本論文旨在討論複空間上的特殊拉格朗日子流行的各種建構方法。這些方法主要自R. Harvey, B. Lawson, D. Joyce, R. Bryanl 以及M. Haskins這幾位數學家發展及研究. 本文側重於擁有不同類型對稱位的特殊拉格朗日子流行的結構方法, 其中包指於n維環面群及特殊正交群不變的例子，以及直紋特殊拉格朗日子流行。除此之外,本論文也會討論以上建構方法所給出的具體制子。最後, 本文亦會討論一種可以建構擁有高虧格鏈的特殊拉格朗日子流行的方法。 / This thesis gives a survey on constructions of special Lagrangian submanifolds in C[superscript n]. These construction methods are mainly studied by Harvey and Lawson, D. Joyce, R. Bryant and M. Haskins. We mainly focus on special Lagrangian submanifolds with different kinds of symmetries. These include local constructions of T[superscript n] --and SO(n)--invariant examples, and ruled examples. We also discuss explicit examples arose from those constructions. Besides local constructions, a global gluing construction of special Lagrangian submanifolds with high genus links is also discussed. / Detailed summary in vernacular field only. / Lam, Yi Chun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 97-99). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Preliminaries --- p.17 / Chapter 2.1 --- Special Lagrangian Geometry --- p.17 / Chapter 2.2 --- Special Legendrian Links --- p.23 / Chapter 2.3 --- Harmonic Maps --- p.25 / Chapter 2.4 --- Moment Maps --- p.28 / Chapter 2.5 --- Evolution Equation --- p.30 / Chapter 2.6 --- Background Materials from Analysis --- p.33 / Chapter 3 --- T[superscript n] invariant Special Lagrangian Submanifolds --- p.36 / Chapter 3.1 --- Basic Example --- p.36 / Chapter 3.2 --- U(1)[superscript n-2] invariant Special Lagrangian Cones in C[superscript n] --- p.37 / Chapter 3.2.1 --- General Construction --- p.37 / Chapter 3.2.2 --- Reduction of O.D.E. System --- p.40 / Chapter 3.2.3 --- The 3--dimensional Case --- p.48 / Chapter 4 --- SO(n)--invariant Special Lagrangian Submanifolds --- p.52 / Chapter 4.1 --- Basic Example --- p.52 / Chapter 4.2 --- Special Lagrangian Submanifolds with Fixed Loci --- p.54 / Chapter 4.2.1 --- Reduction to P.D.E --- p.55 / Chapter 4.2.2 --- The 3--Dimensional Case --- p.63 / Chapter 5 --- Ruled Special Lagrangian Submanifolds --- p.65 / Chapter 5.1 --- Normal Bundle of a Submanifold --- p.65 / Chapter 5.2 --- Twisted Normal Bundles --- p.69 / Chapter 5.3 --- The 3Dimensional Case --- p.72 / Chapter 5.3.1 --- Construction of Special Lagrangian Ruled 3-folds --- p.71 / Chapter 5.3.2 --- Explicit Examples --- p.75 / Chapter 5.4 --- Twisted Special Lagrangian Cones --- p.78 / Chapter 6 --- Other Constructions --- p.81 / Chapter 6.1 --- Analysis on U(1)invariant Special Lagrangian submanifolds --- p.81 / Chapter 6.1.1 --- Non-singular Solutions --- p.83 / Chapter 6.1.2 --- Existence of Singular Solutions --- p.84 / Chapter 6.1.3 --- Properties of Singular Solutions --- p.86 / Chapter 6.2 --- Harmonic Maps and Minimal Immersions --- p.87 / Chapter 6.3 --- Construction of Special Lagrangian 3-folds with High Genus Links --- p.92 / Bibliography --- p.97

by Hao-min Zhou. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references. / Abstract --- p.1 / Introduction --- p.3 / Paper I --- p.8 / Paper II --- p.28

We have directed much attention towards developing quantum trajectory methods which can accurately predict the transmission probabilities for a variety of quantum mechanical barrier scattering processes. One promising method involves solving the complex quantum Hamilton-Jacobi equation with the Derivative Propagation Method (DPM). We present this method, termed complex valued DPM (CVDPM(n)). CVDPM(n) has been successfully employed in the Lagrangian frame to accurately compute transmission probabilities on 'thick' one dimensional Eckart and Gaussian potential surfaces. CVDPM(n) is able to reproduce accurate results with a much lower order of approximation than is required by real valued quantum trajectory methods, from initial wave packet energies ranging from the tunneling case (E[subscript o]=0) to high energy cases (twice the barrier height). We successfully extended CVDPM(n) to two-dimensional problems (one translational degree of freedom representing an Eckart or Gaussian barrier coupled to a vibrational degree of freedom) in the Lagrangian framework with great success. CVDPM helps to explain why barrier scattering from "thick" barriers is a much more well posed problem than barrier scattering from "thin" barriers. Though results in these two cases are in very good agreement with grid methods, the search for an appropriate set of initial conditions (termed an 'isochrone) from which to launch the trajectories leads to a time-consuming search problem that is reminiscent of the rootsearching problem from semi-classical dynamics. In order to circumvent the isochrone problem, we present CVDPM(n) equations of motion which are derived and implemented in the arbitrary Lagrangian-Eulerian frame for a metastable potential as well as the Eckart and Gaussian surfaces. In this way, the isochrone problem can be circumvented but at the cost of introducing other computational difficulties. In order to understand why CVDPM may give better transmission probabilities than real valued counterparts, much attention we have been studying and applying numerical analytic continuation techniques to visualize complex-extended wave packets as well as the complex-extended quantum potential. Numerical analytic continuation techniques have also been used to analytically continue a discrete real-valued potential into the complex plane for CVDPM with very promising results.

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In this research effort, we consider nondifferentiable optimization (NDO) problems that arise in several applications in science, engineering, and management, as well as in the context of other mathematical programming approaches such as Dantzig-Wolfe decomposition, Benders decomposition, Lagrangian duality, penalty function methods, and minimax problems. The importance and necessity of having effective solution methods for NDO problems has long been recognized by many scientists and engineers. However, the practical use of NDO techniques has been somewhat limited, principally due to the lack of computationally viable procedures, that are also supported by theoretical convergence properties, and are suitable for solving large-scale problems. In this research, we present some new algorithms that are based on popular computationally effective strategies, while at the same time, do not compromise on theoretical convergence issues. First, a new variable target value method (VTVM) is introduced that has an e-convergence property, and that differs from other known methods in that it does not require any prior assumption regarding bounds on an optimum or regarding the solution space. In practice, the step-length is often calculated by using an estimate of the optimal objective function value. For general nondifferentiable optimization problems, however, this may not be readily available. Hence, we design an algorithm that does not assume the possibility of having an a prior estimate of the optimal objective function value. Furthermore, along with this new step-length rule, we present a new subgradient deflection strategy in which a selected subgradient is rotated optimally toward a point that has an objective function value less than the incumbent target value. We also develop another deflection strategy based on Shor’s space dilation algorithm, so that the resulting direction of motion turns out to be a particular convex combination of two successive subgradients and we establish suitable convergence results. In the second part of this dissertation, we consider Lagrangian dual problems. Our motivation here is the inadequacy of the simplex method or even interior point methods to obtain quick, near-optimal solutions to large linear programming relaxations of certain discrete or combinatorial optimization problems. Lagrangian dual methods, on the other hand, are quite well equipped to deal with complicating constraints and ill-conditioning problems. However, available optimization methods for such problems are not very satisfactory, and can stall far from the optimal objective function value for some problems. Also, there is no practical implementation strategy for recovering a primal optimal solution. This is a major shortcoming, even if the method is used only for bounding purposes in the context of a branch and bound scheme. With this motivation, we present new primal convergence theorems that generalize existing results on recovering primal optimal solutions, and we describe a scheme for embedding these results within a practical primal-dual Lagrangian optimization procedure. / Ph. D.