Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

Multistabilities and symmetry-broken one-color and two-color states in closely coupled single-mode lasers

We theoretically investigate the dynamics of two mutually coupled, identical single-mode semi-conductor lasers. For small separation and large coupling between the lasers, symmetry-broken one-color states are shown to be stable. In this case the light outputs of the lasers have significantly different intensities while at the same time the lasers are locked to a single common frequency. For intermediate coupling we observe stable symmetry-broken two-color states, where both lasers lase simultaneously at two optical frequencies which are separated by up to 150 GHz. Using a five-dimensional model, we identify the bifurcation structure which is responsible for the appearance of symmetric and symmetry-broken one-color and two-color states. Several of these states give rise to multistabilities and therefore allow for the design of all-optical memory elements on the basis of two coupled single-mode lasers. The switching performance of selected designs of optical memory elements is studied numerically.

Studies in asymmetric and heterocyclic synthesis: I. chiral ketones II. Quinolones III. Trifluoromethylated pyrones

This thesis is split into three sections based on three different areas of research. In the first section, investigations into the α-alkylation of ketones using a novel chiral auxiliary is reported. This chiral auxiliary was synthesised containing a pyrrolidine ring in the chiral arm and was applied in the preparation of α-alkylated ketones which were obtained in up to 92% ee and up to 63% yield over two steps. Both 3-pentanone and propiophenone based ketones were used in the investigation with a variety of both alkyl and benzyl based electrophiles. The novel chiral auxiliary was also successful when applied to Michael and aldol reactions. A diamine precursor en route to the chiral auxiliary was also applied as an organocatalyst in a Michael reaction, with the product obtained in excellent enantioselectivity. In the second section, investigations into potential anti-quorum sensing molecules are reported. The bacteria Pseudomonas aeruginosa is an antibiotic-resistant pathogen that demonstrates cooperative behaviours and communicates using small chemical molecules in a process termed quorum sensing. A variety of C-3 analogues of the quorum sensing molecules used by P. aeruginosa were synthesised. Expanding upon previous research within the group, investigations were carried out into alternative protecting group strategies of 2-heptyl-4-(1H)- quinolone with the aim of improving the yields of products of cross-coupling reactions. In the third section, investigations into fluorination and trifluoromethylation of 2-pyrones, pyridones and quinolones is reported. The incorporation of a fluorine atom or a trifluoromethyl group into a molecule is important in pharmaceutical drug discovery programmes as it can lead to increased lipophilicity and bioavailability, however late-stage incorporation is rarely reported. Both direct fluorination and trifluoromethylation were attempted. Eight trifluoromethylated 2-pyrones, five trifluoromethylated 2-pyridones and a trifluoromethylated 2-quinolone were obtained in a late-stage synthesis from their respective iodinated precursors using methyl fluorosulfonyldifluoroacetate as a trifluoromethylating reagent.

The development of novel chiral tethers for controlling intramolecular aryl-aryl coupling and their application in the synthesis of gomisin M1 and analogues

The primary focus of this thesis was the development of a novel chiral tether that could be used to control axial chirality around a newly formed aryl-aryl bond, and the extension of this methodology to the model synthesis of gomisin M1. In chapter 1, a review detailing the use of chiral tethers in the synthesis of atropisomers is discussed. The use of a variety of chiral molecules including 1,2-diols, 1,3-diols and other diol-based tethers, as well as amine-based and miscellaneous tethers are detailed. In chapter 2, the rationale behind the design of our novel molecular tethers, along with the subsequent synthesis of three chiral 1,3-diol-based tethers, is outlined. The method by which the enantiopurity of these diols was determined is also reviewed. This chapter also includes the attempted Mitsunobu and intramolecular couplings in the model synthesis of BINOL. Chapter 3 discusses the synthesis of suitable aryl halide substrates, and their employment in the attempted tether-controlled asymmetric model synthesis of gomisin M1. A comprehensive investigation into the attempted intramolecular biaryl coupling of these tethered substrates is also included. The non-stereoselective model synthesis of gomisin M1 is outlined in chapter 4. The installation of the desired biaryl linkage and the subsequent attempted intramolecular McMurry couplings are discussed. The impact of different protecting groups in the molecule on the intramolecular McMurry reaction is also outlined. Chapter 5 details the full experimental procedures, including spectroscopic and analytical data for the compounds prepared during this research.