• 대한수학회보
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• 제53권4호
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• pp.1087-1094
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• 2016

On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, after derivng an interesting curvature identity, we show that the conjecture is true in dimension three and four when g is weakly Einstein. In higher dimensional case $n{\geq}5$, we also show that the conjecture is true under an additional Ricci curvature bound. Moreover, we prove that the manifold is isometric to a standard n-sphere when it is n-dimensional weakly Einstein and the kernel of the linearized scalar curvature operator is nontrivial.

• 한국광학회지
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• 제18권1호
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• pp.19-23
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• 2007

원리적인 다양한 장점에도 불구하고 취급이 복잡하고 고차 연립방정식으로 표현되는 현실적 제한으로 인하여 실제 설계과정에서 잘 적용하지 않는 자이델 3차 수차를 이용하여 구한 코마수차 계수식으로부터 근사적인 제로조건을 만족하는 유한 물체거 리를 갖는 코마수차가 제거된 2반사경계의 초기설계에 유용한 곡률선형방정식을 유도하고 그 특징을 조사한다. 즉 주경과 부경의 곡률, 주경과 부경사이의 거리, 유효초점거리로 표현된 변형된 코마수차계수로부터 코마수차계수가 제거되는 조건에서 설계변수를 구하기 위해 전산수치해석 후 나온 데이터를 기반으로 주경과 부경사이의 선형관계가 나타나는 곡률선형방정식을 구하는 것이다. 이는 유한 물점의 코마수차가 보정된 2 반사경계에서 약간의 대수적인 계산만으로 최적화의 초기 입력 데이터를 손쉽게 구할 수 있는 것을 의미한다.

• 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.

• 한국정밀공학회:학술대회논문집
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• 한국정밀공학회 2003년도 춘계학술대회 논문집
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• pp.806-809
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• 2003

In this paper, extracting design information from arbitrary aspherical lens shape in reverse engineering is introduced. Deformation terms and sphere data equation with various variables compose asphere equation. Aspherical lens shape is expressed with complicated polynomial expression that includes deformation terms and sphere data. Deformation term and vertex curvature have direct influence on a geometric shape and an optical characteristics of aspherical lens. Hence, extracting these information mean that design information could be derived and analyzed from shape data of arbitrary aspherical lens. Furthermore, sharing designer's experience and knowledge for aspherical lens design could be expected.

• International Journal of CAD/CAM
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• 제9권1호
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• pp.103-109
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• 2010

Recently, various conformal geometric methods have been presented for non-rigid surface matching and registration. This work proposes to improve the robustness of conformal geometric methods to the boundaries by incorporating the symmetric information of the input surface. We presented two symmetric conformal mapping methods, which are based on solving Riemann-Cauchy equation and curvature flow respectively. Experimental results on geometric data acquired from real life demonstrate that the symmetric conformal mapping is insensitive to the boundary occlusions. The method outperforms all the others in terms of robustness. The method has the potential to be generalized to high genus surfaces using hyperbolic curvature flow.

• 한국해안해양공학회지
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• 제19권6호
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• pp.521-532
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• 2007

완경사 파랑식의 유도에 Galerkin방법을 사용하여 수심 의존함수에 대한 Sturm-Liouville 미분식을 엄밀하게 구성하였다. 구한 방정식의 종속변수에 대한 전형적인 변수변환으로 수심, 해저경사 그리고 해저곡률에 대한 항들로 구성된 변형 Helmholtz식을 얻었다. 수치실험을 통해 이 항들이 지형에 의한 파랑변형에 주요한 역할을 보이고 이들의 상대적인 크기에 의해 수정 완경사 파랑식(MMSE)에 비해 완경사 파랑식(MSE)의 적용성이 제한됨을 입증하였다.

• 대한수학회보
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• 제55권2호
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• pp.331-340
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• 2018

Using the comparison of differential equations involving Ricci and scalar curvatures obtained by Eschenburg and O'Sullivan, the scalar curvatures of level hypersurfaces of geodesic spheres are compared.

• 호남수학학술지
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• 제34권2호
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• pp.273-278
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• 2012

Using a comparison of the Jacobi differential equations instead of volume comparison by Itokawa in [3], the focal radius is estimated under the suitable Ricci and mean curvature conditions.

• 충청수학회지
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• 제31권4호
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• pp.363-368
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• 2018

The aim of this work is to derive a partial differential equation that explains the movement of vortex lines on a vortex trajectory surface in a three dimensional incompressible inviscid flow.

• 호남수학학술지
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• 제43권3호
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• pp.561-570
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• 2021

In this paper, applying the bifurcation method and topological analysis, we investigate the global structures of solutions for one-dimensional Minkowski-curvature problems under certain behavior of nonlinear term near zero.