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Notice of the Final Oral Examination for the Degree of Master of Science of

FREDDIE MULLIN

BEd (University of Victoria, 2013)
BSc (University of Victoria, 2011)
BA (University of Victoria, 2011)

“Studies on Math Education and Trigraph Homomorphisms”

Department of Mathematics and Statistics
Thursday, December 14, 2023
3:30 P.M.
Virtual Defence

Supervisory Committee:
Dr. Gary MacGillivray, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Jane Butterfield, Department of Mathematics and Statistics, UVic (Co-Supervisor)
Dr. Richard Brewster, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Jacobus Swarts, Mathematics Department, Vancouver Island University

Chair of Oral Examination:
Dr. Marlea Clarke, Department of Political Science, UVic

Abstract
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 deciding 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 analyse the effectiveness of the UVic Pretest. First we compare the current online UVic Pretest with the past paper Pretest in terms of their respective effectiveness in identifying students who are ready for Calculus I. We also analyse the current online UVic 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 UVic Pretest and identify questions that are particularly useful to the test.

Notice of the Final Oral Examination for the Degree of Master of Science of

J. ETHAN WILLIAMS
BSc (University of Victoria, 2021)

“Eternal Domination Problems”

Department of Mathematics and Statistics

Wednesday, December 13, 2023
10:30 A.M.
David Turpin Building
Room A203

Supervisory Committee:
Dr. Gary MacGillivray, Department of Mathematics and Statistics, University of Victoria
(Co-Supervisor) Dr. Richard Brewster, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:
Dr. William Klostermeyer, School of Computing, University of North Florida

Chair of Oral Examination:
Dr. Matt Moffitt, Department of Chemistry, UVic

Abstract
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.

See announcement and abstract (PDF)

Notice of the Final Oral Examination for the Degree of Master of Science

Hanan Abousaleh
BSc (University of Victoria, 2021)

Examining Committee

Supervisory Committee
Dr. Julie Zhou, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Michelle Miranda, Department of Mathematics and Statistics, UVic (Member)

External Examiner
Dr. Nilanjana Roy, Department of Economics, UVic

Chair of Oral Examination
Dr. Hong-Chuan Yang, Department of Electrical and Computer Engineering, UVic

Abstract
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.

The Final Oral Examination for the Degree of Master of Science

(Department of Mathematics and Statistics)

ZIYI LYU

University of Waterloo, 2021 (B.Math)

Investigating the Relationship between Bayes Factors and Credible Intervals

Friday, September 22nd, 2023 2:00 P.M

Online

Supervisory Committee:
Dr. Farouk Nathoo, Department of Mathematics and Statistics, UVic (Supervisor)
Dr. Min Tsao, Department of Mathematics and Statistics, UVic (Member)

Chair of Oral Examination:
Dr. Ryan Budney, Department of Mathematics and Statistics, UVic

Reviewers
Supervisory Committee
Dr. Anthony Quas, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Christopher Bose, Department of Mathematics and Statistics, UVic (Member)
Dr. Sean Chester, Department of Computer Science, UVic (Outside Member)

External Examiner
Dr. Ian Morris, School of Mathematical Sciences, Queen Mary University of London

Chair of Oral Examination
Dr. Michel Lefebvre, Department of Physics and Astronomy, UVic

Abstract
A linear cocycle is an object that arises naturally in the study of dynamical systems and statistics. Oseledets’ theorem [21] guarantees a decomposition of X into fast and slow subspaces. This dissertaton is a study of strongly measurable cocycles over an invertible ergodic system acting on a separable Banach space, including a proof of this theorem in this infinite dimensional setting.

The Final Oral Examination
for the Degree of
Master of Science
(Department of Mathematics and Statistics)

Guojun Ma
2020 University of Alberta B.Sc.
A meta-analysis of genome-wide associations with body mass index
August 21, 2023
2:00pm
Online

Supervisory Committee:
Dr. Yu-Ting Chen, Department of Mathematics and Statistics, UVic (Supervisor)
Dr. Xuekui Zhang, Department of Mathematics and Statistics, UVic (Co-Supervisor)

Chair of Oral Examination:
Dr. Trefor Bazett, Department of Mathematics and Statistics, UVic

Reviewers
Supervisory Committee
Dr. Xuekui Zhang, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Li Xing, Department of Mathematics and Statistics, UVic (Member)

External Examiner
Dr. Ibrahim Numanagic, Department of Computer Science, UVic

Chair of Oral Examination
Dr. Catherine Leéger, Department of French, UVic

Abstract
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 showed that our approach increases recall and precision while decreasing the detection time for most types of sequencing data.

Notice of the Final Oral Examination for the Degree of Master of Science of

TUAN VIET DAO
BA (Gustavus Adolphus College, 2017)

“Estimating the Size of the COVID-19 Population in British Columbia Using the Stratified Petersen Estimator”

Department of Mathematics and Statistics

Monday, August 14, 2023
10:00 A.M.
Virtual Defence

Supervisory Committee:
Dr. Laura Cowen, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Junling Ma, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:
Dr. Lloyd Elliott, Department of Statistics and Actuarial Science, Simon Fraser University

Chair of Oral Examination:
Dr. Tim Pelton, Department of Curriculum and Instruction, UVic

Abstract
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.

See announcement and abstract (PDF).

BSc (University of Victoria, 2021)

Notice of the Final Oral Examination for the Degree of Master of Science
Topic: Saturation Problems on Graphs

Mathematics and Statistics

Date and location:
Wednesday, July 19, 2023 10:00 A.M.
David Strong Building Room C108

Reviewers

Supervisory Committee
Dr. Natasha Morrison, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Kieka Mynhardt, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner
Dr. Antonio Girão, Mathematical Institute, University of Oxford

Chair of Oral Examination
Dr. Tao Lu, Department of Electrical and Computer Engineering, UVic

Abstract 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 from , 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 Girão, 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.

BSc (University of Regina, 2021)
Notice of the Final Oral Examination for the Degree of Master of Science
Topic: Cross-Sperner Systems

Department of Mathematics and Statistics

Date and location
Tuesday, July 18, 2023
10:00 A.M.
David Strong Building, Room C108

Reviewers
Supervisory Committee
Dr. Natasha Morrison, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Kieka Mynhardt, Department of Mathematics and Statistics, UVic (Member)
External Examiner
Dr. Karen Meagher, Department of Mathematics and Statistics, University of Regina
Chair of Oral Examination
Dr. Shemine Gulamhusein, School of Child and Youth Care, UVic

See announcement (PDF)

Fahim Alam
BSc (Shahjalal University of Science and Technology, Bangladesh, 2017)

Notice of the Final Oral Examination for the Degree of Master of Science

Topic
A New Numerical Approach to Solve 1D Viscous Plastic Sea Ice Momentum Equation

Department of Mathematics and Statistics
Date and location
Monday, June 26, 2023
9:30 A.M.
David Strong Building, Room C128

Supervisory Committee
Dr. Boualem Khouider, Department of Mathematics and Statistics, University of Victoria(Supervisor)
Dr. David Goluskin, Department of Mathematics and Statistics, UVic (Member)

External Examiner
Dr. Alex Bihlo, Department of Mathematics and Statistics, Memorial University ofNewfoundland

Chair of Oral Examination
Dr. Randy Scharien, Department of Geography, UVic

Abstract
While there has been a colossal effort in the ongoing decades, the ability to simulate ocean icehas fallen behind various parts of the climate system and most Earth System Models areunable to capture the observed adversities of Arctic sea ice, which is, as it were, attributed toour frailty to determine sea ice dynamics. Viscous Plastic rheology is the most by and largerecognized model for sea ice dynamics and it is expressed as a set of partial differentialequations that are hard to tackle numerically. Using the 1D sea ice momentum equation as aprototype, we use the method of lines based on Euler’s backward method. This results in anonlinear PDE in space only. At that point, we apply the Damped Newton’s method which hasbeen 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 linearequation with discontinuous coeffi cients during Newton iteration. To overcome this diffi culty,we use the Finite element method to solve the linear PDE at each Newton iteration. In thispaper, we show that with the adequate smoothing and re-scaling of the linear equation,convergence can be guaranteed and the numerical solution indeed converges effi ciently to thecontinuum solution unlike other numerical approaches that typically solve an alternate set ofequations and avoid the diffi culty of the Newton method for a large nonlinear algebraicsystem. The fi nite element solver failed to converge when the original setting of the smoothedSIME with a smoothing constant K = 2.8 x 10⁸ was used. A much smaller constant of K=100 wasnecessary. The large smoothing constant K leads to an ill conditioned mass matrix.