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Register a new userSome Minimal Shape Decompositions Are Nice
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Added by
Kevin R. Vixie
Publication date
2015
fields
Informatics Engineering
and research's language is
English
Authors
Kevin R. Vixie
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No Arabic abstract
In some sense, the world is composed of shapes and words, of continuous things and discrete things. The recognition and study of continuous objects in the form of shapes occupies a significant part of the effort of unraveling many geometric questions. Shapes can be rep- resented with great generality by objects called currents. While the enormous variety and representational power of currents is useful for representing a huge variety of phenomena, it also leads to the problem that knowing something is a respectable current tells you little about how nice or regular it is. In these brief notes I give an intuitive explanation of a result that says that an important class of minimal shape decompositions will be nice if the input shape (current) is nice. These notes are an exposition of the paper by Ibrahim, Krishnamoorthy and Vixie which can be found on the arXiv:1411.0882 and any reference to these notes, should include a reference to that paper as well.
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We study nice nilpotent Lie algebras admitting a diagonal nilsoliton metric. We classify nice Riemannian nilsolitons up to dimension $9$. For general signature, we show that determining whether a nilpotent nice Lie algebra admits a nilsoliton metric reduces to a linear problem together with a system of as many polynomial equations as the corank of the root matrix. We classify nice nilsolitons of any signature: in dimension $leq 7$; in dimension $8$ for corank $leq 1$; in dimension $9$ for corank zero.
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We describe a new algorithm, the $(k,\\ell)$-pebble game with colors, and use\nit obtain a characterization of the family of $(k,\\ell)$-sparse graphs and\nalgorithmic solutions to a family of problems concerning tree decompositions of\ngraphs. Spe
cial instances of sparse graphs appear in rigidity theory and have\nreceived increased attention in recent years. In particular, our colored\npebbles generalize and strengthen the previous results of Lee and Streinu and\ngive a new proof of the Tutte-Nash-Williams characterization of arboricity. We\nalso present a new decomposition that certifies sparsity based on the\n$(k,\\ell)$-pebble game with colors. Our work also exposes connections between\npebble game algorithms and previous sparse graph algorithms by Gabow, Gabow and\nWestermann and Hendrickson.\n
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