Covering collections and a challenge problem of Serre

We answer a challenge of Serre by showing that every rational point on the projective curve X$^4$ + Y$^4$ = 17 Z$^4$ is of the form ($\pm$1, $\pm$2, 1) or ($\pm$2, $\pm$1, 1). Our approach builds on recent ideas from both Nils Bruin and the authors on the application of covering collections and Chabauty arguments to curves of high rank. This is the only value of c$\le$81 for which the Fermat quartic X$^4$ + Y$^4$ = c Z$^4$ cannot be solved trivially, either by local considerations or maps to elliptic curves of rank 0, and it seems likely that our approach should give a method of attack for other nontrivial values of c

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form $Y^2 = f_3 X^6 + f_2 X^4 + f_1 X^2 + f_0$, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field Q(a). If each of these elliptic curves has rank less than the degree of Q(a) : Q, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over Q(a) on the elliptic curves, for which x is in Q. This in turn allows us to find all Q-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over Q), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over Q

Accurate real-time evolution of electron densities and ground-state properties from generalized Kohn-Sham theory

The exact static and time-dependent Kohn-Sham (KS) exchange-correlation potential is extremely challenging to approximate as it is a local multiplicative potential that depends on the electron density everywhere in the system. The KS approach can be generalized by allowing part of the potential to be spatially nonlocal. We take this nonlocal part to be that of unrestricted Hartree-Fock theory. The additional local correlation potential in principle ensures that the single-particle density exactly equals the many-body density. In our case, the local correlation potential is predominantly nearsighted in its dependence on the density and hence an (adiabatic) local-density approximation to this potential yields accurate ground-state properties and real-time densities for one-dimensional test systems

Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular jacobians of genus 2 curves

This paper provides empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. The second of these conjectures relates six quantities associated to a Jacobian over the rational numbers. One of these six quantities is the size of the Shafarevich-Tate group. Unable to compute that, we computed the five other quantities and solved for the last one. In all 32 cases, the result is very close to an integer that is a power of 2. In addition, this power of 2 agrees with the size of the 2-torsion of the Shafarevich-Tate group, which we could compute

Political and Media Discourses about Integrating Refugees in the UK

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.This article addresses political and media discourses about integrating refugees in the UK in the context of the “refugee crisis”. A discursive psychological approach is presented as the best way to understand what talk about the concept is used to accomplish in these debates. A large corpus of political discussions (13 hours of debate featuring 146 politicians) and 960 newspaper articles from the UK were discourse analysed. The analysis identified five dilemmas about integration: Integration is positive and necessary, but challenging; Host communities are presented as welcoming, but there are limits to their capacity; Refugees are responsible for integration, but host communities need to provide support; Good refugees integrate, bad ones don't; Refugees are vulnerable and are skilled. All are used to warrant the inclusion or exclusion of refugees. The responsibility of western nations to support refugees is therefore contingent on the refugees behaving in specific ways

Jekyll and Hyde: men's constructions of feminism and feminists

Research and commentary on men's responses to feminism has demonstrated the range of ways in which men have mobilised both against and for feminist principles. This paper argues that further analyses of men's responses require a sophisticated theory of discourse acknowledging the fragmented and contradictory nature of representation. A corpus of men's talk on feminism and feminists was studied to identify the pervasive patterns in men's accounting and regularities in rhetorical organisation. Material from two samples of men was included: a sample of white middle-class 17-18 year old school students and a sample of 60 interviews with a more diverse sample of older men aged 20 to 64. Two interpretative repertoires of feminism and feminists were identified. These set up a 'Jekyll and Hyde' binary and positioned feminism along with feminists very differently as reasonable versus extreme and monstrous. Both repertoires tended to be deployed together and the paper explores the ideological and interactional consequences of typical deployments along with the identity work accomplished by the men as they positioned themselves in relation to these

A qualitative study of older adults' perspectives on initiating exercise and mindfulness practice.

BackgroundMindfulness practice and exercise are ways by which older adults can improve and maintain their physical, emotional and cognitive health.MethodsThis single-site qualitative study gathered insights of older adults' perceptions about initiating and maintaining mindfulness and exercise practices. We carried out focus groups with 41 adults aged 65-85 who had recently initiated Mindfulness Based Stress Reduction (MBSR), structured exercise, or their combination as part of participation in a clinical trial. We used a semi-structured interview to ask them open-ended questions regarding the benefits, barriers and facilitators of participating in mindfulness and/or exercise interventions. The interview also included questions regarding translation of these practices into community settings as well as the long-term maintenance potential of these practices.ResultsOlder adults indicated that the mindfulness training increased their awareness and self-reflection and fostered a more self-accepting attitude. Furthermore, they improved their self-care habits and reported having better familial and social relationships. The main barrier for both the exercise and Mindfulness group was time management. The social benefits and sense of community were some of the primary motivators for older adults in the exercise and/or MBSR interventions. However, the research on how to motivate older adults to initiate healthy behavioral changes also needs to be answered. The benefits of exercise and MBSR are a motivation in and of themselves, as indicated by some of the participants.ConclusionsThis study indicates that mindfulness training and exercise can serve as tools to cultivate important health lifestyle qualities among older adults, who are in the midst of mental, social, emotional and physical change. If it were not for the purpose of the research or the incentives provided by the research team, these older adults may have never started the healthy behavioral changes. From the responses, this may indicate that older adults may need more incentives to begin and maintain behavioral changes other than for their own health benefit

Management of acute dental pain: a practical approach for primary health care providers

Reproduced with permission from Australian Prescriber The document attached has been archived with permission from the publisher/copyright holderA detailed history and examination will identify the cause of dentally-related pain in most emergency situations. Sharp, shooting pain can be caused by inflammation in the pulp or exposure of the dentine. Dull throbbing pain has several causes including ulcerative gingivitis, dental caries and food impaction. Simple treatment will usually alleviate the symptoms until patients can be seen by a dentist. Prescription of antibiotics is usually not indicated.John Wetherell, Lindsay Richards, Paul Sambrook, Grant Townsen