Theses (Mathematics and Statistics)

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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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    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.
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    Algebraic Cycles on Products of Generically Smooth Quadrics
    (2022-12-21) Guangzhao, Zhu; Scully, Stephen
    In this thesis, we study rationality questions for algebraic cycles (modulo 2) on products of generically smooth projective quadrics over fields of any characteristic. More specifically, let $X_{1},\cdots,X_{n}$ be generically smooth projective quadrics over a field $F$ with anisotropic totally singular part. We study the image $\overline{Ch}(X_{1} \times \cdots \times X_{n})$ of the scalar-extension homomorphism $Ch(X_{1} \times \cdots \times X_{n}) \to Ch((X_{1} \times \cdots \times X_{n})_{\overline{F}})$, where $\overline{F}$ denotes a fixed algebraic closure of $F$ and $Ch$ denotes the total Chow group modulo $2$. This has been studied extensively in the case where $X_{1},\cdots,X_{n}$ are smooth by A.Vishik, N.Karpenko, A.Merkurjev and others. Our goal is to extend the existing theory to include the case of singular but generically smooth quadrics in characteristic $2$. Here we follow recent work of Karpenko, who has considered the special case where $X_{1}=\cdots=X_{n}$. First, we show that the image $\overline{Ch}(X_{1} \times \cdots \times X_{n})$ inherits a ring structure and an action of Steenrod operations from the mod-2 Chow ring of the smooth locus of $X_{1} \times \cdots \times X_{n}$. Using the ring structure, we then introduce and study a composition of rational correspondences (modulo 2) for products of generically smooth projective quadrics, laying foundations for an investigation of non-totally singular quadratic forms in any characteristic by algebraic-geometric methods. In this direction, we introduce a new discrete invariant for such forms, which we call the \emph{rational correspondence type}. This extends the \emph{motivic decomposition type} previously defined by Vishik for non-degenerate forms. We extend several well-known results on Vishik's invariant to our more general setting. These include a number of restrictions imposed by splitting pattern invariants, as well as results that relate the rational correspondence types of different forms in situations where their associated quadrics can be related via suitable Chow correspondences. Using these results, we compute the rational correspondence type for certain families of forms, including generic forms of even dimension, and so-called quasi-strongly excellent forms. In the final part of the thesis, we show that the deepest result of Vishik on the motivic decomposition type, the so-called \emph{excellent connections theorem}, remains valid for arbitrary non-totally singular forms of dimension at most $9$. We also apply our methods to the study of a conjecture of Hoffmann and Laghribi on the classification of singular Pfister neighbours, and to the study of the isotropy behaviour of quadratic forms over function fields of quadrics.
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    Boundary Independent Broadcasts in Graphs
    (2022-12-08) Hoepner, Jules; MacGillivray, Gary; Mynhardt, Christina
    A \textit{broadcast} on a connected graph $G$ with vertex set $V(G)$ is a function $f:V(G)\rightarrow \{0, 1, ..., \text{diam}(G)\}$ such that $f(v)\leq e(v)$, where $e(v)$ denotes the eccentricity of $v$. A vertex $v$ is said to be \textit{broadcasting} if $f(v)>0$. The \textit{cost} of $f$ is $\sigma(f)=\sum_{v\in V(G)}f(v)$, or the sum of the strengths of the broadcasts on the set of broadcasting vertices $V_f^+=\{v\in V(G)\,:\,f(v)>0\}$. A vertex $u$ \textit{hears} $f$ from $v\in V_f^+$ if $d_G(u, v)\leq f(v)$. The broadcast $f$ is \textit{hearing independent} if no broadcasting vertex hears another. If, in addition, any vertex $u$ that hears $f$ from multiple broadcasting vertices satisfies $f(v)\leq d_G(u, v)$ for all $v\in V_f^+$, the broadcast is said to be \textit{boundary independent.} The minimum cost of a maximal boundary independent broadcast on $G$, called the \textit{lower bn-independence number}, is denoted $i_{bn}(G)$. The \textit{lower h-independence number} $i_h(G)$ is defined analogously for hearing independent broadcasts. We prove that $i_{bn}(G)\leq i_h(G)$ for all graphs $G$, and show that $i_h(G)/i_{bn}(G)$ is bounded, finding classes of graphs for which the two parameters are equal. For both parameters, we show that the lower bn-independence number (h-independence number) of an arbitrary connected graph $G$ equals the minimum lower bn-independence number (h-independence number) among those of its spanning trees. We further study the maximum cost of boundary independent broadcasts, denoted $\alpha_{bn}(G)$. We show $\alpha_{bn}(G)$ can be bounded in terms of the independence number $\alpha(G)$, and prove that the maximum bn-independent broadcast problem is NP-hard by a reduction from the independent set problem to an instance of the maximum bn-independent broadcast problem. With particular interest in caterpillars, we investigate bounds on $\alpha_{bn}(T)$ when $T$ is a tree in terms of its order and the number of vertices of degree at least 3, known as the \textit{branch vertices} of $T$. We conclude by describing a polynomial-time algorithm to determine $\alpha_{bn}(T)$ for a given tree $T$.
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    Random forests on trees
    (2022-09-02) Xiao, Ben; Ray, Gourab
    This thesis focuses a mathematical model from statistical mechanics called the Arboreal gas. The Arboreal gas on a graph $G$ is Bernoulli bond percolation on $G$ with the conditioning that there are no ``loops". This model is related to other models such as the random cluster measure. We mainly study the Arboreal gas and a related model on the $d$-ary wired tree which is simply the $d$-ary wired tree with the leaves identified as a single vertex. Our first result is finding a distribution on the infinite $d$-ary tree that is the weak limit in height $n$ of the Arboreal gas on the $d$-ary wired tree of height $n$. We then study a similar model on the infinite $d$-ary wired tree which is Bernoulli bond percolation with the conditioning that there is at most one loop. In this model, we only have a partial result which proves that the ratio of the partition function of the one loop model in the wired tree of height $n$ and the Arboreal gas model in the wired tree of height $n$ goes to $0$ as $n \rightarrow \infty$. This allows us to prove certain key quantities of this model is actually the same as analogues of that quantity in the Arboreal gas on the $d$-ary wired tree, under an additional assumption.
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    Tight Bounds on 3-Neighbor Bootstrap Percolation
    (2022-08-31) Romer, Abel; Dukes, Peter; Noel, Jonathan
    Consider infecting a subset $A_0 \subseteq V(G)$ of the vertices of a graph $G$. Let an uninfected vertex $v \in V(G)$ become infected if $|N_G(v) \cap A_0| \geq r$, for some integer $r$. Define $A_t = A_{t-1} \cup \{v \in V(G) : |N_G(v) \cap A_{t-1}| \geq r \},$ and say that the set $A_0$ is \emph{lethal} under $r$-neighbor percolation if there exists a $t$ such that $A_t = V(G)$. For a graph $G$, let $m(G,r)$ be the size of the smallest lethal set in $G$ under $r$-neighbor percolation. The problem of determining $m(G,r)$ has been extensively studied for grids $G$ of various dimensions. We define $$m(a_1, \dots, a_d, r) = m\left (\prod_{i=1}^d [a_i], r\right )$$ for ease of notation. Famously, a lower bound of $m(a_1, \dots, a_d, d) \geq \frac{\sum_{j=1}^d \prod_{i \neq j} a_i}{d}$ is given by a beautiful argument regarding the high-dimensional ``surface area" of $G = [a_1] \times \dots \times [a_d]$. While exact values of $m(G,r)$ are known in some specific cases, general results are difficult to come by. In this thesis, we introduce a novel technique for viewing $3$-neighbor lethal sets on three-dimensional grids in terms of lethal sets in two dimensions. We also provide a strategy for recursively building up large lethal sets from existing small constructions. Using these techniques, we determine the exact size of all lethal sets under 3-neighbor percolation in three-dimensional grids $[a_1] \times [a_2] \times [a_3]$, for $a_1,a_2,a_3 \geq 11$. The problem of determining $m(n,n,3)$ is discussed by Benevides, Bermond, Lesfari and Nisse in \cite{benevides:2021}. The authors determine the exact value of $m(n,n,3)$ for even $n$, and show that, for odd $n$, $$\ceil*{\frac{n^2+2n}{3}} \leq m(n,n,3) \leq \ceil*{\frac{n^2+2n}{3}} + 1.$$ We prove that $m(n,n,3) = \ceil*{\frac{n^2+2n}{3}}$ if and only if $n = 2^k-1$, for some $k >0$. Finally, we provide a construction to prove that for $a_1,a_2,a_3 \geq 12$, bounds on the minimum lethal set on the the torus $G = C_{a_1} \square C_{a_2} \square C_{a_3}$ are given by $$2 \le m(G,3) - \frac{a_1a_2 + a_2a_3 + a_3a_1 -2(a_1+a_2+a_3)}{3} \le 3.$$