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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.