The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy σ = (σψ, σ∧), where the second component takes value in operators on the infinite model cone of the local wedges. In general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the elliptcity of the principal edge symbol σ∧ which includes the (in general not explicitly known) number of additional conditions on the edge of trace and potential type. We focus here on these queations and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet-Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich-Dynin formula for edge boundary value problems, and we establish relations of elliptic operators for different weights, via the spectral flow of the underlying conormal symbols.

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The edge city is often criticized as being a center of placelessness. It is the devaluation and commodification of place in contemporary approaches to urban design and planning that is stifling the prosperity of place identity and subsequently the ability for edge cites to create 'a sense of place'. It is probable that a broadened understanding of the situational context and the role of human experience in place making can suggest alternatives to current practices that reduce place to location. Capturing the essence of place inspires superior strategies for producing place identity and a grasp on the meaning of how recovering ‘a sense of place’ is fundamental in turning edge cities from consumable space into real and lasting places.

High turbine inlet temperature becomes necessary for increasing thermal efficiency of modern gas turbines. To prevent failure of turbine components, advance cooling technologies have been applied to different portions of turbine blades. The detailed film cooling effectiveness distributions along a rotor blade has been studied under combined effects of upstream trailing edge unsteady wake with coolant ejection by the pressure sensitive paint (PSP). The experiment is conducted in a low speed wind tunnel with a five blade linear cascade and exit Reynolds number is 370,000. The density ratios for both blade and trailing edge coolant ejection range from 1.5 to 2.0. Blade blowing ratios are 0.5 and 1.0 on suction surface and 1.0 and 2.0 on pressure surface. Trailing edge jet blowing ratio and Strouhal number are 1.0 and 0.12, respectively. Results show the unsteady wake reduces overall effectiveness. However, the unsteady wake with trailing edge coolant ejection enhances overall effectiveness. Results also show that the overall effectiveness increases by using heavier coolant for ejection and blade film cooling. Leading edge film cooling has been investigated using PSP. There are two test models: seven and three-row of film holes for simulating vane and blade, respectively. Four film holes’ configurations are used for both models: radial angle cylindrical holes, compound angle cylindrical holes, radial angle shaped holes, and compound angle shaped holes. Density ratios are 1.0 to 2.0 while blowing ratios are 0.5 to 1.5. Experiments were conducted in a low speed wind tunnel with Reynolds number 100,900. The turbulence intensity near test model is about 7%. The results show the shaped holes have overall higher effectiveness than cylindrical holes for both designs. As increasing density ratio, density effect on shaped holes becomes evident. Radial angle holes perform better than compound angle holes as increasing blowing and density ratios. Increasing density ratio generally increases overall effectiveness for all configurations and blowing ratios. One exception occurs for compound angle and radial angle shaped hole of three-row design at lower blowing ratio. Effectiveness along stagnation row reduces as increasing density ratio due to coolant jet with insufficient momentum caused by heavier density coolant, shaped hole, and stagnation row.

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A superconducting Transition Edge Sensor (TES) can be used to make the most sensitive thermometer which operates in a very narrow temperature range. The thin film bi-layer fabrication details are discussed as well as application in condensed matter physics. These include: measurement of quasi-adiabatic latent heat of superconducting transition, cobalt thermometry and photon detection.

Image compression is a means to perform transmission or storage of visual data in the most economical way. Though many algorithms have been reported, research is still needed to cope with the continuous demand for more efficient transmission or storage. This research work explores and implements polynomial fitting techniques as means to perform block-based lossy image compression. In an attempt to investigate nonpolynomial models, a region-based scheme is implemented to fit the whole image using bell-shaped functions. The idea is simply to view an image as a 3D geographical map consisting of hills and valleys. However, the scheme suffers from high computational demands and inferiority to many available image compression schemes. Hence, only polynomial models get further considerations. A first order polynomial (plane) model is designed to work in a multiplication- and division-free (MDF) environment. The intensity values of each image block are fitted to a plane and the parameters are then quantized and coded. Blocking artefacts, a common drawback of block-based image compression techniques, are reduced using an MDF line-fitting scheme at blocks’ boundaries. It is shown that a compression ratio of 62:1 at 28.8dB is attainable for the standard image PEPPER, outperforming JPEG, both objectively and subjectively for this part of the rate-distortion characteristics. Inter-block prediction can substantially improve the compression performance of the plane model to reach a compression ratio of 112:1 at 27.9dB. This improvement, however, slightly increases computational complexity and reduces pipelining capability. Although JPEG2000 is not a block-based scheme, it is encouraging that the proposed prediction scheme performs better in comparison to JPEG 2000, computationally and qualitatively. However, more experiments are needed to have a more concrete comparison. To reduce blocking artefacts, a new postprocessing scheme, based on Weber’s law, is employed. It is reported that images postprocessed using this scheme are subjectively more pleasing with a marginal increase in PSNR (<0.3 dB). The Weber’s law is modified to perform edge detection and quality assessment tasks. These results motivate the exploration of higher order polynomials, using three parameters to maintain comparable compression performance. To investigate the impact of higher order polynomials, through an approximate asymptotic behaviour, a novel linear mapping scheme is designed. Though computationally demanding, the performances of higher order polynomial approximation schemes are comparable to that of the plane model. This clearly demonstrates the powerful approximation capability of the plane model. As such, the proposed linear mapping scheme constitutes a new approach in image modeling, and hence worth future consideration.

In this thesis we take a specialized approach to edge-colouring by focusing exclusively on multigraphs with high chromatic index. The bulk of our results can be classified into three categories. First, we prove results which aim to characterize those multigraphs achieving known upper bounds. For example, Goldberg's Theorem says that χ'≤ Δ+1+(Δ-2}/(g₀+1) (where χ' denotes chromatic index, Δ denotes maximum degree, and g₀ denotes odd girth). We characterize this bound by proving that for a connected multigraph G, χ'= Δ+1+(Δ-2}/(g₀+1) if and only if G=μC_g₀ and (g₀+1)|2(μ-1) (where μ denotes maximum edge-multiplicity). Our second category of results are new upper bounds for chromatic index in multigraphs, and accompanying polynomial-time edge-colouring algorithms. Our bounds are all approximations to the famous Seymour-Goldberg Conjecture, which asserts that χ'≤ max{⌈ρ⌉, Δ+1} (where ρ=max{(2|E[S]|)/(|S|-1): S⊆V, |S|≥3 and odd}). For example, we refine Goldberg's classical Theorem by proving that χ'≤ max{⌈ρ⌉, Δ+1+(Δ-3)/(g₀+3)}. Our third category of results are characterizations of high chromatic index in general, with particular focus on our approximation results. For example, we completely characterize those multigraphs with χ'> Δ+1+(Δ-3)/(g₀+3). The primary method we use to prove results in this thesis is the method of Tashkinov trees. We first solidify the theory behind this method, and then provide general edge-colouring results depending on Tashkinov trees. We also explore the limits of this method, including the possibility of vertex-colouring graphs which are not line graphs of multigraphs, and the importance of Tashkinov trees with regard to the Seymour-Goldberg Conjecture.

This report contains a description on how it is possible to guide a robot along an edge, by using a camera mounted on the robot. If stereo matching is used to calculate 3Dcoordinates of an object or an edge, it requires two images from different known positions and orientations to calculate where it is. In the image analysis in this project, the Canny edge filter has been used. The result from the filter is not useful directly, because it finds too many edges and it misses some pixels. The Canny edge result must be sorted and finally filled up before the final calculations can be started. This additional work with the image decreases unfortunately the accuracy in the calculations. The accuracy is estimated through comparison between measured coordinates of the edge using a coordinate measuring machine and the calculated coordinates. There is a deviation of up to three mm in the calculated edge. The camera calibration has been described in earlier thesis so it is not mentioned in this report, although it is a prerequisite of this project.

A superconducting Transition Edge Sensor (TES) can be used to make the most sensitive thermometer which operates in a very narrow temperature range. The thin film bi-layer fabrication details are discussed as well as application in condensed matter physics. These include: measurement of quasi-adiabatic latent heat of superconducting transition, cobalt thermometry and photon detection.