Theses (Mathematics and Statistics)

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    Predictive modelling, Anomaly Detection, and Empirical Extraction of Variation Patterns within Longitudinal Data
    (2024) You, Shuai; Zhang, Xuekui; Lesperance, Mary
    This dissertation represents a comprehensive exploration and enhancement of statistical methodologies, addressing complex challenges in transdisciplinary data analysis. It integrates novel techniques across various domains to bridge gaps in existing algorithms, focusing on advancing multitask prediction and anomaly detection. Significant contributions involve extended stacking algorithms for survival and longitudinal data prediction, an innovative unsupervised learning algorithm for real-time spectral anomaly identification and classification, and applications of longitudinal data analytics in multi-decade geospatial marine pollution monitoring data, including paralytic and amnesic shellfish toxins and fecal coliform bacteria. This dissertation emphasizes the collective contributions of these interconnected research findings, aiming to advance predictive modelling methodologies and anomaly detection across disciplines.
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    Pseudoku: A Sudoku Adjacency Algebra and Fractional Completion Threshold
    (2024) Nimegeers, Kate; Dukes, Peter
    The standard Sudoku puzzle is a 9 × 9 grid partitioned into 3 × 3 square boxes and partially filled with symbols from the set {1, 2, ..., 9}, with the goal of the puzzle being to complete the grid so that each symbol appears once and only once in each row, column, and box. We study generalized Sudoku puzzles, set on an n × n grid with cells partitioned into n boxes (sometimes called cages) of height h and width w such that hw = n. Throughout this work, these generalized Sudoku are referred to as (h, w)-Sudoku when h and w are significant, but as simply Sudoku otherwise. The goal of solving a partially filled (h, w)-Sudoku puzzle remains the same; complete the Sudoku by assigning placements in the grid to each symbol from {1, 2, ..., hw} so that each symbol appears once and only once in each row, column, and box. This thesis is specifically concerned with establishing conditions which guarantee a fractional Sudoku completion. A fractional Sudoku completion is an assignment of a set of weights to each symbol-cell incidence, representing the proportion of the symbol for that specific cell. The total weight of symbols for each cell must sum to one, and the sum of the weights for each symbol must be exactly one across the cells from each row, column, and box. These conditions still require a balanced distribution of symbols throughout the grid, but with considerably more flexibility than the typical Sudoku conditions. In order to apply graph theoretic techniques to the problem, we develop a 4-partite graph representation, GP , for a partial Sudoku, P . The 4 parts correspond to the rows, columns, symbols, and boxes of P , and the edges of GP indicate the conditions for a completed Sudoku that remain unsatisfied in P . We then introduce the concept of a tile: a 4-vertex subgraph of GP , which represents a valid symbol placement in P . Completing P is equivalent to decomposing the edges of GP into these tiles. We then use an edge-tile inclusion matrix to relate the existence of such a decomposition to the existence of an solution vector with {0, 1} entries for a specific linear system. It is here that we move to the fractional setting through a relaxation of what constitutes an acceptable solution to the linear system - specifically, we are satisfied with solution vectors for which all entries are non-negative. To find conditions that guarantee such a solution exists we study the Gram matrix of the edge-tile inclusion matrix for the empty (h, w)-Sudoku, denoted M. We show that M is symmetric and that each element of M corresponds to a pair of edges in the graph representation Ghw of the empty (h, w)-Sudoku grid. We then leverage the inherent symmetry of equivalence relations between these edges to establish a Sudoku adjacency algebra which contains M . This allows us to explicitly construct a generalized inverse for M . This generalized inverse, along with some applied perturbation theory, is used to show that given large enough h and w, the linear system for any sufficiently sparse partial (h, w)-Sudoku is a minor perturbation of the linear system for the empty (h, w) Sudoku, and therefore allows a fractional completion. After presenting this main result, we take a brief detour to consider the unique case of Sudoku puzzles with thin boxes, examining how fixing the box width variable w while allowing height h to grow asymptotically influences the density conditions necessary for fractional completion. We also give an overview of our exploratory use of the Schur complement for matrix decomposition. Although this method didn’t directly feed into our primary results, it was instrumental in the discovery of the equivalence relations we used to construct our Sudoku adjacency algebra. Finally, we explore the potential applicability of our methodologies to certain Sudoku variants and acknowledge the limitations inherent in our approach. In the appendices, we provide additional resources that complement the main body of our work. In Appendix A, we give a factorization of the Sudoku matrix M and its eigenvectors as Kronecker products for readers who wish to more directly compare our methodology to algebraic graph theory work done on Sudoku by other researchers. Appendix B presents a series of interactive and educational activities designed to introduce students to the basic principles of Latin squares in a fun spy-themed setting.
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    Mobile Guards’ Strategies for Graph Surveillance and Protection
    (2024) Virgile, Virgélot; MacGillivray, Gary; Mynhardt, C. M.
    In this dissertation, we study the “one guard moves” model of both the eternal domination game and the eviction game. We investigate the computational complexity of deciding whether k guards can respond to any sequence of attacks on an n-vertex graph G in both games. We show that this decision problem is EXPTIME-complete when neither G nor k is fixed, and when the initial configuration of the guards is given in both cases. We further show that in the case of the eternal domination game, if the guards can choose their initial configuration and the graph is directed, the decision problem remains EXPTIME-complete. We present an algorithm that decides the problem in time O(kn^(k+2)) for both games, marking a significant improvement over the previously fastest known algorithm which has time complexity O(n^(2k+2)). Our algorithm further determines the maximum number of attacks (potentially infinite) the guards can defend from each configuration. We study the relationship between the eternal domination number of a graph and its clique covering number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt, and disprove a conjecture of Klostermeyer and MacGillivray (The Fundamental Conjecture [Eternal Domination: Criticality and Reachability, Discuss. Math. Graph Theory 37 (2017), no. 1, 63–77]). We prove that the smallest graph having its eternal domination number less than its clique covering number has ten vertices. We also demonstrate that for any integer k>=2, there exist infinitely many graphs having domination number and eternal domination number equal to k containing dominating sets which are not eternal dominating sets. In addition, we show that there exists a function f such that for any integer k>=1, any graph with independence number k has eviction number at most f(k). We further show that the eviction number of cographs can be computed in polynomial time. Finally, we study the length of both games when played on an n-vertex graph on which are located k guards; that is, the maximum number of turns required before a winner can be decided.
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    Studies on math education and weak trigraph homomorphisms
    (2024-01-03) Mullin, Freddie; MacGillivray, Gary; Butterfield, Jane
    This thesis is comprised of two parts: (i) a study of homomorphisms of weak trigraphs and (ii) an analysis of the effectiveness of the University of Victoria (UVic) Department of Mathematics and Statistics’ Pretest. In the first part of the thesis, we study homomorphisms of weak trigraphs. Results analogous to those for graph homomorphisms are developed. In particular, we determine the complexity of decid- ing whether there is a weak trigraph homomorphism of a weak trigraph G to a weak trigraph H, the complexity of deciding whether a given weak trigraph has a weak trigraph homomorphism to a proper subgraph (the complexity of deciding whether it is not a core) and describe an efficient algorithm based on consistency checking that determines whether there is a weak trigraph homomorphism from a given cactus weak trigraph to a fixed weak trigraph H. In the second part of the thesis, we analyze the effectiveness of the UVic Pretest. First we compare the current online pretest with the past paper pretest in terms of their respective effectiveness in identifying students who are ready for Calculus I. We also analyze the current online pretest in greater detail, to identify which precalculus skills are most likely to predict success on that test itself. Finally, we use odd ratios to categorize each question on the online pretest and identify questions that are particularly useful to the test.
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    The Heisenberg Spectral Triple and Associated Zeta Functions
    (2024-01-03) Steed, Brendan; Emerson, Heath; Putnam, Ian
    The construction of Butler, Emerson, and Schultz [2] produced a certain spectral triple, which they called the Heisenberg cycle, by way of the quantum mechanical annihilation and creation operators, d/dx ± x, along with their relationships to the harmonic oscillator, -d^2/dx^2 + x^2; Where all of these operators are de fined (initially) to act on smooth functions over R. In particular, their Heisenberg cycle was over a crossed-product generated by the natural translation action on the (commutative) C*-algebra of uniformly continuous, bounded, functions on R. In this thesis, we generalize the Heisenberg cycle of Butler, Emerson, and Schultz to allow for the construction of a spectral triple over a crossed-product generated by the natural translation action on the C*-algebra of uniformly continuous, bounded, functions on a Euclidean space, V , of arbitrary finite dimension n. For such a generalization, the annihilation and creation operators are replaced using the exterior derivative and codifferential, exterior and interior multiplication by a certain differential 1-form, and the relationship these four operators have to the n-dimensional harmonic oscillator acting on differential forms. Similarly to [2], we will show that our generalized Heisenberg cycle provides a new way of producing spectral triples over crossed-products of the form C(M) ⋊_α Γ, where Γ is a discrete subgroup of V and α : V x M →M is a smooth V -action on a compact manifold M. In Chapter 1, we introduce the problem and briefly discuss some historical background behind Alain Connes program of noncommutative geometry, as well as touch on some elementary constructions in multi-linear algebra. Chapter 2 is where we de ne the classes of differential forms which appear most frequently in this thesis. Therein, we also rigorously de fine the operators mentioned in the paragraph above, and use them to produce the so-called Dirac-Heisenberg which will be associated to our generalization of the Heisenberg cycle. For the first half of Chapter 3, we discuss some basic C*-algebra theory and introduce the crossed-product native to the Heisenberg cycle. In the latter half of that chapter, we verify that our Heisenberg cycle satis es the conditions of a spectral triple, compute an integral formula for the resulting ζ-functions, and show how one uses the Heisenberg cycle to produce spectral triples over crossed-products generated by smooth actions of V on compact manifolds.
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    Eternal Domination Problems
    (2023-12-19) Williams, Ethan; MacGillivray, Gary; Brewster, Richard
    Consider placing mobile guards on the vertices of a graph. The vertices are then attacked by an assailant, requiring you to move guards to the attacked vertices. What is the minimum number of guards you need in order to be able to defend against any sequence of attacks? This question is the basis for the eternal domination problem. In this thesis we investigate this problem and introduce new parameters related to it. These new parameters arise from changing three of the assumptions made when defining the game. Specifically we assume that any number of guards can move when defending against an attack; only one attack needs to be defended against at a time; and that any number of guards can occupy a vertex. Changing these assumptions gives rise to the maneuver, invasion, and stacking numbers respectively. We investigate these parameters throughout this thesis, especially as they relate to trees. Additionally, we tackle the related problem of eternal Roman domination, which is based on the topic which originally gave rise to the eternal domination problem. We establish a best possible upper bound for this parameter over all graphs. Finally, we present exponential time algorithms for solving all of these problems, as well as a host of other related problems.
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    Innovative CVX-based algorithms for Optimal Design Problems on Discretized Regions
    (2023-12-04) Abousaleh, Hanan; Zhou, Julie
    We focus on a class of optimization problems known as optimal design problems, where the goal is to select design points optimally with respect to some criterion of interest. For regression models, the optimality criterion is based on the statistical model itself and is often a function of the information matrix. We solve A-, D-, and EI-optimal design problems in this thesis. The CVX program in MATLAB is a modelling tool and solver for convex optimization problems. As with other numerical methods in the literature, formulating an optimal design problem in a CVX-compatible way requires a discrete design space. We develop a CVX-based algorithm to solve optimal design problems on large and irregular discrete spaces for multiple regression models. The algorithm uses innovative rules to add several design points at each iteration, and clusters nearby points together at the end of iteration. Furthermore, we provide useful guidelines for discretizing irregular regions. These are based on derived theoretical properties which relate optimal designs on continuous and discrete design spaces. Several numerical examples and their MATLAB codes are presented for A-, D-, and EI-optimal designs for both linear and generalized linear models. The optimal designs found via the CVX solver are better than those presented in the literature. In addition, our guidelines to discretizing design spaces improve the efficiency of optimal designs, especially over irregular regions. We find that our iterative procedure overcomes the bottlenecks of typical sequential and multiplicative algorithms.
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    Some results on linear dynamical systems
    (2023-09-07) Lee, George; Quas, Anthony
    A linear cocycle is an object that arises naturally in the study of dynamical systems and statistics. Oseledets’ multiplicative ergodic theorem [22] guarantees a decompo- sition of a linear space of states into equivariant subspaces that grow logarithmically at rates corresponding to the Lyapunov exponents. Theorem 66 is the main result of this thesis, a semi-invertible version of this theorem: the ergodic system is invertible, the state space is a separable Banach space and the cocycle is strongly measurable and forward integrable, but with no invertability or injectivity assumptions. Past results have implicitly or explicitly made extra assumptions on the underlying cocycle, but in this work no injectivity or separability of the dual is assumed, and no prior version of the result or other overly unconstructive machinery is used in obtaining the direct sum decomposition of the state spacs
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    Iimi: A novel automated workflow for plant virus diagnostics from high-throughput sequencing data
    (2023-08-31) Ning, Haochen; Zhang, Xuekui
    Several workflows have been developed for the diagnostic testing of plant viruses using high-throughput sequencing methods. Most of these workflows require considerable expertise and input from the analyst to perform and interpret the data when deciding on a plant’s disease status. The most common detection methods use workflows based on de novo assembly and/or read mapping. Existing virus detection software mainly uses simple deterministic rules for decision-making, requiring a certain level of understanding of virology when interpreting the results. This can result in inconsistencies in data interpretation between analysts which can have serious ramifications. To combat these challenges, we developed an automated workflow using machine-learning methods, decreasing human interaction while increasing recall, precision, and consistency. Our workflow involves sequence data mapping, feature extraction, and machine learning model training. Using real data, we compared the performance of our method with other popular approaches and show our approach increases recall and precision while decreasing the detection time for most types of sequencing data.
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    Estimating the Size of the COVID-19 Population in British Columbia Using the Stratified Petersen Estimator
    (2023-08-30) Dao, Viet; Cowen, Laura; Ma, Junling
    The presence of undetected COVID-19 cases is a known phenomenon. Mathematical modelling techniques, such as capture-recapture, provide a reliable method for estimating the true size of the infected population. Treating a positive SARS-CoV-2 diagnostic test result as the initial capture and a hospital admission with a COVID-19-related diagnosis code as the recapture, we developed a Lincoln-Petersen model with temporal stratification, taking into account factors that influence the occurrence of captures. Applying this model to repeated patient encounter data collected at the provincial level in British Columbia, we estimated the number of COVID-19 cases among males aged 35 or older during the first week of March 2021. Our analysis revealed that the true number of cases ranged from 4.94 to 9.18 times greater than the number of detected cases.
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    Saturation Problems on Graphs
    (2023-08-25) Ogden, Shannon; Morrison, Natasha; Mynhardt, Kieka
    In this thesis, we consider two variations on classical saturation problems in extremal graph theory: rainbow saturation and weak saturation. An edge-coloured graph G is rainbow if every edge in G receives a distinct colour. Given a graph H, an edge-coloured graph G is H-rainbow-saturated if G does not contain a rainbow copy of H, but the addition of any non-edge to G, in any colour, creates a rainbow copy of H. The rainbow saturation number of H, denoted by rsat(n,H), is the minimum number of edges in an H-rainbow saturated graph on n vertices. In Chapter 2, we prove that, like ordinary saturation numbers, the rainbow saturation number of every graph H is linear in n. This result confirms a conjecture of Girao, Lewis, and Popielarz. In Chapter 3, we consider a specific type of weak saturation known as r-bond bootstrap percolation. In the r-bond bootstrap percolation process on a graph G, we start with a set of initially infected edges of G, and consider all other edges in G to be healthy. At each subsequent step in the process, the infection spreads to a healthy edge if at least one of its endpoints is incident with at least r infected edges. Once an edge is infected, it remains infected indefinitely. If a set of initially infected edges will eventually infect all of E(G), we refer to it as an r-percolating set of G. Define m_e(G,r) to be the minimum number of edges in an r-percolating set of G. Recently, Hambardzumyan, Hatami, and Qian introduced a clever new polynomial method, which they used to provide recursive formulas for m_e(G,r) when G is either a d-dimensional torus or a d-dimensional grid. We push this polynomial method further, in order to determine m_e(G,r) for certain other graphs G. In particular, we provide recursive formulas for m_e(G,r) when G is a Cartesian product of stars or a Cartesian product of joined cycles (cycles with a single chord). We also give upper and lower bounds on m_e(G,r) when G is a Cartesian product of a tree with any graph H, and examine the conditions under which these bounds match.
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    The Edwards–Sokal Coupling for the Potts Higher Lattice Gauge Theory on Z^d
    (2023-08-25) Shklarov, Yakov; Ray, Gourab; Quas, Anthony Nicholas
    The Edwards–Sokal coupling of the standard Potts model with the FK–Potts (random-cluster) bond percolation model can be generalized to arbitrary-dimension cells. In particular, the Potts lattice gauge theory on Z^d has a graphical representation as a plaquette percolation measure. We systematically develop these previously-known results, using the frameworks of cubical (simplicial) homology and discrete Fourier analysis. We show that, in the finite-volume setting, the Wilson loop expectation of a higher cycle γ is equal to the probability that γ is a homological boundary in the higher FK–Potts model. We also prove the strong FKG property of the higher FK–Potts model. These results culminate in a simple proof for the existence of infinite-volume limits in the higher Potts model and, in certain cases, of their invariance under translations and other symmetries. Additionally, we thoroughly examine the behavior of boundary conditions as they relate to the Edwards–Sokal coupling, for the purpose of understanding the higher Potts Gibbs states. In particular, we discuss spatial Markov properties and conditioning in the higher FK–Potts model, and generalize to more general boundary conditions the FKG property, the aforementioned identity for Wilson loop expectations, and a result about monotonicity in the coupling strength parameter. Also, we prove a theorem regarding the sharpness of thresholds of increasing symmetric events for the higher FK–Potts model with periodic boundary conditions. In the final section, we describe some matrix-based sampling algorithms. Lastly, we prove a new characterization of the ground states of the random-cluster model, motivated by the problem of understanding the ground states in the higher FK–Potts model.
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    Cross-Sperner Systems
    (2023-08-22) Kuperus, Akina; Morrison, Natasha
    Two sets $A$ and $B$ are \emph{comparable} if $A \subseteq B$ or $B \subseteq A$. A collection of families $(\F_{1}, \F_{2} , \cdots , \F_{k}) \in \mathcal{P}([n])^k$ is \emph{cross-Sperner} if there is no pair $i \not= j$ for which some $F_i \in \F_i$ is comparable to some $F_j \in \F_j$. Two natural measures of the `size' of such systems are the sum $\sum_{i = 1}^k |\F_i|$ and the product $\prod_{i = 1}^k |\F_i|$. Let $\s(n,k)$ be the maximum size of the sum measure for a cross-Sperner system $(\F_{1} , \cdots , \F_{k}) \in \mathcal{P}([n])^k$, and let $\maxprod(n,k)$ be the maximum size of the product measure for a cross-Sperner system $(\F_{1} , \cdots , \F_{k}) \in \mathcal{P}([n])^k$. We prove new upper and lower bounds on $\s(n,k)$ and $\maxprod(n,k)$ for general $n$ and $k \ge 2$ which improve considerably on the previous best bounds. In this thesis we prove that \[\left(\frac{2^n}{ek}\right)^k \le \maxprod(n,k) \le \left(1+\frac{1}{k}\right)\left(\frac{2^n}{2k}\right)^k, \] and \[2^{n} - \frac{3}{\sqrt{2}}\sqrt{2^n k} + 2(k-1) ~\le~ \sigma(n,k) ~\le~ 2^{n} - 2\sqrt{2^n (k-1)} + 2(k-1).\] In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patk\'{o}s, and Sz\'{e}csi from 2011. To prove these bounds, we exploit a connection between cross-Sperner systems and the \emph{comparability number} of a family of sets. Define the comparability number of a family $\F \subseteq \ps{n}$ to be the number of sets comparable to $\F$. Then define $c(n,m)$ to be the minimum comparability number of a family $\F \subseteq \ps{n}$ where $|\F| =m$. We prove that for $1 \leq m \leq 2^n$, \[c(n,m) \geq 2^{n/2 + 1}\sqrt{m} - m.\] This thesis includes joint work with Natasha Morrison, Natalie Behague, and Ashna Wright.
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    Topological Data Analysis: Persistent Homology of Uniformly Distributed Points
    (2023-08-09) Sohal, Ranjit; Budney, Ryan
    Topological Data Analysis (TDA) is a branch of computational topology that provides methods to extract qualitative information from high dimensional, noisy, and incomplete data. TDA combines techniques from various fields, such as algebraic topology, computational geometry, algorithms, statistics, and graph theory. Persistent Homology (PH), based on homology theory from algebraic topology, is the principal tool used in TDA; PH tracks the evolution of topological features of the data across multiple scales through persistent homological bars, which represent the creation (birth) and disappearance (death) of these features. These bars are graphically depicted through persistence diagrams and persistence barcodes. The challenge in using PH for the analysis of noisy real-world data is to separate the bars generated by noise from the bars that provide meaningful topological information of the underlying geometric object from which the data is sampled; this problem remains unresolved despite various proposed techniques. A limited number of papers analyzed the PH of noise by considering points in R^d generated using probability distributions. This thesis introduces persistent homology concentrating on the computational side, and it examines the birth and death times of persistent homology bars generated by Vietoris-Rips complexes of uniformly distributed points in three spaces: a unit interval, a unit square, and a unit cube. Through numerical simulations, it is identified that the birth and death times of the persistent homology bars adhere to distinct statistical distributions, whose precise nature varies according to the space from which the points are sampled and the homological dimension of the persistent homology bars; the research examines the behaviour of their parameters as the number of points increases, providing insights into the persistent homology of noise and laying the groundwork for further research.
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    A new numerical approach to solve 1D Viscous Plastic Sea Ice Momentum Equation
    (2023-07-20) Alam, Fahim; Khouider, Boualem
    While there has been a colossal effort in the ongoing decades, the ability to simulate ocean ice has fallen behind various parts of the climate system and most Earth System Models are unable to capture the observed adversities of Arctic sea ice, which is, as it were, attributed to our frailty to determine sea ice dynamics. Viscous Plastic rheology is the most by and large recognized model for sea ice dynamics and it is expressed as a set of partial differential equations that are hard to tackle numerically. Using the 1D sea ice momentum equation as a prototype, we use the method of lines based on Euler's backward method. This results in a nonlinear PDE in space only. At that point, we apply the Damped Newton’s method which has been introduced in Looper and Rapetti et al. and used and generalized to 2D in Saumier et al. to solve the Monge-Ampere equation. However, in our case, we need to solve 2nd order linear equation with discontinuous coefficients during Newton iteration. To overcome this difficulty, we use the Finite element method to solve the linear PDE at each Newton iteration. In this paper, we show that with the adequate smoothing and re-scaling of the linear equation, convergence can be guaranteed and the numerical solution indeed converges efficiently to the continuum solution unlike other numerical approaches that typically solve an alternate set of equations and avoid the difficulty of the Newton method for a large nonlinear algebraic system. The finite element solver failed to converge when the original setting of the smoothed SIME with a smoothing constant $K=2.8 \times 10^8$ was used. A much smaller constant of K=100 was necessary. The large smoothing constant K leads to an ill conditioned mass matrix.
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    Background Connectivity-Understanding the brain's functional organization
    (2023-06-22) Holmes, Mikayla; Miranda, Michelle F.
    Task-state fMRI (tfMRI) and rest-state fMRI (rfMRI) surface data from the Human Connectome Project (HCP) was examined with the goal of better understanding the nature of background activation signatures and how they compare to the functional connectivity of a brain at rest. In this paper we use a hybrid---decomposition and seed-based---approach to calculate functional connectivity of both rfMRI data and the estimated residual data from a Bayesian spatiotemporal model. This model accounts for local and global spatial correlations within the brain by applying two levels of data decomposition methods. Moreover, long-memory temporal correlations are taken into account by using the Haar discrete wavelet transform. Modifications applied to the original spatiotemporal model that facilitate the use of surface and volumetric (whole-brain) data -- in the CIFTI file format -- are what make this analysis novel. Motor task data from the HCP is modelled, followed by an analysis of the residuals, which provide details regarding the brain's background functional connectivity. These residual connectivity patterns are assessed using a manual procedure and through studying the induced covariance matrix of the model's error term. When we compare these activation signatures to those found for the same subject at rest we found that regions within the subcortex displayed strong connections in both states. Regions associated with the default mode network also displayed statistically significant connectivity while the subject was at rest. In contrast, the pre-central ventral and mid-cingulate regions had strong functional patterns in the background activation signatures that were not present in the rest-state data. This modelling technique combined with a hybrid approach to assessing functional activation signatures provides valuable insights into the role background connections play in the brain. Moreover, it is easily adaptable which allows for this research to be extended across a variety of tasks and at a multi-subject level.
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    Convex Optimization Methods for Bounding Lyapunov Exponents
    (2023-05-01) Oeri, Hans; Goluskin, David
    In dynamical systems, the stability of orbits is quantified by Lyapunov exponents (LEs), which are computed from the average rate of divergence of trajectories. We develop techniques for computing sharp upper bounds on the largest LE over trajec- tories using methods from convex optimization, which have previously been used to compute sharp bounds on the time averages of scalar quantities on bounded orbits of dynamical systems. For discrete-time dynamics we develop an optimization-based approach for computing sharp bounds on the geometric mean of scalar quantities. We therefore express LEs as infinite-time averages and as geometric means in continuous- time systems and discrete-time systems, respectively, and then derive optimization problems whose solutions give sharp bounds on LEs. When the system’s dynamics is governed by a polynomial vector field, the problems can be relaxed to computa- tionally tractable sum-of-squares (SOS) whose solutions also give sharp bounds on LEs. An approach for the practical implementation of a sequence of SOS feasibility problems whose solutions converge to the maximal LE of discrete systems is provided. We explain how symmetries can be used to simplify and generalize the optimization problems in both continuous-time and discrete-time systems. We conclude by dis- cussing the extension of the techniques developed here to the problem of bounding the sum of the leading LEs. Tractable SOS programs are derived for some special cases of this problem. The applicability of all the techniques developed here is shown by applying them to various explicit examples. For some systems we numerically compute sharp bounds that agree with the the maximal LEs, and for some we prove analytic bounds on maximal LEs by solving the optimization problems by hand.
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    A Uniqueness Theorem for C*-algebras of Hausdorff Étale Groupoids
    (2023-04-27) Goerke, Gavin; Laca, Marcelo; Eagle, Christopher
    In this thesis we study the ideal intersection property for inclusions of C*-algebras C*(H)↪C*(G) induced from a family of open subgroupoids {H} of a locally compact Hausdorff étale groupoid G. For such a family of open subgroupoids we define the notion of relative topological principality and we show that if G is relatively topologically principal to {H} then a representation of C*(G) is faithful if and only if the restriction of the representation to each of the subalgebras C*(H) is faithful. This gives a new method of verifying injectivity of representations of reduced groupoid C*-algebras. As an application of our result we prove a uniqueness theorem for C*-algebras of left cancellative small categories which generalizes a theorem of Marcelo Laca and Camila Sehnem for Toeplitz algebras of group embeddable monoids.
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    Numerical Blowup Solutions for Boundary Value Models
    (2023-01-30) Hoang, Duyen; Ibrahim, Slim; Khouider, Boualem
    In this thesis, we discuss several numerical methods to approximate singular solutions for some partial differential equations such as Burgers’ equation, Prandtl’s equations, and the inviscid primitive equations. The numerical solutions we obtain for Burgers’ equation and Prandtl’s equations are compared with the existing analytical and numerical solutions in the literature. We observe the singularity formation in the numerical solutions to Burgers’ equation and Prandtl’s equations in finite time. For the inviscid primitive equations with the initial data are close to a suitable rescale of a smooth blowup profile proven by Collot, Ibrahim, and Lin in [7], we compare the numerical solution to the theoretical blowup profile. The solution we obtain from the numerical scheme follows the profile, but the difference between the numerical and analytical profiles is quite significant closer to the blowup time. We then examine the stability of the numerical solutions by considering a small perturbation for the initial data. The gap between the perturbed and unperturbed solutions reduces as we choose smaller perturbation. However, this gap grows as it approaches the blowup time, and the stability of the numerical solutions remains in doubt.
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    The i-Graph and Other Variations on the γ-Graph
    (2023-01-03) Teshima, Laura Elizabeth; Mynhardt, C. M.; Brewster, R. C.
    In graph theory, reconfiguration is concerned with relationships among solutions to a given problem. For a graph G, the γ-graph of G, G(γ), is the graph whose vertices correspond to the minimum dominating sets of G, and where two vertices of G(γ) are adjacent if and only if their corresponding dominating sets in G differ by exactly two adjacent vertices. We present several variations of the γ-graph including those using identifying codes, locating-domination, total-domination, paired-domination, and the upper domination number. For each, we show that for any graph H, there exist infinitely many graphs whose γ-graph variant is isomorphic to H. The independent domination number i(G) is the minimum cardinality of a maximal independent set of G. The i-graph of G, denoted I(G), is the graph whose vertices correspond to the i-sets of G, and where two i-sets are adjacent if and only if they differ by two adjacent vertices. In contrast to the parameters mentioned above, we show that not all graphs are i-graph realizable. We build a series of tools to show that known i-graphs can be used to construct new i-graphs and apply these results to build other classes of i-graphs, such as block graphs, hypercubes, forests, and unicyclic graphs. We determine the structure of the i-graphs of paths and cycles, and in the case of cycles, discuss the Hamiltonicity of their i-graphs. We also construct the i-graph seeds for certain classes of line graphs, a class of graphs known as theta graphs, and maximal planar graphs. In doing so, we characterize the line graphs and theta graphs that are i-graphs.