• 대한수학회보
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• 제52권5호
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• pp.1661-1668
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• 2015

We study the Gauss map G of ruled surfaces in the 3-dimensional Euclidean space $\mathbb{E}^3$ with respect to the so called Cheng-Yau operator ${\Box}$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only ruled surfaces with Gauss map G satisfying ${\Box}G=AG$ for some $3{\times}3$ matrix A are the flat ones. Furthermore, we show that the only ruled surfaces with Gauss map G satisfying ${\Box}G=AG$ for some nonzero $3{\times}3$ matrix A are the cylindrical surfaces.

• 충청수학회지
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• 제22권4호
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• pp.771-776
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• 2009

Let (H, g) denote the upper half plane in $R^2$ with the Riemannian metric g := ($(dx)^2$ + $(dy)^2$)$/y^2$. First of all we get a necessary and sufficient condition for a diffeomorphism $\phi$ of (H, g) to be a harmonic map. And, we obtain the fact that if a diffeomorphism $\phi$ of (H, g) is a harmonic function, then the following facts are equivalent: (1) $\phi$ is a harmonic map; (2) $\phi$ is an affine transformation; (3) $\phi$ is an isometry (motion).

• 대한수학회보
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• 제50권3호
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• pp.935-949
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• 2013

In this paper, we study surfaces in $\mathb{E}^3$ whose Gauss map G satisfies the equation ${\Box}G=f(G+C)$ for a smooth function $f$ and a constant vector C, where ${\Box}$ stands for the Cheng-Yau operator. We focus on surfaces with constant Gaussian curvature, constant mean curvature and constant principal curvature with such a property. We obtain some classification and characterization theorems for these kinds of surfaces. Finally, we give a characterization of surfaces whose Gauss map G satisfies the equation ${\Box}G={\lambda}(G+C)$ for a constant ${\lambda}$ and a constant vector C.

• 대한수학회논문집
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• 제32권3호
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• pp.715-724
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• 2017

We study the Gauss map G of helicoidal surfaces in the 3-dimensional Euclidean space ${\mathbb{E}}^3$ with respect to the so called Cheng-Yau operator ${\square}$ acting on the functions defined on the surfaces. As a result, we completely classify the helicoidal surfaces with Gauss map G satisfying ${\square}G=AG$ for some $3{\times}3$ matrix A.

• Kyungpook Mathematical Journal
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• 제57권1호
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• pp.133-144
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• 2017

In this paper, we study ruled surfaces in 3-dimensional Euclidean and Minkowski space in terms of their Gauss map. We obtain classification theorems for these type of surfaces whose Gauss map G satisfying ${\Box}G=f(G+C)$ for a constant vector $C{\in}{\mathbb{E}}^3$ and a smooth function f, where ${\Box}$ denotes the Cheng-Yau operator.

• 호남수학학술지
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• 제27권1호
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• pp.115-129
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• 2005

In this paper, we give a digital graph-theoretical approach of the study of digital images with relation to a simplicial complex. Thus, a digital graph $G_k$ with some k-adjacency in ${\mathbb{Z}}^n$ can be recognized by the simplicial complex spanned by $G_k$. Moreover, we demonstrate that a graphically $(k_0,\;k_1)$-continuous map $f:G_{k_0}{\subset}{\mathbb{Z}}^{n_0}{\rightarrow}G_{k_1}{\subset}{\mathbb{Z}}^{n_1}$ can be converted into the simplicial map $S(f):S(G_{k_0}){\rightarrow}S(G_{k_1})$ with relation to combinatorial topology. Finally, if $G_{k_0}$ is not $(k_0,\;3^{n_0}-1)$-homotopy equivalent to $SC^{n_0,4}_{3^{n_0}-1}$, a graphically $(k_0,\;k_1)$-continuous map (respectively a graphically $(k_0,\;k_1)$-isomorphisim) $f:G_{k_0}{\subset}{\mathbb{Z}}^{n_0}{\rightarrow}G_{k_1}{\subset}{\mathbb{Z}^{n_1}$ induces the group homomorphism (respectively the group isomorphisim) $S(f)_*:{\pi}_1(S(G_{k_0}),\;v_0){\rightarrow}{\pi}_1(S(G_{k_1}),\;f(v_0))$ in algebraic topology.

• 생명과학회지
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• 제13권3호
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• pp.280-290
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• 2003

RAPD 연관지도를 RFLP 연관지도와 합병을 하는 것은 각각의 유전 marker들의 단점을 서로 보완하여 세밀화된 유전자 지도작성을 용이하게 할 수 있다. 본 연구는 Essex와 PI 437654의 $F_2$ 및 $F_3$ 후대계통들을 재료로 하여 작성된 RAPD 연관지도를 콩의 RFLP 연관지도와 합병을 함에 있어서 나타난 몇가지 특징들을 기술하고자 함을 목적으로 하는 바 그 특징들은 아래와 같이 요약된다. 1. RAPD 연관지도상에서의 RFLP probe들의 위치가 RFLP 연관지도상에서의 위치와 부분적으로 변동된 현상이 나타났다. RAPD 연관그룹 L.G.C-3을 RFLP 연관그룹 a1 및 a2와 합병하는 과정에서 pSAC3와 pA136, 그리고 pA170/EcoRV와 pB170/HindIII이 서로 반대방향으로 위치하였다. pK400은 RFLP 연관지도상에서는 pA96-1과 pB172의 사이에 위치한 반면 RAPD 연관지도상에서는 i locus와 pA85 사이에 위치하였다. 2. RAPD 연관지도상에서의 두 marker들간의 간격이 RFLP 연관지도상에서의 간격보다 멀어진 현상이 두드러지게 나타났다. pA890과 pK493간의 간격은 RAPD 연관그룹 L.G.C-1에서는 48.6 cM이었던 반면 RFLP 연관 그룹상에서는 단지 13.3 cM으로 나타났다. 또한 pB32-2와 pA670, pA670과 pA668사이의 간격은 RAPD 연관그룹 L.G.C-2에서는 50.9 cM과 31.7 cM이었던 반면, RFLP 연관지도상에서의 간격은 각각 35.9 cM과 13.5 cM으로 나타났다. 3. 하나의 RFLP probe로부터 두개 이상의 다형화 현상을 나타낸 marker들이 동일한 연관그룹이나 다른 연관그룹에 위치하는 현상이 나타났다. 제한효소 HindIII로 절단된 probe pK418은 세개의 marker를 나타내었는데, 그 중 하나는 L.G.C-20에 위치하였으며, 다른 두개는 L.G.C-4에 위치하였다. 위에 나타난 특징들은 RAPD 연관지도는 intraspecific cross의 후대계통들을 재료로 하여 작성된 반면 RFLP 연관지도는 interspecific cross의 후대계통들을 재료로 하여 작성된 결과에선 비롯된 차이점 때문인 것으로 추측된다.

• 대한수학회지
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• 제53권5호
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• pp.1149-1165
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• 2016

In this paper, we show that there exists no left invariant Riemannian metric h on the Heisenberg group H such that (H, h) is a symmetric Riemannian manifold, and there does not exist any H-invariant metric $\bar{h}$ on the Heisenberg manifold $H/{\Gamma}$ such that the Riemannian connection on ($H/{\Gamma},\bar{h}$) is a Yang-Mills connection. Moreover, we get necessary and sufficient conditions for a group homomorphism of (SU(2), g) with an arbitrarily given left invariant metric g into (H, h) with an arbitrarily given left invariant metric h to be a harmonic and an affine map, and get the totality of harmonic maps of (SU(2), g) into H with a left invariant metric, and then show the fact that any affine map of (SU(2), g) into H, equipped with a properly given left invariant metric on H, does not exist.

• Kyungpook Mathematical Journal
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• 제62권1호
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• pp.167-177
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• 2022

In this manuscript, we handle a tubular surface whose Gauss map G satisfies the equality L₁G = f(G + C) for the Cheng-Yau operator L₁ in Galilean 3-space 𝔾₃. We give an example of a tubular surface having L₁-harmonic Gauss map. Moreover, we obtain a complete classification of tubular surface having L₁-pointwise 1-type Gauss map of the first kind in 𝔾₃ and we give some visualizations of this type surface.

• 대한수학회보
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• 제46권4호
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• pp.673-689
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• 2009

We analyze $MAP_1,\;MAP_2$/G/1 finite queues with service scheduling function dependent upon queue lengths. The customers are classified into two types. The arrivals of customers are assumed to be the Markovian Arrival Processes (MAPs). The service order of customers in each buffer is determined by a service scheduling function dependent upon queue lengths. Methods of embedded Markov chain and supplementary variable give us information for queue length of two buffers. Finally, the performance measures such as loss probability and mean waiting time are derived. Some numerical examples also are given with applications in telecommunication networks.