Negative Correlation Properties for Matroids
| dc.comment.hidden | the abstract compiles in latex with standard packages and no macros. | en |
| dc.contributor.author | Erickson, Alejandro | |
| dc.date.accessioned | 2008-12-23T18:50:42Z | |
| dc.date.available | 2008-12-23T18:50:42Z | |
| dc.date.issued | 2008-12-23T18:50:42Z | |
| dc.date.submitted | 2008 | |
| dc.description.abstract | 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 $$P(\{X:e,f\in X\}) \leq P(\{X:e\in X\})P(\{X:f\in X\})$$ for certain measures, $P$, on the ground set. Let $\mathcal M$ be a matroid. Let $(y_g:g\in E)$ be a weighting of the ground set and let $${Z = \sum_{X}\left( \prod_{x\in X} y_x\right) }$$ be the polynomial which generates Z-sets, were Z $\in \{$ 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 $Z_e$ indicate partial derivative. Then $\mathcal M$ is Z-Rayleigh if $Z_eZ_f-ZZ_{ef}\geq 0$ for every positive evaluation of the $y_g$s. 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. | en |
| dc.identifier.uri | http://hdl.handle.net/10012/4165 | |
| dc.language.iso | en | en |
| dc.pending | false | en |
| dc.publisher | University of Waterloo | en |
| dc.subject | negative correlation | en |
| dc.subject | matroid | en |
| dc.subject | Rayleigh | en |
| dc.subject | balance | en |
| dc.subject | negative association | en |
| dc.subject | paving | en |
| dc.subject.program | Combinatorics and Optimization | en |
| dc.title | Negative Correlation Properties for Matroids | en |
| dc.type | Master Thesis | en |
| uws-etd.degree | Master of Mathematics | en |
| uws-etd.degree.department | Combinatorics and Optimization | en |
| uws.peerReviewStatus | Unreviewed | en |
| uws.scholarLevel | Graduate | en |
| uws.typeOfResource | Text | en |