We describe a lower bound for the rank of any real matrix in which all diagonal entries are significantly larger in absolute value than all other entries, and discuss several applications of this result to the study of problems in Geometry, Coding Theory, Extremal Finite Set Theory and Probability. This is partly a survey, containing a unified approach for proving various known results, but it contains several new results as well.

Ingo Wegener passed away on 26 November 2008, after a three-year struggle with brain cancer. This is a tremendous loss for all those who knew him and worked with him. Ingo is survived by his wife, Christa.

After studying Mathematics in Bielefeld and holding a post in Frankfurt am Main, in 1987 Ingo became a full professor of Computer Science, in Efficient Algorithms and Complexity Theory, at the Technische Universität Dortmund, a position he held until his death.

Random geometric graphs have been one of the fundamental models for reasoning about wireless networks: one places n points at random in a region of the plane (typically a square or circle), and then connects pairs of points by an edge if they are within a fixed distance of one another. In addition to giving rise to a range of basic theoretical questions, this class of random graphs has been a central analytical tool in the wireless networking community.

For many of the primary applications of wireless networks, however, the underlying environment has a large number of obstacles, and communication can only take place among nodes when they are close in space and when they have line-of-sight access to one another – consider, for example, urban settings or large indoor environments. In such domains, the standard model of random geometric graphs is not a good approximation of the true constraints, since it is not designed to capture the line-of-sight restrictions.

Here we propose a random-graph model incorporating both range limitations and line-of-sight constraints, and we prove asymptotically tight results for k-connectivity. Specifically, we consider points placed randomly on a grid (or torus), such that each node can see up to a fixed distance along the row and column it belongs to. (We think of the rows and columns as ‘streets’ and ‘avenues’ among a regularly spaced array of obstructions.) Further, we show that when the probability of node placement is a constant factor larger than the threshold for connectivity, near-shortest paths between pairs of nodes can be found, with high probability, by an algorithm using only local information. In addition to analysing connectivity and k-connectivity, we also study the emergence of a giant component, as well an approximation question, in which we seek to connect a set of given nodes in such an environment by adding a small set of additional ‘relay’ nodes.

We show that many classical decision problems about1-counter ω-languages, context free ω-languages, or infinitary rational relations, are Π½ -complete, hence located at the second level of the analytical hierarchy, and “highly undecidable”.In particular, the universalityproblem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, and theunambiguity problem are all Π½ -complete for context-free ω-languages or for infinitary rational relations. Topological and arithmetical properties of1-counter ω-languages, context free ω-languages, or infinitary rational relations, are also highly undecidable.These very surprising results provide the first examples of highly undecidable problems about the behaviour of verysimple finite machines like 1-counter automata or 2-tapeautomata.