by Chu Kin Fung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 73-75). / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Preliminaries --- p.6 / Chapter 2.1 --- Basic Notations --- p.6 / Chapter 2.2 --- Function Spaces --- p.8 / Chapter 3 --- Examples of Exactly Solvable Cases --- p.19 / Chapter 4 --- Existence and Uniqueness of Solutions --- p.29 / Chapter 5 --- Properties of Solutions --- p.41 / Chapter 6 --- Perturbation Expansion --- p.49

by Chi-cheong Szeto. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 162-165). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgment --- p.ii / Table of Contents --- p.iii / Chapter Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Time series prediction --- p.1 / Chapter 1.2 --- Forecasting models --- p.1 / Chapter 1.2.1 --- Networks using time delays --- p.2 / Chapter 1.2.1.1 --- Model description --- p.2 / Chapter 1.2.1.2 --- Limitation --- p.3 / Chapter 1.2.2 --- Networks using context units --- p.3 / Chapter 1.2.2.1 --- Model description --- p.3 / Chapter 1.2.2.2 --- Limitation --- p.6 / Chapter 1.2.3 --- Layered fully recurrent networks --- p.6 / Chapter 1.2.3.1 --- Model description --- p.6 / Chapter 1.2.3.2 --- Our selection and motivation --- p.8 / Chapter 1.2.4 --- Other models --- p.8 / Chapter 1.3 --- Learning methods --- p.8 / Chapter 1.3.1 --- First order and second order methods --- p.9 / Chapter 1.3.2 --- Nonlinear least squares methods --- p.11 / Chapter 1.3.2.1 --- Levenberg-Marquardt method ´ؤ our selection and motivation --- p.13 / Chapter 1.3.2.2 --- Levenberg-Marquardt method - algorithm --- p.13 / Chapter 1.3.3 --- "Batch mode, semi-sequential mode and sequential mode of updating" --- p.15 / Chapter 1.4 --- Jacobian matrix calculations in recurrent networks --- p.15 / Chapter 1.4.1 --- RTBPTT-like Jacobian matrix calculation --- p.15 / Chapter 1.4.2 --- RTRL-like Jacobian matrix calculation --- p.17 / Chapter 1.4.3 --- Comparison between RTBPTT-like and RTRL-like calculations --- p.18 / Chapter 1.5 --- Computation complexity reduction techniques in recurrent networks --- p.19 / Chapter 1.5.1 --- Architectural approach --- p.19 / Chapter 1.5.1.1 --- Recurrent connection reduction method --- p.20 / Chapter 1.5.1.2 --- Treating the feedback signals as additional inputs method --- p.20 / Chapter 1.5.1.3 --- Growing network method --- p.21 / Chapter 1.5.2 --- Algorithmic approach --- p.21 / Chapter 1.5.2.1 --- History cutoff method --- p.21 / Chapter 1.5.2.2 --- Changing the updating frequency from sequential mode to semi- sequential mode method --- p.22 / Chapter 1.6 --- Motivation for using block-diagonal Hessian matrix --- p.22 / Chapter 1.7 --- Objective --- p.23 / Chapter 1.8 --- Organization of the thesis --- p.24 / Chapter Chapter 2 --- Learning with the block-diagonal Hessian matrix --- p.25 / Chapter 2.1 --- Introduction --- p.25 / Chapter 2.2 --- General form and factors of block-diagonal Hessian matrices --- p.25 / Chapter 2.2.1 --- General form of block-diagonal Hessian matrices --- p.25 / Chapter 2.2.2 --- Factors of block-diagonal Hessian matrices --- p.27 / Chapter 2.3 --- Four particular block-diagonal Hessian matrices --- p.28 / Chapter 2.3.1 --- Correlation block-diagonal Hessian matrix --- p.29 / Chapter 2.3.2 --- One-unit block-diagonal Hessian matrix --- p.35 / Chapter 2.3.3 --- Sub-network block-diagonal Hessian matrix --- p.35 / Chapter 2.3.4 --- Layer block-diagonal Hessian matrix --- p.36 / Chapter 2.4 --- Updating methods --- p.40 / Chapter Chapter 3 --- Data set and setup of experiments --- p.41 / Chapter 3.1 --- Introduction --- p.41 / Chapter 3.2 --- Data set --- p.41 / Chapter 3.2.1 --- Single sine --- p.41 / Chapter 3.2.2 --- Composite sine --- p.42 / Chapter 3.2.3 --- Sunspot --- p.43 / Chapter 3.3 --- Choices of recurrent neural network parameters and initialization methods --- p.44 / Chapter 3.3.1 --- "Choices of numbers of input, hidden and output units" --- p.45 / Chapter 3.3.2 --- Initial hidden states --- p.45 / Chapter 3.3.3 --- Weight initialization method --- p.45 / Chapter 3.4 --- Method of dealing with over-fitting --- p.47 / Chapter Chapter 4 --- Updating methods --- p.48 / Chapter 4.1 --- Introduction --- p.48 / Chapter 4.2 --- Asynchronous updating method --- p.49 / Chapter 4.2.1 --- Algorithm --- p.49 / Chapter 4.2.2 --- Method of study --- p.50 / Chapter 4.2.3 --- Performance --- p.51 / Chapter 4.2.4 --- Investigation on poor generalization --- p.52 / Chapter 4.2.4.1 --- Hidden states --- p.52 / Chapter 4.2.4.2 --- Incoming weight magnitudes of the hidden units --- p.54 / Chapter 4.2.4.3 --- Weight change against time --- p.56 / Chapter 4.3 --- Asynchronous updating with constraint method --- p.68 / Chapter 4.3.1 --- Algorithm --- p.68 / Chapter 4.3.2 --- Method of study --- p.69 / Chapter 4.3.3 --- Performance --- p.70 / Chapter 4.3.3.1 --- Generalization performance --- p.70 / Chapter 4.3.3.2 --- Training time performance --- p.71 / Chapter 4.3.4 --- Hidden states and incoming weight magnitudes of the hidden units --- p.73 / Chapter 4.3.4.1 --- Hidden states --- p.73 / Chapter 4.3.4.2 --- Incoming weight magnitudes of the hidden units --- p.73 / Chapter 4.4 --- Synchronous updating methods --- p.84 / Chapter 4.4.1 --- Single λ and multiple λ's synchronous updating methods --- p.84 / Chapter 4.4.1.1 --- Algorithm of single λ synchronous updating method --- p.84 / Chapter 4.4.1.2 --- Algorithm of multiple λ's synchronous updating method --- p.85 / Chapter 4.4.1.3 --- Method of study --- p.87 / Chapter 4.4.1.4 --- Performance --- p.87 / Chapter 4.4.1.5 --- Investigation on long training time: analysis of λ --- p.89 / Chapter 4.4.2 --- Multiple λ's with line search synchronous updating method --- p.97 / Chapter 4.4.2.1 --- Algorithm --- p.97 / Chapter 4.4.2.2 --- Performance --- p.98 / Chapter 4.4.2.3 --- Comparison of λ --- p.100 / Chapter 4.5 --- Comparison between asynchronous and synchronous updating methods --- p.101 / Chapter 4.5.1 --- Final training time --- p.101 / Chapter 4.5.2 --- Computation load per complete weight update --- p.102 / Chapter 4.5.3 --- Convergence speed --- p.103 / Chapter 4.6 --- Comparison between our proposed methods and the gradient descent method with adaptive learning rate and momentum --- p.111 / Chapter Chapter 5 --- Number and sizes of the blocks --- p.113 / Chapter 5.1 --- Introduction --- p.113 / Chapter 5.2 --- Performance --- p.113 / Chapter 5.2.1 --- Method of study --- p.113 / Chapter 5.2.2 --- Trend of performance --- p.115 / Chapter 5.2.2.1 --- Asynchronous updating method --- p.115 / Chapter 5.2.2.2 --- Synchronous updating method --- p.116 / Chapter 5.3 --- Computation load per complete weight update --- p.116 / Chapter 5.4 --- Convergence speed --- p.117 / Chapter 5.4.1 --- Trend of inverse of convergence speed --- p.117 / Chapter 5.4.2 --- Factors affecting the convergence speed --- p.117 / Chapter Chapter 6 --- Weight-grouping methods --- p.125 / Chapter 6.1 --- Introduction --- p.125 / Chapter 6.2 --- Training time and generalization performance of different weight-grouping methods --- p.125 / Chapter 6.2.1 --- Method of study --- p.125 / Chapter 6.2.2 --- Performance --- p.126 / Chapter 6.3 --- Degree of approximation of block-diagonal Hessian matrix with different weight- grouping methods --- p.128 / Chapter 6.3.1 --- Method of study --- p.128 / Chapter 6.3.2 --- Performance --- p.128 / Chapter Chapter 7 --- Discussion --- p.150 / Chapter 7.1 --- Advantages and disadvantages of using block-diagonal Hessian matrix --- p.150 / Chapter 7.1.1 --- Advantages --- p.150 / Chapter 7.1.2 --- Disadvantages --- p.151 / Chapter 7.2 --- Analysis of computation complexity --- p.151 / Chapter 7.2.1 --- Trend of computation complexity of each calculation --- p.154 / Chapter 7.2.2 --- Batch mode of updating --- p.155 / Chapter 7.2.3 --- Sequential mode of updating --- p.155 / Chapter 7.3 --- Analysis of storage complexity --- p.156 / Chapter 7.3.1 --- Trend of storage complexity of each set of variables --- p.157 / Chapter 7.3.2 --- Trend of overall storage complexity --- p.157 / Chapter 7.4 --- Parallel implementation --- p.158 / Chapter 7.5 --- Alternative implementation of weight change constraint --- p.158 / Chapter Chapter 8 --- Conclusions --- p.160 / References --- p.162

Law, Man Fai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 84-86). / Text in English; abstract also in Chinese. / Law, Man Fai. / Contents --- p.i / Abstract --- p.iii / Acknowledgement --- p.iv / List of Figures --- p.v / List of Tables --- p.ix / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- Spectral Representation of Composite Materials --- p.4 / Chapter 2.1 --- Spectral method --- p.6 / Chapter 2.2 --- General properties --- p.10 / Chapter 2.3 --- Duality --- p.12 / Chapter 2.4 --- Dilute limit --- p.13 / Chapter 2.5 --- Effective-medium approximation (EMA) --- p.15 / Chapter 2.6 --- Moment expansions --- p.16 / Chapter Chapter 3. --- Local Field Effect and Depolarization Factor --- p.19 / Chapter 3.1 --- Isotropic homogeneous media --- p.20 / Chapter 3.2 --- Linear anisotropic homogeneous media --- p.22 / Chapter 3.3 --- Inhomogeneous anisotropic media --- p.24 / Chapter Chapter 4. --- Simulation on Correlated Microstructures --- p.26 / Chapter 4.1 --- Solving the nonlinear impedance networks --- p.27 / Chapter 4.2 --- Models of correlated microstructure --- p.29 / Chapter 4.2.1 --- Two-site correlated microstruture --- p.29 / Chapter 4.2.2 --- Environment correlated microstructure --- p.38 / Chapter 4.3 --- Conclusions --- p.45 / Chapter Chapter 5. --- Simulation on Anisotropic Microstructure --- p.46 / Chapter 5.1 --- Solving anisotropic impedance networks --- p.50 / Chapter 5.2 --- Simulation and results --- p.50 / Chapter 5.2.1 --- Parallel response --- p.51 / Chapter 5.2.2 --- Perpendicular response --- p.57 / Chapter 5.2.3 --- Unpolarized response --- p.61 / Chapter 5.3 --- Conclusions --- p.65 / Chapter Chapter 6. --- Conclusions --- p.66 / Appendix A. Symbolic Simulation --- p.67 / Chapter A.1 --- Formalism --- p.67 / Chapter A.2 --- Scaling Properties --- p.69 / Chapter A.3 --- The simulation --- p.71 / Appendix B. Fluctuation of Local Field in Composite Materials --- p.74 / Chapter B.l --- Simulations and Results --- p.74 / Appendix C. Lattice Animals in Correlated Network --- p.82 / Bibliography --- p.84

"As a part of an effective SETAR (self-exciting threshold autoregressive) mod- eling methodology, it is important to identify processes exhibiting SETAR-type non- linearity. A number of tests of nonlinearity have been developed in the literature, including those of Keenan (1985), Petruccelli and Davies (1986), Tsay (1986, 1989), Luukkonen (1988), and Chan and Tong (1990). However, it has recently been shown that all these tests perform poorly for SETAR-type nonlinearity detection in the presence of outliers. In this project we develop an improved test for SETAR-type nonlinearity in time series. The test is an outlier-robust variant of the Petruccelli and Davies (1986) test based on the cumulative sums of ordered weighted residuals from generalized maximum likelihood fits (which we call CUSUM-GM). The properties of the proposed CUSUM-GM test are illustrated by means of Monte Carlo simulations. The merits, in terms of size and power, of the proposed test are evaluated relative to the test based on ordered residuals from the ordinary least squares fit (which we call CUSUM-LS) and also to that of other tests for nonlinearity developed in literature. The simulations are run for uncontaminated data and for data contaminated with additive and innovational outliers. The simulation study strongly supports the validity of the proposed robust CUSUM-GM test, particularly in situations in which outliers might be a problem."

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1979. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Vita. / Includes bibliographical references. / Ph.D.

本博士论文对于有结构但又有相当一般性的约束条件下的非线性优化问题给出了系统性研究。比较经典的例子包指球面约束下的多重线性函数优化问题。这些模型已被广泛应用于数值线性代数、材料科学、量子物理学、信号处理、语音识别、生物医学工程以及控制论等。本论文着重探讨一类特定的方法来解这些广义模型，即块变量改进方法。具体地说，我们构造了一类块坐标下降型搜索算法来解带块变量结构的非线性优化问题。这类算法通过每次迭代中只更新一块变量以达到最大限度的目标函数值的改进(因而，这一新搜索算法命名为最优块改进算法(简称为MBI)。之后，我们重点研究了该算法在求解众多领域中实际问题的潜在能力。首先，这一算法可以直接应用于生物信息学中聚类基因表达数据的一种新模型的设计及求解。接着，我们把注意力转移到球面约束下的齐次多项式优化问题，此问题与张量的最优秩-1 逼近问题相关。对于这一优化问题， MBI 算法通常可以在较少的计算时间内找到全局最优解。第三，我们继续深入研究多项式优化问题，在双协半正定的新概念下建立了齐次多项式优化问题与其多重线性优化问题关系的一般性结果。最后，我们在Tucker 分解的框架下给出了求解高阶张量的最优多重线性秩的逼近问题的方法，并提出一种新的模型和算法来解未知变量数的Tucker 分解问题。本论文讨论并试验了一些应用实例，数值实验表明所提出的算法分别对于求解以上这些问题是可行并有效的。 / In this thesis we present a systematic analysis for optimization of a general nonlinear function, subject to some fairly general constraints. A typical example includes the optimization of a multilinear tensor function over spherical constraints. Such models have found wide applications in numerical linear algebra, material sciences, quantum physics, signal processing, speech recognition, biomedical engineering, and control theory. This thesis is mainly concerned with a specific approach to solve such generic models: the block variable improvement method. Specifically, we establish a block coordinate descent type search method for nonlinear optimization, which accepts only a block update that achieves the maximum improvement (hence the name of our new search method: maximum block improvement (MBI)). Then, we focus on the potential capability of this method for solving problems in various research area. First, we demonstrate that this method can be directly used in designing a new framework for co-clustering gene expression data in the area of bioinformatics. Second, we turn our attention to the spherically constrained homogeneous polynomial optimization problem, which is related to best rank-one approximation of tensors. The MBI method usually finds the global optimal solution at a low computational cost. Third, we continue to consider polynomial optimization problems. A general result between homogeneous polynomials and multi-linear forms under the concept of co-quadratic positive semidefinite is established. Finally, we consider the problem of finding the best multi-linear rank approximation of a higher-order tensor under the framework of Tucker decomposition, and also propose a new model and algorithms for computing Tucker decomposition with unknown number of components. Some real application examples are discussed and tested, and numerical experiments are reported to reveal the good practical performance and efficiency of the proposed algorithms for solving those problems. / Detailed summary in vernacular field only. / Chen, Bilian. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 86-98). / Abstract also in Chinese. / Abstract --- p.iii / Acknowledgements --- p.vii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Overview --- p.1 / Chapter 1.2 --- Notations and Preliminaries --- p.5 / Chapter 1.2.1 --- The Tensor Operations --- p.6 / Chapter 1.2.2 --- The Tensor Ranks --- p.9 / Chapter 1.2.3 --- Polynomial Functions --- p.11 / Chapter 2 --- The Maximum Block Improvement Method --- p.12 / Chapter 2.1 --- Introduction --- p.12 / Chapter 2.2 --- MBI Method and Convergence Analysis --- p.14 / Chapter 3 --- Co-Clustering of Gene Expression Data --- p.19 / Chapter 3.1 --- Introduction --- p.19 / Chapter 3.2 --- A New Generic Framework for Co-Clustering Gene Expression Data --- p.22 / Chapter 3.2.1 --- Tensor Optimization Model of The Co-Clustering Problem --- p.22 / Chapter 3.2.2 --- The MBI Method for Co-Clustering Problem --- p.23 / Chapter 3.3 --- Algorithm for Co-Clustering 2D Matrix Data --- p.25 / Chapter 3.4 --- Numerical Experiments --- p.27 / Chapter 3.4.1 --- Implementation Details and Some Discussions --- p.27 / Chapter 3.4.2 --- Testing Results using Microarray Datasets --- p.30 / Chapter 3.4.3 --- Testing Results using 3D Synthesis Dataset --- p.32 / Chapter 4 --- Polynomial Optimization with Spherical Constraint --- p.34 / Chapter 4.1 --- Introduction --- p.34 / Chapter 4.2 --- Generalized Equivalence Result --- p.37 / Chapter 4.3 --- Spherically Constrained Homogeneous Polynomial Optimization --- p.41 / Chapter 4.3.1 --- Implementing MBI on Multilinear Tensor Form --- p.42 / Chapter 4.3.2 --- Relationship between Homogeneous Polynomial Optimization over Spherical Constraint and Tensor Relaxation Form --- p.43 / Chapter 4.3.3 --- Finding a KKT point for Homogeneous Polynomial Optimization over Spherical Constraint --- p.45 / Chapter 4.4 --- Numerical Experiments on Randomly Simulated Data --- p.47 / Chapter 4.4.1 --- Multilinear Tensor Function over Spherical Constraints --- p.49 / Chapter 4.4.2 --- Tests of Another Implementation of MBI --- p.49 / Chapter 4.4.3 --- General Polynomial Function over Quadratic Constraints --- p.51 / Chapter 4.5 --- Applications --- p.53 / Chapter 4.5.1 --- Rank-One Approximation of Super-Symmetric Tensors --- p.54 / Chapter 4.5.2 --- Magnetic Resonance Imaging --- p.55 / Chapter 5 --- Logarithmically Quasiconvex Optimization --- p.58 / Chapter 5.1 --- Introduction --- p.58 / Chapter 5.2 --- Logarithmically Quasiconvex Optimization --- p.60 / Chapter 5.2.1 --- A Simple Motivating Example --- p.61 / Chapter 5.2.2 --- Co-Quadratic Positive Semide nite Tensor Form --- p.61 / Chapter 5.2.3 --- Equivalence at Maxima --- p.64 / Chapter 6 --- The Tucker Decomposition and Generalization --- p.68 / Chapter 6.1 --- Introduction --- p.68 / Chapter 6.2 --- Convergence of Traditional Tucker Decomposition --- p.71 / Chapter 6.3 --- Tucker Decomposition with Unknown Number of Components --- p.73 / Chapter 6.3.1 --- Problem Formulation --- p.74 / Chapter 6.3.2 --- Implementing the MBI Method on Tucker Decomposition with Unknown Number of Components --- p.75 / Chapter 6.3.3 --- A Heuristic Approach --- p.79 / Chapter 6.4 --- Numerical Experiments --- p.80 / Chapter 7 --- Conclusion and Recent Developments --- p.83 / Bibliography --- p.86

時間序列之變點鬥檻模型是一種非線性的模型。此論文探討有關該模型之參數估計，同時對其參數估計作出統計分析。我們運用了遺傳式計算機運算來估計這些參數及對其作出研究。我們利用了MDL來對比不同的變點門檻模型，同時我們也利用了MDL來選取對應的變點門檻模型。 / This article considers the problem of modeling non-linear time series by using piece-wise TAR model. The numbers of change points, the numbers of thresholds and the corresponding order of AR in each piecewise TAR segments are assumed unknown. The goal is to nd out the “best“ combination of the number of change points, the value of threshold in each time segment, and the underlying AR order for each threshold regime. A genetic algorithm is implemented to solve this optimization problem and the minimum description length principle is applied to compare various segmented TAR. We also show the consistency of the minimal MDL model selection procedure under general regularity conditions on the likelihood function. / Detailed summary in vernacular field only. / Tang, Chong Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 45-47). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 2 --- Minimum Description Length for Pure TAR --- p.4 / Chapter 2.1 --- Model selection using Minimum Description Length for Pure TAR --- p.4 / Chapter 2.1.1 --- Derivation of Minimum Description Length for Pure TAR --- p.5 / Chapter 2.2 --- Optimization Using Genetic Algorithms (GA) --- p.7 / Chapter 2.2.1 --- General Description --- p.7 / Chapter 2.2.2 --- Implementation Details --- p.9 / Chapter 3 --- Minimum Description Length for TAR models with structural change --- p.13 / Chapter 3.1 --- Model selection using Minimum Description Length for TAR models with structural change --- p.13 / Chapter 3.1.1 --- Derivation of Minimum Description Length for TAR models with structural change --- p.14 / Chapter 3.2 --- Optimization Using Genetic Algorithms --- p.17 / Chapter 4 --- Main Result --- p.20 / Chapter 4.1 --- Main results --- p.20 / Chapter 4.1.1 --- Model Selection using minimum description length --- p.21 / Chapter 5 --- Simulation Result --- p.24 / Chapter 5.1 --- Simulation results --- p.24 / Chapter 5.1.1 --- Example of TAR Model Without Structural Break --- p.24 / Chapter 5.1.2 --- Example of TAR Model With Structural Break I --- p.26 / Chapter 5.1.3 --- Example of TAR Model With Structural Break II --- p.29 / Chapter 6 --- An empirical example --- p.33 / Chapter 6.1 --- An empirical example --- p.33 / Chapter 7 --- Consistency of the CLSE --- p.36 / Chapter 7.1 --- Consistency of the TAR parameters --- p.36 / Chapter 7.1.1 --- Consistency of the estimation of number of threshold --- p.36 / Chapter 7.1.2 --- Consistency of the change point parameters --- p.43 / Bibliography --- p.45

In this dissertation I suggest a new (regularized) weighted quantile regression estimation approach for nonlinear regression models and double threshold ARCH (DTARCH) models. I allow the number of parameters in the nonlinear regression models to be fixed or diverge. The proposed estimation method is robust and efficient and is applicable to other models. I use the adaptive-LASSO and SCAD regularization to select parameters in the nonlinear regression models. I simultaneously estimate the AR and ARCH parameters in the DTARCH model using the proposed weighted quantile regression. The values of the proposed methodology are revealed. / Keywords: Weighted quantile regression, Adaptive-LASSO, High dimensionality, Model selection, Oracle property, SCAD, DTARCH models. / Under regularity conditions, I establish asymptotic distributions of the proposed estimators, which show that the model selection methods perform as well as if the correct submodels are known in advance. I also suggest an algorithm for fast implementation of the proposed methodology. Simulations are conducted to compare different estimators, and a real example is used to illustrate their performance. / Jiang, Xuejun. / Adviser: Xinyuan Song. / Source: Dissertation Abstracts International, Volume: 73-01, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 86-92). / 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, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web.

by Siu Wing Hon. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references. / by Siu Wing Hon. / Acknowledgement --- p.i / Abstract --- p.ii / Publication List --- p.iv / Chapter 1 --- Introduction --- p.1 / References --- p.6 / Chapter 2 --- Real-Space Renormalization Group (RG) Theory in Electrical Conduction --- p.9 / Chapter 2.1 --- Scale Invariance --- p.10 / Chapter 2.2 --- Critical Exponents --- p.14 / Chapter 2.3 --- Alternative View-Point of RG Theory --- p.15 / References --- p.18 / Chapter 3 --- "Weakly Nonlinear Composites: Critical Behavior, Flicker Noise and Crossover Behavior" --- p.19 / Chapter 3.1 --- Introduction --- p.19 / Chapter 3.2 --- Formalism --- p.20 / Chapter 3.3 --- Critical Exponents by RG Method --- p.22 / Chapter 3.4 --- Connection to Flicker Noise Problem and Crossover Behavior --- p.25 / Chapter 3.5 --- Discussions and Conclusions --- p.27 / References --- p.28 / Chapter 4 --- Critical Behavior of Strongly Nonlinear Composites --- p.30 / Chapter 4.1 --- Introduction --- p.30 / Chapter 4.2 --- Formalism --- p.31 / Chapter 4.3 --- Applications of RG Theory to Strongly Nonlinear Composites --- p.32 / Chapter 4.4 --- Connections with Links-Nodes-Blobs picture --- p.36 / Chapter 4.5 --- Discussions and Conclusions --- p.39 / References --- p.41 / Chapter 5 --- "Enhanced Nonlinear Response of Superconductor-Normal-conductor Composite Wires, Strips and Rods" --- p.43 / Chapter 5.1 --- Introduction --- p.43 / Chapter 5.2 --- Formalism --- p.45 / Chapter 5.3 --- Linear and Nonlinear Responses of Composite Wires --- p.46 / Chapter 5.4 --- Linear and Nonlinear Response of Composite Strips --- p.49 / Chapter 5.5 --- Linear and Nonlinear Responses of Composite Rods --- p.56 / Chapter 5.6 --- Scaling Behaviors --- p.59 / Chapter 5.7 --- Discussions and Conclusions --- p.63 / References --- p.64 / Chapter 6 --- Renormalized Effective Medium Theory for Weakly Nonlinear Composites --- p.66 / Chapter 6.1 --- Introduction --- p.66 / Chapter 6.2 --- Weakly Nonlinear Conductance Network --- p.69 / Chapter 6.3 --- Simulation --- p.70 / Chapter 6.4 --- Effective Medium Approximation --- p.76 / Chapter 6.5 --- Renormalized Effective Medium Approximation --- p.79 / Chapter 6.6 --- Discussion and Conclusions --- p.81 / References --- p.83 / Chapter 7 --- Conclusions --- p.86 / Chapter A --- Derivation of Voltage-Summation Formulas --- p.88 / Chapter B --- Effective Linear and Nonlinear Response of 2 x 2 cell --- p.92 / Chapter C --- Duality Symmetry in 2D Network --- p.97 / Chapter D --- Derivation of Effective-Medium Approximation --- p.99

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science in Engineering. Johannesburg, 2016 / An investigation and analysis of a collection of different techniques, for estimating the states of nonlinear systems, was undertaken. It was found that most of the existing literature on the topic could be organized into several groups of nonlinear observer design techniques, of which each group follows a specific concept and slight variations thereof. From out of this investigation it was discovered that a variation of the adaptive observer could be successfully applied to numerous nonlinear systems, given only limited output information. This particular technique formed the foundation on which a design procedure was developed in order to asymptotically estimate the states of nonlinear systems of a certain form, using only partial state information available. Lyapunov stability theory was used to prove the validity of this technique, given that certain conditions and assumptions are satisfied. A heuristic procedure was then developed to get a linearized model of the error transient behaviour that could form the upper bounds of the transient times of the observer. The technique above, characterized by a design algorithm, was then applied to three well-known nonlinear systems; namely the Lorenz attractor, the Rössler attractor, and the Van Der Pol oscillator. The results, illustrated through numerical simulation, clearly indicate that the technique developed is successful, provided all assumptions and conditions are satisfied. / MT2017