• home
• research
• events
• graduate dissertations

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

MATHEW LEWIS
BSc (University of Victoria, 2021)

“Large-scale structures in two-dimensional Rayleigh–Bénard convection driven by fixed heat fluxes”

Department of Mathematics and Statistics
Friday, May 3, 2024
10:00 A.M.
David Strong Building
Room C114
https://uvic.zoom.us/j/82433823321

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

External Examiner:
Dr. Philipp Vieweg, Department of Applied Mathematics and Theoretical Physics, Cambridge University

Chair of Oral Examination:
Dr. Timothy Iles, Department of Pacific and Asian Studies, UVic

Abstract
Existence of large-scale structures in two-dimensional Rayleigh–Bénard convection is investigated in the case of boundaries that are no-slip and have fixed heat fluxes. Direct numerical simulations are carried out using the code Dedalus, which implements spectral methods. Simulations are carried out in a horizontally periodic domain, primarily with a horizontal period 20 times the layer height. The large-scale structure of interest is a pair of wide convection rolls. After finding one such two-roll state at fixed values of the Rayleigh number R and Prandtl number Pr, the parameters are varied slowly in time to find two-roll states elsewhere in the R-Pr plane. Loss of a two-roll state occurs by transition to a four-roll state, which is detected using several criteria. The R-Pr plane is divided into one region where we have found two-roll states that persist, and one region where we have not. Along part of the boundary between these regions the two-roll states are steady, suggesting that their break-up is a linear instability. Elsewhere in the R-Pr plane the boundary is hard to locate precisely because the two-roll states are unsteady and can display metastable behaviour. The two-roll regime is found only when R is sufficiently small and Pr is sufficiently large, and these two-roll states are further classified as steady or unsteady. Contrasting our findings with simulations in the literature that have different boundary conditions and/or are three dimensional, we find that existence or nonexistence of large-scale structures is substantially affected by both thermal and velocity boundary conditions and by dimension. A simple model with one fitting parameter is found to capture the middle region of a wide roll at various parameter values, and partial results are presented towards using this model to understand the region in the R-Pr plane at which two-roll states are found.

The Final Oral Examination
for the Degree of

Master of Science
(Department of Mathematics and Statistics)

Xinning Liu BS.C (North China University of Technology (NCUT), 2022)

“On the Prediction Accuracy of Models Chosen by Constrained Minimum Criterion”

Thursday, April 18, 2024
4:00 P.M.
Virtual Defence: Zoom link

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

Chair of Oral Examination:
Dr. Xuekui Zhang, Department of Mathematics and Statistics, UVic

Notice of the Final Oral Examination for the Degree of Doctor of Philosophy of

SKYE DORE-HALL

MSc (University of Victoria, 2021)
BSc Hons. (Kwantlen Polytechnic University, 2018)

“Comparisons of Ramp Functions and Michaelis-Menten Functions in Biochemical Dynamical Systems”

Department of Mathematics and Statistics
Friday, April 12, 2024
9:00 A.M.
Clearihue Building Room B021

Join Zoom Meeting
https://uvic.zoom.us/j/87408735591

Supervisory Committee:
Dr. Roderick Edwards, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Junling Ma, Department of Mathematics and Statistics, UVic (Member)
Dr. Stephanie Willerth, Department of Mechanical Engineering, UVic (Outside Member)

External Examiner:
Prof. Jean-Luc Gouzé, French National Institute for Research in Digital Science and Technology

Chair of Oral Examination:
Dr. Tao Wang, Department of Economics, UVic

Abstract
Analysis of nonlinear dynamical systems, such as those modeled using Michaelis-Menten kinetics, can be difficult. Thus, it is natural to consider whether such systems can be simplified in a way that facilitates analysis while preserving qualitative behaviour. Previously, we showed that when the Michaelis-Menten terms in a model of plant metabolism are replaced by piecewise linear approximations called ramp functions, the qualitative behaviour of the model is maintained. We then defined a limited class of systems containing ramp functions called biochemical ramp systems and studied their properties, including the existence and stability of equilibria and global flow.

Here, we expand on our previous work by reforming the definition of a biochemical ramp system to describe a wider class of systems. We study the properties of several types of biochemical ramp systems that were previously not covered by the definition, and show that their qualitative behaviour is similar to that of their Michaelis-Menten counterparts. We then introduce concepts from chemical reaction network theory, such as the Deficiency Zero and Deficiency One Theorems, and explain how they are applicable to the analysis of biochemical ramp functions, but cannot be applied to the corresponding Michaelis-Menten systems. In the last chapter, we show that when ramp functions are used in systems that do not fall under the expanded definition of a biochemical ramp system, there can be qualitative differences in behaviour between these ramp systems and their Michaelis-Menten counterparts. We end with a look at periodic behaviour in ramp systems by studying a version of the Lotka-Volterra predator-prey model containing ramp functions.

Notice of the Final Oral Examination for the Degree of Doctor of Philosophy of

SHUAI YOU
BSc (University of British Columbia, 2019)

“Predictive modelling, Anomaly Detection, and Empirical Extraction of Variation Patterns within Longitudinal Data”

Department of Mathematics and Statistics

Wednesday, March 13, 2024
2:30 P.M.
Virtual Defence https://uvic.zoom.us/j/86576342841

Supervisory Committee:
Dr. Mary Lesperance, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Xuekui Zhang, Department of Mathematics and Statistics, UVic (Co-Supervisor)
Dr. Youlian Pan, Department of Mathematics and Statistics, UVic (Member)
Dr. Tao Lu, Department of Electrical and Computer Engineering, UVic (Outside Member)

External Examiner:
Dr. Depeng Jiang, Department of Community Health Sciences, University of Manitoba

Chair of Oral Examination:
Dr. Jamie Lawson, Department of Political Science, UVic

Abstract
This thesis 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 extending stacking algorithms for survival and longitudinal data prediction, a generalizable workflow, and an innovative unsupervised learning algorithm for real-time spectral anomaly identification and classification. This thesis emphasizes the collective contributions of these interconnected research findings, aiming to advance predictive modeling methodologies and anomaly detection across disciplines.

See announcement and abstract.

Announcement (PDF)

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.