# Department of Mathematics and Statistics Graduate Theses

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Item Quasiregular Ellipticity and CohomologyMorris, Ethan H.; Mathematics and Statistics; Mansouri, Abdol-RezaWe study the topological consequences of quasiregular ellipticity. A manifold is quasiregularly elliptic if it supports the existence of a quasiregular map- ping from Rd, a function characterized by bounded distortion. Quasiregular mappings are a generalization of conformal maps that share some of their useful properties but allow for the consideration of a richer family of func- tions in higher dimensions. This thesis offers a comprehensive examination of several recent results related to topological constraints on quasiregularly elliptic manifolds.Item A Mathematical Model for Tick Transmission of Lyme Disease: Does Questing Style Impact the Competition Between Two Strains of Infection?Wasser, Alexandra J.; Mathematics and Statistics; Day, TroyBorrelia burgdorferi is the bacteria responsible for Lyme disease and there are multiple strains of B. burgdorferi that exist. There is a perception that all ticks carry Lyme disease and all bacteria responsible for the disease are equal. Each strain of B. burgdorferi can be characterised by varying persistence in a host. By investigating the questing behaviours of ticks, an analysis is conducted to explore the conditions conducive to different strains. A model is constructed to reproduce the dynamics of the life cycle of the tick and the relationship with hosts to replicate natural interactions between the tick and a host species. The model construction captures the intra-seasonal dynamics using differential equations and inter-seasonal dynamics using a set of discrete recursions. Using an invasion analysis, the conditions that allow for mutant invasion and environmental factors that favour invasion are investigated. The questing styles, both synchronous and asynchronous, are replicated using mathematical techniques. It is found that the relative fitness of competing strains is not impacted by questing style but rather the absolute fitness of a strain is impacted. A conclusion can be drawn that chronic infection is predominantly found in asynchronous systems and with the worsening state of climate change, the persistence of acute infection is less likely.Item Generative Adversarial Networks Based on a General Parameterized Family of Generator Loss FunctionsVeiner, Justin; Mathematics and Statistics; Alajaji, Fady; Gharesifard, BahmanThis thesis introduces a unifying parameterized generator loss function for generative adversarial networks (GANs). We establish an equilibrium theorem for our resulting GAN system under a canonical discriminator in terms of the so-called Jensen-$f$-divergence, a natural generalization of the Jensen-Shannon divergence to the $f$-divergence. We also show that our result recovers as special cases several GANs from the literature, including the original GAN, least square GAN (LSGAN), $\alpha$-GAN and others. Finally, we systematically conduct experiments on three image datasets for different manifestations of our GAN system to illustrate their performance and stability.Item F-tests for Frequency Modulated Time Series Using Multitaper MethodsOtt, Ben; Mathematics and Statistics; Takahara, Glen; Burr, WesleyWe propose three new semiparametric multitaper tests for the detection of modulated line components where the modulation is assumed to be created by a polynomial of degree P. The first test, F_4, is a modification of the F-test found in [3] and uses a set of weights to obtain a better in-band concentration for a wider range of tapers. The second test, F_4', builds on the aforementioned test by fixing a problem that arises when choosing to evaluate the F-test with an even number of tapers. We derive the distributions for these two tests along with an examination of the final test based on an aggregation of F_4' tests, where the aggregation is over different multitaper orders. We derive an upper bound for the distribution of the Aggregate F-test statistic, and via simulation, we approximate the F-test's confidence level needed to obtain a pre-specified Type 1 error of the Aggregated test. These new F-tests are compared to the F-test in [3], and [30] via simulation comparing the probability of detecting a frequency-modulated signal in different types of noise and modulation widths. We also examine the functionality of these new test statistics through a study of the SoHO GOLF instrument data.Item Reinforcement and Preferential Attachment Models via Pólya UrnsSingh, Somya; Mathematics and Statistics; Alajaji, Fady; Gharesifard, BahmanIn this thesis, we devise two different types of discrete-time stochastic models using modified Pólya urn schemes. The first set of models concerns interacting contagion networks constructed using two-colour (red and black) finite memory Pólya urns in which reinforcing balls are removed M time steps after being added (where M is the “memory” of the urn). The urns interact in the sense that the probability of drawing a red ball (which represents an infectious state or an opinion) for a given urn, not only depends on the ratio of red balls in that urn but also on the ratio of red balls in other urns in a network representing the interconnections, hence accounting for the effect of spatial contagion. The finite memory reinforcement provides a diminishing effect of past draws which represents curing of an infection in an epidemic spread model, or lessening influence of a popular opinion in a social network. We examine the stochastic properties of the underlying Markov draw process and construct a class of dynamical systems to approximate the asymptotic marginal distributions. We also design a consensus achieving connected network of agents via two-color finite memory Pólya urns. The interaction between urns is time-varying and is represented via “super-urns” which combine for each node its own urn with its neighbouring urns. We obtain the consensus value in terms of the network’s reinforcement parameters and memory. In the second part of this thesis, we introduce a novel preferential attachment model using the draw variables of a modified Pólya urn with an expanding number of colors, notably capable of modeling influential opinions (in terms of vertices of high degree) as the graph evolves. Unlike the Barabási-Albert model, the color-coded vertices in conjunction with the time-varying reinforcing parameter in our model allows for the vertices added (born) later in the process to potentially attain a high degree in a way that is not captured by the former. We study the degree count of the vertices in the graphs generated via our model by analyzing the draw vectors of the underlying stochastic process. Furthermore, we compare our model with the Barabási-Albert model via simulations