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

If X is a complex with a triangulation and if P is a homeomorph of a polyhedron in X with respect to this triangulation, then P is tame in X if there is a homeomorphism h of X onto itself and another triangulation of X in which h(P) is a polyhedron. A function from one complex X into a complex is called tame and is said to preserve tame sets if for each tame set PcX, f(P) is tame. Tame local homeomorphisms from triangulated n-manifolds into triangulated n-manifolds and tame light open maps of 2-manifolds into themselves are homeomorphisms. Connected complexes are compact if and only if every tame map of the complex into itself has a polyhedral image. Tame linear maps of Euclidean spaces and tame simplicial maps on triangulated n-manifolds with boundaries are homeomorphisms if their images are of dimension greater than one. Functions from polyhedra into topological spaces which take tame arcs onto sets consisting of finite number of components have images of, at most, a finite number of components. If the function and its inverse takes tame sets onto tame sets then the image is connected, provided its image is in a complex. If the function is from a topological space into a polyhedron, then it is continuous if and only if its inverse takes tame arcs onto closed sets. Finally a function from a complex to a complex is continuous if its inverse takes tame sets onto tame sets. A function from an n-manifold into an n-manifold which has an image of dimension greater than one and which takes arcs onto arcs or points is a homeomorphism. A function from a compact triangulated n-manifold into a topological space which takes tame arcs onto arcs or points and whose image is not an arc or point is a homeomorphism. A function from a triangulated n-manifold into an n-manifold which takes tame arcs onto arcs or points and whose image is of dimension greater than one is a homeomorphism. A function from a triangulated n-manifold into a triangulated n-manifold which takes tame arcs onto connected tame sets such that the image of no tame arc contains a triod is a homeomorphism if its image set is not a point, arc or simple closed curve. Finally there are tame maps which raise the dimension of sets. And there are 1:1 maps which do not preserve tame sets. A K-R manifold is a n-manifold with boundary whose interior is Eⁿ and whose boundary is Eⁿ⁻¹. A 1:1 map of a 2-dimensional K-R manifold onto a 2-dimensional K-R manifold is a homeomorphism. / Doctor of Philosophy

A topological space X is invertible at p ∈ X if for every· neighborhood U of p in X, there is a homeomorphism h on X onto X such that h(X - U) ⊆ U. X is continuously invertible at p ∈ X if for every neighborhood U of p in X there is an isotopy {h_t , 0 ≤ t ≤ 1, on X onto X such that h₁(X - U) ⊆ U. It is proved that, if X is a locally compact space which is invertible at a point p which has an open cone neighborhood, and if the inverting homeomorphisms may be taken to be the identity at p, then X is continuously invertible at p. A locally compact Hausdorff space X, invertible at two or more points which have open cone neighborhoods in X, is characterized as a suspension. A locally compact Hausdorff space X which is invertible at exactly one point p, which has an open cone neighborhood U such that U - p has two components, while X - p is connected, is characterized as a suspension with suspension points identified. Let Cⁿ be an n-conplex with invert point p. Let U be an open cone neighborhood of p in Cⁿ, and let L be the link of U in Cⁿ. Then it is shown that H_p(Cⁿ) is isomorphic to a subgroup of H_p-1(L). Invertibility properties of the i-skeleton of an n-complex are discussed, for i < n. Also, a method is described by which an n-complex which is invertible at certain points may be expressed as the union of subcomplexes, ca.ch of which is invertible at the same points. One-complexes with invert points are characterized as either a suspension over a finite set of points or a union of simple closed curves [n above ⋃ and i = 1 below that symbol], such that Sᵢ ⋂ Sⱼ = p, i ≠ j. It is proved that, if C² may be expressed as the monotone union of closed 2-cells. Also if the link of an open cone neighborhood of an invert point in a 2 - complex C² is planar, C² may be embedded in E³. / Doctor of Philosophy

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This study considers an infinite orthotropic plate on a frictionless elastic foundation subjected to a transverse point load. We are concerned with the stress distribution in the vicinity of the load application. Based on the assumption of a uniform stress distribution under the load, a Fourier Transform technique is utilized to formulate and solve the problem in the transform domain. The inverse transformation is carried out numerically, via the Gaussian Quadrature scheme, to obtain the real response. Symmetry of the transform response due to the material orthotropy has been used to reduce the effort involved in performing the integration. Results indicated a similar symmetric behavior for the real stress distribution, and satisfied the boundary conditions of the problem. The transverse equilibrium is also verified by summing up the reaction forces exerted by the foundation. / M.S.

In this thesis, we set out to provide an enhanced set of techniques for determining the eigenvalues of the Laplacian in polygonal domains. Currently, finite-element methods provide a numerical means by which we can approximate these eigenvalues with ease. However, we would like a more analytic method which may allow us to avoid a basic parameter sweep in finite-element software such as COMSOL to determine what could possibly be an "optimal" distribution of eigenvalues. The hope is that this would allow us to draw conclusions about the acoustic quality of a pentagonally-shaped room. First, we find the eigenvalues using a common finite-element method through COMSOL Multiphysics. We then examine another method which makes use of conformal maps and Schwarz-Christoffel transformations with the prospect that it might provide a more analytic understanding of the calculation of these eigenvalues and possibly allow for variation of certain parameters. This method, as far as we could find, had not yet been developed on the pentagon. We end up carrying this method through nearly all of the steps necessary in finding these eigenvalues. We find that the finite-element method is not only easier to use, but is also more efficient in terms of computing power. / Master of Science

Fast Fourier transforms are introduced as a.unique digital data handling tool for square wave voltammetry. Theoretical square wave voltammogra.ms were examined to study the effect of experimental parameters in the spatial frequency domain. A new method was developed to determine truncation frequencies. A₀, the first spatial frequency, was shown to be an effective measure of peak height and its use in calibration curves was demonstrated. The real and imaginary spatial frequencies were shown to be sensitive measures of peak position. FFT square wave voltammetry was developed. The technique performs the current differencing scheme in the spatial frequency domain over spatial frequencies which are noise free. The technique was verified experimentally with solutions of lead (II) in HCl and thallium (I) in HNO₃. FFT linear parameter estimation analysis was introduced as a new digital data manipulation technique for fused peak analytical systems. The algorithm was tested theoretically and experimentally on various ratios of lead (II) to thallium (I) in nitric acid where the peak separation is approximately 70 millivolts. The algorithm performs the calculation over noise free spatial frequencies and significantly reduces the complexity of the calculation relative to the conventional case. / Ph. D.

In the main hypothesis for this paper, H and K are Hilbert spaces, F:H->K is a function with continuour second Fréchet differential such that dF(x)dF(x)* is onto for all x in H.

No / Unitary transformations in an angular momentum Hilbert space H(2j + 1), are considered. They are expressed as a finite sum of the displacement operators (which play the role of SU(2j + 1) generators) with the Weyl function as coefficients. The Chinese remainder theorem is used to factorize large qudits in the Hilbert space H(2j + 1) in terms of smaller qudits in Hilbert spaces H(2ji + 1). All unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits.

In recent years program transformation technology has matured into a practical solution for many software reengineering and migration tasks. FermaT, an industrial strength program transformation system, has demonstrated that legacy systems can be successfully transformed into efficient and maintainable structured C or COBOL code. Its core, a transformation engine, is based on mathematically proven program transformations and ensures that transformed programs are semantically equivalent to its original state. Its engine facilitates a Wide Spectrum Language (WSL), with low-level as well as high-level constructs, to capture as much information as possible during transformation steps. FermaT’s methodology and technique lack in provision of concurrent migration and analysis. This provision is crucial if the transformation process is to be further automated. As the constraint based program migration theory has demonstrated, it is inefficient and time consuming, trying to satisfy the enormous computation of the generated transformation sequence search-space and its constraints. With the objective to solve the above problems and to extend the operating range of the FermaT transformation system, this thesis proposes a Parallel Transformations Framework which makes parallel transformations processing within the FermaT environment not only possible but also beneficial for its migration process. During a migration process, many thousands of program transformations have to be applied. For example a 1 million line of assembler to C migration takes over 21 hours to be processed on a single PC. Various approaches of search, prediction techniques and a constraint-based approach to address the presented issues already exist but they solve them unsatisfactorily. To remedy this situation, this dissertation proposes a framework to extend transformation processing systems with parallel processing capabilities. The parallel system can analyse specified parallel transformation tasks and produce appropriate parallel transformations processing outlines. To underpin an automated objective, a formal language is introduced. This language can be utilised to describe and outline parallel transformation tasks whereas parallel processing constraints underpin the parallel objective. This thesis addresses and explains how transformation processing steps can be automatically parallelised within a reengineering domain. It presents search and prediction tactics within this field. The decomposition and parallelisation of transformation sequence search-spaces is outlined. At the end, the presented work is evaluated on practical case studies, to demonstrate different parallel transformations processing techniques and conclusions are drawn.