ﻻ يوجد ملخص باللغة العربية
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.
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
We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension $n$ up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for $nleq9$. On every nilpotent Lie algebra of dimension $leq 7$, we deter
We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invari
We describe a new algorithm, the $(k,ell)$-pebble game with colors, and use it obtain a characterization of the family of $(k,ell)$-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special inst
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