Generalised time functions and finiteness of the Lorentzian distance

We show that finiteness of the Lorentzian distance is equivalent to the existence of generalised time functions with gradient uniformly bounded away from light cones. To derive this result we introduce new techniques to construct and manipulate achronal sets. As a consequence of these techniques we obtain a functional description of the Lorentzian distance extending the work of Franco and Moretti.Comment: 22 pages. Some imprecisions clarified compared to first versio

A Correspondence Between Distances and Embeddings for Manifolds: New Techniques for Applications of the Abstract Boundary

We present a one-to-one correspondence between equivalence classes of embeddings of a manifold (into a larger manifold of the same dimension) and equivalence classes of certain distances on the manifold. This correspondence allows us to use the Abstract Boundary to describe the structure of the `edge' of our manifold without resorting to structures external to the manifold itself. This is particularly important in the study of singularities within General Relativity where singularities lie on this `edge'. The ability to talk about the same objects, e.g., singularities, via different structures provides alternative routes for investigation which can be invaluable in the pursuit of physically motivated problems where certain types of information are unavailable or difficult to use.Comment: 23 page

Generalizations of the Abstract Boundary singularity theorem

The Abstract Boundary singularity theorem was first proven by Ashley and Scott. It links the existence of incomplete causal geodesics in strongly causal, maximally extended spacetimes to the existence of Abstract Boundary essential singularities, i.e., non-removable singular boundary points. We give two generalizations of this theorem: the first to continuous causal curves and the distinguishing condition, the second to locally Lipschitz curves in manifolds such that no inextendible locally Lipschitz curve is totally imprisoned. To do this we extend generalized affine parameters from $C^1$ curves to locally Lipschitz curves.Comment: 24 page

Reducing unwarranted variation in the delivery of high quality hip fracture services in England and Wales (REDUCE): protocol for a mixed-methods study

Introduction Substantial variation in the delivery of hip fracture care, and patient outcomes persists between hospitals, despite established UK national standards and guidelines. Patients’ outcomes are partly explained by patient-level risk factors, but it is hypothesised that organisational-level factors account for the persistence of unwarranted variation in outcomes. The mixed-methods REducing unwarranted variation in the Delivery of high qUality hip fraCture services in England and Wales (REDUCE) study, aims to determine key organisational factors to target to improve patient care. Methods and analysis Quantitative analysis will assess the outcomes of patients treated at 172 hospitals in England and Wales (2016–2019) using National Hip Fracture Database data combined with English Hospital Episodes Statistics; Patient Episode Database for Wales; Civil Registration (deaths) and multiple organisational-level audits to characterise each service provider. Statistical analyses will identify which organisational factors explain variation in patient outcomes, and typify care pathways with high-quality consistent patient outcomes. Documentary analysis of 20 anonymised British Orthopaedic Association hospital-initiated peer-review reports, and qualitative interviews with staff from four diverse UK hospitals providing hip fracture care, will identify barriers and facilitators to care delivery. The COVID-19 pandemic has posed a major challenge to the resilience of services and interviews will explore strategies used to adapt and innovate. This system-wide understanding will inform the development, in partnership with key national stakeholders, of an ‘Implementation Toolkit’ to inform and improve commissioning and delivery of hip fracture services. Ethics and dissemination This study was approved: quantitative study by London, City and East Research Ethics Committee (20/LO/0101); and qualitative study by Faculty of Health Sciences University of Bristol Research Ethics Committee (Ref: 108284), National Health Service (NHS) Health Research Authority (20/HRA/71) and each NHS Trust provided Research and Development approval. Findings will be disseminated through scientific conferences, peer-reviewed journals and online workshops