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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.
    Journal of Number Theory 10/2014; 143:248–261. DOI:10.1016/j.jnt.2014.04.008
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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.
    Annals of Human Genetics 11/2013; DOI:10.1111/ahg.12043
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    ABSTRACT: Certain modified gravity theories predict the existence of an additional, nonconformally coupled scalar field. A disformal coupling of the field to the cosmic microwave background (CMB) is shown to affect the evolution of the energy density in the radiation fluid and produces a modification of the distribution function of the CMB, which vanishes if photons and baryons couple in the same way to the scalar. We find the constraints on the couplings to matter and photons coming from the measurement of the CMB temperature evolution and from current upper limits on the μ distortion of the CMB spectrum. We also point out that the measured equation of state of photons differs from w_{γ}=1/3 in the presence of disformal couplings.
    Physical Review Letters 10/2013; 111(16):161302. DOI:10.1103/PhysRevLett.111.161302
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    ABSTRACT: A general fact about language is that subject relative clauses are easier to process than object relative clauses. Recently, several self-paced reading studies have presented surprising evidence that object relatives in Chinese are easier to process than subject relatives. We carried out three self-paced reading experiments that attempted to replicate these results. Two of our three studies found a subject-relative preference, and the third study found an object-relative advantage. Using a random effects bayesian meta-analysis of fifteen studies (including our own), we show that the overall current evidence for the subject-relative advantage is quite strong (approximate posterior probability of a subject-relative advantage given the data: 78-80%). We argue that retrieval/integration based accounts would have difficulty explaining all three experimental results. These findings are important because they narrow the theoretical space by limiting the role of an important class of explanation-retrieval/integration cost-at least for relative clause processing in Chinese.
    PLoS ONE 10/2013; 8(10):e77006. DOI:10.1371/journal.pone.0077006
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    ABSTRACT: There is increasing experimental evidence that exposure to low doses of infection may 'prime' the immune response of invertebrate hosts, giving them greater protection against future infection. This form of immune memory is not compatible with the 'acquired immunity' modelled by the classic Susceptible-Infected-Recovered (SIR) epidemiological model, but instead requires the development of an alternative Susceptible-Primed-Infected (SPI) framework. Some initial theoretical work has explored the epidemiological and evolutionary dynamics of the SPI model, but these have assumed hosts exist in a constant environment. In reality, natural invertebrate-disease systems will be subject to significant environmental variation. Here, I use bifurcation analysis using numerical continuation software, complemented with numerical simulations, to investigate the effects of seasonal forcing on the already complex epidemiological dynamics of the SPI model. I show that multi-year cycles, quasi-periodicity, chaos, and multiple stability may all result, and highlight the importance not just of the forcing amplitude, but also the ecological and epidemiological background, for complex dynamics to emerge.
    Bulletin of Mathematical Biology 08/2013; 75(11). DOI:10.1007/s11538-013-9889-3
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    ABSTRACT: We exploit model-based Bayesian inference methodologies to analyse lung tumour-derived methylation data from a CpG island in the O6-methylguanine-DNA methyltransferase (MGMT) promoter. Interest is in modelling the changes in methylation patterns in a CpG island in the first exon of the promoter during lung tumour development. We propose four competils of methylation state propagation based on two mechanisms. The first is the location-dependence mechanism in which the probability of a gain or loss of mehylation at a CpG within the promoter depends upon its location in the CpG sequence. The second mechanism is that of neighbour-dependence in which gain or loss of methylation at a CpG depends upon the methylation status of the immediately preceding CpG. Our data comprises of the methylation status at 12 CpGs near the 5' end of the CpG island in two lung tumour samples for both alleles of a nearby polymorphism. We use approximate Bayesian computation, a computationally intensive rejection-sampling algorithm to infer model parameters and compare models without the need to evaluate the likelihood function. We compare the four proposed models using two criteria: the approximate Bayes factors and the distribution of the Euclidean distance between the summary statistics of the observed and simulated datasets. Our model-based analysis demonstrates compelling evidence for both location and neighbour dependence in the process of aberrant DNA methylation of this MGMT promoter CpG island in lung tumours. We find equivocal evidence to support the hypothesis that the methylation patterns of the two alleles evolve independently.
    Journal of Theoretical Biology 07/2013; 336. DOI:10.1016/j.jtbi.2013.07.019
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    ABSTRACT: A key feature of the vertebrate adaptive immune system is acquired immune memory, whereby hosts launch a faster and heightened response when challenged by previously encountered pathogens, preventing full infection. Here, we use a mathematical model to explore the role of ecological and epidemiological processes in shaping selection for costly acquired immune memory. Applying the framework of adaptive dynamics to the classic SIR (Susceptible-Infected-Recovered) epidemiological model, we focus on the conditions that may lead hosts to evolve high levels of immunity. Linking our work to previous theory, we show how investment in immune memory may be greatest at long or intermediate host lifespans depending on whether immunity is long lasting. High initial costs to gain immunity are also found to be essential for a highly effective immune memory. We also find that high disease infectivity and sterility, but intermediate virulence and immune period, increase selection for immunity. Diversity in host populations through evolutionary branching is found to be possible but only for a limited range of parameter space. Our model suggests that specific ecological and epidemiological conditions have to be met for acquired immune memory to evolve.
    Ecology and Evolution 07/2013; 3(7):2223-32. DOI:10.1002/ece3.611
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    ABSTRACT: The details of the mechanism(s) responsible for the observed heating and dynamics of the solar atmosphere still remain a mystery. Magnetohydrodynamic (MHD) waves are thought to play a vital role in this process. Although it has been shown that incompressible waves are ubiquitous in off-limb solar atmospheric observations their energy cannot be readily dissipated. We provide here, for the first time, on-disk observation and identification of concurrent MHD wave modes, both compressible and incompressible, in the solar chromosphere. The observed ubiquity and estimated energy flux associated with the detected MHD waves suggest the chromosphere is a vast reservoir of wave energy with the potential to meet chromospheric and coronal heating requirements. We are also able to propose an upper bound on the flux of the observed wave energy that is able to reach the corona based on observational constraints, which has important implications for the suggested mechanism(s) for quiescent coronal heating.
    Nature Communications 06/2013; 3. DOI:10.1038/ncomms2324
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    ABSTRACT: This paper presents an overview of the developments that have been made towards the design of an inline rheometer that has the capabilities for monitoring in real time the viscous constitutive parameters of non-Newtonian fluids in a pipe flow. This has potential applications for a wide range of fluids, including hydrocolloid solutions and polymer solutions. This is of relevance to many industries, for example the pharmaceutical, lubrication, food and printing industries. The use of mathematical algorithms for inferring rheological parameters from properties of flow field statistics is explored. Particular focus is given to the development of a flow cell rheometer containing a T-junction geometry with the capacity to induce a range of shear rates in the vicinity of the bend, and a distribution of elongational viscosities along the back-wall. Such features create an information-rich flow field that is beneficial for the development of a rheometer with a fast response time that is suitable for commercial purposes.
    Advances in Colloid and Interface Science 06/2013; 206. DOI:10.1016/j.cis.2013.05.006
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    ABSTRACT: While invertebrates lack the machinery necessary for 'acquired immunity', there is increasing empirical evidence that exposure to low levels of disease may 'prime' an invertebrate's immune response, increasing its defence to subsequent exposure. Despite this increasing empirical data, there has been little theoretical attention paid to immune priming. Here, we investigate the evolution of immune priming, focusing on the role of the unique feedbacks generated by a newly developed susceptible-primed-infected epidemiological model. Contrasting our results with previous models on the evolution of acquired immunity, we highlight that there are important implications to the evolution of immunity through priming owing to these different epidemiological feedbacks. In particular, we find that in contrast to acquired immunity, priming is strongly selected for at high as well as intermediate pathogen virulence. We also find that priming may be greatest at either intermediate or high host lifespans depending on the severity of disease. Furthermore, hosts faced with more severe pathogens are more likely to evolve diversity in priming. Finally, we show when the evolution of priming leads to the exclusion of the pathogens or hosts experiencing population cycles. Overall the model acts as a baseline for understanding the evolution of priming in host-pathogen systems.
    Journal of The Royal Society Interface 03/2013; 10(80):20120887. DOI:10.1098/rsif.2012.0887
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