On sets of integers whose shifted products are powers

AbstractLet N be a positive integer and let A be a subset of {1,…,N} with the property that aa′+1 is a pure power whenever a and a′ are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A|≪(logN)2/3(loglogN)1/3

Population prevalence, incidence, and predictors of atrial fibrillation in the Renfrew/Paisley study

<b>OBJECTIVES</b> Though atrial fibrillation (AF) is an important cause of cardiovascular morbidity, there are few large epidemiological studies of its prevalence, incidence, and risk factors. The epidemiological features of AF are described in one of the largest population cohorts ever studied. <b>METHODS</b> The prevalence and incidence of AF were studied in the Renfrew/Paisley population cohort of 15 406 men and women aged 45-64 years living in the west of Scotland. This cohort was initially screened between 1972 and 1976 and again between 1977 and 1979. Incident hospitalisations with AF in the 20 year period following initial screening were also studied. <b>RESULTS</b> The population prevalence of AF in this cohort was 6.5 cases/1000 examinations. Prevalence was higher in men and older subjects. In those who were rescreened, the four year incidence of AF was 0.54 cases/1000 person years. Radiological cardiomegaly was the most powerful predictor of new AF (adjusted odds ratio 14.0). During 20 year follow up, 3.5% of this cohort was discharged from hospital with a diagnosis of AF; the rate of incident hospitalisation for AF was 1.9 cases/1000 person years. Radiological cardiomegaly (adjusted odds ratio 1.46) and systolic blood pressure (adjusted odds ratio 2.1 for ≥ 169 mm Hg) were independent predictors of this outcome. <b>CONCLUSIONS</b> Data from one of the largest epidemiological studies ever undertaken confirm that AF has a large population prevalence and incidence, even in middle aged people. More important, it was shown that the long term incidence of hospitalisation related to AF is high and that two simple clinical measurements are highly predictive of incident AF. These findings have important implications for the prevention of AF

Carbohydrate carbon sources induce loss of flocculation of an ale-brewing yeast strain

Aims: To identify the nutrients that can trigger the loss of flocculation under growth conditions in an ale-brewing strain, Saccharomyces cerevisiae NCYC 1195. Methods and Results: Flocculation was evaluated using the method of Soares, EX. and Vroman, A. [Journal of Applied Microbiology (2003) 95, 325]. Yeast growth with metabolizable carbon sources (glucose, fructose, galactose, maltose or sucrose) at 2% (w/v), induced the loss of flocculation in yeast that had previously been allowed to flocculate. The yeast remained flocculent when transferred to a medium containing the required nutrients for yeast growth and a sole nonmetabolizable carbon source (lactose). Transfer of flocculent yeast into a growth medium with ethanol (4% v/v), as the sole carbon source did not induce the loss of flocculation. Even the addition of glucose (2% w/v) or glucose and antimycin A (0.1 mg lˉ¹) to this culture did not bring about loss of flocculation. Cycloheximide addition (15 mglˉ¹) to glucose-growing cells stopped flocculation loss. Conclusions: Carbohydrates were the nutrients responsible for stimulating the loss of flocculation in flocculent yeast cells transferred to growing conditions. The glucose-induced loss of flocculation required de novo protein synthesis. Ethanol prevented glucose-induced loss of flocculation. This protective effect of ethanol was independent of the respiratory function of the yeast. Significance and Impact of the Study: This work contributes to the elucidation of the role of nutrients in the control of the flocculation cycle in NewFlo phenotype yeast strains.Instituto Politécnico do Porto (IPP) - Fundo de Apoio à Investigação - Project P24/96 , P24/97.Programa Plurianual de Unidades de I&D-CIEA/ ISEP

Second-order corrections to noncommutative spacetime inflation

We investigate how the uncertainty of noncommutative spacetime affects on inflation. For this purpose, the noncommutative parameter $\mu_0$ is taken to be a zeroth order slow-roll parameter. We calculate the noncommutative power spectrum up to second order using the slow-roll expansion. We find corrections arisen from a change of the pivot scale and the presence of a variable noncommutative parameter, when comparing with the commutative power spectrum. The power-law inflation is chosen to obtain explicit forms for the power spectrum, spectral index, and running spectral index. In cases of the power spectrum and spectral index, the noncommutative effect of higher-order corrections compensates for a loss of higher-order corrections in the commutative case. However, for the running spectral index, all higher-order corrections to the commutative case always provide negative spectral indexes, which could explain the recent WMAP data.Comment: 15 pages, no figure, version published in PR

Second-order corrections to slow-roll inflation in the brane cosmology

We calculate the power spectrum, spectral index, and running spectral index for the RS-II brane inflation in the high-energy regime using the slow-roll expansion. There exist several modifications. As an example, we take the power-law inflation by choosing an inverse power-law potential. When comparing these with those arisen in the standard inflation, we find that the power spectrum is enhanced and the spectral index is suppressed, while the running spectral index becomes zero as in the standard inflation. However, since second-order corrections are rather small, these could not play a role of distinguishing between standard and brane inflations.Comment: 6 page