A function is called quasiperiodic if its fundamental frequencies are linearly independent over the rationals. With appropriate parameters, the sliding window point clouds of such functions can be shown to be dense in tori with dimension equal to the number of independent frequencies. In this paper, we develop theoretical and computational techniques to study the persistent homology of such sets. Specifically, we provide parameter optimization schemes for sliding windows of quasiperiodic functions, and present theoretical lower bounds on their Rips persistent homology. The latter leverages a recent persistent K\"{u}nneth formula. The theory is illustrated via computational examples and an application to dissonance detection in music audio samples.

翻译：如果一个函数的基本频率线性地独立于理性值之上,则该函数被称为准周期值。如果有适当的参数,这些函数的滑动窗口点云可以显示其密度在千里方块中,其维度与独立频率数相等。在本文中,我们开发理论和计算技术,以研究这些组的持久性同质性。具体地说,我们为准周期函数的滑动窗口提供参数优化方案,并在它们提取的直径上提出理论下限。后者利用了最近的持久性 K\"{u}neth 公式。该理论通过计算实例和音乐音频样本中不协调检测的应用加以说明。