四维超立方格的重合

Coincidence isometries of a shifted square lattice

We consider the coincidence problem for the square lattice that is translated by an arbitrary vector. General results are obtained about the set of coincidence isometries and the coincidence site lattices of a shifted square lattice by identifying the square lattice with the ring of Gaussian integers. To illustrate them, we calculate the set of coincidence isometries, as well as generating functions for the number of coincidence site lattices and coincidence isometries, for specific examples.

Coincidences of a shifted hexagonal lattice and the hexagonal packing

A geometric study of twin and grain boundaries in crystals and quasicrystals is achieved via coincidence site lattices (CSLs) and coincidence site modules (CSMs), respectively. Recently, coincidences of shifted lattices and multilattices (i.e. finite unions of shifted copies of a lattice) have been investigated. Here, we solve the coincidence problem for a shifted hexagonal lattice. This result allows us to analyze the coincidence isometries of the hexagonal packing by viewing the hexagonal packing as a multilattice.

CSLs of the root lattice $\mathbf{A_4}$

Recently, the group of coincidence isometries of the root lattice $A_4$ has been determined providing a classification of these isometries with respect to their coincidence indices. A more difficult task is the classification of all CSLs, since different coincidence isometries may generate the same CSL. In contrast to the typical examples in dimensions $d \leq 3$, where coincidence isometries generating the same CSL can only differ by a symmetry operation, the situation is more involved in 4 dimensions. Here, we find coincidence isometries that are not related by a symmetry operation but never…

Well-rounded sublattices and coincidence site lattices

A lattice is called well-rounded, if its lattice vectors of minimal length span the ambient space. We show that there are interesting connections between the existence of well-rounded sublattices and coincidence site lattices (CSLs). Furthermore, we count the number of well-rounded sublattices for several planar lattices and give their asymptotic behaviour.

The coincidence problem for shifted lattices and crystallographic point packings

A coincidence site lattice is a sublattice formed by the intersection of a lattice $Γ$ in $\mathbb{R}^d$ with the image of $Γ$ under a linear isometry. Such a linear isometry is referred to as a linear coincidence isometry of $Γ$. Here, we consider the more general case allowing any affine isometry. Consequently, general results on coincidence isometries of shifted copies of lattices, and of crystallographic point packings are obtained. In particular, we discuss the shifted square lattice and the diamond packing in detail.

超立方格$Z^n$的符合等距指数

Indices of Coincidence Isometries of the Hyper Cubic Lattice $Z^n$

考虑超立方格点$\mathbb{Z}^{n}$的重合等距指数的计算问题。本文详细分析了有理正交矩阵的标准形，得到了$\mathbb{Z}^{n}$的某些重合等距的指数的显式表达式。这些公式推广了关于$n\le 4$的已知结果。

Multiplicativity in the theory of coincidence site lattices

Coincidence Site Lattices (CSLs) are a well established tool in the theory of grain boundaries. For several lattices up to dimension $d=4$, the CSLs are known explicitly as well as their indices and multiplicity functions. Many of them share a particular property: their multiplicity functions are multiplicative. We show how multiplicativity is connected to certain decompositions of CSLs and the corresponding coincidence rotations and present some criteria for multiplicativity. In general, however, multiplicativity is violated, while supermultiplicativity still holds.

根格$A_4$的符合旋转

Coincidence rotations of the root lattice $A_4$

考虑了根格$A_4$的重合格点，用狄利克雷级数生成函数表示了对应的重合旋转统计量.通过将$A_4$嵌入到具有丰富算术结构的icosian环中，这是可能的，最近（arXiv：math.MG/0702448）导致了$A_4$的相似子格的分类。

重合对称群的结构

Structures of Coincidence Symmetry Groups

通过描述一组生成元，研究了$n$维欧几里得空间中任意$n$维格点的重合对称群的结构.特别注意的是重合等距子群（由这些重合对称的正交群的元素形成的子群）。将讨论由晶格矢量定义的反射产生符合等距群的条件，并给出根据晶格矢量定义的反射分解符合等距群的任意元素的算法。

计算模重合

Computing modular coincidences

计算模重合可以显示R^d中的点格上所支持的给定替换系统是否由模型集组成。我们证明了这个问题的可计算性，并确定了所需迭代次数的上界。主要工具是一个计算模重合的简单算法，它本质上是Dekking重合到多维的推广，并证明了这种推广的Dekking重合和模重合的等价性。作为结果，我们还得到了模重合存在的一些条件。在单独的一节中，并在整个文章中，给出了一些例子。

体心立方晶格的多重CSL

Multiple CSLs for the body centered cubic lattice

普通重合点晶格（CSL）被定义为晶格$Γ$与其自身的旋转副本$RΓ$的相交。它们对晶界的分类很有用，自六十年代中期以来得到了广泛的研究。最近的兴趣转向所谓的多个CSL，即给定格$Γ$的$n$旋转副本的交叉，特别是与格量化器有关。在这里，我们考虑多个CSL的三维体心立方晶格。本文讨论了重合指数的谱及其重数，特别证明了重数是一个乘性函数，并在某些特殊情况下给出了重数的显式表达式。

相似子模与重合位置模

Similar submodules and coincidence site modules

本文将格的相似与重合等距理论推广到一般自由 Z-模及 S-模，证明商群 OS(M)/OC(M) 为 Abel 群，并给出 CSL 与相似子格指数间的关系。

三维空间中的重合座模

Coincidence site modules in 3-space

利用实代数数域上类数为1的四元数代数的极大序模，重新讨论了重合格问题及其在欧氏空间Z-模上的推广，并统一地证明了各种结果和定理.

维度$d\le 4$中重合问题的求解

Solution of the coincidence problem in dimensions $d\le 4$

离散点集$\mathcal{S}$，例如格或准周期Delone集，除了它们的对称性之外，可以允许某些等距$R$，使得$\mathcal{S}\cap R\mathcal{S}$是有限密度的$\mathcal{S}$的子集。这就是所谓的重合等距线。它们在理解和分类晶体和准晶体中的晶界和孪晶方面很重要。本文的目的是在数学环境中引入相应的重合问题，并演示如何在2维、3维和4维中用代数方法解决该问题。明确地处理来自晶体和准晶的各种实例，特别是具有$H_2$、$H_3$和$H_4$类型的非晶体学点群的（超）立方晶格和准晶。我们推导出所有线性重合等距的参数化，确定相应的重合指数（重合点密度的倒数，也称为$\varSigma$ -因子），最后将它们的统计量封装在合适的狄利克雷级数生成函数中.

多重N重对称平面符合

Multiple planar coincidences with N-fold symmetry

平面重合点格和具有N重对称性的模在基于分圆域的公式中得到了很好的理解，特别是对于类数情况，它们在相应的整数环中作为某些主理想出现。我们将这种方法扩展到多个巧合，适用于三重或多重路口。特别是，我们给出了明确的结果的频谱，组合和渐近性质的狄利克雷级数生成函数。