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

WANMENG WANG

BSc (University of Manitoba, 2022)

AdaptVarLM: A Linear Regression Model for Covariate-Dependent Non-Constant Error Variance

Department of Mathematics and Statistics

Friday, August 16, 2024

11:00 A.M. Virtual Defence

Supervisory Committee:
Dr. Xuekui Zhang, Department of Mathematics and Statistics, 番茄社区 (Supervisor)
Dr. Li Xing, Department of Mathematics and Statistics, UVic (Member)
Dr. Xiaojian Shao, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Ke Xu, Department of Economics, UVic

Chair of Oral Examination:
Dr. Amanda Bates, Department of Biology, UVic
Dr. Robin G. Hicks, Dean, Faculty of Graduate Studies

Abstract
In biological research, traditional multiple regression models assume homoscedasticity- constant variance of error terms-an assumption that is difficult to maintain in complex biological data. This thesis introduces AdaptVarLM, a novel linear regression model specialized in dealing with non-constant error variance dependent on one covariate. AdaptVarLM integrates an auxiliary linear relationship between the logarithmic variance of the error term and a specific explanatory variable, and uses maximum likelihood estimation (MLE) in the iterative updating process to improve the parameter estimation accuracy. By modelling non-constant error variance, AdaptVarLM outperforms the traditional regression model in capturing the complex variability inherent in biological data. Applying to the study of Alzheimer's disease, AdaptVarLM detects genetically linked genes associated with the disease and error variance. The results of analyzing both bulk and single-cell data validate the effectiveness of AdaptVarLM in detecting significant genes.

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

XIAOXIAO MA
MSc (Tianjin University, 2020)
BSc (Tianjin University, 2017)

On Bilevel Programs and Minimax Problems

Department of Mathematics and Statistics

Monday, August 12, 2024
9:00 A.M.
Virtual Defence

Supervisory Committee:
Dr. Jane Ye, Department of Mathematics and Statistics, 番茄社区 (Supervisor)
Dr. Julie Zhou, Department of Mathematics and Statistics, UVic (Member)
Dr. Yang Shi, Department of Mechanical Engineering, UVic (Outside Member)

External Examiner:
Dr. Patrick Mehlitz, Department of Mathematics and Computer Science, University of Marburg

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

Abstract
Second-order optimality conditions usually offer more precise insights into local optimality compared to their first-order counterparts. Concurrently, there has been a growing prevalence of bilevel programs and minimax problems in recent years. In our research, we intricately explore second-order optimality conditions within the realm of bilevel programs and minimax problems.

First, we provide a comprehensive exploration of second-order combined approaches for bilevel problems. Building on the well-known first-order combined approach, the research introduces novel techniques that incorporate lower-level second-order information to overcome the difficulty of the constraint qualification for bilevel problems. By characterizing lower-level optimal solutions using both first and second-order necessary optimality conditions, together with the value function constraint, we give some new single-level reformulations for bilevel problems for which the important partial calmness condition can be more likely to hold.

We then focus on the introduction and analysis of calm local minimax points, which is an appropriate local notion for nonconvex-nonconcave nonsmooth minimax problems. We study the properties of calm local minimax points, establishing their strong connections with existing optimality concepts. We provide a comprehensive exploration of first-order and second-order sufficient and necessary optimality conditions for calm local minimax points.

See announcement and abstract.

See announcement and abstract (PDF).

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

KHASHAYAR NESHAT TAHERZADEH
MSc (Sharif University of Technology, 2019)
BSc (Azad University, 2016)

鈥淚nvariant conic optimization with basis-dependent cones: scaled diagonally dominant matrices and real *-algebra decomposition鈥�

Department of Mathematics and Statistics
Wednesday, July 17, 2024 9:00 A.M.
Engineering and Computer Science Building Room 130

Supervisory Committee:
Dr. David Goluskin, Department of Mathematics and Statistics, 番茄社区 (Supervisor)
Dr. Heath Emerson, Department of Mathematics and Statistics, UVic (Member)
External Examiner:
Dr. Cordian Riener, Department of Mathematics and Statistics, University of Troms酶
Chair of Oral Examination:
Dr. Violeta Iosub, Department of Chemistry, UVic

Abstract
Symmetry reduction for a semidefinite program (SDP) with symmetries makes computational solution of the SDP easier by decomposing the semidefiniteness constraint into multiple smaller semidefineness constraints. This decomposition requires changing to a symmetry-adapted basis that block diagonalizes the matrix variable, but this does not change the optimum value of the SDP because the semidefinite cone is basis-independent. For other cones that are basis-dependent, if optimization problems over those cones have symmetries one can still change to a symmetry-adapted basis that block diagonalizes the matrix. However, this change of basis generally changes the constraint cone and can change the optimum. In this thesis we develop a framework for determining when symmetry reduction for basis-dependent conic optimization makes the optimum increase, decrease, or stay the same. The aim is to determine this using general features such as the symmetry group of the optimization problem, without having to solve the problem computationally. We then use our framework to prove various results of this type for scaled diagonally dominant programs (SDDPs), which are convex optimization problems over the cone of scaled diagonally dominant matrices. These results depend on the orbital structure of the underlying representation of invariant SDDPs. Using the regular representation, we demonstrate that analysis of SDDPs of any size can be confined to a smaller SDDP that is invariant under a particular representation. Our approach uses real *-algebra decomposition of equivariant maps, which is not needed for existing symmetry reduction of SDPs. Because polynomial optimization problems with sum-of-squares and sum-of-binomial-squares can be represented as SDPs and SDDPs, respectively, our results on SDDPs have implications for polynomial optimization. Using several polynomial optimization problems as examples, we give computational results that illustrate our theorems. For polynomial optimization subject to sum-of-binomial-squares, our examples included cases in which symmetry reduction causes the optimum to increase, decrease, or stay the same.

See abstract and announcement (PDF).

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

MATHEW LEWIS
BSc (番茄社区, 2021)

鈥淟arge-scale structures in two-dimensional Rayleigh鈥揃茅nard convection driven by fixed heat fluxes鈥�

Department of Mathematics and Statistics
Friday, May 3, 2024
10:00 A.M.
David Strong Building
Room C114

Supervisory Committee:
Dr. David Goluskin, Department of Mathematics and Statistics, 番茄社区 (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鈥揃茅nard 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)

鈥淥n the Prediction Accuracy of Models Chosen by Constrained Minimum Criterion鈥�

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

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 (番茄社区, 2021)
BSc Hons. (Kwantlen Polytechnic University, 2018)

鈥淐omparisons of Ramp Functions and Michaelis-Menten Functions in Biochemical Dynamical Systems鈥�

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

Join Zoom Meeting

Supervisory Committee:
Dr. Roderick Edwards, Department of Mathematics and Statistics, 番茄社区 (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 Gouz茅, 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.