The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\mathbb F_p(t)$, when $p$ is prime and ...$r\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\mathbb F_q(t^1/d)$.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2014.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 75-76).
Faltings' theorem ...states that curves of genus g > 2 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the upper bound on the number of rational points, XI, paragraph2, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry to show that we can also give an effective bound on the number of rational points outside of the special set of Symd X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g > d.
by Jennifer Mun Young Park.
Ph. D.
Under the generalized age replacement policy, the system is replaced either at the predetermined age or upon failure if its corresponding repair time exceeds the threshold, whichever comes first. In ...this paper, we investigate the optimal choice of the pre‐determined preventive replacement age for a nonwarranted system, which minimizes the expected cost rate during the life cycle of the system from the customer's perspective under certain cost structures. Furthermore, we discuss several properties of such a generalized age replacement policy in comparison with the traditional age replacement policy. An efficiency, which represents the fractional time that the system is on, is defined under the proposed generalized age replacement policy and its monotonicity properties are investigated as well. The main objective of this study is to investigate the advantageous features of the generalized age replacement policy over the traditional age replacement policy with regard to the availability of the repairable system. Assuming that the system deteriorates with age, we illustrate our proposed optimal policies numerically and observe the impact of relevant parameters on the optimal preventive replacement age.
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Evolution of the Brassica species has been recursively affected by polyploidy events, and comparison to their relative, Arabidopsis thaliana, provides means to explore their genomic complexity.
A ...genome-wide physical map of a rapid-cycling strain of B. oleracea was constructed by integrating high-information-content fingerprinting (HICF) of Bacterial Artificial Chromosome (BAC) clones with hybridization to sequence-tagged probes. Using 2907 contigs of two or more BACs, we performed several lines of comparative genomic analysis. Interspecific DNA synteny is much better preserved in euchromatin than heterochromatin, showing the qualitative difference in evolution of these respective genomic domains. About 67% of contigs can be aligned to the Arabidopsis genome, with 96.5% corresponding to euchromatic regions, and 3.5% (shown to contain repetitive sequences) to pericentromeric regions. Overgo probe hybridization data showed that contigs aligned to Arabidopsis euchromatin contain ~80% of low-copy-number genes, while genes with high copy number are much more frequently associated with pericentromeric regions. We identified 39 interchromosomal breakpoints during the diversification of B. oleracea and Arabidopsis thaliana, a relatively high level of genomic change since their divergence. Comparison of the B. oleracea physical map with Arabidopsis and other available eudicot genomes showed appreciable 'shadowing' produced by more ancient polyploidies, resulting in a web of relatedness among contigs which increased genomic complexity.
A high-resolution genetically-anchored physical map sheds light on Brassica genome organization and advances positional cloning of specific genes, and may help to validate genome sequence assembly and alignment to chromosomes.All the physical mapping data is freely shared at a WebFPC site (http://lulu.pgml.uga.edu/fpc/WebAGCoL/brassica/WebFPC/; Temporarily password-protected: account: pgml; password: 123qwe123.
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