We investigate weighted finite automata over strings and strong bimonoids. Such algebraic structures satisfy the same laws as semirings except that no distributivity laws need to hold. We define two different behaviors and prove precise characterizations for them if the underlying strong bimonoid satisfies local finiteness conditions. Moreover, we show that in this case the given weighted autom...

We show that when certain cohomological finiteness conditions hold, then the dimensions of the cohomology groups of modules over a finite dimensional algebra are given by polynomials. This applies in particular to quantum complete intersections, and among these all exterior algebras.

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This conjecture, known as the “finiteness conjecture”, is...

The Jacobson-Bourbaki Theorem for division rings was formulated in terms of corings by Sweedler in [14]. Finiteness conditions hypotheses are not required in this new approach. In this paper we extend Sweedler’s result to simple artinian rings using a particular class of corings, comatrix corings. A Jacobson-Bourbaki like correspondence for simple artinian rings is then obtained by duality.

By explicit solution of the one-loop finiteness conditions for gauge and quartic scalarboson self-interaction coupling constants, a particular class of grand unified theories with vanishing Yukawa couplings as well as vanishing one-loop renormalization-group beta functions is constructed.

For HTN planning, we formally characterize and classify four kinds of problem spaces in which each node represents a planning problem or subproblem. Two of the problem spaces are searched by current HTN planning algorithms; the other two problem spaces are new. This enables us to provide: • Sufficient (and in one case, necessary) conditions for finiteness of each kind of problem space. The cond...

We discuss residual finiteness and several related separability conditions for the class of monoid acts, namely weak subact separability, strong complete separability. For each these four conditions, we investigate which monoids have property that all their (finitely generated) acts satisfy condition. In particular, prove that: over a finite are completely separable (and hence other three condi...