• Journal for History of Mathematics
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• v.29 no.6
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• pp.335-351
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• 2016

Spherical convexity may be defined in different ways. It depends on which statement we take as a definition among several statements that can be all used as a definition of convexity of subsets in an affine space. In this article, we consider this question from various perspectives. We compare several different definitions of spherical convexity which are found in mathematical papers. In particular, we focus our discussion on the definitions of J. P. $Benz{\acute{e}}cri$ and N. H. Kuiper who built a solid foundation for theory of convex bodies and convex affine(projective) structures on manifolds.

• The Journal of the Acoustical Society of Korea
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• v.41 no.6
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• pp.637-646
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• 2022

Sub-arrays of arbitrary conformal array have different geometric shape through steering direction, thus the beam patterns of sub-arrays are always non-uniform. In this paper, we apply the beam pattern synthesis method using convex optimization into the conformal array, and shows the improvement of uniformity of beam performance. The simulation is performed with the conformal array of cut-sphere shape. As a result, the standard deviation of 3 dB beamwidth in elevation is greatly reduced but the directivity index is also reduced. To alleviate this trade-off, we propose a convex optimization using a shading function.

• The Korean Journal of Applied Statistics
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• v.33 no.2
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• pp.115-122
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• 2020

K-means clustering uses a spherical or elliptical metric to group data points; however, it does not work well for non-convex data such as the concentric circles. Spectral clustering, based on graph theory, is a generalized and robust technique to deal with non-standard type of data such as non-convex data. Results obtained by spectral clustering often outperform traditional clustering such as K-means. In this paper, we review spectral clustering and show important issues in spectral clustering such as determining the number of clusters K, estimation of scale parameter in the adjacency of two points, and the dimension reduction technique in clustering high-dimensional data.