We study elliptic diff erential and pseudo-differential operators with periodic coefficients. For a wide class of such operators we prove the Bethe-Sommerfeld conjecture, i.e. that the spectrum can have only finitely many gaps. We also study the integrated density of states of periodic Schroedinger operators and prove a lower bound for its variance in the high energy regime. This results in the lower bound for the non-integrated density of states of such operators.

Steady, two-dimensional flows of a single layer of inviscid fluid discharging through an aperture are treated in the hodograph or velocity plane on a rectangular grid. The following problems are considered individually: irrotational planar and axisymmetric flow of air through a nozzle, incompressible flow through an aperture with bottom topography and lastly rotating flow of a uniform potential vorticity fluid issuing from a passage on the wall. The rotating case differs from other cases in that three parameters are required to describe the solutions instead of two. In all cases the problems are formulated so that flows range from subcritical to supercritical including choked flow. The rectangular domain for the supercritical problems results from the way the information travels in the hodograph plane in the region that is image of the flow that occurs around the lip of the nozzle wall. Supercritical jets are solved up to a short distance away from the aperture, hence shocks that occur further downstream are avoided although limiting lines develop in the vicinity of the exit plane depending on the strength of the topography. The equations governing irrotational flows are expressed in terms of the Legendre potential and those for the rotating flow are expressed in terms of the streamfunction. Solutions of these equations are computed using standard finite-differences approximations. Knowledge of the characteristics directions and the corresponding compatibility equations in the supercritical region of the domain is not required which demonstrates the robustness of solving in the hodograph plane. All that is necessary is that the general direction in which information propagates is perceived so that explicit or implicit finite-differences can be employed.

The motion of a free boundary separating two immiscible fluids in an unbounded Hele-Shaw cell is considered. In the one-phase problem, a viscous fluid is separated from an inviscid fluid by a simple closed boundary. Preliminaries for a complex variable technique are presented by which the one-phase problem can be solved explicitly via conformal mappings. The Schwarz function of the boundary plays a major role giving rise to the so called Schwarz function equation which governs the evolution of exact solutions. The Schwarz function approach is used to study the stability of a translating elliptical bubble due to a uniform background flow, and the stability of a blob (or bubble) subject to an external electric field. The one-phase problem of a translating free boundary and of a free boundary subject to an external field are studied numerically. A boundary integral method is formulated in the complex plane by considering the Cauchy integral formula and the complex velocity of a fluid particle on the free boundary. In the case of a free boundary subject to an external electric field due to a point charge, it is demonstrated that a stable steady state is achieved for appropriate charge strength. The method is also employed to study breakup of a single translating bubble in which the Schwarz function singularities (shown to be stationary) of the initial boundary play an important role. The two-phase problem is also considered, where the free boundary now separates two viscous fluids, and the construction of exact solutions is studied. The one-phase numerical model is enhanced, where a boundary integral method is formulated to accommodate the variable pressure in both viscous phases. Some numerical experiments are presented with a comparison to analytical results, in particular for the case where the free boundary is driven by a uniform background flow.

Interactive dynamical ocean and atmosphere models are commonly used for predictions on seasonal timescales, but initialisation of such systems is problematic. In this thesis, idealised coupled models of the El Ni~no Southern Oscillation phenomenon are used to explore potential new initialisation methods. The basic ENSO model is derived using the two-strip concept for tropical ocean dynamics, together with a simple empirical atmosphere. A hierarchy of models is built, beginning with a basic recharge oscillator type model and culminating in a general n-box model. Each model is treated as a dynamical system. An important step is the 10-box model, in which the seasonal cycle is introduced as an extension of the phase space by two dimensions, which paves the way for more complex and occasionally chaotic behaviour. For the simplest 2-box model, analytic approximate solutions are described and used to investigate the parameter dependence of regimes of behaviour. Model space is explored statistically and parametric instability is found for the 10-box and upward versions: while it is by no means a perfect simulation of the real world phenomena, some regimes are found which have features similar to those observed. Initialisation is performed on a system from the n-box model (with n = 94), using dimensional reduction via two separate methods: a linear singular value decomposition approach and a nonlinear slow manifold (approximate inertial manifold) type reduction. The influence of the initialisation methods on predictive skill is tested using a perfect model approach. Data from a model integration are treated as observation, which are perturbed randomly on large and small spatial scales, and used as initial states for both reduced and full model forecasts. Integration of the reduced models provides a continuous initialisation process, ensuring orbits remain close to the attractor for the duration of the forecasts. From sets of ensemble forecasts, statistical measures of skill are calculated. Results are found to depend on the dimensionality of the reduced models and the type of initial perturbations used, and model reduction is found to result in a slight improvement in skill from the full model in each case, as well as a significant increase in the maximum timestep.

Let X be the scheme P1Qp \ {0,1, ∞} We can assign a fundamental group to each rational basepoint on this scheme. These groups are non-canonically isomorphic, so they need not have isomorphic Galois actions. We study a description of this map from points to groups with Galois action, in terms of non-abelian cohomology. Using this description, we see that the fundamental groups associated to different basepoints are not isomorphic.

Flows in flexible tubes and vessels have been studied extensively in the past with particular application to the cardiovascular and respiratory systems. However there have been few treatments of the lower urinary tract, which consists of the bladder and urethra. This thesis concentrates specifically on the urethra with the aim of giving insight into the evolving flow characteristics within the vessel and mechanical responses of the vessel which give rise to fluid structure interactions. Urethral modelling is an important area of research given the social and economic costs involved in lower urinary tract dysfunction. In the modelling, examination is given to slow and fast opening vessels where certain exact analytical solutions are found along with numerical results. Following this, fast and slow responses of the walls of the vessels are considered, where the response is defined as the relative change in cross-sectional area for relatively varying transmural pressure. These features are important for pathologies that alter the characteristics of the vessel wall such as bladder outlet obstruction. A change in the distensibility along the vessel resulting from pathologies or normal transition through the various sections of the urethra is studied both in terms of developing jump conditions based on a localised Euler region and also over a comparatively short length scale giving rise to the Burgers equation; small amplitude instabilities are studied through the derivation of the KdV equation. Following on from these mostly two-dimensional treatments, three-dimensional systems are then studied. Consideration is given to the secondary flow effects driven by the tortuosity of a vessel in three dimensions. We study cases of three-dimensional constriction, with main interest in the effects of benign prostate hyperplasia or urethral stricture on the flow, where pressure drops are demonstrated. Finally an appendix deals with the effects concerned with a wide population, focusing on an allied problem of consumer choice.

The aim of this thesis is to contribute to the understanding of genuine cusp forms on the group SL2=Q(i), from a computational point of view. We show, via the generalised Eichler-Shimura-Harder isomorphism, that a genuine cusp form of cohomological type exists at level SL2(Z[i]; 4)SL2(Z). We show, by calculating cohomology groups, that such a form exists at weight (2; 2). Finally, we compute the genuine quotient of the Hecke algebra acting on representations of SL2(Q2(i)) containing non-zero SL2(Z2[i]; 4)SL2(Z2)- xed vectors. When such a representation $ corresponds to an unrami ed representation of SL2(Q2(i)), we show that the space of SL2(Z2[i]; 4)SL2(Z2)- xed vectors in $ is a sum of two 1-dimensional components. We determine which 1-dimensional representations arise in this way.

This thesis is centred around computations in the derived module category of finitely generated lattices over the integral group ring of a finite group G. Building upon the representability of the cohomology functor in the derived module category in dimensions greater than 0, we give a new characterisation of the cohomology of lattices in terms of their G-invariants, only having the syzygies of the trivial lattice to keep track of dimension. With the example of the dihedral group of order 6 we show that this characterisation significantly simplifies computations in cohomology. In particular, we determine the Bieberbach groups, that is, the fundamental groups of compact at Riemannian manifolds, with dihedral holonomy group of order 6. Furthermore, we give an interpretation of the cup product in the derived module category and show that it arises naturally as the composition of morphisms. Inspired by the graded-commutativity of the cup product in singular cohomology we give a sufficient condition for the cohomology ring of a lattice to be graded-commutative in dimensions greater than 0.

The thesis is concerned with the modelling of urinary motion during bladder collapse and is mathematically based. The bladder model as a collapsing vessel is developed as a step towards complementing use of nomograms. Urine motion inside is taken as unsteady flow of incompressible fluid, while the dimensions and collapse rate of the vessel are prescribed using data which is close to realistic biological values. Evolutions of velocities, volume ow rates and effects of the collapse rate are examined. An introduction is made which includes lower urinary tract urodynamics as well as the unique feature that the bladder changes shape and size substantially compared with other vessels. An investigation of simple two-dimensional shapes takes place in chapter two, along with limiting cases for thin vessels. Chapter three analyses simple axisymmetric shapes especially the sphere because of its relevance in addition to its fundamental nature. Development of a numerical scheme is addressed in the next chapter to tackle more complex shapes through the boundary element method and an iterative finite difference scheme with emphasis on flexibility of approach. An extension to more advanced structures of the vessel is constructed in chapter five by means of a concise boundary condition and shape definition. Chapter six takes the work a step further as the approach is applied to axisymmetric configurations. While in an appendix, an extension to implement full viscous effects is then inspected. Finally, further work is discussed in the conclusion.

This thesis is based on five articles, four of which have been published in the Journal of Mathematical Physics, Physical Review D, Modern Physics Letters A and Journal of High Energy Physics. The fifth has been submitted to Mathematika. In these works we study several distinct problems within the broad subject area of Mathematical Physics. The common feature is that all these works deal with rotations of one form or another. In particular, we show an equivalence between the massless and massive Dirac equations and models based on the concept of rotating material points. We also solve an open problem in Einstein-Cartan theory, namely, we find a natural matter source for a non-trivial spin angular momentum tensor. Finally, we construct a complete class of non-standard (non-local) spinor field theories and examine their possible applications in Cosmology.