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The Final Oral Examination for the Degree of

Master of Science
(Department of Mathematics and Statistics)

Farbod Esmaeili
Shahid Beheshti University, B.Sc. in Statistics (2022)

“A Comparative Review of Stock Market Forecasting Models with a Simulation Study on GARCH Dynamics”

December 12, 2024
3:00 pm
Zoom: https://uvic.zoom.us/j/83893006823

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. Peter Dukes, Department of Mathematics and Statistics, UVic

The Final Oral Examination for the Degree of Master of Science

(Department of Mathematics and Statistics)

Naresh Neupane
MA, University of Victoria (2020)
BSc, Southeastern Louisiana University (2016)

“Optimal Designs for Quadratic, Cubic, Quartic and Quintic Polynomial Models in Balls”

December 13, 2024
1:30 pm
DTB A203

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

Chair of Oral Examination: Dr. Junling Ma, Department of Mathematics and Statistics, UVic

Examining Committee

Supervisory Committee
Dr. Rod Edwards, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Anthony Quas, Department of Mathematics and Statistics, UVic (Member)

External Examiner
Dr. Bastien Fernandez, Laboratoire de Probabilités Statistique and Modélisation, French National Centre for Scientific Research

Chair of Oral Examination
Dr. Eva Kwoll, Department of Geography, UVic

Meeting link: https://uvic.zoom.us/j/87448763654

Abstract
Electronic circuitry based on chaotic Glass networks, a type of piecewise smooth dynamical system, has recently been proposed as a potential design for true random number generators. Glass networks are good designs due to their potential for chaotic behaviour and because their analytic tractability allows us here to propose a method for approximating their entropy, a measure of irregularity in dynamical systems. We discuss some of the historical developments that led to the interest in the model that we consider within the context of random number generation. Additionally, we discuss a method for converting a Glass network’s governing piecewise-smooth differential equations into discrete-time dynamical systems, and then into symbolic dynamical systems. We also detail how the symbolic entropy of the given Glass network is bounded above by the entropy of the symbolic dynamical system formed from its transition graph, a type of directed graph that represents the possible transitions in phase space between regions not containing discontinuities. We then extend previous results by detailing our new method of refining the transition graph to be a more accurate depiction of the true system’s dynamics, making use of more specific information about trapping regions in phase space. Refinements come in the form of splitting nodes and duplicating/partitioning edges on the transition graph and removing those that are never realized by the continuous dynamics. We show that refinements can be done to arbitrary levels and in the limit as the level of refinement goes to infinity, the entropy of the refined transition graphs converges to the true entropy of the system. Along with this, since it is not possible to calculate the limiting value, approximation is necessary. Doing this by hand is tedious and difficult, so as a result, we also detail here an algorithm we devised that automates the refinement process, allowing for approximation (from above) of symbolic entropy. Various examples are considered throughout and we also discuss how numerical simulation can be used to non-rigorously estimate symbolic entropy, as an independent (approximate) verification of our results. Finally, we detail some unfinished and future work which could extend our results further, along with alternative methods to achieve similar and potentially even stronger results. With our results and algorithm, using upper bounds on a Glass network’s symbolic representation’s entropy is now a viable method for assessing the irregularity of its dynamics.

Zoom meeting.

Programme for the Degree of Master of Science

(Department of Mathematics and Statistics)

Yingjie HOU

BSc. (University of Liverpool, 2013)
MSc. (Imperial College London, 2014)

"A literature review of batch effect removal methods for scRNA-seq Data Analysis"

Tuesday, October 15, 2024
1:00 P.M.

Virtual Defence

Supervisory Committee:
Dr. Xuekui Zhang, Department of Mathematics and Statistics, UVic (Supervisor)
Dr. Ke Xu, Department of Economics, UVic (Member)

Chair of Oral Examination:
Dr. Min Tsao, Department of Mathematics and Statistics, UVic

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, University of Victoria (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, University of Victoria (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)

“Invariant conic optimization with basis-dependent cones: scaled diagonally dominant matrices and real *-algebra decomposition”

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, University of Victoria (Supervisor)
Dr. Heath Emerson, Department of Mathematics and Statistics, UVic (Member)
External Examiner:
Dr. Cordian Riener, Department of Mathematics and Statistics, University of Tromsø
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).