Zeitschrift für verallgemeinerte Lügentheorie und -anwendungen

To compute any physical quantity for a random particle, the mathematical shape of the particle must be known. The mathematics for regular particles like spheres and ellipsoids is simple. Mathematically characterising the shape of random particles with realistic shapes had not been done previously. However, since around the year 2002, a method that combines X-ray computed tomography and spherical harmonic analysis has been developed to give analytical, differentiable mathematical functions for the three-dimensional shape of star-shape particles, which are a broad class of particles covering most industrial particles of interest, ranging from micrometre scale to millimetre scale particles.

On multiple spatiotemporal scales, living systems exhibit complex yet organised behaviour. To investigate the nature of multiscale coordination in living systems, a meaningful and systematic method of quantifying the complex dynamics is required, which is a challenge in both theoretical and empirical domains. The current work demonstrates how combining approaches from computational algebraic topology and dynamical systems can assist us in meeting this challenge. We concentrate on the application of multiscale topological analysis to coordinated rhythmic processes in particular. First, theoretical arguments are presented to demonstrate why certain topological features and their scale dependency are critical for comprehending complex collective dynamics. Second, we propose using persistent homology to capture such dynamically relevant topological information.