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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.

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

MIKAYLA HOLMES
BSc (University of Victoria, 2021)

Background Connectivity: Understanding the brain's functional organization

Department of Mathematics and Statistics

Friday, May 19, 2023 10:00 A.M.
David Strong Building Room C128

Supervisory Committee:
Dr. Michelle Miranda, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Mary Lesperance, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Jodie Gawryluk, Department of Psychology, UVic

Chair of Oral Examination:
Dr. Raad Nashmi, Department of Biology, UVic

Abstract
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. 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.

The Final Oral Examination
for the Degree of

Master of Science
(Department of Mathematics and Statistics)

Jianping Yu
Phd in Math, Chinese Academy of Sciences

“Hierarchical Model for Evaluation of Physician Care in the ICU”

April 26th, 2023
10:00am
On zoom

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

Chair of Oral Examination:
Dr. Laura Cowen, Department of Mathematics and Statistics, UVic

See announcement (pdf).

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

HANS OERI

MSc (Simon Fraser University, 2018)
BSc (Simon Fraser University, 2016)

“Convex Optimization Methods for Bounding Lyapunov Exponents”

Department of Mathematics and Statistics

Friday, April 14, 2023
10:00 A.M.
Virtual Defence

Supervisory Committee:
Dr. David Goluskin, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Anthony Quas, Department of Mathematics and Statistics, UVic (Member)
Dr. Nishant Mehta, Department of Computer Science, UVic (Outside Member)

External Examiner:
Dr. Milan Korda, Laboratoire d'analyse et d'architectures des systèmes, Centre national de la recherche scientifique, France

Chair of Oral Examination:
Dr. Daniela Constantinescu, Department of Mechanical Engineering, UVic

Abstract
In dynamical systems, the stability of orbits is quantified by Lyapunov exponents, which are computed from the average rate of divergence of trajectories. We develop techniques for computing sharp upper bounds on the maximal LE 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 computationally tractable SOS programs 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 discussing 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.

See announcement (pdf)

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

Jack THOMAS
BSc (University of Victoria, 2019)

"Response to Improving abundance estimation by combining capture-recapture and presence-absence data: example with a large carnivore"

Tuesday, April 11, 2023
9:30 A.M.
David Strong Building C128

Supervisory Committee:
Dr. Laura Cowen, Department of Mathematics and Statistics, UVic (Co-Supervisor)
Dr. Simon Bonner, Department of Statistical and Actuarial Sciences, UWO (Co-Supervisor)

Chair of Oral Examination:
Dr. Farouk Nathoo, Department of Mathematics and Statistics, UVic

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

Meifan LIN
BMath, University of Waterloo, 2021

“On linear models selected by the constrained minimum criterion”

April 5th 10:00 a.m.
Conducted Virtually

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. Junling Ma, Department of Mathematics and Statistics, UVic