Variations on a theorem of Davenport concerning abundant numbers

Let \sigma(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/\sigma(n) possesses a continuous distribution function. In other words, the limit D(u):= \lim_{x\to\infty} \frac{1}{x}\sum_{n \leq x,~n/\sigma(n) \leq u} 1 exists for all u \in [0,1] and varies continuously with u. We study the behavior of the sums \sum_{n \leq x,~n/\sigma(n) \leq u} f(n) for certain complex-valued multiplicative functions f. Our results cover many of the more frequently encountered functions, including \varphi(n), \tau(n), and \mu(n). They also apply to the representation function for sums of two squares, yielding the following analogue of Davenport's result: For all u \in [0,1], the limit $$\tilde{D}(u):= \lim_{R\to\infty} \frac{1}{\pi R}\#\{(x,y) \in \Z^2: 0<x^2+y^2 \leq R \text{ and } \frac{x^2+y^2}{\sigma(x^2+y^2)} \leq u\}$$ exists, and \tilde{D}(u) is both continuous and strictly increasing on [0,1]

Beyond resistance: social factors in the general public response to pandemic influenza

Background: Influencing the general public response to pandemics is a public health priority. There is a prevailing view, however, that the general public is resistant to communications on pandemic influenza and that behavioural responses to the 2009/10 H1N1 pandemic were not sufficient. Using qualitative methods, this paper investigates how members of the general public respond to pandemic influenza and the hygiene, social isolation and other measures proposed by public health. Going beyond the commonly deployed notion that the general public is resistant to public health communications, this paper examines how health individualism, gender and real world constraints enable and limit individual action. Methods: In-depth interviews (n = 57) and focus groups (ten focus groups; 59 individuals) were conducted with community samples in Melbourne, Sydney and Glasgow. Participants were selected according to maximum variation sampling using purposive criteria, including: 1) pregnancy in 2009/2010; 2) chronic illness; 3) aged 70 years and over; 4) no disclosed health problems. Verbatim transcripts were subjected to inductive, thematic analysis. Results: Respondents did not express resistance to public health communications, but gave insight into how they interpreted and implemented guidance. An individualistic approach to pandemic risk predominated. The uptake of hygiene, social isolation and vaccine strategies was constrained by seeing oneself 'at risk' but not 'a risk' to others. Gender norms shape how members of the general public enact hygiene and social isolation. Other challenges pertained to over-reliance on perceived remoteness from risk, expectation of recovery from infection and practical constraints on the uptake of vaccination. Conclusions: Overall, respondents were engaged with public health advice regarding pandemic influenza, indicating that the idea of public resistance has limited explanatory power. Public communications are endorsed, but challenges persist. Individualistic approaches to pandemic risk inhibit acting for the benefit of others and may deepen divisions in the community according to health status. Public communications on pandemics are mediated by gender norms that may overburden women and limit the action of men. Social research on the public response to pandemics needs to focus on the social structures and real world settings and relationships that shape the action of individuals

Care staff intentions to support adults with an intellectual disability to engage in physical activity: An application of the Theory of Planned Behaviour

Researchers suggest that people with an intellectual disability (ID) undertake less physical activity than the general population and many rely, to some extent, on others to help them to access activities. The Theory of Planned Behaviour (TPB) model was previously found to significantly predict the intention of care staff to facilitate a healthy diet in those they supported. The present study examined whether the TPB was useful in predicting the intentions of 78 Scottish care staff to support people with ID to engage in physical activity. Regression analyses indicated that perceived behavioural control was the most significant predictor of both care staff intention to facilitate physical activity and reported physical activity levels of the people they supported. Attitudes significantly predicted care staff intention to support physical activity, but this intention was not itself significantly predictive of reported activity levels. Increasing carers' sense of control over their ability to support clients' physical activity may be more effective in increasing physical activity than changing their attitudes towards promoting activit

Evaluating Violent Conduct in Sport: A Hierarchy of Vice

The landscape of sport shows conspicuous discursive and material disparities between the responses to openly violent on-field transgressors and the responses to other kinds of transgressor, most notably drug-users. The former gets off significantly lighter in terms of ideological framing and formal punishment. The latter – and drug-users in particular – are typically demonised and heavily punished, whilst the former are regularly lionised, dramatized, celebrated and punished less severely. The preceding disparities cannot be upheld from the standpoint of morality in general or from that of a Broad Internalist sport ethic. Consideration of the consequences, actions, motives and vices involved in the respective categories fails to support them. Nor is support provided by the notion that sports are tests of the physical skills and virtues that the obstacles presented are designed to foster and promote, and behaviour that threatens the opportunity to exercise those excellences or have competitions determined by them should be the subject of critical moral scrutiny. Openly violent on-field transgression does not fare at all well by the yardstick of Broad Internalism. Robust investigation of and ultimate change in the values underpinning the disparities is warranted

Mutagenic Studies of a Unique Cysteine Ligase Enzyme

BshC is the final enzyme in a three step pathway for the synthesis of bacillithiol, a compound that enables resistance to fosfomycin in Gram-positive bacteria. BshC is unique from other enzymes of its kind because of an additional ADP binding site and inactivity when studied in the laboratory. To explore BshC function, several site-directed mutants have been selected within the ADP binding pocket. Fluorescence assays have been utilized on the wild-type BshC and one mutant, Y510Q. We determined that Y510Q does not bind ATP as effectively as wild-type BshC. These fluorescence assays will be utilized on W506L, E384A, and H386A mutants and structural analysis of all the mutants will be initiated. Gaining more understanding of the structure of these mutants and how they bind ATP will give a better understanding of how BshC binds its substrate, which will allow the development of inhibitors to combat fosfomycin resistance