• 대한수학회논문집
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• 제9권3호
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• pp.731-737
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• 1994

Let $(\Omega, F, P)$ be a probability space with F a $\sigma$-algebra of subsets of the measure space $\Omega$ and P a probability measures on $\Omega$. Suppose $a > 0$ and let $(F_t)_{t \in [0,a]}$ be an increasing family of sub-$\sigma$- algebras of F. If $r > 0$, let $J = [-r, 0]$ and $C(J, R^n)$ the Banach space of all continuous paths $\gamma : J \to R^n$ with the sup-norm $\Vert \gamma \Vert_C = sup_{s \in J} $\mid$\gamma(x)$\mid$$ where $$\mid$\cdot$\mid$$ denotes the Euclidean norm on $R^n$. Let E and F be separable real Banach spaces and L(E,F) be the Banach space of all continuous linear maps $T : E \to F$ with the norm $\Vert T \Vert = sup {$\mid$T(x)$\mid$_F : x \in E, $\mid$x$\mid$_E \leq 1}$.

• 충청수학회지
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• 제15권2호
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• pp.57-66
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• 2003

The various deformation spaces associated with maximal geometric structures on closed oriented 3-manifolds was studied in [2], leaving out the geometry of $\mathbb{R}^3$. In this paper, we study the Weil spaces and Teichm$\ddot{u}$ller spaces of non-orientable 3-dimensional flat Riemannian manifolds. In particular, we find the Teichm$\ddot{u}$ller spaces are homeomorphic to the Euclidean spaces $\mathbb{R}^4$ or $\mathbb{R}^3$ depending on the holonomy group $\mathbb{Z}_2$ or $\mathbb{Z}_2{\times}\mathbb{Z}_2$ respectively.

• 대한수학회보
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• 제38권4호
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• pp.753-761
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• 2001

We introduce the notion of Gauss map of pointwise 1-type on ruled surfaces in the Euclidean 3-space for which vector valued functions is neither trivial nor it extends or coincides with the usual notion of 1-type, in general. We characterize the minimal helicoid in terms of it and give a complete classification of the ruled surfaces with pointwise 1-type Gauss map.

• 대한수학회지
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• 제33권3호
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• pp.609-624
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• 1996

Recently, Giannessi [9] firstly introduced the vector-valued variational inequalities in a real Euclidean space. Later Chen et al. [5] intensively discussed vector-valued variational inequalities and vector-valued quasi variationl inequalities in Banach spaces. They [4-8] proved some existence theorems for the solutions of vector-valued variational inequalities and vector-valued quasi-variational inequalities. Lee et al. [14] established the existence theorem for the solutions of vector-valued variational inequalities for multifunctions in reflexive Banach spaces.

• 대한수학회지
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• 제52권6호
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• pp.1337-1346
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• 2015

We examine the relationship of the shape operator of a surface of Euclidean 3-space with its Gauss map of pointwise 1-type. Surfaces with constant mean curvature and right circular cones with respect to some properties of the shape operator are characterized when their Gauss map is of pointwise 1-type.

• 대한수학회보
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• 제54권6호
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• pp.2081-2089
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• 2017

In this paper we generalize 3-dimensional Bour's Theorem to the case of 4-dimension. We proved that a helicoidal surface in $\mathbb{R}^4$ is isometric to a family of surfaces of revolution in $\mathbb{R}^4$ in such a way that helices on the helicoidal surface correspond to parallel circles on the surfaces of revolution. Moreover, if the surfaces are required further to have the same Gauss map, then they are hyperplanar and minimal. Parametrizations for such minimal surfaces are given explicitly.

• 호남수학학술지
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• 제35권4호
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• pp.647-655
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• 2013

We obtain $C^{\infty}$-continuous paths of explicit Riemannian metrics $g_t$, $0{\leq}t$ < ${\varepsilon}$, whose scalar curvatures $s(g_t)$ decrease, where $g_0$ is a flat metric, i.e. a metric with vanishing curvature. Most of them can exist on tori of dimension ${\geq}3$. Some of them yield scalar curvature decrease on a ball in the Euclidean space.

• 한국수학사학회지
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• 제20권4호
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• pp.49-60
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• 2007

본 논문에서는 초등기하에 나오는 도형의 합동의 개념으로부터 자연스럽게 얻어지는 3차원 유클리드 공간에 있는 매끄러운 곡면의 강성의 개념을 소개하고 컴팩트 곡면의 강성이론의 발전과정과 그 일반화를 살펴본다.

• East Asian mathematical journal
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• 제31권3호
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• pp.351-356
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• 2015

In this paper, we study holomorphic discs in a domain with a plurisubharmonic peak function at a boundary point. The aim is to describe boundary behavior of holomorphic discs in convex finite type domains in the complex Euclidean space in term of a special local neigh-borhood system at a boundary point.

• 대한수학회보
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• 제53권2호
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• pp.461-477
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• 2016

This work considers a particular type of swept surface named canal surfaces in Euclidean 3-space. For such a kind of surfaces, some interesting and important relations about the Gaussian curvature, the mean curvature and the second Gaussian curvature are found. Based on these relations, some canal surfaces are characterized.