Highlights d There is extreme variation in the probability of food finding in vertebrate species d Decreasing probability increases inter-individual variation in foraging duration d This variability can result in individual or breeding ruin d Apex predators are most likely to incur ruin under changing environmental conditions
For fast-flying birds, the ability to respond to wind during landing is critical, as errors can lead to injury or even death. Nonetheless, landing ability, and its ecological significance, remain unstudied. We show that for auks, 60% of attempts to land at their cliff nests fail in a strong breeze (80% in near-gale winds). This is most likely because wind interferes with the ability to maintain flight control in the last phase of landing. Their extreme flight costs mean that the energetic penalty for multiple landing attempts is high. We propose that exposure, and ability to respond to, such conditions will influence the suitability of breeding habitat. In support of this (i) auk colonies appear to be orientated away from prevailing winds and (ii) landing success within colonies is higher on crowded ledges with more airspace for manoeuvring. More generally, the interplay between wind and flight capacities could impact breeding distributions across species and scales.
We consider the Bohr correspondence limit of the Schrödinger wave function for an atomic elliptic state. We analyze this limit in the context of Nelson's stochastic mechanics, exposing an underlying deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This solves the long standing problem of obtaining Kepler's laws of planetary motion in a quantum mechanical setting. In this quantum mechanical setting, local mild instabilities occur in the Keplerian orbit for eccentricities greater than 1 ͱ 2 which do not occur classically.
We summarise a selection of results on the inviscid limit of the stochastic Burgers equation emphasising geometric properties of the caustic, Maxwell set and Hamilton-Jacobi level surfaces and relating these results to a discussion of stochastic turbulence. We show that for small viscosities there exists a vortex filament structure near to the Maxwell set. We discuss how this vorticity is directly related to the adhesion model for the evolution of the early universe and include new explicit formulas for the distribution of mass within the shock.
The correspondence limit of the atomic elliptic state in three dimensions is discussed in terms of Nelson’s stochastic mechanics. In previous work we have shown that this approach leads to a limiting Nelson diffusion, and here we discuss in detail the invariant measure for this process and show that it is concentrated on the Kepler ellipse in the plane z=0. We then show that the limiting Nelson diffusion generator has a spectral gap; thereby proving that in the infinite time limit the density for the limiting Nelson diffusion will converge to its invariant measure. We also include a summary of the Cheeger and Poincaré inequalities, both of which are used in our proof of the existence of the spectral gap.
Education in the UK is failing to provide the increases in the numbers of school-leavers with science and mathematics qualifications required by industry, business and the research community to assure the UK's future economic competitiveness" (The Royal Society, 2008, p17). Furthermore, the proportion of students in Wales following mathematics courses post 16 is lower than in England (GSR, 2014). In particular, although the situation has improved, fewer students in Wales choose to study for the Further Mathematics (FM) A-level. This paper explores the reasons behind student choices in studying mathematics between the ages of 16 and 18, with a particular focus on the FM A-level, in order to make recommendations about how to increase participation in FM. Phase One of the study used a questionnaire to access the opinions of students studying mathematically based courses in sixth forms and colleges to explore the reasons behind their choices and the factors influencing their progression in mathematics. In Phase Two, small focus groups of students in selected schools and colleges were interviewed to enrich the questionnaire data and provide further insight into their decisions. The study identified a lack of information from peers, siblings, parents and teachers about FM as a factor restricting choice. Current models of delivery contribute to the false perception that the FM A-level is harder than the Mathematics A-level and only suitable for the most talented mathematicians. We suggest: developing teachers' knowledge and skills so that whenever possible students can be offered FM as a fully timetabled subject; promoting FM to parents; and establishing student champions to encourage participation.
We consider a proto-ring nebula of a gas giant such as Neptune as a cloud of gas/dust particles whose behaviour is governed by the stochastic mechanics associated to the Kepler problem. This leads to a stochastic Burgers-Zeldovich type model for the formation of planetesimals involving a stochastic Burgers equation with vorticity which could help to explain the turbulent behaviour observed in ring systems. The Burgers fluid density and the distribution of the mass M(T) of a spherical planetesimal of radius δ are computed for times T. For circular orbits, sufficient conditions on certain time averages of δ2 are given ensuring that VarM(T) ∼ 0 as T ∼ ∞. Some applications are given to the satellites of Jupiter and Saturn, in particular giving a possible explanation of the equal mass families of satellites.
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