Efficient and rigorous source finding techniques are needed for the upcoming large data sets from telescopes like MeerKAT, LSST and the SKA. Most of the current source-finding algorithms lack full statistical rigor. Typically these algorithms use some form of thresholding to find sources, which leads to contamination and missed sources. Ideally we would like to use all the available information when performing source detection, including any prior knowledge we may have. Bayesian statistics is the obvious approach as it allows precise statistical interrogations of the data and the inclusion of all available information. In this thesis, we implement nested sampling and Monte Carlo Markov Chain (MCMC) techniques to develop a new Bayesian source finding technique called FireFly. FireFly employs a technique of switching ‘on’ and ‘off’ sources during sampling to deal with the fact that we don’t know how many true sources are present. It therefore tackles one of the critical questions in source finding, which is estimating the number of real sources in the image. We compare FireFly against a Bayesian evidence-based search method and show on simulated astronomical images that FireFly outperforms the evidence-based approach. We further investigate two implementations of FireFly: the first with nested sampling and the second with MCMC. Our results show that MCMC FireFly has better computational scaling than the nested sampling version FireFly but the nested sampling version of FireFly appears to perform somewhat better than MCMC FireFly. Future work should examine how best to quantify FireFly performance and extend the formalism developed here to deal with multiwavelength data.

Beyond gravitational evidence for dark matter, a set of search techniques are employed in the present thesis within the particle dark matter paradigm. Under the possibility of dark matter annihilating into particles of the Standard Model of Particle Physics, we study the products of annihilation with cosmic-ray detectors, such as AMS, Fermi-LAT and PAMELA, and radio telescopes, such as the SKA. In this work, we focus on the positron fraction measured in the Solar System due to dark matter annihilating in the dark matter galactic halo, but also on radio signals from the Milky Way and dwarf spheroidal galaxies. Our main purpose is to constrain the dark matter parameter space under the light of the latest experimental data for cosmic-rays and the new sensitivities reached in radio astronomy. Furthermore, we discuss some of the most promising locations and synchrotron frequencies to search for dark matter with masses around the TeV scale. The analysis presented in this thesis lies in setting constraints on modelindependent dark matter. However, some specific dark matter candidates in the context of extra-dimensional theories are considered as well. Indeed, brane fluctuations, dubbed branons, are new degrees of freedom appearing in flexible brane-world models. These new fields behave as standard weakly interacting massive particles with a significant associated thermal relic density and would explain dark matter observational features.

This thesis lies in the field of constant mean curvature (cmc) hypersurfaces and specifically cmc 1/2 surfaces in the three-manifold H 2 × R. The value 1/2 is the critical mean curvature for H 2 × R, in that there do no exist closed cmc surfaces with mean curvature 1/2 or less. Daniel and Hauswirth have constructed a one-parameter family of complete, cmc 1/2 annuli that are symmetric about a reflection in the horizontal place H 2 × {0}, the horizontal catenoids. In this thesis we prove that these catenoids converge to a singular limit of two tangent horocylinders as the neck size tends to zero. We discuss the analytic gluing construction that this fact suggests, which would create a multitude of cmc 1/2 surfaces with positive genus. The main result of the thesis concerns a key step in such an analytic gluing construction. We construct families of cmc 1/2 annuli with boundary, whose single end is asymptotic to an end of a horizontal catenoid. We produce these families by solving the mean curvature equation for normal graphs off the end of a horizontal catenoid. This is a non-linear boundary value problem, which we solve by perturbative methods. To do so we analyse the linearised mean curvature operator, known as the Jacobi operator. We show that on carefully chosen weighted H¨older spaces the Jacobi operator can be inverted, modulo a finite-dimensional subspace, and provided the neck size of the horizontal catenoid is sufficiently small. Using these linear results we solve the boundary value problem for the mean curvature equation by a contraction mapping argument.

The accelerated expansion of spacetime intuitively points to the existence of new, unknown energy fields pervading the universe, but it is has also spurred the growth of the research field of modified gravity theories. Of these, f(R) theories of gravity is the first and simplest modification to General Relativity, and have been studied extensively for their astrophysical and cosmological predictions. Power law f(R) modifications have been shown to exhibit desirable characteristics, producing the late time accelerated expansion as well as satisfying local tests of gravity. However, there is wide degeneracy among models in this class, and they are known to suffer from cosmological instabilities, which could lead to curvature singularities at finite times. This thesis addresses questions directly relating to model degeneracy and sudden singularities. Cosmologies and cosmological perturbations, resulting from a general broken power law modification to GR are generated, studied and evolved. Simulations are performed using 1+3 space time decomposition of the field equations and a dynamical systems approach to f(R) cosmology. The parameter space of this model, which includes the HuSawicki [6], Starobinsky [96] and Miranda [7] f(R) forms as subclasses, is investigated. It is found that there are regions in the parameter space which are completely singular and bound by continuous curves. We also investigate regions of the parameter space in which the attractive nature of gravity is preserved, and find that these regions intersect. The results of a Markov Chain Monte Carlo analysis significantly narrowed the viable region of the exponent parameter space of the general power law f(R) model. Current cosmological distance data; SNIa (Union 2), BAO (6dFGS, BOSS, SDSS, WiggleZ) as well as the LRG power spectrum (SDSS DR9), were used to obtain these constraints. The best fits are compared with the ΛCDM model, and leads to the conclusion that this class is still a candidate for the gravitational interaction.

Yes / Purpose To assess the impact of a six-week ‘Maths on the Move’ (MOTM) physically active learning programme on primary school children’s physical activity (PA) levels and maths performance. Method Randomised control trial. Year 5 children’s PA was assessed using accelerometry for five consecutive school days at baseline and during the final intervention week (final sample: n=97, age M=9.61±0.29, 52.6% female). Two maths performance tests were used; one assessing mathematical content taught during MOTM and one assessing math’s fluency (MASSAT). Both tests were conducted at baseline and following the intervention (week seven). Focus groups were conducted in week seven with intervention children (n=12), randomly choosing an even split of children classified with pre-intervention low or high PA levels. Results On average, during a typical 45-49 minutes MOTM lesson, children obtained an additional five minutes of moderate-to-vigorous PA and 5.7 minutes of light PA counteracted by a reduction of 9.5 minutes of time spent sedentary compared to children that remained in the classroom (control condition). The maths attainment test performance significantly improved over time for children in the MOTM compared to the control (+6.1 versus +0.9, p≤0.0001, d=1.507). No significant improvements were found in the MASSAT total score. Seven emerging themes were derived from the child focus groups. Children felt the MOTM sessions resulted in social and environmental improvements, which improved learning during the sessions. Children described the MOTM sessions as enjoyable, fun, engaging and invigorating – resulting in positive associations to learning and activity. Conclusion Collectively, the findings identify the MOTM programme improves pupil’s PA levels, academic outcomes and identifies pupil’s willingness, enjoyment, and engagement.

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This study investigated whether pupils hold personal beliefs and attitudes which could affect their performance in mathematics lessons in such a way as to either facilitate or impede learning. There were four parts to the study which took place over three years. In the first part, personal constructs about all school subjects were elicited from a group of pupils in their first year of comprehensive school. The interviews were recorded and provided background data for the study. One year later, the same pupils were asked to rate eighteen mathematics topics on the constructs of like/dislike; easy/difficult and useful/not useful. The interviews were again recorded and used to develop categories of pupil beliefs. These were used to develop a number of questions which were later put to the same group. Six weeks later the pupils divided into groups of three and took part in videorecorded problem solving sessions. This provided triangulated observational and oral data to corroborate or refute data from other parts of the study. Finally, approximately one year later, each pupil was asked the questions developed from the second interview categories. These were posed in an open ended form and were also used to develop belief categories. These final categories provided the information on which to compare the beliefs of the study group pupils. The basis for comparison was the pupils' mathematical setting and their positions in yearly examinations. Data from across the study were used to provide case studies of three pupils. The main conclusions were that beliefs and attitudes do affect mathematics performance, but that the effect was not the same for high and low settings; that problem solving ability correlated poorly with setting, and that for individuals it was necessary to examine a constellation of beliefs rather than any single ones.

This thesis discusses the effect of an obstruction on the evolution of the Rayleigh-Taylor instability in a confined geometry at low Atwood numbers. Laboratory experiments are the principal method of investigation, though these data are supplemented with implicit large eddy simulations (ILES). The laboratory data are captured using an innovative laser scanning system which is able to simultaneously record density and velocity data in 3D. A new approach for calculating density data from laser induced fluorescence measurements is developed and demonstrated. The technique is used to improve the accuracy of the density measurement from laser induced fluorescence, by correcting for the damage to dye caused by the laser. The introduction of an obstacle at the height of the initial interface results in dramatic changes to the dynamics of mixing, even when this obstacle is only a few percent of the domain width. Two obstructed scenarios are considered. In both of these an obstruction is placed on the interface between an upper heavy layer and lower light layer. In the first case, a single horizontal opening connects the upper and lower layers. A bidirectional flow exchanges fluid through the opening, establishing a circulation cell in each layer. These cells exist quasi-steadily for long periods, constantly recirculating and mixing the fluid in each layer. This acts to increase the time required for mixing compared with the classical unobstructed case, but results in a more uniformly mixed final stratification. The second case has two horizontal openings, one either side of the obstruction. This results in markedly different dynamics. The flow through each of the openings switches back and forth between being bidirectional (as with the single opening case) and unidirectional, with unidirectional exchange reversing direction with a constant period. These results are consistent with the ILES data. For both of these cases a wide range of analytical techniques are used to connect the new obstructed dynamics with previously conducted research, such as calculating the molecular mixing fraction, energetics and mixing efficiency. A multistage mixing process is identified, unique for cases with an obstruction. For the single opening case a hierarchy of models are developed that accurately capture the density change of each layer for both the experimental and numerical data. The effect of changing the aspect ratio of the domain is investigated using ILES, from which different dynamical regimes are observed, discussed and analysed.