To each discrete translationally periodic bar-joint framework $C$ in $R^d$ we associate a matrix-valued function $Phi_C(z)$ defined on the $d$-torus. The rigid unit mode spectrum $Omega(C)$ of $C$ is defined in terms of the multi-phases of phase-periodic infinitesimal flexes and is shown to correspond to the singular points of the function $z o ank Phi_C(z)$ and also to the set of wave vectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium the determinant of $Phi_C(z)$ is defined and gives rise to a unique multi-variable polynomial $p_C(z_1,dots ,z_d)$. For ideal zeolites the algebraic variety of zeros of $p_C(z)$ on the $d$-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealised framework rigidity and flexibility and in particular leads to an explicit formula for the number of supercell-periodic floppy modes. In the case of certain zeolite frameworks in dimensions $2$ and $3$ direct proofs are given to show the maximal floppy mode property (order $N$). In particular this is the case for the cubic symmetry sodalite framework and other idealised zeolites.
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机译:对于$ bR ^ d $中每个离散的平移周期性bar-joint框架$ C $,我们将在$ d $ -torus上定义的矩阵值函数$ Phi_ C(z)$关联起来。 $ C $的刚性单位模式谱$ Omega( C)$是根据相位周期无穷小挠度的多相定义的,并且显示为对应于函数$ z to 的奇异点秩 Phi_ C(z)$以及在长波长范围内具有消失能量的谐波激励的波矢量集合。对于麦克斯韦计数平衡的晶体框架,定义了行列式$ Phi_ C(z)$并产生唯一的多元多项式$ p_ C(z_1, dots,z_d)$。对于理想的沸石,$ d $ -torus上$ p_ C(z)$的零代数变化与RUM光谱重合。矩阵函数与理想化框架的刚性和灵活性的其他方面有关,并且特别是导致了超级单元周期软盘模式数量的明确公式。对于某些尺寸为$ 2 $和$ 3 $的沸石骨架,将给出直接的证明以显示最大的软盘模式属性(定额$ N $)。立方对称方钠石骨架和其他理想化沸石尤其如此。
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