Thesis (Ph. D.)--University of Hong Kong, 2008. / Also available in print.

Thesis (Ph. D.)--University of Hong Kong, 2009. / Includes bibliographical references (leaves 80-91). Also available in print.

Master of Science / Department of Mathematics / Bacim Alali / Nonlocal vector calculus, introduced in generalizes differential operators' calculus to nonlocal calculus of integral operators. Nonlocal vector calculus has been applied to many fields including peridynamics, nonlocal diffusion, and image analysis. In this report, we present a vector calculus for nonlocal operators such as a nonlocal divergence, a nonlocal gradient, and a nonlocal Laplacian. In Chapter 1, we review the local (differential) divergence, gradient, and Laplacian operators. In addition, we discuss their adjoints, the divergence theorem, Green's identities, and integration by parts. In Chapter 2, we define nonlocal analogues of the divergence and gradient operators, and derive the corresponding adjoint operators. In Chapter 3, we present a nonlocal divergence theorem, nonlocal Green's identities, and integration by parts for nonlocal operators. In Chapter 4, we establish a connection between the local and nonlocal operators. In particular, we show that, for specific integral kernels, the nonlocal operators converge to their local counterparts in the limit of vanishing nonlocality.

We develop a graph structure for feature vectors in machine learning, which we denote as a feature network (FN); this is different from sample-based networks, in which nodes simply represent samples. FNs reveal the underlying relationship among feature vector components and re-represent features as functions on a network. Our study focuses on using FN structures to extract underlying information and thus improve machine learning performance. Upon the representation of feature vectors as such functions, so-called graph signal processing, or graph functional analytic techniques can be implemented, consisting of analytic operations including differentiation and integration of feature vectors. Our motivation originated from a study using infrared spectroscopy data, where domain experts prefer using the second derivative information rather than the original data; this is an illustration of the potential power of understanding the underlying feature structure. We begin by developing a classification method based on the premise that is assuming data from different classes (e.g., different cancer subtypes) will have distinct underlying graph structures, for graphs consisting of genes as nodes and gene covariances as edges. That is, a feature vector from one class will tend to be "smooth" on the related FN, and "fluctuate" in the other FNs. This method, using an entirely new set of features from standard ones, on its own proves to somewhat outperform SVM and KNN in classifying cancer subtypes in infrared spectroscopy data and gene expression data. We are effectively also projecting high-dimensional data into a low dimensional representation of graph smoothness, providing a unique way of data visualization. Additionally, FNs represent new ways of thinking about data. With a graph structure for feature vectors, graphical functional analysis can be used to extract various types of information not apparent in the original feature vectors. Specifically, operations such as calculus, Fourier transforms, and convolutions can be performed on the graph vertex domain. We introduce a family of calculus-like operators in reproducing kernel Hilbert spaces for feature vector regularization to deal with two types of data deficiency, which we designate as noise and blurring. Such operations are generalized from widely used ones in computer vision. The derivative operations on feature vectors provide additional information by amplifying differences between highly correlated features. Integrating feature vectors smooths and denoises them. Applications show that those denoising and deblurring operators can improve classification algorithms. The feature network with deep learning can be naturally extended to graph convolutional networks. We proposed a deep multiscale clustering structure with small learning complexity on general graph distance structures. This framework substantially reduces the number of parameters, and it allows the introduction of general machine learning algorithms such as SVM to feed-forward in this deep structure.

In this dissertation, we study the existence and nonexistence of positive radial solutions for classes of quasilinear elliptic equations and systems in a ball with Dirichlet boundary conditions. Our nonlinearities are asymptotically p-linear at infinity and are allowed to be singular at zero with non-positone structure, which have not been considered in the literature. In the one parameter single equation problem, we are able to show the existence of a positive radial solution with precise lower bound estimate for a certain range of the parameter. We also extend the study to a class of asymptotically p-linear system with two parameters and in the presence of singularities. We establish the existence of a positive solution with a precise lower bound estimate when the product of the parameters is in a certain range. Necessary and sufficient conditions for the existence of a positive solution are also obtained for both the single equation and system under additional assumptions. Our approach is based on the Schauder Fixed Point Theorem.

In this report the connection between p-harmonic functions and the tug-of-war game is presented and some applications are mentioned. Moreover, sufficient background information of solutions to the p-Laplace equation is given. And to finish, an example of the game is given with simulations. / I denna rapport presenteras kopplingen mellan p-harmoniska funktioner och spelet tug-of-war och några applikationer nämns. Utöver så ges tillräckligt med infomration om lösningar till p-Laplace ekvation. Som avslutning, ges ett exempel på spelet tillsammans med simuleringar.

In this thesis, we set out to provide an enhanced set of techniques for determining the eigenvalues of the Laplacian in polygonal domains. Currently, finite-element methods provide a numerical means by which we can approximate these eigenvalues with ease. However, we would like a more analytic method which may allow us to avoid a basic parameter sweep in finite-element software such as COMSOL to determine what could possibly be an "optimal" distribution of eigenvalues. The hope is that this would allow us to draw conclusions about the acoustic quality of a pentagonally-shaped room. First, we find the eigenvalues using a common finite-element method through COMSOL Multiphysics. We then examine another method which makes use of conformal maps and Schwarz-Christoffel transformations with the prospect that it might provide a more analytic understanding of the calculation of these eigenvalues and possibly allow for variation of certain parameters. This method, as far as we could find, had not yet been developed on the pentagon. We end up carrying this method through nearly all of the steps necessary in finding these eigenvalues. We find that the finite-element method is not only easier to use, but is also more efficient in terms of computing power. / Master of Science

Some eigenvalue inequalities for Klein-Gordon operators and fractional Laplacians restricted to a bounded domain are proved. Such operators became very popular recently as they arise in many problems ranging from mathematical finance to crystal dislocations, especially relativistic quantum mechanics and symmetric stable stochastic processes. Many of the results obtained here are concerned with finding bounds for some functions of the spectrum of these operators. The subject, which is well developed for the Laplacian, is examined from the spectral theory perspective through some of the tools used to prove analogous results for the Laplacian. This work highlights some important results, sparking interest in constructing a similar theory for Klein-Gordon operators. For instance, the Weyl asymptotics and semiclassical bounds for the Klein-Gordon operator are developed. As a result, a Berezin-Li-Yau type inequality is derived and an improvement of the bound is proved in a separate chapter. Other results involving some universal bounds for the Klein-Gordon Hamiltonian with an external interaction are also obtained.

Dans cette thèse, nous donnons des résultats d’existence, de non-existence, d’unicité et de multiplicité de solutions pour des équations aux dérivées partielles avec croissance critique dans le gradient. Les principales méthodes utilisées dans nos preuves sont des arguments variationnels, la théorie des sous et sur-solutions, des estimations à priori et la théorie de la bifurcation. La thèse se compose de six chapitres. Dans le chapitre 0 nous introduisons le sujet de thèse et nous présentons les résultats principaux. Le chapitre 1 porte sur l’´étude d’une équation du type p-Laplacien avec croissance critique dans le gradient et dépendant d’un paramètre. En fonction de l’intervalle où se trouve le paramètre, nous obtenons l’existence et l’unicité d’une solution ou nous montrons l’existence et la multiplicité de solutions. Dans les chapitres 2 et 3, nous poursuivons notre étude dans le cas où l’opérateur utilisé est le Laplacien mais, contrairement au chapitre 1, nous étudions le cas où les coefficients changent de signe. Nous obtenons à nouveau des résultats d’existence et de multiplicité de solutions. Dans le chapitre 4, nous étudions des problèmes nonlocaux du type Laplacien fractionnaire avec différents termes de gradient non-local. Nous montrons des résultats d’existence et de non-existence de solutions pour différentes équations de ce type. Finalement, dans le chapitre 5 nous présentons quelques problèmes ouverts liés au contenu de la thèse et des perspectives de recherche. / In this thesis, we provide existence, non-existence, uniqueness and multiplicity results for partial differential equations with critical growth in the gradient. The principal techniques employed in our proofs are variational techniques, lower and upper solution theory, a priori estimates and bifurcation theory. The thesis consists of six chapters. In chapter 0, we introduce the topic of the thesis and we present the main results. Chapter 1 deals with a p-Laplacian type equation with critical growth in the gradient. This equation will depend on a real parameter. Depending on the interval where this parameter lives, we obtain the existence and uniqueness of one solution or we prove the existence and multiplicity of solutions. In chapters 2 and 3, we continue our study in the case where the operator is the Laplacian. However, unlike chapter 1, we study the case where the coefficient functions may change sign. We obtain again existence and multiplicity results. In chapter 4, we study non-local problems of fractional Laplacian type with different non-local gradient terms. We prove existence and non-existence results for different equations of this type. Finally, in chapter 5, we present some open problems related to the content of the thesis and some research perspectives.

Master of Science / Department of Electrical and Computer Engineering / Christopher L. Lewis / This thesis presents a novel approach for detecting robust and scale invariant interest points in images. The detector accurately and efficiently approximates the Laplacian of Gaussian using an optimal set of weighted box filters that take advantage of integral images to reduce computations. When combined with state-of-the art descriptors for matching, the algorithm performs better than leading feature tracking algorithms including SIFT and SURF in terms of speed and accuracy.