• Bulletin of the Korean Mathematical Society
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• v.24 no.1
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• pp.31-37
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• 1987

I want to focus on developments in the areas of general relativity and gauge theory. The topics to be considered are the singularity theorms of Hawking and Penrose, the positivity of mass, instantons on the four-dimensional sphere, and the string picture of quantum gravity. I should mention that I will not have time do discuss either classical mechanics or symplectic structures. This is especially unfortunate, because one of the roots of differential geometry is planted firmly in mechanics, Cf. [GS]. The French geometer Elie Cartan first formulated his invariant approach to geometry in a series of papers on affine connections and general relativity, Cf. [C]. Cartan was trying to recast the Newtonian theory of gravity in the same framework as Einstein's theory. From the historical perspective it is significant that Cartan found relativity a convenient framework for his ideas. As about the same time Hermann Weyl in troduced the idea of gauge theory into geometry for purposes much different than those for which it would ultimately prove successful, Cf. [W]. Weyl wanted to unify gravity with electromagnetism and though that a conformal structure would fulfill thel task but Einstein rebutted this approach.

• Water Engineering Research
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• v.6 no.2
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• pp.49-61
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• 2005

This paper presents the numerical study to examine characteristics of fluid flow and solute transport in a rough fracture subject to effective normal stresses. The aperture distribution is generated by using the self-affine fractal model. In order to represent a nonlinear relationship between the supported normal stress and the fracture aperture, we combine a simple mechanical model with the local flow model. The solute transport is simulated using the random walk particle following algorithm. Results of numerical simulations show that the flow is significantly affected by the geometry of aperture distribution varying with the effective normal stress level while it is slightly affected by the fractal dimension that determines the degree of the fracture surface roughness. However, solute transport is influenced by the effective normal stress as well as the fracture surface roughness.

• Proceedings of the Korean Society for Technology of Plasticity Conference
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• 2009.10a
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• pp.33-37
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• 2009

Finite-element methods are used to study non-adhesive, frictionless rough contact of elastic and plastic solids. Roughness on spherical surfaces is realized by self-affine fractal. True contact area between the rough surfaces and flat rigid surfaces increases with power law under external normal loads. The power exponent is sensitive to surface roughness as well as the curvature of spherical geometry. Surface contact pressures are analyzed and compared for the elastic and plastic solids. Distributions of local contact pressure are shown dependent on the surface roughness and the yield stress of plastic solids.

• Journal of the Korea Computer Graphics Society
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• v.9 no.3
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• pp.15-21
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• 2003

An algorithm is developed for augmenting a real video with virtual graphics objects without computing Euclidean information. Real motion of the camera is obtained in affine space by a direct linear method using image matches. Then, virtual camera is provided by determining the locations of four basis points in two input images as initialization process. The four pairs of 2D location and its 3D affine coordinates provide Euclidean orthographic projection camera through the whole video sequence. Our method has the capability of generating views of objects shaded by virtual light sources, because we can make use of all the functions of the graphics library written on the basis of Euclidean geometry. Our novel formulation and experimental results with real video sequences are presented.

• Journal of the Korean Mathematical Society
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• v.33 no.4
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• pp.1123-1128
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• 1996

Let A be an (nonassociative) algebra with multiplication xy over a field F, and denote by $A^-$ the algebra with multiplication [x, y] = xy - yx$ defined on the vector space A. If $A^-$ is a Lie algebra, then A is called Lie-admissible. Lie-admissible algebras arise in various topics, including geometry of invariant affine connections on Lie groups and classical and quantum mechanics(see [2, 5, 6, 7] and references therein).

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.849-866
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• 2024

In this paper, we study the rectifying curves in multiplicative Euclidean space of dimension 3, i.e., those curves for which the position vector always lies in its rectifying plane. Since the definition of rectifying curve is affine and not metric, we are directly able to perform multiplicative differential-geometric concepts to investigate such curves. By several characterizations, we completely classify the multiplicative rectifying curves by means of the multiplicative spherical curves.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.51 no.5
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• pp.212-219
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• 2002

In this paper, the epipolar geometry, genera17y used as a pin-hole camera model, is newly adapted to our proposed method that enables the affine reconstruction of the 3D object from two projected views. The proposed method models the projective projection of inherent X-ray imaging system, obviates the need to attach artifirially constructed material on the body, and requires none of the prior-knowledge regarding to intrinsic and extrinsic parameters of two X-ray imaging systems. The optimum numerical solution is obtained by applying the least mean square estimator to corresponding points on two projected X-ray planes. The performance of this proposed method is Quantitatively analyzed using computer synthesized model of Cochlear implantation electrodes. In simulated experiments, the propnsed method is insensitive to the added random noise, the scaling factor change, the center point change, and rotational angular change between two projection planes, as well as enables the stable 3D reconstruction in least square sense even in worst testing cases.

• Proceedings of the KSRS Conference
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• 2005.10a
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• pp.182-185
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• 2005

Satellite cameras are calibrated before launch in detail and in general, but it cannot be guaranteed that the geometry is not changing during launch and caused by thermal influence of the sun in the orbit. Modem satellite imaging systems are based on CCD-line sensors. Because of the required high sampling rate the length of used CCD-lines is limited. For reaching a sufficient swath width, some CCD-lines are combined to a longer virtual CCD-line. The images generated by the individual CCD-lines do overlap slightly and so they can be shifted in x- and y-direction in relation to a chosen reference image just based on tie points. For the alignment and difference in scale, control points are required. The resulting virtual image has only negligible errors in areas with very large difference in height caused by the difference in the location of the projection centers. Color images can be related to the joint panchromatic scenes just based on tie points. Pan-sharpened images may show only small color shifts in very mountainous areas and for moving objects. The direct sensor orientation has to be calibrated based on control points. Discrepancies in horizontal shift can only be separated from attitude discrepancies with a good three-dimensional control point distribution. For such a calibration a program based on geometric reconstruction of the sensor orientation is required. The approximations by 3D-affine transformation or direct linear transformation (DL n cannot be used. These methods do have also disadvantages for standard sensor orientation. The image orientation by geometric reconstruction can be improved by self calibration with additional parameters for the analysis and compensation of remaining systematic effects for example caused by a not linear CCD-line. The determined sensor geometry can be used for the generation? of rational polynomial coefficients, describing the sensor geometry by relations of polynomials of the ground coordinates X, Y and Z.

• Proceedings of the Korean Society of Surveying, Geodesy, Photogrammetry, and Cartography Conference
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• 2007.04a
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• pp.135-140
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• 2007

Due to the expensiveness of IKONOS Pro and Precision Products, it is attractive to use the low-cost IKONOS Geo Product with vendor-provided RPCs to produce highly accurate mapping products. The imaging geometry of IKONOS high-resolution imagery is described by RFs instead of rigorous sensor models. This paper presents four different models defined respectively in object space and image space to improve the accuracies of the RF-derived ground coordinates. The four models include the offset model, the scale & offset model, the affine model and the 2nd-order polynomial model. Different configurations of ground control points (GCPs) are carefully examined to evaluate the effect of the GCPs arrangement on the accuracy of ground coordinates. The experiment also evaluates the effect of different cartographic parameters such as the number location, and accuracy of GCPs on the accuracy of geopositioning.

• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.647-673
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• 2002

There are many models to study topological $R^2$-planes. Unlike topological $R^2$-planes, it is difficult to find models to study topological R$^3$)-spaces. If an 4-dimensional affine plane intersects with R$^3$, we are able to get a geometrical structure on R$^3$ which is similar to R$^3$-space, and called $R^2$-divisible R$^3$-space. Such spatial geometric models is useful to study topological R$^3$-spaces. Hence, we introduce some classes of topological $R^2$-divisible R$^3$-spaces which are induced from 4-dimensional anne planes.