Zero-sum problems in finite cyclic groups

The purpose of this thesis is to investigate some open problems in the area of combinatorial number theory referred to as zero-sum theory. A zero-sequence in a finite cyclic group G is said to have the basic property if it is equivalent under group automorphism to one which has sum precisely IGI when this sum is viewed as an integer. This thesis investigates two major problems, the first of which is referred to as the basic pair problem. This problem seeks to determine conditions for which every zero-sequence of a given length in a finite abelian group has the basic property. We resolve an open problem regarding basic pairs in cyclic groups by demonstrating that every sequence of length four in Zp has the basic property, and we conjecture on the complete solution of this problem. The second problem is a 1988 conjecture of Kleitman and Lemke, part of which claims that every sequence of length n in Zn has a subsequence with the basic property. If one considers the special case where n is an odd integer we believe this conjecture to hold true. We verify this is the case for all prime integers less than 40, and all odd integers less than 26. In addition, we resolve the Kleitman-Lemke conjecture for general n in the negative. That is, we demonstrate a sequence in any finite abelian group isomorphic to Z2p (for p ~ 11 a prime) containing no subsequence with the basic property. These results, as well as the results found along the way, contribute to many other problems in zero-sum theory.

Plant rhizosphere specificity and variability in the insect and plant adhesins, Madl and Mad2, within the genus Metarhizium suggest plant adaptation as an evolutionary force

Metarhizium is a soil-inhabiting fungus currently used as a biological control agent against various insect species, and research efforts are typically focused on its ability to kill insects. In section 1, we tested the hypothesis that species of Metarhizium are not randomly distributed in soils but show plant rhizosphere-specific associations. Results indicated an association of three Metarhizium species (Metarhizium robertsii, M. brunneum and M. guizhouense) with the rhizosphere of certain types of plant species. M. robertsii was the only species that was found associated with grass roots, suggesting a possible exclusion of M. brunneum and M. guizhouense, which was supported by in vitro experiments with grass root exudate. M. guizhouense and M. brunneum only associated with wildflower rhizosphere when co-occurring with M. robertsii. With the exception of these co-occurrences, M. guizhouense was found to associate exclusively with the rhizosphere of tree species, while M. brunneum was found to associate exclusively with the rhizosphere of shrubs and trees. These associations demonstrate that different species of Metarhizium associate with specific plant types. In section 2, we explored the variation in the insect adhesin, Madl, and the plant adhesin, Mad2, in fourteen isolates of Metarhizium representing seven different species. Analysis of the transcriptional elements within the Mad2 promoter region revealed variable STRE, PDS, degenerative TATA box, and TATA box-like regions. Phylogenetic analysis of 5' EF-Ia, which is used for species identification, as well as Madl and Mad2 sequences demonstrated that the Mad2 phylogeny is more congruent with 5' EF-1a than Madl. This suggests Mad2 has diverged among Metarhizium lineages, contributing to clade- and species-specific variation. While other abiotic and biotic factors cannot be excluded in contributing to divergence, it appears that plant associations have been the driving factor causing divergence among Metarhizium species.

Rings, Group Rings, and Their Graphs

We associate some graphs to a ring R and we investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of the graphs associated to R. Let Z(R) be the set of zero-divisors of R. We define an undirected graph ᴦ(R) with nonzero zero-divisors as vertices and distinct vertices x and y are adjacent if xy=0 or yx=0. We investigate the Isomorphism Problem for zero-divisor graphs of group rings RG. Let Sk denote the sphere with k handles, where k is a non-negative integer, that is, Sk is an oriented surface of genus k. The genus of a graph is the minimal integer n such that the graph can be embedded in Sn. The annihilating-ideal graph of R is defined as the graph AG(R) with the set of ideals with nonzero annihilators as vertex such that two distinct vertices I and J are adjacent if IJ=(0). We characterize Artinian rings whose annihilating-ideal graphs have finite genus. Finally, we extend the definition of the annihilating-ideal graph to non-commutative rings.

Variability in the Insect and Plant Adhesins, Mad1 and Mad2, within the Fungal Genus Metarhizium Suggest Plant Adaptation as an Evolutionary Force

Several species of the insect pathogenic fungus Metarhizium are associated with certain plant types and genome analyses suggested a bifunctional lifestyle; as an insect pathogen and as a plant symbiont. Here we wanted to explore whether there was more variation in genes devoted to plant association (Mad2) or to insect association (Mad1) overall in the genus Metarhizium. Greater divergence within the genus Metarhizium in one of these genes may provide evidence for whether host insect or plant is a driving force in adaptation and evolution in the genus Metarhizium. We compared differences in variation in the insect adhesin gene, Mad1, which enables attachment to insect cuticle, and the plant adhesin gene, Mad2, which enables attachment to plants. Overall variation for the Mad1 promoter region (7.1%), Mad1 open reading frame (6.7%), and Mad2 open reading frame (7.4%) were similar, while it was higher in the Mad2 promoter region (9.9%). Analysis of the transcriptional elements within the Mad2 promoter region revealed variable STRE, PDS, degenerative TATA box, and TATA box-like regions, while this level of variation was not found for Mad1. Sequences were also phylogenetically compared to EF-1a, which is used for species identification, in 14 isolates representing 7 different species in the genus Metarhizium. Phylogenetic analysis demonstrated that the Mad2 phylogeny is more congruent with 59 EF-1a than Mad1. This would suggest that Mad2 has diverged among Metarhizium lineages, contributing to clade- and species-specific variation, while it appears that Mad1 has been largely conserved. While other abiotic and biotic factors cannot be excluded in contributing to divergence, these results suggest that plant relationships, rather than insect host, have been a major driving factor in the divergence of the genus Metarhizium.

Variability in the Insect and Plant Adhesins, Mad1 and Mad2, within the Fungal Genus Metarhizium Suggest Plant Adaptation as an Evolutionary Force

Several species of the insect pathogenic fungus Metarhizium are associated with certain plant types and genome analyses suggested a bifunctional lifestyle; as an insect pathogen and as a plant symbiont. Here we wanted to explore whether there was more variation in genes devoted to plant association (Mad2) or to insect association (Mad1) overall in the genus Metarhizium. Greater divergence within the genus Metarhizium in one of these genes may provide evidence for whether host insect or plant is a driving force in adaptation and evolution in the genus Metarhizium. We compared differences in variation in the insect adhesin gene, Mad1, which enables attachment to insect cuticle, and the plant adhesin gene, Mad2, which enables attachment to plants. Overall variation for the Mad1 promoter region (7.1%), Mad1 open reading frame (6.7%), and Mad2 open reading frame (7.4%) were similar, while it was higher in the Mad2 promoter region (9.9%). Analysis of the transcriptional elements within the Mad2 promoter region revealed variable STRE, PDS, degenerative TATA box, and TATA box-like regions, while this level of variation was not found for Mad1. Sequences were also phylogenetically compared to EF-1a, which is used for species identification, in 14 isolates representing 7 different species in the genus Metarhizium. Phylogenetic analysis demonstrated that the Mad2 phylogeny is more congruent with 59 EF-1a than Mad1. This would suggest that Mad2 has diverged among Metarhizium lineages, contributing to clade- and species-specific variation, while it appears that Mad1 has been largely conserved. While other abiotic and biotic factors cannot be excluded in contributing to divergence, these results suggest that plant relationships, rather than insect host, have been a major driving factor in the divergence of the genus Metarhizium.