This work discusses the problem of exploration of an unknown environment using a collaborative group of simple agents. While this problem was known to be of a non-polynomial time complexity, it was speculated in the past that in grid domains the completion time of this problem is much lower (although analytic proofs were not available hitherto). In this work we present a preliminary result conc...

We give short new geometric proofs of theorems of Bogley and Pride and of Serre. Both follow quickly from a global fixed point theorem. The Bogley–Pride theorem concerns aspherical relative group presentations and was applied in [5] to the multivariable adjunction problem. Serre’s theorem is a basic result concerning group cohomology and finite subgroups.

The aim of this paper is to give a survey of results and ideas concerning orbits of operators and related notions of weak and polynomial orbits. These concepts are closely related to the invariant subspace/subset problem. Most of the proofs are not given in full details, we rather try to indicate the basic ideas. The central problems in the field are also formulated. Mathematics Subject Classif...