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Symmetry 2018, 10(5), 159; https://doi.org/10.3390/sym10050159

The Local Theory for Regular Systems in the Context of t-Bonded Sets

1
University of Texas Rio Grande Valley, Brownsville, TX 78520, USA
2
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow 119991, Russia
The work of N. P. Dolbilin is supported by the Russian Science Foundation under Grant 14-50-00005.
*
Author to whom correspondence should be addressed.
Received: 14 February 2018 / Revised: 17 April 2018 / Accepted: 7 May 2018 / Published: 14 May 2018
(This article belongs to the Special Issue Mathematical Crystallography)
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Abstract

The main goal of the local theory for crystals developed in the last quarter of the 20th Century by a geometry group of Delone (Delaunay) at the Steklov Mathematical Institute is to find and prove the correct statements rigorously explaining why the crystalline structure follows from the pair-wise identity of local arrangements around each atom. Originally, the local theory for regular and multiregular systems was developed with the assumption that all point sets under consideration are ( r , R ) -systems or, in other words, Delone sets of type ( r , R ) in d-dimensional Euclidean space. In this paper, we will review the recent results of the local theory for a wider class of point sets compared with the Delone sets. We call them t-bonded sets. This theory, in particular, might provide new insight into the case for which the atomic structure of matter is a Delone set of a “microporous” character, i.e., a set that contains relatively large cavities free from points of the set. View Full-Text
Keywords: Delone sets; regular systems; crystals, t-bonded sets; clusters; cluster counting function Delone sets; regular systems; crystals, t-bonded sets; clusters; cluster counting function
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Bouniaev, M.; Dolbilin, N. The Local Theory for Regular Systems in the Context of t-Bonded Sets. Symmetry 2018, 10, 159.

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