论文标题
Delone集中的本地团体
Local Groups in Delone Sets
论文作者
论文摘要
在本文中，我们证明，在任意的DeLone设置$ x $ 3D $空间中的$ x $中，从$ x $中的所有点的子集$ x_6 $，本地组的轴轴的轴不超过6个，也是Delone集合。在这里，在x $点$ x \点的本地组下，意味着对称组$ s_x（2r）$ c_x $ c_x（2r）的$ x $，radius $ 2r $，其中$ r $（根据DeLone的“空镜头理论”，是“空spery Sphere”的理论）是最大的“空球”，这是$ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x。 主要结果似乎是绝对通用的DeLone集对第一个严格证明的陈述，这意味着对具有强大晶体学限制的DeLONO集合的实质性陈述。例如，对shtogrin的重要观察是对DeLone组的本地组的界限，即$ 2R $ -CLUSTERS立即从主要定理开始。 在论文中，建议了“晶体内核猜想”（猜想1）及其两个较弱的版本（猜想2和3）。根据猜想1，在非常任意的Delone集中，只有局部晶体学轴（2,3,4或6）的点不可避免地构成了集合的重要组成部分。这些猜想显着概括了著名的晶体学陈述，即3D晶格中（全局）5倍对称性的不可能。
In the paper, we prove that in an arbitrary Delone set $X$ in $3D$ space, the subset $X_6$ of all points from $X$ at which local groups have axes of the order not greater than 6 is also a Delone set. Here, under the local group at point $x\in X$ is meant the symmetry group $S_x(2R)$ of the cluster $C_x(2R)$ of $x$ with radius $2R$, where $R$ (according to Delone's theory of the 'empty sphere') is the radius of the largest 'empty' ball, that is, the largest ball free of points of $X$. The main result seems to be the first rigorously proved statement on absolutely generic Delone sets which implies substantial statements for Delone sets with strong crystallographic restrictions. For instance, an important observation of Shtogrin on the boundedness of local groups in Delone sets with equivalent $2R$-clusters immediately follows from the main theorem. In the paper, the 'crystalline kernel conjecture' (Conjecture 1) and its two weaker versions (Conjectures 2 and 3) are suggested. According to Conjecture 1, in a quite arbitrary Delone set, points with locally crystallographic axes (of order 2,3,4, or 6) only inevitably constitute an essential part of the set. These conjectures significantly generalize the famous statement of Crystallography on the impossibility of (global) 5-fold symmetry in a 3D lattice.
