晶体框架、矩阵值函数与刚性算子
Crystal Frameworks, Matrix-valued Functions and Rigidity Operators
摘要：
本文介绍和评述了最近关于\textit{crystal frameworks}的无穷小动力学，即$\mathbb{R}^d$中的周期离散键结结构的研究工作，其中包括对$ d=2,3,...$的研究。我们讨论了有限杆-关节框架理论的基本对象刚性矩阵、刚性算子、矩阵函数表示和低能声子。在石英和沸石等材料晶体中，这些声子被称为刚性单元模（RUM），并且与刚性单元的相对运动有关，例如石英的四面体多面体键结模型中的~SiO $_4$四面体。我们还介绍了半无限晶体框架，双晶体框架和相关的多变量Toeplitz算子。
英文摘要：
An introduction and survey is given of some recent work on the infinitesimal dynamics of \textit{crystal frameworks}, that is, of translationally periodic discrete bond-node structures in $\mathbb{R}^d$, for $ d=2,3,...$. We discuss the rigidity matrix, a fundamental object from finite bar-joint framework theory, rigidity operators, matrix-function representations and low energy phonons. These phonons in material crystals, such as quartz and zeolites, are known as rigid unit modes, or RUMs, and are associated with the relative motions of rigid units, such as ~SiO$_4$ tetrahedra in the tetrahedral polyhedral bond-node model for quartz. We also introduce semi-infinite crystal frameworks, bi-crystal frameworks and associated multi-variable Toeplitz operators.
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