• Journal of Biomedical Engineering Research
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• v.28 no.6
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• pp.817-824
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• 2007

Tractography using Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) is a method to determine the architecture of axonal fibers in the central nervous system by computing the direction of the principal eigenvector in the white matter of the brain. However, the fiber tracking methods suffer from the noise included in the diffusion tensor images that affects the determination of the principal eigenvector. As the fiber tracking progresses, the accumulated error creates a large deviation between the calculated fiber and the real fiber. This problem of the DT-MRI tractography is known mathematically as the ill-posed problem which means that tractography is very sensitive to perturbations by noise. To reduce the noise in DT-MRI measurements, a tensor-valued median filter which is reported to be denoising and structure-preserving in fiber tracking, is applied in the tractography. In this paper, we proposed the modified gradient descent method which converges fast and accurately to the optimal tensor-valued median filter by changing the step size. In addition, the performance of the modified gradient descent method is compared with others. We used the synthetic image which consists of 45 degree principal eigenvectors and the corticospinal tract. For the synthetic image, the proposed method achieved 4.66%, 16.66% and 15.08% less error than the conventional gradient descent method for error measures AE, AAE, AFA respectively. For the corticospinal tract, at iteration number ten the proposed method achieved 3.78%, 25.71 % and 11.54% less error than the conventional gradient descent method for error measures AE, AAE, AFA respectively.

• Journal of the Korean Mathematical Society
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• v.35 no.4
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• pp.1045-1060
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• 1998

Recently, Chung and et al. ([11], 1991c) introduced a new concept of a manifold, denoted by ＊g-SE $X_{n}$ , in Einstein's n-dimensional ＊g-unified field theory. The manifold ＊g-SE $X_{n}$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor ＊ $g^{λν}$ through the SE-connection which is both Einstein and semi-symmetric. In this paper, they proved a necessary and sufficient condition for the unique existence of SE-connection and presented a beautiful and surveyable tensorial representation of the SE-connection in terms of the tensor ＊ $g^{λν}$. This paper is the first part of the following series of two papers: I. The SE-curvature tensor of ＊g-SE $X_{n}$ II. The contracted SE-curvature tensors of ＊g-SE $X_{n}$ In the present paper we investigate the properties of SE-curvature tensor of ＊g-SE $X_{n}$ , with main emphasis on the derivation of several useful generalized identities involving it. In our subsequent paper, we are concerned with contracted curvature tensors of ＊g-SE $X_{n}$ and several generalized identities involving them. In particular, we prove the first variation of the generalized Bianchi's identity in ＊g-SE $X_{n}$ , which has a great deal of useful physical applications.tions.

• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.287-311
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• 2023

In this paper, we introduce the notion of cyclic structure Jacobi semi-symmetric real hypersurfaces in the complex hyperbolic quadric Q^m* = SO⁰_2,m/SO₂SO_m. We give a classifiction of when real hypersurfaces in the complex hyperbolic quadric Q^m* having 𝔄-principal or 𝔄-isotropic unit normal vector fields have cyclic structure Jacobi semi-symmetric tensor.

• Journal of the Korean Mathematical Society
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• v.39 no.2
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• pp.193-203
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• 2002

Weyl structure can be viewed as generalizations of Riemannian metrics. We study Weyl structures which are critical points of the squared L$^2$ norm functional of the full curvature tensor, defined on the space of Weyl structures on a compact 4-manifold. We find some relationship between these critical Weyl structures and the critical Riemannian metrics. Then in a search for homogeneous critical structures we study left-invariant metrics on some solv-manifolds and prove that they are not critical.

• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.661-679
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• 1995

The concept of the structure conformal vector field C on a Sasakian manifold M is defined. The existence of such a C on M is determined by an exterior differential system in involution. In this case M is a foliate manifold and the vector field C enjoys the property to be exterior concurrent. This allows to prove some interesting properties of the Ricci tensor and Obata's theorem concerning isometries to a sphere. Different properties of the conformal Lie algebra induced by C are also discussed.

• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1441-1456
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• 2017

We consider a 3-dimensional Riemannian manifold M with a circulant metric g and a circulant structure q satisfying $q^3=id$. The structure q is compatible with g such that an isometry is induced in any tangent space of M. We introduce three classes of such manifolds. Two of them are determined by special properties of the curvature tensor. The third class is composed by manifolds whose structure q is parallel with respect to the Levi-Civita connection of g. We obtain some curvature properties of these manifolds (M, g, q) and give some explicit examples of such manifolds.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.641-652
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• 1998

Chung and et al. ([2].1991) introduced a new concept of a manifold, denoted by $^{\ast}g-SEX_n$, in Einstein's n-dimensional $^{\ast}g$-unified field theory. The manifold $^{\ast}g-SEX_n$ is a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $^{\ast}g^{\lambda \nu}$ through the SE-connection which is both Einstein and semi-symmetric. In this paper, they proved a necessary and sufficient condition for the unique existence of SE-connection and sufficient condition for the unique existence of SE-connection and presented a beautiful and surveyable tensorial representation of the SE-connection in terms of the tensor $^{\ast}g^{\lambda \nu}$. Recently, Chung and et al.([3],1998) obtained a concise tensorial representation of SE-curvature tensor defined by the SE-connection of $^{\ast}g-SEX_n$ and proved deveral identities involving it. This paper is a direct continuations of [3]. In this paper we derive surveyable tensorial representations of constracted curvature tensors of $^{\ast}g-SEX_n$ and prove several generalized identities involving them. In particular, the first variation of the generalized Bianchi's identity in $^{\ast}g-SEX_n$, proved in theorem (2.10a), has a great deal of useful physical applications.

• Proceedings of the Korean Society For Composite Materials Conference
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• 2003.04a
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• pp.184-187
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• 2003

Complete prediction of second order permeability tensor for three dimensional circular braided preform is critical to understand the resin transfer molding process of composites. The permeability can be predicted by considering resin flow through the multi-axial fiber structure. In this study, permeability tensor for a 3-D circular braided preform is calculated by solving a boundary problem of a periodic unit cell. Flow field through the unit cell is obtained by using a 3-D finite volume method (FVM) and Darcy's law is utilized to obtain permeability tensor. Flow analysis for two cases that a fiber tow is regarded as impermeable solid and permeable porous medium is carried out respectively. It is found that the flow within the intra-tow region of the braided preform is negligible if inter-tow porosity is relatively high but the flow through the tow must be considered when the porosity is low. To avoid checkerboard pressure field and improve the efficiency of numerical computation, a new interpolation function for velocity variation is proposed on the basis of analytic solutions. Permeability of the braided preform is measured through a radial flow experiment and compared with the permeability predicted numerically.

• Honam Mathematical Journal
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• v.38 no.1
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• pp.39-45
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• 2016

In this paper, we investigate several properties on a weakly symmetric structure on a Riemannian manifold.

• Proceedings of the KSME Conference
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• 2005.11a
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• pp.177.2-177.2
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• 2005