• Bulletin of the Korean Mathematical Society
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• v.45 no.3
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• pp.523-561
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• 2008

The purpose of this paper is to survey the use of the important method of scaling in analysis, and particularly in complex analysis. Applications are given to the study of automorophism groups, to canonical kernels, to holomorphic invariants, and to analysis in infinite dimensions. Current research directions are described and future paths indicated.

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1057-1073
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• 2020

In this paper, we study some of the factorization aspects of rational multicyclic monoids, that is, additive submonoids of the nonnegative rational numbers generated by multiple geometric sequences. In particular, we provide a complete description of the rational multicyclic monoids M that are hereditarily atomic (i.e., every submonoid of M is atomic). Additionally, we show that the sets of lengths of certain rational multicyclic monoids are finite unions of multidimensional arithmetic progressions, while their unions satisfy the Structure Theorem for Unions of Sets of Lengths. Finally, we realize arithmetic progressions as the sets of distances of some additive submonoids of the nonnegative rational numbers.

• The Pure and Applied Mathematics
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• v.12 no.4 s.30
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• pp.275-287
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• 2005

Geometric invariants are basic tools for geometric processing and computer vision. In this paper, we give a linear approximation for the differentiation of the binormal vector field of a space curve by using the forward and backward differences of discrete binormal vectors. Two kind of discrete torsion, say, back-ward torsion $T_b$ and forward torsion $T_f$ can be defined by the dot product of the (backward and forward) discrete differentiation of binormal vectors that are linear approximations of torsion. Using Frenet formula and Taylor series expansion, we give error estimations for the discrete torsions. We also give numerical tests for a curve. Notably the average of $T_b$ and $T_f$ looks more stable in errors.

• Journal of Broadcast Engineering
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• v.20 no.6
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• pp.912-920
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• 2015

Unlike general texture-based feature description algorithms, Locally Likely Arrangement Hashing (LLAH) algorithm describes a feature based on the geometric relationship between its neighbors. Thus, even in poor-textured scenes or large camera pose changes, it can successfully describe and track features and enables to implement augmented reality. This paper aims to optimize the performance of LLAH algorithm for tracking random dots (= features) with Gaussian noise. For this purpose, images with different number of features and magnitude of Gaussian noise are prepared. Then, the performance of LLAH algorithm according to the conditions: the number of neighbors, the type of geometric invariants, and the distance between features, is analyzed, and the optimal conditions are determined. With the optimal conditions, each feature could be matched and tracked in real-time with a matching rate of more than 80%.

• Proceedings of the IEEK Conference
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• 2002.07b
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• pp.1141-1144
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• 2002

The research on computer graphics(CG) has been actively studied and developed. Namely, many surface/solid models have been proposed in the field of computer aided geometric design as well as the one of CG. Since it is difficult to visualize the complex shape exactly, an approximation by generating a set of meshes is usually used. Therefore it is important to guarantee the quality of the approximation in consideration of the computational cost. In this paper, a mesh generation algorithm will be proposed for a surface defined by Lie algebra. The proposed algorithm considers the quality in the meaning of validation of invariants obtained by the mesh, using automatic differentiation.

• Proceedings of the IEEK Conference
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• 1999.06a
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• pp.639-642
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• 1999

There exist geometrically invariant relations in single-view images under a specific geometrical structure. This invariance may be utilized for 3D object recognition. Two types of invariants are compared in terms of the robustness to the variation of the feature points. Deviation of the invariant relations are measured by adding random noise to the feature point location. Zhu’s invariant requires six points on adjacent planes having two sets of four coplanar points, whereas the Kaist method requires four coplanar points and two non-coplanar points. Experimental results show that the latter method has the advantage in choosing feature points while suffering from weak robustness to the noise.

• Proceedings of the IEEK Conference
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• 2000.07b
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• pp.1103-1106
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• 2000

Recently, the research on computer graphics (CG) has been actively studied and developed. Namely, many surface/solid models have been proposed in the field of computer aided geometric design as well as the one of CG. Since it is difficult to visualize the complex shape exactly, an approximation by generating a set of meshes is usually used. Therefore it is important to guarantee the quality of the approximation in consideration of the computational cost. In this paper, a mesh generation algorithm will be proposed for a surface defined by linear tie algebra The proposed algorithm which considers the quality in the meaning of validation of invariants obtained by the mesh.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.333-349
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• 2019

The paper study the curvature theory of a line-trajectory of constant Disteli-axis, according to the invariants of the axodes of moving body in spatial motion. A necessary and sufficient condition for a line-trajectory to be a constant Disteli-axis is derived. From which new proofs of the Disteli's formulae and concise explicit expressions of the inflection line congruence are directly obtained. The obtained explicit equations degenerate into a quadratic form, which can easily give a clear insight into the geometric properties of a line-trajectory of constant Disteli-axis with the theory of line congruence. The degenerated cases of the Burmester lines are discussed according to dual points having specific trajectories.

• Kyungpook Mathematical Journal
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• v.61 no.4
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• pp.791-803
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• 2021

Earlier investigators have made detailed studies of geometric properties such as integrability, partial integrability, and invariants, such as the fundamental 2-form, of some canonical f-structures, such as f³ ± f = 0, on the frame bundle FM. Our aim is to study metallic structures on the frame bundle: polynomial structures of degree 2 satisfying F² = pF +qI where p, q are positive integers. We introduce a tensor field F_α, α = 1, 2…, n on FM show that it is a metallic structure. Theorems on Nijenhuis tensor and integrability of metallic structure F_α on FM are also proved. Furthermore, the diagonal lifts g^D and the fundamental 2-form Ω_α of a metallic structure F_α on FM are established. Moreover, the integrability condition for horizontal lift F_α^H of a metallic structure F_α on FM is determined as an application. Finally, the golden structure that is a particular case of a metallic structure on FM is discussed as an example.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1299-1320
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• 2023

In the present paper, firstly we obtain the general expression of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres, pseudo hyperbolic hyperspheres and null hyper-cones whose centers lie on a non-null curve with non-null Frenet vector fields in E⁴₁ and give their some geometric invariants such as unit normal vector fields, Gaussian curvatures, mean curvatures and principal curvatures. Also, we give some results about their flatness and minimality conditions and Weingarten canal hypersurfaces. Also, we obtain these characterizations for tubular hypersurfaces in E⁴₁ by taking constant radius function and finally, we construct some examples and visualize them with the aid of Mathematica.