by Mei Long. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 93-98). / Abstracts in English and Chinese. / Table of Contents --- p.v / Abstract --- p.viii / Acknowledgements --- p.x / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Satisfiability Problem --- p.1 / Chapter 1.2 --- Motivation of the Research --- p.1 / Chapter 1.3 --- Overview of the Thesis --- p.2 / Chapter 2 --- Satisfiability Problem --- p.4 / Chapter 2.1 --- Satisfiability Problem --- p.5 / Chapter 2.1.1 --- Basic Definition --- p.5 / Chapter 2.1.2 --- Phase Transitions --- p.5 / Chapter 2.2 --- History --- p.6 / Chapter 2.3 --- The Basic Search Algorithm --- p.8 / Chapter 2.4 --- Some Improvements to the Basic Algorithm --- p.9 / Chapter 2.4.1 --- Satz by Chu-Min Li --- p.9 / Chapter 2.4.2 --- Heuristics and Local Search --- p.12 / Chapter 2.4.3 --- Relaxation --- p.13 / Chapter 2.5 --- Benchmarks --- p.14 / Chapter 2.5.1 --- Specific Problems --- p.14 / Chapter 2.5.2 --- Randomly Generated Problems --- p.14 / Chapter 2.6 --- Software and Internet Information for SAT solving --- p.16 / Chapter 2.6.1 --- Stochastic Local Search Algorithms (incomplete) --- p.16 / Chapter 2.6.2 --- Systematic Search Algorithms (complete) --- p.16 / Chapter 2.6.3 --- Some useful Links to SAT Related Sites --- p.17 / Chapter 3 --- Integer Programming Formulation for Logic Problem --- p.18 / Chapter 3.1 --- SAT Problem --- p.19 / Chapter 3.2 --- MAXSAT Problem --- p.19 / Chapter 3.3 --- Logical Inference Problem --- p.19 / Chapter 3.4 --- Weighted Exact Satisfiability Problem --- p.20 / Chapter 4 --- Integer Programming Formulation for SAT Problem --- p.22 / Chapter 4.1 --- From 3-CNF SAT Clauses to Zero-One IP Constraints --- p.22 / Chapter 4.2 --- Integer Programming Model for 3-SAT --- p.23 / Chapter 4.3 --- The Equivalence of the SAT and the IP --- p.23 / Chapter 4.4 --- Example --- p.24 / Chapter 5 --- Integer Solvability of Linear Programs --- p.27 / Chapter 5.1 --- Unimodularity --- p.27 / Chapter 5.2 --- Totally Unimodularity --- p.28 / Chapter 5.3 --- Some Results on Recognition of Linear Solvability of IP --- p.32 / Chapter 6 --- TU Based Matrix Research Results --- p.33 / Chapter 6.1 --- 2x2 Matrix's TU Property --- p.33 / Chapter 6.2 --- Extended Integer Programming Model for SAT --- p.34 / Chapter 6.3 --- 3x3 Matrix's TU Property --- p.35 / Chapter 7 --- Totally Unimodularity Based Branching-and-Bound Algorithm --- p.38 / Chapter 7.1 --- Introduction --- p.38 / Chapter 7.1.1 --- Enumeration Trees --- p.39 / Chapter 7.1.2 --- The Concept of Branch and Bound --- p.42 / Chapter 7.2 --- TU Based Branching Rule --- p.43 / Chapter 7.2.1 --- How to sort variables based on 2x2 submatrices --- p.43 / Chapter 7.2.2 --- How to sort the rest variables --- p.45 / Chapter 7.3 --- TU Based Bounding Rule --- p.46 / Chapter 7.4 --- TU Based Branch-and-Bound Algorithm --- p.47 / Chapter 7.5 --- Example --- p.49 / Chapter 8 --- Numerical Result --- p.57 / Chapter 8.1 --- Experimental Result --- p.57 / Chapter 8.2 --- Statistical Results of ILOG CPLEX --- p.59 / Chapter 9 --- Conclusions --- p.61 / Chapter 9.1 --- Contributions --- p.61 / Chapter 9.2 --- Future Work --- p.62 / Chapter A --- The Coefficient Matrix A for Example in Chapter 7 --- p.64 / Chapter B --- The Detailed Numerical Information of Solution Process for Exam- ple in Chapter 7 --- p.66 / Chapter C --- Experimental Result --- p.67 / Chapter C.1 --- "# of variables: 20, # of clauses: 91" --- p.67 / Chapter C.2 --- "# of variables: 50, # of clauses: 218" --- p.70 / Chapter C.3 --- # of variables: 75，# of clauses: 325 --- p.73 / Chapter C.4 --- "# of variables: 100, # of clauses: 430" --- p.76 / Chapter D --- Experimental Result of ILOG CPLEX --- p.80 / Chapter D.1 --- # of variables: 20´ة # of clauses: 91 --- p.80 / Chapter D.2 --- # of variables: 50，#of clauses: 218 --- p.83 / Chapter D.3 --- # of variables: 75，# of clauses: 325 --- p.86 / Chapter D.4 --- "# of variables: 100, # of clauses: 430" --- p.89 / Bibliography --- p.93

本論文會討論有關於二維的耗散準地轉方程，特別是有關於存在性及規律性的問題。有關的討論主要取決於該方程的分數冪。這份論文中將會介紹一些最近有關耗散準地轉方程的結果。 / This paper is discussing about problems in the 2-D Dissipative Quasi-Geostrophic equation, mainly the existence and regularity results, depending on the fractional power. We will introduce the recent results in this topics. / Detailed summary in vernacular field only. / Kwan, Danny. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 65-68). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Main Results in QG --- p.9 / Chapter 2.1 --- Definitions --- p.9 / Chapter 2.2 --- Subcritical Case (γ>[1/2]) --- p.10 / Chapter 2.3 --- Critical Case (γ=[1/2]) --- p.10 / Chapter 2.4 --- Supercritical Case (γ<[1/2]) --- p.11 / Chapter 3 --- The Proofs of Main Results --- p.12 / Chapter 3.1 --- Some Previous Results --- p.12 / Chapter 3.2 --- Subcritical Case (γ>[1/2]) --- p.14 / Chapter 3.2.1 --- Proof of Theorem 1 --- p.14 / Chapter 3.2.2 --- Proof of Theorem 2 --- p.19 / Chapter 3.2.3 --- Proof of Corollary 1 --- p.25 / Chapter 3.2.4 --- Summary for Subcritical Case --- p.27 / Chapter 3.3 --- Critical Case (γ=[1/2]) --- p.28 / Chapter 3.3.1 --- Proof of Theorem 3 --- p.28 / Chapter 3.3.2 --- Proof of Theorem 4 --- p.36 / Chapter 3.3.3 --- Summary for Critical Case --- p.41 / Chapter 3.4 --- Supercritical Case (γ<[1/2]) --- p.41 / Chapter 3.4.1 --- Proof of Theorem 5 --- p.41 / Chapter 3.4.2 --- Proof of Theorem 6 --- p.50 / Chapter 3.4.3 --- Proof of Theorem 7 --- p.54 / Chapter 3.4.4 --- Summary for Supercritical Case --- p.64 / Chapter 4 --- Further Development --- p.65

In this thesis, we consider the following two important subjects in the modern variational analysis for the corresponding nonconvex/nonmonotone and nonsmooth cases: geometric results and the variational inequality problem. By using the variational technique, we first present several nonsmooth (nonconvex) geometric results (including an approximate projection result, an extended extremal principle, nonconvex separation theorems, a nonconvex generalization of the Bishop-Phelps theorem and a separable point result) which extend some fundamental theorems in linear functional analysis, convex analysis and optimization theory. Then, by transforming the variational inequality problem into equivalent optimization problems, we establish some error bound result for the nonsmooth and nonmonotone variational inequality problem. / Variational arguments are classical techniques whose use can be traced back to the early development of the calculus of variations. Rooted in the physical principle of least action they have evolved greatly in connection with applications in optimization theory and optimal control. Recently, the discovery of modern variational principles and nonsmooth analysis further expand the range of applications of these techniques and give a new way for extending some geometric results in linear functional analysis and convex analysis. / Li, Guoyin. / "August 2007." / Adviser: Kung-Fu Ng. / Source: Dissertation Abstracts International, Volume: 69-02, Section: B, page: 1043. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 80-86). / 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. / Abstract in English and Chinese. / School code: 1307.

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Nonlocal phenomena are ubiquitous in nature. The nonlocal models investigated in this thesis use integration in replace of differentiation and provide alternatives to the classical partial differential equations. The nonlocal interaction kernels in the models are assumed to be as general as possible and usually involve finite range of nonlocal interactions. Such settings on one hand allow us to connect nonlocal models with the existing classical models through various asymptotic limits of the modeling parameter, and on the other hand enjoy practical significance especially for multiscale modeling and simulations. To make connections with classical models at the discrete level, the central theme of the numerical analysis for nonlocal models in this thesis concerns with numerical schemes that are robust under the changes of modeling parameters, with mathematical analysis provided as theoretical foundations. Together with extensive discussions of linear nonlocal diffusion and nonlocal mechanics models, we also touch upon other topics such as high order nonlocal models, nonlinear nonlocal fracture models and coupling of models characterized by different scales.

Neural data often span multiple indices, such as neuron, experimental condition, trial, and time, resulting in a tensor or multidimensional array. Standard approaches to neural data analysis often rely on matrix factorization techniques, such as principal component analysis or nonnegative matrix factorization. Any inherent tensor structure in the data is lost when flattened into a matrix. Here, we analyze datasets from primary motor cortex from the perspective of tensor analysis, and develop a theory for how tensor structure relates to certain computational properties of the underlying system. Applied to the motor cortex datasets, we reveal that neural activity is best described by condition-independent dynamics as opposed to condition-dependent relations to external movement variables. Motivated by this result, we pursue one further tensor-related analysis, and two further dynamical systems-related analyses. First, we show how tensor decompositions can be used to denoise neural signals. Second, we apply system identification to the cortex- to-muscle transformation to reveal the intermediate spinal dynamics. Third, we fit recurrent neural networks to muscle activations and show that the geometric properties observed in motor cortex are naturally recapitulated in the network model. Taken together, these results emphasize (on the data analysis side) the role of tensor structure in data and (on the theoretical side) the role of motor cortex as a dynamical system.

Tensors, a.k.a. multi-dimensional arrays, arise naturally when modeling higher-order objects and relations. Among ubiquitous applications including image processing, collaborative filtering, demand forecasting and higher-order statistics, there are two recurring themes in general: tensor recovery and tensor decomposition. The first one aims to recover the underlying tensor from incomplete information; the second one is to study a variety of tensor decompositions to represent the array more concisely and moreover to capture the salient characteristics of the underlying data. Both topics are respectively addressed in this thesis. Chapter 2 and Chapter 3 focus on low-rank tensor recovery (LRTR) from both theoretical and algorithmic perspectives. In Chapter 2, we first provide a negative result to the sum of nuclear norms (SNN) model---an existing convex model widely used for LRTR; then we propose a novel convex model and prove this new model is better than the SNN model in terms of the number of measurements required to recover the underlying low-rank tensor. In Chapter 3, we first build up the connection between robust low-rank tensor recovery and the compressive principle component pursuit (CPCP), a convex model for robust low-rank matrix recovery. Then we focus on developing convergent and scalable optimization methods to solve the CPCP problem. In specific, our convergent method, proposed by combining classical ideas from Frank-Wolfe and proximal methods, achieves scalability with linear per-iteration cost. Chapter 4 generalizes the successive rank-one approximation (SROA) scheme for matrix eigen-decomposition to a special class of tensors called symmetric and orthogonally decomposable (SOD) tensor. We prove that the SROA scheme can robustly recover the symmetric canonical decomposition of the underlying SOD tensor even in the presence of noise. Perturbation bounds, which can be regarded as a higher-order generalization of the Davis-Kahan theorem, are provided in terms of the noise magnitude.

Submitted by Repositorio Tarefa (tarefarepositorio@gmail.com) on 2018-09-11T00:11:40Z No. of bitstreams: 2 TituloDeniseIlidio.pdf: 10074 bytes, checksum: 051a02493cf99b99197f680aeebab230 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-09-11T00:11:40Z (GMT). No. of bitstreams: 2 TituloDeniseIlidio.pdf: 10074 bytes, checksum: 051a02493cf99b99197f680aeebab230 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-09-03 / This paper deals with the analysis, calculation and modeling of the vacuum system / Este trabalho trata sobre a análise, o cálculo e a modelagem do sistema a vácuo

This study examined the effects of using graphing calculators with a numerical approach designed by the researcher on students learning of limits and derivatives in an Applied Calculus course at a community college. The purposes of this study were to investigate the following: (1) students achievement in solving limit problems (Skills, Concepts, and Applications) with a numerical approach compared to that of students who solved limit problems with a traditional approach (primarily an algebraic approach); and (2) students achievement in solving derivative problems (Skills, Concepts, and Applications) with a numerical approach compared to that of students who solved derivative problems with a traditional approach (primarily an algebraic approach). Students (n = 93) in all four daytime sections of an Applied Calculus course in a community college participated in the study during the spring 2005 semester. One of two MWF sections and one of two TR sections served as the treatment groups; the other two sections served as the control groups. Two instructors other than the researcher participated in the study. Instructor A taught one treatment group (a TR section) and one control group (a MWF section); instructor B taught one treatment group (a MWF section) and one control group (a TR section). Dependent variables were achievement to solve skill, concept, and application limit problems and skill, concept, and application derivative problems, measured by two teacher-made tests. A pretest administered on the first day of class determined that no significant difference existed between the groups on prerequisite algebra skills. Separate ANCOVA tests were conducted on the skill, concept, and application portions of each of the limit and derivative exams. Data analyses revealed the following: (1) there was no significant difference found on the skill portion of the limit topic (unit 1 exam) due to instruction or to instructor; (2) there was a significant difference found on the concept portion of the limit topic due to instruction and to instructor; (3) there was a significant difference found on the application portion of the limit topic due to instruction but not due to instructor; (4) the interaction effects between instructor and instruction were not significant on the skill, concept, and application portions of the limit topic; (5) there was a significant difference found on the skill portion of the derivative topic (unit 2 exam) due to instruction but not due to instructor; (6) there was a significant difference found on the concept portion of the derivative topic due to instruction and to instructor; (7) there was a significant difference found on the application portion of the derivative topic due to instruction but not due to instructor; and (8) the interaction effects between instructor and instruction were not significant on the skill, concept, and application portions of the derivative topic. All significant differences were in favor of the treatment group.

The study examined classroom instructional practices and teacher's professed conceptions about teaching and learning college calculus in relationship to the implementation of scientific-programmable-graphics (SPG) calculators. The study occurred at a university not affiliated with any reform project. The participants were not the catalysts seeking to implement calculus reform, but expressed a willingness to teach the first quarter calculus course with the SPG calculator. The research design was based on qualitative methods using comparative case studies of five teachers. Primary data were collected through pre-school interviews and weekly classroom observations with subsequent interviews. Teachers' profiles were established describing general conceptions of teaching calculus, instructional practices, congruence between conceptions and practice, conceptions about teaching using SPG calculators, instructional practice with SPG calculators, and the relationship of conceptions and practice with SPG calculators. Initially, all the teachers without prior experience using SPG calculators indicated concern and skepticism about the usefulness of the technology in teaching calculus and were uncertain how to utilize the calculator in teaching the calculus concepts. During the study the teachers became less skeptical about the calculator's usefulness and found it effective for illustrating graphs. Some of the teachers' exams included more conceptual and graphically-oriented questions, but were not significantly different from traditional exams. Findings indicated the college teachers' conceptions of teaching calculus were generally consistent with their instructional practice when not constrained by time. The teachers did not perceive a dramatic change in their instructional practices. Rather, the new graphing approach curriculum and technology were assimilated into the teachers' normal teaching practices. No major shifts in the role of the teachers were detected. Two teachers demonstrated slight differences in their roles when the SPG calculators were used in class. One was a consultant to the students as they used the SPG calculators; the other became a fellow learner as the students presented different features on the calculator. Use of the calculator was influenced by several factors: inexperience with the calculator, time constraints, setting up the classroom display calculator, preferred teaching styles and emphasis, and a willingness to risk experimenting with established teaching practices and habits. / Graduation date: 1996