• Tunnel and Underground Space
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• v.27 no.1
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• pp.39-49
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• 2017

The factor of safety (FOS) is commonly used as an index to quantitatively state the degree of safety of various rock structures. Therefore it is important to understand the definition and characteristics of the adopted FOS because the calculated FOS may be different according to the definition of FOS even if it is estimated under the same stress condition. In this study, four local factors of safety based on maximum shear stress, maximum shear strength, stress invariants, and maximum principal stress were defined using the Mohr-Coulomb and Hoek-Brown failure criteria. Then, the variation characteristics of each FOS along five stress paths were investigated. It is shown that the local FOS based on the shear strength, which is widely used in the stability analysis of rock structures, results in a higher FOS value than those based on the maximum principal stress and the stress invariants. This result implies that the local FOS based on the maximum shear stress or the stress invariants is more necessary than the local FOS based on the shear strength when the conservative rock mechanics design is required. In addition, it is shown that the maximum principal stresses at failure may reveal a large difference depending on the stress path.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.871-878
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• 2007

We define a numerical invariant $row_{CM}(A)$ over Cohen-Macaulay local ring A, which is related to rows of the presenting matrices of maximal Cohen-Macaulay modules without free summands. We show that $row(A)=row_{CM}(A)$ for a Cohen-Macaulay(not necessarily Gorenstein) local ring A.

• Journal of the Korean Mathematical Society
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• v.56 no.4
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• pp.1063-1081
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• 2019

In this paper, we formulate a new local move on virtual knot diagram, called arc shift move. Further, we extend it to another local move called region arc shift defined on a region of a virtual knot diagram. We establish that these arc shift and region arc shift moves are unknotting operations by showing that any virtual knot diagram can be turned into trivial knot using arc shift (region arc shift) moves. Based upon the arc shift move and region arc shift move, we define two virtual knot invariants, arc shift number and region arc shift number respectively.

• Journal of Korea Multimedia Society
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• v.1 no.2
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• pp.173-182
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• 1998

As a result of remarkable developments in multimedia technology, the image database system that can efficiently retrieve image data becomes a core technology of information-oriented society. In this paper, we proposed the 2-phase Image Retrieval System considering both color and shape information as the method of image features extraction for content-based image data retrieval. At the first level, to get color information, with improving and extending the indexing method using color distribution characteristic suggested by Striker et al., i.e. the indexing method considering local color distribution characteristics, the system roughly classifies images through the improved method. At the second level, the system finally retrieves the most similar image from the image queried by the user using the shape information about the image groups classified at the first level. To extract the shape information, we use the Improved Moment Invariants (IMI) that manipulates only the pixels on the edges of objects in order to overcome two main problems of the existing Moment Invariant methods large amount of processing and rotation sensitiveness which can frequently be seen in the Directive Histogram Intersection technique suggested by Jain et al. Experiments have been conducted on 300 automobile images. And we could obtain the more improved results through the comparative test with other methods.

• Journal of the Korea Society of Computer and Information
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• v.5 no.2
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• pp.60-70
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• 2000

As a result of remarkable developments in computer technology, the image retrieval system that can efficiently retrieve image data becomes a core technology of information-oriented society. In this paper, we implemented the Hierarchical Image Retrieval System for content-based image data retrieval. At the first level, to get color information, with improving the indexing method using color distribution characteristic suggested by Striker et al., i.e. the indexing method considering local color distribution characteristics, the system roughly classifies images through the improved method. At the second level, the system finally retrieves the most similar image from the image queried by the user using the shape information about the image groups classified at the first level. To extract the shape information, we use the Improved Moment Invariants(IMI) that manipulates only the pixels on the edges of objects in order to overcome two main problems of the existing Moment Invariant methods large amount of processing and rotation sensitiveness which can frequently be seen in the Directive Histogram Intersection technique suggested by Jain et al. Experiments have been conducted on 300 automobile images And we could obtain the more improved results through the comparative test with other methods.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.875-884
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• 2016

Let x : $^{Mn-1}{\rightarrow}{\mathbb{R}}^n$ ($n{\geq}4$) be an umbilical free hyper-surface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. We denote the Laguerre scalar curvature by R and the trace-free Laguerre tensor by ${\tilde{L}}:=L-{\frac{1}{n-1}}tr(L)g$. In this paper, we prove a local classification result under the assumption of parallel Laguerre form and an inequality of the type $${\parallel}{\tilde{L}}{\parallel}{\leq}cR$$ where $c={\frac{1}{(n-3){\sqrt{(n-2)(n-1)}}}$ is appropriate real constant, depending on the dimension.

• The Pure and Applied Mathematics
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• v.25 no.2
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• pp.95-113
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• 2018

A mathematical knot is an embedded circle in ${\mathbb{R}}^3$. A fundamental problem in knot theory is classifying knots up to its numbers of crossing points. Knots are often distinguished by using a knot invariant, a quantity which is the same for equivalent knots. Knot polynomials are one of well known knot invariants. In 2006, J. Przytycki showed the effects of a n - move (a local change in a knot diagram) on several knot polynomials. In this paper, the authors review about knot polynomials, especially Jones polynomial, and give an alternative proof to a part of the Przytychi's result for the case n = 3 on the Jones polynomial.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.34 no.1
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• pp.37-53
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• 2021

In this paper we define scaled visual curvature and visual Frenet frame that can be visually accepted for discrete space curves. Scaled visual curvature is relatively simple compared to multi-scale visual curvature and easy to control the influence of noise. We adopt scaled minimizing directions of height functions on each neighborhood. Minimizing direction at a point of a curve is a direction that makes the point a local minimum. Minimizing direction can be given by a small noise around the point. To reduce this kind of influence of noise we exmine the direction whether it makes the point minimum in a neighborhood of some size. If this happens we call the direction scaled minimizing direction of C at p ∈ C in a neighborhood Br(p). Normal vector of a space curve is a second derivative of the curve but we characterize the normal vector of a curve by an integration of minimizing directions. Since integration is more robust to noise, we can find more robust definition of discrete normal vector, visual normal vector. On the other hand, the set of minimizing directions span the normal plane in the case of smooth curve. So we can find the tangent vector from minimizing directions. This lead to the definition of visual tangent vector which is orthogonal to the visual normal vector. By the cross product of visual tangent vector and visual normal vector, we can define visual binormal vector and form a Frenet frame. We examine these concepts to some discrete curve with noise and can see that the scaled visual curvature and visual Frenet frame approximate the original geometric invariants.