Back to Search
Start Over
Delone sets in ℝ3: Regularity Conditions
- Source :
- Journal of Mathematical Sciences. 248:743-761
- Publication Year :
- 2020
- Publisher :
- Springer Science and Business Media LLC, 2020.
-
Abstract
- A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.
- Subjects :
- Statistics and Probability
Discrete mathematics
Euclidean space
Applied Mathematics
General Mathematics
010102 general mathematics
Delone set
01 natural sciences
Identity (music)
010305 fluids & plasmas
Set (abstract data type)
0103 physical sciences
Homogeneous space
Mathematics::Metric Geometry
0101 mathematics
Symmetry (geometry)
Orbit (control theory)
Link (knot theory)
Mathematics
Subjects
Details
- ISSN :
- 15738795 and 10723374
- Volume :
- 248
- Database :
- OpenAIRE
- Journal :
- Journal of Mathematical Sciences
- Accession number :
- edsair.doi...........48b7746b066535a503ebf4f6ff4a766a