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The PUL-integral is a McShane type of definition in which the notion of a partition of unity is of great importance. It was first introduced by Kurzweil and Jarnik. Recently, Boonpogkrong revisited this definition and presented its, relatively, simplified approach. In this paper, a Henstock-Kurzweil approach of this integral including its fundamental properties will be presented.

Homan introduced the variational Kurzweil-Hentock-Stieltjes integral on a real-valued function and presented some of its properties. In this paper, we dened the variational Kurzweil-Henstock-Stieltjes integral on a compact interval in Rn. Fundamental properties such as uniqueness, linearity property and monotonocity property of both the integrand and integrator, additivity and integrability over a subinterval are provided. In addition, a characterization of the variational Kurzweil-Henstock-Stieltjes integral is established by formulating the Cauchy Criterion.

Kurzweil-Henstock integral is a generalization of the Reimann integral. In this paper, we established the definition of Kurzweil-Henstock-Stieltjes integral on \(\mathbb{R}\)n via gauge type approach where integrand and integrator are all real-valued functions defined on a compact interval in \(\mathbb{R}\)n. Moreover, the Cauchy Criterion is established. To this end, some underlying simple properties of this integral are studied, specifically, uniqueness, linearity, monotonocity, integrability over a subset, and additivity. Results gathered in this paper may serve as a foundation to some related studies such as the notion of convergence with respect to this integral, and the formulation of the Saks-Henstock Lemma.

Background Obtaining accurate estimates of the risk of COVID-19-related death in the general population is challenging in the context of changing levels of circulating infection. Methods We propose a modelling approach to predict 28-day COVID-19-related death which explicitly accounts for COVID-19 infection prevalence using a series of sub-studies from new landmark times incorporating time-updating proxy measures of COVID-19 infection prevalence. This was compared with an approach ignoring infection prevalence. The target population was adults registered at a general practice in England in March 2020. The outcome was 28-day COVID-19-related death. Predictors included demographic characteristics and comorbidities. Three proxies of local infection prevalence were used: model-based estimates, rate of COVID-19-related attendances in emergency care, and rate of suspected COVID-19 cases in primary care. We used data within the TPP SystmOne electronic health record system linked to Office for National Statistics mortality data, using the OpenSAFELY platform, working on behalf of NHS England. Prediction models were developed in case-cohort samples with a 100-day follow-up. Validation was undertaken in 28-day cohorts from the target population. We considered predictive performance (discrimination and calibration) in geographical and temporal subsets of data not used in developing the risk prediction models. Simple models were contrasted to models including a full range of predictors. Results Prediction models were developed on 11,972,947 individuals, of whom 7999 experienced COVID-19-related death. All models discriminated well between individuals who did and did not experience the outcome, including simple models adjusting only for basic demographics and number of comorbidities: C-statistics 0.92–0.94. However, absolute risk estimates were substantially miscalibrated when infection prevalence was not explicitly modelled. Conclusions Our proposed models allow absolute risk estimation in the context of changing infection prevalence but predictive performance is sensitive to the proxy for infection prevalence. Simple models can provide excellent discrimination and may simplify implementation of risk prediction tools.

This publication lists, by major institution, 32 population information resources in the United States. Each entry provides name and location of center or library, person in charge; date established; number of staff; size of collection and acquisition rate; area of specialization; publications circulated; services offered, and whether or not the facility is equipped to handle outside requests. Personnel and organization indices are provided. (ED 050 961 is a preliminary listing of information sources outside the United States.) (JLB)

Background: Early descriptions of patients admitted to hospital during the COVID-19 pandemic showed a lower prevalence of asthma and chronic obstructive pulmonary disease (COPD) than would be expected for an acute respiratory disease like COVID-19, leading to speculation that inhaled corticosteroids (ICSs) might protect against infection with severe acute respiratory syndrome coronavirus 2 or the development of serious sequelae. We assessed the association between ICS and COVID-19-related death among people with COPD or asthma using linked electronic health records (EHRs) in England, UK. Methods: In this observational study, we analysed patient-level data for people with COPD or asthma from primary care EHRs linked with death data from the Office of National Statistics using the OpenSAFELY platform. The index date (start of follow-up) for both cohorts was March 1, 2020; follow-up lasted until May 6, 2020. For the COPD cohort, individuals were eligible if they were aged 35 years or older, had COPD, were a current or former smoker, and were prescribed an ICS or long-acting β agonist plus long-acting muscarinic antagonist (LABA–LAMA) as combination therapy within the 4 months before the index date. For the asthma cohort, individuals were eligible if they were aged 18 years or older, had been diagnosed with asthma within 3 years of the index date, and were prescribed an ICS or short-acting β agonist (SABA) only within the 4 months before the index date. We compared the outcome of COVID-19-related death between people prescribed an ICS and those prescribed alternative respiratory medications: ICSs versus LABA–LAMA for the COPD cohort, and low-dose or medium-dose and high-dose ICSs versus SABAs only in the asthma cohort. We used Cox regression models to estimate hazard ratios (HRs) and 95% CIs for the association between exposure categories and the outcome in each population, adjusted for age, sex, and all other prespecified covariates. We calculated e-values to quantify the effect of unmeasured confounding on our results. Findings: We identified 148 557 people with COPD and 818 490 people with asthma who were given relevant respiratory medications in the 4 months before the index date. People with COPD who were prescribed ICSs were at increased risk of COVID-19-related death compared with those prescribed LABA–LAMA combinations (adjusted HR 1·39 [95% CI 1·10–1·76]). Compared with those prescribed SABAs only, people with asthma who were prescribed high-dose ICS were at an increased risk of death (1·55 [1·10–2·18]), whereas those given a low or medium dose were not (1·14 [0·85–1·54]). Sensitivity analyses showed that the apparent harmful association we observed could be explained by relatively small health differences between people prescribed ICS and those not prescribed ICS that were not recorded in the database (e value lower 95% CI 1·43). Interpretation: Our results do not support a major role for regular ICS use in protecting against COVID-19-related death among people with asthma or COPD. Observed increased risks of COVID-19-related death can be plausibly explained by unmeasured confounding due to disease severity. Funding: UK Medical Research Council.