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Motivated by the theory of Kleinian groups and by the theory of continued fractions, we study semigroups of Möbius transformations. Like Kleinian groups, semigroups have limit sets, and indeed each semigroup is equipped with two limit sets. We find that limit sets have an internal structure with features similar to the limit sets of Kleinian groups and the Julia sets of iterates of analytic functions. We introduce the notion of a semidiscrete semigroup, and find that this property is akin to the discreteness property for groups. We study semigroups of Möbius transformations that fix the unit disc, and lay the foundations of a theory for such semigroups. We consider the composition sequences generated by such semigroups, and show that every such composition sequence converges pointwise in the open unit disc to a constant function whenever the identity element does not lie in the closure of the semigroup. We establish various results that have counterparts in the theory of Fuchsian groups. For example we show that aside from a certain exceptional family, any finitely-generated semigroup S is semidiscrete precisely when every two-generator semigroup contained in S is semidiscrete. We show that the limit sets of a nonelementary finitely-generated semidiscrete semigroup are equal (and non-trivial) precisely when the semigroup is a group. We classify two-generator semidiscrete semigroups, and give the basis for an algorithm that decides whether any two-generator semigroup is semidiscrete. We go on to study finitely-generated semigroups of Möbius transformations that map the unit disc strictly within itself. Every composition sequence generated by such a semigroup converges pointwise in the open unit disc to a constant function. We give conditions that determine whether this convergence is uniform on the closed unit disc, and show that the cases where convergence is not uniform are very special indeed.

This thesis is concerned with the effect that boundary layer instabilities have on laminar-turbulent transition over an aircraft wing. The receptivity analysis of the two-dimensional marginal separation flow with respect to suction/blowing is considered. The solution of the linearised perturbation equation is sought in the form of perturbations that are periodic in time. Numerical results are obtained and it is observed that for large enough frequencies Tollmien-Schlicting wave packets begin to form downstream of a source of perturbations. A receptivity analysis of an incompressible steady two-dimensional marginally separated laminar boundary layer with respect to three-dimensional unsteady perturbations is then considered. An asymptotic theory of this flow is constructed on the basis of an analysis of the Navier-Stokes equations at large Reynolds numbers by means of matched asymptotic expansions. Two particular cases are considered, one in which we assume that the solution is periodic in both time $T$ and the spanwise coordinate $Z$, and the second where the solution is periodic in time only. As a result an integro-differential equation is derived and studied numerically by means of a spectral method. The next mathematical formulation is the study of marginal separation theory for a swept wing. In this formulation, an additional equation for the spanwise velocity component $w$ is needed. Upon solving the triple-deck equations, an algebraic Fourier transform equation is obtained and solved numerically. Finally, the last chapter concerns itself with a numerical global stability of the two-dimensional unsteady marginal separation equation.

This thesis is motivated by two associated obstacles we face for the solution and analysis of master equation models of gene transcription. First, the master equation – a differential-difference equation that describes the time evolution of the probability distribution of a discrete Markov process – is difficult to solve and few approaches for solution are known, particularly for non-stationary systems. Second, we lack a general framework for solving master equations that promotes explicit comprehension of how extrinsic processes and variation affect the system, and physical intuition for the solutions and their properties. We address the second obstacle by deriving the exact solution of the master equation under general time-dependent assumptions for transcription and degradation rates. With this analytical solution we obtain the general properties of a broad class of gene transcription models, within which solutions and properties of specific models may be placed and understood. Furthermore, there naturally emerges a decoupling of the discrete component of the solution, common to all transcription models of this kind, and the continuous, model-specific component that describes uncertainty of the parameters and extrinsic variation. Thus we also address the first obstacle, since to solve a model within this framework one needs only the probability density for the extrinsic component, which may be non-stationary. We detail its physical interpretations, and methods to calculate its probability density. Specific models are then addressed. In particular we solve for classes of multistate models, where the gene cycles stochastically between discrete states. We use the insights gained from these approaches to deduce properties of several other models. Finally, we introduce a quantitative characterisation of timescales for multistate models, to delineate “fast” and “slow” switching regimes. We have thus demonstrated the power of the obtained general solution for analytically predicting gene transcription in non-stationary conditions.

The celebrated Cwikel-Lieb-Rozenblum inequality gives an upper estimate for the number of negative eigenvalues of Schrödinger operators in dimension three and higher. The situation is much more difficult in the two dimensional case. There has been significant progress in obtaining upper estimates for the number of negative eigenvalues of two dimensional Schrödinger operators on the whole plane. In this thesis, we present upper estimates of the Cwikel-Lieb-Rozenblum type for the number of eigenvalues (counted with multiplicities) of two dimensional Schrödinger operators lying below the essential spectrum in terms of the norms of the potential. The problem is considered on the whole plane with different supports of the potential (in particular, sets of dimension 2 (0; 2)) and on a strip with various boundary conditions. In both cases, the estimates involve weighted L1 norms and Orlicz norms of the potential.

The long-term motion of the Earth's mantle is of considerable interest to geologists and geodynamists in explaining the evolution of the planet and its internal and surface history. The inaccessible nature of the mantle necessitates the use of computer simulations to further our understanding of the processes underlying the motion of tectonic plates. Numerical methods employed to solve the equations describing this motion lead to linear systems of a size which stretch the current capabilities of supercomputers to their limits. Progress towards the satisfactory simulation of this process is dependent upon the use of new mathematical and computational ideas in order to bring the largest problems within the reach of current computer architectures. In this thesis we present an implementation of the discontinuous Galerkin method, coupled to a more traditional finite element method, for the simulation of this system. We also present an a posteriori error estimate for the convection-diffusion equation without reaction, using an exponential fitting technique and artificial reaction to relax the restrictions upon the derivative of the convection field that are usually imposed within the existing literature. This error bound is used as the basis of an h-adaptive mesh refinement strategy. We present an implementation of the calculation of this bound alongside the simulation and the indicator, in a parallelised C++ code, suitable for use in a distributed computing setting. Finally, we present an implementation of the discontinuous Galerkin method into the community code ASPECT, along with an adaptivity indicator based upon the proven a posteriori error bound. We furnish both implementations with numerical examples to explore the applicability of these methods to a number of circumstances, with the aim of reducing the computational cost of large mantle convection simulations.

This thesis presents and develops two tools which can be used to work with lower bounds of operators. One tool in working with lower bounds is invertible extensions. They allow one to turn a lower bound of an operator into the norm of an inverse operator. Some results giving extensions are known for single operators and certain other semigroup representations. Chapter 3 includes some positive new results for operators on Hilbert space and also certain unbounded operators. An example shows that a uniform lower bound for the powers of an operator does not give an extension with power bounded inverse. Variations are given for generators of C0-semigroups, and for operators on Hilbert space. A result by Read, which gives an extension with minimal spectrum raises the question how the lower bounds of the original operator are related to the resolvent bounds of the extension. Another tool which is developed in this thesis is a reduction using semi- norms. A seminorm can place a different emphasis on elements and even neglect some. In this way, we can shape a Banach space to attain properties that we impose. This idea is used to define maximal parts in Chapter 4. They are identified in the context of contractivity and expansiveness of a bounded operator, and in the context of dissipativity and accretivity for certain unbounded operators. Applications are an improvement of a theorem by Batty and Tomilov which characterises embeddings into hyperbolic C0-semigroups, and a generalisation of a theorem by Goldberg and Smith leading to a characterisation of generators of C0-semigroups which have an extending group with bounded inverses.

The main results of my work contribute to the mathematical study of a stability mechanism common to both the Vlasov–Poisson equation and the Kuramoto equation. These kinetic models come from very different areas of physics: the Vlasov–Poisson equation models plasmas and the Kuramoto equation models synchronisation behaviour. The stability was first described by Landau in 1946 and is a subtle behaviour, because the damping only happens in a suitably weak sense. In fact, the models are not dissipative and cannot be stable in a strong topology. Instead, the so-called Landau damping happens through phase mixing. My contributions include a simplified linear analysis for the Vlasov–Poisson equation around the spatially homogeneous state. For the Kuramoto equation, I cover the linear analysis around general stationary states and show nonlinear stability results with algebraic and exponential decay. Moreover, I show how the mean-field estimate by Dobrushin can be improved around the incoherent state. In addition, I study how a kinetic system can reach a thermal equilibrium. This is modelled by adding a dissipative term, which by itself drives the system to a local equilibrium. In hypocoercivity theory, the complementary effect of the transport operator is used to show exponential decay to a global equilibrium. In particular, I show how a probabilistic treatment can complement the standard hypocoercivity theory, which constructs equivalent norms, and I discuss the necessity of the geometric control condition for the spatially degenerate kinetic Fokker–Planck equation. Finally, I study the possible discretisation of the velocity variable for kinetic equations. For the numerical stability, Hermite functions are a suitable choice, because their differentiation matrix is skew-symmetric. However, so far a fast expansion algorithm has been lacking and this is addressed in this work.