Theses (Mathematics and Statistics)
Permanent URI for this collection
Browse
Recent Submissions
Item Graph-theoretic and chemical properties of anionic fullerenes(2025) Slobodin, Aaron; MacGillivray, Gary; Myrvold, W. J.A fullerene is an all-carbon molecule with a polyhedral structure where each atom is bonded to three others and each face is either a pentagon or a hexagon. Fullerenes correspond to cubic planar graphs whose faces have sizes 5 or 6. The p-anionic Clar number of a fullerene G is equal to p + h, where h is maximized over all choices of p + h independent faces (p pentagons and h hexagons) the deletion of whose vertices results in a graph that admits a perfect matching. This definition is motivated by the chemical observation that pentagonal rings can accommodate an extra electron, so that the pentagons of a fullerene p-anion compete with the hexagons to host ‘Clar sextets’ of six electrons, and pentagons preferentially acquire the excess electrons of the anion. Tight upper bounds are established for the p-anionic Clar number of fullerenes for p > 0. The upper bounds are derived via graph theoretic arguments and new results on minimal cyclic-k-edge cuts in IPR fullerenes (fullerenes that have all pentagons pairwise disjoint). These bounds are shown to be tight by infinite families of fullerenes that achieve them. A fullerene G is said to be k-anionic-resonant if the deletion of the vertices of any k independent pentagons in G results in a graph that admits a perfect matching. We prove necessary conditions for a fullerene to be 2-anionic-resonant and provide structural properties of fullerenes (should they exist) for which our necessary conditions are not sufficient. Chemical aspects of the anionic Clar model and its utility are also explored in this work. These include the central question of the general comparison between predictions of the anionic Clar model and qualitative molecular-orbital theory for relative stability of charged fullerenes. H\"{u}ckel accounts of stability, including the chemical concepts of total pi energy, resonance energy, HOMO-LUMO gap and Coulson bond order, and the unified perspective offered by CSI (the recently defined charge stabilization index) are used in this analysis.Item Quasirandom forcing in regular tournaments(2025) Simbaqueba Marin, Lina; Noel, JonathanThe study of quasirandom forcing in various discrete structures has been a well-known problem in Extremal Combinatorics since 1987. In this work, we study quasirandom forcing in the case of tournaments. We say that a tournament H forces quasirandomness if in every quasirandom sequence (T_n)_{n\in \mathbb{N}} of tournaments of increasing order, the density of H in T_n asymptotically equals its expected value. In contrast to the analogous problem in graphs, it was shown that there exists only one non-transitive tournament that forces quasirandomness. To obtain a richer family of tournaments with this property, we propose a variant of it, restricting the definition of quasirandom forcing to only nearly regular sequences of tournaments (T_n)_{n\in \mathbb{N}}. We characterize all tournaments on at most 5 vertices that force quasirandomness under this new setting, obtaining that 11 out of 16 tournaments on four or five vertices are quasirandom forcing in sequences of nearly regular tournaments.Item Deterministic and stochastic modelling of infectious diseases in the early stages(2025) Wang, Manting; Ma, Junling; Van den Driessche, PaulineDuring the early stages of an epidemic, case counts typically grow exponentially, influenced by disease transmissibility, contact patterns, and implemented control measures. Understanding this exponential growth and disentangling the effects of various interventions are critical for public health decision-making. This dissertation investigates the dynamics of the early stages of an epidemic under control measures, addressing two key topics: evaluating the effectiveness of contact tracing and estimating the exponential growth rate of cases. Contact tracing is a key public health measure to reduce disease transmission. However, due to limited public health capacity, it is mostly effective during the early stage when the case counts are low. In Chapter 2, I develop a novel modelling framework to track contacts in a randomly mixed population. This approach borrows the idea of edge dynamics from network models to track contacts included in a compartmental SIR model for an epidemic spreading. Using COVID-19 as a case study, I evaluate the effectiveness of contact tracing during the early stage when multiple control measures were implemented in Chapter 3. I conduct a simulation study to determine the necessary dataset for parameter estimation. I find that new case counts, cases identified through contact tracing (or voluntary testing), and symptomatic onset counts are necessary for parameter identification. Finally, I apply our models to the early stages of the COVID-19 pandemic in Ontario, Canada. Chapters 4 and 5 focus on reliably estimating the exponential growth rate during the early stages of an outbreak, a key measure of the speed of disease spread. To establish a suitable likelihood function for accurate growth rate estimation, I derive the probability generating function for new cases using a linear stochastic SEIR model and obtain formulas for its mean and variance in Chapter 4. Numerical simulations show that the binomial or negative binomial distribution closely approximates the distribution of new cases. To determine the most appropriate method for estimating the growth rate, I compare the performance of the negative binomial regression model and the hidden Markov model (HMM) in Chapter 5. My results show that the 95% credible intervals produced by the HMM have a higher probability of covering the true growth rate.Item Entropy bounds for glass networks(2024) Wild, Benjamin Walter; Edwards, RoderickElectronic 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.Item Invariant conic optimization with basis-dependent cones: Scaled diagonally dominant matrices and real *-algebra decomposition(2024) Neshat Taherzadeh, Khashayar; Goluskin, DavidSymmetry reduction for a semidefinite program (SDP) with symmetries makes computational solution of the SDP easier by decomposing the semidefiniteness constraint into multiple smaller semidefiniteness 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 include cases in which symmetry reduction causes the optimum to increase, decrease, or stay the same.Item AdaptVarLM: A linear regression model for covariate-dependent non-constant error variance(2024) Wang, Wanmeng; Zhang, XuekuiIn 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.Item A non-local reaction advection-diffusion model for self-interacting species(2024) Yue, Zongzhi; Lewis, Mark; Ibrahim, SlimIn biological models, advection is inherently a non-local process. In this thesis, we proposed a natural extension of the non-local advection-diffusion model in [7] to include the reaction term (birth and death process). This thesis begins with an investigation of the well-posedness and existence of travelling wave solutions for this non-local reaction-advection-diffusion (RAD) equation. We prove the local-in-time existence and positivity of solutions under H³(R) initial conditions and provide a continuation criterion of the equation. Subsequently, we explore the existence of travelling wave solutions of this non-local RAD using a combination of perturbation methods, Fredholm operator theory, and Banach's fixed point theorem. Our analysis reveals that such solutions exist when the non-local advection term is small. Finally, we simulate the travelling wave solution to verify our theoretical findings and speculate the solution with a large advection term.Item Development of a disease analytic model for estimating the hidden population using the stratified-Petersen estimator(2024) Ma, Siying; Cowen, Laura Louise ElizabethThe COVID-19 pandemic brought the need for novel disease analytic models capable of estimating the true number of infections, including those that evaded detection. Statistical methods, such as the stratified-Petersen estimator, provide effective ways in wildlife population modelling to estimate hard-to-reach population size. We developed a novel disease analytic model to estimate the levels of underreported COVID-19 cases and the true population size based on the idea of developing a Bayesian version of the stratified-Petersen estimator under a state-space formulation using individual-level capture-recapture data. We obtained the capture events from individuals’ electronic health records and treated the occurrence of positive SARS-CoV-2 diagnostic test results and 2020 COVID-19-related hospitalizations as the tagging and recapture processes. Applying this model to the data from the Northern Health Authority region in British Columbia, Canada in 2020 by using a Bayesian Markov chain Monte Carlo (MCMC) approach, we found that the estimate of the size of the COVID-19 population (Nˆ = 2, 967) is 1.58 (95% CI: (1.53, 1.63)) times greater than the observed cases (nobs = 1, 880), which is a comparable result to those reported in other studies.Item Optimization problems with variational inequality constraints(1995) Ye, Xiang YangOptimization problems with variational inequality constraints are a class of mathematical programming problems which have variational inequalities as constraints. Due to the implicit hierarchical structure of the constraint region induced by variational inequalities, these problems are usually very difficult to solve. Nevertheless, they have very important applications in the areas such as economics, operations research and engineering. This thesis is devoted to necessary optimality conditions for such problems. Using the penalty methods and nonsmooth analysis technique, two types of necessary optimality conditions are derived.Item Mega-dose vitamins and minerals for the treatment of breast cancer : a comparison study of treated nonmetastatic patients versus two control groups.(2000) Zhao, YangrDr. Hoffer has been a practising psychiatrist in Victoria since 1967. During this time he has treated more than 200 female breast cancer patients with niacin, beta-carotene, selenium, vitamin C, Q10 and zinc. We compared the outcomes of Dr. Hoffer's nonmetastatic patients who were diagnosed from 1989 to 1996 with those from two sets of controls. The first set of controls was obtained as matched controls. The second set of controls includes all the female nonmetastatic breast cancer patients diagnosed at the Victoria office of the BC Cancer Agency between 1989 and 1996 who did not take large doses of vitamins as far as can be known. Using the Cox proportional hazards model and the aligned rank test we found that breast cancer patients' life times for Dr. Hoffer's patients are shorter than those of the control patients.Item Item Application of Bayesian networks to testing theory(2001) Yamada, ShinichiWe investigate Bayesian Networks and Learning Bayesian Networks of Probabilistic Expert Systems. We apply those methods to Testing Theory. Bayesian Networks are simple and flexible graphical representation schemes and can be combined with other analytical tools. In this thesis we examine the application of Bayesian Networks to the analysis of complicated real events.Item Subdifferentials and their applications(1997) Wu, ZiliTo deal with nonsmooth phenomena in mathematics and optimization, various kinds of subdifferentials have been introduced in the literature over the last two decades. Among them are the Clarke generalized gradient, the limiting subgradient, and the proximal subgradient. In this thesis, we provide some conditions that guarantee the nonemptiness of the proximal subgradient, and the sum rule for proximal subgradients to hold. We have also found a class of functions f that can be recovered up to a constant by the proximal subgradient of J or - J. Finally we provide some sufficient conditions for the existence of error bounds in terms of the above three subdifferentials.Item Modelling and analysis of center of pressure data(2003) Webber, Adam MatthewCenter of Pressure (COP) data has been used in the study of postural control for several decades. This study has usually been limited to basic measures of the COP trajectory, such as velocity. In the last decade, attempts have been made to extract more meaningful physiological parameters through the use of mathematical modelling of the postural control system. Beginning with stabilogram-diffusion analysis and focusing on the Pinned Polymer, Inverted Pendulum and FARIMA models of quiet stance, this thesis reviews some of the attempts get physiological estimates from COP data, and applies the three models to a subject group consisting of Down syndrome and non-Down syndrome individuals. These models have not previously been applied to Down syndrome subjects. The resulting estimates from each model are examined, and the strengths and weaknesses of each are discussed, as well as possibilities for future work.Item SYMDIP : a computer program for symbolic differentiation and integration(1976) Wang, Kek-WanThe goal of this thesis is the design and implementation of a computer program capable of computing symbolic derivatives and indefinite integrals. The program must be suitable for implementation on the University of Victoria IBM 370/145 computer with the VS-1 operating system. In particular, the program is required to satisfy both of the following conditions: (a) It should be able to solve a differentiation problem or an integration problem using less than 1 minute of CPU time and less than 256K bytes of main storage when run on the university computer. (b) Its design should make it suitable f or use as a teaching tool in a typica l first year calculus class. The program, SYMDIP (symbolic differentiation and integration package), meets all of these basic design goals. Tests using a set of 30 problems taken from the 1975-76 fall and spring examination papers for Mathematics 130 show that all but one of these 30 problems can be successfully solved by the program. To illustrate that the package is capable of solving many problems which are beyond the capabilities of a first year student, a set of 18 more complicated differentiation and integration problems are also solved by SYMDIP. The user states his problem in a simple and natural way as follows: DIFFERENTIATE "function" WITH RESPECT TO "variable" or INTEGRATE "function" WITH RESPECT TO "variable". "Function" is the function to be differentiated or integrated and it is written using the FORTRAN notation. "Variable" is any single alphabetic character of the user's choice. The complete program is composed of 3 separate job steps the analyzer, the processor, and the writer. These job steps are named in accordance with their respective functions within the package. The analyzer and the writer are written in the SNOBOL4 language while the main body of the program, the processor, is written using the algebraic programming language ALTRAN. Because SYMDIP requires 354K bytes of main storage space for execution, an overlay structure is constructed to be used with the program. Storage requirements of SYMDIP are reduced to 256K bytes of main memory in this way. Design of this structure i s complicated by the relatively large number (approximately 300) of subprocedures involved. The average user is completely unaware of both the overlay structure and the separate job steps because he activates ~1e package using a simple catalogued procedure.Item Aspects of order, relative primeness and quotient ring structure for polynomials over integer rings(2001) Walshe, Bridget AnneMany aspects of polynomials over finite fields have been studied. In this thesis we prove results for polynomials over integer rings that are analogous to known results regarding polynomials over finite fields. A definition of relatively prime for two polynomials over an integer ring is given. Linear algebra and the theory of resultants are used to give two proofs for necessary and sufficient conditions for two polynomials to be relatively prime over certain integer rings We then examine the quotient ring formed by the ring of polynomials over an integer ring mod a monic polynomial f. The existence of an order for certain polynomials over the integers mod n is exhibited and a bound is given for the maximum order of polynomials over the integers mod 2k. Finally, we prove theorems that can be used to simplify the calculation of the order of particular polynomials.Item The Sitnikov problem for manev type potentials(2003) Verjinschi, BogdanThis thesis focuses on how the Sitnikov problem changes if the classical Newtonian potential is replaced by a relativist potential of the Manev type. Considering initially the movement of the two primaries only, we study how replacing the potential will affect their trajectories. We establish the condition for closed trajectories for the movement of the primaries. The Sitnikov problem for Manev-type potential is a relativistic perturbation of the classical Newtonian potential, a particular case of the Manev three-body problem with two primaries moving on precessional ellipses and a third negligible mass that oscillates on an axis passing through their centre of mass, perpendicularly to their plane of motion. We use Melnikov's theory to prove the existence of transverse homoclinic orbits for at least a discrete set of initial conditions, showing that in those cases the motion is very complicated, thus extending a result obtained by Dankowikz and Holmes in the classical Newtonian case.Item Least absolute error regression(1998) Tung, Yi-YuanRegression analysis is usually based on the method of least squares . However, the least squares estimator is highly sensitive to outliers and can be far from optimal when the errors are non-normal in distribution. As an alternative to the least squares estimator, the least absolute error estimator has attracted much attention during the past thirty years with the development of several efficient algorithms based on its connections to linear programming, least absolute error regression can be put into practical use. In this thesis, we review some basic theoretical properties of least absolute error regression, examine the asymptotic efficiency of the least absolute error estimator relative to the least squares estimator for a general family of error distribution, and demonstrate its use for practical data analysis in a variety of situations using computer routines written in S-Plus .Item The derivation of the BBGKY-hierarchy for the hard sphere system(1989) Tie, JingzhiWe study the time evolution of a system of N identical hard spheres in R3 and present a derivation of the BBGKY-hierarchy for the joint distributions of k spheres ( k = l, ... , N ). Previous derivations tacitly assumed that the unknowns had enough regularities for the lower-dimensional integrals appearing in the hierarchy to make sense. A rigorous argument due to Illner & Pulvirenti [7] shows that if the initial measure in phase space is continuous along trajectories and has a suitable decay at space infinity, then a weak version of the BBGKY-hierarchy holds. Here a rigorous proof of the uniqueness of the solution of the weak form is given for a sufficiently regular initial value. The uniqueness leads to the equivalence of the weak and mild versions of the hierarchy.Item Factorization of matrices into matrices with prescribed eigenvalues(1990) Tang, KunikyoNecessary and sufficient conditions that a singular square complex matrix can be written as a product of two square matrices with prescribed eigenvalues are given. We will also use our results together with some old results to give new proofs or shorter proofs of some known theorems such as RadJavi's theorem about characterization of products of four Hermitlan matrices and Wu's theorem about products of four positive semidefinite matrices.