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We show how to interpret logarithmic spiral tilings as one-dimensional particle systems undergoing inelastic collapse. By deforming the spirals appropriately, we can simulate collisions among particles with distinct or varying coefficients of restitution. Our geometric constructions provide a strikingly simple illustration of a widely studied phenomenon in the physics of dissipative gases: the collapse of inelastic particles.
Cao, X. (Ed.)We investigate random walks in graphs whose edges change over time as a function of the current probability distribution of the walk. We show that such systems can be chaotic and can exhibit ``hyper-torpid" mixing. Our main result is that, if each graph is strongly connected, then the dynamics is asymptotically periodic almost surely.
PertInInt: An Integrative, Analytical Approach to Rapidly Uncover Cancer Driver Genes with Perturbed Interactions and FunctionalitiesA major challenge in cancer genomics is to identify genes with functional roles in cancer and uncover their mechanisms of action. We introduce an integrative framework that identifies cancer-relevant genes by pinpointing those whose interaction or other functional sites are enriched in somatic mutations across tumors. We derive analytical calculations that enable us to avoid time-prohibitive permutation-based significance tests, making it computationally feasible to simultaneously consider multiple measures of protein site functionality. Our accompanying software, PertInInt, combines knowledge about sites participating in interactions with DNA, RNA, peptides, ions, or small molecules with domain, evolutionary conservation, and gene-level mutation data. When applied to 10,037 tumor samples, PertInInt uncovers both known and newly predicted cancer genes, while additionally revealing what types of interactions or other functionalities are disrupted. PertInInt’s analysis demonstrates that somatic mutations are frequently enriched in interaction sites and domains and implicates interaction perturbation as a pervasive cancer-driving event.