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Differential geometry is the study of curved spaces using the techniques of calculus. It is a mainstay of undergraduate mathematics education and a cornerstone of modern geometry. It is also the language used by Einstein to express general relativity, and so is an essential tool for astronomers and theoretical physicists. This introductory textbook originates from a popular course given to third year students at Durham University for over twenty years, first by the late L. M. Woodward and later by John Bolton (and others). It provides a thorough introduction by focusing on the beginnings of the subject as studied by Gauss: curves and surfaces in Euclidean space. While the main topics are the classics of differential geometry - the definition and geometric meaning of Gaussian curvature, the Theorema Egregium, geodesics, and the Gauss–Bonnet Theorem - the treatment is modern and student-friendly, taking direct routes to explain, prove and apply the main results. It includes many exercises to test students' understanding of the material, and ends with a supplementary chapter on minimal surfaces that could be used as an extension towards advanced courses or as a source of student projects.
To assess if there is a difference in salt intake (24 h urine collection and dietary recall) and dietary sources of salt (Na) on weekdays and weekend days.
A cross-sectional study of adults who provided one 24 h urine collection and one telephone-administered 24 h dietary recall.
Community-dwelling adults living in the State of Victoria, Australia.
Adults (n 598) who participated in a health survey (53·5 % women; mean age 57·1 (95 % CI 56·2, 58·1) years).
Mean (95 % CI) salt intake (dietary recall) was 6·8 (6·6, 7·1) g/d and 24 h urinary salt excretion was 8·1 (7·8, 8·3) g/d. Mean dietary and 24 h urinary salt (age-adjusted) were 0·9 (0·1, 1·6) g/d (P=0·024) and 0·8 (0·3, 1·6) g/d (P=0·0017), respectively, higher at weekends compared with weekdays. There was an indication of a greater energy intake at weekends (+0·6 (0·02, 1·2) MJ/d, P=0·06), but no difference in Na density (weekday: 291 (279, 304) mg/MJ; weekend: 304 (281, 327) mg/MJ; P=0·360). Cereals/cereal products and dishes, meat, poultry, milk products and gravy/sauces accounted for 71 % of dietary Na.
Mean salt intake (24 h urine collection) was more than 60 % above the recommended level of 5 g salt/d and 8–14 % more salt was consumed at weekends than on weekdays. Substantial reductions in the Na content of staple foods, processed meat, sauces, mixed dishes (e.g. pasta), convenience and takeaway foods are required to achieve a significant consistent reduction in population salt intake throughout the week.
We have used ground-penetrating radar (GPR) to observe englacial structural control upon the development of an esker formed during a high-magnitude outburst flood (jökulhlaup). The surge-type Skeiðarárjökull, an outlet glacier of the Vatnajökull ice cap, Iceland, is a frequent source of jökulhlaups. The rising-stage waters of the November 1996 jökulhlaup travelled through a dense network of interconnected fractures that perforated the margin of the glacier. Subsequent discharge focused upon a small number of conduit outlets. Recent ice-marginal retreat has exposed a large englacial esker associated with one of these outlets. We investigated structural controls on esker genesis in April 2006, by collecting >2.5km of GPR profiles on the glacier surface up-glacier of where the esker ridge has been exposed by meltout. In lines closest to the exposed esker ridge, we interpret areas of englacial horizons up to ~30m wide and ~10–15m high as an up-glacier continuation of the esker sediments. High-amplitude, dipping horizons define the base of esker materials across many lines. Similar dipping surfaces deeper in the profiles suggest that: (1) the dipping surfaces beneath the esker are englacial tephera bands; (2) floodwaters were initially discharged along structurally controlled englacial surfaces (tephra bands); (3) the rapid increase in discharge resulted in hydrofracturing; (4) establishment of preferential flow paths resulted in conduit development along the tephra bands due to localized excavation of surrounding glacier ice; and (5) sedimentation took place within the new accommodation space to form the englacial structure melting out to produce the esker.
We present 1 and 100 MHz ground-based radar data from the onset region of Rutford Ice Stream, West Antarctica, which indicate the form and internal structure of isochrones. In the flow-parallel lines, modelled isochrone patterns reproduce the gross pattern of the imaged near-surface layers, assuming steady-state flow velocity from GPS records and the current accumulation rate for the last 200 years. We interpret this as indicating overall stability in flow in the onset region of Rutford Ice Stream throughout this period. However, in the cross-flow lines some local variability in accumulation is seen in areas close to the ice-stream margin where a number of tributaries converge towards the ice-stream onset zone. Episodic surface lowering events are observed followed by rapid fill episodes. The fill events indicate deposition towards the northwest, most likely generated by storm winds, which blow at an oblique angle to ice flow. More problematic is explaining the generation of episodic surface lowering in this area. We speculate this may be due to: changing ice-flow direction in the complex tributary area of the onset zone; a change in basal sediments or sedimentary landforms; a change in basal melt rates or water supply; or episodic lake drainage events in the fjord systems of the Ellsworth Subglacial Highlands. The study highlights the difficulty of assessing flow stability in the complex onset regions of West Antarctic ice streams.
Deformational structures at the surge-type glacier Kongsvegen, Svalbard, are displayed at the glacier surface and on a grounded cliff section at the terminus. A 300 m × 65 m grid of 200 MHz ground-penetrating radar (GPR) profiles has been collected adjacent to the cliff section in order to identify englacial structure.Two sub-horizontal reflectors have been imaged; the upper is interpreted as the glacier bed, and represents a transition between glacier ice and frozen subglacial sediments; while the lower is interpreted as a transition between frozen and unfrozen subglacial sediment. Dipping reflectors, corresponding to sediment-filled features on the cliff and glacier surface, do not cross the glacier bed. A small number of reflectors, interpreted as thrust faults, are visible below the bed reflector. A model is developed for structural development, which suggests that ice built up in a reservoir zone during quiescence. During the surge, ice propagated rapidly from this reservoir, creating a zone of compression which resulted in thrusting. Subsequently an extensional flow regime resulted in extensive fracture of the ice. We suggest dilated sediment was evacuated into these extensional crevasses from the glacier bed, accelerating surge termination.