In the Skyrme model, nuclei are described as topological solitons known as Skyrmions. To make contact with nuclear data one must quantise these Skyrmions; most calculations to date have used rigid body quantisation, where the Skyrmions are allowed to rotate but remain rigid. The method reproduces some experimental results for light nuclei but there are some contradictions with data. In this thesis we study a more sophisticated quantisation scheme where the Skyrmions may deform, called vibrational quantisation, in the hope of fixing some of these problems. Vibrational quantisation is applied to the dodecahedral $B=7$ Skyrmion, which models Lithium-$7$. Using rigid body quantisation, the Skyrme model predicts a spin $\frac{7}{2}$ ground state while in reality the Lithium-$7$ nucleus ground state has spin $\frac{3}{2}$. We show that a quantisation which includes a $5$-dimensional vibrational manifold of deformed Skyrme configurations remedies this problem, giving the correct ground state spin. Further, the model leads to a robust prediction that the ground state of the nucleus has a larger root mean square matter radius than the second quantum state, in contrast with standard nuclear models. We consider the vibrational modes of the tetrahedral $B=16$ Skyrmion, to describe Oxygen-$16$. Motivated by Skyrme dynamics, a special $2$-dimensional submanifold of configurations is constructed. We study the manifold in detail by modelling it as a $6$-punctured sphere with constant negative curvature. The Schr\"odinger equation is solved on the sphere and the results give an excellent fit to the experimental energy spectrum. The model describes an energy splitting between certain states with equal spins but opposite parities, which is hard to explain in other models. We also find the first ever isospin $0$, spin-parity $0^-$ state in the Skyrme model. A method to calculate electromagnetic transition rates between states is formulated and then applied to our system. By considering a special type of Skyrme configuration, where a single Skyrmion orbits a large core, we show that the Skyrme model can reproduce a classical spin-orbit force due to the structure of the Skyrme fields. We quantise this model to try and find out if the classical picture holds quantum mechanically.

The transform-quantisation stage is one of the most complex operations in the state-of-the-art High Efficiency Video Coding (HEVC) standard, accounting for 11–41% share of the encoding complexity. This study aims to reduce its complexity, making it suitable for dedicated hardware accelerated architectures. Adopted methods include multiplier-free approach, Multiple-Constant Multiplication architectural designs, and exploiting useful properties of the well-known Discrete Cosine Transform. Besides, an approximation scheme was introduced to represent the original HEVC transform and quantisation matrix elements with more hardware-friendly integers. Out of several derived approximation alternatives, an approximated transform matrix (T16) and its downscaled version (ST16) were further evaluated. An approximated quantisation multipliers matrix (Q) and its combination with one transform matrix (ST16 + Q) were also assessed in HEVC reference software, HM-13.0, using test video sequences of High Definition (HD) quality or higher. Their hardware architectures were designed in IEEE-VHDL language targeting a Xilinx Virtex-6 Field Programmable Gate Array technology to estimate resource savings over original HEVC transform and quantisation. T16, ST16, Q, and ST16 + Q approximated transform or/and quantisation matrices provided average Bjøntegaard-Delta bitrate differences of 1.7%, 1.7%, 0.0%, and 1.7%, respectively, in entertainment scenario and 0.7%, 0.7%, -0.1%, and 0.7%, respectively, in interactive scenario against HEVC. Conversely, around 16.9%, 20.8%, 21.2%, and 25.9% hardware savings, respectively, were attained in the number of Virtex-6 slices compared with HEVC transform or/and quantisation. The developed architecture designs achieved a 200 MHz operating frequency, enabling them to support the encoding of Quad Full HD (3840 × 2160) videos at 60 frames per second. Comparing T16 and ST16 with similar designs in the literature yields better hardware efficiency measures (0.0687 and 0.0721, respectively, in mega sample/second/slice). The presented approximated transform and quantisation matrices may be applicable in a complexity-reduced HEVC encoding on hardware platforms with non-detrimental coding performance degradations.

The Skyrme model enables us to approximate nuclei via topological solitons known as Skyrmions. The B = 4 Skyrmion is of particular importance as its symmetry and stability means that multiple B = 4 Skyrmions can combine with each other to form larger B = 4N Skyrmions. In this thesis we investigate the properties of these B = 4N Skyrmions and compare them with results found in the wider nuclear physics community. We go beyond rigid body quantization and develop a formalism of using vibrational quantisation to generate the energy spectrum of the Oxygen−16 nucleus. The Oxygen−16 nucleus is treated as an arrangement of four B = 4 Skyrmions, whose dynamics enable us to create a 2−dimensional manifold of B = 16 configurations. We solve the Schrödinger equation on this manifold and discover new states previously not found in the B = 16 sector of the Skyrme model. We compare these states with those found experimentally and find that there is a excellent it to the energy spectrum. In order to apply vibrational quantization to a wider range of nuclei we create a novel approximation for Skyrmions and the interactions between them. By generating Skyrmions with Gaussian sources we find analytic expressions for the pion fields and interaction energies of Skyrmions, with particular focus on the B = 1 and B = 4 Skyrmions, and show how this could be applied to vibrational quantization and the clustering of B = 4 Skyrmions. B = 4N nuclei are the only nuclei with zero spin and isospin, which means that their electric charge density is proportional to their baryon density. This simplification makes these nuclei particularly susceptible to investigation via electron scattering. We develop a classical averaging method to calculate the Patterson function and the form factor for a B = 4N nucleus and make comparisons with experimental data. We also discover a way of using the baryon density directly to approximate the locations of zeroes and stationary points of the form factor.

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Many real-world optimisation problems are discrete in nature. Although recent rapid developments in computer technologies are steadily increasing the speed of computations, the size of an instance of a hard discrete optimisation problem solvable in prescribed time does not increase linearly with the computer speed. This calls for the development of new solution methodologies for solving larger instances in shorter time. Furthermore, large instances of discrete optimisation problems are normally impossible to solve to optimality within a reasonable computational time/space and can only be tackled with a heuristic approach. In this thesis the development of so called matheuristics, the heuristics which are based on the mathematical formulation of the problem, is studied and employed within the variable neighbourhood search framework. Some new variants of the variable neighbourhood searchmetaheuristic itself are suggested, which naturally emerge from exploiting the information from the mathematical programming formulation of the problem. However, those variants may also be applied to problems described by the combinatorial formulation. A unifying perspective on modern advances in local search-based metaheuristics, a so called hyper-reactive approach, is also proposed. Two NP-hard discrete optimisation problems are considered: 0-1 mixed integer programming and clustering with application to colour image quantisation. Several new heuristics for 0-1 mixed integer programming problem are developed, based on the principle of variable neighbourhood search. One set of proposed heuristics consists of improvement heuristics, which attempt to find high-quality near-optimal solutions starting from a given feasible solution. Another set consists of constructive heuristics, which attempt to find initial feasible solutions for 0-1 mixed integer programs. Finally, some variable neighbourhood search based clustering techniques are applied for solving the colour image quantisation problem. All new methods presented are compared to other algorithms recommended in literature and a comprehensive performance analysis is provided. Computational results show that the methods proposed either outperform the existing state-of-the-art methods for the problems observed, or provide comparable results. The theory and algorithms presented in this thesis indicate that hybridisation of the CPLEX MIP solver and the VNS metaheuristic can be very effective for solving large instances of the 0-1 mixed integer programming problem. More generally, the results presented in this thesis suggest that hybridisation of exact (commercial) integer programming solvers and some metaheuristic methods is of high interest and such combinations deserve further practical and theoretical investigation. Results also show that VNS can be successfully applied to solving a colour image quantisation problem.

Speech signal processing for voice communications can be characterised in terms of silence compression, noise reduction, and speech compression. The limit in the channel bandwidth of voice communication systems requires efficient compression of speech and silence signals while retaining the voice quality. Silence compression by means of both voice activity detection (VAD) and comfort noise generation could present transparent speech-quality while substantially lowering the transmission bit-rate, since pause regions between talk spurts do not include any voice information. Thus, this thesis proposes smoothed likelihood ratio-based VAD, designed on the basis of a behavioural analysis and improvement of a statistical model-based voice activity detector. Input speech could exhibit noisy signals, which could make the voice communication fatiguing and less intelligible. This task can be alleviated by noise reduction as a preprocessor for speech coding. Noise characteristics in speech enhancement are adapted typically during the pause regions classified by a voice activity detector. However, VAD errors could lead to over- or under- estimation of the noise statistics. Thus, this thesis proposes mixed decision-based noise adaptation based on a integration of soft and hard decision-based methods, defined with the speech presence uncertainty and VAD result, respectively. At low bit-rate speech coding, the sinusoidal model has been widely applied because of its good nature exploiting the phase redundancy of speech signals. Its performance, however, can be severely smeared by mis-estimation of the pitch. Thus, this thesis proposes a robust pitch estimation technique based on the autocorrelation of spectral amplitudes. Another important parameter in sinusoidal speech coding is the spectral magnitude of the LP-residual signal. It is, however, not easy to directly quantise the magnitudes because the dimensions of the spectral vectors are variable from frame to frame depending on the pitch. To alleviate this problem, this thesis proposes mel-scale-based dimension conversion, which converts the spectral vectors to a fixed dimension based on mel-scale warping. A predictive coding scheme is also employed in order to exploit the inter-frame redundancy between the spectral vectors. Experimental results show that each proposed technique is suitable for enhancing speech quality for voice communications. Furthermore, an improved speech coder incorporating the proposed techniques is developed. The vocoder gives speech quality comparable to TIA/EIA IS-127 for noisy speech whilst operating at lower than half the bit-rate of the reference coder. Key words: voice activity detection, speech enhancement, pitch, spectral magnitude quantisation, low bit-rate coding.

On construit de nouvelles connexions quantiques intégrables dans fibrés vectoriels au-dessus d'espaces de modules de surfaces de Riemann et de leurs généralisations sauvages, en utilisant deux approches différentes. Premièrement, on utilise la quantification par déformation pour construire de nouvelles connexions intégrables à partir d'Hamiltoniennes d'isomonodromie irrégulières, dans l'esprit de Reshetikhin de la dérivation de la connexion de Knizhnik-Zamolodchikov à partir des Hamiltoniennes de Schlesinger. Deuxièmement, on construit une version complexe de la connexion de Hitchin pour la quantification géométrique de l'espace de modules de Hitchin sur une surface de genre un, par rapport au groupe SL(2,C) et à des polarisations Kähleriennes, en complémentant l'approche par polarisations réelles de Witten. Finalement, on utilise la transformée de Bargmann pour dériver une formule pour la connexion de Hitchin-Witten dans le fibré vectoriel des sections holomorphes, et pour transformer l'action de Hitchin en une transformée sur l'espace de Segal--Bargmann, basée sur les états cohérents. / We construct new flat quantum connections on vector bundles over moduli spaces of Riemann surfaces and their wild generalisations, using two different approaches. Firstly, we use deformation quantisation to construct new flat connections from irregular isomonodromy Hamiltonians, in the spirit of Reshetikhin's derivation of the Knizhnik-Zamolodchikov connection from the Schlesinger Hamiltonians. Secondly, we construct a complex version of the Hitchin connection for the geometric quantisation of the Hitchin moduli space over a surface of genus one, with respect to the group SL(2,C) and to Kähler polarisations, complementing Witten's real polarisation approach. Finally, we use the Bargmann transform to derive a formula for the connection of Hitchin-Witten on the vector bundle of holomorphic sections, and to turn Hitchin's action into a transform on the Segal--Bargmann space, which relies on coherent states.

Thesis (PhD)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: Please refer to full text for abstract. / AFRIKAANSE OPSOMMING: Sien asb volteks vir opsomming.

In this work we review the basic principles of the theory of the relativistic bosonic string through the study of the action functionals of Nambu-Goto and Polyakov and the techniques required for their canonical, light-cone, and path-integral quantisation. For this purpose, we briefly review the main properties of the gauge symmetries and conformal field theory involved in the techniques studied. / Neste trabalho fazemos uma revisão dos princípios básicos da teoria da corda bosônica relativística através do estudo dos funcionais ação de Nambu-Goto e de Polyakov e das técnicas necessárias para sua quantização canônica, no cone de luz e usando integrais de trajetória. Para tanto apresentamos uma pequena revisão das principais propriedades das simetrias de calibre a da teoria de campos conforme envolvidas nas técnicas estudadas.