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# Mathematics - Theses, Dissertations, and other Required Graduate Degree Essays

Receive updates for this collection## 2-Regular Digraphs on Surfaces

A 2-regular digraph is one where every vertex has in-degree and out-degree 2. This thesis focuses on surface embeddings of 2-regular digraphs, one where the underlying graph is embedded in a surface and all faces are bounded by directed closed walks. Immersion acts as a natural minor-like containment relation for embedded 2-regular digraphs and this theory is linked to undirected graph minors by way of the directed medial graph. We parallel the theory of undirected graphs in surfaces by proving analogues of Whitney's Theorem and Tutte's peripheral cycles theorem for 2-regular digraphs in the sphere. Then, using a notion of branch-width and a 2-regular digraph grid theorem by Johnson, we prove that for each fixed surface S, the 2-regular digraphs embeddable in S are characterized by a finite list of immersion obstructions. We then present the current state of the art with regard to classification of obstructions for surfaces: Johnson characterized the sphere obstructions, we classify all projective plane obstructions, and we have a computer assisted partial list of obstructions for the torus and Klein bottle. We also consider two open problems in the world of undirected graphs on surfaces and resolve their analogues in the world of 2-regular digraphs on surfaces. The first conjecture by Negami we resolve in the affirmative, that a 2-regular digraph has a finite planar cover if and only if it is projective planar. The second, the strong embedding conjecture, is resolved in the negative and we provide an infinite family of well connected counter-examples.

## Aspects of the Arithmetic of Uniquely Trigonal Genus Four Curves: Arithmetic Invariant Theory and Class Groups of Cubic Number Fields

In this thesis, we study the family of uniquely trigonal genus 4 curves via their connection to del Pezzo surfaces of degree 1. We consider two different aspects of these curves. First, we show how to construct any uniquely trigonal genus 4 curve whose Jacobian variety has fully rational 2-torsion and whose trigonal morphism has a prescribed totally ramified fibre. Using this construction, we find an infinite family of cubic number fields whose class group has 2-rank at least 8. We also consider genus 4 curves that are superelliptic of degree 3, and prove sharp results on the size of the 2-torsion subgroup of their Jacobian varieties over the rationals. Our second result is a contribution to arithmetic invariant theory. We consider the moduli space of uniquely trigonal genus 4 curves whose trigonal morphism has a marked ramification point, originally studied by Coble, from a modern perspective.We then show under a technical hypothesis how to construct an assignment from the rational points on the Jacobian variety of such a curve to a fixed orbit space that is independent of the curve. The assignment is compatible with base extensions of the field, and over an algebraically closed field our assignment reduces to the assignment of a marked (in the aforementioned sense) uniquely trigonal genus 4 curve to its moduli point. The orbit space is constructed from the split algebraic group of type E8.

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## Connectedness of Certain Random Graphs

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## The genus of generalized random and quasirandom graphs

The genus of a graph $G$ is the minimum integer $h$ such that $G$ has an embedding in some surface (closed compact 2-manifold) $S_h$ of genus $h$. In this thesis, we will discuss the genus of generalized random and quasirandom graphs. First, by developing a general notion of random graphs, we determine the genus of generalized random graphs. Next, we approximate the genus of dense generalized quasirandom graphs. Based on analysis of minimum genus embeddings of quasirandom graphs, we provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. More precisely, we provide an algorithm that for a given (dense) graph $G$ of order $n$ and given $\varepsilon>0$, returns an integer $g$ such that $G$ has an embedding into a surface of genus $g$, and this is $\varepsilon$-close to a minimum genus embedding in the sense that the minimum genus $\g(G)$ of $G$ satisfies: $\g(G)\le g\le (1+\varepsilon)\g(G)$. The running time of the algorithm is $O(f(\varepsilon) n^2)$, where $f(\cdot)$ is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) of genus $g$. This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time $O(f_1(\varepsilon)\,n^2)$. The last part of the thesis studies the genus of complete $3$-uniform hypergraphs, which is a special case of genus of random bipartite graphs, and also a natural generalization of Ringel--Youngs Theorem. Embeddings of a hypergraph $H$ are defined as the embeddings of its associated Levi graph $L_H$ with vertex set $V(H)\cup E(H)$, in which $v\in V(H)$ and $e\in E(H)$ are adjacent if and only if $v$ and $e$ are incident in $H$. The construction in the proof may be of independent interest as a design-type problem.

## Parameter estimation and uncertainty quantification applied to advection-diffusion problems arising in atmospheric source inversion

In this thesis we present a method to obtain an efficient algorithm to perform parameter estimation with uncertainty quantification of mathematical models that are complex and computationally expensive. We achieve this with a combination of emulation of the mathematical model using Gaussian processes and Bayesian statistics and inversion for the parameter estimation and uncertainty quantification. In particular we apply these ideas to a source inversion problem in atmospheric dispersion. We explain the theory and ideas behind each relevant part of the process in the emulation and parameter estimation. The concepts and methodology presented in this work are general and can be applied to a wide range of problems where it is necessary to estimate parameters but the underlying mathematical model is expensive, rendering more classical approaches unfeasible. To validate the concepts used, we perform a parameter estimation study in a model that is relatively cheap to compute and whose parameter values are known in advance. Finally we perform a parameter estimation withuncertainty quantification of a much more expensive atmospheric dispersion model using real data from a lead-zinc smelter in Trail, British Columbia. The parameter estimation includes approximating high-dimensional integrals with Markov chain Monte Carlo methods and solving the source inversion problem in atmospheric dispersion using the Bayesian framework.

## On the Nikolaevskiy equation and the fractal dimension of its attractor

We investigate the attractor of the Nikolaevskiy equation, a sixth-order partial differential equation (PDE) containing a small parameter whose solutions exhibit spatiotemporal chaos with strong scale separation. We first prove well-posedness and regularity of the solutions, and derive asymptotic bounds on their derivatives, to put the subsequent results on a firm footing. The rest of the work focuses on showing that the dynamical system associated with the Nikolaevskiy equation possesses an attractor with a finite fractal dimension. Bounds on this dimension are both derived analytically and computed numerically, paying particular attention to their scaling with the parameters. We describe the numerical methods, and present computational results that include the scaling of various norms of the solutions, as well as of the power spectrum and the spectrum of Lyapunov exponents of the PDE.

## The unrestricted linear fractional assignment problem

The linear fractional assignment problem (LFAP) is a well-studied combinatorial optimization problem with various applications. It attempts to minimize ratio of two linear functions subject to standard assignment constraints. When the denominator of the objective function is positive, LFAP is solvable in polynomial time. However, when the denominator of the objective function is allowed to take positive and negative values, LFAP is known to be NP-hard. In this thesis, we present two 0-1 programming formulations of LFAP and compare their relative merits using experimental analysis. We also show that LFAP can be solved as a polynomialy bounded sequence of constrained assignment problems. Experimental results using this methodology are also given. Finally, some local search heuristics are developed and analyzed their efficiency using systematic experimental analysis. Our algorithms are able to solve reasonably large size problems within acceptable computational time.

## Game of Cops and Robbers on Eulerian Digraphs

Cops and Robbers is a well-known pursuit game played on a graph. There are two players, one controls the cops and the other controls the robber, who take turns moving along edges of the graph. The goal of the cops is to capture the robber, which is accomplished if a cop occupies the same vertex as the robber. The main question is to determine the minimum number of cops that can guarantee the robber’s capture on the given graph. This problem has been widely studied for the case of undirected graphs, but very little attention has been given to finding the cop number of digraphs. In the thesis we focus on this game on Eulerian digraphs, viewed as an extension of the game on undirected graphs. Some preliminary results, which were obtained for the special case of 4-regular quadrangulations of the torus and the Klein bottle, show that there is a possibility to develop rich results in this area.

## Modelling tuberculosis in South Africa

Tuberculosis ranks alongside HIV/AIDS as one of the deadliest infectious diseases in the world, killing 1.4 million people in 2015. The World Health Organization and the Stop TB Partnership set targets to achieve a 90% reduction in incidence and a 95% reduction in mortality from the 2015 values by 2035. We investigate the global dynamics of a compartmental system of ordinary differential equations that models tuberculosis. A more detailed model is then calibrated to the HIV negative TB endemic in South Africa and used to evaluate the 2035 reduction targets. Optimal care and control strategies for fixed budgets are also identified. Model projections for South Africa show the mortality targets can be met through combined interventions, however due to relapse and latent progression the incidence targets are unrealistic. To minimize incidence in 20 years, funding should be prioritized into linking drug susceptible cases (over multi-drug resistant cases) onto care.

## Spectral differentiation: Integration and inversion

Pseudospectral differentiation matrices suffer from large round-off error, and give rise to illconditioned systems used to solve differential equations numerically. This thesis presents two types of matrices designed to precondition these systems and improve robustness towards this round-off error for spectral methods on Chebyshev-Gauss-Lobatto points. The first of these is a generalization of a pseudospectral integration matrix described by Wang et al. [18]. The second uses this integration matrix to construct the matrix representing the inverse operator of the differential equation. Comparison is made between expected and calculated eigenvalues. Both preconditioners are tested on several examples. In many cases, accuracy is improved over the standard methodology by several orders of magnitude. Using these matrices on general sets of points is briefly discussed.