The exact scalar wave solution for light scattering from a general dielectric disk array is found. The exact solution as well as a numerical solution is also given for an array of three dielectric disks, whose centers are placed on the vertices of an equilateral triangle. The various (differential, total and averaged total) cross sections and the poles of the scattering matrix are given. These results are analyzed in part by considering the effects which arise from the geometry of the system, and in part by comparison with the results (cross sections and poles) for a system with an identical arrangement of three hard (perfectly conducting) disks and with a system of one dielectric disk. This analysis helps rule out structure that arise from the chaos, which is very likely to exist, in the classical (geometric) limit of the three (or more) dielectric disk system. In the future after I study the geometric (classical) and physics (semiclassical) regimes of this system, I plan to reanalyse the exact wave solution in an attempt to discover the traces of the chaos present in the system. Time dependent Hartree-Fock theory (TDHF), with improved formulation and improved computer capabilities, is used to repeat the calculations of charge capture for the He²⁺ on He collision. The results of these calculations allow us to discuss the effects of various numerical truncations and to establish with certainty the viability and the accuracy of TDHF in its application to ion-atom collisions. Initially, we had hoped to find chaos in the TDHF problem, as it is nonlinear. However, due to the complexity and computational difficulties present in the TDHF problem, a simpler scattering system of light scattering from dielectric disk arrays was chosen for study.

The <i>mixed postman problem</i> consists of finding a minimum cost tour of a mixed graph <i>M</i> = (<i>V</i>,<i>E</i>,<i>A</i>) traversing all its edges and arcs at least once. We prove that two well-known linear programming relaxations of this problem are equivalent. The <i>extra cost</i> of a mixed postman tour <i>T</i> is the cost of <i>T</i> minus the cost of the edges and arcs of <i>M</i>. We prove that it is <i>NP</i>-hard to approximate the minimum extra cost of a mixed postman tour. A related problem, known as the <i>windy postman problem</i>, consists of finding a minimum cost tour of an undirected graph <i>G</i>=(<i>V</i>,<i>E</i>) traversing all its edges at least once, where the cost of an edge depends on the direction of traversal. We say that <i>G</i> is <i>windy postman perfect</i> if a certain <i>windy postman polyhedron O</i> (<i>G</i>) is integral. We prove that series-parallel undirected graphs are windy postman perfect, therefore solving a conjecture of Win. Given a mixed graph <i>M</i> = (<i>V</i>,<i>E</i>,<i>A</i>) and a subset <i>R</i> &#8838; <i>E</i> &#8746; <i>A</i>, we say that a mixed postman tour of <i>M</i> is <i>restricted</i> if it traverses the elements of <i>R</i> exactly once. The <i>restricted mixed postman problem</i> consists of finding a minimum cost restricted tour. We prove that this problem is <i>NP</i>-hard even if <i>R</i>=<i>A</i> and we restrict <i>M</i> to be planar, hence solving a conjecture of Veerasamy. We also prove that it is <i>NP</i>-complete to decide whether there exists a restricted tour even if <i>R</i>=<i>E</i> and we restrict <i>M</i> to be planar. The <i>edges postman problem</i> is the special case of the restricted mixed postman problem when <i>R</i>=<i>A</i>. We give a new class of valid inequalities for this problem. We introduce a relaxation of this problem, called the <i>b-join problem</i>, that can be solved in polynomial time. We give an algorithm which is simultaneously a 4/3-approximation algorithm for the edges postman problem, and a 2-approximation algorithm for the extra cost of a tour. The <i>arcs postman problem</i> is the special case of the restricted mixed postman problem when <i>R</i>=<i>E</i>. We introduce a class of necessary conditions for <i>M</i> to have an arcs postman tour, and we give a polynomial-time algorithm to decide whether one of these conditions holds. We give linear programming formulations of this problem for mixed graphs arising from windy postman perfect graphs, and mixed graphs whose arcs form a forest.

Optical Coherence Tomography (OCT) is a coherence-gated, micrometer-resolution imaging technique that focuses a broadband near-infrared laser beam to penetrate into optical scattering media, e.g. biological tissues. The OCT resolution is split into two parts, with the axial resolution defined by half the coherence length, and the depth-dependent lateral resolution determined by the beam geometry, which is well described by a Gaussian beam model. The depth dependence of lateral resolution directly results in the defocusing effect outside the confocal region and restricts current OCT probes to small numerical aperture (NA) at the expense of lateral resolution near the focus. Another limitation on OCT development is the presence of a mixture of speckles due to multiple scatterers within the coherence length, and other random noise. Motivated by the above two challenges, a multiple scattering model based on Rytov approximation and Gaussian beam optics is proposed for the OCT setup. Some previous papers have adopted the first Born approximation with the assumption of small perturbation of the incident field in inhomogeneous media. The Rytov method of the same order with smooth phase perturbation assumption benefits from a wider spatial range of validity. A deconvolution method for solving the inverse problem associated with the first Rytov approximation is developed, significantly reducing the defocusing effect through depth and therefore extending the feasible range of NA.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Approximation algorithms employing Monte Carlo methods, across application domains, often require as a subroutine the estimation of the mean of a random variable with support on [0,1]. One wishes to estimate this mean to within a user-specified error, using as few samples from the simulated distribution as possible. In the case that the mean being estimated is small, one is then interested in controlling the relative error of the estimate. We introduce a new (epsilon, delta) relative error approximation scheme for [0,1] random variables and provide a comparison of this algorithm's performance to that of an existing approximation scheme, both establishing theoretical bounds on the expected number of samples required by the two algorithms and empirically comparing the samples used when the algorithms are employed for a particular application.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

In this thesis, a discrete Markov chain model for analyzing memory interference in multiprocessors, is presented.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

No description available.

We investigate semiclassical contributions to correlation functions in N = 1 supersymmetric gauge theories. Our principal example is the gluino condensate, which signals the breaking of chiral symmetry, and should be exactly calculable, according to a persymmetric non-renormalisation theorem. However, the two calculational approaches previously employed, SCI and WCI methods, yield different values of the gluino condensate. We describe work undertaken to resolve this discrepancy, involving a new type of calculation in which the space is changed from R(^4) to the cylinder R(3) x S(1) This brings control over the coupling, and supersymmetry ensures that we are able to continue to large radii and extract answers relevant to R(^4). The dominant semiclassical configurations on the cylinder are all possible combinations of various types of fundamental monopoles. One specific combination is a periodic instanton, so monopoles are the analogue of the instanton partons that have been conjectured to be important at strong coupling. Other combinations provide significant contributions that are neglected in the SCI approach. Monopoles are shown to generate a superpotential that determines the quantum vacuum, where the theory is confining. The gluino condensate is calculated by summing the direct contributions from all fundamental monopoles. It is found to be in agreement with the WCI result for any classical gauge group, whereas the values for the exceptional groups have not been calculated before. The ADS superpotential, which describes the low energy dynamics of matter in a supersymmetric gauge theory, is derived using monopoles for all cases where instantons do not contribute. We report on progress made towards a two monopole calculation, in an attempt to quantify the missed contributions of the SCI method. Unfortunately, this eventually proved too complicated to be feasible.