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

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Reproduced with permission from Australian Prescriber The document attached has been archived with permission from the publisher/copyright holderA detailed history and examination will identify the cause of dentally-related pain in most emergency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

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

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

Narrative skills in adolescents with a history of SLI in relation to non-verbal IQ scores

There is a debate about whether the language of children with primary language disorders and normal cognitive levels is qualitatively different from those with language impairments who have low or borderline non-verbal IQ (NVIQ). As children reach adolescence, this distinction may be even harder to ascertain, especially in naturalistic settings. Narrative may provide a useful, ecologically valid way in which to assess the language ability of adolescents with specific language impairment (SLI) who have intact or lowered NVIQ and to determine whether there is any discernable difference in every day language. Nineteen adolescents with a history of SLI completed two narrative tasks: a story telling condition and a conversational condition. Just under half the group (n = 8) had non-verbal IQs of 85. The remaining 11 had NVIQs in the normal range or above. Four areas of narrative (productivity, syntax, cohesion and performance) were assessed. There were no differences between the groups on standardized tests of language. However, the group with low NVIQ were poorer on most aspects of narrative, suggesting that cognitive level is important, even when language is the primary disorder. The groups showed similar patterns of differences between story telling and conversational narrative. It was concluded that adolescents with a history of SLI and poor cognitive levels have poorer narrative skills than those with normal range NVIQ even though these may not be detected by standardized assessment. Their difficulties present as qualitatively similar to those with normal range NVIQ and narratives appear impoverished rather than inaccurate

'When I click "ok" I become Sassy – I become a girl.' Young people and gender identity: Subverting the ‘body’ in massively multi-player online role-playing games

This article is available open access through the publisher’s website through the link below. Copyright @ 2012 Taylor & Francis.This article explores young people's practices in the virtual spaces of online gaming communities. Based on a five-year ethnographic study of virtual worlds, it considers how young people construct and maintain identities within virtual social systems. In particular, the article discusses digital gender practices and considers the potential that these games offer for their young users to engage in alternate gender identities. We argue that these digital spaces offer spaces for the imagination and can enhance agency and, potentially, resistance. However, digital identity is simultaneously no ‘liberated space’ and it incorporates norms and practices that often mirror those of the material world. We argue that this ‘porosity’ is an important tool through which young people come to understand gender identity

A new measure of fear of falling: psychometric properties of the fear of falling questionnaire revised (FFQ-R)

BACKGROUND: Although fear of falling is prevalent among older adults recovering from hip fracture, current instruments are inadequate due to focus on specific situations and measurement of self-efficacy rather than fear. METHODS: The authors revised and tested a form of the Fear of Falling Questionnaire with three groups of older adults: 405 recovering from hip fracture, 89 healthy community-dwelling, and 42 with severe fear of falling. Test-retest reliability was evaluated in a subsample of 16 hip fracture patients. Internal consistency was compared across all groups Construct validity was established through factor analysis, convergent validity with a measure of fall-related self-efficacy, and discriminant validity with measures of depression and affect. RESULTS: A revised two-factor, 6-item scale appears to have adequate psychometric properties. Scores were lower among the healthy comparison group relative to the hip fracture and fear of falling groups. Cronbach’s alphas ranged from .72–.83, with test-retest reliability of .82. Correlations with a measure of fall-related self-efficacy were moderate for the hip fracture group (.42) and high with the healthy comparison (.68) and fear of falling (.70) groups. Correlations with depression, negative, and positive affect were low to moderate. CONCLUSIONS: The Fear of Falling Questionnaire - Revised shows promise as a self-report measure of fear of falling, and is one of the first to be tested in older adults recovering from hip fracture. Advantages are that it is global rather than situation-specific and measures fear rather than self-efficacy. Future research on this scale is recommended in other older adult samples for whom fear of falling is relevant

Jockeying for position: the construction of masculine identities

In this paper we examine the construction of masculine identities within a real-life social situation. Using data from an extensive series of interviews with small groups of sixth-form (17-18-year-old) students attending a UK-based, single-sex independent school, the analysis looks at the action orientation of different constructions of identity. More specifically, it focuses upon how the identity talk of one particular group of students were oriented towards managing their subordinate status within the school. In a number of instances the identity of the `new man' was adopted as a strategy of resistance. However, it was found that the more common strategy involved buying back into values embodied within a more traditional definition of masculinity

Linear independence in linear systems on elliptic curves

Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with $\operatorname{char} k \nmid N$. For $P \in C$, let $s_P$ be a rational function with divisor $N \cdot P - N \cdot O$. We ask whether the $N$ functions $s_P$ are linearly independent. For generic $(E,C)$, we prove that the answer is yes. We bound the number of exceptional $(E,C)$ when $N$ is a prime by using the geometry of the universal generalized elliptic curve over $X_1(N)$. The problem can be recast in terms of sections of an arbitrary degree $N$ line bundle on $E$.Comment: 10 page