Thesis (Ph.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / This thesis studies the question of control of Maxwell's equations in a medium with positive conductivity by means of boundary surface currents. Two types of domains and media are considered in connection with this question. First is a bounded simply connected star-shaped domain in R^3 which is made up of a heterogeneous medium with small conductivity, with controls being applied over the entire boundary. Using the Hilbert Uniqueness Method of Lions, the exact boundary controllability over a sufficiently long time period is established for this case, provided the conductivity is small enough to satisfy a certain technical inequality. It is also found that the requirement for the conductivity term to be very small remains in place even if the medium considered is homogenous. In order to remove this constraint, a special domain type is considered next - a cube - made up of a homogenous medium where the conductivity is allowed to take on any non-negative value. An additional restriction imposed here in order to make this problem more suitable for practical implementations is that the controls are applied over only one face of the cube. Employing the Method of Moments the spectral controllability is established for this case. It is also established that the exact controllability fails for this geometry regardless of the size of the conductivity term. This thesis will also consider the question of reconstructing the source of electromagnetic radiation, which is related to the controllability problem. / 2031-01-02

Analisa-se numericamente o comportamento das soluções da equação X(t) + X(t) = f(X(t-)) para f(X) = A X (l-X), em função dos parâmetros , A. Constroem-se as curvas de duplicação de período no espaço de parâmetros para uma determinada condição inicial, que assegura um determinado tipo de solução assintótica (pertencente ao \"ramo 1\"). Verifica-se a conjectura de que a \"rota para o caos\" neste ramo 1\", possa ser a rota de Feigenbaum. Realça-se o fato de que esta equação, para alguns valores de , A, possui diversos atratores. Estuda-se a organização das soluções globais e limitadas da equação acima em \"ramos\" (certos domínios de soluções), e faz-se uma análise das relações entre as soluções dos diversos \"ramos\". Constata-se que uma cascata de duplicação de período no ramo 1, implica em cascatas de duplicação, ao menos parciais, em outros ramos. Para a equação acima com f(X) = A X (l-X), apresentam-se algumas soluções sob a forma de série, parcialmente computáveis sobre a reta, e faz-se uma aplicação de um resultado acerca da estabilidade do ramo 1 no caso f(X) = A sen(X-C), que corresponde a uma equação da ótica. / Numerical analysis are made of the behavior of the solutions of the equation X(t) + X(t) = f(X(t-)) for f(X) = A X (1 - X), as function of the parameters , A. Period-doubling bifurcation curves are constructed in the parameter space for some particular initial conditions, that insures a certain asymptotic behavior of the solutions (it belongs to \"branch 1\"). It is verified the conjecture that the \"route to chaos\" in the \"branch 1\" may be the Feigenbaum\'s route. The organization of the global and bounded solutions of the above equation in branches (certain domains of solutions) is studied. An analysis is made of the relations between solutions belonging to different branches. It is verified that the existence of a full period-doubling cascade in the branch 1 implies the existence, at least partially, of period-doubling cascade in other branches. It is noted that, for some values of (, A), the equation has many attractors. Some series expansions of solutions of the above equation are presented. These series expantions may be partially computed on the set R. An application of a result about the \"stability\" of branch 1 is made for the case f(X) = A sin(X-C), used to describe an optical system.

在这篇论文中，我们考虑二维中的Gray-Scott 模型核心问题的解: / [ 附圖]. / 对于足够小的ε，我们会构造一个“多个凸 的解。这些解的“凸 会分布在一个正多边形的顶点上。在这个解的U 方向上，经过一个合适的放缩之后，它会看起来像下列方程的唯一对称解： / [ 附圖]. / 此外，我们同时也会构造单个“凸 和两个“凸 的解。 / In this thesis, we consider solutions to the core problem for Gray-Scott model in R²: / [With mathematic formula]. / We construct multi-bump solutions for this problem for all sufficiently small ε. The centers of these bumps are located at the vertices of a regular polygon, andthey resemble, after a suitable scaling in their U-coordinate, the unique radial solution of / [With mathematic formula]. / The solutions with one single bump and two bumps are also constructed. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Yip, Chit Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 45-46). / Abstracts also in Chinese. / Chapter 1 --- Introduction: Derivation of the Core Problem --- p.6 / Chapter 2 --- One-dimensional core problem --- p.12 / Chapter 3 --- Main results on two-dimensional core problem --- p.19 / Chapter 4 --- Proof of Theorem 3.1 --- p.22 / Chapter 4.1 --- Estimate for S₁({U+03A6}) --- p.24 / Chapter 4.2 --- Estimate for N₁({U+03A6}) --- p.25 / Chapter 4.3 --- Estimate for S₁({U+03A6}₁) - S₁({U+03A6}₂) --- p.26 / Chapter 4.4 --- Estimate for N₁({U+03A6}₁) - N₁({U+03A6}₂) --- p.26 / Chapter 5 --- Proof of Theorem 3.2 --- p.29 / Chapter 5.1 --- Estimate for S₂({U+03A6}) --- p.33 / Chapter 5.2 --- Estimate for N₂({U+03A6}) --- p.34 / Chapter 5.3 --- Estimate for S₂({U+03A6}₁) - S₂({U+03A6}₂) --- p.34 / Chapter 5.4 --- Estimate for N₂({U+03A6}₁) - N₂({U+03A6}₂) --- p.35 / Chapter 5.5 --- The reduced problem --- p.35 / Chapter 6 --- Proof of Theorem 3.3 --- p.40 / Chapter 6.1 --- Invariance under permutations --- p.41 / Chapter 6.2 --- Reducing number of equations for regular polygons --- p.42 / Bibliography --- p.45

Cheung, Chun Ngai. / "August 2011." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 55-57). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Coding the Lorenz knots --- p.10 / Chapter 2.1 --- Lorenz words --- p.10 / Chapter 3 --- Lorenz links and positive braids --- p.15 / Chapter 3.1 --- Lorenz braids --- p.15 / Chapter 3.2 --- Properties of Lorenz links as the closure of a positive braid --- p.17 / Chapter 4 --- T-Links and the braid index --- p.28 / Chapter 4.1 --- T-links --- p.30 / Chapter 4.2 --- Symmetries --- p.35 / Chapter 4.3 --- Trip number and the braid index --- p.39 / Chapter 5 --- Modular knots --- p.47 / Chapter 5.1 --- The Modular flow --- p.47 / Chapter 5.2 --- Modular Knots --- p.49 / Bibliography --- p.55

Cheung Ka Luen. / "December 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 95-101). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Finally, we get various estimates on the rupture set of the solution to the Thin Film type equations. / In the first part of the thesis, we focus on the semilinear equations with supercritical growth, and give upper bounds on the Hausdorff dimension of the singular sets for borderline solution. As a result, we can prove that the positive borderline solution must blow up in finite time. / Secondly, for the semilinear equations with critical growth, we apply a fundamental e-regularity property to illustrate the concentration phenomenon for the positive borderline solution when time goes to infinity. More precisely, we show that the lost energy can be counted exactly by the standard bubbles. / Du, Shizhong. / Adviser: Kai-Seng Chou. / Source: Dissertation Abstracts International, Volume: 70-09, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 113-119). / 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.

線性丢番圖方程系統（LDEs）連同其子類--子集和問題--在現實世界有大量且重要的應用。可是當線性番圖方程系統的整數解集被限定在一個有界的多面體中，則此系統屬于NP類問題。同理其子類--子集和問題--也同屬于NP類問題。另一方面，密度（density）接近或等于一的子集和問題已在文獻中被證實爲最難的一類子集和問題，並且現有的針對此最難子集和問題的所有解決方案都不能達到令人滿意的成功率。因此，在這篇論文中，我們旨在提出有效的算法來求解線性番圖方程系統以及其子類問題--密度爲一的子集和問題。 / 我們在此論文中的研究包括：1）基于格理論和LLL算法的性質，采用並改良針對LDEs的格表達式（lattice formulations）；2）提出針對子集和問題的分解（disaggregation）技術；3）創造性地將分解技術與格表達式整合在一起，從而提高求解密度爲一的子集和問題的成功率。 / 數值實驗顯示，我們提出的新整合算法對提高密度爲一的子集和問題的成功率有著顯著的效果。比如，針對維數分別爲20,30和40的密度爲一的子集和問題，對各個維數隨機産生的100個問題，我們的新整合算法均可將成功率提高到100%。同時，針對新整合算法的理論分析顯示，能將短且非0-1的整數解切割掉的分解在達到新整合算法的顯著實驗效果中起到了關鍵作用。 / While systems of linear Diophantine equations (LDEs) with bounded feasible set, including subset sum problem as its special subclass, find wide, and often significant, real-world applications, they unfortunately belong to the NP class in general. Furthermore, the literature has revealed that subset sum problems with their density close to one constitute the hardest subclass of subset sum problems and all the existing solution methods do not perform to a satisfactory level (with low success ratio) even when the problem size is only medium. / We take the challenge in this thesis to investigate lattice formulations for systems of LDEs in which LLL basis reduction algorithm (LLL algorithm) is utilized, propose disaggregation techniques for subset sum problems, and develop a powerful integration of disaggregation techniques with lattice formulations in solving feasible subset sum problems. / More specifically, the contributions in this thesis can be classified into three parts: i) we propose two revised lattice formulations of Aardal et al. (2000) for systems of LDEs to enhance further the computational capability of the LLL algorithm; ii) we study properties related to disaggregation of a single LDE and investigate thoroughly disaggregation schemes based on modular transformations; and iii) we develop a novel version of LLL algorithm by integrating modular disaggregation into the solution process. Promising numerical results have been achieved when applying our newly proposed LLL algorithm in tackling hard subset sum problems with density close to one. For instance, the success ratio can be raised to 100% for 100 randomly generated hard subset sum problems with dimensions 20, 30, and 40, respectively. We carry out theoretical study for possible driving force behind the success of our new algorithm, including dimension reduction of the solution space, information recovering of LDEs, and mechanism in cutting off short non-binary integer solutions when attaching disaggregation with LLL algorithm. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Lu, Bojun. / Thesis (Ph.D.) Chinese University of Hong Kong, 2014. / Includes bibliographical references (leaves 143-153). / Abstracts also in Chinese.

by Wing-Fai Ng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 7-8). / Abstract --- p.1-2 / Introduction --- p.3-6 / References --- p.7-8 / Paper I --- p.9-32 / Paper II --- p.33-60

Shi-Zhong Du. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 56-58). / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- A Nondegenerate Cahn-Hilliard Type Equation --- p.11 / Chapter 2.1 --- A priori estimation --- p.13 / Chapter 2.2 --- Long time existence --- p.19 / Chapter 3 --- The Thin Film Type Equation --- p.31 / Chapter 3.1 --- Positivity for n>4 --- p.31 / Chapter 3.2 --- Improved entropy estimates --- p.36 / Chapter 4 --- Finite Speed of Propagation --- p.43 / Chapter 4.1 --- Finite speed of propagation --- p.44 / Chapter 4.2 --- The regularity of free boundary --- p.54

Yao, Wei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 52-58). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Orbital stability without linear optical lattice --- p.12 / Chapter 2.1 --- Properties of single-spike bound states --- p.12 / Chapter 2.2 --- Proof of theorem 1.0.1 --- p.20 / Chapter 3 --- Orbital stability with linear optical lattice --- p.30 / Chapter 3.1 --- Properties of single-spike bound states --- p.30 / Chapter 3.2 --- Proof of theorem 1.0.2 --- p.33 / Chapter 3.3 --- Proof of theorem 1.0.3 --- p.34 / Chapter 3.4 --- Proof of theorem 1.0.4 --- p.37 / Chapter 4 --- Uniqueness for semilinear elliptic systems --- p.42 / Chapter 4.1 --- Proof of theorem 1.0.5 --- p.42 / Chapter 5 --- Appendix --- p.44 / Chapter 5.1 --- Appendix A --- p.44 / Chapter 5.2 --- Appendix B --- p.45 / Chapter 5.3 --- Appendix C --- p.47