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Asymptotic formulae are established for the number of natural numbers mm with largest square-free divisor not exceeding mϑ{m}^{{\vartheta }}, for any fixed positive parameter ϑ{\vartheta }. Related counting functions are also considered.

We interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely solution counting in older partial progress on Vinogradov's mean value theorem corresponds to in Fourier decoupling theory. [ABSTRACT FROM AUTHOR]

Exclusion netting systems are effective in various contexts and are increasingly used to control crop pests. However, factors affecting pollination management under nets are poorly known. The pollination effectiveness of commercial bumble bee hives of Bombus impatiens Cresson (Hymenoptera: Apidae) was studied for apple production under exclusion netting systems in a research orchard located in Quebec, Canada during 2016–2017. Sixteen single-row plots of apple trees (plot length: 18.5 m, cultivar GingerGold) were subjected to one of the following four treatments during bloom: (1) introduction of a bumble bee hive placed at the end of the row, under nets; (2) introduction of a bumble bee hive placed in the middle of the row, under nets; (3) negative control with no pollinators, under nets and (4) agronomic control with nearby bee hives (<50 m), without nets. Resulting post-harvest fruit quality (e.g., fruit weight, size, number and distribution of seeds) was evaluated, as well as correlations between bumble bee visitation rates and fruit quality parameters were evaluated. Results suggest that bumble bees provided adequate pollination under exclusion netting systems and that resulting fruit quality was equivalent to that of apple fruit conventionally pollinated by honey bees and wild bees community (bumble bees and other bees) in the orchard environment. Positioning bumble bee hives in the middle of the row provided better fruit load homogeneity in pollinated trees. Additional discussion on bumble bees as apple pollinators and on pollen distribution methods is also included. [ABSTRACT FROM AUTHOR]

Abstract Background In acute-care hospitals, no evidence of a protective effect of healthcare worker (HCW) vaccination on hospital-acquired influenza (HAI) in patients has been documented. Our study objective was to ascertain the effectiveness of influenza vaccination of HCW on HAI among patients. Methods A nested case-control investigation was implemented in a prospective surveillance study of influenza-like illness (ILI) in a tertiary acute-care university hospital. Cases were patients with virologically-confirmed influenza occurring ≥ 72 h after admission, and controls were patients with ILI presenting during hospitalisation with negative influenza results after nasal swab testing. Four controls per case, matched per influenza season (2004-05, 2005-06 and 2006-07), were randomly selected. Univariate and multivariate conditional logistic regression models were fitted to assess factors associated with HAI among patients. Results In total, among 55 patients analysed, 11 (20%) had laboratory-confirmed HAI. The median HCW vaccination rate in the units was 36%. The median proportion of vaccinated HCW in these units was 11.5% for cases vs. 36.1% for the controls (P = 0.11); 2 (20%) cases and 21 (48%) controls were vaccinated against influenza in the current season (P = 0.16). The proportion of ≥ 35% vaccinated HCW in short-stay units appeared to protect against HAI among patients (odds ratio = 0.07; 95% confidence interval 0.005-0.98), independently of patient age, influenza season and potential influenza source in the units. Conclusions Our observational study indicates a shielding effect of more than 35% of vaccinated HCW on HAI among patients in acute-care units. Investigations, such as controlled clinical trials, are needed to validate the benefits of HCW vaccination on HAI incidence in patients.