• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1989-2000
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• 2013

We show that for any given closed orientable 3-manifold M with a Heegaard surface of genus g, any positive integers b and n, there exists a knot K in M which admits a (g, b)-bridge splitting of distance greater than n with respect to the Heegaard surface except for (g, b) = (0, 1), (0, 2).

• Journal of Biomedical Engineering Research
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• v.45 no.4
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• pp.179-186
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• 2024

This paper investigates the impact of Riemannian Procrustes Analysis (RPA) on enhancing the classification performance of SPD-Net when applied to EEG signals across different sessions and subjects. EEG signals, known for their inherent individual variability, are initially transformed into Symmetric Positive Definite (SPD) matrices, which are naturally represented on a Riemannian manifold. To mitigate the variability between sessions and subjects, we employ RPA, a method that geometrically aligns the statistical distributions of these matrices on the manifold. This alignment is designed to reduce individual differences and improve the accuracy of EEG signal classification. SPD-Net, a deep learning architecture that maintains the Riemannian structure of the data, is then used for classification. We compare its performance with the Minimum Distance to Mean (MDM) classifier, a conventional method rooted in Riemannian geometry. The experimental results demonstrate that incorporating RPA as a preprocessing step enhances the classification accuracy of SPD-Net, validating that the alignment of statistical distributions on the Riemannian manifold is an effective strategy for improving EEG-based BCI systems. These findings suggest that RPA can play a role in addressing individual variability, thereby increasing the robustness and generalization capability of EEG signal classification in practical BCI applications.

• The Pure and Applied Mathematics
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• v.11 no.4
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• pp.309-321
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• 2004

For a complete open Riemannian manifold, the ideal boundary consists of points at infinity. The so-called Busemann-functions play the role of distance functions for points at infinity. We study the similarity and difference between Busemann-functions and ordinary distance functions.

• Bulletin of the Korean Mathematical Society
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• v.48 no.1
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• pp.67-77
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• 2011

Let M be a compact orientable closed 3-manifold, and F a non-separating incompressible closed surface in M. Let M' = M - ${\eta}(F)$, where ${\eta}(F)$ is an open regular neighborhood of F in M. In the paper, we give a lower bound of genus of self-amalgamation of minimal Heegaard splitting $V'\;{\cup}_{S'}\;W'$ of M' under some conditions on the distance of the Heegaard splitting.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.515-520
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• 2019

We give new evidence that "complicated" Heegaard surfaces behave like incompressible surfaces. More precisely, suppose that a closed connected orientable 3-manifold M contains a closed connected incompressible surface F which separates M into two (connected) components $M_1$ and $M_2$. Let S be a Heegaard surface of M. Our result is that if the Hempel distance of S is at least four, then S is isotoped so that $S{\cap}M_i$ is incompressible for each i = 1, 2.

• Proceedings of the Korean Statistical Society Conference
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• 2003.05a
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• pp.31-36
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• 2003

Using a short time expansion of the fundamental solution of heat equation by analysis of Wiener functional with the help of Malliavin calculus, we obtain the asymptotic expansion of the mean distance of Brownian motion on Riemannian manifolds.

• The Journal of the Korea Contents Association
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• v.7 no.3
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• pp.9-16
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• 2007

This paper describe methodology that is distributed on 2-dimensional plane to much high-dimensional facial motion datas using Isomap algorithm, and user interface techniques to control facial expressions by selecting expressions while user navigates this space in real-time. Isomap algorithm is processed of three steps as follow; first define an adjacency expression of each expression data, and second, calculate manifold distance between each expressions and composing expression spaces. These facial spaces are created by calculating of the shortest distance(manifold distance) between two random expressions. We have taken a Floyd algorithm for it. Third, materialize multi-dimensional expression spaces using Multidimensional Scaling, and project two dimensions plane. The smallest adjacency distance to define adjacency expressions uses Pearson Correlation Coefficient. Users can control facial expressions of 3-dimensional avatar by using user interface while they navigates two dimension spaces by real-time.

• Clean Technology
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• v.18 no.4
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• pp.410-418
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• 2012

The performance of the ammonia injection gun (AIG) used for maximizing the utilization of reducing agent in the selective catalytic reduction (SCR) system is decided by several parameters such as the pattern of flow distribution, geometry of the air distribution manifold (ADM) and the array and geometry of nozzles. In the study, the uniformity of jet flows from the nozzles in AIG was analyzed statistically by using the computational fluid dynamics (CFD) method to evaluate the role of design parameters on the performance of the SCR system. The uniformity of jet flows from the nozzles is being deteriorated with increasing the supplying flow rate to the AIG. Distribution rates to each branch pipe become lower with decreasing distance to the header, and flow rates from nozzle are also reduced with decreasing distance to the header. The uniformity of jet flows from nozzles becomes stable significantly when the ratio of summative area of nozzles to each sectional area of the branch pipe is below 0.5.

• Proceedings of the Korean Information Science Society Conference
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• 2005.07b
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• pp.748-750
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• 2005

Isomap [1] is a manifold learning algorithm, which extends classical multidimensional scaling (MDS) by considering approximate geodesic distance instead of Euclidean distance. The approximate geodesic distance matrix can be interpreted as a kernel matrix, which implies that Isomap can be solved by a kernel eigenvalue problem. However, the geodesic distance kernel matrix is not guaranteed to be positive semidefinite. In this paper we employ a constant-adding method, which leads to the Mercer kernel-based Isomap algorithm. Numerical experimental results with noisy 'Swiss roll' data, confirm the validity and high performance of our kernel Isomap algorithm.

• Journal of Institute of Convergence Technology
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• v.10 no.1
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• pp.7-12
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• 2020

We describe a efficient surface reconstruction method that reconstructs a 3D manifold polygonal mesh approximately passing through a set of 3D oriented points. Our algorithm includes 3D convex hull, octree data structure, signed distance function (SDF), and marching cubes. The 3D convex hull provides us with a fast computation of SDF, octree structure allows us to compute a minimal distance for SDF, and marching cubes lead to iso-surface generation with SDF. Our approach gives us flexibility in the choice of the resolution of the reconstructed surface, and it also enables to use on low-level PCs with minimal peak memory usage. Experimenting with publicly available scan data shows that we can reconstruct a polygonal mesh from point cloud of sizes varying from 10,000 ~ 1,000,000 in about 1~60 seconds.