• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.279-286
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• 2019

Complex Grassmann manifolds $G_{n,k}$ are a generalization of complex projective spaces and have many important features some of which are captured by the $Pl{\ddot{u}}cker$ embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$ where $N=\(^n_k\)$. The problem of existence of cross sections of fibrations can be studied using the Gottlieb group. In a more generalized context one can use the relative evaluation subgroup of a map to describe the cohomology of smooth fiber bundles with fiber the (complex) Grassmann manifold $G_{n,k}$. Our interest lies in making use of techniques of rational homotopy theory to address problems and questions involving applications of Gottlieb groups in general. In this paper, we construct the Sullivan minimal model of the (complex) Grassmann manifold $G_{n,k}$ for $2{\leq}k<n$, and we compute the rational evaluation subgroup of the embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$. We show that, for the Sullivan model ${\phi}:A{\rightarrow}B$, where A and B are the Sullivan minimal models of ${\mathbb{C}}P^{N-1}$ and $G_{n,k}$ respectively, the evaluation subgroup $G_n(A,B;{\phi})$ of ${\phi}$ is generated by a single element and the relative evaluation subgroup $G^{rel}_n(A,B;{\phi})$ is zero. The triviality of the relative evaluation subgroup has its application in studying fibrations with fibre the (complex) Grassmann manifold.

• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.623-669
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• 2018

In Donaldson-Thomas theory, moduli spaces are locally the critical locus of a holomorphic function defined on a complex manifold. In this paper, we develop a theory of critical virtual manifolds which are the gluing of critical loci of holomorphic functions. We show that a critical virtual manifold X admits a natural semi-perfect obstruction theory and a virtual fundamental class $[X]^{vir}$ whose degree $DT(X)=deg[X]^{vir}$ is the Euler characteristic ${\chi}_{\nu}$(X) weighted by the Behrend function ${\nu}$. We prove that when the critical virtual manifold is orientable, the local perverse sheaves of vanishing cycles glue to a perverse sheaf P whose hypercohomology has Euler characteristic equal to the Donaldson-Thomas type invariant DT(X). In the companion paper, we proved that a moduli space X of simple sheaves on a Calabi-Yau 3-fold Y is a critical virtual manifold whose perverse sheaf categorifies the Donaldson-Thomas invariant of Y and also gives us a mathematical theory of Gopakumar-Vafa invariants.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1285-1296
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• 2010

We give criterions for a submanifold to be an extrinsic sphere and to be a totally geodesic submanifold by observing some Frenet curves of order 2 on the submanifold. We also characterize constant isotropic immersions into arbitrary Riemannian manifolds in terms of Frenet curves of proper order 2 on submanifolds. As an application we obtain a characterization of Veronese embeddings of complex projective spaces into complex projective spaces.

• Bulletin of the Korean Mathematical Society
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• v.29 no.2
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• pp.237-243
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• 1992

Let (M, g, J) be a closed Kahler manifold of complex dimension m > 1. We denote by Spec(M,g) the spectrum of the real Laplace-Beltrami operator. DELTA. acting on functions on M. The following characterization problem on the spectral rigidity of the complex projective space (CP$^{m}$ , g$_{0}$ , J$_{0}$ ) with the standard complex structure J$_{0}$ and the Fubini-Study metric g$_{0}$ has been attacked by many mathematicians : if (M,g,J) and (CP$^{m}$ ,g$_{0}$ ,J$_{0}$ ) are isospectral then is it true that (M,g,J) is holomorphically isometric to (CP$^{m}$ ,g$_{0}$ ,J$_{0}$ )\ulcorner In [BGM], [LB], it is proved that if (M,J) is (CP$^{m}$ , J$_{0}$ ) then the answer to the problem is affirmative. Tanno ([Ta]) has proved that the answer is affirmative if m .leq. 6. Recently, Wu([Wu]) has showed in a more general sense that if (M, g) and (CP$^{m}$ ,g$_{0}$ ) are (-4/m)-isospectral, m .geq. 4, and if the second betti number b$_{2}$(M) is equal to b$_{2}$(CP$^{m}$ ).

• Journal of KIISE:Software and Applications
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• v.36 no.11
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• pp.960-966
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• 2009

A manifold is used to represent a relationship between high-dimensional data samples in low-dimensional space. In human pose estimation, it is created in low-dimensional space for processing image and 3D body configuration data. Manifold learning is to build a manifold. But it is vulnerable to silhouette variations. Such silhouette variations are occurred due to view-change, person-change, distance-change, and noises. Representing silhouette variations in a single manifold is impossible. In this paper, we focus a silhouette variation problem occurred by view-change. In previous view invariant pose estimation methods based on manifold learning, there were two ways. One is modeling manifolds for all view points. The other is to extract view factors from mapping functions. But these methods do not support one by one mapping for silhouettes and corresponding body configurations because of unsupervised learning. Modeling manifold and extracting view factors are very complex. So we propose a method based on triple manifolds. These are view manifold, pose manifold, and body configuration manifold. In order to build manifolds, we employ biased manifold learning. After building manifolds, we learn mapping functions among spaces (2D image space, pose manifold space, view manifold space, body configuration manifold space, 3D body configuration space). In our experiments, we could estimate various body poses from 24 view points.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.1009-1038
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• 1996

R.L. Bishop and S.I. Goldberg [3] introduced the notion of totally real bisectional curvature B(X, Y) on a Kaehler manifold M. It is determined by a totally real plane [X, Y] and its image [JX, JY] by the complex structure J. where [X, Y] denotes the plane spanned by linealy independent vector fields X, and Y. Moreover the above two planes [X, Y] and [JX, JY] are orthogonal to each other. And it is known that two orthonormal vectors X and Y span a totally real plane if and only if X, Y and JY are orthonormal.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.409-421
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• 2018

Let X be a complex manifold of dimension n $n{\geqslant}2$ and let ${\Omega}{\Subset}X$ be a weakly q-convex domain with smooth boundary. Assume that E is a holomorphic line bundle over X and $E^{{\otimes}m}$ is the m-times tensor product of E for positive integer m. If there exists a strongly plurisubharmonic function on a neighborhood of $b{\Omega}$, then we solve the ${\bar{\partial}}$-problem with support condition in ${\Omega}$ for forms of type (r, s), $s{\geqslant}q$ with values in $E^{{\otimes}m}$. Moreover, the solvability of the ${\bar{\partial}}_b$-problem on boundaries of weakly q-convex domains with smooth boundary in $K{\ddot{a}}hler$ manifolds are given. Furthermore, we shall establish an extension theorem for the ${\bar{\partial}}_b$-closed forms.

• Journal of Power System Engineering
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• v.9 no.2
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• pp.19-25
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• 2005

Empirical experiments have been undertaken to investigate the effects of Intake Pulsating Flow on volumetric efficiency in a diesel engine. Waves occurs in the manifolds of engine owing to the periodic nature of the induction and exhaust processes caused by piston motion. During induction process, as waves travel both directions, they are reflected and interacted each other and pressure waves are transmitted through it. Hence, the flow become more complex and unsteady flow. These pressure waves act upon intake pulsating flow and affects on volumetric efficiency. In this paper the effects of change in length of induction pipes and wide range of engine speed on volumetric efficiency was examined and evaluated. It was found that volumetric efficiency was affected by intake pulsating flow with engine speed and the pipe length. The results obtained were considered by adopting a theory of wave action.

• 한국전산유체공학회:학술대회논문집
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• 2005.10a
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• pp.229-234
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• 2005

Following the recent trend in the automotive manufacturing technologies, the product design subject to the die casting becomes more and more complex. The requirement of the die design becomes more demanding than ever before. In some cases the product's shape can have multiple slender manifolds. In such cases, design of the inlet and outlet parts of the die is very important in the whole manufacturing process. The main issues required for the qualified products are to attain gentle and uniform flow of the molten liquid within the passages of the die. To satisfy such issues, the inlet cylinder ('bed cylinder' in this paper) must be as large as possible and simultaneously the outlet opening at the end of each passage must be as small as possible. However these in turn obviously bring additional manufacturing costs caused by re-melting of the bed cylinder and increased power due to the small outlet-openings. The purpose of this paper is to develop effective simulation methods of calculation for fluid flows in multiple columns, which mimic the actual complex design, and to get some useful information which can give some contributions to the die-casting industry. We have used a commercial code CFX in the numerical simulation. The primary parameter involved is the size of the air-vent. We will show how the very small opening of the outlet, i.e. the air-vent, can be treated with the aid of the porous model provided in the code. To check the validity of the numerical results we have also conducted a simple experiment by using water.

• Journal of Mechanical Science and Technology
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• v.20 no.10
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• pp.1765-1772
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• 2006

High-pressure die casting such as thixocasting and rheocasting is an effective process in the manufacturing automotive parts. Following the recent trend in the automotive manufacturing technologies, the product design subject to the die casting becomes more and more complex. Simultaneously the injection speed is also designed to be very high to establish a short cycletime. Thus, the requirement of the die design becomes more demanding than ever before. In some cases the product's shape can have multiple slender manifolds. In such cases, design of the inlet and outlet parts of the die is very important in the whole manufacturing process. The main issues required for the qualified products are to attain gentle and uniform flow of the molten liquid within the passages of the die. To satisfy such issues, the inlet cylinder ('bed cylinder' in this paper) must be as large as possible and simultaneously the outlet opening at the end of each passage must be as small as possible. However these in turn obviously bring additional manufacturing costs caused by re-melting of the bed cylinder and increased power due to the small outlet-openings. The purpose of this paper is to develop effective simulation methods of calculation for fluid flows in multiple columns, which mimic the actual complex design, and to get some useful information which can give some contributions to the die-casting industry. We have used a commercial code CFX in the numerical simulation. The primary parameter involved is the size of the bed cylinder. We will show how the very small opening of the outlet can be treated with the aid of the porous model provided in the code. To check the validity of the numerical results we have also conducted a simple experiment by using water.