42 research outputs found

    String rewriting for Double Coset Systems

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    In this paper we show how string rewriting methods can be applied to give a new method of computing double cosets. Previous methods for double cosets were enumerative and thus restricted to finite examples. Our rewriting methods do not suffer this restriction and we present some examples of infinite double coset systems which can now easily be solved using our approach. Even when both enumerative and rewriting techniques are present, our rewriting methods will be competitive because they i) do not require the preliminary calculation of cosets; and ii) as with single coset problems, there are many examples for which rewriting is more effective than enumeration. Automata provide the means for identifying expressions for normal forms in infinite situations and we show how they may be constructed in this setting. Further, related results on logged string rewriting for monoid presentations are exploited to show how witnesses for the computations can be provided and how information about the subgroups and the relations between them can be extracted. Finally, we discuss how the double coset problem is a special case of the problem of computing induced actions of categories which demonstrates that our rewriting methods are applicable to a much wider class of problems than just the double coset problem.Comment: accepted for publication by the Journal of Symbolic Computatio

    The Early Royal Society and Visual Culture

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    Recent studies have fruitfully examined the intersection between early modern science and visual culture by elucidating the functions of images in shaping and disseminating scientific knowledge. Given its rich archival sources, it is possible to extend this line of research in the case of the Royal Society to an examination of attitudes towards images as artefacts –manufactured objects worth commissioning, collecting and studying. Drawing on existing scholarship and material from the Royal Society Archives, I discuss Fellows’ interests in prints, drawings, varnishes, colorants, images made out of unusual materials, and methods of identifying the painter from a painting. Knowledge of production processes of images was important to members of the Royal Society, not only as connoisseurs and collectors, but also as those interested in a Baconian mastery of material processes, including a “history of trades”. Their antiquarian interests led to discussion of painters’ styles, and they gradually developed a visual memorial to an institution through portraits and other visual records.AH/M001938/1 (AHRC

    Polygenic Risk Scores for Prediction of Breast Cancer and Breast Cancer Subtypes

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    Stratification of women according to their risk of breast cancer based on polygenic risk scores (PRSs) could improve screening and prevention strategies. Our aim was to develop PRSs, optimized for prediction of estrogen receptor (ER)-specific disease, from the largest available genome-wide association dataset and to empirically validate the PRSs in prospective studies. The development dataset comprised 94,075 case subjects and 75,017 control subjects of European ancestry from 69 studies, divided into training and validation sets. Samples were genotyped using genome-wide arrays, and single-nucleotide polymorphisms (SNPs) were selected by stepwise regression or lasso penalized regression. The best performing PRSs were validated in an independent test set comprising 11,428 case subjects and 18,323 control subjects from 10 prospective studies and 190,040 women from UK Biobank (3,215 incident breast cancers). For the best PRSs (313 SNPs), the odds ratio for overall disease per 1 standard deviation in ten prospective studies was 1.61 (95%CI: 1.57-1.65) with area under receiver-operator curve (AUC) = 0.630 (95%CI: 0.628-0.651). The lifetime risk of overall breast cancer in the top centile of the PRSs was 32.6%. Compared with women in the middle quintile, those in the highest 1% of risk had 4.37- and 2.78-fold risks, and those in the lowest 1% of risk had 0.16- and 0.27-fold risks, of developing ER-positive and ER-negative disease, respectively. Goodness-of-fit tests indicated that this PRS was well calibrated and predicts disease risk accurately in the tails of the distribution. This PRS is a powerful and reliable predictor of breast cancer risk that may improve breast cancer prevention programs.NovartisEli Lilly and CompanyAstraZenecaAbbViePfizer UKCelgeneEisaiGenentechMerck Sharp and DohmeRocheCancer Research UKGovernment of CanadaArray BioPharmaGenome CanadaNational Institutes of HealthEuropean CommissionMinistère de l'Économie, de l’Innovation et des Exportations du QuébecSeventh Framework ProgrammeCanadian Institutes of Health Researc