• International Journal of CAD/CAM
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• v.9 no.1
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• pp.103-109
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• 2010

Recently, various conformal geometric methods have been presented for non-rigid surface matching and registration. This work proposes to improve the robustness of conformal geometric methods to the boundaries by incorporating the symmetric information of the input surface. We presented two symmetric conformal mapping methods, which are based on solving Riemann-Cauchy equation and curvature flow respectively. Experimental results on geometric data acquired from real life demonstrate that the symmetric conformal mapping is insensitive to the boundary occlusions. The method outperforms all the others in terms of robustness. The method has the potential to be generalized to high genus surfaces using hyperbolic curvature flow.

• Honam Mathematical Journal
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• v.27 no.4
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• pp.655-663
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• 2005

One of the basic questions concerning harmonic map is on the existence of harmonic maps satisfying a certain condition. Rigidity of a certain harmonic map can be considered as an answer for this kinds of questions. In this article, we study a rigidity property of harmonic diffeomorphisms under the condition that the inverse map is also harmonic. We show that every such a harmonic diffeomorphism is totally geodesic or conformal in two dimensional case.

• Journal of the Korean Mathematical Society
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• v.60 no.1
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• pp.71-90
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• 2023

We construct generalized Cauchy-Riemann equations of the first order for a pair of two ℝ-valued functions to deform a minimal graph in ℝ³ to the one parameter family of the two dimensional minimal graphs in ℝ⁴. We construct the two parameter family of minimal graphs in ℝ⁴, which include catenoids, helicoids, planes in ℝ³, and complex logarithmic graphs in ℂ². We present higher codimensional generalizations of Scherk's periodic minimal surfaces.

• Journal of the Korean Mathematical Society
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• v.38 no.5
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• pp.1069-1105
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• 2001

Regular maps and hypermaps are cellular decompositions of closed surfaces exhibiting the highest possible number of symmetries. The five Platonic solids present the most familar examples of regular maps. The gret dodecahedron, a 5-valent pentagonal regular map on the surface of genus 5 discovered by Kepler, is probably the first known non-spherical regular map. Modern history of regular maps goes back at least to Klein (1878) who described in [59] a regular map of type (3, 7) on the orientable surface of genus 3. In its early times, the study of regular maps was closely connected with group theory as one can see in Burnside’s famous monograph [19], and more recently in Coxeter’s and Moser’s book [25] (Chapter 8). The present-time interest in regular maps extends to their connection to Dyck\`s triangle groups, Riemann surfaces, algebraic curves, Galois groups and other areas, Many of these links are nicely surveyed in the recent papers of Jones [55] and Jones and Singerman [54]. The presented survey paper is based on the talk given by the author at the conference “Mathematics in the New Millenium”held in Seoul, October 2000. The idea was, on one hand side, to show the relationship of (regular) maps and hypermaps to the above mentioned fields of mathematics. On the other hand, we wanted to stress some ideas and results that are important for understanding of the nature of these interesting mathematical objects.

• The Bulletin of The Korean Astronomical Society
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• v.38 no.2
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• pp.101.2-101.2
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• 2013

We present a new computational fluid dynamic (CFD) simulation code. The code employs the moving and polyhedral unstructured mesh scheme, which is known as a superior approach to the conventional SPH (smoothed particle hydrodynamics) and AMR (adaptive mesh refinement) schemes. The code first generates unstructured meshes by the Voronoi tessellation at every time step, and then solves the Riemann problem for surfaces of every Voronoi cell to update the hydrodynamic states as well as to move former generated meshes. For the second-order accuracy, the MUSCL-Hancock scheme is implemented. To increase efficiency for generating Voronoi tessellation we also develop the incremental expanding method, by which the CPU time is turned out to be just proportional to the number of particles, i.e., O(N). We will discuss the applications of our code in the context of cosmological simulations as well as numerical experiments for galaxy formation.

• Bulletin of the Korean Mathematical Society
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• v.42 no.4
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• pp.817-828
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• 2005

In this note we show that there is an anti-symplectic involution $\sigma\;:\;X\;\to\;X$ on a simply-connected, closed, non-Kahler and symplectic 4-manifold X with a disjoint union of Riemann surfaces ${\amalg}^n_{i=1}{\Sigma}_i,\;n\;{\ge}\;2$ as a fixed point set. Also we show that its quotient X/$\sigma$ is homeomorphic to $\mathbb{CP}^2{\sharp}r\mathbb{CP}^2$ but not diffeomorphic to $\mathbb{CP}^2{\sharp}r\mathbb{CP}^2,\;r\;=\;b_2^-(X/{\sigma})$.

• The Bulletin of The Korean Astronomical Society
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• v.39 no.2
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• pp.65.2-65.2
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• 2014

We present a new hydrodynamic simulation code based on the Voronoi tessellation for estimating the density precisely. The code employs both of Lagrangian and Eulerian description by adopting the movable mesh scheme, which is superior to the conventional SPH (smoothed particle hydrodynamics) and AMR (adaptive mesh refinement) schemes. The code first generates unstructured meshes by the Voronoi tessellation at every time step, and then solves the Riemann problem for all surfaces of each Voronoi cell so as to update the hydrodynamic states as well as to move current meshes. Besides, the IEM (incremental expanding method) is devised to compute the Voronoi tessellation to desired degree of speed, thereby the CPU time is turned out to be just proportional to the number of particles, i.e., O(N). We discuss the applications of our code in the context of cosmological simulations as well as numerical experiments for galaxy formation.

• Journal for History of Mathematics
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• v.28 no.2
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• pp.103-115
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• 2015

Contemporary differential geometry owes much to the theory of connections on the bundles over manifolds. In this paper, following the work of Gauss on surfaces in 3 dimensional space and the work of Riemann on the curvature tensors on general n dimensional Riemannian manifolds, we will investigate how differential geometry had been developed from mid 19th century to early 20th century through lives and mathematical works of Christoffel, Ricci-Curbastro and Levi-Civita. Christoffel coined the Christoffel symbol and Ricci used the Christoffel symbol to define the notion of covariant derivative. Levi-Civita completed the theory of absolute differential calculus with Ricci and discovered geometric meaning of covariant derivative as parallel transport.

• Journal of the Korean Society for Nondestructive Testing
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• v.32 no.3
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• pp.296-301
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• 2012

This paper describes an analytic method for infrared thermography to detect surface cracks in thin plates. Traditional thermographic method uses the spatial contrast of a thermal field, which is often corrupted by noise in the experiment induced mainly by emissivity variations of target surfaces. This study developed a robust analytic approach to crack detection for thermography using the holomorphic function of a temperature field in thin plate under steady-state thermal conditions. The holomorphic function of a simple temperature field was derived for 2-D heat flow in the plate from Cauchy-Riemann conditions, and applied to define a contour integral that varies depending on the existence and strength of singularity in the domain of integration. It was found that the contour integral at each point of thermal image reduced the noise and temperature variation due to heat conduction, so that it provided a clearer image of the singularity such as cracks.

• Journal of applied mathematics & informatics
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• v.28 no.5_6
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• pp.1315-1322
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• 2010

Let D be a plane domain whose boundary consists of n components and $C_1$, $C_2$ two boundary components of D. We consider the family $F_1$ of conformal mappings f satisfying f(D) $\subset$ {1 < |w| < ${\mu}(f)$}, $f(C_1)=\{|w|=1\}$, $f(C_2)=\{|w|={\mu}(f)\}$. There are conformal mappings $g_0$, $g_1({\in}F_1)$ onto a radial and a circular slit annulus respectively. We obtain the following theorem, $$\{{\mu}(f)|f\;{\in}\;F_1\}=\{\mu|\mu(g_1)\;{\leq}\;{\mu}\;{\leq}\;{\mu}(g_0)\}$$. And we consider the family $F_n$ of conformal mappings $\tilde{f}$ from D onto a covering surfaces of the Riemann sphere satisfying some conditions. We obtain the following theorems, {$\mu|1$ < ${\mu}\;{\leq}\;{\mu}(g_1)$} ${\subset}\;\{{\mu}(\tilde{f})|\tilde{f}\;{\in}\;F_2\}\;{\subset}\;\{{\mu}(\tilde{f})|\tilde{f}\;{\in}\;F_n\}$ and ${\mu}(\tilde{f})\;{\leq}\;{\mu}(g_0)^n$.