Mikhail Bouniaev | Searcharxiv

arXiv2018

On the Origin of Crystallinity: a Lower Bound for the Regularity Radius of Delone Sets

The local theory of regular or multi-regular systems aims at finding sufficient local conditions for a Delone set $X$ to be a regular or multi-regular system. One of the main goals is to estimate the regularity radius $\hatρ_d$ for Delone sets $X$ in terms of the radius $R$ of the largest "empty ball" for $X$. The present paper establishes the lower bound $\hat{ρ_d}\geq 2dR$ for all $d$, which is linear in $d$. The best previously known lower bound had been $\hatρ_d\geq 4R$ for $d\geq 2$. The proof of the new lower bound is accomplished through explicit constructions of Delone sets with mutually equivalent $(2dR-\varepsilon)$-clusters, which are not regular systems.

math.MG↗

arXiv2016

Local Theory of t-bonded sets

The local theory for regular and multi-regular systems was developed in the assumption that these systems are Delone sets, or (r;R)-systems. The requirement for a set to be a (r;R)-system particularly implies that any two points in a Delone set can be connected by a sequence of points from the set with sequel interpoint distances bounded by 2R. In the terminology we adopted in this paper, it means that a Delone set is a 2R-bonded set. Meanwhile, there are crystals, e.g. zeolites, whose atomic structure is multi-regular microporous point set. In these structures there are cavities that are relatively large compared to the "length"of bonds between atomes. In other words, the parameter R in this Delone set significantly exceeds a natural link parameter. For a better description of such "microporous"structures it is worthwhile to take into consideration a parameter that represents atomic bonds within the matter. In the paper we generalize some results of the local theory to the sets that we called t-bonded sets even without making an assumption that a set is a Delone set.

math.MG↗