59 research outputs found
On the density of the odd values of the partition function, II: An infinite conjectural framework
We continue our study of a basic but seemingly intractable problem in integer
partition theory, namely the conjecture that is odd exactly of
the time. Here, we greatly extend on our previous paper by providing a
doubly-indexed, infinite framework of conjectural identities modulo 2, and show
how to, in principle, prove each such identity. However, our conjecture remains
open in full generality.
A striking consequence is that, under suitable existence conditions, if any
-multipartition function is odd with positive density and
(mod 3), then is also odd with positive density. These are all facts
that appear virtually impossible to show unconditionally today.
Our arguments employ a combination of algebraic and analytic methods,
including certain technical tools recently developed by Radu in his study of
the parity of the Fourier coefficients of modular forms.Comment: 14 pages. To appear in the J. of Number Theor
On the density of the odd values of the partition function
The purpose of this note is to introduce a new approach to the study of one
of the most basic and seemingly intractable problems in partition theory,
namely the conjecture that the partition function is equidistributed
modulo 2.
Our main result will relate the densities, say , of the odd values
of the -multipartition functions , for several integers . In
particular, we will show that if for some , then (assuming it exists) ; that is,
itself is odd with positive density. Notice that, currently, the best
unconditional result does not even imply that is odd for
values of . In general, we conjecture that for all
odd, i.e., that similarly to the case of , all multipartition functions
are in fact equidistributed modulo 2.
Our arguments will employ a number of algebraic and analytic methods, ranging
from an investigation modulo 2 of some classical Ramanujan identities and
several other eta product results, to a unified approach that studies the
parity of the Fourier coefficients of a broad class of modular form identities
recently introduced by Radu.Comment: Several changes with respect to the 2015 version. 18 pages. To appear
in the Annals of Combinatoric
Services just for men? Insights from a national study of the well men services pilots.
Men continue to have a lower life expectancy in most countries compared to women. Explanations of this gendered health inequality tend to focus on male risk taking, unhealthy lifestyle choices and resistance to seeking help from health services. In the period 2005-2008 the Scottish Government funded a nationwide community health promotion programme aimed at improving men's health, called Well Men Service Pilots (henceforth WMS)
Can Monkeys Choose Optimally When Faced with Noisy Stimuli and Unequal Rewards?
We review the leaky competing accumulator model for two-alternative forced-choice decisions with cued responses, and propose extensions to account for the influence of unequal rewards. Assuming that stimulus information is integrated until the cue to respond arrives and that firing rates of stimulus-selective neurons remain well within physiological bounds, the model reduces to an Ornstein-Uhlenbeck (OU) process that yields explicit expressions for the psychometric function that describes accuracy. From these we compute strategies that optimize the rewards expected over blocks of trials administered with mixed difficulty and reward contingencies. The psychometric function is characterized by two parameters: its midpoint slope, which quantifies a subject's ability to extract signal from noise, and its shift, which measures the bias applied to account for unequal rewards. We fit these to data from two monkeys performing the moving dots task with mixed coherences and reward schedules. We find that their behaviors averaged over multiple sessions are close to optimal, with shifts erring in the direction of smaller penalties. We propose two methods for biasing the OU process to produce such shifts
Dimethyl fumarate in patients admitted to hospital with COVID-19 (RECOVERY): a randomised, controlled, open-label, platform trial
Dimethyl fumarate (DMF) inhibits inflammasome-mediated inflammation and has been proposed as a treatment for patients hospitalised with COVID-19. This randomised, controlled, open-label platform trial (Randomised Evaluation of COVID-19 Therapy [RECOVERY]), is assessing multiple treatments in patients hospitalised for COVID-19 (NCT04381936, ISRCTN50189673). In this assessment of DMF performed at 27 UK hospitals, adults were randomly allocated (1:1) to either usual standard of care alone or usual standard of care plus DMF. The primary outcome was clinical status on day 5 measured on a seven-point ordinal scale. Secondary outcomes were time to sustained improvement in clinical status, time to discharge, day 5 peripheral blood oxygenation, day 5 C-reactive protein, and improvement in day 10 clinical status. Between 2 March 2021 and 18 November 2021, 713 patients were enroled in the DMF evaluation, of whom 356 were randomly allocated to receive usual care plus DMF, and 357 to usual care alone. 95% of patients received corticosteroids as part of routine care. There was no evidence of a beneficial effect of DMF on clinical status at day 5 (common odds ratio of unfavourable outcome 1.12; 95% CI 0.86-1.47; p = 0.40). There was no significant effect of DMF on any secondary outcome
Dimethyl fumarate in patients admitted to hospital with COVID-19 (RECOVERY): a randomised, controlled, open-label, platform trial
Dimethyl fumarate (DMF) inhibits inflammasome-mediated inflammation and has been proposed as a treatment for patients hospitalised with COVID-19. This randomised, controlled, open-label platform trial (Randomised Evaluation of COVID-19 Therapy [RECOVERY]), is assessing multiple treatments in patients hospitalised for COVID-19 (NCT04381936, ISRCTN50189673). In this assessment of DMF performed at 27 UK hospitals, adults were randomly allocated (1:1) to either usual standard of care alone or usual standard of care plus DMF. The primary outcome was clinical status on day 5 measured on a seven-point ordinal scale. Secondary outcomes were time to sustained improvement in clinical status, time to discharge, day 5 peripheral blood oxygenation, day 5 C-reactive protein, and improvement in day 10 clinical status. Between 2 March 2021 and 18 November 2021, 713 patients were enroled in the DMF evaluation, of whom 356 were randomly allocated to receive usual care plus DMF, and 357 to usual care alone. 95% of patients received corticosteroids as part of routine care. There was no evidence of a beneficial effect of DMF on clinical status at day 5 (common odds ratio of unfavourable outcome 1.12; 95% CI 0.86-1.47; p = 0.40). There was no significant effect of DMF on any secondary outcome
Oral bisphosphonate use and age-related macular degeneration: retrospective cohort and nested case-control study
Our objective here was to determine whether oral bisphosphonate (BP) use is associated with the incidence of age-related macular degeneration (AMD). We performed a population-based study using electronic health records from UK primary care (Clinical Practice Research Datalink). A cohort of 13,974 hip fracture patients (1999–2013) was used to conduct (1) a propensity score–matched cohort analysis and (2) a nested case–control analysis. Hip fracture patients were aged ≥50 years without AMD diagnosis before hip fracture date or in the first year of follow-up. Among 6208 matched patients and during 22,142 person-years of follow-up, 57 (1.8%) and 42 (1.4%) AMD cases occurred in BP users and non-BP users, respectively. The survival analysis model did not provide significant evidence of a higher risk of AMD in BP users (subhazard ratio: 1.60; 95% confidence interval (CI): 0.95–2.72; P = 0.08), although there was a significant increased risk among BP users with high medication possession ratio (MPR) (top quartile) relative to non-BP users (odds ratio: 5.08, 95% CI: 3.11–8.30; P < 0.001, respectively). Overall, oral BP use was not associated with an increased risk of AMD in this cohort of hip fracture patients, although the risk increased significantly with higher MPR. More data are needed to confirm these findings
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