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    ABSTRACT: We introduce an $\mathfrak{F}$-valued generalization of the Virasoro algebra, called the Frobenius-Virasoro algebra $\mathfrak{vir_F}$, where $\mathfrak{F}$ is a Frobenius algebra over $\mathbb{R}$. We also study Euler equations on the regular dual of $\mathfrak{vir_F}$, including the $\mathfrak{F}$-$\mathrm{KdV}$ equation and the $\mathfrak{F}$-$\mathrm{CH}$ equation and the $\mathfrak{F}$-$\mathrm{HS}$ equation, and discuss their Hamiltonian properties.
    Journal of Geometry and Physics 02/2014; 86. DOI:10.1016/j.geomphys.2014.08.002
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    ABSTRACT: A key paper in modelling patient recruitment in multi-centre clinical trials is that of Anisimov and Fedorov. They assume that the distribution of the number of patients in a given centre in a completed trial follows a Poisson distribution. In a second stage, the unknown parameter is assumed to come from a Gamma distribution. As is well known, the overall Gamma-Poisson mixture is a negative binomial. For forecasting time to completion, however, it is not the frequency domain that is important, but the time domain and that of Anisimov and Fedorov have also illustrated clearly the links between the two and the way in which a negative binomial in one corresponds to a type VI Pearson distribution in the other. They have also shown how one may use this to forecast time to completion in a trial in progress. However, it is not just necessary to forecast time to completion for trials in progress but also for trials that have yet to start. This suggests that what would be useful would be to add a higher level of the hierarchy: over all trials. We present one possible approach to doing this using an orthogonal parameterization of the Gamma distribution with parameters on the real line. The two parameters are modelled separately. This is illustrated using data from 18 trials. We make suggestions as to how this method could be applied in practice. Copyright © 2013 John Wiley & Sons, Ltd.
    Statistics in Medicine 12/2013; 32(30). DOI:10.1002/sim.5979
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    ABSTRACT: There is growing interest in links between poor health and socio-environmental inequalities (e.g. inferior housing, crime and industrial emissions) under the environmental justice agenda. The current project assessed associations between soil metal content, air pollution (NO2/PM10) and deprivation and health (respiratory case incidence) across Glasgow. This is the first time that both chemical land quality and air pollution have been assessed citywide in the context of deprivation and health for a major UK conurbation. Based on the dataset 'averages' for intermediate geography areas, generalised linear modelling of respiratory cases showed significant associations with overall soil metal concentration (p = 0.0367) and with deprivation (p < 0.0448). Of the individual soil metals, only nickel showed a significant relationship with respiratory cases (p = 0.0056). Whilst these associations could simply represent concordant lower soil metal concentrations and fewer respiratory cases in the rural versus the urban environment, they are interesting given (1) possible contributions from soil to air particulate loading and (2) known associations between airborne metals like nickel and health. This study also demonstrated a statistically significant correlation (-0.213; p < 0.05) between soil metal concentration and deprivation across Glasgow. This highlights the fact that despite numerous regeneration programmes, the legacy of environmental pollution remains in post-industrial areas of Glasgow many decades after heavy industry has declined. Further epidemiological investigations would be required to determine whether there are any causal links between soil quality and population health/well-being. However, the results of this study suggest that poor soil quality warrants greater consideration in future health and socio-environmental inequality assessments.
    Environmental Geochemistry and Health 11/2013; 36(2). DOI:10.1007/s10653-013-9565-4
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    ABSTRACT: Genetic diversity in multigene families is shaped by multiple processes, including gene conversion and point mutation. Because multi-gene families are involved in crucial traits of organisms, quantifying the rates of their genetic diversification is important. With increasing genomic data, there is an increasing need for quantitative approaches that integrate the molecular evolution of gene families with their higher-scale function. In this study, we integrate a stochastic simulation framework with population genetics theory, namely the diffusion approximation, to investigate the dynamics of genetic diversification in a gene family. Duplicated genes can diverge and encode new functions as a result of point mutation, and become more similar through gene conversion. To model the evolution of pairwise identity in a multigene family, we first consider all conversion and mutation events in a discrete manner, keeping track of their details and times of occurrence, secondly, we consider only the infinitesimal effect of these processes on pairwise identity accounting for random sampling of genes and positions. The purely stochastic approach is closer to biological reality and is based on many explicit parameters, such as conversion tract length and family size, but is more challenging analytically. The population genetics approach is an approximation accounting implicitly for point mutation and gene conversion, only in terms of per-site average probabilities. Comparison of these two approaches across a range of parameter combinations reveals that they are not entirely equivalent, but that for certain relevant regimes they do match. As an application of this modelling framework, we consider the distribution of nucleotide identity among VSG genes of African trypanosomes, representing the most prominent example of a multi-gene family mediating parasite antigenic variation and within-host immune evasion.
    Journal of Theoretical Biology 10/2013; 341. DOI:10.1016/j.jtbi.2013.10.001
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    ABSTRACT: We use constructions in monoid and group theory to exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms and the category of partially ordered groups and group homomorphisms such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups.We introduce the new notion of a lazy homomorphism for a function f between partially ordered monoids such that f(m ○ m′) ≤ f(m) ○ f(m′).Every monoid can be endowed with the discrete partial ordering (m ≤ m′ if and only if m=m′), so our constructions provide a way of embedding monoids into groups. A simple counterexample (the two-element monoid with a non-trivial idempotent) and some calculations show that one can never hope for such an embedding to be a monoid homomorphism, so the price paid for injecting a monoid into a group is that we must weaken the notion of a homomorphism to this new notion of a lazy homomorphism.The computational significance of this is that a monoid is an abstract model of computation – or at least of ‘operations’ – and, similarly, a group models reversible computations/operations. With this reading, the adjunction with its injective unit gives a systematic high-level way of faithfully translating an irreversible system into a ‘lazy’ reversible one.Informally, but perhaps informatively, we can describe this work as follows: we give an abstract analysis of how we can sensibly add ‘undo’ (in the sense of ‘control-Z’).
    Mathematical Structures in Computer Science 10/2013; 23(05). DOI:10.1017/S0960129512000849
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    ABSTRACT: In a previous article, we tackled the question "How are we related?" for the simple case of one explanatory variable or covariate. Now we can move onto a natural extension, with still only one response variable, but more than one explanatory variable.
    Journal of Small Animal Practice 10/2013; 54(10):541-546. DOI:10.1111/jsap.12131
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    ABSTRACT: SUMMARY The associations with weather and bathing water quality on infectious intestinal disease (IID) were investigated using data from two Scottish NHS Board areas. Monthly counts of viral and non-viral gastrointestinal infections were modelled as a smooth function of temperature, relative humidity and average monthly counts of faecal indicator organisms, respectively, adjusting for season and long-term trend effects. Strong seasonal patterns were observed for each group of pathogens. Peak viral gastrointestinal infection was in May while that of non-viral gastrointestinal infections was in July. A statistically significant negative association existed between weather (temperature and humidity) and viral infection. Average levels of non-viral gastrointestinal infections increased as temperature and relative humidity increased. Increasing levels of faecal indicator organisms in bathing waters were also associated with an increase in the average number of viral and non-viral gastrointestinal infections at the ecological level. Future climate change and prolonged precipitation events may result in increasing levels of faecal indicator organisms in bathing waters leading to likely increases in IIDs.
    Epidemiology and Infection 09/2013; 142(06):1-11. DOI:10.1017/S0950268813002148
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    ABSTRACT: Concentrations of nutrient nitrogen (N) and phosphorus (P) are elevated in rivers across large areas of Europe (European Nitrogen Assessment (ENA), Sutton et al., 2011). Environmental policies have been implemented over the past 20years with the aim of reducing nitrogen inputs to surface waters. However, environmental and ecological status is still below set targets (ENA, Sutton et al., 2011). Identification of patterns in long-term change for nutrient trends in hydrological catchments in England & Wales is required to assess impacts of nutrient management policy and provide better evidence for future policy. Such information could provide essential evidence for supporting policy by combining information from the wider catchment, rather than relying on the analysis of data from individual sites. Surface water quality is subject to considerable spatial and short-period temporal variability, reflecting variability in loading and dilution. This makes it difficult to determine temporal trends at individual monitoring sites with relatively sparse sampling. Here we apply spatiotemporal statistical additive models for both nitrogen and phosphorus in river networks across England & Wales to investigate the overall pattern of nutrient concentrations in these river surface waters over the past 20-40years. Concentrations of Orthophosphate (OP) have generally decreased over time for many of the Large Hydrological Areas with a seasonal pattern highlighting one peak in the summer months. Over the past ten years, Total Oxidised Nitrogen (Nitrate+Nitrite, TON) concentrations have generally been slowly decreasing or fairly constant. However, prior to 2000, concentrations were generally on an upward trend. The seasonal pattern highlights one trough in the summer months. The highest levels for OP and TON broadly occur in the same general areas across England & Wales. On average, over time, the lowest values are evident in the north-west and south-west (particularly for OP) and highest values are evident in the Midlands, Anglian and Southern regions.
    Science of The Total Environment 08/2013; 466-467C:914-923. DOI:10.1016/j.scitotenv.2013.07.113
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    ABSTRACT: More than 20 human genetic diseases are associated with inheriting an unstable expanded DNA simple sequence tandem repeat, for example, CTG (cytosine-thymine-guanine) repeats in myotonic dystrophy type 1 (DM1) and CAG (cytosine-adenine-guanine) repeats in Huntington disease (HD). These sequences mutate by changing the number of repeats not just between generations, but also during the lifetime of affected individuals. Levels of somatic instability contribute to disease onset and progression but as changes are tissue-specific, age- and repeat length-dependent, interpretation of the level of somatic instability in an individual is confounded by these considerations. Mathematical models, fitted to CTG repeat length distributions derived from blood DNA, from a large cohort of DM1-affected or at risk individuals, have recently been used to quantify inherited repeat lengths and mutation rates. Taking into account age, the estimated mutation rates are lower than predicted among individuals with small alleles (inherited repeat lengths less than 100 CTGs), suggesting that these rates may be suppressed at the lower end of the disease-causing range. In this study, we propose that a length-specific effect operates within this range and tested this hypothesis using a model comparison approach. To calibrate the extended model, we used data derived from blood DNA from DM1 individuals and, for the first time, buccal DNA from HD individuals. In a novel application of this extended model, we identified individuals whose effective repeat length, with regards to somatic instability, is less than their actual repeat length. A plausible explanation for this distinction is that the expanded repeat tract is compromised by interruptions or other unusual features. We quantified effective length for a large cohort of DM1 individuals and showed that effective length better predicts age of onset than inherited repeat length, thus improving the genotype-phenotype correlation. Under the extended model, we removed some of the bias in mutation rates making them less length-dependent. Consequently, rates adjusted in this way will be better suited as quantitative traits to investigate cis- or trans-acting modifiers of somatic mosaicism, disease onset and progression.
    Journal of The Royal Society Interface 08/2013; 10(88):20130605. DOI:10.1098/rsif.2013.0605
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    ABSTRACT: In a previous article, we asked the simple question "Are we related?" and used scatterplots and correlation coefficients to provide an answer. In this article, we will take this question and reword it to "How are we related?" and will demonstrate the statistical techniques required to reach a conclusion.
    Journal of Small Animal Practice 05/2013; 54(6). DOI:10.1111/jsap.12068
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