This paper presents various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv) 1 4 improving on previous lower bounds. For polygons with v vertices, we show that psd rank cannot exceed 4 dv/6e which in turn shows that the psd ran...

We provide a counterexample to a recent conjecture that the minimum rank of every sign pattern matrix can be realized by a rational matrix. We use one of the equivalences of the conjecture and some results from projective geometry. As a consequence of the counterexample we show that there is a graph for which the minimum rank over the reals is strictly smaller than the minimum rank over the rat...

Some old and new constructions of the tilde geometry (the flag transitive connected triple cover of the unique generalized quadrangle W(2) of order (2, 2)) are discussed. Using them, we prove some properties of that geometry. In particular, we compute its generating rank, we give an explicit description of its universal projective embedding and we determine its homogeneous embeddings.

We study a ring of weak Jacobi forms indexed by integral lattices rank two. find an explicit finite set generators this and give dimension formula for two lattice index.