This thesis introduces three new model-free tests for isotropy, equal distribution and random superposition in non-rectangular windows respectively. For isotropy, a bootstrap-type test is proposed. The corresponding test statistic assesses the discrepancy between the uniform distribution and the empirical normalised reduced second-order moment measure of a sector of fixed radius with increasing central angle. The null distribution of the discrepancy is then estimated by stochastic reconstruction, which generates bootstrap-type samples of point patterns that resemble the spatial structure of the given pattern. The new test is applicable for small sample sizes and is shown to have more robust powers to different choices of user-chosen parameter when compared with the asymptotic chi-squared test by Guan et al. (2006) in our simulation. For equal distribution, a model-free asymptotic test is introduced. The proposed test statistic compares the discrepancy between the empirical second-order product densities of the observed point patterns at some pre-chosen lag vectors. Under certain mild moment conditions and a weak dependence assumption, the limiting null distribution of the test statistic is the chi-squared distribution. Simulation results show that the new test is more powerful than the permutation test by Hahn (2012) for comparing point patterns with similar structures but different distributions. The new test for random superposition is a modification of the toroidal shift test by Lotwick and Silverman (1982). The idea is to extrapolate the pattern observed in a non-rectangular window to a larger rectangular region by the stochastic reconstruction so that the toroidal shift test can be applied. Simulation results show that the powers of the test applied to patterns with extrapolated points are remarkably higher than those of the test applied to the largest inscribed rectangular windows, with only slightly increased type I error rates. Real data sets are used to illustrate the advantages of the tests developed in this thesis over the existing tests in the literature.

We give asymptotics near the boundary for the distribution of the first exit time of the isotropic alpha-stable Levy process on bounded Lipschitz sets in real euclidean space. These asymptotics bear some relation to the existence of limits in the Yaglom sense of alpha-stable processes. Our approach relies on the uniform integrability of the ratio of Green functions on bounded Lipschitz sets. We use bounds for the heat remainder to give the first two terms in the small time asymptotic expansion of the trace of the heat kernel of unimodal Levy processes satisfying some weak scaling conditions on bounded Lipschitz domains.

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Investigations of machining optical surfaces into brittle materials using an ultra precision machine tool are presented. The newly developed ultra precision NION machine is evaluated to gain a good appreciation of its operating performance. The machining accuracy capability of this machine is established by careful measurement of its motion accuracy, thermal and dimensional stability and loop stiffness. Corroboration of these measurements are provided by assessment of surfaces which were produced in soft 'easily machined' metal materials. It was found that surfaces smooth to -1 nm Ra could be produced on the NION machine and with a form error of less than 100 nm P-V. The main source of figure error, approximately 80 nm, was found to be caused by the synchronous axial error motion of the workhead spindle. Other elements of the machine, including thermal effects, incurred less than 25 nm of additional figure error. Assessment of the diamond turning process for the producing optical surfaces made in a number of important optical materials, which are ostensibly brittle, were undertaken. Turning tests were carried out to establish the relative difficulty for machining optical surfaces in these materials and to define the most important parameters which affect the attained surface quality. Assessment of the produced surfaces was based on their roughness quality, surface morphology and residual stress condition. It was found that diamond tool edge quality degraded with total cut distance. Tool cut distance was found to be a major influence on achievable material removal rate before micro-fractures became present at the surface. Surface quality and residual stress condition were also greatly influenced by the overall tool cut distance. Diamond grinding trials were also carried out using the NION machine tool. These grinding trials were carried out using a mode of grinding which permits complex shape optical surfaces to be produced. Various grinding technologies were employed to establish the optimum methods. Selected grinding trials were carried out to establish the dominate parameters affecting the optical quality. Assessment of the machined surfaces was in regard of their surface roughness, residual stress and severity of sub-surface micro cracking. It was found that grinding wheel specification was a major influence on surface quality and sub-surface damage. The level of residual stress associated with 'ductile' mode grinding was not found to prohibit its application toward the direct manufacture of optical elements. Selection of grinding parameters which ensured the grain depth of cut, GDOC, parameter did not exceed the materials critical depth, dc, allowed glass surfaces to be ground to 1-2 nm Ra. These ground glass surfaces appeared free of any surface fractures. Sub-surface assessments did however reveal small levels of micro-fractures hidden below the surface. Discussion of both machining processes is provided. Available material removal rates for each process is given when cutting a number of important optical materials. Conclusions regarding the production of both Infrared and visible wavelength optics using the NION machine tool are provided. Recommendations for future work to improve both the understanding of the processes and the effectiveness of applying the processes are suggested.

The recent developments in process plant design have made it desirable that a better understanding of transient compressor performance within the process plant be gained. The model outlined in this thesis is capable of representing most types of process plants. Further the number of degress of freedom, plus the dynamic flexibility of a poly- tropic analysis allow any system transient to be simulated, including compressor surge. A computer program has been developed from the model and validated using a small centrifugal compressor. The results showing successful simulation of compressor transient behaviour (including surge) are given. The model was then applied to study the transient behaviour of a natural Gas Transmission station. The model successfully highlighted compressor surge problems under certain operative conditions. This surge phenomenon is present in the trans- mission station and a possible solution to the problem was suggested due to the better understanding of dynamic response of station.

The recent years, the aerospace industry in particular has embraced and actively pursued the development of stronger high performance materials, namely nickel based superalloys and hardwearing steels. This has resulted in a need for a more efficient method of machining, and this need was answered with the advent of High Efficiency Deep Grinding (HEDG). This relatively new process using Cubic Boron Nitride (CBN) electroplated grinding wheels has been investigated through experimental and theoretical means applied to two widely used materials, M50 bearing steel and IN718 nickel based superalloy. It has been shown that this grinding method using a stiff grinding centre such as the Edgetek 5-axis machine is a viable process. Using a number of experimental designs, produced results which were analysed using a variety of methods including visual assessment, sub-surface microscopy and surface analysis using a Scanning Electron Microscope (SEM), residual stress measurement using X-Ray Diffraction (XRD) techniques, Barkhausen Noise Amplitude (BNA) measurements, surface roughness and Vickers micro-hardness appraisal. It has been shown that the fundamentals of the HEDG process have been understood through experimental as well as theoretical means and that through the various thermal models used, grinding temperatures can be predicted to give more control over this dynamic process. The main contributions to knowledge are made up of a number of elements within the grinding environment, the most important being the demonstration of the HEDG effect, explanation of the phenomenon and the ability to model the process. It has also been shown that grinding is a dynamic process and factors such as wheel wear will result in a continuous change in the optimum grinding conditions for a given material and wheel combination. With the significance of these factors recognised, they can be accounted for within an industrial adaptive control scenario with the process engineer confident of a more efficient use of time and materials to produce a higher quality product at lower cost.

In this thesis, we study some geometrical aspects of metric measure spaces (Rn, psi1/2 , mu)where mu is a locally finite regular Borel measure and a metric on psi1/2 which arises from a continuous negative definite function psi : Rn &rarr; R which satisfies psi(xi) &ge; 0 with psi(xi) = 0. This study is motivated by the investigation of a transition density estimate for pure jump processes on a general metric measure space. To gain a better insight into the behaviour of transition functions of symmetric Levy processes in this general setting, it seems desirable to understand geometrical properties of their underlying state spaces. More precisely, we show completeness of the metric spaces (Rn, psi1/2) and study under which circumstances open balls Bpsi(x,r), x &isin; Rn, r > 0, with respect to this metric are convex. Moreover, we focus on conditions of the metric measure spaces (Rn,psi1/2 ,mu) for the balls to satisfy the volume growth property [equation] for mu-almost all x &isin; Rn, 0 < r < R and a constant Cpsi(x,R)&ge;1. Finally, we show that the homogeneity property of a metric measure space can be applied to our case and provide some results associated with the construction of a Hajlasz-Sobolc space over (Rn,psi1/2, lambda(n)),where lambda(n) denotes the n-dirnensional Lebesgue measure.

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