Related papers: Local Groups in Delone Sets

In the paper we present a proof of the local criterion for crystalline structures which generalizes the local criterion for regular systems. A Delone set is called a crystal if it is invariant with respect to a crystallgraphic group.…

Delone sets are discrete point sets $X$ in $\mathbb{R}^d$ characterized by parameters $(r,R)$, where (usually) $2r$ is the smallest inter-point distance of $X$, and $R$ is the radius of a largest ``empty ball" that can be inserted into the…

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 $\hat{\rho}_d$ for…

This paper characterizes when a Delone set X is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the hetereogeneity of their distribution. Let N(T) count the number of…

Delone sets are locally finite point sets, such that (a) any two points are separated by a given minimum distance, and (b) there is a given radius so that every ball of that radius contains at least one point. Important examples include the…

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…

We complete the proof of the upper bound $\hat\rho_3\leq 10R$ for the regularity radius of Delone sets in three-dimensional Euclidean space. Namely, summing up the results obtained earlier, and adding the missing cases, we show that if all…

A Delone set in $\mathbb{R}^n$ is a set such that (a) the distance between any two of its points is uniformly bounded below by a strictly positive constant and such that (b) the distance from any point to the remaining points in the set is…

Let $S$ be a non-empty subset of a group $G$. We say $S$ is product-free if $S\cap SS=\varnothing$, and $S$ is locally maximal if whenever $T$ is product-free and $S\subseteq T$, then $S=T$. Finally $S$ fills $G$ if $G^*\subseteq S \sqcup…

The absolute sets of local systems on a smooth complex algebraic variety are the subject of a conjecture of N. Budur and B. Wang based on an analogy with special subvarieties of Shimura varieties. An absolute set should be the…

Let $G$ be a finite group and $S$ a subset of $G$. Then $S$ is product-free if $S \cap SS = \emptyset$, and complete if $G^{\ast} \subseteq S \cup SS$. A product-free set is locally maximal if it is not contained in a strictly larger…

We address the problem of determining finite subsets of Delone sets $\varLambda\subset\R^d$ with long-range order by $X$-rays in prescribed $\varLambda$-directions, i.e., directions parallel to non-zero interpoint vectors of $\varLambda$.…

A new locally averaged density for sphere packing in R^3 is defined by a proper combination of the local cell (Voronoi cell) and Delaunay decompositions (\S 1.2.2), using only a single layer of surrounding spheres. Local packings attaining…

For a general radially symmetric, non-increasing, non-negative kernel $h\in L ^ 1 _{loc} ( R ^ d)$, we study the rigidity of measurable sets in $R ^ d$ with constant nonlocal $h$-mean curvature. Under a suitable "improved integrability"…

Let $G$ be a finite group and $S$ a subset of $G$. Then $S$ is {\em product-free} if $S \cap SS = \emptyset$, and $S$ {\em fills} $G$ if $G^{\ast} \subseteq S \cup SS$. A product-free set is locally maximal if it is not contained in a…

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of…

Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behaviour at larger…

We prove Suslin's local-global principle for principal congruence subgroups of Chevalley groups. Let $G$ be a Chevalley--Demazure group scheme with a root system $\Phi\ne A_1$ and $E$ its elementary subgroup. Let $R$ be a ring and $I$ an…

We consider a group $G$ acting on a local dendrite $X$ (in particular on a graph). We give a full characterization of minimal sets of $G$ by showing that any minimal set $M$ of $G$ (whenever $X$ is different from a dendrite) is either a…

The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N = 1054. In the predecessor to this paper…