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It is a well-known conjecture that if a regular graph G of order 2n has degree d(G) satisfying d(G) ^ n, then G is the union of edge-disjoint 1-factors. It is well known that this conjecture is true for d(G) equal to 2n —1 or 2n — 2. We show here that it is true for d(G) equal to2n — 3, In — 4, or2n — 5. We also show that it is true for </(G)>$|K(G)|.
Motivated by recent interest in the possible spatial clustering of rare diseases, the paper develops an approach to the assessment of spatial clustering based on the second-moment properties of a labelled point process. The concept of no spatial clustering is identified with the hypothesis that in a realisation of a stationary spatial point process… (More)
We consider the problem of detecting and describing space-time interaction in point process data. We extend existing second-order methods for purely spatial point process data to the spatial-temporal setting. This extension allows us to estimate space-time interaction as a function of spatial and temporal separation, and provides a useful reinterpretation… (More)
BACKGROUND Many patients with primary biliary cirrhosis (PBC) are asymptomatic at the time of diagnosis. However, because most studies of asymptomatic PBC have been small and from tertiary centres, asymptomatic PBC remains poorly characterised. AIMS To describe the features and progression of initially asymptomatic PBC patients. METHODS Follow up by… (More)
The incidence of primary biliary cirrhosis (PBC) varies widely between regions. However, little is known about variation within regions and the degree to which this may reflect environmental risk factors. The aim of this study was to describe the spatial distribution of cases of PBC in a defined region of Northeast England over a defined period, and to… (More)