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This thesis presents new results on probabilistic error analysis. By freeing ourselves of the burden of having to account for every possible worst-case scenario, we provide error bounds that are more informative than the long standing worst-case bounds. We also see in practice how these new bounds can guide the development of new algorithms. We begin by studying stochastic rounding, a rounding mode which rounds a computed result to the next larger or smaller floating-point number randomly. We compare stochastic rounding with round-to-nearest, finding some similarities but also a large number of properties that hold for round-to-nearest but fail to hold for stochastic rounding. Importantly, we show that the rounding errors produced by stochastic rounding are mean zero, mean independent random variables. Using this fact, we build on earlier probabilistic error analysis to show that for a wide range of inner product based linear algebra computations, stochastic rounding provides backward error bounds where we can take the square root of the dimensional constants in the worst-case bounds. This has been a well known rule of thumb for some time, but we show it to be a rule for stochastic rounding. Expanding on this work for inner-product based algorithms, we investigate orthogonal transformations. Using the same model of rounding errors as that which stochastic rounding satisfies, we show that for Householder QR factorization, and other related algorithms, we see the same square rooting of the dimensional constant in the backward error bound. The analysis makes use of a matrix concentration inequality, where previous probabilistic error analyses employed scalar concentration inequalities. We also derive a new backward error formula for QR factorization, used in our numerical experiments which validate our bounds. Finally, we consider the randomized SVD, a well known method for computing low rank matrix approximations. We perform a complete rounding error analysis of the fixed-rank problem, the most straightforward version of the randomized SVD where we have a priori specified a target rank. This is complementary to the existing literature providing error bounds on this procedure in exact arithmetic. While our initial analysis is worst-case, we use our previous probabilistic results to refine these error bounds. Using these refined bounds, we propose a mixed-precision version of the algorithm that offers potential speedups by gradually reducing the precision during the execution of the algorithm.

Enzymes are biological catalysts which are crucial for life and are fundamental in many important processes in organisms. The temperature dependence of enzyme-catalysed reactions is important for homeostasis, and until recently the drop-off in enzyme-catalysed reaction rates at higher temperatures has been attributed to protein denaturation. It has been shown experimentally that rate drop-off occurs in the absence of denaturation. Macromolecular rate theory (MMRT) does not assume that enthalpy and entropy changes are independent of temperature, introducing an activation heat capacity term, ∆C_p^‡ to understand the curvature of reaction rates, even in the absence of protein denaturation. Molecular dynamics (MD) simulations can provide insight into dynamical behaviours of proteins and have previously been used for the calculation of thermodynamic properties including heat capacities. In this thesis, MD simulations are used to calculate ∆C_p^‡ for two different enzyme-substrate complexes. The results show that a negative ∆C_p^‡ value is generally obtained from MD calculations for these enzymes, which agrees with the experimentally observed curvature for the enzyme-catalysed reaction rates. Principal component analysis, Kullback-Leibler divergence and hidden Markov state modelling techniques are all utilised to analyse the origin of the dynamical changes in the protein chain leading to a negative ∆C_p^‡ term. The work in this thesis shows that MD simulations can be used to provide insight into MMRT and the dynamical properties of enzymes, and points to potential future work in understanding how the optimum temperature for enzyme-catalysed reactions is influenced by the dynamics of the protein chain of an enzyme.

This thesis critically explores the life of Brother Walfrid (Andrew Kerins), one of the most significant Irish immigrants to Britain, in relation to his contribution to Catholic education and charity, as well as organised sport in Scotland in the late nineteenth century. Despite knowledge around him as a prime founder of Celtic FC, Walfrid's story remains largely untold. This research investigates the 'real' Brother Walfrid - the man separated from myth - along with his legacy among the multi-generational Irish Catholic community in Scotland, and elsewhere. This study analyses the origins of Andrew Kerins beginning with his Sligo birth and departure for Glasgow. Drawing on surviving historical and archived documents, the effects of An Gorta Mor (the Great Hunger in Irish) on Kerins and his family are reflected upon. Further, new information drawn from contemporaneous publications and interviews provided the source material on this epochal figure for the Irish Catholic diaspora. The socio-economic circumstances that provided the conditions for the emergence of Celtic FC - a unique representation of the Irish diaspora in world sport - are explicated through this biography of Brother Walfrid's lived experience. Critically, it also seeks to understand and explore Walfrid's role, motivations and achievements as a Marist Brother: especially with respect to his importance to Catholic religious, educational, charitable and cultural identities in Scotland. Walfrid's integral role in the creation of Celtic FC is critically reappraised along with his faith and charity work amongst the poor and marginalised in Glasgow. By producing a full account of the life of Brother Walfrid, what emerges is a more substantive insight into a figure of totemic historical significance for the Irish Catholic diaspora, in Scotland and beyond. Through the combination of faith, charity and football, Walfrid's life illustrates a historic contribution to Irish Catholicism in nineteenth century Britain.

'Autism Plus' is a concept which relates to individuals with Autism Spectrum Condition (ASC) and co-occurring conditions (Gillberg & Fernell, 2014). Three studies reported in this thesis examined evidence for a key premise of Autism Plus, namely that individuals with Autism Plus experience poorer social and independent living outcomes and greater support needs compared to those with 'Autism Only' (i.e. those with ASC but no co-occurring diagnoses). Study One was a secondary analysis of survey data from secondary data analysis of 404 adults with autistic disorder (n = 82), Asperger's/High-Functioning Autism (n = 236) and other ASCs (n = 86). A series of chi-square analyses comparing employment, relationship, residential and independent travel outcomes and service pattern use between those with Autism Plus and Autism Only. No group differences were found in employment or travel outcomes, though contrary to the premise of interest here, those with Autism Plus were found to be more likely to live independently and to be in long-term relationships. However, in line with the premise, those with Autism Plus were more likely to have used support services in the six months prior to data collection. In Study Two, logistic regression analyses focusing on the same sample explored whether Autism Plus may better predict poorer adult outcomes as part of a larger model accounting for age and autistic symptom severity. Findings indicated that Autism Plus was not a useful predictor of employment or independent travel outcomes as part of this model, though again indicated that those with Autism Plus were more likely to live independently and to be in long-term relationships. Other key findings indicated that outcomes were better amongst older individuals and those with milder autistic symptoms. Study Three aimed to gain greater insight into the ways in which co-occurring diagnoses could influence the lives of those with ASC through interviews with adults with Autism Plus. Analysis of accounts of lived experience using Interpretative Phenomenological Analysis (IPA) supported the view that the participants' co-occurring symptoms added considerable disruption and difficulty to their lives, and at times could have a greater impact than their ASC symptoms. There was also evidence that within the context of Autism Plus, the severity of a co-occurring condition may be as important as its presence. Overall, the evidence reported here indicates that current conceptualisation of Autism Plus is limited in the extent to which it can predict those with ASC most likely to experience poor outcomes, though may be helpful in establishing those most likely to require support. Given the findings from Study Two and Three, it is recommended that a modified version of Autism Plus, which considers Autism Plus as an important component of a larger model and acknowledges co-occurring symptom severity, should be explored further.