On Discreteness of Commensurators

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dense subgroups with non-empty domain of discontinuity have discrete commensurators. This generalizes a Theorem of Greenberg for Kleinian groups. We then prove that for all finitely generated, Zariski dense, infinite covolume discrete subgroups of $Isom ({\mathbb{H}}^3)$, commensurators are discrete. Together these prove discreteness of commensurators for all known examples of finitely generated, Zariski dense, infinite covolume discrete subgroups of $Isom(X)$ for $X$ a symmetric space of non-compact type.Comment: 14 pages, 1 figur

Predicting knee osteoarthritis risk in injured populations

Background Individuals who suffered a lower limb injury have an increased risk of developing knee osteoarthritis. Early diagnosis of osteoarthritis and the ability to track its progression is challenging. This study aimed to explore links between self-reported knee osteoarthritis outcome scores and biomechanical gait parameters, whether self-reported outcome scores could predict gait abnormalities characteristic of knee osteoarthritis in injured populations and, whether scores and biomechanical outcomes were related to osteoarthritis severity via Spearman's correlation coefficient. Methods A cross-sectional study was conducted with asymptomatic participants, participants with lower-limb injury and those with medial knee osteoarthritis. Spearman rank determined relationships between knee injury and outcome scores and hip and knee kinetic/kinematic gait parameters. K-Nearest Neighbour algorithm was used to determine which of the evaluated parameters created the strongest classifier model. Findings Differences in outcome scores were evident between groups, with knee quality of life correlated to first and second peak external knee adduction moment (0.47, 0.55). Combining hip and knee kinetics with quality of life outcome produced the strongest classifier (1.00) with the least prediction error (0.02), enabling classification of injured subjects gait as characteristic of either asymptomatic or knee osteoarthritis subjects. When correlating outcome scores and biomechanical outcomes with osteoarthritis severity only maximum external hip and knee abduction moment (0.62, 0.62) in addition to first peak hip adduction moment (0.47) displayed significant correlations. Interpretation The use of predictive models could enable clinicians to identify individuals at risk of knee osteoarthritis and be a cost-effective method for osteoarthritis screening

Have Centuries of Inefficient Fishing Sustained a Wild Oyster Fishery: a Case Study

The native European flat oyster (Ostrea edulis) has declined throughout its range, due to over-exploitation, a situation mirrored in oyster stocks globally. There are three remaining oyster fisheries in England (Fal, Solent, and Thames Estuary). The Fal oyster fishery though employs traditional methods, using hand-hauled dredges from rowing punts or under sail and is home to the last commercial sailing fleet in Europe. Against a backdrop of temporary closures to protect dwindling stocks in the Solent and Thames Estuary, this study considers whether the longevity of the Fal oyster fishery is linked to the traditional methods that have been employed for centuries. Using GPS tracking in combination with on board observers, we demonstrate that dredging under sail is inefficient compared to more modern mechanically powered methods that are utilised elsewhere. A review of historical landings suggests that both overall landings and fishing effort have declined. The fishery appears to have gone through cycles of over-exploitation and one closure due to disease. However, the key to the long-term survival of the Fal oyster fishery may be linked to the traditional method of dredging. It is estimated that a switch from traditional methods to modern techniques would result in a greater than 9 fold increase in effort per season. The data presented highlight this unique fishery as a counterfactual to the increases in power seen in commercial fisheries over the last century and serve as a reference point for future studies