We determine the mixing time (up to a constant factor) of the Markov chain whose state space consists of n “dots” on the unit interval, wherein a dot is selected uniformly at random and moved to a uniformly random point between its two neighbors. The method involves a two-step coupling for the upper bound, and an unusual probabilistic secondmoment argument for the lower.

Localized cell shape change initiates epithelial folding, while neighboring cell invagination determines the final depth of an epithelial fold. The mechanism that controls the extent of invagination remains unknown. During Drosophila gastrulation, a higher number of cells undergo invagination to form the deep posterior dorsal fold, whereas far fewer cells become incorporated into the initially ...

Given a linear isometry A0 : Rn → Rn of finite order on Rn , a general 〈A0〉-invariant closed subset M of Rn is considered with Lipschitz boundary. Under suitable topological restrictions the existence of A0-invariant geodesics of M is proven.