Amicable pairs and aliquot cycles for elliptic curves

An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good reduction for E satisfying #E(F_p) = q and #E(F_q) = p. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We show that there exist elliptic curves with arbitrarily long aliqout cycles, but that CM elliptic curves (with j not 0) have no aliqout cycles of length greater than two. We give conjectural formulas for the frequency of amicable pairs. For CM curves, the derivation of precise conjectural formulas involves a detailed analysis of the values of the Grossencharacter evaluated at a prime ideal P in End(E) having the property that #E(F_P) is prime. This is especially intricate for the family of curves with j = 0.Comment: 53 page

ASTEROID: A new clinical stereotest on an autostereo 3D tablet

Purpose: To describe a new stereotest in the form of a game on an autostereoscopic tablet computer designed to be suitable for use in the eye clinic and present data on its reliability and the distribution of stereo thresholds in adults. Methods: Test stimuli were four dynamic random-dot stereograms, one of which contained a disparate target. Feedback was given after each trial presentation. A Bayesian adaptive staircase adjusted target disparity. Threshold was estimated from the mean of the posterior distribution after 20 responses. Viewing distance was monitored via a forehead sticker viewed by the tablet's front camera, and screen parallax was adjusted dynamically so as to achieve the desired retinal disparity. Results: The tablet must be viewed at a distance of greater than ∼35 cm to produce a good depth percept. Log thresholds were roughly normally distributed with a mean of 1.75 log10 arcsec = 56 arcsec and SD of 0.34 log10 arcsec = a factor of 2.2. The standard deviation agrees with previous studies, but ASTEROID thresholds are approximately 1.5 times higher than a similar stereotest on stereoscopic 3D TV or on Randot Preschool stereotests. Pearson correlation between successive tests in same observer was 0.80. Bland-Altman 95% limits of reliability were ±0.64 log10 arcsec = a factor of 4.3, corresponding to an SD of 0.32 log10 arcsec on individual threshold estimates. This is similar to other stereotests and close to the statistical limit for 20 responses. Conclusions: ASTEROID is reliable, easy, and portable and thus well-suited for clinical stereoacuity measurements. Translational Relevance: New 3D digital technology means that research-quality psychophysical measurement of stereoacuity is now feasible in the clinic

Impact of exercise-based cardiac rehabilitation in patients with heart failure (ExTraMATCH II) on mortality and hospitalisation:an individual-patient data meta-analysis of randomised trials

This is the author accepted manuscript. The final version is available from Wiley via the DOI in this recordAIMS: To undertake an individual patient data (IPD) meta-analysis to assess the impact of exercise-based cardiac rehabilitation (ExCR) in patients with heart failure (HF) on mortality and hospitalisation, and differential effects of ExCR according to patient characteristics: age, sex, ethnicity, New York Heart Association functional class, ischaemic aetiology, ejection fraction, and exercise capacity. METHODS AND RESULTS: Randomised trials of exercise training for at least 3 weeks compared with no exercise control with 6-month follow-up or longer, providing IPD time to event on mortality or hospitalisation (all-cause or HF-specific). IPD were combined into a single dataset. We used Cox proportional hazards models to investigate the effect of ExCR and the interactions between ExCR and participant characteristics. We used both two-stage random effects and one-stage fixed effect models. IPD were obtained from 18 trials including 3912 patients with HF with reduced ejection fraction. Compared to control, there was no statistically significant difference in pooled time to event estimates in favour of ExCR although confidence intervals (CIs) were wide [all-cause mortality: hazard ratio (HR) 0.83, 95% CI 0.67-1.04; HF-specific mortality: HR 0.84, 95% CI 0.49-1.46; all-cause hospitalisation: HR 0.90, 95% CI 0.76-1.06; and HF-specific hospitalisation: HR 0.98, 95% CI 0.72-1.35]. No strong evidence was found of differential intervention effects across patient characteristics. CONCLUSION: Exercise-based cardiac rehabilitation did not have a significant effect on the risk of mortality and hospitalisation in HF with reduced ejection fraction. However, uncertainty around effect estimates precludes drawing definitive conclusions.This work is supported by UK National Institute for Health Research funding (HTA 15/80/30)

Elevated <scp>CO₂</scp> interacts with nutrient inputs to restructure plant communities in phosphorus‐limited grasslands

AbstractGlobally pervasive increases in atmospheric CO2 and nitrogen (N) deposition could have substantial effects on plant communities, either directly or mediated by their interactions with soil nutrient limitation. While the direct consequences of N enrichment on plant communities are well documented, potential interactions with rising CO2 and globally widespread phosphorus (P) limitation remain poorly understood. We investigated the consequences of simultaneous elevated CO2 (eCO2) and N and P additions on grassland biodiversity, community and functional composition in P‐limited grasslands. We exposed soil‐turf monoliths from limestone and acidic grasslands that have received &gt;25 years of N additions (3.5 and 14 g m−2 year−1) and 11 (limestone) or 25 (acidic) years of P additions (3.5 g m−2 year−1) to eCO2 (600 ppm) for 3 years. Across both grasslands, eCO2, N and P additions significantly changed community composition. Limestone communities were more responsive to eCO2 and saw significant functional shifts resulting from eCO2–nutrient interactions. Here, legume cover tripled in response to combined eCO2 and P additions, and combined eCO2 and N treatments shifted functional dominance from grasses to sedges. We suggest that eCO2 may disproportionately benefit P acquisition by sedges by subsidising the carbon cost of locally intense root exudation at the expense of co‐occurring grasses. In contrast, the functional composition of the acidic grassland was insensitive to eCO2 and its interactions with nutrient additions. Greater diversity of P‐acquisition strategies in the limestone grassland, combined with a more functionally even and diverse community, may contribute to the stronger responses compared to the acidic grassland. Our work suggests we may see large changes in the composition and biodiversity of P‐limited grasslands in response to eCO2 and its interactions with nutrient loading, particularly where these contain a high diversity of P‐acquisition strategies or developmentally young soils with sufficient bioavailable mineral P.</jats:p

The evolution and ecology of multiple antipredator defences

Prey seldom rely on a single type of antipredator defence, often using multiple defences to avoid predation. In many cases, selection in different contexts may favour the evolution of multiple defences in a prey. However, a prey may use multiple defences to protect itself during a single predator encounter. Such “defence portfolios” that defend prey against a single instance of predation are distributed across and within successive stages of the predation sequence (encounter, detection, identification, approach (attack), subjugation and consumption). We contend that at present, our understanding of defence portfolio evolution is incomplete, and seen from the fragmentary perspective of specific sensory systems (e.g., visual) or specific types of defences (especially aposematism). In this review, we aim to build a comprehensive framework for conceptualizing the evolution of multiple prey defences, beginning with hypotheses for the evolution of multiple defences in general, and defence portfolios in particular. We then examine idealized models of resource trade-offs and functional interactions between traits, along with evidence supporting them. We find that defence portfolios are constrained by resource allocation to other aspects of life history, as well as functional incompatibilities between different defences. We also find that selection is likely to favour combinations of defences that have synergistic effects on predator behaviour and prey survival. Next, we examine specific aspects of prey ecology, genetics and development, and predator cognition that modify the predictions of current hypotheses or introduce competing hypotheses. We outline schema for gathering data on the distribution of prey defences across species and geography, determining how multiple defences are produced, and testing the proximate mechanisms by which multiple prey defences impact predator behaviour. Adopting these approaches will strengthen our understanding of multiple defensive strategies.</p

Search for W~1Z~2\widetilde{W}_1\widetilde{Z}_2 Production via Trilepton Final States in ppˉp\bar{p} collisions at s=1.8\sqrt{s}=1.8 TeV

We have searched for associated production of the lightest chargino, $\widetilde{W}_1$, and next-to-lightest neutralino, $\widetilde{Z}_2$, of the Minimal Supersymmetric Standard Model in $p\bar{p}$ collisions at \mbox{$\sqrt{s}$ = 1.8 TeV} using the \D0 detector at the Fermilab Tevatron collider. Data corresponding to an integrated luminosity of 12.5$\pm 0.7$ \ipb were examined for events containing three isolated leptons. No evidence for $\widetilde{W}_1\widetilde{Z}_2$ pair production was found. Limits on $\sigma(\widetilde{W}_1\widetilde{Z}_2)$Br$(\widetilde{W}_1\to l\nu\widetilde{Z}_1)$Br$(\widetilde{Z}_2\to l\bar{l}\widetilde{Z}_1)$ are presented.Comment: 17 pages (13 + 1 page table + 3 pages figures). 3 PostScript figures will follow in a UUEncoded, gzip'd, tar file. Text in LaTex format. Submitted to Physical Review Letters. Replace comments - Had to resumbmit version with EPSF directive