The main result gives sufficient conditions for the existence of a rational point on certain intersections of two quadrics in P<SUP>4</SUP>, assuming two major hypotheses. These assumptions are the finiteness of the Tate-Shafarevitch group III of an elliptic curve over Q and the Schinzel Hypothesis. The intersection of the respective variety with a hyperplane in general position is a curve <I>D</I> of genus 1. We choose these varieties in such a fashion that the Jacobian of thus curve has exactly one rational 2-torsion point. In this situation, the 2-Selmer group of the Jacobian must have at least four elements, one of which is represented by the curve <I>D</I> on the variety. If it has exactly four, then either <I>D</I> has a rational point or it is one of 2 elements of the 2-primary component of III. By a theorem of Cassels, this is impossible if we assume III to be finite. The Schinzel Hypothesis is needed to carry out the explicit calculations to derive conditions for the 2-Selmer group to have exactly four elements. The fourth chapter supplies the details to a sketch of Y.I. Manin. It derives an algorithm to decide whether an arbitrary elliptic curve <I>E </I>over Q has a rational point from the conjecture of Birch and Swinnerton-Dyer. It is shown that this assumption implies the existence of an upper bound on the height of a rational point on <I>E. </I>Since it is easily seen that there are only finitely many points with bounded height, this reduces the decision procedure to a finite computation.

The main objective is to provide a wide range of approaches that yield useful understanding for the different scenarios of waves in locally stratified media. A classification system is constructed, where the physical problems are put into categories with common mathematical models and problems. This provides a convenient tool for identifying which methods and approaches to employ for a particular physical problem. The bulk of the analysis is centred around two methods or theories - multiple scale homogenisation and generalised ray theory. Multiple scale homogenisation is relevant for long-wave propagation and finding the effective medium of the layered structure. The Backus average for finely layered media is derived in a new way, where multiple scale techniques and the propagator matrix formulation are combined to form a general framework for most linear waves in locally stratified media. With this framework, one can easily derive known formulae for effective media, including the anisotropic effective mass density for acoustic waves. Since the original Backus average was derived for systems close to the static limit, and the anisotropic mass density is a low frequency dynamic effect, acoustic waves are analysed in more detail. The technique is also applied to more complicated media, and it is shown how effective stiffness tensor can be found for piezoelectric media. However, the key feature of this framework is that higher order terms are easily calculated. Nevertheless, the higher order terms are sensitive to the distance propagated, and the effective medium solution is only valid for short distances of propagation. This is in agreement with previous simulations. The validity of Backus averaging or accuracy of effective medium theory is investigated for the one dimensional case. Using Magnus series and manipulation of exponential operators, a new relationship between the effective medium and exact solutions are derived. Based on this relation, the same observations are made as in previous numerical work. Bounds for the factor between the effective and exact solutions are also calculated.

There are two principle problems to address. Firstly, there is a greater degree of freedom in model approximation within the context on an inner-loop, as something less than closed-loop stability on C⁺ is required. The standard notions which under-pin the <i>H</i>_∞-loopshaping framework, such as coprime factor and graph representations, readily generalise to connected subsets of C. However, to assess the fitness of any candidate approximation regarding inner-loop design we require a generalisation of the <i>v</i>-gap metric. This is obtained by removing the artificial dependence upon the existence of normalised coprime factorisations and working instead with a pointwise normalisation on the boundary. Secondly, we desire a means of synthesising a controller with a nested structure, that preserves the robustness guarantees associated with the conventional case. On the occasion that the inner-loop is implemented in observer form, we obtain a simple inequality relating the stability margins associated with a 2-loop nested feedback system. The guarantee this provides for the sequential approach to design is too weak to be of use. Two new methods are therefore developed which exploit the inequality to achieve a robustness guarantee. To complete this work, the generalised <i>v</i>-gap and a rigorous approach to nested feedback system synthesis are both applied to the design of a full flight control system for the USAF Have Dash II air-air missile.

This thesis is in two parts. The first part is about a bifurcation of a type of invariant set called a Cantorus. Cantori are most familiar as the Denjoy counter-example in the theory of circle homeomorphisms, but this example is not very smooth. More representative, smooth examples of Cantori occur in non-invertible circle mappings and in area preserving twist maps. In these systems, as the parameters are varied, Cantori can be created or destroyed or change their structure. In other words, they undergo bifurcation. In the first chapter I study a particular bifurcation which destroys a Cantorus in a family of bi-modal circle maps. I show that this bifurcation takes place in a way that is reminiscent of a saddle-node bifurcation destroying a pair of fixed points. The second part of this thesis describes attempts to construct physically realistic examples of uniformly hyperbolic systems. The theory of uniform hyperbolicity is one of the great results from nonlinear dynamics. It provides many powerful tools for analysing dynamical systems, and even though people now realise that a lot of interesting systems are not uniformly hyperbolic, the study of these systems consists, to a large extent, of taking the ideas and methods developed for uniformly hyperbolic systems and then pushing them a little bit further. There are, of course, many standard examples of uniformly hyperbolic systems. The Smale Horseshoe, hyperbolic toral automorphisms and geodesic flows on surfaces of negative curvature to name but a few. All of these examples are, however, rather mathematical and abstract. It is all very well having a nice theory, but is it something that a Physicist could go and measure in a lab? I wanted to demonstrate the physical relevance of uniform hyperbolicity by finding a system that was as "real" as possible and which was provably uniformly hyperbolic. In fact I found two such systems. In Chapter 2 I give the construction of a time periodic flow in the plane which has a global attractor that is a uniformly hyperbolic Plykin attractor. The techniques used here could be applied more generally to construct other folding and stretching flows.

In this thesis we will present a comprehensive set of formalisms for comparing, evolving, and constraining primordial non-Gaussian models through the CMB bispectrum. First, we introduce the idea of a shape function for characterising the primordial non-Gaussianity. The shape function can also be used to construct a correlator between the models which we use to group the space of possible models into four main classes: equilateral, squeezed, flattened, and scale dependent. Next, we use a common property of the shape function to create a method for calculating, without approximation, the CMB bispectrum from a general primordial model. There are two techniques we use to speed up the calculation. The first is to use the flat sky approximation for large <i>l</i>, and the second is to exploit the smoothness of the reduced bispectrum to calculate the bispectrum first on a sparse grid then interpolate to obtain the remaining points. We then discuss methods for calculating estimators by decomposing the bispectrum, either today or at primordial times, into the product of eigenmodes. First we deal with the primordial bispectrum and describe how the decomposition can be used to both constrain primordial models and to estimate the primordial bispectrum from observations. Then we repeat the analysis for the CMB bispectrum and describe how this process can be used to constrain models, but this time allowing for the inclusion of late time effects. It also presents a method for generating maps with an arbitrary bispectrum and power spectrum.

This dissertation attempts a resolution of the apparent tension between objectivity and conventionality in modern physics. It is argued that the physical sciences, though dependent on convention, may nevertheless produce objective representations of reality. In demonstrating this, a view of representation is introduced which explicitly includes the human subjects between whom representation actually takes place; this view is termed, 'representation as performance.' These human subjects are, as ever, sources of subjective ambiguity in representation. Nevertheless, representation may still be substantially objective. It has been suggested that objectivity may be conceived in terms of group theoretical invariance. Rejecting Hermann Weyl's well-known proposal along these lines, a new notion, 'objectivity of alignment,' is introduced which rehabilitates the notion of objectivity as related to invariance. Even within representations which are objective in this sense, however, remaining ambiguities present themselves which must be resolved through various kinds of conventional choice. Two case studies illustrate this argument, relating to the 'twin paradox' of special relativity and the localization of a single particle in relativistic quantum mechanics. It is shown that objective representations of the twin paradox rely on the invariant, 'proper time,' but that even such objective representations contain certain conventional ambiguities. To show this, a novel scheme for unifying the various versions of the twin paradox is presented and elucidated. In the case of localizing a quantum state, it is shown that the choice of how to recover objectivity is also one of convention, made on the basis of a number of criteria. In the process it is also shown that Hegerfeldt nonlocality can be understood as arising from what is termed the 'Jericho effect;' in addition, a direct evaluation of the Hegerfeldt integral is provided. Thus in both case studies objectivity and conventionality need not be held permanently in opposition.

The majority of this work is concerned with the <i>local</i>-linear stability of the incompressible boundary-layer flows over rotating spheres and rotating cones; convective and absolute instabilities are investigated and the effects of viscosity and streamline-curvature are included in each analysis. Preliminary investigations into the linear <i>global</i>-mode behaviour of the rotating-disk, rotating-cone and rotating-sphere boundary layers are also presented. The local rotating-sphere analyses are conducted at various latitudes from the axis of rotation (<i>q</i>), and the local rotating-cone analyses are conducted at points along cones of various half-angles (<i>ψ</i>), in each case convective and absolute instabilities are found within specific parameter spaces. The predictions of the Reynolds number, vortex angle and vortex speed at the onset of convective instability are consistent with existing experimental measurements for both boundary-layer types. Axial flow is found to stabilize each boundary layer with respect to convective and absolute instabilities. The global behaviour of the boundary-layer flows over rotating disks, cones and spheres is considered by taking into account the slowly varying basic state along each body surface. The locations of saddle points in the absolute frequency are determined which give the leading-order estimate of the global frequency. For the rotating disk and rotating cones, the global frequency indicates the disturbances in the boundary-layer flow are globally damped; and for rotating spheres, the global frequency indicates the boundary layer may support neutrally stable global modes when a region of absolute instability exists.

No description available.

Numerical general relativity has typically been used in studying 1+1 dimensional spherically symmetric spacetimes, and more recently, 3+1 dimensional spacetimes without any symmetries. In this thesis, an intermediate case - 2+1+1-dimensional axisymmetric spacetimes - are studied. In the first part, the equations of general relativity are developed for a general axisymmetric spacetime. This uses the Geroch reduction, and this is generalised to find evolution equations for the matter terms. The conditions for regularity on the symmetry axis are derived for several types of tensor, and this is used to help define appropriate variables for numerical evolution. The characteristic structure of both the geometry and the matter evolution systems is given. Next, the numerical methods used to solve the equations are described. The elliptic constraint equations are solved using multigrid, and the hyperbolic evolution equations are evolved using High Resolution Shock Capturing Methods, using WENO-3 and the Maquina flux solver. The iterative Crank Nicholson method is also described as an alternative method of evolving the geometry equations. The third part of the thesis considers a 1+1-dimensional plane-symmetric spacetime, with perfect fluid matter. Several versions of the equations, with different characteristic structures, are derived. A linearised version of the equations shows that discontinuities may be present in the second derivatives of metric terms, and this is backed up by a variety of numerical results evolved using the non-linear equations. These numerical results also show that the numerical methods used to evolve the geometry equations must be able to deal with discontinuities if the expected order of accuracy is to be maintained. Finally, results from the axisymmetric code are presented. In vacuum, Brill waves are briefly studied, as well as some test problems. With a perfect fluid, the standard shocktube problem is used to test the code, and perturbed, rotating neutron stars are studied.

First-principles calculations based on density-functional theory are important for the study of a wide range of systems. However, conventional density-functional-theory calculations are very expensive since the computational effort and memory requirement scale as the cube and square of the system size, respectively. Recently there has been a surge of activity to investigate linear-scaling methods (where the computational effort and memory requirement scale linearly with the system size), all of which use localised basis sets in their implementations. One localised basis set proposed for linear-scaling methods, the spherical-wave basis set, is interesting because while sharing some of the properties (such as the concept of energy cutoff) with the popular extended plane-wave basis set, it possesses other advantages such as each basis function being fully localised within a sphere. Even though this basis set has been used to implement a linear-scaling method which has been tested on bulk crystalline silicon, the basis set itself has not yet been studied. This dissertation investigates the accuracy and properties of this localised basis set using an iterative matrix diagonalisation approach, which frees us from having to consider other sources of error introduced in linear-scaling methods. The matrix diagonalisation approach requires an efficient solution of the generalised eigenvalue problems <I>Hx = ^ΕSx</I>. I have proposed a new and efficient iterative conjugate-gradient method to obtain the lowest few eigenvalues and corresponding eigenvectors of the generalised eigenvalue problem. This method exhibits linear convergence. A preconditioning scheme which uses the kinetic energy matrix is introduced to improve the convergence of the solutions. The scheme is controlled by a single parameter whose optimal value may be chosen automatically.