Applied Mathematics

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This is the collection for the University of Waterloo's Department of Applied Mathematics.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Now showing 1 - 20 of 509

Contributions to the study of general relativistic shear-free perfect fluids: an approach involving Cartan's equivalence method, differential forms and symbolic computation
(University of Waterloo, 1993) Lang, Jérôme Michel
It has been conjectured that general relativistic shear-free perfect ﬂuids with a barotropic equation of state, and such that the energy density, µ, and the pressure, p, satisfy µ + p ̸= 0, cannot simultaneously be rotating and expanding (or contracting). A survey of the known results about this conjecture is included herein. We show that the conjecture holds true under either of the following supplementary conditions: 1) the Weyl tensor is purely magnetic with respect to the ﬂow velocity vector or 2) dp/dµ = −1/3. Any hypersurface-homogeneous shear-free perfect ﬂuid which is not space-time homogeneous and whose acceleration vector is not parallel to the vorticity vector belongs to one of three invariantly deﬁned classes, labelled A, B and C. It is found that the Petrov types which are allowed in each class are as follows: for class A, type I only; for class B, types I, II and III; and for class C, types I, D, II and N. Two-dimensional pseudo-Riemannian space-times are classiﬁed in a manner similar to that of the Karlhede classiﬁcation of four-dimensional general-relativistic space-times. In an appendix, the forms diﬀerential forms package for the Maple program is described.
Multi-Resolution Approximate Inverses
(University of Waterloo, 1999) Bridson, Robert
This thesis presents a new preconditioner for elliptic PDE problems on unstructured meshes. Using ideas from second generation wavelets, a multi-resolution basis is constructed to effectively compress the inverse of the matrix, resolving the sparsity vs. quality problem of standard approximate inverses. This finally allows the approximate inverse approach to scale well, giving fast convergence for Krylov subspace accelerators on a wide variety of large unstructured problems. Implementation details are discussed, including ordering and construction of factored approximate inverses, discretization and basis construction in one and two dimensions, and possibilities for parallelism. The numerical experiments in one and two dimensions confirm the capabilities of the scheme. Along the way I highlight many new avenues for research, including the connections to multigrid and other multi-resolution schemes.
A Probabilistic Approach to Image Feature Extraction, Segmentation and Interpretation
(University of Waterloo, 2000) Pal, Christopher Joseph
This thesis describes a probabilistic approach to imagesegmentation and interpretation. The focus of the investigation is the development of a systematic way of combining color, brightness, texture and geometric features extracted from an image to arrive at a consistent interpretation for each pixel in the image. The contribution of this thesis is thus the presentation of a novel framework for the fusion of extracted image features producing a segmentation of an image into relevant regions. Further, a solution to the sub-pixel mixing problem is presented based on solving a probabilistic linear program. This work is specifically aimed at interpreting and digitizing multi-spectral aerial imagery of the Earth's surface. The features of interest for extraction are those of relevance to environmental management, monitoring and protection. The presented algorithms are suitable for use within a larger interpretive system. Some results are presented and contrasted with other techniques. The integration of these algorithms into a larger system is based firmly on a probabilistic methodology and the use of statistical decision theory to accomplish uncertain inference within the visual formalism of a graphical probability model.
On the Solution of the Hamilton-Jacobi Equation by the Method of Separation of Variables
(University of Waterloo, 2000) Bruce, Aaron
The method of separation of variables facilitates the integration of the Hamilton-Jacobi equation by reducing its solution to a series of quadratures in the separable coordinates. The case in which the metric tensor is diagonal in the separable coordinates, that is, orthogonal separability, is fundamental. Recent theory by Benenti has established a concise geometric (coordinate-independent) characterisation of orthogonal separability of the Hamilton-Jacobi equation on a pseudoRiemannian manifold. It generalises an approach initiated by Eisenhart and developed by Kalnins and Miller. Benenti has shown that the orthogonal separability of a system via a point transformation is equivalent to the existence of a Killing tensor with real simple eigen values and orthogonally integrable eigenvectors. Applying a moving frame formalism, we develop a method that produces the orthogonal separable coordinates for low dimensional Hamiltonian systems. The method is applied to a two dimensional Riemannian manifold of arbitrary curvature. As an illustration, we investigate Euclidean 2-space, and the two dimensional surfaces of constant curvature, recovering known results. Using our formalism, we also derive the known superseparable potentials for Euclidean 2-space. Some of the original results presented in this thesis were announced in [8, 9, 10].
Contributions to the Study of the Validity of Huygens' Principle for the Non-self-adjoint Scalar Wave Equation on Petrov Type D Spacetimes
(University of Waterloo, 2000) Chu, Kenneth
This thesis makes contributions to the solution of Hadamard's problem through an examination of the question of the validity of Huygens'principle for the non-self-adjoint scalar wave equation on a Petrov type D spacetime. The problem is split into five further sub-cases based on the alignment of the Maxwell and Weyl principal spinors of the underlying spacetime. Two of these sub-cases are considered, one of which is proved to be incompatible with Huygens' principle, while for the other, it is shown that Huygens' principle implies that the two principal null congruences of the Weyl tensor are geodesic and shear-free. Furthermore, an unpublished result of McLenaghan regarding symmetric spacetimes of Petrov type D is independently verified. This result suggests the possible existence of counter-examples of the Carminati-McLenaghan conjecture.
Two- and Three-Dimensional Coding Schemes for Wavelet and Fractal-Wavelet Image Compression
(University of Waterloo, 2001) Alexander, Simon
This thesis presents two novel coding schemes and applications to both two- and three-dimensional image compression. Image compression can be viewed as methods of functional approximation under a constraint on the amount of information allowable in specifying the approximation. Two methods of approximating functions are discussed: Iterated function systems (IFS) and wavelet-based approximations. IFS methods approximate a function by the fixed point of an iterated operator, using consequences of the Banach contraction mapping principle. Natural images under a wavelet basis have characteristic coefficient magnitude decays which may be used to aid approximation. The relationship between quantization, modelling, and encoding in a compression scheme is examined. Context based adaptive arithmetic coding is described. This encoding method is used in the coding schemes developed. A coder with explicit separation of the modelling and encoding roles is presented: an embedded wavelet bitplane coder based on hierarchical context in the wavelet coefficient trees. Fractal (spatial IFSM) and fractal-wavelet (coefficient tree), or IFSW, coders are discussed. A second coder is proposed, merging the IFSW approaches with the embedded bitplane coder. Performance of the coders, and applications to two- and three-dimensional images are discussed. Applications include two-dimensional still images in greyscale and colour, and three-dimensional streams (video).
Complex Bases, Number Systems and Their Application to Fractal-Wavelet Image Coding
(University of Waterloo, 2002) Piché, Daniel Guy
This thesis explores new approaches to the analysis of functions by combining tools from the fields of complex bases, number systems, iterated function systems (IFS) and wavelet multiresolution analyses (MRA). The foundation of this work is grounded in the identification of a link between two-dimensional non-separable Haar wavelets and complex bases. The theory of complex bases and this link are generalized to higher dimensional number systems. Tilings generated by number systems are typically fractal in nature. This often yields asymmetry in the wavelet trees of functions during wavelet decomposition. To acknowledge this situation, a class of extensions of functions is developed. These are shown to be consistent with the Mallat algorithm. A formal definition of local IFS on wavelet trees (LIFSW) is constructed for MRA associated with number systems, along with an application to the inverse problem. From these investigations, a series of algorithms emerge, namely the Mallat algorithm using addressing in number systems, an algorithm for extending functions and a method for constructing LIFSW operators in higher dimensions. Applications to image coding are given and ideas for further study are also proposed. Background material is included to assist readers less familiar with the varied topics considered. In addition, an appendix provides a more detailed exposition of the fundamentals of IFS theory.
The de Broglie-Bohm Causal Interpretation of Quantum Mechanics and its Application to some Simple Systems
(University of Waterloo, 2003) Colijn, Caroline
The de Broglie-Bohm causal interpretation of quantum mechanics is discussed, and applied to the hydrogen atom in several contexts. Prominent critiques of the causal program are noted and responses are given; it is argued that the de Broglie-Bohm theory is of notable interest to physics. Using the causal theory, electron trajectories are found for the conventional Schrödinger, Pauli and Dirac hydrogen eigenstates. In the Schrödinger case, an additional term is used to account for the spin; this term was not present in the original formulation of the theory but is necessary for the theory to be embedded in a relativistic formulation. In the Schrödinger, Pauli and Dirac cases, the eigenstate trajectories are shown to be circular, with electron motion revolving around the z-axis. Electron trajectories are also found for the 1s-2p0 transition problem under the Schrödinger equation; it is shown that the transition can be characterized by a comparison of the trajectory to the relevant eigenstate trajectories. The structures of the computed trajectories are relevant to the question of the possible evolution of a quantum distribution towards the standard quantum distribution (quantum equilibrium); this process is known as quantum relaxation. The transition problem is generalized to include all possible transitions in hydrogen stimulated by semi-classical radiation, and all of the trajectories found are examined in light of their implications for the evolution of the distribution to the standard distribution. Several promising avenues for future research are discussed.
Well-Posedness of Boundary Control Systems
(Society for Industrial and Applied Mathematics, 2003-01-01) Cheng, Ada; Morris, Kirsten
Continuity of the input/output map for boundary control systems is shown through the system transfer function. Our approach transforms the question of continuity of the input/output map of a boundary control system to uniform boundedness of the solution to a related elliptic problem. This is shown for a class of boundary control systems with Dirichlet, Neumann, or Robin boundary control.
Impulsive Control and Synchronization of Chaos-Generating-Systems with Applications to Secure Communication
(University of Waterloo, 2004) Khadra, Anmar
When two or more chaotic systems are coupled, they may exhibit synchronized chaotic oscillations. The synchronization of chaos is usually understood as the regime of chaotic oscillations in which the corresponding variables or coupled systems are equal to each other. This kind of synchronized chaos is most frequently observed in systems specifically designed to be able to produce this behaviour. In this thesis, one particular type of synchronization, called impulsive synchronization, is investigated and applied to low dimensional chaotic, hyperchaotic and spatiotemporal chaotic systems. This synchronization technique requires driving one chaotic system, called response system, by samples of the state variables of the other chaotic system, called drive system, at discrete moments. Equi-Lagrange stability and equi-attractivity in the large property of the synchronization error become our major concerns when discussing the dynamics of synchronization to guarantee the convergence of the error dynamics to zero. Sufficient conditions for equi-Lagrange stability and equi-attractivity in the large are obtained for the different types of chaos-generating systems used. The issue of robustness of synchronized chaotic oscillations with respect to parameter variations and time delay, is also addressed and investigated when dealing with impulsive synchronization of low dimensional chaotic and hyperchaotic systems. Due to the fact that it is impossible to design two identical chaotic systems and that transmission and sampling delays in impulsive synchronization are inevitable, robustness becomes a fundamental issue in the models considered. Therefore it is established, in this thesis, that under relatively large parameter perturbations and bounded delay, impulsive synchronization still shows very desired behaviour. In fact, criteria for robustness of this particular type of synchronization are derived for both cases, especially in the case of time delay, where sufficient conditions for the synchronization error to be equi-attractivity in the large, are derived and an upper bound on the delay terms is also obtained in terms of the other parameters of the systems involved. The theoretical results, described above, regarding impulsive synchronization, are reconfirmed numerically. This is done by analyzing the Lyapunov exponents of the error dynamics and by showing the simulations of the different models discussed in each case. The application of the theory of synchronization, in general, and impulsive synchronization, in particular, to communication security, is also presented in this thesis. A new impulsive cryptosystem, called induced-message cryptosystem, is proposed and its properties are investigated. It was established that this cryptosystem does not require the transmission of the encrypted signal but instead the impulses will carry the information needed for synchronization and for retrieving the message signal. Thus the security of transmission is increased and the time-frame congestion problem, discussed in the literature, is also solved. Several other impulsive cryptosystems are also proposed to accommodate more solutions to several security issues and to illustrate the different properties of impulsive synchronization. Finally, extending the applications of impulsive synchronization to employ spatiotemporal chaotic systems, generated by partial differential equations, is addressed. Several possible models implementing this approach are suggested in this thesis and few questions are raised towards possible future research work in this area.
Probabilistic Properties of Delay Differential Equations
(University of Waterloo, 2004) Taylor, S. Richard
Systems whose time evolutions are entirely deterministic can nevertheless be studied probabilistically, i. e. in terms of the evolution of probability distributions rather than individual trajectories. This approach is central to the dynamics of ensembles (statistical mechanics) and systems with uncertainty in the initial conditions. It is also the basis of ergodic theory--the study of probabilistic invariants of dynamical systems--which provides one framework for understanding chaotic systems whose time evolutions are erratic and for practical purposes unpredictable. Delay differential equations (DDEs) are a particular class of deterministic systems, distinguished by an explicit dependence of the dynamics on past states. DDEs arise in diverse applications including mathematics, biology and economics. A probabilistic approach to DDEs is lacking. The main problems we consider in developing such an approach are (1) to characterize the evolution of probability distributions for DDEs, i. e. develop an analog of the Perron-Frobenius operator; (2) to characterize invariant probability distributions for DDEs; and (3) to develop a framework for the application of ergodic theory to delay equations, with a view to a probabilistic understanding of DDEs whose time evolutions are chaotic. We develop a variety of approaches to each of these problems, employing both analytical and numerical methods. In transient chaos, a system evolves erratically during a transient period that is followed by asymptotically regular behavior. Transient chaos in delay equations has not been reported or investigated before. We find numerical evidence of transient chaos (fractal basins of attraction and long chaotic transients) in some DDEs, including the Mackey-Glass equation. Transient chaos in DDEs can be analyzed numerically using a modification of the "stagger-and-step" algorithm applied to a discretized version of the DDE.
The Dynamics of Inhomogeneous Cosmologies
(University of Waterloo, 2004) Lim, Woei Chet
In this thesis we investigate cosmological models more general than the isotropic and homogeneous Friedmann-Lemaître models. We focus on cosmologies with one spatial degree of freedom, whose matter content consists of a perfect fluid and the cosmological constant. We formulate the Einstein field equations as a system of quasilinear first order partial differential equations, using scale-invariant variables. The primary goal is to study the dynamics in the two asymptotic regimes, i. e. near the initial singularity and at late times. We highlight the role of spatially homogeneous dynamics as the background dynamics, and analyze the inhomogeneous aspect of the dynamics. We perform a variety of numerical simulations to support our analysis and to explore new phenomena.
Synchronization in Heterogeneous Networks of Hippocampal Interneurons
(University of Waterloo, 2005) Bazzazi, Hojjat
The hippocampus is one of the most intensely studied brain structures and the oscillatory activity of the hippocampal neurons is believed to be involved in learning and memory consolidation. Therefore, studying rhythm generation and modulation in this structure is an important step in understanding its function. In this thesis, these phenomena are studied via mathematical models of networks of hippocampal interneurons. The two types of neural networks considered here are homogenous and heterogenous networks. In homogenous networks, the input current to each neuron is equal, while in heterogenous networks, this assumption is relaxed and there is a specified degree of heterogeneity in the input stimuli. A phase reduction technique is applied to the neural network model of the hippocampal interneurons and attempts are made to understand the implications of heterogeneity to the existence and stability of the synchronized oscillations. The Existence of a critical level of heterogeneity above which the synchronized rhythms are not stable is established, and linear analysis is applied to derive expressions for estimating the perturbations in the network frequency and timing of the neural spikes. The mathematical techniques developed in this thesis are general enough to be applied to models describing other types of neurons not considered here. Possible biological implications include the application of high frequency local stimulation to alleviate the synchronous neural oscillations in pathological conditions such as epilepsy and Parkinson's disease and the possible role of heterogeneity in controlling the rhythm frequency and switching between various cognitive states.
The modeling of blood rheology in small vessels
(University of Waterloo, 2005) Scott, Matthew
Blood is a dense suspension of flexible red blood cells. In response to a background flow, these cells distribute themselves non-uniformly throughout the vessel. As a result, material properties that are well defined in homogeneous fluids, such as viscosity, are no longer so, and depend upon the flow geometry along with the particle properties. Using a simple model that accounts for the steady-state particle distribution in vessel flow, we derive an expression for the effective viscosity of blood and the suspension flow velocity field in a pressure-driven tube flow.

We derive the steady-state particle distribution from a conservation equation with convective flux arising from particle deformation in the flow. We then relate the particle microstructure to the overall flow through a generalized Newtonian stress-tensor, with the particle volume fraction appearing in the expression for the local viscosity. Comparing with experimental data, we show that the model quantitatively reproduces the observed rheology of blood in tube flow.

We reconsider the problem in an alternate geometry corresponding to the flow between two concentric cylinders. The steady-state particle distribution, suspension velocity field and the measured effective viscosity are all very different from their counterparts in tube flow, casting serious doubt upon the practice of using data from a Couette viscometer to parameterize constitutive models applied to vascular blood flow.

Finally, we calculate the effect of random fluctuations in the particle velocity on the averaged behaviour of the particle conservation equation. Using a smoothing method for linear stochastic differential equations, we derive a correction to the free Einstein-Stokes diffusion coeffcient that is due to the interaction of the particles with their neighbours.
Multiscale Methods in Image Modelling and Image Processing
(University of Waterloo, 2005) Alexander, Simon
The field of modelling and processing of 'images' has fairly recently become important, even crucial, to areas of science, medicine, and engineering. The inevitable explosion of imaging modalities and approaches stemming from this fact has become a rich source of mathematical applications.

'Imaging' is quite broad, and suffers somewhat from this broadness. The general question of 'what is an image?' or perhaps 'what is a natural image?' turns out to be difficult to address. To make real headway one may need to strongly constrain the class of images being considered, as will be done in part of this thesis. On the other hand there are general principles that can guide research in many areas. One such principle considered is the assertion that (classes of) images have multiscale relationships, whether at a pixel level, between features, or other variants. There are both practical (in terms of computational complexity) and more philosophical reasons (mimicking the human visual system, for example) that suggest looking at such methods. Looking at scaling relationships may also have the advantage of opening a problem up to many mathematical tools.

This thesis will detail two investigations into multiscale relationships, in quite different areas. One will involve Iterated Function Systems (IFS), and the other a stochastic approach to reconstruction of binary images (binary phase descriptions of porous media). The use of IFS in this context, which has often been called 'fractal image coding', has been primarily viewed as an image compression technique. We will re-visit this approach, proposing it as a more general tool. Some study of the implications of that idea will be presented, along with applications inferred by the results. In the area of reconstruction of binary porous media, a novel, multiscale, hierarchical annealing approach is proposed and investigated.
Detectability of Singularly Perturbed Systems
(University of Waterloo, 2005) Vu, Leonard Phong
A form of detectability, known as the input-output-to-state stability property, for singularly perturbed systems is examined in this work. This work extends the result of a paper by Christofides & Teel wherein they presented a notion of total stability for input-to-state stability with respect to singular perturbations. Analyzing singularly perturbed systems with outputs we show that if the boundary layer system is uniformly globally asymptotically stable and the reduced system is input-output-to-state stable with respect to disturbances, then these properties continue to hold, up to an arbitrarily small offset, for initial conditions in an arbitrarily large compact set and sufficiently small singular perturbation parameter over the time interval for which disturbances, their derivatives, and outputs remain in an arbitrarily large compact set. An application of the result is presented where we analyze the stability of a circuit with a nonlinear element through the measurement of only one of the variables of interest.
Mutual Information Based Methods to Localize Image Registration
(University of Waterloo, 2005) Wilkie, Kathleen Patricia
Modern medicine has become reliant on medical imaging. Multiple modalities, e. g. magnetic resonance imaging (MRI), computed tomography (CT), etc. , are used to provide as much information about the patient as possible. The problem of geometrically aligning the resulting images is called image registration. Mutual information, an information theoretic similarity measure, allows for automated intermodal image registration algorithms.

In applications such as cancer therapy, diagnosticians are more concerned with the alignment of images over a region of interest such as a cancerous lesion, than over an entire image set. Attempts to register only the regions of interest, defined manually by diagnosticians, fail due to inaccurate mutual information estimation over the region of overlap of these small regions.

This thesis examines the region of union as an alternative to the region of overlap. We demonstrate that the region of union improves the accuracy and reliability of mutual information estimation over small regions.

We also present two new mutual information based similarity measures which allow for localized image registration by combining local and global image information. The new similarity measures are based on convex combinations of the information contained in the regions of interest and the information contained in the global images.

Preliminary results indicate that the proposed similarity measures are capable of localizing image registration. Experiments using medical images from computer tomography and positron emission tomography demonstrate the initial success of these measures.

Finally, in other applications, auto-detection of regions of interest may prove useful and would allow for fully automated localized image registration. We examine methods to automatically detect potential regions of interest based on local activity level and present some encouraging results.
Hydrodynamic Stability of Free Convection from an Inclined Elliptic Cylinder
(University of Waterloo, 2006) Finlay, Leslie
The steady problem of free convective heat transfer from an isothermal inclined elliptic cylinder and its stability is investigated. The cylinder is inclined at an arbitrary angle with the horizontal and immersed in an unbounded, viscous, incompressible fluid. It is assumed that the flow is laminar and two-dimensional and that the Boussinesq approximation is valid. The full steady Navier-Stokes and thermal energy equations are transformed to elliptical co-ordinates and an asymptotic analysis is used to find appropriate far-field conditions. A numerical scheme based on finite differences is then used to obtain numerical solutions. Results are found for small to moderate Grashof and Prandtl numbers, and varying ellipse inclinations and aspect ratios.

A linear stability analysis is performed to determine the critical Grashof number at which the flow loses stability. Comparisons are made with long-time unsteady solutions.
Stability of Hybrid Singularly Perturbed Systems with Time Delay
(University of Waterloo, 2006) Alwan, Mohamad
Hybrid singularly perturbed systems (SPSs) with time delay are considered and exponential stability of these systems is investigated. This work mainly covers switched and impulsive switched delay SPSs . Multiple Lyapunov functions technique as a tool is applied to these systems. Dwell and average dwell time approaches are used to organize the switching between subsystems (modes) so that the hybrid system is stable. Systems with all stable modes are first discussed and, after developing lemmas to ensure existence of growth rates of unstable modes, these systems are then extended to include, in addition, unstable modes. Sufficient conditions showing that impulses contribute to yield stability properties of impulsive switched systems that consist of all unstable subsystems are also established. A number of illustrative examples are presented to help motivate the study of these systems.
Error-Tolerant Coding and the Genetic Code
(University of Waterloo, 2006) Gutfraind, Alexander
The following thesis is a project in mathematical biology building upon the so-called "error minimization hypothesis" of the genetic code. After introducing the biological context of this hypothesis, I proceed to develop some relevant information-theoretic ideas, with the overall goal of studying the structure of the genetic code. I then apply the newfound understanding to an important question in the debate about the origin of life, namely, the question of the temperatures in which the genetic code, and life in general, underwent their early evolution.

The main advance in this thesis is a set of methods for calculating the primordial evolutionary pressures that shaped the genetic code. These pressures are due to genetic errors, and hence the statistical properties of the errors and of the genome are imprinted in the statistical properties of the code. Thus, by studying the code it is possible to reconstruct, to some extent, the primordial error rates and the composition of the primordial genome. In this way, I find evidence that the fixation of the genetic code occurred in organisms which were not thermophiles.

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