# Graduate dissertations

Title: Pseudoku: A Sudoku Adjacency Algebra and Fractional Completion Threshold

Speaker: Kate Nimegeers, University of Victoria

Date and time:
26 Feb 2024,
10:30am -
11:30am

Location: DSB C124

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Title: Mobile Guards’ Strategies for Graph Surveillance and Protection

Speaker: Virgelot Virgile, University of Victoria

Date and time:
13 Feb 2024,
8:30am -
9:30am

Location: via Zoom

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Title: Studies on Math Education and Trigraph Homomorphisms

Speaker: Freddie Mullen, University of Victoria

Date and time:
14 Dec 2023,
3:30pm -
4:30pm

Location: via Zoom

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

FREDDIE MULLIN

BEd (University of Victoria, 2013)

BSc (University of Victoria, 2011)

BA (University of Victoria, 2011)

“Studies on Math Education and Trigraph Homomorphisms”

Department of Mathematics and Statistics

Thursday, December 14, 2023

3:30 P.M.

Virtual Defence

Supervisory Committee:

Dr. Gary MacGillivray, Department of Mathematics and Statistics, University of Victoria (Supervisor)

Dr. Jane Butterfield, Department of Mathematics and Statistics, UVic (Co-Supervisor)

Dr. Richard Brewster, Department of Mathematics and Statistics, UVic (Member)

External Examiner:

Dr. Jacobus Swarts, Mathematics Department, Vancouver Island University

Chair of Oral Examination:

Dr. Marlea Clarke, Department of Political Science, UVic

Abstract

This thesis is comprised of two parts: (i) a study of homomorphisms of weak trigraphs and
(ii) an analysis of the effectiveness of the University of Victoria (UVic) Department of
Mathematics and Statistics’ Pretest. In the first part of the thesis, we study homomorphisms
of weak trigraphs. Results analogous to those for graph homomorphisms are developed. In
particular, we determine the complexity of deciding whether there is a weak trigraph
homomorphism of a weak trigraph G to a weak trigraph H, the complexity of deciding whether
a given weak trigraph has a weak trigraph homomorphism to a proper subgraph (the
complexity of deciding whether it is not a core) and describe an efficient algorithm based on
Consistency Checking that determines whether there is a weak trigraph homomorphism from
a given cactus weak trigraph to a fixed weak trigraph H. In the second part of the thesis, we
analyse the effectiveness of the UVic Pretest. First we compare the current online UVic
Pretest with the past paper Pretest in terms of their respective effectiveness in identifying
students who are ready for Calculus I. We also analyse the current online UVic Pretest in
greater detail, to identify which precalculus skills are most likely to predict success on that
test itself. Finally, we use odd ratios to categorize each question on the online UVic Pretest
and identify questions that are particularly useful to the test.

Title: Eternal Domination Problems

Speaker: Ethan Williams, Universiity of Victoria

Date and time:
13 Dec 2023,
10:30am -
11:30am

Location: DTB A203 and Zoom

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

J. ETHAN WILLIAMS

BSc (University of Victoria, 2021)

“Eternal Domination Problems”

Department of Mathematics and Statistics

Wednesday, December 13, 2023

10:30 A.M.

David Turpin Building

Room A203

Supervisory Committee:

Dr. Gary MacGillivray, Department of Mathematics and Statistics, University of Victoria

(Co-Supervisor)
Dr. Richard Brewster, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:

Dr. William Klostermeyer, School of Computing, University of North Florida

Chair of Oral Examination:

Dr. Matt Moffitt, Department of Chemistry, UVic

Abstract

Consider placing mobile guards on the vertices of a graph. The vertices are then attacked by an assailant, requiring you to move guards to the attacked vertices. What is the minimum number of guards you need in order to be able to defend against any sequence of attacks? This question is the basis for the eternal domination problem. In this thesis we investigate this problem and introduce new parameters related to it.

These new parameters arise from changing three of the assumptions made when defining the game. Specifically we assume that any number of guards can move when defending against an attack; only one attack needs to be defended against at a time; and that any number of guards can occupy a vertex. Changing these assumptions gives rise to the maneuver, invasion, and stacking numbers respectively. We investigate these parameters throughout this thesis, especially as they relate to trees.

Additionally, we tackle the related problem of eternal Roman domination, which is based on the topic which originally gave rise to the eternal domination problem. We establish a best possible upper bound for this parameter over all graphs. Finally, we present exponential time algorithms for solving all of these problems, as well as a host of other related problems.

Title: The Heisenberg Spectral Triple and Associated Zeta Functions

Speaker: Brendan Steed, University of Victoria

Date and time:
27 Nov 2023,
10:00am -
11:00am

Location: David Turpin Building, Room A203

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Title: Innovative CVX-based Algorithms for Optimal Design Problems on Discretized Regions

Speaker: Hanan Abousaleh, University of Victoria

Date and time:
23 Nov 2023,
1:00pm -
2:00pm

Location: David Turpin Building, Room A203

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

Hanan Abousaleh

BSc (University of Victoria, 2021)

Examining Committee

Supervisory Committee

Dr. Julie Zhou, Department of Mathematics and Statistics, University of Victoria (Supervisor)

Dr. Michelle Miranda, Department of Mathematics and Statistics, UVic (Member)

External Examiner

Dr. Nilanjana Roy, Department of Economics, UVic

Chair of Oral Examination

Dr. Hong-Chuan Yang, Department of Electrical and Computer Engineering, UVic

Abstract

We focus on a class of optimization problems known as optimal design problems, where the goal is to select design points optimally with respect to some criterion of interest. For regression models, the optimality criterion is based on the statistical model itself and is often a function of the information matrix. We solve A-, D-, and EI-optimal design problems in this thesis. The CVX program in MATLAB is a modelling tool and solver for convex optimization problems. As with other numerical methods in the literature, formulating an optimal design problem in a CVX-compatible way requires a discrete design space. We develop a CVX-based algorithm to solve optimal design problems on large and irregular discrete spaces for multiple regression models. The algorithm uses innovative rules to add several design points at each iteration, and clusters nearby points together at the end of iteration. Furthermore, we provide useful guidelines for discretizing irregular regions. These are based on derived theoretical properties which relate optimal designs on continuous and discrete design spaces. Several numerical examples and their MATLAB codes are presented for A-, D-, and EI-optimal designs for both linear and generalized linear models. The optimal designs found via the CVX solver are better than those presented in the literature. In addition, our guidelines to discretizing design spaces improve the efficiency of optimal designs, especially over irregular regions. We find that our iterative procedure overcomes the bottlenecks of typical sequential and multiplicative algorithms.

Title: Investigating the Relationship between Bayes Factors and Credible Intervals

Speaker: Zoe Lyu, University of Victoria

Date and time:
22 Sep 2023,
2:00pm -
3:00pm

Location: via Zoom

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The Final Oral Examination for the Degree of
Master of Science

(Department of Mathematics and Statistics)

ZIYI LYU

University of Waterloo, 2021 (B.Math)

Investigating the Relationship between Bayes Factors and Credible Intervals

Friday, September 22nd, 2023 2:00 P.M

Online

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. Ryan Budney, Department of Mathematics and Statistics, UVic