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    ABSTRACT: We prove that if a prime ℓ>3ℓ>3 divides pk−1pk−1, where p is prime, then there is a congruence modulo ℓ , like Ramanujan's mod 691 congruence, for the Hecke eigenvalues of some cusp form of weight k and level p . We relate ℓ to primes like 691 by viewing it as a divisor of a partial zeta value, and see how a construction of Ribet links the congruence with the Bloch–Kato conjecture (theorem in this case). This viewpoint allows us to give a new proof of a recent theorem of Billerey and Menares. We end with some examples, including where p=2p=2 and ℓ is a Mersenne prime.
    No preview · Article · Oct 2014 · Journal of Number Theory
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    ABSTRACT: We study a coarse-grained asymptotic equation which describes deformation of vortex lattices [Smirnov & Chukbar 2001]. It reads φt=φxxφyy–φ2xy, where φ denotes displacement of vortex locations. This equation is valid for a lattice with short-ranged interaction, e.g. geostrophic vortices with a screened potential. New self-similar blow-up solutions with infinite total energy are found. We ask whether or not finite-time blow-up can take place developing from smooth initial data with finite energy. The numerical simulations show finite-time blow-up in such a way that φ∈H3 but φ∉H4.
    Full-text · Article · Dec 2013
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    ABSTRACT: Genome-wide association studies have successfully identified associations between common diseases and a large number of single nucleotide polymorphisms (SNPs) across the genome. We investigate the effectiveness of several statistics, including p-values, likelihoods, genetic map distance and linkage disequilibrium between SNPs, in filtering SNPs in several disease-associated regions. We use simulated data to compare the efficacy of filters with different sample sizes and for causal SNPs with different minor allele frequencies (MAFs) and effect sizes, focusing on the small effect sizes and MAFs likely to represent the majority of unidentified causal SNPs. In our analyses, of all the methods investigated, filtering on the ranked likelihoods consistently retains the true causal SNP with the highest probability for a given false positive rate. This was the case for all the local linkage disequilibrium patterns investigated. Our results indicate that when using this method to retain only the top 5% of SNPs, even a causal SNP with an odds ratio of 1.1 and MAF of 0.08 can be retained with a probability exceeding 0.9 using an overall sample size of 50,000.
    Full-text · Article · Nov 2013 · Annals of Human Genetics
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