• Journal of the Korean Mathematical Society
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• v.57 no.5
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• pp.1299-1322
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• 2020

We consider closed, star-shaped, admissible hypersurfaces in ℝⁿ⁺¹ expanding along the flow Ẋ = |X|^α-1 F^-β, α ≤ 1, β > 0, and prove that for the case α ≤ 1, β > 0, α + β ≤ 2, this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a round sphere after rescaling. Besides, for the case α ≤ 1, α + β > 2, if furthermore the initial closed hypersurface is strictly convex, then the strict convexity is preserved during the evolution process and the flow blows up at finite time.

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1407-1419
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• 2021

The Iwasawa decomposition NAK of the Lie group G = SO₀(2, 1) with a left invariant metric produces Riemannian submersions G → N\G, G → A\G, G → K\G, and G → NA\G. For each of these, we calculate the curvature of the base space and the lifting of a simple closed curve to the total space G. Especially in the first case, the base space has a constant curvature 0; the holonomy displacement along a (null-homotopic) simple closed curve in the base space is determined only by the Euclidean area of the region surrounded by the curve.

• Communications of the Korean Mathematical Society
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• v.13 no.3
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• pp.545-560
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• 1998

Let ø and A be denoted by the structure tensor field of type (1,1) and by the shape operator of a real hypersurface in a complex space form $M_{n}$ (c), c $\neq$ 0 respectively. The main purpose of this paper is to prove that if a real hypersurface in $M_{n}$ (c) satisfies $R_{ξ}$ øA = $AøR_{ξ}$, then the structure vector field ξ is principal, where $R_{ξ}$ / is the Jacobi operator with respect to ξ.

• Communications of the Korean Mathematical Society
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• v.29 no.1
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• pp.141-153
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• 2014

In this article, we apply the weak maximum principle in order to obtain a suitable characterization of the complete linearWeingarten hypersurfaces immersed in locally symmetric Riemannian manifold $N^{n+1}$. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or hypersurface is an isoparametric hypersurface with two distinct principal curvatures one of which is simple.

• Bulletin of the Korean Mathematical Society
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• v.42 no.2
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• pp.279-294
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• 2005

We study a real hypersurface M satisfying $L_{\xi}S=0\;and\;R_{\xi}S=SR_{\xi}$ in a complex hyperbolic space $H_n\mathbb{C}$, where S is the Ricci tensor of type (1,1) on M, $L_{\xi}\;and\;R_{\xi}$ denotes the operator of the Lie derivative and the Jacobi operator with respect to the structure vector field e respectively.

• Communications of the Korean Mathematical Society
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• v.31 no.3
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• pp.647-656
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• 2016

In this paper, we define the generalized golden shaped hypersurfaces in Lorentz space forms. Based on the classification of proper semi-Riemannian hypersurfaces in semi-Riemannian real space forms, we obtain the whole families of the generalized golden shaped hypersurfaces in Lorentz space forms.

• The Mathematical Education
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• v.20 no.3
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• pp.23-26
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• 1982

Many interesting properties of a space curve C in E$^3$ may be investigated by means of the concept of spherical indicatrix of tangent, principal normal, or binormal, to C. The purpose of the present paper is to derive the representations of the Frenet frame field., curvature, and torsion of spherical indicatrix to C in terms of the quantities associated with C. Furthermore, several interesting properties of spherical indicatrix are found in the present paper.

• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.789-799
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• 2022

We establish a sharp integral inequality related to compact (without boundary) linear Weingarten hypersurfaces (immersed) in a locally symmetric Einstein manifold and we apply it to characterize totally umbilical hypersurfaces and isoparametric hypersurfaces with two distinct principal curvatures, one which is simple, in such an ambient space. Our approach is based on the ideas and techniques introduced by Alías and Meléndez in [4] for the case of hypersurfaces with constant scalar curvature in an Euclidean round sphere.

• Journal of Korea Multimedia Society
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• v.11 no.4
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• pp.423-433
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• 2008

The surface curvatures extracted from the face contain the most important personal facial information. In particular, the face shape using the depth information represents personal features in detail. In this paper, we develop a method for recognizing the range face images by combining the multiple face regions using fuzzy integral. For the proposed approach, the first step tries to find the nose tip that has a protrusion shape on the face from the extracted face area and has to take into consideration of the orientated frontal posture to normalize. Multiple areas are extracted by the depth threshold values from reference point, nose tip. And then, we calculate the curvature features: principal curvature, gaussian curvature, and mean curvature for each region. The second step of approach concerns the application of eigenface and Linear Discriminant Analysis(LDA) method to reduce the dimension and classify. In the last step, the aggregation of the individual classifiers using the fuzzy integral is explained for each region. In the experimental results, using the depth threshold value 40 (DT40) show the highest recognition rate among the regions, and the maximum curvature achieves 98% recognition rate, incase of fuzzy integral.

• Journal of the Korean Wood Science and Technology
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• v.30 no.2
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• pp.87-94
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• 2002

This study was carried out to evaluate bending property on principal domestic species such as sargent cherry(Prunus sargentii), bitter wood(Picrasma quassioides), horn beam(Carpinus laxiflora), cork oak(Quercus variabilis), birch(Betula schmidtii), painted maple(Acer mono), basswood(Tilia amurensis), red pine(Pinus densiflora), pitch pine(Pinus rigtda), royal pawlonia(Paulownia tomentosa) by microwave heating. In this study, radius of curvature(ROC) for bending process was classified by radius of curvature(ROC) of bending plate such as 4 cm, 6 cm, and 10 cm, and thickness of metal-strap(TMS) was 0.6 mm and 0.8 mm. Bending process was successfully operated for 100 percent in bitter wood, horn beam, birch and painted maple. On the other hand, there was a success rate of 58 percent in sargent cherry and 83 percent in cork oak and 29 percent in basswood and 8 percent in royal pawlonia which is the worst bending property. All specimens of basswood and royal pawlonia were broken at 4 cm of ROC. Success rate of bending property was shown 44 percent in red pine and 56 percent in pitch pine. TMS has an effect on only drying speed in drying process than difficulty and facility of bending property. It was considered that the thinner TMS in drying process is the faster in drying speed of bent wood.