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

MACKENZIE CARR

BSc Hons (Acadia University, 2018)

Enumerating Digitally Convex Sets in Graphs

Department of Mathematics and Statistics

Wednesday, June 24, 2020 10:00 A.M
Remote Defence

Supervisory Committee: Dr. Kieka Mynhardt, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
Dr. Ortrud Oellermann, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)
External Examiner: Dr. Danielle Cox, Department of Mathematics, Mount Saint Vincent University
Chair of Oral Examination: Dr. Catherine Costigan, Department of Psychology, UVic

Abstract

Given a finite set V , a convexity, C , is a collection of subsets of V that contains both the empty set and the set V and is closed under intersections. The elements of C are called convex sets. We can define several different convexities on the vertex set of a graph. In particular, the digital convexity, originally proposed as a tool for processing digital images, is defined as follows: a subset S _ V (G) is digitally convex if, for every v 2 V (G), we have N[v] _ N[S] implies v 2 S. Or, in other words, each vertex v that is not in the digitally convex set S needs to have a private neighbour in the graph with respect to S. In this thesis, we focus on the generation and enumeration of digitally convex sets in several classes of graphs. We establish upper bounds on the number of digitally convex sets of 2-trees, k-trees and simple clique 2-trees, as well as conjecturing a lower bound on the number of digitally convex sets of 2-trees and a generalization to k-trees. For other classes of graphs, including powers of cycles and paths, and Cartesian products of complete graphs and of paths, we enumerate the digitally convex sets using recurrence relations. Finally, we enumerate the digitally convex sets of block graphs in terms of the number of blocks in the graph, rather than in terms of the order of the graph.

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

RONGZHENG XIE

BSc (Swansea University, 2014)

“A population approach to systems of Izhikevich neurons: can neuron interaction cause bursting?”

Department of Mathematics and Statistics

Thursday, April 23, 2020 8:30 A.M.
Remote Defence

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

External Examiner: Dr. Julien Modolo, Inserm -LTSI, Université de Rennes

Chair of Oral Examination: Dr. Caterina Valeo, Department of Mechanical Engineering, UVic

Abstract
In 2007, Modolo and colleagues derived a population density equation for a population of Izhekevich neurons. This population density equation can describe oscillations in the brain that occur in Parkinson’s disease. Numerical simulations of the population density equation showed bursting behaviour even though the individual neurons had parameters that put them in the tonic firing regime. The bursting comes from neuron interactions but the mechanism producing this behaviour was not clear. In this thesis we study numerical behaviour of the population density equation and then use a combination of analysis and numerical simulation to analyze the basic qualitative behaviour of the population model by means of a simplifying assumption: that the initial density is a Dirac function and all neurons are identical, including the number of inputs they receive, so they remain as a point mass over time. This leads to a new ODE model for the population. For the new ODE system, we define a Poincaré map and then to describe and analyze it under conditions on model parameters that are met by the typical values adopted by Modolo and colleagues. We show that there is a unique fixed point for this map and that under changes in a bifurcation parameter, the system transitions from fast tonic firing, through an interval where bursting occurs, the number of spikes decreasing as the bifurcation parameter increases, and finally to slow tonic firing.

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

JOSEPH ANTHONY HORAN

MSc (University of Victoria, 2015)
BMath (University of Waterloo, 2013)

“Spectral gap and asymptotics for a family of cocycles of Perron-Frobenius operators”

Department of Mathematics and Statistics

Friday, April 17, 2020 9:30 A.M.
Remote Defence

Supervisory Committee:
Dr. Christopher Bose, Department of Mathematics and Statistics, University of Victoria (CoSupervisor)
Dr. Anthony Quas, Department of Mathematics and Statistics, UVic (Co-Supervisor)
Dr. Sue Whitesides, Department of Computer Science, UVic (Outside Member)

External Examiner:
Dr. Jérôme Buzzi, Département de Mathématiques d’Orsay, Université Paris-Saclay

Chair of Oral Examination:
Dr. Elisabeth Gugl, Department of Economics, UVic

Abstract
At its core, a dynamical system is a set of things and rules for how they change. In the study of dynamical systems, we often ask questions about long-term or average phenomena: whether or not there is an equilibrium for the system, and if so, how quickly the system approaches that equilibrium. These questions are more challenging in the non-autonomous (or random) setting, where the rules change over time. The main goal of this dissertation is to develop new tools with which to study random dynamical systems, and demonstrate their application in a nontrivial context. We prove a new Perron-Frobenius theorem for cocycles of bounded linear operators which preserve and sometimes contract a cone in a Banach space; this new theorem provides an explicit upper bound for the second-largest Lyapunov exponent of the cocycle, which determines how quickly the system approaches its equilibrium-like state. Using this theorem and other tools (including a new Lasota-Yorke-type inequality for Perron-Frobenius operators for use with a family of maps), we show that a class of cocycles of piecewise linear maps has a Lyapunov spectral gap (hence answering the equilibrium question in the affirmative), and we moreover have an explicit lower bound on the spectral gap. We also prove asymptotics for a family of cocycles arising from a perturbation of a fixed map with two invariant densities; we obtain a linear upper bound for the second-largest Lyapunov exponent, and the bound is sharp, in the sense that there are members of this family of perturbations where the second-largest Lyapunov exponent is linear in the perturbation parameter. The sharpness example is studied through an in-depth determinant-free linear algebra computation for Markov operators.

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

MATTHEW PARKER
BSc (University of Victoria, 2016)

“N-mixture models with auxiliary populations and for large population abundances”

Department of Mathematics and Statistics

Friday, April 17, 2020 9:00 A.M.
Remote Defence

Supervisory Committee:
Dr. Laura Cowen, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Julie Zhou, Department of Mathematics and Statistics, UVic (Member)

External Examiner:
Dr. Simon Bonner, Department of Biology, Western University

Chair of Oral Examination:
Dr. Hossein Nassaji, Department of Linguistics, UVic

Abstract
The key results of this thesis are (1) an extension of N-mixture models to incorporate the additional layer of obfuscation brought by observing counts from a related auxiliary population (rather than the target population), (2) an extension of N-mixture models to allow for grouped counts, the purpose being two-fold: to extend the applicability of Nmixtures to larger population sizes, and to allow for the use of coarse counts in N-mixture models, (3) a new R package allowing the easy application of the new Nmixture models, (4) a new R package allowing for optimization of multi-parameter functions using arbitrary precision arithmetic, which was a necessary tool for optimization of the likelihood in large population abundance N-mixture models, as well as (5) simulation studies validating the new grouped count models and comparing them to the classic N-mixtures models.

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

CHRIS BRUCE BSc (University of Victoria, 2014)

“C*-algebras from actions of congruence monoids”

Department of Mathematics and Statistics

Thursday, April 16, 2020 10:00 A.M.
Remote Defence

Supervisory Committee:
Dr. Marcelo Laca, Department of Mathematics and Statistics, University of Victoria (Supervisor)
Dr. Ian Putnam, Department of Mathematics and Statistics, UVic (Member)
Dr. Michel Lefebvre, Department of Physics and Astronomy, UVic (Outside Member)

External Examiner:
Dr. Judith A. Packer, Department of Mathematics, University of Colorado at Boulder

Chair of Oral Examination:
Dr. David Berg, Department of Chemistry, UVic

We initiate the study of a new class of semigroup C*-algebras arising from number theoretic considerations; namely, we generalize the construction of Cuntz, Deninger, and Laca by considering the left regular C*-algebras of ax+b-semigroups from actions of congruence monoids on rings of algebraic integers in number fields. Our motivation for considering actions of congruence monoids comes from class field theory and work on Bost–Connes type systems. We give two presentations and a groupoid model for these algebras, and establish a faithfulness criterion for their representations. We then explicitly compute the primitive ideal space, give a semigroup crossed product description of the boundary quotient, and prove that the construction is functorial in the appropriate sense. These C*-algebras carry canonical time evolutions, so that our construction also produces a new class of C*-dynamical systems. We classify the KMS (equilibrium) states for this canonical time evolution, and show that there are several phase transitions whose complexity depends on properties of a generalized ideal class group. We compute the type of all high temperature KMS states, and consider several related C*-dynamical systems.

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

JESSE JOHNSON
BSc (University of Victoria, 2010)
New and Existing Results on Circular Words
Department of Mathematics and Statistics

Wednesday, April 15, 2020 10:30 A.M
Remote Defence

Supervisory Committee: Dr. James Currie, Department of Mathematics and Statistics, University of Victoria
(Co-Supervisor) Dr. Gary MacGillivray, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner: Dr. Jacobus Swarts, Department of Mathematics, Vancouver Island University

Chair of Oral Examination: Dr. Gina Harrison, Department of Educational Psychology and Leadership Studies, UVic

Abstract
A brief history on circular words, also known as necklaces, is given, from their early conception to present results. We introduce the concept of level word, that being a word containing a nearly equal amount of each letter. We show the existence of a level circular word on three letters avoiding squares and having for all lengths greater than 18. This is accomplished through a modification of Shur’s construction of square-free circular words. A word on two letters is called a Frankel-Simpson word if the only squares it contains are 00, 11, and 0101. Using the result mentioned above and several computer searches, we show the existence of circular Frankel Simpson words for every length greater than 84.

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

Sofiia ROMANOVSKA, BSC

Evaluating citizen science Golden Eagle migration data collected by the Rocky Mountain Eagle Research Foundation, Mount Lorette, Alberta 1993-2018

April 9, 2020 10:00am

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

Chair of Oral Examination:
Dr. Michelle Miranda, Department of Mathematics and Statistics, UVic

See announcement and abstract (PDF).

See announcement (PDF).