Mikhail Bouniaev

Geometrical Problems Related to Crystals, Fullerenes, and Nanoparticle Structure

This paper focuses on three groups of geometrical problems, closely related to material sciences in general and particularly to crystal/quasicrystal structures along with their formations and fullerenes. Some new results in mathematics are presented and discussed, for example, in section one, new estimates of minimum radius of local identity that guarantee that a Delone set is a point regular set. New results related to locally rigid packings are discussed in section two. One of the goals of the paper is to establish some internal (mathematically) and external (applications to material science) connections between research agendas of various studies in geometry and material sciences.

Regular and Multi-regular t-bonded Systems

The concept of t-bonded sets was briefly introduced by the second author in 1976 under the name of dconnected sets, though it has not received due consideration. This concept is a generalization of the concept of Delone (r,R)-systems. In light of the developments in the local theory for crystals that occurred since 1976 and demands in chemistry and crystallography, we believe the local theory for t-bonded sets deserves to be developed to describe materials whose atomic structures is multi-regular “microporous” point set. For a better description of such “microporous” structures it is worthwhile to take into consideration a parameter that represents atomic bonds within the matter. The overarching goal of this paper is to prove that analogous local conditions that guarantee that a Delone set is a regular (or multi-regular) system also guarantee that a t-bonded set is a regular (or multi-regular) t-bonded system.

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

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.

Two Groups of Geometrical Problems Related to Study of Fullerenes & Crystals

The paper focuses on two groups of geometrical problems closely related to formations of crystal/quasi-crystal structures, and fullerenes. In section one we discuss a minimum radius of local identity that guarantee that a Delone set is a point regular set, and prove the local theorem for crystals. New results related to locally rigid packings are discussed in section two. One of the goals of the paper is to establish some internal (mathematically), and external (applications to material science), connections between research agendas of various studies in material sciences, and two classes of geometric problem related to Delone sets and packings.