AI 中文总结
该研究将矩阵谱序列与有理数序列关联，通过数值实验发现与尖点形式傅里叶展开得到的序列相关的特定处理矩阵谱具有相干行为。
AI 中文摘要
我们将矩阵谱序列与有理数序列关联，并通过数值实验对其进行研究。与多个有理数序列相关的谱序列的初始段似乎呈现出几何规律性。对称函数理论中的恒等式将算术序列 $\{h_n\}_{n \ge 0} \\ (h_0 = 1)$ 的元素表示为 $h_n = |J_{h,n}|/n!$，其中 $|\cdot|$ 为行列式，$J_{h,n}$ 是特定矩阵。在此背景下，我们对由尖点形式的傅里叶展开得到的序列 $\{a(n)\}_{n \ge 1}$ 进行了数值研究。在观测范围内，某些“处理过的”矩阵序列 $\{J^{(c)}_{a,n}\}_{n=1,2,3,...}$ 的谱表现出相干行为，即当 $n$ 足够大时，$J^{(c)}_{a,n}$ 的特征值最小模的图像似乎会振荡或形成近似非负斜率的直线。
英文摘要
We attach sequences of matrix spectra to sequences of rational numbers and study them with numerical experiments. The initial segments of the spectrum sequences associated to each of several rational sequences appear to exhibit geometric regularities. Identities originating in the theory of symmetric functions express members of arithmetic sequences $\{h_n\}_{n \ge 0} \thinspace (h_0 = 1)$ as $h_n = |J_{h,n}|/n !$ for particular matrices $J_{h,n}$, where $|\cdot|$ is the determinant. In this setting we studied numerically sequences $\{a(n)\}_{n \ge 1}$ obtained from the Fourier expansions of cusp forms. Within the range of our observations, the sequences of spectra of certain "treated" matrices $\{J^{(c)}_{a,n}\}_{n=1,2,3,...}$ exhibit coherent behavior, meaning that the plots of the minimum moduli among the eigenvalues of the $J^{(c)}_{a,n}$ appear to oscillate or to form approximately straight lines of non-negative slope once $n$ is large enough.
Comments22 figures, 10 tables
