No abstract

Let P be a Poisson process of intensity one in the infinite plane R 2 . We surround each point x of P by the open disc of radius r centred at x. Now let S n be a fixed disc of area n, and let C r (S n ) be the set of discs which intersect S n . Write E k r for the event that C r (S n ) is a k-cover of S n , and F k r for the event that C r (S n ) may be partitioned into k disjoint single covers of S n . We prove thatlog n , and that this result is best possible. We also give improved estimates for P(E k r ). Finally, we study the obstructions to k-partitionability in more detail. As part of this study, we prove a classification theorem for (deterministic) covers of R 2 with half-planes that cannot be partitioned into two single covers.

Following Etherington, Hoge and Parkes, we consider a network consisting of (approximately) N transceivers in the plane R 2 distributed randomly with density given by a Gaussian distribution about the origin, and assume each transceiver can communicate with all other transceivers within distance s. We give bounds for the distance from the origin to the furthest transceiver connected to the origin, and that of the closest transceiver that is not connected to the origin.

No abstract