Weak hypergraph regularity and linear hypergraphs

We consider conditions which allow the embedding of linear hypergraphs of fixed size. In particular, we prove that any k-uniform hypergraph H of positive uniform density contains all linear k-uniform hypergraphs of a given size. More precisely, we show that for all integers l >= k >= 2 and every d > 0 there exists Q > 0 for which the following holds: if His a sufficiently large k-uniform hypergraph with the property that the density of H induced on every vertex subset of size on is at least d, then H contains every linear k-uniform hypergraph F with l vertices. The main ingredient in the proof of this result is a counting lemma for linear hypergraphs, which establishes that the straightforward extension of graph epsilon-regularity to hypergraphs suffices for counting linear hypergraphs. We also consider some related problems. (C) 2009 Elsevier Inc. All rights reserved.

Diagonal forms, linear algebraic methods and Ramsey-type problems

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.

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.

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.

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.

Fast modularisation and aomic decomposition of ontologies using axiom dependency hypergraphs

In this paper we define the notion of an axiom dependency hypergraph, which explicitly represents how axioms are included into a module by the algorithm for computing locality-based modules. A locality-based module of an ontology corresponds to a set of connected nodes in the hypergraph, and atoms of an ontology to strongly connected components. Collapsing the strongly connected components into single nodes yields a condensed hypergraph that comprises a representation of the atomic decomposition of the ontology. To speed up the condensation of the hypergraph, we first reduce its size by collapsing the strongly connected components of its graph fragment employing a linear time graph algorithm. This approach helps to significantly reduce the time needed for computing the atomic decomposition of an ontology. We provide an experimental evaluation for computing the atomic decomposition of large biomedical ontologies. We also demonstrate a significant improvement in the time needed to extract locality-based modules from an axiom dependency hypergraph and its condensed version.

Components of learning and assessment in linear algebra

Linear algebra provides theory and technology that are the cornerstones of a range of cutting edge mathematical applications, from designing computer games to complex industrial problems, as well as more traditional applications in statistics and mathematical modelling. Once past introductions to matrices and vectors, the challenges of balancing theory, applications and computational work across mathematical and statistical topics and problems are considerable, particularly given the diversity of abilities and interests in typical cohorts. This paper considers two such cohorts in a second level linear algebra course in different years. The course objectives and materials were almost the same, but some changes were made in the assessment package. In addition to considering effects of these changes, the links with achievement in first year courses are analysed, together with achievement in a following computational mathematics course. Some results that may initially appear surprising provide insight into the components of student learning in linear algebra.