by Zheng Yu-kun. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Includes bibliographies.

Thesis. 1977. Sc.D.--Massachusetts Institute of Technology. Dept. of Materials Science and Engineering. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Includes bibliographical references. / Sc.D.

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本論文給出了在鏡像對稱中對非拉格朗日A-膜的Fourier型的變換。 / SYZ構想斷言，鏡像對稱應該來自於一種在卡拉比-丘流形上逐纖維的Fourier-Mukai變換。在半平坦卡拉比-丘流形上的拉格朗日A-膜的情形下，這已經被實現。然而， Kapustin和Orlov指出，對於一些特殊的卡拉比-丘流形， A-膜範疇應該加上某些額外的物件。他們稱這些額外的物件為餘迷向A-膜。在半平坦卡拉比-丘流形的情況下，我們需要加入一些在每個纖維上是楊-米爾斯的A-膜以及B-膜。 / 我們首先推廣Nahm變換到環面上的楊-米爾斯叢。這也可以看作一種Fourier型的變換。然後我們在半平坦卡拉比-丘流形上實施逐纖維的這種Nahm變換。我們在一些半平坦卡拉比丘流形上構造了一些新的B-膜的例子。這些B-膜限制到每一個纖維環面上都是環面上的楊-米爾斯叢。並且我們驗證了在這種逐纖維的變換下，他們恰好就是Kapustin和Orlov所提出的餘迷向A 膜。 / This thesis gives the construction of Fourier type transformations in mirror symmetry for non-Lagrangian A-branes. / The SYZ proposal asserts that mirror symmetry should come from a fiberwise Fourier-Mukai transformation along torus fibrations on Calabi-Yau manifolds. This can be realized explicitly for Lagrangian A-branes in semi-flat case. However, Kapustin and Orlov pointed out that for certain Calabi-Yau manifolds some extra objects called coisotropic A-branes should be added into the category of A-branes. In semi-flat cases, we need to include A-and B-branes which are Yang-Mills along fibers. / We first generalize the Nahm transformation to Yang-Mills line bundles over tori which can also be regarded as a Fourier type transformation. Then we carry out a family version of this transformation for semi-flat Calabi-Yau manifolds. More precisely, we construct a new class of B-branes in semi-flat Calabi-Yau manifolds which are Yang-Mills line bundles when restricted to each fiber torus. And we show that this fiberwise transformation of these B-branes produce the coisotropic A-branes predicted by Kapustin and Orlov. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Zhang, Yi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 61-62). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Fourier-Mukai Transformation --- p.10 / Chapter 2.1 --- A torus case --- p.10 / Chapter 2.1.1 --- Moduli space of flat U(1) bundles over T --- p.11 / Chapter 2.1.2 --- Poincare line bundle P --- p.12 / Chapter 2.1.3 --- Definition of the Fourier-Mukai Transformation for a torus --- p.13 / Chapter 2.1.4 --- Some concrete computations --- p.14 / Chapter 2.2 --- Semi-flat Calabi-Yau case --- p.15 / Chapter 2.2.1 --- Semi-flat Calabi-Yau manifolds and semi-flat branes --- p.15 / Chapter 2.2.2 --- Fourier-Mukai transformation for semi-flat branes --- p.18 / Chapter 3 --- Coisotropic A-branes --- p.23 / Chapter 3.1 --- Why Lagrangian branes are not enough? --- p.23 / Chapter 3.2 --- An example --- p.27 / Chapter 4 --- Nahm transformation --- p.29 / Chapter 4.1 --- Spinor bundle and the Dirac operator --- p.30 / Chapter 4.1.1 --- Clifford algebra and spin group --- p.30 / Chapter 4.1.2 --- Spinor bundle --- p.33 / Chapter 4.1.3 --- Dirac operator --- p.36 / Chapter 4.2 --- Nahm transformation for a torus (T, g) --- p.38 / Chapter 4.3 --- Fourier-Mukai transformation for coisotropic A-branes --- p.53

We study mirror symmetry via Fourier-Mukai-type transformations, which we call SYZ mirror transformations, in view of the ground-breaking Strominger-Yau-Zaslow Mirror Conjecture which asserted that the mirror symmetry for Calabi-Yau manifolds could be understood geometrically as a T-duality modified by suitable quantum corrections. We apply these transformations to investigate a case of mirror symmetry with quantum corrections, namely the mirror symmetry between the A-model of a toric Fano manifold X¯ and the B-model of a Landau-Ginzburg model (Y, W). Here Y is a noncompact Kahler manifold and W : Y → C is a holomorphic function. We construct an explicit SYZ mirror transformation which realizes canonically the isomorphism QH*X&d1; ≅Ja cW between the quantum cohomology ring of X¯ and the Jacobian ring of the function W. We also show that the symplectic structure oX¯ of X¯ is transformed to the holomorphic volume form eWOY of ( Y, W). Concerning the Homological Mirror Symmetry Conjecture, we exhibit certain correspondences between A-branes on X¯ and B-branes on (Y, W) by applying the SYZ philosophy. / Chan, Kwok Wai. / Adviser: Nai Chung Conan Leung. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3536. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 52-56). / 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.

For discretization method, which provides an effective way to handle the problem, we focus on constructing a discretized version of continuous failure times. The major observation is that discrete failure times with finite values will attain their exact values although only ranks are given, as long as sample size is large enough. The score function can be asymptotically approximated by a sum of independent random variables. Consistency, asymptotic normality and efficiency of estimator can be obtained by using standard results for estimating equations, given some milder and more feasible conditions than those of martingale method. / In this thesis, some finite sample properties of marginal likelihood will be established. We prove under some regular conditions the score function of the marginal likelihood is a martingale, and prove the marginal likelihood satisfies some properties enjoyed by the standard likelihood method, although only use the relative ranks to make inference of the parameter instead of the full information. / In this thesis, the author studies some asymptotic properties of the marginal maximum likelihood estimate (marginal MLE) for general transformation models. The general transformation model is an important class of models for survival times and is nontrivially more general model than the linear transformation model (Gu, Sun and Zuo, 2005). By using marginal likelihood, we obtain estimator of regression parameter which does not depend on its baseline survival function, a property enjoyed by the Cox regression model. The major obstacle for the general transformation models is the resulting estimation function is complicated and usually has no closed analytic expression. Gu etc. (2005) proposed Markov chain Monte Carlo (MCMC) stochastic approximation algorithm to solve the marginal MLE. / Moreover, a discussion is given for the question of asymptotic properties for the proposed marginal MLE based on two different methods: martingale method and discretization method. For martingale method, emphasis is given to the role of martingale limit theory and results presented are primarily theoretical. / We also demonstrate some important transformation models do satisfy our conditions and thus show their consistency, asymptotic normality and efficiency for the first time. / Huang Bin. / "Dec 2005." / Source: Dissertation Abstracts International, Volume: 67-11, Section: B, page: 6484. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 63-69). / 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.

Yu Wei-jian. / Thesis submitted in: November 2000. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 70-71). / Text in English; abstracts in English and Chinese. / Yu Wei-jian. / TITLE PAGE --- p.i / THESIS COMMITTEE --- p.ii / ABSTRACT (English) --- p.iii / ABSTRACT (Chinese) --- p.iv / ACKNOWLEDGEMENTS --- p.v / TABLE OF CONTENTS --- p.vi / LIST OF FIGURES --- p.viii / LIST OF TABLES --- p.ix / Chapter CHAPTER 1 --- INTRUDUCTION --- p.1 / Chapter Section 1.1 --- Background of the BN System --- p.1 / Chapter 1.1.1 --- Desirable Properties of c-BN --- p.1 / Chapter 1.1.2 --- Phases in the BN System --- p.2 / Chapter 1.1.3 --- Phase Diagram of BN --- p.4 / Chapter 1.1.4 --- Synthesis Techniques for c-BN Films --- p.5 / Chapter 1.1.5 --- Characterization of BN Films --- p.6 / Chapter Section 1.2 --- Background of Theory --- p.7 / Chapter Section 1.3 --- Objectives --- p.9 / Chapter 1.3.1 --- Determination of Stable State of BN --- p.9 / Chapter 1.3.2 --- Phonon-dispersion Relations of BN --- p.9 / Chapter 1.3.3 --- "Phase (p, T) Diagram" --- p.10 / Chapter 1.3.4 --- Transformation Paths in Direct Compressions --- p.10 / Chapter Section 1.4 --- Roadmap --- p.11 / Chapter CHAPTER 2 --- METHODS --- p.12 / Chapter Section 2.1 --- Density Functional Theory (DFT) for E0 Calculation --- p.12 / Chapter Section 2.2 --- Direct Force-constant Method --- p.16 / Chapter Section 2.2 --- Quasi-harmonic Approximation --- p.26 / Chapter CHAPTER 3 --- RESULTS --- p.27 / Chapter Section 3.1 --- Stable State of BN --- p.27 / Chapter Section 3.2 --- Phonon-dispersion Relations --- p.29 / Chapter Section 3.3 --- "Phase (p, T) Diagram of BN" --- p.36 / Chapter Section 3.4 --- Transformation Paths via Direct Compression --- p.44 / Chapter 3.4.1 --- Direct Compression of h-BN and r-BN --- p.46 / Chapter 3.4.2 --- Direct Compression of t-BN --- p.50 / Chapter Section 3.5 --- Energy Barriers in the Transformation of h-BN to c-BN --- p.52 / Chapter CHAPTER 4 --- DISCUSSION --- p.58 / Chapter Section 4.1 --- Transition States in the Transformation of h-BN to c-BN --- p.58 / Chapter Section 4.2 --- Phonon-dispersion Relations --- p.60 / Chapter Section 4.3 --- Phase Diagrams --- p.62 / Chapter Section 4.4 --- Future Studies --- p.63 / Chapter 4.4.1 --- Cubic BN Films Formation --- p.63 / Chapter 4.4.1.1 --- Nanoarches Nucleation --- p.64 / Chapter 4.4.1.2 --- Growth: Interfaces Between h-BN {0001} and c-BN Planes --- p.64 / Chapter 4.4.2 --- Transformation Paths of w-BN to c-BN and h-BN to r-BN --- p.65 / Chapter CHAPTER 5 --- CONCLUSION --- p.66 / APPENDIX: Mechanistic Models in c-BN Films Formation --- p.67 / REFERENCES --- p.70

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Physics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Vita. / Includes bibliographical references. / by Lawrence Maxwell Krauss. / Ph.D.

A topological group is an abstract group which is also a topological space and in which the group operation are continuous. In group theory the algebraic binary operation of passage to a limit is studied in a similar manner. The two fundamental mathematical concepts of binary operation and passage to a limit are united and interrelated in the concept of topological group. The concept of topological groups arose from the study of continuous transformations. However, topological groups can be studied quite independently from continuous transformations and the latter can be presented as applications of topological groups. The first person to consider topological groups was Lie, but he was concerned with groups defined by analytic operations. Later, around 1900-1910 other men, beginning with Hilbert and Brouwer, studied more general topological groups. The topological group is then -- from a logical point of view only -- a combination of the abstract group and the topological space. Hence, the first and second chapters of this paper will be devoted to the concept of abstract group and topological space respectively, while the third and main chapter will utilize these two concepts in the formation and study of the topological group. Our main source of information will be Leon Pontrjagin's book, "Topological Groups" (1939); however, our approach will be somewhat broader and we will include results from other sources and our own investigations. In order to avoid making this paper to lengthy for its purpose, we will prove only some of the theorems. The rest of them will be simply stated and often followed by a sketch or an outline of the proof. The major definitions and theorems and all the examples will be numbered consecutively as they appear. For instance "Theorem 2.5" is the fifth numbered item in the second chapter.

The use and application of geometric transformations is a fundamental standard for the Common Core State Standards. This study was developed to determine current high school teachers’ prior mathematical content knowledge and develop their content knowledge of transformations and their applications. The design of this study was guided by the questions: “Why is there a level of reluctance amongst secondary teachers when it comes to teaching geometric transformations?” and “How can their content knowledge become deepened to apply geometric transformations to prove that two figures are congruent?” The study provided teachers a chance to gain experience with transformations and use transformations to develop an understanding of congruence and similarity. The teachers’ work with transformations also enhanced their understanding of how transformations are the foundation for Euclidean geometry and begin to lay a foundation for the basics of rigid motion in the plane, with or without the use of coordinates. The results supported the claim that teachers’ transformation content knowledge needs to be deepened overall and in particular with respect to the application of transformations to prove that two figures are congruent. The results also showed that, with an increase of understanding of the mathematical properties of transformations, teachers are better prepared to teach them in their classrooms.