Separable subgroups have bounded packing

In this note, we prove that separable subgroups have bounded packing in ambient groups. The notion bounded packing was introduced by Hruska and Wise and in particular, our result answers positively a question of theirs, asking whether each subgroup of a virtually polycyclic group has the bounded packing property.Comment: 2 pages, to appear Proc. Amer. Math. So

Connectivity-Preserving Swarm Teleoperation With A Tree Network

During swarm teleoperation, the human operator may threaten the distance-dependent inter-robot communications and, with them, the connectivity of the slave swarm. To prevent the harmful component of the human command from disconnecting the swarm network, this paper develops a constructive strategy to dynamically modulate the interconnections of, and the locally injected damping at, all slave robots. By Lyapunov-based set invariance analysis, the explicit law for updating that control gains has been rigorously proven to synchronize the slave swarm while preserving all interaction links in the tree network. By properly limiting the impact of the user command rather than rejecting it entirely, the proposed control law enables the human operator to guide the motion of the slave swarm to the extent to which it does not endanger the connectivity of the swarm network. Experiment results demonstrate that the proposed strategy can maintain the connectivity of the tree network during swarm teleoperation