Lehmer’s question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spaces L(n, 1), n > 0. Lehmer’s question for Perron polynomials is equivalent to one about generalized growth rates of words under injective free group endomorphisms.

We discuss a variant of multi-task n-vehicle exploration problem. Instead of requiring an optimal permutation of vehicles in every group, the new problem asks all vehicles in a group to arrive at a same destination. It can also be viewed as to maximize every processor’s average profit, given n tasks, and each task’s consumetime and profit. Meanwhile, we propose a new kind of partition problem i...

I. Review: An obstruction theory. — Let Θn be the group of diffeomorphism classes of oriented smooth homotopy spheres of dimension n. Let Diff(Sn−1) be the simplicial group of orientation preserving diffeomorphisms Sn−1 → Sn−1. For n > 5, the natural homomorphism from π0Diff(Sn−1) to Θn is surjective by Smale’s h–cobordism theorem, and injective by Cerf’s pseudo–isotopy theorem. It is easily se...