Algebraic and combinatorial aspects of sandpile monoids on directed graphs

The sandpile group of a graph is a well-studied object that combines ideas from algebraic graph theory, group theory, dynamical systems, and statistical physics. A graph's sandpile group is part of a larger algebraic structure on the graph, known as its sandpile monoid. Most of the work on sandpiles so far has focused on the sandpile group rather than the sandpile monoid of a graph, and has also assumed the underlying graph to be undirected. A notable exception is the recent work of Babai and Toumpakari, which builds up the theory of sandpile monoids on directed graphs from scratch and provides many connections between the combinatorics of a graph and the algebraic aspects of its sandpile monoid. In this paper we primarily consider sandpile monoids on directed graphs, and we extend the existing theory in four main ways. First, we give a combinatorial classification of the maximal subgroups of a sandpile monoid on a directed graph in terms of the sandpile groups of certain easily-identifiable subgraphs. Second, we point out certain sandpile results for undirected graphs that are really results for sandpile monoids on directed graphs that contain exactly two idempotents. Third, we give a new algebraic constraint that sandpile monoids must satisfy and exhibit two infinite families of monoids that cannot be realized as sandpile monoids on any graph. Finally, we give an explicit combinatorial description of the sandpile group identity for every graph in a family of directed graphs which generalizes the family of (undirected) distance-regular graphs. This family includes many other graphs of interest, including iterated wheels, regular trees, and regular tournaments.Comment: v2: Cleaner presentation, new results in final section. Accepted for publication in J. Combin. Theory Ser. A. 21 pages, 5 figure

Validity and reliability of an incremental double poling protocol in cross-country skiers

This study determined the validity and reliability of an incremental double poling protocol performed on a Concept II ski-ergometer and validated this against an existing treadmill ski-striding protocol. Ten well-trained male cross-country skiers (age: 19 ± 1.4 y; height: 182 ± 72 cm; body mass: 76.0 ± 10.8 kg, whole body VO2Peak: 5.2 ± 1.0 L.min-1; upper body VO2Peak: 4.6 ± 1.0 L.min-1; upper body:lower body ratio: 87.2 ± 5.6%) performed four VO2Peak tests; one treadmill ski-striding test and three double poling ski-ergometer tests over five days. Test-retest reliability of the ski-ergometer protocol was determined for maximal oxygen consumption (VO2peak). The ski-ergometer test showed excellent reliability for VO2Peak (L.min-1) (coefficient of variation [CV] = 1.9%, 95% confidence limit [95% CL] [1.2, 4.7]; r = 1.00, [0.96, 1.00]) and UBPPeak (W) (CV = 1.4%, [0.9, 3.4]; r = 1.00, [0.97, 1.00]). Very strong correlations existed between the ski-ergometer and the ski-striding protocol for VO2Peak (r = 0.95, [0.76, 0.99]). The upper body ski-ergometer test provided valid and reliable measurements of ski-specific upper body aerobic power in well-trained male cross-country skiers

Scanning the horizon: emerging hospital-wide technologies and their impact on critical care

This commentary represents a selective survey of developments relevant to critical care. Selected themes include advances in point-of-care diagnostic testing, glucose control, novel microbiological diagnostics and infection control measures, and developments in information technology that have implications for intensive care. The latter encompasses an early example of an artificially intelligent clinical decision support mechanism, the introduction of a national health care information technology programme (UK NPfIT) and its implications, and exotic threats to patient safety due to emergent behaviour in complex information systems

Ensuring an Essential Supply of Allied Health Professions (AHP) Placements: Using Crowdsourcing to Develop a National Call to Action

Sustainable growth in the Allied Health Professions (AHP) workforce is an ambition of the United Kingdom’s NHS Long Term Plan. However historically, access to good quality placements has been a barrier to increasing pre-registration training numbers. This article focuses on work carried out by Health Education England (HEE) to gain insights on the impact of the COVID-19 pandemic on capacity. Using a pragmatic, embedded mixed-methods approach, insights were gathered using an online workshop, crowdsourcing, open for two weeks in the summer of 2020. AHP placement stakeholders could vote, share ideas or comment. Descriptive data were extracted, and comments made were analysed using inductive thematic analysis. Participants (N = 1,800) made over 8,500 comments. The themes identified included: diversity of placement opportunity, improved placement coordination, a more joined-up system, supervision models and educator capacity. Alongside considering the challenges to placement capacity, several areas of innovative practice owing to the pandemic were highlighted. Generated insights have shaped the aims and objectives of the Health Education (HEE) pre-registration AHP student practice learning programme for 2020/2021 and beyond. The COVID-19 pandemic has disrupted the delivery of AHP placements. In the absence of face-to-face activities, crowdsourcing provided an online data collection tool offering stakeholders an opportunity to engage with the placement capacity agenda and share learning. Findings have shaped the HEE approach to short-term placement recovery and long-term growth