• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.375-382
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• 2017

We exhibit a one-to-one correspondence between 3-colored graphs and subarrangements of certain hyperplane arrangements denoted ${\mathcal{J}}_n$, $n{\in}{\mathbb{N}}$. We define the notion of centrality of 3-colored graphs, which corresponds to the centrality of hyperplane arrangements. Via the correspondence, the characteristic polynomial ${\chi}{\mathcal{J}}_n$ of ${\mathcal{J}}_n$ can be expressed in terms of the number of central 3-colored graphs, and we compute ${\chi}{\mathcal{J}}_n$ for n = 2, 3.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1281-1289
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• 2016

Richard Stanley suggested the problem of finding combinatorial proofs of formulas for counting regions of certain hyperplane arrangements defined by hyperplanes of the form $x_i=0$, $x_i=x_j$, and $x_i=2x_j$ that were found using the finite field method. We give such proofs, using embroidered permutations and linear extensions of posets.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.185-190
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• 1989

In [4], J. Leray introduced the notion of partial hyperbolicity to characterize the operators for which the non-characteristic Cauchy problem is solvable in the Geverey class for any data which are holomorphic in a part of variables x"=(x$_{2}$,..,x$_{l}$ ) in the initial hyperplane x$_{1}$=0. A linear partial differential operator is called partially hyperbolic modulo the linear subvarieties S:x"=constant if the equation P$_{m}$(x, .zeta.$_{1}$, .xi.')=0 for .zeta.$_{1}$ has only real roots when .xi.'is real and .xi."=0, where P$_{m}$ is the principal symbol of pp. Limiting to the case of operators with constant coefficients, A. Kaneko proposed a new sharper condition when S is a hyperplane [3]. In this paper, we generalize this condition to the case of general linear subvariety S and show that it is sufficient for the solvability of Cauchy problem for the hyperfunction Cauchy data which contains variables parallel to S as holomorphic parameters.blem for the hyperfunction Cauchy data which contains variables parallel to S as holomorphic parameters.

• East Asian mathematical journal
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• v.16 no.2
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• pp.205-214
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• 2000

In this paper, we shall give new characterizations of best approximation element in linear 2-normed spaces in terms of bounded linear 2-functionals and 2-hyperplanes.

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.29-34
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• 1993

J. Leray [7] proposed a sufficient condition ofr the solvability of the Cauchy problem on the initial hyperplane x$_{1}$=0 with Cauchy data which are holomorphic with respect to the variables parallel to some analytic subvariety S of the initial hyperplane. Limiting the problem to the case of operators with constant coefficients, A. Kaneko [2] proposed a new sharper sufficient condition. Later we generalized this condition and showed that it is necessary and sufficient for the solvability of the Cauchy problem for the hyperfunction Cauchy data and the distribution Cauchy data which contain variables parallel to S as holomorphic parameters in [5, 6]. In this paper, we extend the results in [6] to the case of operators with variable coefficients and show that it is sufficient for the solvability of the Cauchy problem for the hyperfunction Cauchy data. Our main theorem can be considered as an example of a deep theorem on micro-hyperbolic systems by Kashiwara-Schapira [4] and we give a direct proof based on an elementary sweeping out procedure developed in Kaneko [3].

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1485-1500
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• 2021

A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curves that are also slant helices in three-dimensional space as geodesics of circular cones. In addition, we consider curves that lie on a moving hyperplane normal to (i) one of the normal vector fields of the Frenet frame and to (ii) a rotation minimizing vector field along the curve. The former class is characterized in terms of the constancy of a certain vector field normal to the curve, while the latter contains spherical and plane curves. Finally, we establish a formal mapping between rectifying curves in an (m + 2)-dimensional space and spherical curves in an (m + 1)-dimensional space.

• 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.

• Journal of IKEEE
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• v.13 no.2
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• pp.140-149
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• 2009

Traffic anomaly detection is one of important technology that should be considered in network security and administration. In this paper, we propose an abnormal traffic detection mechanism that includes traffic monitoring and traffic analysis. We develop analytical passive monitoring system called WISE-Mon which can inspect traffic behavior. We establish a criterion by analyzing the characteristics of a traffic training set. To detect abnormal traffic, we derive a hyperplane by using Fisher linear discriminant and chi-square distribution as well as the analyzed characteristics of traffic. Our mechanism can support reliable results for traffic anomaly detection and is compatible to real-time detection. In addition, since the trend of traffic can be changed as time passes, the hyperplane has to be updated periodically to reflect the changes. Accordingly, we consider the self-learning algorithm which reflects the trend of the traffic and so enables to increase the pliability of detection probability. Numerical results are presented to validate the accuracy of proposed mechanism. It shows that the proposed mechanism is reliable and relevant for traffic anomaly detection.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.361-373
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• 2022

We study the factoriality of a nodal quartic hypersurface V₄ in ℙ⁴ when there is a hyperplane in ℙ⁴ containing all the nodes of V₄. As an application, we obtain new examples of irrational quartic 3-folds.