Splines are piecewise polynomial functions of a given order of smoothness r. Given complex delta the set of splines of degree less than or equal to d forms a vector space and is denoted by Sr d(delta). For a simplicial complex delta, Strang conjectured a lower bound on the dimension of spline space Srd(delta) and it is known that the equality holds for sufficiently large d. It is called the dimension formula. In this dissertation, we approach the study of splines from the viewpoint of algebraic geometry. This dissertation follows the works of Lau and Stiller. They introduced the conformality conditions which lead to the machinery of sheaves and cohomology which provided a powerful type of generalization of linear algebra. First, we try to analyze effects in the dimensions of spline spaces when we remove or add certain faces in the given complex. We define the cofactor spaces and cofactor maps from the given complexes and use them to interpret the changes in the dimensions of spline spaces. Second, given polyhedral complex delta, we break it into two smaller complexes delta1 and delta2 which are usually easier to handle. We will find conditions for delta1 and delta2 which guarantee that the dimension formula holds for the original complex delta. Next, we use the previous splitting method on certain types of triangulations. We explain how to break the given triangulation and show what kind of simple complexes we end up with. Finally, we study the "2r+1" conjecture on a certain triangulation. The "2r+1" conjecture is that the dimension formula holds on any triangulation for d >/= 2r + 1. We know that the conjecture is sharp because the dimension formula fails on a certain triangulation for d = 2r, but we do not know if it holds on the same triangulation when d = 2r + 1. It is related to a Toeplitz matrix.

An examination of the possibility of applying pulse pressure metalorganic chemical vapour deposition (PP-MOCVD) to conformal coating and an investigation of PP-MOCVD processing parameters were undertaken using the deposition of thin, conformal titanium dioxide (TiO₂) on 3-D featured and non-featured substrates. The characterisation of the conformality and wettability analysis of thin TiO₂ was carried out using titanium tetraisopropoxide (TTIP) dissolved in toluene as a precursor and featured silicon (Si) and silicon nitride (Si₃N₄) as substrates. The features on the substrates were in micro- and nano-scale with the aspect ratio up to 2:1. The processing parameters investigated were temperatures between 400 and 600°C, reactor base pressures from 50 to 200 Pa, injection volumes between 50 and 250 µl, precursor concentrations in the range of 0.15 to 0.50 mol% and pulsing times from 10 to 20 sec. The surface morphology and thickness were examined using a scanning electron microscope (SEM). The composition of the films was qualitatively identified by energy dispersive X-ray spectroscopy (EDS). X-ray diffraction (XRD) and Raman spectroscopy were used to analyse the phase and grain size. The surface roughness and grain size were evaluated using atomic force microscopy (AFM). The optical properties were characterised using UV-VIS light spectroscopy. The anti-sticking characteristic was examined by wettability analysis, measuring the contact angle of the film with water. The research examined the relationships between processing parameters and growth rate, conformality, surface roughness, grain size, phase and water contact angle. A new measurement for thin film conformality was derived based on a statistical analysis of a large number of film thickness measurements on a fracture surface over the lithographed features. The best conformality of 0.95 was obtained for micro-scale features at the lowest temperature in the range of investigation, 400℃, with pulse exposure characterised by a base pressure of 100 Pa, TTIP concentration of 0.50 mol%, injection volume of 50 µl and pulsing time of 10 sec. Conformality for micro-scale features was in the range of 0.82 to 0.97 over a wide range of deposition temperatures. Conformality was as low as 0.45 over nano-scale structures at the higher exposure rate. The conformality decreased as the temperature and precursor concentration increased. The precursor injection volume was found to have minor influences on conformality. The growth rate increased as the temperature increased and reached the maximum at the deposition temperature of 450℃ with the precursor concentration of 0.50 mol% and injection volume of 100 µl. The base pressure and relaxation time had slight influences on the growth rate over the deposition temperature range of 400 to 500℃. The growth rate was increased as the precursor concentration and precursor injection volume increased. The deposited TiO₂ films exhibited columnar growth and anatase phase. The base pressure and pulsing time had no obvious effects on grain size and surface roughness. The grain size decreased as the deposition temperature increased. The surface roughness increased as the deposition temperature increased. Contact angles of over 100° were found with conformality of over 0.80. The variation in contact angle was related to the surface morphology of the deposited films. The contact angle increased as the grain size decreased. High wettability was found for films in the mid-range of pulse exposure, in this study at pulse exposure of 53, or at high deposition temperature, in this case at 600°C. The as-deposited TiO₂ thin films were hydrophobic depending on the surface morphology, surface roughness and grain size.

It is well-known that the basic properties of a bivariate spline space such as dimension and approximation order depend on the geometric structure of the partition. The dependence of geometric structure results in the fact that the dimension of a C1 cubic spline space over an arbitrary triangulation becomes a well-known open problem. In this paper, by employing a new group of smoothness conditions and conformality conditions, we determine the dimension of bivariate C1 cubic spline spaces over a so-called even stratified triangulation.

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Increase in demand for uniform ceramic coatings on larger industrial components have led to a need for a PP-MOCVD coating system scale up. The objective of this thesis is to develop a fully functional coating system operating in the PP-MOCVD regime that is able to deposit thin film ceramic coatings on commercial or industrial components with complex 3D geometries. This can be achieved by applying engineering and vacuum science theories, coupled with the established fundamentals of PP-MOCVD. A larger system was designed and assembled around the boundaries set by the dimensions and geometry of a stainless steel water pump impellor acting as the base substrate. Most of the components were sourced off the shelf from vacuum and fluid specialists. Components which were unavailable for various reasons were designed, and machined in-house by the departmental workshop. Initial test depositions were conducted using small stainless steel disk substrates, heated using a resistive heater similar to the one utilised on the research scale system. The test depositions were performed with the heater and substrate combination placed in strategic locations. This is to test the overall uniformity of precursor flux in the chamber volume. The resulting coating uniformity on the disk surfaces were fair but problems such as the large collection of unreacted precursor on the chamber viewport and valve timing issues had to be addressed. Before making any improvements to the system, each of the process areas leading to a successful deposition needed to be understood. Five process areas were developed: ‘Liquid Delivery’, ‘Atomization’, ‘Evaporation’, ‘Transport and Reactor Geometry’, and ‘Droplet Management’. Each of the process areas were analysed individually and changes were made to push for a maximum evaporation efficiency. xviii The improved system provided opportunities to perform depositions that were once not possible for PP-MOCVD. Two sets of deposition tests were designed and conducted. Firstly, the improvements were justified with a series of depositions using flat stainless steel plates with dimensions 65x65x5mm. The other set of 3D case study depositions involve observing the effects of the operational parameters of PP-MOCVD on the uniformity and penetration depths of the coatings into different sized macro blind trenches. Five geometric setup conditions were used to justify the improvements made to the system. These are: ‘Substrate positioned in the direct line of spray’, ‘Use of an unheated receptor’, ‘Use of a heated receptor’, ‘Use of an unheated receptor with a non-axial substrate setup’, and “Choked Flow’. As expected, the uniformity of the coatings on both sides of the plate varied significantly when the substrate is placed over the line of sight of the precursor spray. Similarly, the coating produced under the induced choked flow condition resulted in low conformality. The introduction of an unheated receptor plate resulted in an increase in uniformity on both sides of the plate. Further prove that PP-MOCVD is geometry independent is provided by the deposition made with the non-axial substrate placement resulting in a coating of similar result to the unheated receptor. The use of a heated receptor provided a source for a secondary evaporation of the larger precursor droplets collected resulting in an increase in coating thickness while maintaining good conformality. The effects of temperature, pressure, injection volume, and concentration were explored in the final case study. With maximum depths of 50mm, the macro blind trenches has an aspect ratio of 1:1 and cross-sectional areas of 3x3mm, 9x9mm, and 15x15mm. The final results show that as the temperature rises, the depth penetrated into the trench decreases. This could be due to the change in rate limiting steps as homogeneous reactions begin to increase at higher temperatures. Similar trends were observed with increasing pressure. As the pressure difference between the volume of the trenches and the rest of the chamber decreases, the push needed to xix force the precursor down the trench also decreases, resulting in less depth penetration. The effects of injection volume and concentration observed, can be explained by how much precursor molecules are present during one pulse cycle. The more that is available at any given time, the more likely a reaction will occur and deeper the penetration will get. Of course a ceiling or a limit exists where the molecules in the chamber will get evacuated without being reacted. The future work made possible as a result of the scaled up system are proposed. These include a scale up of the operational parameters to suit any given substrate geometry, improvements to the heating source to achieve greater thermal uniformity, further improvements to the overall system accessibility, and performing other depositions using different substrate materials and precursor types.