Linear relations of zeroes of the zeta-function

This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function. The main application is an alternative disproof to Mertens' conjecture. We show that $\limsup M(x)x^{-1/2} \geq 1.6383$ and that $\liminf M(x)x^{-1/2}\leq -1.6383$.Comment: 12 pages, 2 figures, 2 tables. Version 2: some typos corrected. To appear in Math. Com

Parity of transversals of Latin squares

We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2 \bmod 4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving $E_1,\dots, E_n$, where $E_i$ is the number of diagonals of a given Latin square that contain exactly $i$ different symbols. Let $A(i\mid j)$ denote the matrix obtained by deleting row $i$ and column $j$ from a parent matrix $A$. Define $t_{ij}$ to be the number of transversals in $L(i\mid j)$, for some fixed Latin square $L$. We show that $t_{ab}\equiv t_{cd}\bmod2$ for all $a,b,c,d$ and $L$. Also, if $L$ has odd order then the number of transversals of $L$ equals $t_{ab}$ mod 2. We conjecture that $t_{ac} + t_{bc} + t_{ad} + t_{bd} \equiv 0 \bmod 4$ for all $a,b,c,d$. In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a $k$-regular bipartite graph on $2n$ vertices is divisible by $4$ when $n$ is odd and $k\equiv0\bmod 4$. We also show that $${\rm per}\, A(a \mid c)+{\rm per}\, A(b \mid c)+{\rm per}\, A(a \mid d)+{\rm per}\, A(b \mid d) \equiv 0 \bmod 4$$ for all $a,b,c,d$, when $A$ is an integer matrix of odd order with all row and columns sums equal to $k\equiv2\bmod4$

Biangular vectors

viii, 133 leaves ; 29 cmThis thesis introduces unit weighing matrices, a generalization of Hadamard matrices. When dealing with unit weighing matrices, a lot of the structure that is held by Hadamard matrices is lost, but this loss of rigidity allows these matrices to be used in the construction of certain combinatorial objects. We are able to fully classify these matrices for many small values by defining equivalence classes analogous to those found with Hadamard matrices. We then proceed to introduce an extension to mutually unbiased bases, called mutually unbiased weighing matrices, by allowing for different subsets of vectors to be orthogonal. The bounds on the size of these sets of matrices, both lower and upper, are examined. In many situations, we are able to show that these bounds are sharp. Finally, we show how these sets of matrices can be used to generate combinatorial objects such as strongly regular graphs and association schemes

Protein-coding gene promoters in Methanocaldococcus (Methanococcus) jannaschii

Although Methanocaldococcus (Methanococcus) jannaschii was the first archaeon to have its genome sequenced, little is known about the promoters of its protein-coding genes. To expand our knowledge, we have experimentally identified 131 promoters for 107 protein-coding genes in this genome by mapping their transcription start sites. Compared to previously identified promoters, more than half of which are from genes for stable RNAs, the protein-coding gene promoters are qualitatively similar in overall sequence pattern, but statistically different at several positions due to greater variation among their sequences. Relative binding affinity for general transcription factors was measured for 12 of these promoters by competition electrophoretic mobility shift assays. These promoters bind the factors less tightly than do most tRNA gene promoters. When a position weight matrix (PWM) was constructed from the protein gene promoters, factor binding affinities correlated with corresponding promoter PWM scores. We show that the PWM based on our data more accurately predicts promoters in the genome and transcription start sites than could be done with the previously available data. We also introduce a PWM logo, which visually displays the implications of observing a given base at a position in a sequence

The Distribution of Toxoplasma gondii Cysts in the Brain of a Mouse with Latent Toxoplasmosis: Implications for the Behavioral Manipulation Hypothesis

reportedly manipulates rodent behavior to enhance the likelihood of transmission to its definitive cat host. The proximate mechanisms underlying this adaptive manipulation remain largely unclear, though a growing body of evidence suggests that the parasite-entrained dysregulation of dopamine metabolism plays a central role. Paradoxically, the distribution of the parasite in the brain has received only scant attention. at six months of age and examined 18 weeks later. The cysts were distributed throughout the brain and selective tropism of the parasite toward a particular functional system was not observed. Importantly, the cysts were not preferentially associated with the dopaminergic system and absent from the hypothalamic defensive system. The striking interindividual differences in the total parasite load and cyst distribution indicate a probabilistic nature of brain infestation. Still, some brain regions were consistently more infected than others. These included the olfactory bulb, the entorhinal, somatosensory, motor and orbital, frontal association and visual cortices, and, importantly, the hippocampus and the amygdala. By contrast, a consistently low incidence of tissue cysts was recorded in the cerebellum, the pontine nuclei, the caudate putamen and virtually all compact masses of myelinated axons. Numerous perivascular and leptomeningeal infiltrations of inflammatory cells were observed, but they were not associated with intracellular cysts. distribution stems from uneven brain colonization during acute infection and explains numerous behavioral abnormalities observed in the chronically infected rodents. Thus, the parasite can effectively change behavioral phenotype of infected hosts despite the absence of well targeted tropism

Parity of transversals in latin squares

Non UBCUnreviewedAuthor affiliation: Monash UniversityGraduat