Phd records
https://feeds.library.caltech.edu/people/Wong-Wing-Hong-Tony/Phd.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 31 Jan 2024 20:02:34 +0000Diagonal Forms, Linear Algebraic Methods and Ramsey-Type Problems
https://resolver.caltech.edu/CaltechTHESIS:05312013-153531964
Authors: {'items': [{'email': 'tonywhwong@yahoo.com.hk', 'id': 'Wong-Wing-Hong-Tony', 'name': {'family': 'Wong', 'given': 'Wing Hong Tony'}, 'show_email': 'YES'}]}
Year: 2013
DOI: 10.7907/5B5A-Q252
<p>This thesis focuses mainly on linear algebraic aspects of combinatorics. Let N_t(H) be an incidence matrix with edges versus all subhypergraphs of a complete hypergraph that are isomorphic to H. Richard M. Wilson and the author find the general formula for the Smith normal form or diagonal form of N_t(H) for all simple graphs H and for a very general class of t-uniform hypergraphs H.</p>
<p>As a continuation, the author determines the formula for diagonal forms of integer matrices obtained from other combinatorial structures, including incidence matrices for subgraphs of a complete bipartite graph and inclusion matrices for multisets.</p>
<p>One major application of diagonal forms is in zero-sum Ramsey theory. For instance, Caro's results in zero-sum Ramsey numbers for graphs and Caro and Yuster's results in zero-sum bipartite Ramsey numbers can be reproduced. These results are further generalized to t-uniform hypergraphs. Other applications include signed bipartite graph designs.</p>
<p>Research results on some other problems are also included in this thesis, such as a Ramsey-type problem on equipartitions, Hartman's conjecture on large sets of designs and a matroid theory problem proposed by Welsh.</p>https://thesis.library.caltech.edu/id/eprint/7801