Generalized U-factorization in Commutative Rings with Zero-Divisors

Recently substantial progress has been made on generalized factorization techniques in integral domains, in particular $\tau$-factorization. There has also been advances made in investigating factorization in commutative rings with zero-divisors. One approach which has been found to be very successful is that of U-factorization introduced by C.R. Fletcher. We seek to synthesize work done in these two areas by generalizing $\tau$-factorization to rings with zero-divisors by using the notion of U-factorization.Comment: 16 pages, to appear in Rocky Mountain Journal of Mathematic

Generalized Irreducible Divisor Graphs

In 1988, I. Beck introduced the notion of a zero-divisor graph of a commutative rings with $1$. There have been several generalizations in recent years. In particular, in 2007 J. Coykendall and J. Maney developed the irreducible divisor graph. Much work has been done on generalized factorization, especially $\tau$-factorization. The goal of this paper is to synthesize the notions of $\tau$-factorization and irreducible divisor graphs in domains. We will define a $\tau$-irreducible divisor graph for non-zero non-unit elements of a domain. We show that by studying $\tau$-irreducible divisor graphs, we find equivalent characterizations of several finite $\tau$-factorization properties.Comment: 17 pages, 2 figures, to appear in Communications in Algebr

The antibiotic sensitivity patterns and plasmid DNA content of gram-negative anaerobic bacteria isolated in Palmerston North, New Zealand : a thesis presented in partial fulfilment of the requirements for the degree of Masters in Science at Massey University

One hundred and seven Gram-negative bacteria, including 65 Bacteroides species, 28 fusobacteria and 14 veillonellae were isolated from 17 oral infections treated in two dental surgeries in Palmerston North. These bacteria, plus 37 isolates belonging to the B. fragilis group received from Palmerston North hospital, were surveyed for their antibiotic sensitivity levels, and their plasmid DNA content. The hospital isolates of the B. fragilis group were found to have sensitivity levels comparable with those of B. fragilis group isolates reported in the literature recently. The oral isolates were more sensitive to penicillin, cefoxitin, and tetracycline than isolates of the same species reported in the literature. Half the hospital isolates had plasmids, which were all between 8.5 and 2.7 kilobases (kb) in size except for one 60, and one 43 kb plasmid. Comparatively few of the oral anaerobes had plasmids. One Fusobacterium russii isolate had four plasmids, and five Bacteroides isolates had one plasmid each. These five Bacteroides isolates came from two specimens, R5 and R6. Restriction enzyme analysis of all plasmids revealed that the three 5.6 kb plasmids from sample R5 may be related to a group of 5.8 kb plasmids harboured by four of the hospital isolates. Two different species of Bacteroides isolated from sample R5 harboured the 5.6 kb plasmid, and two species of the B. fragilis group bacteria harboured the 5.8 kb plasmid. Plasmid DNA isolated from two tetracycline resistant hospital isolates was used to transform restriction negative E. coli to a low level of tetracycline resistance

Cell-Like Equivalences and Boundaries of CAT(0) Groups

In 2000, Croke and Kleiner showed that a CAT(0) group G can admit more than one boundary. This contrasted with the situation for word hyperbolic groups, where it was well-known that each such group admitted a unique boundary---in a very stong sense. Prior to Croke and Kleiner's discovery, it had been observed by Geoghegan and Bestvina that a weaker sort of uniquness does hold for boundaries of torsion free CAT(0) groups; in particular, any two such boundaries always have the same shape. Hence, the boundary really does carry significant information about the group itself. In an attempt to strengthen the correspondence between group and boundary, Bestvina asked whether boundaries of CAT(0) groups are unique up to cell-like equivalence. For the types of space that arise as boundaries of CAT(0) groups, this is a notion that is weaker than topological equivalence and stronger than shape equivalence. In this paper we explore the Bestvina Cell-like Equivalence Question. We describe a straightforward strategy with the potential for providing a fully general positive answer. We apply that strategy to a number of test cases and show that it succeeds---often in unexpectedly interesting ways.Comment: 21 pages, 5 figure

Examples of Non-Rigid CAT(0) Groups from the Category of Knot Groups

C Croke and B Kleiner have constructed an example of a CAT(0) group with more than one visual boundary. J Wilson has proven that this same group has uncountably many distinct boundaries. In this article we prove that the knot group of any connected sum of two non-trivial torus knots also has uncountably many distinct CAT(0) boundaries.Comment: 16 pages, 2 figure