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.

This paper is a review of the major literature dealing with transformations of random variates which achieve variance stabilization and approximate normalization. The subject can be said to have been initiated by a genetical paper of R. A. Fisher (1922) which uses the angular transformation Φ = 2 arcsin√p to deal with the analysis of proportions p with E(p) = P. Here it turns out that Var Φ is almost independent of P and so stabilizes the variance. Some fourteen years later Bartlett introduced the so-called square-root transformation which achieves variance stabilization for variates following a Poisson distribution. These two transformations and their ramifications in theory and application are fully discussed. along with refinements introduced by later writers, notably Curtiss (1943) and Anscombe (1948). Another important transformation discussed is one which refers to an analysis of observations on to a logarithmic scale, and here there are uses in analysis of variance situations and theoretical problems in the field of estimation: in the case of the latter, the work of D. J. Finney (1941) is considered in some detail. The asymptotic normality of the transformation is also considered. Transformations primarily designed to bring about ultimate normality in distribution are also included. In particular, there is reference to work on the chi-square probability integral (Fisher), (Wilson and Hilferty (1931)) and the logarithmic transformation of a correlation coefficient (Fisher (1921)). Other miscellaneous topics referred include i. the probability integral transformation (Probits), with applications in bioassay: ii. applications of transformation theory to set up approximate confidence intervals for distribution parameters (BIom (1954)): iii. transformations in connection with the interpretation of so-called 'ranked' data. / M.S.

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 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.

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.

The purpose of this investigation is to make more detailed studies of transformations. Fourteen compounds have been examined by high temperature X-ray diffraction for this purpose. The investigations have been carried out in such a way as to reveal: 1. the existence of transformations, 2. the influence of polarizability on thermal expansion, 3. the anisotropy of expansion, and 4. the discontinuity of thermal expansion.