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See announcement and abstract (PDF).

See announcement and abstract (PDF).

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

Xiaolin Huang
BSc (Washington University, 2019)

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. You Liang, Department of Mathematics, Toronto Metropolitan University

Chair of Oral Examination
Dr. Kirstin Lane, School of Exercise Science, Physical and Health Education, UVic

Abstract
COVID-19 has affected the lives of people worldwide. This thesis includes two studies on the response to COVID-19 using statistical methods. The first study explores the impact of lockdown timing on COVID-19 transmission across US counties. We used Functional Principal Component Analysis to extract COVID-19 transmission patterns from county-wise case counts, and used machine learning methods to identify risk factors, with the timing of lockdowns being the most significant. In particular, we found a critical time point for lockdowns, as lockdowns implemented after this time point were associated with significantly more cases and faster spread. The second study proposes an adaptive sample pooling strategy for efficient COVID-19 diagnostic testing. When testing a cohort, our strategy dynamically updates the prevalence estimate after each test, and uses the updated information to choose the optimal pool size for the subsequent test. Simulation studies showed that our strategy reduces the number of tests required to test a cohort compared to traditional pooling strategies.

Reviewers

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

External Examiner
Dr. Quyuan Lin, Department of Mathematics, University of California Santa Barbara

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

Abstract
This thesis is devoted to the motion of the incompressible and inviscid ow which is axisymmetric and swirl-free in a cylinder, where the hydrostatic approximation is made in the axial direction. It addresses the problem of local existence and uniqueness in the spaces of analytic functions for the Cauchy problem for the inviscid primitive equations, also called the hydrostatic incompressible Euler equations, on a cylinder, under some extra conditions. Following the method introduced by Kukavica-Temam-Vicol-Ziane in Int. J. Differ. Equ. 250 (2011) , we use the suitable extension of the Cauchy-Kowalewski theorem to construct locally in time, unique and real-analytic solution, and find the explicit rate of decay of the radius of real- analiticity. Furthermore, this thesis discusses the problem of finite-time blowup of the solution of the system of equations. Following a part of the method introduced by Wong in Proc Am Math Soc. 143 (2015), we prove that the first derivative of the radial velocity blows up in time, using primary functional analysis tools for a certain class of initial data. Taking the solution frozen at r = 0, we can apply an a priori estimate on the second derivative of the pressure term, to derive a Ricatti type inequality.

Nazim Acar

BSc (Uludağ Universitesi, 1998)

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

Reviewers

Supervisory Committee
Dr. Christopher Bose, Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Ahmed Sourour, Mathematics and Statistics, University of Victoria (Member)

External Examiner
Dr. Shafiqul Islam, School of Mathematical and Computational Sciences, University of Prince Edward Island

Chair of Oral Examination
Dr. Richard Keeler, Department of Physics, UVic

Abstract
A Pythagorean Triple (PT) is a triple of positive integers (a, b, c), that satisfies a² + b² = c². By requiring two of the entries being relatively prime, (a, b, c) becomes a Primitive Pythagorean Triple (PPT). This removes trivially equivalent PTs. Following up on the unpublished paper by D. Romik [1] we develop a sequence of mappings and show how each PPT has a unique path starting from one of the two initial nodes (3, 4, 5), (4, 3, 5). We explain a way of generating the PPTs through paper folding. Using a various techniques from dynamics we show how these mappings can be carried over to their conjugates in the first unit arc x² + y² = z², x, y ≥ 0 and the unit interval [0, 1]. Under these mappings and through the conjugacies we show that the PPTs, the pair of rational points on the first unit arc and the rational numbers on the unit interval correspond to each other with the forward orbits exhibiting similar behavior. We identify infinite, σ-finite invariant measures for one-dimensional systems. With the help of the developed conjugacies we extend the dynamics of the PPTs to the continued fraction expansion of the real numbers in the unit interval and show a connection to the Euclidean algorithm. We show that the dynamical system is conservative and ergodic.