A linear bound on the k-rendezvous time for primitive sets of NZ matrices

A set of nonnegative matrices is called primitive if there exists a product of these matrices that is entrywise positive. Motivated by recent results relating synchronizing automata and primitive sets, we study the length of the shortest product of a primitive set having a column or a row with k positive entries, called its k-rendezvous time (k-RT}), in the case of sets of matrices having no zero rows and no zero columns. We prove that the k-RT is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We provide two upper bounds on the k-RT: the second is an improvement of the first one, although the latter can be written in closed form. We then report numerical results comparing our upper bounds on the k-RT with heuristic approximation methods.Comment: 27 pages, 10 figur

Orienting polyhedral parts by pushing

A common task in automated manufacturing processes is to orient parts prior to assembly. We consider sensorless orientation of an asymmetric polyhedral part by a sequence of push actions, and show that is it possible to move any such part from an unknown initial orientation into a known final orientation if these actions are performed by a jaw consisting of two orthogonal planes. We also show how to compute an orienting sequence of push actions.We propose a three-dimensional generalization of conveyor belts with fences consisting of a sequence of tilted plates with curved tips; each of the plates contains a sequence of fences. We show that it is possible to compute a set-up of plates and fences for any given asymmetric polyhedral part such that the part gets oriented on its descent along plates and fences