• Journal of the Chungcheong Mathematical Society
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• v.31 no.4
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• pp.369-379
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• 2018

Let ${\mathbb{H}}_3$ be the 3-dimensional Heisenberg group equipped with a left-invariant metric. In this paper we calculate the volumes of geodesic balls in ${\mathbb{H}}_3$. Let $B_e(R)$ be the geodesic ball with center e (the identity of ${\mathbb{H}}_3$) and radius R in ${\mathbb{H}}_3$. Then, the volume of $B_e(R)$ is given by $$Vol(B_e(R))={\frac{\pi}{6}}\{-16R+(R^2+6){\sin}\;R+(R^3+10R){\cos}\;R+(R^4+12R^2){\int\nolimits_0^R}\;{\frac{{\sin}\;t}{t}}dt\}$$.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• v.9 no.1
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• pp.685-688
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• 2005

In this study, an efficient algorithm for the numerical search of the geodesic path of the creeping wave on a doubly curved surface is developed. The ellipsoid as a doubly curved surface is studied because of its three dimensional nature in that it can be used to simulate the body of an aircraft, or a missile body. Numerical result of the geodesic path on an ellipsoid is given.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.3
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• pp.349-363
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• 2019

Let ${\mathbb{H}}^5$ be the 5-dimensional Heisenberg group equipped with a left-invariant metric. In this paper we calculate the volumes of geodesic balls in ${\mathbb{H}}^5$. Let $B_e(R)$ be the geodesic ball with center e (the identity of ${\mathbb{H}}^5$) and radius R in ${\mathbb{H}}^5$. Then, the volume of $B_e(R)$ is given by $${\hfill{12}}Vol(B_e(R))\\{={\frac{4{\pi}^2}{6!}}{\left(p_1(R)+p_4(R){\sin}\;R+p_5(R){\cos}\;R+p_6(R){\displaystyle\smashmargin{2}{\int\nolimits_0}^R}{\frac{{\sin}\;t}{t}}dt\right.}\\{\left.{\hfill{65}}{+q_4(R){\sin}(2R)+q_5(R){\cos}(2R)+q_6(R){\displaystyle\smashmargin{2}{\int\nolimits_0}^{2R}}{\frac{{\sin}\;t}{t}}dt}\right)}$$ where $p_n$ and $q_n$ are polynomials with degree n.

• Journal for History of Mathematics
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• v.14 no.1
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• pp.109-114
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• 2001

Using Fermi coordinate and Jacobi operator on Kahler manifold, we characterize partially conformal geodesic transformation in Kahler geometry.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.2
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• pp.81-86
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• 2001

In this paper, we present the proof of the property for interior metric space and geodesic space.

• Journal of Korean Association for Spatial Structures
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• v.8 no.4
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• pp.47-55
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• 2008

This paper deals with basic theories and some numerical results on structural optimization for geodesic dome. First of all, the space efficiency of geodesic dome is investigated by using the ratio of icosahedron's surface area to the internal volume enclosed by it. The procedure how to create the geodesic dome is also provided in systematic way and implemented and utilized into the design optimization code ISADO-OPT. The mathematical programming technique is introduced to find out the optimum pattern of member size of geodesic dome against a point load. In this study, total weight of structure is considered as the objective function to be minimized and the displacement occurred at loading point and member stresses of geodesic dome are used as the constraint functions. The finite difference method is used to calculate the design sensitivity of objective function with respect to design variables. The SLP, SQP and MFDM available in the optimizer DoT is used to search optimum member size patterns of geodesic dome. It is found to be that the optimum member size pattern can be efficiently obtained by using the proposed design optimization technique and numerical results can be used as benchmark test as a basic reference solution for design optimization of dome structures.

• Communications of the Korean Mathematical Society
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• v.21 no.3
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• pp.491-496
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• 2006

In the paper, we study an n-dimensional compact Riemannian manifold (M, g) with the property that the lengthes of the images $c({\mathbb{R}})$ in M of any geodesic curves $c({\mathbb{R}}){\mapsto}M$ are finite.

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.411-418
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• 2006

In this paper, we show that a short simple closed geodesic in quaternionic hyperbolic 2-manifold has an embedded tubular neighborhood whose width only depends on the length of the geodesic.

• Bulletin of the Korean Mathematical Society
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• v.55 no.2
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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.

• Honam Mathematical Journal
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• v.22 no.1
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• pp.83-90
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• 2000

We establish a sufficient condition for a simple closed curve in the Euclidean plain $\mathbb{R}^2$, the unit sphere $S^2$ or the hyperbolic plane $H^2$ to be the boundary of a metric ball.