• Journal of the Korean Mathematical Society
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• v.45 no.4
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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.

• Journal of the Korea Society of Computer and Information
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• v.12 no.2 s.46
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• pp.83-91
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• 2007

This paper describes an algorithm to compute and visualize a topologically accurate trilinear isosurface from three dimensional volumetric image via cubic cell decomposition. An isosurface is often used for visualizing a three dimensional volumetric image. An isosurface defined in each cubic cell of the volume is triangulated in order to be visualized in a computer. However, most isosurface extraction methods generate a triangulated isosurface which may not be topologically equivalent to the ideal trilinear isosurface. We propose a method to decide a correct connectivity of a trilinear isosurface in a cubic cell and perform appropriate cell decomposition according to the decision. Using the method, we can extract isosurface triangles from the cells generated by the decomposition. We prove that this method generates a triangulated isosurface which is topologically equivalent to the trilinear isosurface. We implemented our proposed algorithm and the result shows it can generate topologically accurate trilinear isosurface.

• Proceedings of the Membrane Society of Korea Conference
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• 1999.07a
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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.