On Pinned Billiard Balls and Foldings

We consider systems of “pinned balls,” that is, balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times of possible pseudo-collisions for different pairs of pinned balls are chosen in an exogenous way. We give an explicit upper bound for the maximum number of pseudo-collisions for a system of n pinned balls in a d-dimensional space, in terms of n, d, and the locations of ball centers. We also study foldings, that is, mappings that formalize the idea of folding a piece of paper along a crease. We prove a theorem on foldings that implies the number of pseudo-collisions of a finite number of pinned balls must be finite.