# Graduate dissertations

Title: Sarvate-Beam Group Divisible Designs and Related Multigraph Decomposition Problems

Speaker: Joanna Niezen, University of Victoria

Date and time:
11 Sep 2020,
11:00am -
12:00pm

Location: Remote Defence

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Title: A Novel Method for Differential Expression Analysis of Single Cell RNA-seq Data

Speaker: Qincheng Lu, University of Victoria

Date and time:
19 Aug 2020,
2:30pm -
3:30pm

Location: Remote Defense

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The Final Oral Examination for the

Degree of Master of Science

of

Qincheng LU

LL.B. (Fudan University, 2018)

“A Novel Method for Differential Expression Analysis of Single Cell RNA-seq Data”

Department of Mathematics and Statistics

Wednesday, August 19th, 2020 2:30 P.M.

Conducted Remotely

Supervisory Committee:

Dr. Xuekui Zhang, Department of Mathematics and Statistics, UVic (Supervisor)

Dr. Min Tsao, Department of Mathematics and Statistics, UVic (Member)

Chair of Oral Examination:

Dr. Mary Lesperance, Department of Mathematics and Statistics, UVic

Title: Obstructions for Local Tournament Orientation Completions

Speaker: Kevin Hsu, University of Victoria

Date and time:
11 Aug 2020,
10:00am -
11:00am

Location: Remote Defense

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

KEVIN HSU

BSc (Honours) (University of Victoria, 2018)

“Obstructions for Local Tournament Orientation Completions”

Department of Mathematics and Statistics

Tuesday, August 11, 2020

10:00 A.M.

Conducted Remotely

Supervisory Committee:

Dr. Jing Huang, Department of Mathematics and Statistics, University of Victoria (Supervisor)

Dr. Gary MacGillivray, Department of Mathematics and Statistics, UVic (Member)

External Examiner:

Dr. Yubao Guo, Department of Mathematics, RWTH Aachen University

Chair of Oral Examination:

Dr. Michael McGuire, Department of Electrical and Computer Engineering, UVic

Abstract

The orientation completion problem for a hereditary class C of oriented graphs asks whether a given partially oriented graph can be completed to a graph belonging to C. This problem was introduced recently and is a generalization of several existing problems, including the recognition problem for certain classes of graphs and the representatation extension problem for proper interval graphs. A local tournament is an oriented graph in which the in-neighbourhood as well as the out-neighbourhood of each vertex induces a tournament. Local tournaments are a well-studied class of oriented graphs that generalize tournaments and their underlying graphs are intimately related to proper circular-arc graphs. Proper interval graphs are precisely those which can be oriented as acyclic local tournaments. The orientation completion problems for the class of local tournaments and the class of acyclic local tournaments have been shown to be polynomial time solvable. In this thesis, we characterize the partially oriented graphs that can be completed to local tournaments by finding a complete list of obstructions. These are in a sense the minimal partially oriented graphs that cannot be completed to local tournaments. We also determine the minimal partially oriented graphs that cannot be completed to acyclic local tournaments.

Title: Directional constraint qualiﬁcations and optimality conditions with application to bilevel programs

Speaker: Kuang Bai, University of Victoria

Date and time:
07 Jul 2020,
10:00am -
11:00am

Location: Remote Defense

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

KUANG BAI

MSc (Dalian University of Technology, 2016)

BSc (Lanzhou University, 2013)

“Directional constraint qualiﬁcations and optimality conditions with application to bilevel programs”

Department of Mathematics and Statistics

Tuesday, July 7, 2020 10:00 A.M. Remote Defence

Supervisory Committee: Dr. Jane Ye, Department of Mathematics and Statistics, University of Victoria (Supervisor)

Dr. David Goluskin, Department of Mathematics and Statistics, UVic (Member)

Dr. Yang Shi, Department of Mechanical Engineering, UVic (Outside Member)

External Examiner: Dr. Tim Hoheisel, Department of Mathematics and Statistics, McGill University

Chair of Oral Examination: Dr. Helen Kurki, Department of Anthropology, UVic

Abstract

The main purpose of this dissertation is to investigate directional constraint qualiﬁcations and necessary optimality conditions for nonsmooth set-constrained mathematical programs. First, we study suﬃcient conditions for metric subregularity of the set-constrained system. We introduce the directional version of the quasi-/pseudo-normality as a suﬃcient condition for metric subregularity, which is weaker than the classical quasi/pseudo-normality, respectively. Then we apply our results to complementarity and Karush-Kuhn-Tucker systems. Secondly, we study directional optimality conditions of bilevel programs. It is wellknown that the value function reformulation of bilevel programs provides equivalent singlelevel optimization problems which are nonsmooth and never satisfy the usual constraint qualiﬁcations such as the Mangasarian-Fromovitz constraint qualiﬁcation (MFCQ). We show that even the ﬁrst-order suﬃcient condition for metric subregularity (which is generally weaker than MFCQ) fails at each feasible point of bilevel programs. We introduce the directional Clarke calmness condition and show that under the directional Clarke calmness condition, the directional necessary optimality condition holds. We perform directional sensitivity analysis of the value function and propose the directional quasi-normality as a suﬃcient condition for the directional Clarke calmness.

Title: Enumerating Digitally Convex Sets in Graphs

Speaker: Mackenzie Carr, University of Victoria

Date and time:
24 Jun 2020,
10:00am -
11:00am

Location: Remote Defense

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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.

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

Speaker: Rongzheng, Xie

Date and time:
23 Apr 2020,
8:30am -
9:30am

Location: Remote Defense

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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.