A Completeness Study on Certain 2×22\times2 Lax Pairs Including Zero Terms

We expand the completeness study instigated in [J. Math. Phys. 50 (2009), 103516, 29 pages] which found all $2\times2$ Lax pairs with non-zero, separable terms in each entry of each Lax matrix, along with the most general nonlinear systems that can be associated with them. Here we allow some of the terms within the Lax matrices to be zero. We cover all possible Lax pairs of this type and find a new third order equation that can be reduced to special cases of the non-autonomous lattice KdV and lattice modified KdV equations among others

Discrete Lax pairs, reductions and hierarchies

The term `Lax pair' refers to linear systems (of various types) that are related to nonlinear equations through a compatibility condition. If a nonlinear equation possesses a Lax pair, then the Lax pair may be used to gather information about the behaviour of the solutions to the nonlinear equation. Conserved quantities, asymptotics and even explicit solutions to the nonlinear equation, amongst other information, can be calculated using a Lax pair. Importantly, the existence of a Lax pair is a signature of integrability of the associated nonlinear equation. While Lax pairs were originally devised in the context of continuous equations, Lax pairs for discrete integrable systems have risen to prominence over the last three decades or so and this thesis focuses entirely on discrete equations. Famous continuous systems such as the Korteweg de Vries equation and the Painleve equations all have integrable discrete analogues, which retrieve the original systems in the continuous limit. Links between the different types of integrable systems are well known, such as reductions from partial difference equations to ordinary difference equations. Infinite hierarchies of integrable equations can be constructed where each equation is related to adjacent members of the hierarchy and the order of the equations can be increased arbitrarily. After a literature review, the original material in this thesis is instigated by a completeness study that finds all possible Lax pairs of a certain type, including one for the lattice modified Korteweg de Vries equation. The lattice modified Korteweg de Vries equation is subsequently reduced to several q-discrete Painleve equations, and the reductions are used to form Lax pairs for those equations. The series of reductions suggests the presence of a hierarchy of equations, where each equation is obtained by applying a recursion relation to an earlier member of the hierarchy, this is confirmed using expansions within the Lax pairs for the q-Painleve equations. Lastly, some explorations are included into fake Lax pairs, as well as sets of equivalent nonlinear equations with similar Lax pairs

Defining childhood severe falciparum malaria for intervention studies.

Background Clinical trials of interventions designed to prevent severe falciparum malaria in children require a clear endpoint. The internationally accepted definition of severe malaria is sensitive, and appropriate for clinical purposes. However, this definition includes individuals with severe nonmalarial disease and coincident parasitaemia, so may lack specificity in vaccine trials. Although there is no “gold standard” individual test for severe malaria, malaria-attributable fractions (MAFs) can be estimated among groups of children using a logistic model, which we use to test the suitability of various case definitions as trial endpoints. Methods and Findings A total of 4,583 blood samples were taken from well children in cross-sectional surveys and from 1,361 children admitted to a Kenyan District hospital with severe disease. Among children under 2 y old with severe disease and over 2,500 parasites per microliter of blood, the MAFs were above 85% in moderate- and low-transmission areas, but only 61% in a high-transmission area. HIV and malnutrition were not associated with reduced MAFs, but gastroenteritis with severe dehydration (defined by reduced skin turgor), lower respiratory tract infection (clinician's final diagnosis), meningitis (on cerebrospinal fluid [CSF] examination), and bacteraemia were associated with reduced MAFs. The overall MAF was 85% (95% confidence interval [CI] 83.8%–86.1%) without excluding these conditions, 89% (95% CI 88.4%–90.2%) after exclusions, and 95% (95% CI 94.0%–95.5%) when a threshold of 2,500 parasites/μl was also applied. Applying a threshold and exclusion criteria reduced sensitivity to 80% (95% CI 77%–83%). Conclusions The specificity of a case definition for severe malaria is improved by applying a parasite density threshold and by excluding children with meningitis, lower respiratory tract infection (clinician's diagnosis), bacteraemia, and gastroenteritis with severe dehydration, but not by excluding children with HIV or malnutrition

Losing the discursive battle but winning the ideological war: who holds Thatcherite values now?

In what ways, if at all, do past ideologies shape the values of subsequent generations of citizens? Are public attitudes in one period shaped by the discourses and constructions of an earlier generation of political leaders? Using Thatcherism – one variant of the political New Right of the 1980s – as the object of our enquiries, this paper explores the extent to which an attitudinal legacy is detectable amongst the citizens of the UK some 40 years after Margaret Thatcher first became Prime Minister. Our paper, drawing on survey data collected in early 2019 (n = 5,781), finds that younger generations express and seemingly embrace key tenets of her and her governments’ philosophies. Yet at the same time, they are keen to describe her government’s policies as having ‘gone too far’. Our contribution throws further light on the complex and often covert character of attitudinal legacies. One reading of the data suggests that younger generations do not attribute the broadly Thatcherite values that they hold to Thatcher or Thatcherism since they were socialised politically after such values had become normalised.We would like to thank the ESRC for their generous funding (as award number ES/P002862/1)

Lattice modified KdV hierarchy from a Lax pair expansion

We produce a hierarchiy of integrable equations by systematically adding terms to the Lax pair for the lattice modified KdV equation. The equations in the hierarchy are related to one aonother by recursion relations. These recursion relations are solved explicitly so that every equation in the hierarchy along with its Lax pair is known

How data visualisation using historical medical journals can contribute to current debates around antibiotic use and antimicrobial resistance in primary care

BackgroundThe early years of antibiotic use in primary care (c1950-1969) has received little attention. Medical journals provide a rich source for studying historic healthcare practitioners’ views and interests, with the potential to inform contemporary debate around issues of overuse and antimicrobial resistance. AimsPilot study to test the application of digital methods to interrogate historical medical journal data in relation to antibiotic use.Methods / ApproachMeta-data and scanned articles were extracted from the online British Journal of General Practice (BJGP) archive from inception (1953) to 1969. Searchable text was generated using an application called ABBYY optical character recognition, and Python used to generate data visualisations exploring (1) how BJGP changed during the period, (2) mentions of terms ‘antibiotic(s)’, ‘penicillin’, ‘resistance/resistant’ and mapping when and where they occurred.Results / EvaluationFrom 1953-1969, BJGP expanded in terms of number of annual issues (4 to 17) and annual pages (&lt;25 to &gt;1100). Heatmap visualisations were used to facilitate understanding of the frequency with which use of the term ‘antibiotic(s)’ occurred. By 1969 an article mentioning ‘antibiotic(s)’ was published monthly. Bigram searches found ‘treatment’ and ‘therapy’ to be the two most common terms that appeared with ‘antibiotic(s)’. The fourth and seventh most common terms were ‘resistant’ (first appearing in 1955) and ‘resistance’ (1962).ConclusionsThis pilot work shows that primary care publications increased considerably between 1953-1969. Articles on antibiotics featured frequently in relation to therapeutic intervention, and concerns around resistance occurred at an early stage. This approach provides new insights into how attitudes and behaviours around antibiotic use by primary care have evolved over time. It may also have the potential to inform study of the future use of antibiotics in primary care. <br/