• 대한수학회지
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• 제45권4호
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• pp.965-976
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• 2008

In this article, we constructively prove that on a surface S with genus g$\geq$2, there exit maximal geodesic laminations with 7g-7,...,9g-9 leaves. Thus, S can have ideal cell-decompositions (i.e., S can be (ideally) triangulated by maximal geodesic laminations) with 7g-7,...,9g-9 (ideal) 1-cells. Once there is a triangulation for a compact surface, the Euler characteristic for the surface can be calculated as the alternating sum F-E+V, where F, E, and V denote the number of faces, edges, and vertices, respectively. We also prove that the same formula holds for the ideal cell decompositions.

• 한국컴퓨터정보학회논문지
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• 제12권2호
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• pp.83-91
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• 2007

본 논문에서는 삼차원 볼륨이미지로부터 정육면체형 셀의 분해를 통해 위상학적으로 정확한 삼중선형 등위면을 계산하고 가시화하는 알고리즘을 제시하였다. 등위면은 삼차원 볼륨이미지를 가시화하는데 자주 사용되며, 볼륨데이타의 각 정육면체형 셀안의 등위면을 삼각형의 집합으로 나타냄으로써 컴퓨터에서 가시화하기 쉽게 된다. 그러나, 대부분의 기존 등위면 계산 방식들은 이상적인 형태인 삼중선형 등위면의 위상을 보존하지 못하여 정확성이 떨어지는 단점을 가지고 있다. 본 논문은 삼차원 볼륨이미지를 구성하는 정육면체형 셀에서 일어날 수 있는 모든 등위면 형태에 대하여 정확한 삼중선형 등위면의 위상을 결정하고, 각 경우에 따라 적절한 정육면체 셀의 분해를 실시한 뒤, 분해된 셀들에 대하여 등위면을 나타내는 삼각형들을 추출하여, 위상학적으로 정확한 삼중선형 등위면을 만들어낸다. 마지막으로, 본 논문에서 제안된 알고리즘을 구현한 결과는 정확한 삼중선형을 가시화할 수 있음을 보여준다.

• 한국막학회:학술대회논문집
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• 한국막학회 1999년도 The 7th Summer Workshop of the Membrane Society of Korea
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• pp.39-42
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• 1999

Perfluoropolymers represent the ultimate resistance to hostile chemical environments and high service temperature, attributed to the presence of fluorine in the polymer backbone, i.e. to the high bond energy of C-F and C-C bonds of fluorocarbons. Copolymers of Tetrafluoroethylene (TEE) and 2, 2, 4Trifluoro-5Trifluorometoxy- 1, 3Dioxole (TTD), commercially known as HYFLON AD, are amorphous perfluoropolymers with glass transition temperature (Tg)higher than room temperature, showing a thermal decomposition temperature exceeding 40$0^{\circ}C$. These polymer systems are highly soluble in fluorinated solvents, with low solution viscosities. This property allows the preparation of self-supported and composite membranes with desired membrane thickness. Symmetric and asymmetric perfluoropolymer membranes, made with HYFLON AD, have been prepared and evaluated. Porous and not porous symmetric membranes have been obtained by solvent evaporation with various processing conditions. Asymmetric membranes have been prepared by th wet phase inversion method. Measure of contact angle to distilled water have been carried out. Figure 1 compares experimental results with those of other commercial membranes. Contact angles of about 120$^{\circ}$for our amorphous perfluoropolymer membranes demonstrate that they posses a high hydrophobic character. Measure of contact angles to hexandecane have been also carried out to evaluate the organophobic character. Rsults are reported in Figure 2. The observed strong organophobicity leads to excellent fouling resistance and inertness. Porous membranes with pore size between 30 and 80 nanometers have shown no permeation to water at pressures as high as 10 bars. However high permeation to gases, such as O2, N2 and CO2, and no selectivities were observed. Considering the porous structure of the membrane, this behavior was expected. In consideration of the above properties, possible useful uses in th field of gas- liquid separations are envisaged for these membranes. A particularly promising application is in the field of membrane contactors, equipments in which membranes are used to improve mass transfer coefficients in respect to traditional extraction and absorption processes. Gas permeation properties have been evaluated for asymmetric membranes and composite symmetric ones. Experimental permselectivity values, obtained at different pressure differences, to various single gases are reported in Tab. 1, 2 and 3. Experimental data have been compared with literature data obtained with membranes made with different amorphous perfluoropolymer systems, such as copolymers of Perfluoro2, 2dimethyl dioxole (PDD) and Tetrafluorethylene, commercialized by the Du Pont Company with the trade name of Teflon AF. An interesting linear relationship between permeability and the glass transition temperature of the polymer constituting the membrane has been observed. Results are descussed in terms of polymer chain structure, which affects the presence of voids at molecular scale and their size distribution. Molecular Dyanmics studies are in progress in order to support the understanding of these results. A modified Theodoru- Suter method provided by the Amorphous Cell module of InsightII/Discover was used to determine the chain packing. A completely amorphous polymer box of about 3.5 nm was considered. Last but not least the use of amorphous perfluoropolymer membranes appears to be ideal when separation processes have to be performed in hostile environments, i.e. high temperatures and aggressive non-aqueous media, such as chemicals and solvents. In these cases Hyflon AD membranes can exploit the outstanding resistance of perfluoropolymers.