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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

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

Supervisory Committee
Dr. Slim Ibrahim, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Boualem Khouider, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner
Dr. David Muraki, Department of Mathematics, Simon Fraser University

Chair of Oral Examination
Dr. Graham Voss, Department of Economics, UVic

Abstract
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, 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.

See announcement and abstract (PDF)

See announcement and abstract PDF.