• Communications of the Korean Mathematical Society
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• v.30 no.3
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• pp.253-267
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• 2015

Given a complex submanifoldM of the projective space $\mathbb{P}$(T), the hyperplane system R on M characterizes the projective embedding of M into $\mathbb{P}$(T) in the following sense: for any two nondegenerate complex submanifolds $M{\subset}\mathbb{P}$(T) and $M^{\prime}{\subset}\mathbb{P}$(T'), there is a projective linear transformation that sends an open subset of M onto an open subset of M' if and only if (M,R) is locally equivalent to (M', R'). Se-ashi developed a theory for the differential invariants of these types of systems of linear differential equations. In particular, the theory applies to systems of linear differential equations that have symbols equivalent to the hyperplane systems on nondegenerate equivariant embeddings of compact Hermitian symmetric spaces. In this paper, we extend this result to hyperplane systems on nondegenerate equivariant embeddings of homogeneous spaces of the first kind.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.21-30
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• 2021

In this paper, we prove some rigidity results about embedded minimal hypersurface M ⊂ ℝn+1 with compact ∂M that has one end which is regular at infinity. We first show that if M ⊂ ℝn+1 meets a hyperplane in a constant angle ≥ ��/2, then M is part of an n-dimensional catenoid. We show that if M meets a sphere in a constant angle and ∂M lies in a hemisphere determined by the hyperplane through the center of the sphere and perpendicular to the limit normal vector nM of the end, then M is part of either a hyperplane or an n-dimensional catenoid. We also show that if M is tangent to a C2 convex hypersurface S, which is symmetric about a hyperplane P and nM is parallel to P, then M is also symmetric about P. In special, if S is rotationally symmetric about the xn+1-axis and nM = en+1, then M is also rotationally symmetric about the xn+1-axis.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.211-221
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• 2024

Parallel conics have interesting area and chord properties. In this paper, we study such properties of conics and conic hypersurfaces. First of all, we characterize conics in the plane with respect to the above mentioned properties. Finally, we establish some characterizations of hypersurfaces with centrally symmetric hyperplane sections.

• East Asian mathematical journal
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• v.17 no.2
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• pp.325-337
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• 2001

We consider the special hypersurface of the first approximate Matsumoto metric with $b_i(x)={\partial}_ib$ being the gradient of a scalar function b(x). In this paper, we consider the hypersurface of the first approximate Matsumoto space with the same equation b(x)=constant. We are devoted to finding the condition for this hypersurface to be a hyperplane of the first or second kind. We show that this hypersurface is not a hyper-plane of third kind.

• Journal of KIISE:Software and Applications
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• v.27 no.4
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• pp.422-431
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• 2000

This paper proposes a new codebook generation method, called a PCA-Based VQ, that incorporates the PCA (Principal Component Analysis) technique into VQ (Vector Quantization) codebook design. The PCA technique reduces the data dimensions by transforming input image vectors into the feature vectors. The cluster of feature vectors in the transformed domain is bisectioned into two subclusters by an optimally chosen partitioning hyperplane. We expedite the searching of the optimal partitioning hyperplane that is the most time consuming process by considering that (1) the optimal partitioning hyperplane is perpendicular to the first principal axis of the feature vectors, (2) it is located on the equilibrium point of the left and right cluster's distortions, and (3) the left and right cluster's distortions can be adjusted incrementally. This principal axis bisectioning is successively performed on the cluster whose difference of distortion between before and after bisection is the maximum among the existing clusters until the total distortion of clusters becomes as small as the desired level. Simulation results show that the proposed PCA-based VQ method is promising because its reconstruction performance is as good as that of the SOFM (Self-Organizing Feature Maps) method and its codebook generation is as fast as that of the K-means method.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.905-913
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• 2023

Suppose that M is a strictly convex hypersurface in the (n + 1)-dimensional Euclidean space 𝔼ⁿ⁺¹ with the origin o in its convex side and with the outward unit normal N. For a fixed point p ∈ M and a positive constant t, we put 𝚽_t the hyperplane parallel to the tangent hyperplane 𝚽 at p and passing through the point q = p - tN(p). We consider the region cut from M by the parallel hyperplane 𝚽_t, and denote by I_p(t) the (n + 1)-dimensional volume of the convex hull of the region and the origin o. Then Schneider's characterization theorem for ellipsoids states that among centrally symmetric, strictly convex and closed surfaces in the 3-dimensional Euclidean space 𝔼³, the ellipsoids are the only ones satisfying I_p(t) = 𝜙(p)t, where 𝜙 is a function defined on M. Recently, the characterization theorem was extended to centrally symmetric, strictly convex and closed hypersurfaces in 𝔼ⁿ⁺¹ satisfying for a constant 𝛽, I_p(t) = 𝜙(p)t^𝛽. In this paper, we study the volume I_p(t) of a strictly convex and complete hypersurface in 𝔼ⁿ⁺¹ with the origin o in its convex side. As a result, first of all we extend the characterization theorem to strictly convex and closed (not necessarily centrally symmetric) hypersurfaces in 𝔼ⁿ⁺¹ satisfying I_p(t) = 𝜙(p)t^𝛽. After that we generalize the characterization theorem to strictly convex and complete (not necessarily closed) hypersurfaces in 𝔼ⁿ⁺¹ satisfying I_p(t) = 𝜙(p)t^𝛽.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.12 no.5
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• pp.2375-2382
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• 2011

The user clustering for web navigation pattern discovery is very useful to get preference and behavior pattern of users for web pages. In addition, the information by the user clustering is very essential for web personalization or customer grouping. In this paper, an algorithm for clustering the web navigation path of users is proposed and then some special navigation patterns can be recognized by the algorithm. The proposed algorithm has two clustering phases. In the first phase, all paths are classified into k-groups on the bases of the their similarities. The initial solution obtained in the first phase is not global optimum but it gives a good and feasible initial solution for the second phase. In the second phase, the first phase solution is improved by revising the k-means algorithm. In the revised K-means algorithm, grouping the paths is performed by the hyperplane instead of the distance between a path and a group center. Experimental results show that the proposed method is more efficient.

• Honam Mathematical Journal
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• v.43 no.4
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• pp.679-689
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• 2021

The first author suggested an exact volume formula of the hypercubes [0, 1]ⁿ clipped by several hyperplanes expressed directly in terms of linear coefficients of the hyperplanes. However, it requires awkward assumptions to apply the formula to various situations. We suggest a concrete method to overcome those restrictions for two or three hyperplanes using 𝜖-perturbation, which gives an exact value applicable for any kind of arrangement of hyperplanes with no consideration.

• Kyungpook Mathematical Journal
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• v.58 no.4
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• pp.781-788
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• 2018

In the present paper, we have studied the Finslerian hypersurfaces and generalized ${\beta}$-conformal change of Finsler metric. The relations between the Finslerian hypersurface and the other which is Finslerian hypersurface given by generalized ${\beta}$-conformal change have been obtained. We have also proved that generalized ${\beta}$-conformal change makes three types of hypersurfaces invariant under certain conditions.

• Journal of the Institute of Electronics Engineers of Korea SP
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• v.45 no.6
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• pp.142-147
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• 2008

In this paper, we propose a novel a roach to improve the performance of speech/music classification for the selectable mode vocoder (SMV) of 3GPP2 using the support vector machine (SVM). The SVM makes it possible to build on an optimal hyperplane that is separated without the error where the distance between the closest vectors and the hyperplane is maximal. We first present an effective analysis of the features and the classification method adopted in the conventional SMV. And then feature vectors which are a lied to the SVM are selected from relevant parameters of the SMV for the efficient speech/music classification. The performance of the proposed algorithm is evaluated under various conditions and yields better results compared with the conventional scheme of the SMV.