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.

For a ∇-module M over the ring K[[x]]0 of bounded functions over a p-adic local field K we define the notion of special and generic log-growth filtrations on the base of the power series development of the solutions and horizontal sections. Moreover, if M also admits a Frobenius structure then it is endowed with generic and special Frobenius slope filtrations. We will show that in the case of M a ϕ–∇-module of rank 2, the Frobenius polygon for M and the log-growth polygon for its dual, Mv, coincide, this is proved by showing explicit relationships between the filtrations. This will lead us to formulate some conjectural links between the behaviours of the filtrations arising from the log-growth and Frobenius structures of the differential module. This coincidence between the two polygons was only known for the hypergeometric cases by Dwork.

Given two unital continuous C*-bundles, A and B, over the same compact Hausdorff base space X, we study the continuity properties of their different amalgamated free products over C(X).

We complete the investigation of growth properties of analytic functions connected with the Nevanlinna parametrization of the solutions of an indeterminate strong Hamburger moment problem.

Suppose that G is an abelian group and that A ⊂ G is finite and contains no non-trivial three-term arithmetic progressions. We show that |A+A| »ε|A|(log|A|)⅓−ε.

We study the sharp threshold for blow-up and global existence and the instability of standing wave eiωtuω(x) for the Davey–Stewartson system

in ℝ3, where uω is a ground state. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp criterion for global existence and blow-up of the solutions to (DS), which can be used to show that there exist blow-up solutions of (DS) arbitrarily close to the standing waves.

A problem posed in the early eighteenth century asks for right-angled triangles, each of whose sides exceeds double the area by a perfect square. We summarize known results and find such triangles with the smallest possible standard generators.

We consider semilinear elliptic problems in which the right-hand-side nonlinearity depends on a parameter λ > 0. Two multiplicity results are presented, guaranteeing the existence of at least three non-trivial solutions for this kind of problem, when the parameter λ belongs to an interval (0,λ*). Our approach is based on variational techniques, truncation methods and critical groups. The first result incorporates as a special case problems with concave–convex nonlinearities, while the second one involves concave nonlinearities perturbed by an asymptotically linear nonlinearity at infinity.

Using variational methods, we establish the existence and multiplicity of positive solutions for the following class of problems:

where λ,β∈(0,∞), q∈(1,2*−1), 2*=2N/(N−2), N≥3, V,Z:ℝN→ℝ are continuous functions verifying some hypotheses.

We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Γ is any graph not containing a four-cycle without chords, then the group G(Γ) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Γ) does not have a subgroup isomorphic to a direct product of non-abelian groups. We also obtain corresponding and more general results in the monoid case.

Let Ω ⊂ ℝN be a bounded domain such that 0 ∈ Ω, N ≥ 3, 2*(s) = 2(N − s)/(N − 2), 0 ≤ s < 2, . We obtain the existence of infinitely many solutions for the singular critical problem with Dirichlet boundary condition for suitable positive number λ.

We study a class of second-order nonlinear differential equations on a finite interval with periodic boundary conditions. The nonlinearity in the equations can take negative values and may be unbounded from below. Criteria are established for the existence of non-trivial solutions, positive solutions and negative solutions of the problems under consideration. Applications of our results to related eigenvalue problems are also discussed. Examples are included to illustrate some of the results. Our analysis relies mainly on topological degree theory.

There are only finitely many non-constant holomorphic mappings between two fixed compact Riemann surfaces of genus greater than 1. This result goes under the name of the de Franchis theorem. Having seen that the set of such holomorphic mappings is finite, we naturally want to obtain a bound on its cardinality. It has been known for some time that there exist various bounds depending only on the genera of the surfaces. Here we obtain ‘better’ bounds of the above type, using arguments based on the rigidity of holomorphic mappings and the hyperbolic geometry of surfaces.

In this paper we determine the group of endotrivial modules for certain symmetric and alternating groups in characteristic p. If p = 2, then the group is generated by the class of Ωn(k) except in a few low degrees. If p > 2, then the group is only determined for degrees less than p2. In these cases we show that there are several Young modules which are endotrivial.

We use the compensated compactness method coupled with some basic ideas of kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor to give a refined proof for the existence of global bounded entropy solutions to the Le Roux system. This new method of the reduction of Young measures can be applied to solve other problems.

The theorem referred to in the title is a technical result that is needed for the classification of elliptic and K3 fibrations birational to Fano 3-fold hypersurfaces in weighted projective space. We present a complete proof of the curve exclusion theorem, which first appeared in the author's PhD thesis and has since been relied upon in solutions to many cases of the fibration classification problem. We give examples of these solutions and discuss them briefly.

We prove that if two transcendental meromorphic functions share all limit values from a set of positive linear measure on a rectifiable Jordan arc, then they share all limit values.