Unanticipated demand of Physiotherapist-Deployed Airway Clearance during the COVID-19 Surge 2020 a single centre report.

Bronchial secretion management was not an anticipated clinical problem in patients intubated and ventilated with COVID-19. Yet 63 (62%) of our intubated and ventilated patients demonstrated a moderate or greater sputum load, as recorded by physiotherapists on 5 or more days of the patient's ICU stay. The efficacy of airway clearance in these patients was further compounded by ineffective or absent cough and increased secretion tenacity, dramatically increasing the workload of critical care physiotherapists. We provide data to support the modelling of critical care physiotherapy staffing for future COVID-19 surges.

Communication cost of consensus for nodes with limited memory.

Motivated by applications in wireless networks and the Internet of Things, we consider a model of n nodes trying to reach consensus with high probability on their majority bit. Each node i is assigned a bit at time 0 and is a finite automaton with m bits of memory (i.e., [Formula: see text] states) and a Poisson clock. When the clock of i rings, i can choose to communicate and is then matched to a uniformly chosen node j. The nodes j and i may update their states based on the state of the other node. Previous work has focused on minimizing the time to consensus and the probability of error, while our goal is minimizing the number of communications. We show that, when [Formula: see text], consensus can be reached with linear communication cost, but this is impossible if [Formula: see text] A key step is to distinguish when nodes can become aware of knowing the majority bit and stop communicating. We show that this is impossible if their memory is too low.

Gravitational allocation on the sphere.

Given a collection L of n points on a sphere [Formula: see text] of surface area n, a fair allocation is a partition of the sphere into n parts each of area 1, and each is associated with a distinct point of L. We show that, if the n points are chosen uniformly at random and if the partition is defined by a certain "gravitational" potential, then the expected distance between a point on the sphere and the associated point of L is [Formula: see text] We use our result to define a matching between two collections of n independent and uniform points on the sphere and prove that the expected distance between a pair of matched points is [Formula: see text], which is optimal by a result of Ajtai, Komlós, and Tusnády.