| In pursuit of negatively associated measures, this thesis focuses on certain negative correlation properties in matroids. In particular, the results presented contribute to the search for matroids which satisfy PX:e,f∈X ≤PX:e∈ XP X:f∈X for certain measures, P, on the ground set.;The known elementary results for the B, I and S-Rayleigh properties and two special cases called negative correlation and balance are proved. Furthermore, several new results are discussed. In particular, if a matroid is binary on at most nine elements or paving or rank three, then it is I-Rayleigh if it is B-Rayleigh. Sparse paving matroids are B-Rayleigh. The I-Rayleigh difference for graphs on at most seven vertices is a sum of monomials times squares of polynomials and this same special form holds for all series parallel graphs.;Let M be a matroid. Let (yg : g ∈ E) be a weighting of the ground set and let Z=X x∈Xyx be the polynomial which generates Z-sets, were Z ∈ {B, I, S}. For each of these, the sum is over bases, independent sets and spanning sets, respectively. Let e and f be distinct elements of E and let Ze indicate partial derivative. Then M is Z-Rayleigh if ZeZf -- ZZef ≥ 0 for every positive evaluation of the ygs. |
