The objective of my thesis is to find optimal points and the quantization error for a probability measure defined on a Cantor set. The Cantor set, we have considered in this work, is generated by two self-similar contraction mappings on the real line with distinct similarity ratios. Then we have defined a nonhomogeneous probability measure, the support of which lies on the Cantor set. For such a probability measure first we have determined the n-optimal points and the nth quantization error for n = 2 and n = 3. Then by some other lemmas and propositions we have proved a theorem which gives all the n-optimal points and the nth quantization error for all positive integers n. In addition, we have given some properties of the optimal points and the quantization error for the probability measure. In the end, we have also given a list of n-optimal points and error for some positive integers n. The result in this thesis is a nonhomogeneous extension of a similar result of Graf and Luschgy in 1997. The techniques in my thesis could be extended to discretise any continuous random variable with another random variable with finite range.

On presente une generalisation d'un resultat de Wallin ainsi qu'une caracterisation des ensembles compacts polaires dans un espace de Brelot. Ces resultats se generalisent a un produit de n espaces de Brelot en demontrant la continuite des fonctions multireduites. On en deduit qu'un ensemble localement n polaire est un ensemble in polaire. Des resultats semblebles ont lieu pour une sous-classe des ensembles pluripolaires dans un domaine hyperconvexe et borne de C('n).

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A Hamilton cycle is a cycle which passes through every vertex of a graph. A Hamilton cycle decomposition of a k-regular graph is defined as the partition of the edge set into Hamilton cycles if k is even, or a partition into Hamilton cycles and a 1-factor, if k is odd. Consequently, for 2-regular or 3-regular graphs, finding a Hamilton cycle decompositon is equilvalent to finding a Hamilton cycle. Two classes of graphs are studies in this thesis and both have significant symmetry. The first class of graphs is the 6-regular circulant graphs. These are a king of Cayley graph. Given a finite group A and a subset S ⊆ A, the Cayley Graph Cay(A,S) is the simple graph with vertex set A and edge set {{a, as}|a ∈ A, s ∈ S}. If the group A is cyclic then the graph is called a circulant graph. This thesis proves two results on 6-regular circulant graphs: 1. There is a Hamilton cycle decomposition of every 6-regular circulant graph Cay(Z[subscript n],S) in which S has an element of order n; 2. There is a Hamilton cycle decomposition of every connected 6-regular circulant graph of odd order. The second class of graphs examined in this thesis is a futher generalization of the Generalized Petersen graphs. The Petersen graph is well known as a highly symmetrical graph which does not contain a Hamilton cycle. In 1983 Alspach completely determined which Generalized Petersen graphs contain Hamilton cycles. In this thesis we define a larger class of graphs which includes the Generalized Petersen graphs as a special case. We call this larger class spoked Cayley graphs. We determine which spoked Cayley graphs on Abelian groups are Hamiltonian. As a corollary, we determine which are 1-factorable.

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A system for visual guidance in patient set-up for external-beam radiotherapy procedures was developed using augmented reality. The system uses video cameras to obtain views of the linear accelerator, and the live images are displayed on a monitor in the treatment room. A 3D model of the patient's external surface, obtained from planning CT data, is superimposed onto the treatment couch in the camera images. The augmented monitor can then be viewed, and alignment performed against the virtual contour. The system provides an intuitive method for set-up guidance, and allows non-rigid deformations to patient pose to be visualised. It also allows changes to patient geometry between treatment fractions to become observable, and can remain in operation throughout the treatment procedure, so that patient motion becomes apparent. Coordinate registration between the camera view and the linac is performed using a cube which is aligned with the linac isocentre using room lasers or cone-beam CT. The AR tracking software detects planar fiducial tracking markers attached to the cube faces, and determines their positions in order to perform pose estimation of the 3D model on-screen. Experimental results with an anthropomorphic phantom in a clinical environment have shown that the system can be used to position a rigid-body with a translational error of 3 mm, and a rotational error of 0.19 degrees, 0.06 degrees and 0.27 degrees, corresponding to pitch, roll and yaw respectively. With further developments to optimise the system accuracy and its interface, it could be made into a valuable tool for radiotherapy clinics. The outcome of the project has been encouraging, and has shown that augmented reality for patient set-up guidance has great potential.

A Hamilton cycle is a cycle which passes through every vertex of a graph. A Hamilton cycle decomposition of a k-regular graph is defined as the partition of the edge set into Hamilton cycles if k is even, or a partition into Hamilton cycles and a 1-factor, if k is odd. Consequently, for 2-regular or 3-regular graphs, finding a Hamilton cycle decompositon is equilvalent to finding a Hamilton cycle. Two classes of graphs are studies in this thesis and both have significant symmetry. The first class of graphs is the 6-regular circulant graphs. These are a king of Cayley graph. Given a finite group A and a subset S ⊆ A, the Cayley Graph Cay(A,S) is the simple graph with vertex set A and edge set {{a, as}|a ∈ A, s ∈ S}. If the group A is cyclic then the graph is called a circulant graph. This thesis proves two results on 6-regular circulant graphs: 1. There is a Hamilton cycle decomposition of every 6-regular circulant graph Cay(Z[subscript n],S) in which S has an element of order n; 2. There is a Hamilton cycle decomposition of every connected 6-regular circulant graph of odd order. The second class of graphs examined in this thesis is a futher generalization of the Generalized Petersen graphs. The Petersen graph is well known as a highly symmetrical graph which does not contain a Hamilton cycle. In 1983 Alspach completely determined which Generalized Petersen graphs contain Hamilton cycles. In this thesis we define a larger class of graphs which includes the Generalized Petersen graphs as a special case. We call this larger class spoked Cayley graphs. We determine which spoked Cayley graphs on Abelian groups are Hamiltonian. As a corollary, we determine which are 1-factorable.