We consider a model for random loops on graphs which is inspired by the Feynman–Kac formula for the grand canonical partition function of an ideal gas. We associate to this model a corresponding occupation field, which is a positive random field detailing the total time spent by loops at each vertex. We argue that well known critical phenomena for the ideal gas can be reinterpreted in terms of random variables of this occupation field. We also argue that higher order correlations, such as the existence of off-diagonal long-range order, can only be seen in the occupation field by studying a modified space–time model of loops. We provide an isomorphism theorem for this model to a complex Gaussian field, and derive a version of Symanzik’s formula which describes the ideal gas interacting with a random background environment. Finally we consider the effect of interactions on the gas, and present a large deviations analysis of the cycle distribution of the loop model under two mean field Hamiltonians.

This thesis is concerned with the classification and evaluation of various numerical schemes that are available for computing solutions for fluid-flow problems, and secondly, with the development of an improved numerical discretisation scheme of the finite-volume type for solving steady-state differential equations for recirculating flows with and without sources. In an effort to evaluate the performance of the various numerical schemes available, some standard test cases were used. The relative merits of the schemes were assessed by means of one-dimensional laminar flows and two-dimensional laminar and turbulent flows, with and without sources. Furthermore, Taylor series expansion analysis was also utilised to examine the limitations that were present. The outcome of this first part of the work was a set of conclusions, concerning the accuracy of the numerous schemes tests, vis-a-vis their stability, ease of implementation, and computational costs. It is hoped that these conclusions can be used by `computational fluid-dynamics' practitioners in deciding on an optimum choice of scheme for their particular problem. From the understanding gained during the first part of the study, and in an effort to combine the attributes of a successful discretisation scheme, eg positive coefficients. conservation and the elimination of 'false-diffusion', a new flow-oriented finite-volume numerical scheme was devised and applied to several test cases in order to evaluate its performance. The novel approach in formulating the new CUPID* scheme (for Corner UPw^nDing) underlines the idea of focussing attention at the control-volume corners rather than at the control-volume cell-faces. In two-dimensions, this leads to an eight neighbour influence for the central grid point value, depending on the flow-directions at the corners of the control-volume. In the formulation of the new scheme, false-diffusion is considered from a pragmatic perspective, with emphasis on physics rather than on strict mathematical considerations such as the order of discretisation, etc. The accuracy of the UPSTREAM scheme (for JJPwind in STREAMIines) indicates that although it is formally only first-order accurate, it considerably reduces 'false-diffusion'. Scalar transport calculations (without sources) show that the UPSTREAM scheme predicts bounded solutions which are more accurate than the upwind-difference scheme and the unbounded skew-upstream-difference scheme. Furthermore, for laminar and turbulent flow calculations, improved results are obtained when compared with the performances of the other schemes. The advantage of the UPSTREAM-difference scheme is that all the influence coefficients are always positive and thus the coefficient matrices are suitable for iterative solution procedures. Finally, the stability and convergence characteristics are similar to those of the upwind-difference scheme, eg converged solutions are guaranteed. What cannot be guaranteed, however, is the conservatism of the scheme and it is recommended that future work should be directed towards improving that disadvantage.

<p> In this work we investigate the global symmetries of six-dimensional superconformal field theories (6D SCFTs) via their description in F-theory. We provide computer algebra system routines determining global symmetry maxima for all known 6D SCFTs while tracking the singularity types of the associated elliptic fibrations. We tabulate these bounds for many CFTs including every 0-link based theory. The approach we take provides explicit tracking of geometric information which has remained implicit in the classifications of 6D SCFTs to date. We derive a variety of new geometric restrictions on collections of singularity collisions in elliptically fibered Calabi-Yau varieties and collect data from local model analyses of these collisions. The resulting restrictions are sufficient to match the known gauge enhancement structure constraints for all 6D SCFTs without appeal to anomaly cancellation and enable our global symmetry computations for F-theory SCFT models to proceed similarly. </p><p>

This thesis examines how periodic orbits may be used in acoustics to understand solutions of the Helmholtz equation. A review of the links between ray and wave mechanics is given including WKBJ (Wentzel, Kramers, Brillouin and Jeffreys) and EBK (Einstein, Brillouin, Keller) methods. It is also noted that some mode shapes in chaotic enclosures are scarred by the short periodic orbits. This motivates the proposal of the Mode Scar Hypothesis and the Mode Resonance Function Hypothesis. The trace formula, which is a sum over periodic orbits, approximates the level density for an acoustic enclosure. The trace formula in the concentric annulus domain is derived using a formulation for enclosures with continuous symmetry by Creagh and Littlejohn [1]. Results for the variance of the difference between the true and average mode counts are obtained. A technique called short periodic orbit theory (SPOT) for the approximation of mode shapes devised by Babiˇc and Buldyrev [2] and Vergini [3] is given. SPOT is extended to impedance boundary conditions. SPOT is implemented in the quarter stadium, quadrupole, circle and eccentric annulus enclosures with Dirichlet, Neumann and impedance boundary conditions. Concave enclosures with Dirichlet or Neumann boundary conditions were best approximated using SPOT. A design loop for enclosures is proposed using the periodic orbit ideas given. A model problem is used to provide insight into the effectiveness of these methods. It was found that it was not possible to breakdown all mode shapes in the eccentric annulus into contributions from short periodic orbits.

This thesis is devoted to the application of the Finite Difference Time Domain (FDTD) method for describing light interaction with liquid crystal media. This method has been known in the electromagnetic community since 1966, but has so far only found limited application in liquid crystal optics. The thesis consists of four parts. In the first part of the thesis I extend the conventional FDTD algorithm to be used for studying anisotropic media with continuous spatial variations of the dielectric properties. In the second part I discuss tests of the algorithm and software on various dielectric systems with known response. The third and the fourth parts of the thesis are concerned with the nonlinear interaction between light and liquid crystals. In the third part I develop a FDTD self-consistent algorithm that takes into account the coupling between light and liquid crystal orientation. The algorithm is used to simulate the optical Freedericksz transition in a homeotropic liquid crystal cell. I find that solving the problem self-consistently significantly modifies the main characteristics of the transition. In the fourth part I use the liquid crystal FDTD algorithm to investigate the birth of optical singularities. The results are compared to a recent analytical theory. I find that the analytic theory is only qualitatively useful except in the extreme short wave limit.

We consider strong wave propagation in the generalized compressible Euler equations. Our results include pairwise interactions of nonlinear waves, smooth wave propagation, formation of gradient blowup and several exact examples. In particular, we directly generalize P.Lax’s gradient blowup results for conservation laws with two variables to the generalized compressible Euler equations.

This thesis is concerned with the notions of complementarity and uncertainty encountered in quantum mechanics. Its starting point is an assessment of how these concepts have been represented and illustrated by various writers dating back to their inception. Following the survey a coherent account of the connections and contrasts between complementarity and uncertainty is developed in the context of Mach-Zehnder interferometry. The effect on the interference pattern contrast of path detection via entanglement with a probe system, is explored and a joint unsharp measurement scheme of the complementary pairs, path and interference, described. The Mach-Zehnder set-up proves sufficiently versatile to show that quantum erasure and quantitative quantum erasure constitute instances of joint unsharp measurement of complementary observables. The analysis uses the representation of observables as positive operator valued measures. Path detection and interference observation require different experimental set-ups but can be reconciled in the simultaneous unsharp measurement and preparation. This reconciliation is expressed as an uncertainty relation however the mutually exclusive feature of complementarity is not discarded. It is possible to recover strict complementarity as a limit case of the appropriate uncertainty relation. One motivation for this study is the effort some authors have made in trying to express the founding features of quantum mechanics in the form of a hierarchy of significance. Here it is shown that complementarity and uncertainty have separate identities but are not completely independent of each other. Consequently, establishing a hierarchy of these features within the present formalism of quantum mechanics is not possible.

Thesis (Ph. D.)--University of Canterbury, 2007. / Typescript (photocopy). Includes bibliographical references (p. 127-135). Also available via the World Wide Web.

No description available.

Solar sailing is the use of a thin and lightweight membrane to reflect sunlight and obtain a thrust force on the spacecraft. That is, a sailcraft has a potentially-infinite specific impulse and, therefore, it is an attractive solution to reach mission goals otherwise not achievable, or very expensive in terms of propellant consumption. The recent scientific interest in near-Earth asteroids (NEAs) and the classification of some of those as potentially hazardous asteroids (PHAs) for the Earth stimulated the interest in their exploration. Specifically, a multiple NEA rendezvous mission is attractive for solar-sail technology demonstration as well as for improving our knowledge about NEAs. A preliminary result in a recent study showed the possibility to rendezvous three NEAs in less than ten years. According to the NASA’s NEA database, more than 12,000 asteroids are orbiting around the Earth and more than 1,000 of them are classified as PHA. Therefore, the selection of the candidates for a multiple-rendezvous mission is firstly a combinatorial problem, with more than a trillion of possible combinations with permutations of only three objects. Moreover, for each sequence, an optimal control problem should be solved to find a feasible solar-sail trajectory. This is a mixed combinatorial/optimisation problem, notoriously complex to tackle all at once. Considering the technology constraints of the DLR/ESA Gossamer roadmap, this thesis focuses on developing a methodology for the preliminary design of a mission to visit a number of NEAs through solar sailing. This is divided into three sequential steps. First, two methods to obtain a fast and reliable trajectory model for solar sailing are studied. In particular, a shape-based approach is developed which is specific to solar-sail trajectories. As such, the shape of the trajectory that connects two points in space is designed and the control needed by the sailcraft to follow it is analytically retrieved. The second method exploits the homotopy and continuation theory to find solar-sail trajectories starting from classical low-thrust ones. Subsequently, an algorithm to search through the possible sequences of asteroids is developed. Because of the combinatorial characteristic of the problem and the tree nature of the search space, two criteria are used to reduce the computational effort needed: (a) a reduced database of asteroids is used which contains objects interesting for planetary defence and human spaceflight; and (b) a local pruning is carried out at each branch of the tree search to discard those target asteroids that are less likely to be reached by the sailcraft considered. To reduce further the computational effort needed in this step, the shape-based approach for solar sailing is used to generate preliminary trajectories within the tree search. Lastly, two algorithms are developed which numerically optimise the resulting trajectories with a refined model and ephemerides. These are designed to work with minimum input required by the user. The shape-based approach developed in the first stage is used as an initial-guess solution for the optimisation. This study provides a set of feasible mission scenarios for informing the stakeholders on future mission options. In fact, it is shown that a large number of five-NEA rendezvous missions are feasible in a ten-year launch window, if a solar sail is used. Moreover, this study shows that the mission-related technology readiness level for the available solar-sail technology is larger than it was previously thought and that such a mission can be performed with current or at least near-term solar sail technology. Numerical examples are presented which show the ability of a solar sail both to perform challenging multiple NEA rendezvous and to change the mission en-route.