A thesis submitted to the Faculty of Science University of the Witwatersrand Johannesburg in fulfillment of the requirements for the Master of Science Johannesburg 1997

A three-dimensional parabolic equation (PE) and perturbation approach is used to invert for the depth- and range-dependent geoacoustic characteristics of the seabed. The model assumes that the sound speed profile is the superposition of a known range-independent profile and an unknown depth- and range-dependent perturbation. Using a Green’s function approach, the total measured pressure field in the water column is decomposed into a background field, which is due to the range-independent profile, and a scattered field, which is due to the range-dependent perturbation. When the Born approximation is applied to the resulting integral equation, it can be solved for the range-dependent profile using linear inverse theory. Although the method is focused on inverting for the sound speed profile in the bottom, it can also invert for the sound speed profile in the water column. For simplicity, the sound speed profile in the water column was assumed to be known with a margin of error of ± 5 m/s. The range-dependent perturbation is added to the index of refraction squared n2(r), rather than the sound speed profile c(ro). The method is implemented in both Cartesian (x,y,z) and cylindrical (r,q,z) coordinates with the forward propagation of the field in x and r, respectively. Synthetic data are used to demonstrate the validity of the method [1]. Two inversion methods were combined, a Monte Carlo like algorithm, responsible for a starting approximation of the sound speed profile, and a steepest descent method, that fine-tuned the results. In simulations, the inversion algorithm is capable of inverting for the sound speed profile of a flat bottom. It was tested, for three different frequencies (50 Hz, 75 Hz, and 100 Hz), in a Pekeris waveguide, a range-independent layered medium, and a range-dependent medium, with errors in the inverted sound speed profile of less than 3%. Keywords: Three-dimensional parabolic equation method, geoacoustic inversion, range-dependent sound speed profile, linear inversion, Born approximation, Green’s functions. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2017. / FAU Electronic Theses and Dissertations Collection

A well-known long standing problem in combinatorics and statistical mechanics is to find the generating function for self-avoiding walks (SAW) on a two-dimensional lattice, enumerated by perimeter. A SAW is a sequence of moves on a square lattice which does not visit the same point more than once. It has been considered by more than one hundred researchers in the pass one hundred years, including George Polya, Tony Guttmann, Laszlo Lovasz, Donald Knuth, Richard Stanley, Doron Zeilberger, Mireille Bousquet-Mlou, Thomas Prellberg, Neal Madras, Gordon Slade, Agnes Dit- tel, E.J. Janse van Rensburg, Harry Kesten, Stuart G. Whittington, Lincoln Chayes, Iwan Jensen, Arthur T. Benjamin, and many others. More than three hundred papers and a few volumes of books were published in this area. A SAW is interesting for simulations because its properties cannot be calculated analytically. Calculating the number of self-avoiding walks is a common computational problem. A recently proposed model called prudent self-avoiding walks (PSAW) was first introduced to the mathematics community in an unpublished manuscript of Pra, who called them exterior walks. A prudent walk is a connected path on square lattice such that, at each step, the extension of that step along its current trajectory will never intersect any previously occupied vertex. A lattice path composed of connected horizontal and vertical line segments, each passing between adjacent lattice points. We will discuss some enumerative problems in self-avoiding walks, lattice paths and walks with several step vectors. Many open problems are posted. / by Shanzhen Gao. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.

Nous montrons la validité du formalisme multifractal pour les mesures aléatoires faiblement Gibbs portées par l’ attracteur associé à une dynamique aléatoire C¹ codée par un sous-shift de type fini aléatoire, et expansive en moyenne. Nous établissons également des loi de type 0-∞ pour les mesures de Hausdorff et de packing généralisées des ensembles de niveau de la dimension locale, et calculons les dimensions de Hausdorff et de packing des ensembles de points en lesquels la dimension inférieure locale et la dimension supérieure locale sont prescrites. Lorsque l’attracteur est un ensemble de Cantor de mesure de Lebesgue nulle, nous montrons la validité du formalisme multifractal pour les mesures discrètes obtenues comme inverses de ces mesures faiblement Gibbs. / We establish the validity of the multifractal formalism for random weak Gibbs measures supported on the attractor associated with a C¹ random dynamics coded by a random subshift of finite type, and expanding in the mean. We also prove a 0-∞ law for the generalized Hausdorff and packing measures of the level sets of the local dimension, and we compute the Hausdorff and packing dimensions of the sets of points at which the lower and upper local dimensions are prescribed. In the case that the attractor is a Cantor set of zero Lebesgue measure, we prove the validity of the multifractal formalism for the discrete measures obtained as inverse of these weak Gibbs measures.

L'apprentissage par renforcement est la réponse du domaine de l'apprentissage numérique au problème du contrôle optimal. Dans ce paradigme, un agent informatique apprend à contrôler un environnement en interagissant avec ce dernier. Il reçoit régulièrement une information locale de la qualité du contrôle effectué sous la forme d'une récompense numérique (ou signal de renforcement), et son objectif est de maximiser une fonction cumulante de ces récompenses sur le long terme, généralement modélisée par une fonction dite de valeur. Le choix des actions appliquées à l'environnement en fonction de sa configuration est appelé une politique, et la fonction de valeur quantifie donc la qualité de cette politique donc la qualité de cette politique. Ce parangon est très général, et permet de s'intéresser à un grand nombre d'applications, comme la gestion des flux de gaz dans un complexe sidérurgique, que nous abordons dans ce manuscrit. Cependant, sa mise en application pratique peut être difficile. Notamment, lorsque la description de l'environnement à contrôler est trop grande, une représentation exacte de la fonction de la valeur (ou de la politique) n'est pas possible. Dans ce cas se pose le problème de la généralisation (ou de l'approximation de fonction de valeur) : il faut d'une part concevoir des algorithmes dont la complexité algorithmique ne soit pas trop grande, et d'autre part être capable d'interférer le comportement à suivre pour une configuration de l'environnement inconnue lorsque des situations proches ont déjà été expérimentées. C'est le problème principal que nous traitons dans ce manuscrit, en proposant une approche inspirée du filtrage de Kalman / Reinforcement learning is the response of machine learning to the problem of optimal control. In this paradigm, an agent learns do control an environment by interacting with it. It receives evenly a numeric reward (or reinforcement signal), which is a local information about the quality of the control. The agent objective is to maximize a cumulative function of these rewards, generally modelled as a so-called value function. A policy specifies the action to be chosen in a particular configuration of the environment to be controlled, and thus the value function quantifies the quality of yhis policy. This paragon is very general, and it allows taking into account many applications. In this manuscript, we apply it to a gas flow management problem in the iron and steel industry. However, its application can be quite difficult. Notably, if the environment description is too large, an exact representation of the value function (or of the policy) is not possible. This problem is known as generalization (or value function approximation) : on the one hand, one has to design algorithms with low computational complexity, and on the other hand, one has to infer the behaviour the agent should have in an unknown configuration of the environment when close configurations have been experimented. This is the main problem we address in this manuscript, by introducing a family of algorithms inspired from Kalman filtering

A finalidade deste trabalho e calcular a composição de modelos no FBST (the Full Bayesian Signicance Test) descrito por Borges e Stern [6]. Nosso objetivo foi encontrar um método de aproximação numérica mais eficiente que consiga substituir o método de condensação descrita por Kaplan. Três técnicas foram comparadas: a primeira é a aproximação da convolução de Mellin usando discretização e condensação descrita por Kaplan [11], a segunda é a aproximação da convolução de Mellin usando mistura de exponenciais, descrita por Dufresne [8], para calcular a convolução de Fourier mediante a aproximação de mistura de convoluções exponenciais, usando a estrutura algébrica descrita por Hogg [10], mais a aplicação do operador descrito por Collins [7], para transformar a convolução de Fourier para a convolução de Mellin, a terceira é a aproximação da convolução de Mellin usando mistura de exponenciais, descrita por Dufresne [8], para aproximar diretamente via mistura de exponenciais a convolução de Fourier, mais a aplicação do operador descrito por Collins [7], para transformar a convolução de Fourier para a convolução de Mellin. / The purpose of this work is to calculate the compositional models of FBST (the Full Bayesian Signicance Test) studied by Borges and Stern [6]. The objective of this work was to find an approximation method numerically eficient that can replace the condensation methods described by Kaplan. Three techniques were compared: First, the approximation of Mellin convolution using discretization and condensation described by Kaplan [11], second, the approximation of Mellin convolution using mixtures of exponentials, described by Dufresne [8], to calculate the Fourier convolution by approximation of mixtures of exponential convolutions, using the algebraic structure described by Hogg [10], and then to apply the operator described by Collins [7], to transform the usual convolution to Mellin convolution, third, the approximation of Mellin convolution using mixtures of exponentials, described by Dufresne [8], to calculate the Fourier convolution by direct approximation of mixtures of exponentials, and then to apply the operator described by Collins [7], to transform the usual convolution to Mellin convolution.

by Jones Chui. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaves 62-[66]). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Fundamentals of Artificial Neural Networks --- p.2 / Chapter 1.1.1 --- Processing Unit --- p.2 / Chapter 1.1.2 --- Topology --- p.3 / Chapter 1.1.3 --- Learning Rules --- p.4 / Chapter 1.2 --- Overview of Various Neural Network Models --- p.6 / Chapter 1.3 --- Introduction to the Radial Basis Function Networks (RBFs) --- p.8 / Chapter 1.3.1 --- Historical Development --- p.9 / Chapter 1.3.2 --- Some Intrinsic Problems --- p.9 / Chapter 1.4 --- Objective of the Thesis --- p.10 / Chapter 2 --- Low-dimensional Gaussian RBF networks (LowD RBFs) --- p.13 / Chapter 2.1 --- Architecture of LowD RBF Networks --- p.13 / Chapter 2.1.1 --- Network Structure --- p.13 / Chapter 2.1.2 --- Learning Rules --- p.17 / Chapter 2.2 --- Construction of LowD RBF Networks --- p.19 / Chapter 2.2.1 --- Growing Heuristic --- p.19 / Chapter 2.2.2 --- Pruning Heuristic --- p.27 / Chapter 2.2.3 --- Summary --- p.31 / Chapter 3 --- Application examples --- p.34 / Chapter 3.1 --- Chaotic Time Series Prediction --- p.35 / Chapter 3.1.1 --- Performance Comparison --- p.39 / Chapter 3.1.2 --- Sensitivity Analysis of MSE THRESHOLDS --- p.41 / Chapter 3.1.3 --- Effects of Increased Embedding Dimension --- p.41 / Chapter 3.1.4 --- Comparison with Tree-Structured Network --- p.46 / Chapter 3.1.5 --- Overfitting Problem --- p.46 / Chapter 3.2 --- Nonlinear prediction of speech signal --- p.49 / Chapter 3.2.1 --- Comparison with Linear Predictive Coding (LPC) --- p.54 / Chapter 3.2.2 --- Performance Test in Noisy Conditions --- p.55 / Chapter 3.2.3 --- Iterated Prediction of Speech --- p.59 / Chapter 4 --- Conclusion --- p.60 / Chapter 4.1 --- Discussions --- p.60 / Chapter 4.2 --- Limitations and Suggestions for Further Research --- p.61 / Bibliography --- p.62

Thesis (M.Phil.)--Chinese University of Hong Kong. / Bibliography: leaf 34.

Each aspect of this thesis is motivated by the recent paper of Beresnevich, Dickinson and Velani (BDV03]. Let 'ljJ be a real, positive, decreasing function i.e. an approximation function. Their paper considers a general lim sup set A( 'ljJ), within a compact metric measure space (0, d, m), consisting of points that sit in infinitely many balls each centred at an element ROt of a countable set and of radius 'I/J(130) where 130 is a 'weight' assigned to each ROt. The classical set of 'I/J-well approximable numbers is the basic example. For the set A('ljJ) , [BDV03] achieves m-measure and Hausdorff measure laws analogous to the classical theorems of Khintchine and Jarnik. Our first results obtain an application of these metric laws to the set of 'ljJ-well approximable numbers with restricted rationals, previously considered by Harman (Har88c]. Next, we consider a generalisation of the set of badly approximable numbers, Bad. For an approximation function p, a point x of a compact metric space is in a general set Bad(p) if, loosely speaking, x 'avoids' any ball centred at an element ROt of a countable set and of radius c p(I3Ot) for c = c(x) a constant. In view of Jarnik's 1928 result that dim Bad = 1, we aim to show the general set Bad(p) has maximal Hausdorff dimension. Finally, we extend the theory of (BDV03] by constructing a general lim sup set dependent on two approximation functions, A('ljJll'ljJ2)' We state a measure theorem for this set analogous to Khintchine's (1926a) theorem for the Lebesgue measure of the set of ('l/Jl, 1/12)-well approximable pairs in R2. We also remark on the set's Hausdorff dimension.

Successive Approximation (SAR) analog-to-digital converters are used extensively in biomedical applications such as CAT scan due to the high resolution they offer. Capacitor mismatch in the SAR converter is a limiting factor for its accuracy and resolution. Without some form of calibration, a SAR converter can only achieve 10 bit accuracy. In industry, the CAL-DAC approach is a popular approach for calibrating the SAR ADC, but this approach requires significant test time. This thesis applies the“Split-ADC" architecture with a deterministic, digital, and background self-calibration algorithm to the SAR converter to minimize test time. In this approach, a single ADC is split into two independent halves. The two split ADCs convert the same input sample and produce two output codes. The ADC output is the average of these two output codes. The difference between these two codes is used as a calibration signal to estimate the errors of the calibration parameters in a modified Jacobi method. The estimates are used to update calibration parameters are updated in a negative feedback LMS procedure. The ADC is fully calibrated when the difference signal goes to zero on average. This thesis focuses on the specific implementation of the“Split-ADC" self-calibrating algorithm on a 16 bit, 1 MS/s differential SAR ADC. The ADC can be calibrated with 105 conversions. This represents an improvement of 3 orders of magnitude over existing statistically-based calibration algorithms. Simulation results show that the linearity of the calibrated ADC improves to within ±1 LSB.