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In this paper, we consider the problem of representing planar graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear-time algorithm(More)
On many distributed memory systems, such as workstation clusters or the Intel iPSC/860, the multigrid algorithm suffers from having extensive communication requirements and, in general, it is not very competitive in comparison to the conjugate gradient algorithm. This is in contrast to the sequential problem whereby the multigrid algorithm is very effective(More)
The correlation between sub-band gap absorption and the chemical states and electronic and atomic structures of S-hyperdoped Si have been extensively studied, using synchrotron-based x-ray photoelectron spectroscopy (XPS), x-ray absorption near-edge spectroscopy (XANES), extended x-ray absorption fine structure (EXAFS), valence-band photoemission(More)
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