ with ${\int}_{M}u^2dV=1$, has positive solutions, where a and b are two real constants. We also construct various monotonic quantities under the Ricci-Bourguignon flow and the normalized Ricci-Bourguignon flow. Moreover, we prove that a compact steady breather which evolves under the Ricci-Bourguignon flow should be Ricci-flat.

• 대한수학회논문집
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• 제38권1호
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• pp.257-266
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• 2023

Let G_k,n(ℍ) for 2 ≤ k < n denote the Grassmann manifold of k-dimensional vector subspaces of ℍⁿ. In this paper, we determine the evaluation subgroups of the Plüker embedding $G{_{2,n}}({\mathbb{H}}){\hookrightarrow}{\mathbb{H}}P^{N-1}$, where $N\;=\;({n \atop 2}\)$.

• 대한수학회보
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• 제35권2호
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• pp.195-201
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• 1998

We let (M,g) be a noncompact complete Riemannian manifold of dimension n $\geq$ 3 with scalar curvatue S, which is close to -1. We show the existence of a conformal metric $\bar{g}$, near to g, whose scalar curvature $\bar{S}$ = -1 by gluing solutions of the corresponding partial differential equation on each bounded subsets $K_i$ with ${\bigcup}K_i$ = M.

• 대한수학회보
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• 제35권4호
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• pp.757-767
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• 1998

For a Riemannian manifold $(M^n, g)$ we consider the space $V(M^n, g)$ of all smooth functions on $M^n$ whose Hessian is proportional to the metric tensor $g$. It is well-known that if $M^n$ is a space form then $V(M^n)$ is of dimension n+2. In this paper, conversely, we prove that if $V(M^n)$ is of dimension $\ge{n+1}$, then $M^n$ is a Riemannian space form.

• 대한수학회보
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• 제50권6호
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• pp.1989-2000
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• 2013

We show that for any given closed orientable 3-manifold M with a Heegaard surface of genus g, any positive integers b and n, there exists a knot K in M which admits a (g, b)-bridge splitting of distance greater than n with respect to the Heegaard surface except for (g, b) = (0, 1), (0, 2).

• Korean Journal of Mathematics
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• 제19권4호
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• pp.381-390
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• 2011

This paper is a direct continuation of [1]. In this paper we derive tensorial representations of contracted ES curvature tensors of $g-ESX_n$ and prove several generalized identities involving them. In particular, a variation of the generalized Bianchi's identity in $g-ESX_n$, which has a great deal of useful physical applications, is proved in Theorem (2.9).

• 소성∙가공
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• 제18권2호
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• pp.156-159
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• 2009

Slot coating has been wide spread in photo-resist coating on the glass for liquid crystal display. Die in slot coater consists of manifold and land. Material comes in inlet of the die and flow into the manifold and then flow out through the land. The coating thickness variations along the die length depend upon inside of die design such as manifold and die land. However the coating thickness variations along the moving direction(coating direction) of the coater depend upon the operational conditions of coater as well as die lip design. The coating behaviors including atmospheric air have been investigated in this study. Die geometries considered in this study were nozzle gap and length of the die lip. Coating gap and coating speed were the variables fur coating operational conditions. When the nozzle gap and length of die lip increased climbing effect of PR on the downstream die lip was reduced. Subsequently uniformity of coating thickness improved. Uniformity of coating thickness also enhanced as coating gap and coater speed increased. The uniformity of coating gap was related to the velocity vector distributions on the coating surface.

• Kyungpook Mathematical Journal
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• 제47권3호
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• pp.335-340
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• 2007

Let M be a hyperbolic 3-manifold such that ${\partial}M$ has at least two boundary tori ${\partial}_oM$ and ${\partial}_1M$. Suppose that M contains an essential orientable surface P of genus $g$ with one outer boundary component ${\partial}_oP$, lying in ${\partial}_oM$ and having slope ${\lambda}$ in ${\partial}_oM$, and $p$ inner boundary components ${\partial}_iP$, $i=1$, ${\cdots}$, $p$, each having slope ${\alpha}$ in ${\partial}_1M$. Let ${\beta}$ be a slope in ${\partial}_1M$ and suppose that $M({\beta})$ is toroidal. Let $\hat{T}$ be a minimal essential torus in $M({\beta})$, which means that $\hat{T}$ is pierced a minimal number of times by the core of the ${\beta}$-Dehn filling, among all essential tori in $M({\beta})$. Let $T=\hat{T}{\cap}M$ and denote by $t$ the number of components of ${\partial}T$. In this paper we prove: (i) if $t{\geq}3$, then ${\Delta}({\alpha},{\beta}){\leq}6+\frac{10g-5}{p}$, (ii) If $t=2$, then ${\Delta}({\alpha},{\beta}){\leq}13+\frac{24g-12}{p}$, (iii) If $t=1$, then ${\Delta}({\alpha},{\beta}){\leq}1$.

• 대한수학회보
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• 제45권1호
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• pp.33-37
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• 2008

Let G be a compact abelian Lie group, and M an odd-dimensional closed smooth G-manifold. If the fixed point set $M^G\neq\emptyset$ and dim $M^G=0$, then G has a subgroup H with $G/H{\cong}\mathbb{Z}_2$, the cyclic group of order 2. The tangential representation $\tau_x$(M) of G at $x{\in}M^G$ is also regarded as a representation of H by restricted action. We show that the number of fixed points is even, and that the tangential representations at fixed points are pairwise isomorphic as representations of H.

• 대한기계학회:학술대회논문집
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• 대한기계학회 2005년도 창립60주년 기념 추계 학술대회
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• pp.223.1-223.1
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• 2005