• The Mathematical Education
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• v.50 no.2
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• pp.233-246
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• 2011

The geometry field in the current high school curriculum deals mainly with analytic geometry and the reference to logic geometry leaves much to be desired. This study investigated the construction on a heptagon by using conic sections as one of measures for achieving harmony between analytic geometry and logic geometry in the high school curriculum with the Geometer's Sketchpad(GSP), which is a specialized software prevalent in mathematics education field and is intended to draw an educational suggestion on it.

• Journal for History of Mathematics
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• v.22 no.2
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• pp.99-116
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• 2009

In this paper we study errors related with constructing regular polygons in the book 'Method of Ruler and Compass' written three hundreds years ago. It is well known that regular heptagon and regular nonagon are not constructible using compass and ruler. But in this book construction methods of these regular polygons is suggested. We show that the construction methods are incorrect, it include some errors.

• Proceedings of the Korean Vacuum Society Conference
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• 2010.02a
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• pp.340-340
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• 2010

Various structures of vacancy defects in graphene layers and carbon nanotubes have been reported by high resolution transmission electron microscope (HR-TEM) and those arouse an interest of reconstruction processes of vacancy defects. In this talk, we present reconstruction processes of vacancy defects in a graphene and a carbon nanotube by tight-binding molecular dynamics (TBMD) simulations and by first principles total energy calculations. We found that a structure of a dislocation defect with two pentagon-heptagon (5-7) pairs in graphene becomes more stable than other structures when the number of vacancy units is ten and over. The simulation study of scanning tunneling microscopy reveals that the pentagon-heptagon pair defects perturb the wavefunction of electrons near Fermi level to produce the $\sqrt{3}\;{\times}\;\sqrt{3}$ superlattice pattern, which is in excellent agreement with experiment. It is also observed in our tight-binding molecular dynamics simulation that 5-7 pair defects play a very important role in vacancy reconstruction in a graphene layer and carbon nanotubes.

• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.367-382
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• 2011

We establish a necessary and sufficient condition for a heptagonal knot to be figure-8 knot. The condition is described by a set of Radon partitions formed by vertices of the heptagon. In addition we relate this result to the number of nontrivial heptagonal knots in linear embeddings of the complete graph $K_7$ into $\mathbb{R}^3$.

• Proceedings of the Korean Vacuum Society Conference
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• 2011.02a
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• pp.428-429
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• 2011

Recently, we performed ab initio total energy calculation and tight-binding molecular dynamics (TBMD) simulation to study structures and the reconstruction of native defects in graphene. In the previous study, we predicted by TBMD simulation that a double vacancy in graphene is reconstructed into a 555-777 composed of triple pentagons and triple heptagons [1]. The structural change from pentagon-octagon-pentagon (5-8-5) to 555-777 has been confirmed by recent experiments [2,3] and the detail of the reconstruction process is carefully studied by ab initio calculation. Pentagon-heptagon (5-7) pairs are also found to play an important role in the reconstruction of vacancy in graphene and single wall carbon nanotube [4]. In the TBMD simulation of graphene nanoribbon (GNR), we found the evaporation of carbon atoms from both the zigzag and armchair edges is preceded by the formation of heptagon rings, which serve as a gateway for carbon atoms to escape. In the simulation for a GNR armchair-zigzag-armchair junction, carbon atoms are evaporated row-by-row from the outermost row of the zigzag edge [5], which is in excellent agreement with recent experiments [2, 6]. We also present the recent results on the formation and development of dislocation in graphene. It is found that the coalescence of 5-7 pairs with vacancy defects develops dislocation in graphene and induces the separation of two 5-7 pairs. Our TBMD simulations also show that adatoms are ejected and evaporated from graphene surface due to large strain around 5-7 pairs. It is observed that an adatom wanders on the graphene surface and helps non-hexagonal rings change into stable hexagonal rings before its evaporation.

• Journal of the Semiconductor & Display Technology
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• v.3 no.3
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• pp.23-25
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• 2004

The defect formation energy of boron nitride (BN) nanotubes is investigated using molecular-dynamics simulation. Although the defect with tetragon-octagon pairs (4-88-4) is favored in the flat cap of BN nanotubes, BN clusters, and the growth of BN nanotubes, the formation energy of the 4-88-4 defect is significantly higher than that of the pentagon-heptagon pairs (5-77-5) defect in BN nanotubes. The 5-77-5 defect reduces the effect of the structural distortion caused by the 4-88-4 defect, in spite of homoelemental bonds.

• Proceedings of the Korean Society Of Semiconductor Equipment Technology
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• 2004.05a
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• pp.59-62
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• 2004

The defect formation energy of boron nitride (BN) nanotubes is investigated using molecular-dynamics simulation. Although the defect with tetragon-octagon pairs (4-88-4) is favored in the flat cap of BN nanotubes, BN clusters, and the growth of BN nanotubes, the formation energy of the 4-88-4 defect is significantly higher than that of the pentagon-heptagon pairs (5-77-5) defect in BN nanotubes. The 5-77-5 defect reduces the effect of the structural distortion caused by the 4-88-4 defect, in spite of homoelemental bonds. The instability of the 4-88-4 defect generates the structural transformation into BNNTs with no defect at about 1500 K.

• Proceedings of the Korean Vacuum Society Conference
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• 2010.08a
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• pp.54-55
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• 2010

Vacancy defects in graphene can be created by electron or ion irradiation and those induce ripples which can change the electronic properties of graphene. Recently, the formation of defect structures such as vacancy defects and non-hexagonal rings has been reported in the high resolution transmission electron microscope (HR-TEM) of reduced graphene oxide [1]. In those HR-TEM images, it is noticed that the dislocations with pentagon-heptagon (5-7) pairs are formed and diffuses. Interestingly, it is also observed that two 5-7 pairs are separated and diffuse far away from each other. The separation of 5-7 pairs has been known to be due to their self-diffusion. However, from our tight-binding molecular dynamics simulation, it is found that the separation of 5-7 pairs is due to the diffusion of single vacancy defects and coalescence with 5-7 pairs. The diffusion and coalescence of single vacancy defects is too fast to be observed even in HR-TEM. We also implemented Van der Waals interaction in our tight-binding carbon model to describe correctly bi-layer and multi-layer graphene. The compressibility of graphite along c-axis in our tight-binding calculation is found to be in excellent agreement with experiment. We also discuss the difference between single layer and bi-layer graphene about vacancy diffusion and reconstruction.