Formal power series over non-commuting variables have been investigated as representations of the behavior of automata with multiplicities. Here we introduce and investigate the concepts of aperiodic and of star-free formal power series over semirings and partially commuting variables. We prove that if the semiring K is idempotent and commutative, or if K is idempotent and the variables are non...

In the paper we pose and discuss new Burnside-type problems, where the role of nilpo-tency is replaced by that of solvability.

We propound a Descent Principle by which previously constructed equations over GF(q n)(X) may be deformed to have incarnations over GF(q)(X) without changing their Galois groups, where q = p u > 1 is a power of a prime p and n is a positive integer. Currently this is achieved by starting with a vectorial (= additive) q-polynomial of q-degree m with Galois group GL(m; q) where m is any positive ...

We describe the close relationship between permutation groups and combinatorial species (introduced by A. Joyal, Adv. in Math. 42, 1981, l-82). There is a bijection @ between the set of transitive actions (up to isomorphism) of S, on finite sets and the set of “molecular” species of degree n (up to isomorphism). This bijection extends to a ring isomorphism between B(S,) (the Burnside ring of th...

Using a classical method from physics called Rayleigh’s cutting method, we prove the conjecture of Phillips and Sarnak that there is a universal lower bound L2 > 0 for the lowest eigenvalue of the quotient manifold of a classical Schottky group Γ acting on hyperbolic 3-space H 3. By work of Patterson and Sullivan, this implies that there is a universal upper bound U2 < 2 for the Hausdorff dimen...

The goal of this note is to write down the proofs of results about characters of irreducible representations of finite groups, and specifically, the result (due to Burnside) that each of them vanishes at at least one element of the group. We present various results told to us by various people, and conclude with two theorems by Gallagher, which strengthen or generalize Burnside’s theorem for fi...