Topological representation of matroids from diagrams of spaces

Swartz proved that any matroid can be realized as the intersection lattice of an arrangement of codimension one homotopy spheres on a sphere. This was an unexpected extension from the oriented matroid case, but unfortunately the construction is not explicit. Anderson later provided an explicit construction, but had to use cell complexes of high dimensions that are homotopy equivalent to lower dimensional spheres. Using diagrams of spaces we give an explicit construction of arrangements in the right dimensions. Swartz asked if it is possible to arrange spheres of codimension two, and we provide a construction for any codimension. We also show that all matroids, and not only tropical oriented matroids, have a pseudo-tropical representation. We determine the homotopy type of all the constructed arrangements.Comment: 18 pages, 6 figures. Some more typos fixe

The Diminished Trial

This Article thus highlights the need for a much larger inquiry into not only what is happening to trials but also what is happening in those that remain. Part I draws together statistics that suggest that trials are changing: Short trials (of one day) are becoming relatively more common, while long trials (of twenty days or more) are becoming less common.Part II then identifiesand explores some of the factors that may explain these trends. Finally, a brief conclusion highlights some problems with all the above and issues a call for further study and analysis