The traditional power grid has been designed in a hierarchical fashion, with energy pushed from the large scale production factories towards the end users. With the increasing availability of micro and medium scale generating facilities, the situation is changing. Many end users can now produce energy and share it over the power grid. Of course, end users need incentives to do so and want to act in an open decentralized energy market. In the present work, we offer a novel analysis of the medium and low voltage power grids of the North Netherlands using statistical tools from the complex network analysis field. We use a weighted model based on actual grid data and propose a set of statistical measures to evaluate the adequacy of the current infrastructure for a decentralized energy market. Further, we use the insight gained by the analysis to propose parameters that tie the statistical topological measures to economic factors that influence the attractiveness of participating in such decentralized energy market, thus identifying the important topological parameters to work on to facilitate such open decentralized markets.

We investigate the topological interpretation of modal logic in modern terms, using a new notion of bisimulation. We look at modal logics with interesting topological content, presenting, among others, a new proof of McKinsey and Tarski's theorem on completeness of S4 with respect to the real line, and a completeness proof for the logic of finite unions of convex sets of reals. We conclude with a broader picture of extended modal languages of space, for which the main logical questions are still wide open.