This thesis collates, extends and applies the abstract theory of W*-bundles. Highlights include the standard form for W*-bundles, a bicommutant theorem for W*-bundles, and an investigation of completions, ideals, and quotients of W*-bundles. The Triviality Problem, whether all W*-bundles with fibres isomorphic to the hyperfinite II_1 factor are trivial, is central to this thesis. Ozawa's Triviality Theorem is presented, and property gamma and the McDuff property for W*-bundles are investigated thoroughly. Ozawa's Triviality Theorem is applied to some new examples such as the strict closures of Villadsen algebras and non-trivial C(X)-algebras. The solution to the Triviality Problem in the locally trivial case, obtained by myself and Pennig, is included. A theory of sub-W*-bundles is developed along the lines of Jones' subfactor theory. A sub-W*-bundle encapsulates a tracially continuous family of subfactors in a single object. The basic construction and the Jones tower are generalised to this new setting and the first examples of sub-W*-bundles are constructed.

In the first part of the thesis, the solvability of stochastic differential equations with reflecting boundary conditions is studied. Such equations arise in singular stochastic control problems as a way for determining the optimal strategies. The stochastic differential equations represent homogeneous one-dimensional diffusions while the boundaries are given by c`adl`ag functions. Pathwise solutions are constructed under mild assumptions on the coefficients of the equations. In particular, the solutions are derived as the diffusions’ scale functions composed with appropriately time-changed reflected Brownian motions. Several probabilistic properties are addressed and analysed. In the second part of the thesis, piecewise diffusion approximations of Levy processes are studied. Such approximating processes have been called Itˆo semi-diffusions. While keeping the statistical fit to Levy processes, this class of processes has the analytical tractability of Ito diffusions. At a given time grid, their distribution is the same as the one of the underlying Levy processes. At times outside the grid, they evolve like homogeneous diffusions. The analysis identifies conditions under which Itˆo semi-diffusions can be used as an alternative to Levy processes for modelling financial asset prices. In particular, for a sequence of Itˆo semi-diffusions determined by a given Levy process, conditions for the convergence of their finite-dimensional distributions to the ones of the Levy process are established. Furthermore, for a single Ito semi-diffusion, conditions for the existence of pricing measures are established.

Space weather refers to electromagnetic disturbances in the near-Earth environment as a result of the Sun-Earth interaction. Severe space weather events such as magnetic storms can cause disruption to a wide range of technologies and infrastructure, including communications systems, electronic circuits and power grids. Because of its high potential impact, space weather has been included in the UK National Risk Register since 2011. Space weather monitoring and early magnetic storm detection can be used to mitigate risk in sensitive technological systems. The aim of this project is to investigate the electromagnetic disturbances in the near-Earth environment through developing statistical models that quantifies the variations and uncertainties in the near-Earth magnetic field. Data of the near-Earth magnetic field arise from in-situ satellite measurements and computer model outputs. The Cluster II mission (Escoubet et al., 2001a) has four satellites that provide in-situ measurements of the near-Earth magnetic field at time-varying locations along their trajectories. The computer model consists of an internal part that calculates the magnetic field sourced from Earth itself and an external part that estimates the magnetic field resulting from the Sun-Earth interaction. These magnetic fields, termed as the internal field and the external field, add up to the total magnetic field. Numerical estimates of the internal field and the external field are obtained respectively from the IGRF-11 model (Finlay et al., 2010) and the Tysganenko-96 (T96) model (Tsyganenko, 2013) given the times and the locations as inputs. The IGRF model outputs are invariant to space weather conditions whereas the T96 model outputs change with the input space weather parameters. The time-varying space weather parameters for T96 model include the solar wind ram pressure, the y and the z-components of the interplanetary magnetic field, and the disturbance storm time index. These parameters are the estimated time series of the solar wind conditions at the magnetopause, i.e. the boundary of the magnetosphere on the day-side, and the disturbance level at the Earth’s surface. Real-time values of the T96 model input parameters are available at hourly resolution from https://omniweb.gsfc.nasa.gov/. The overall aim of the thesis is to build spatio-temporal models that can be used to understand uncertainties and constraints leveraged from 3D mathematical models of space weather events. These spatio-temporal models can be then used to help understand the design parameters that need to be varied in building a precise and reliable sensor network. Chapter 1 provides an introduction to space weather in terms of the near-Earth magnetic field environment. Beginning with an overview of the near-Earth magnetic field environment, Chapter 2 describes the sources for generating in-situ satellite measurements and computer model outputs, namely the Cluster II mission, the IGRF model, and the T96 model. The process of sampling the magnetic field data from the different data sources and the space-time dependence in the hourly sampled magnetic field data are also included in this Chapter. Converting the space-time structure in the magnetic field data into a time series structure with a function relating the position in space to time, Chapter 3 explores the temporal variations in the sampled in-situ satellite measurements. Through a hierarchical approach, the satellite measurements are related to the computer model outputs. This chapter proposes statistical methods for dealing with the non-stationary features, temporal autocorrelation, and volatility present in the time series data. With the aim of better characterising the electromagnetic environment around the Earth, Chapter 4 develops time-series models of the near-Earth magnetic field utilising in-situ (CLUSTER) magnetic field data. Regression models linking the CLUSTER satellite observations and two physical models of the magnetic field (T96 and IGRF) are fit to each orbit in the period 2003-2013. The time series of model parameter estimates are then analysed to examine any long term patterns, variations and associations to storm indices. In addition to explaining how the two physical models calibrate with the observed satellite measurements, these statistical models capture the inherent volatility in the magnetic field, and allow us to identify other factors associated with the magnetic field variation, such as the relative position of each satellite relative to the Earth and the Sun. Mixed-effect models that include these factors are constructed for parameters estimated from the regression models for evaluating the performance of the two computer models. Following the calibration of the computer models against the satellite measurements, Chapter 5 investigates how these computer models allow us to investigate the association between the variations in near-Earth magnetic field and storms. To identify the signatures of storm onsets in different locations in the magnetosphere, change-point detection methods are considered for time series magnetic field signals generated from the computer models along various feasible satellite orbits. The detection results inform on potential sampling strategies of the near-Earth magnetic field to be predictive of storms through selecting achievable satellite orbits for placing satellite sensors and detecting changes in the time series magnetic signals. Chapter 6 provides of a summary of the main finding within this thesis, identifies some limitations of the work carried out in the main chapters, and include a discussion of future research. An Appendix provides details of coordinate transformation for converting the time and position dependent magnetic field data into an appropriate coordinate system.

This thesis studies the subgroup structure of mapping class groups. We use techniques that fall into two categories: analysing the group action on a family of simplicial complexes, and investigating regular, finite-sheeted covering spaces. We use the first approach to prove that a wide class of normal subgroups of mapping class groups of punctured surfaces are geometric, that is, they have the extended mapping class group as their group of automorphisms, expanding on work of BrendleMargalit. For example, we determine that every member of the Johnson filtration is geometric. By considering punctured spheres, we also establish the automorphism groups of many normal subgroups of the braid group. The second approach is to relate subgroups of each of the mapping class groups associated to a covering space, namely, the liftable and symmetric mapping class groups. Given that the two surfaces have boundary, we consider covers in which either every mapping class lifts or every mapping class is fibre-preserving. We classify all covers that fall into one of these cases. In Chapter 1 we recall some preliminaries before stating the main results of the thesis. We then extend Brendle-Margalit's definition of complexes of regions to surfaces with punctures. Chapter 2 proves that the automorphism group of a complex of regions is the extended mapping class group, resolving in part a metaconjecture of N. V. Ivanov. In Chapter 3 we construct a complex of regions associated to a general normal subgroup of a mapping class group of a surface with punctures. We then apply the main result of the previous chapter to establish that such a normal subgroup is geometric. Finally, Chapter 4 presents joint work with Tyrone Ghaswala. We give a proof of the Birman-Hilden Theorem for surfaces with boundary and then prove the classifications of regular, finite-sheeted covering spaces of surfaces with boundary discussed above. We conclude by investigating an infinite family of branched covers of the disc. This family induces embeddings of the braid group into mapping class groups. We prove that each of these embeddings maps a standard generator of the braid group to a product of Dehn twists about curves forming a chain, providing an answer to a question of Wajnryb.

The thesis is aimed at asymptotic analysis of the near-surface boundary layers in non-locally elastic solids. The dynamic response of a homogeneous half-space with a traction-free surface is analysed for a nonlocal exponential kernel. A typical wave-length is assumed to be much greater than the length scale associated with internal properties of the elastic medium. The dominant effect of the boundary layer is revealed. The leading order long-wave approximations are shown to coincide with the `local' problem for a half-space having a vertical inhomogeneity localised near the sur-face. An explicit correction to the classical boundary conditions on the surface of a `locally' elastic half-space is obtained by asymptotic analysis of the near-surface behaviour. The order of the derived correction exceeds that of the well-known correction to the governing differential equations of Eringen's model, e.g., see [44]. The obtained refined boundary conditions enable evaluating the interior stress-strain solution outside a narrow boundary layer localised near the surface. As examples, the effect of nonlocal elastic phenomena on the Rayleigh wave speed and also a plane strain problem of a moving load on the surface of a half-space are studied. In addition, a thin layer with a traction-free upper face, subjected to prescribed displacements along its lower face, is investigated. Further, the 3D dynamic equations in nonlocal elasticity for a thin plate are considered, assuming the plate thickness to be much greater than a typical microscale size. The long-wave low-frequency approximations are obtained for both plate bending and extension. Boundary layers characteristic of nonlocal behaviour are revealed near the plate faces. It is established that taking into account the effect of the boundary layers results in first-order corrections to the bending and extensional stiffness in the classical 2D plate theory.

The analysis of sample surveys is one of the key areas in official statistics. An integral part of analysing sample data is choosing appropriate weights for each sample member. These weights can informally be thought of as the number of population members each person in the sample represents. Calibration is a method that adjusts the weights assigned to sample members in order to satisfy (or approximately satisfy) some pre-determined constraints. These are typically based on Census data or other large surveys. The key idea is that estimates formed from the weighted sample should replicate the known values from other sources. This thesis begins with the mathematical formulation of the calibration problem as an optimization problem. Whilst the calibration problem has been defined in existing calibration literature, it has not been clearly formulated as a problem in optimization. New calibration functions are also presented, and an outline of their benefits compared to existing calibration functions given. Much of the calibration literature focuses on so-called hard calibration. This requires an exact matching between the weighted sample data and the pre-determined constraints. However, relaxing this condition can often lead to more “well-behaved” solutions. This is the idea behind soft calibration, which has received less attention in existing literature. In this thesis, soft calibration is formulated using an optimization framework, and also presented as a diagnostic tool for identifying problematic constraints. For many practitioners, the variance (and mean square error) of the estimates obtained is of particular interest. This is the motivation for a new approach to calibration that seeks to directly minimize the mean square error of the calibration estimator. This method is compared with existing calibration techniques, and future research directions for this approach are considered.

This thesis concerns the capacitated arc routing problem (CARP), which can be used as a model of various real-life scenarios such as rubbish collection, snow ploughing, and other situations where an emphasis is placed on providing a certain service along streets. The goal of the CARP is to find a minimum-cost set of routes such that (i) each route starts and ends at the depot, (ii) each task is serviced in one of the routes, and (iii) the total demand in each route does not exceed the capacity. Until recently, the study of the CARP is concentrated on its "static" version, that is, it is assumed that the problem remains unchanged after vehicles start their journeys. However, with today's communication technology, a route planner and drivers can communicate with each other in real time, hence the possibility of amending vehicle routes if deemed necessary or appropriate for changes that may occur in the problem. This motivates the study of a dynamic CARP. This thesis focusses on one type of change in the dynamic CARP, namely the appearance of new tasks. To ensure that a service can be performed smoothly, the ability to update a solution quickly is often preferable to achieving optimality with an excessive amount of computational effort. For this reason, we opt to develop a dynamic CARP solver based on heuristic algorithms. An investigation is conducted to gain more insights about what makes an algorithm improve a solution quickly. Furthermore, factors in the dynamic CARP beyond a solution-seeking algorithm are investigated. This includes the frequency of updating the solution and the idea of instructing vehicles to wait for additional tasks at certain locations. Efforts are focussed on reducing the total distance at the end of the service while ensuring that the service completion time is not excessive.

We say that two sets $A$ and $B$, each of cardinality $m$, form an $m+m$ \emph{sum-and-distance system} $\{A,B\}$ if the sum-and-distance set $A^*B$ comprised of all the absolute values of the sums and distances $a_i\pm b_j$ contains either the consecutive odd integers $\{1,3,5,\ldots 4m^2-1\}$ or with the inclusion of the set elements themselves, the consecutive integers $\{1,2,3,\ldots,2m(m+1)\}$ (an inclusive sum-and-distance system). Sum-and-distance systems can be thought of as a discrete analogue of the union of a Minkowski sum system with a Minkowski difference system. We show that they occur naturally within a traditional reversible square matrix, where conjugation with a specific orthogonal symmetric involution, always reveals a sum-and-distance system within the block structure of the conjugated matrix. Moreover, we show that the block representation is an algebra isomorphism. Building upon results of Ollerenshaw, and Br\'ee, for a fixed dimension $n$, we establish a bijection between the set of sum-and-distance systems and the set of traditional principal reversible square matrices of size $n\times n$. Using the $j$th non-trivial divisor function $c_j (n)$, which counts the total number of proper ordered factorisations of the integer $n= p_1^\ldots p_t^$ into $j$ parts, we prove that the total number of $n+n$ principal reversible square matrices, and so sum-and-distance systems, $N_n$, is given by \[ N_n = \sum_^ \left( c_j(n)^2 +c_(n)c_j(n) \right)=\sum_^ c_j^(n) c_j^(n). \] \[=\sum_^ \left(\sum^j_(-1)^ \prod_^t \right ) \left ( \sum^j_(-1)^ \prod_^t \right), \] where $\Omega(n)=a_1 + a_2 + \ldots + a_t$ is the total number of prime factors (including repeats) of $n$. Further relations between the divisor functions and their Dirichlet series are deduced, as well as a construction algorithm for all sum-and-distance systems of either type. Superalgebra structures relating to the matrix symmetry properties are identified, including those for the reversible and most-perfect square matrices of those considered by Ollerenshaw and Br\'ee. For certain symmetry types, links between the block representation constructed from a sum-and-distance system, and quadratic forms are also established.

There are currently four major formal foundational systems for intuitionistic analysis: LS, CS (both in Troelstra 1977), FIM (Kleene and Vesley 1965) and the derivable FIRM-INT (Moschovakis 2016). All of these systems rely on different universes of choice sequences and different conceptions of what a choice sequence is. There is a strong common ground between these systems as they use the same very restrictive notion of finite information when dealing with these choice sequences { the notion of restricting ourselves to initial segments. This text extends the notion of a choice sequence given in Fletcher (1998) and uses it to construct a generalised system capable of expressing results about intensional properties of choice sequences. This is achieved by constructing a language capable of representing intensional first order restrictions on choice sequences (the language of knowledge states) and their relations to other sequences. This extended system allows us to formulate a notion of lawlessness that evades a series of paradoxes highlighted in Fletcher (1998), allows us to prove a generalised form of open data and offers additional clarity to other key areas of the theory. When a certain set of restrictions are applied to this extended theory (extensionality and a second order restriction on knowledge states) we obtain a system suitable for the foundation of analysis.

If G is a none trivial group and t is an element distinct from G then r(t) =g_1 t^l_1 .........g_kt^l_k= 1 , k ≥1, g_i in G\{1}, l_i in Z\{0} is said to be an equation over G. There has been much research aimed to investigate solvability of such equations over groups and these researches adopted two main approaches. The first considers the properties of G. The other direction which we are following here is to put restrictions on r(t). The results obtained in this direction was of length restriction at first. More resent more results have been obtained and the concept of isolated t-exponent has been introduced and used to study a generalized form of equations of unlimited length. In this study we investigate r(t) which has the generalized form w_1t^l_1w_2t^l_2w_3t^l_3w_4t^l_4 where w_i = g_{i,1} t^m_{i,1} .....t^m_{i,k_i-1}g_{i,k_i}. In Chapter 1 we introduce the concept of equations over groups and we give some of the known results and the geometric method of proof is explained. We also state our main theorem and the main lemma which will be proved in the following chapters and some technical lemmas are proved. In Chapter 2, Chapter 3, and Chapter 4 the Cases I, II, III are discussed and the distribution is shown. In Chapter 5 the main lemma is proved and the proof of the main theorem is completed.