The objective of this thesis is to determine the linear transformations of a Grassmann product space which sends the set of nonzero Grassmann product vectors (also called pure vectors) into itself. Let U be an n-dimensional vector space over a field F and let r be an integer such that 0 ≤ r≤ n. The r-th Grassmann product space will be denoted by Ar(U). Subspaces of Ar(U) consisting entirely of pure vectors are called pure subspaces. With each non-zero pure vector of Ar(U) we associate an r-dimensional subspace of U. By studying the set of subspaces of U corresponding to a basis set of a pure subspace of Ar(U) we are able to deduce the form of this pure subspace. In this way we are able to classify the pure subspaces of Ar(U), arriving at only two essentially distinct types. We next study the maximal pure subspaces, i. e. the pure subspaces which are not contained in larger pure subspaces. They are of importance because the assumptions on the linear transformations under consideration imply that a maximal pure subspace is mapped into another maximal pure subspace. The form of the transformation is now almost completely determined by examining the incidence relations between pairs of maximal pure subspaces before and after the transformation is applied. Some algebraic manipulations are then needed in order to display the form of the transformation completely. With the suitable assumptions, our results state that the transformations under consideration are induced by linear transformations of the vector space U, except possibly when 2r = n. When 2r = n two types of transformations are possible. This arises from the fact that the two types of maximal pure subspaces have the same dimensions, (unlike the situation when 2r ≠ n). One type of transformation (those induced by linear maps of U)does not alter the type of pure subspaces, while the other interchanges the two types. / Science, Faculty of / Mathematics, Department of / Graduate

The anisothermal and isothermal phase transformations in the Ag-20.1 to 27 at% Al alloys were examined. The cooling rates employed during continuous cooling had a distinct effect on the phase transformations. In the 24, 25, 26 and 27 at% Al alloys the low temperature transformations required cooling rates of 0.3-0.6°C per minute in order to obtain the phases predicted from the phase diagram. The 20.1 and 23 at% Al alloys exhibited much less μ-phase at room temperature after cooling at 0.3°C per minute than anticipated from the phase diagram. The μ-phase developed within the (ξ + μ) field in the 24 and 25 at% alloys resembles Widmannsfatten plates. The WidmannstStten structure was also present in the 25 at% alloy at room temperature. During the transformation ξ →(ξ → μ) in the two phase field the parent ξ matrix develops a polygonized substructure. This substructure is interpreted in terms of dislocation rearrangements in the accommodation region adjacent to the μ-phase plates. It .was found that the quenched β-phase exhibits precipitation hardening during low temperature ageing. At 200°C the μ-phase grows in the β-matrix as spherical particles. The radius of these particles is a linear function of time. Thus the β → μ transformation was classified as an interface controlled process. The kinetic data obtained during the isothermal transformation of the quenched ξ-phase into the μ-phase at 160 and 200°C indicatesthat the overall growth rate of this reaction is also interface controlled. However, the growth characteristics and the surface distortion accompanying the ξ → μ transformation indicate that more than one atomic process operates during the ξ → μ reaction. The X-ray and electron microscopy analyses suggest that there is not a random distribution of the Ag and Al atoms in the μ-phase. It .appears that the μ-phase contains layers rich in Ag or Al atoms. The regular stacking of: these layers may give rise to a long range superlattice as suggested by the X-ray diffraction patterns. The designation of the μ-phase as an isomorphic β-Mn structure seems incorrect. Electron transmission microscopy of the.isothermally produced μ + ξ structures in the 24 at% alloy established that there is a crystallographic relationship between the parent and the product. The structure relationship between the Widmannstatten μ-phase and the ξ-matrix in the 25 at% alloy was also obtained using X-ray Laue techniques. ξ-phase single crystals transformed into μ-phase polycrystals. The orientation of the μ-crystallites was not completely random. Although, a high degree of randomness was indicated when using X-ray D.S. techniques on fine rods. "Fragmentation" of the ξ-phase single crystal into a polycrystal takes place during room temperature deformation of the quenched ξ-phase. Relatively small amounts of deformation introduces numerous twin lamellae into the ξ-phase. Although the ductility of the deformed ξ-phase decreased rapidly with increasing deformation, no presence of a strain induced transformation product was detected using X-ray methods. The ξ → μ isothermal transformation bears a strong resemblance to the ξ → ξ transformation in the Ag-Zn alloy. / Applied Science, Faculty of / Materials Engineering, Department of / Graduate

Given a topological space (X,Ʊ), let H(X,Ʊ), be the class of all homeomorphisms of (XƱ ) onto itself. This paper is devoted to study the following problem posed by Everett and Ulam [1], [11] in 1948. When and how a new topology Ʋ can be constructed on X such that H(X,Ʊ) = H(X,Ʋ), i.e., these two topological spaces have exactly the same class of homeomorphisms. Some of the results obtained are original, and other results agree essentially with the work done previously by Yu-Lee Lee [5], [6], [7], [8], [9]. / Science, Faculty of / Mathematics, Department of / Graduate

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This thesis investigates some of the properties of certain transformations. Some properties are considered in general; others, only in the xy-plane.

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In this work we will consider the steady and two-dimensional potential flow of an incompressible fluid past a body without friction. Contrary to common experience, we will show that it is possible to calculate the Lie point symmetries that will leave the boundary value problem invariant. We are able to do this by solving the determining equation for the Lie point symmetries subject to a side condition. The side condition is a consequence of the boundary condition that occurs in the boundary value problem. We will show that solutions of the boundary value problem that were obtained previously using the method of conformal transformations are also group invariant solutions of the boundary value problem. We will also show that every group invariant solution of the boundary value problem can be used to generate new group invariant solutions of the same boundary value problem.

Keywords: General transformation model, Model Misspecification, Marginal likelihood, Markov chain Monte Carlo, Stochastic approximation, Mixed-effects model, Consistency, Asymptotic normality, Discretization technique. / Part II of this thesis concerns studies of mixed-effects general transformation models, i.e. general transformation models incorporating both fixed and random effects, to analyze grouped or clustered data. Rank-based marginal likelihood estimation is proposed. The estimation procedure is baseline-free, a good property enjoyed by the Cox partial likelihood. A three-stage Markov chain Monte Carlo stochastic approximation (MCMC-SA) algorithm is developed to find the maximum marginal likelihood estimation (MMLE). The asymptotic normality is obtained via a discretization procedure. Monte Carlo simulation shows that the MMLE has a good small- and moderate-sample behavior. In the end we illustrate an application of the proposed method to Hong Kong horse racing data. / Since it was first proposed by Dabrowska and Doksum in 1988, there is an explosive growth in both studies and applications of transformation model. Transformation model has many naturally endowed merits such as flexibility and conciseness in modeling lifetime or duration and ranking data involving covariates. However, like many other statistical models, transformation model may suffer the problem of misspecification due to falsely specified error term distribution or omitted covariates. The author investigates the large sample behavior of the rank-based quasi maximum marginal likelihood estimator (QMMLE) when transformation model is misspecified, and shows that owing to model misspecification, the QMMLE converges not to the true value of the parameter of interest, but to a "pseudo-true value" which minimizes the Kullback-Leibler divergence between the true model and the misspecified working model. A robust "sandwich" estimate of variance is proposed. The asymptotic normality of the QMMLE is also proved. Following the steps of White (1982), the appropriate Wald test statistic, Lagrange Multiplier test statistic and Information matrix specification test statistic are proposed. / Ni, Zhongxin. / Adviser: Ming Gao Gu. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3587. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 86-99). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

by Sai-kee Wong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 74-77). / Acknowledgments --- p.ii / List Of Figures --- p.v / List Of Tables --- p.vii / Abstract --- p.viii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivation --- p.1 / Chapter 1.2 --- Objective of the Study --- p.2 / Chapter 1.3 --- Our Approach --- p.3 / Chapter 1.4 --- Original Contributions --- p.4 / Chapter 1.5 --- Organization of this Dissertation --- p.5 / Chapter 2 --- Previous Work --- p.6 / Chapter 2.1 --- Absolution orientation approach --- p.6 / Chapter 2.2 --- Relative orientation approach --- p.7 / Chapter 3 --- Calibration using OTV correspondences --- p.12 / Chapter 3.1 --- Problem Statement --- p.12 / Chapter 3.2 --- Recovering the orientation of an OTV from a single view --- p.14 / Chapter 3.3 --- Recovering the transformation parameters between two views --- p.18 / Chapter 3.3.1 --- Recovering R --- p.19 / Chapter 3.3.2 --- Recovering t --- p.20 / Chapter 3.3.3 --- Summary of all the steps --- p.20 / Chapter 3.3.4 --- Recovering R and t using more than 2 OTVs --- p.21 / Chapter 4 --- Experimental Results --- p.23 / Chapter 4.1 --- Simulated Data Experiments --- p.23 / Chapter 4.1.1 --- Error versus the smallest angle among the projected branches of an OTV --- p.23 / Chapter 4.1.2 --- Comparison with a point correspondence algorithm --- p.24 / Chapter 4.2 --- Real Image Experiment --- p.41 / Chapter 5 --- Error Analysis --- p.52 / Chapter 5.1 --- Translation in x only --- p.55 / Chapter 5.1.1 --- Jacobian Matrix on Rotation --- p.56 / Chapter 5.1.2 --- Jacobian Matrix on Translation --- p.57 / Chapter 5.2 --- "Rotation + translation in x, y, z" --- p.60 / Chapter 5.2.1 --- Jacobian Matrix on Rotation --- p.60 / Chapter 5.2.2 --- Jacobian Matrix on Translation --- p.61 / Chapter 5.3 --- "Rotation + translation in x,y" --- p.64 / Chapter 5.3.1 --- Jacobian Matrix on Rotation --- p.65 / Chapter 5.3.2 --- Jacobian Matrix on Translation --- p.65 / Chapter 6 --- Conclusion and Future work --- p.68 / Chapter Appendix A --- Least-squares Approximation of a set of Rotation Matrices --- p.70 / Chapter Appendix B --- Epipolar Lines independent of the Translation Magnitude --- p.72