Mathematics & Statistics

Permanent URI for this collectionhttps://hdl.handle.net/10315/27569

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Access status: Open Access ,
Elliptic Curves Over Function Fields: A Numerical Investigation of Lower Bounds for Ulmer Curves
(2025-11-11) Stevens, Peter Ryan; Ingram, Patrick
This thesis investigates the ranks of Ulmer curves over the function fields F_p(t), p a prime, with a focus on computational techniques to estimate their group structure. Using SageMath, we implement point-generation algorithms, discriminant checks, and height-pairing computations to produce numerical evidence supporting predicted ranks. We combine brute-force and probabilistic sampling methods, enabling point generation and verification across a range of parameters. These results illustrate the computational challenges in large rank detection, suggest refinements, and contribute to the broader study of function fields.
Access status: Open Access ,
Combinatorics and Modelling of Highly Branched Polymers
(2025-11-11) Palin, Jason; Madras, Neal M.
Highly branched polymers such as dendrimers and hyperbranched polymers have a found a variety of chemical applications owing to their unique structural and functional properties, but in many cases are still lacking sufficient theoretical characterization. This thesis takes marginal steps toward addressing this by firstly studying a combinatorial analog of the Degree of Branching – a quantity used by chemists to classify the extent of branching in polymers -- in the mathematically convenient setting of lattice models of polymers. Next the adsorption behaviour of dendrimers is studied by adapting the Monte Carlo method of Random Sequential Adsorption to dendrimers and applying the model to experimental results of a novel synthetic dendrimer of interest (dendritic Polyglycerol Amine).
Access status: Open Access ,
Forecasting the Next Winning Stock: A Comparative Analysis of Machine Learning Models
(2025-11-11) Fernandez Mendez, Blanca Elvira; Diaz-Rodriguez, Jairo
Stock price prediction is a common and complex problem due to the high volatility of financial markets. This master’s thesis presents a new approach to stock price forecasting by reformulating the problem as a multiclass classification task. The main objective is to predict which stock will yield the highest return the next day within a given set of features. To this end, various statistical and machine learning models are analyzed, with special emphasis on the Transformer model due to its relevance and alignment with the structure of this work. The present study proposes a novel idea to address the problem. Its contributions stand out in an initial exploratory analysis of model performance, as well as in risk minimization in investments, enabling portfolio diversification thanks to the Transformer model.
Access status: Open Access ,
Modeling and Analysis of Transmission Dynamics of Respiratory Infectious Diseases: Co-Circulations, Mutations and Delayed Interventions
(2025-11-11) Majeed, Bushra; Wu, Jianhong
Mathematical models are essential tools for understanding the transmission dynamics of infectious diseases and evaluating control strategies. This thesis develops compartmental mathematical models to investigate key issues observed during the COVID-19 pandemic, including the emergence of variants of concern (VOC) due to mutations, the co-circulation of respiratory pathogens, and the impact of delayed interventions. The first model assesses the effects of mutations, focusing on the emergence of new variants and variant-specific control strategies. The model analysis emphasizes the importance of rapid detection through whole genome sequencing (WGS) to manage outbreaks from two strains effectively. The second model considers concurrent epidemics of COVID-19 and influenza. The model simulates the transmission dynamics of both viruses and optimizes vaccination strategies to minimize strain on healthcare systems by delaying or separating peak infections. Finally, time-dependent removal rates are incorporated into classical SIR models to account for delays in diagnosis and isolation due to limitations of healthcare resources, and our study shows how this delay leads to oscillatory dynamics. This thesis research forms appropriate models, develops theoretical analyses, and provides valuable insights into the complex dynamics of respiratory diseases and offers strategies for managing mutations, co-circulation, and delayed interventions, ultimately improving pandemic preparedness.
Access status: Open Access ,
Integrating Cognitive Factors in Network Models of Epidemiology with Applications to Disease Control
(2025-11-11) Shi, Congjie; Moghadas, Seyed M.
Understanding the interplay between information dissemination, behavioural responses, and disease dynamics remains a critical challenge in network-based epidemiological modelling. While network models offer a powerful framework for capturing individual-level interactions across both physical and virtual spaces, important knowledge gaps persist—particularly in how misinformation and behavioural adaptation jointly shape epidemic outcomes. This dissertation addresses these gaps by developing a novel three-layer network model that integrates information diffusion, cognitive processing, and epidemic transmission. In the first part, we show that protective behaviours driven by information-based decision-making are significantly more effective at suppressing disease spread than imitation--based strategies. We also find that educating and warning individuals to counter misinformation is more effective than network-based sanctions, such as suspending gossip spreaders. The second part explores the structural complexity of the information network, focusing on higher-order interactions represented through hyper-edge topologies. We demonstrate that scale-free information structures sustain prolonged and periodic waves of misinformation, in contrast to the more transient dynamics observed in small-world networks. In the third part, we extend our analysis to vaccination behaviour. Our results highlight the importance of timely misinformation correction in enhancing vaccine uptake and reducing disease burden. We also show that preemptive vaccination strategies significantly improve coverage and mitigate attack rates, even in environments saturated with disinformation. Notably, targeted vaccination approaches, which prioritise highly connected individuals (hubs), consistently outperform random strategies in reducing infections and severe disease outcomes. Together, this dissertation offers a comprehensive framework for examining how complex information-behaviour-epidemic feedbacks shape public health outcomes, and provides actionable insights for designing robust interventions against misinformation and infectious disease spread.
Access status: Open Access ,
Mathematical Modelling Of Electric Double Layers In Electrolytes For Lithium-Ion Batteries
(2025-07-23) Keane, Laura Marie; Moyles, Iain
In this thesis we explore electric double layers (EDLs) in electrolytes for lithium-ion batteries using mathematical modelling tools. We review three standard continuum modelling approaches applied to model electrolytes: dilute theory, moderately concentrated theory, and thermodynamically consistent theory. We implement the thermodynamically consistent formulation to model a solid electrolyte whereby we investigate the structure of the EDLs both from numerical and asymptotic perspectives. We introduce an auxiliary variable to remove singularities from the domain, allowing for standard numerical methods and robust numerical simulations. In our non-dimensionalisation of the model we uncover a length scale representing the true width of these double charge layers. This informs an asymptotic reduction of the model whereby we reveal that the EDL is composed of two distinct regions: a boundary layer and an intermediate layer. The boundary layer exhibits polynomial behaviour while the intermediate layer exhibits exponential behaviour. We refer to the boundary layer as the strong space charge layer, and the intermediate layer as the weak space charge layer. Asymptotic matching between these two layers is non-standard, therefore we introduce a pseudo matching technique to complete the asymptotic solutions. We observe excellent agreement between our numerical simulations and asymptotics. Motivated by these results we apply the thermodynamic formulation to a liquid electrolyte to investigate the differences between the two electrolytes; noting that throughout the literature it is posited that these double charge layers in solid electrolytes are wider than those of the liquid, and that the liquid exhibits exponential behaviour in these layers, without any reference to a polynomial region. Through our numerics we confirm that the layers are wider in the solid, however, via our asymptotics we determine that the structure of these layers in the liquid also displays both polynomial and exponential behaviour. We introduce a parameter into the model to reconcile this thermodynamic model with the standard Poisson-Nernst-Planck (PNP) model, which is widely associated to the observation of exponential behaviour in the double layers. We find that the PNP model becomes ill-posed under the prescribed boundary conditions and suggest ways to rectify that.
Access status: Open Access ,
Spillover Modelling and Dynamics in Multi-Host Pathogens Transmission
(2025-07-23) Tan, Yi; Zhu, Huaiping
Many pathogens of concern to both human and animal populations exhibit a generalist nature of infecting multiple host species. The behavior and transmission dynamics within reservoir hosts not only influence outbreaks within their own population but also contribute to the spillover of pathogens to new target hosts. Although existing works have incorporated spillover transmission into zoonotic models, significant gaps remain in understanding the epidemic or endemic spread of disease in target hosts due to spillover, particularly in epizootic contexts. One typical example is the monkeypox. In this research, by delineating host roles and examining transmission dynamics of monkeypox, we can effectively assess the risk of spillover events and inform mitigation and control strategies. We start with a foundational framework that models monkeypox transmission in a single host species. Two kinds of stochasticity, namely demographic and environmental stochasticity, are incorporated. We find population-size-dependent shift in the relative influence of demographic and environmental stochasticity on disease dynamics. By developing a basic reservoir-target epidemic systems, we observe that the basic reproduction number of the system fails to capture interspecific transmissibility. Our novel threshold derived from the final size relation reflects the influence of spillover processes and intraspecific transmission within target hosts, providing an appropriate measure for quantifying the spillover phenomena. Subsequently, incorporating population demographics allows us to determine the population extinction threshold and the maximum persistence threshold. We further verify that stochasticity in the spillover rate induces Phenomenological bifurcation (P-bifurcation) within the model. These analyses reveal that the spillover rate is the most critical factor influencing the epidemic and endemic prevalence in target hosts. Finally, we evaluate the effectiveness of reservoir control strategies such as quarantine and culling. Our findings indicate that the interactions between spillover events and the implementation of reservoir control strategies lead to complex dynamics due to the higher codimension bifurcations. A novel observation from our analysis and numerical simulations is the existence and collision of two limit cycles generated by distinct endemic equilibria within the system. Our study underscores the importance of controlling spillover events and managing reservoir prevalence as key interventions to mitigate spillover effects on target hosts.
Access status: Open Access ,
Bayesian Methods for Data Integration and High Dimensional Linear Model with Non-Sparsity
(2025-07-23) Zhang, Guan-Lin; Gao, Xin
We address data integration where correlated data are collected across multiple platforms, modeling responses and predictors linearly. We extend this framework by incorporating random errors from sub-Gaussian and sub-exponential distributions. The goal is to identify key predictors across platforms, even as the number of predictors and observations grows indefinitely. Our approach combines marginal response densities from multiple platforms into a composite likelihood and introduces a Bayesian model selection criterion. Under regularity conditions, this criterion consistently selects the true model, even with a diverging model size. When true models differ across platforms, our method recovers the union support of predictors—those relevant in at least one platform. We implement a Monte Carlo Markov Chain (MCMC) algorithm for model selection. Simulations show that integrating multiple platforms improves model selection accuracy. Applied to financial data, our method combines information from three indices, identifying key predictors and yielding a more accurate predictive model with lower mean squared error than single-source models. In high-dimensional regression, sparsity assumptions on regression coefficients often fail when most coefficients are nonzero, causing bias. To address this, we propose Bayesian Grouping-Gibbs Sampling (BGGS), which partitions coefficients into 𝑘 groups, enabling efficient high-dimensional sampling. We explore 𝑘-selection via simulations and recommend an "elbow plot" for optimal determination. Theoretical analysis ensures model selection consistency and bounded prediction error. Numerical experiments confirm BGGS’s advantage in estimation and prediction. Applied to financial data, it effectively identifies robust predictive models.
Access status: Open Access ,
Nonlinear Dynamics, Stochastic Methods, And Predictive Modelling For Infectious Disease: Application To Public Health And Epidemic Forecasting
(2025-04-10) Prashad, Christopher Daniel; Wu, Jianhong
Statistical models must adapt to the evolving nature of many processes over time. This thesis introduces flexible models and statistical methods designed to infer data-generating processes that vary temporally. The primary objective is to develop frameworks for efficient estimation and prediction of both univariate and multivariate time series data. The models considered are general dynamic predictive models with parameters that change over time, featuring time-varying regression coefficients or variance components. These models are capable of accommodating time-dependent covariates and can handle situations where information is incomplete. Several novel enhancements to existing mathematical models are introduced, with a particular focus on online learning and real-time prediction. Efficient Bayesian inference methodology is developed for analyzing the posterior of covariance components of dynamic models sequentially with a closed-form estimation algorithm for real-time online processing. Additionally, an online change detection algorithm for structural breaks is developed, combining the benefits of Kalman filters with sequential Monte Carlo methods. A general and extensible compartmental model for the study of infectious disease data is proposed, with several innovative extensions to established probability models for the analysis of data. Next, we extend the classical SIRS (Susceptible-Infectious-Recovered-Susceptible) model by integrating innovative stochastic mean-reverting transmission processes to more accurately capture the variability observed in real-world epidemic data. Lastly, we provide a methodology that harnesses expansive data sources and feature engineering for analyzing and forecasting peak time and height of epidemic waves, crucial for the planning of public health strategies and interventions. The performance of these inference methodologies is assessed through simulation experiments and real data from clinical, social-demographic, and epidemic domains.
Access status: Open Access ,
Transmission Dynamics And Control Of Cholera In Africa: A Mathematical Modelling Approach
(2025-04-10) Adeniyi, Ebenezer Olayinka; Kong, Jude
Background: Cholera, caused by Vibrio cholerae, is a global health threat, with outbreaks surging since 2021, particularly in Africa. In 2024, over 13 African countries faced outbreaks worsened by climatic events, poverty, and weak healthcare systems. A shortage of vaccines further complicates control efforts. Objective: This study uses data science, machine learning, and modelling to analyze cholera dynamics, identify outbreak drivers, and propose targeted interventions. Methods: A compartmental model with Bayesian estimation analyzed cholera data from eight African countries. Sensitivity analysis identified key transmission parameters, and hierarchical clustering grouped countries by outbreak characteristics. Results: Average R0 was 2.0, ranging from 1.41 (Zimbabwe) to 2.80 (Mozambique). Factors like infection rate and human shedding increased R0, while recovery rate reduced it. Clustering identified three outbreak drivers: natural disasters, conflict, and sanitation issues. Conclusion: Tailored, data-driven interventions are critical for effective cholera management across diverse contexts.
Access status: Open Access ,
Robust Statistical Modeling In Functional Linear Regression
(2025-04-10) Yan Zhang; Wu, Yuehua
Functional linear regression is a prominent field within the domain of functional data analysis, with extensive applications in various domains such as biomedical studies, brain imaging, and chemometrics. However, despite the abundance of literature on functional linear regression, limited attention has been devoted to addressing outliers or heavy-tailed distributions in the data. Consequently, robust statistical analysis remains an underdeveloped practice in this area. The primary objective of this dissertation is to enhance the utilization of robust methods for modeling functional linear regression by primarily focusing on robust estimation techniques, hypothesis testing procedures that are resilient to outliers or heavy-tailed distributions, and robust variable selection methods. First, we consider the problem of robust estimation in partial functional linear models under RKHS framework. The theoretical properties of robust estimation simulation studies are discussed in this chapter. Furthermore, two real data examples are presented to illustrate the performance of the robust procedure. Then, we extend three robust tests: Wald-type, the likelihood ratio-type and F-type in functional linear models. Meanwhile, we investigate the theoretical properties of these robust testing procedures and assess the finite sample properties through the numerical simulation. Finally, we propose a robust variable selection method in multiple functional linear regression and present a novel algorithm for identifying significant functional predictors using a robust group variable inflation factor (VIF) selection procedure. Our methodology is validated through rigorous simulation studies as well as its application to real-world data. To ensure the cohesiveness of this dissertation, Chapter 1 provides an introduction to the research background, mathematical foundations, and primary motivations underlying this study. Chapter 2 presents a comprehensive overview of basis expansion methods for functional data analysis. Lastly, Chapter 6 concludes this dissertation by offering potential avenues for future research.
Access status: Open Access ,
On PCF Polynomials
(2025-04-10) Fraser, Benjamin Alexander; Ingram, Patrick
The author of [27] proves that the set of post-critically finite (PCF) polynomials of given degree is a set of bounded height, up to PGL_2-conjugacy. This result is extended to show that the set of monic polynomials g(z) with rational coefficients of given degree such that there exists a d ≥ 2 such that g(z^d) is PCF, is also a set of bounded height. Note that by fixing the degree of a polynomial and algebraic degree of its coefficients, the set of such PCF polynomials is in fact finite, and computable. Bounds on the coefficients for quartic PCF polynomials with rational coefficients are computed, and a search of the resulting space yields 16 distinct conjugacy classes. Infinite families of PCF polynomials containing each of these distinct conjugacy classes are found, giving a lower bound on the number of such conjugacy classes in terms of degree d.
Access status: Open Access ,
The Mathematics Of Deep Neural Networks With Application In Predicting The Spread Of Infectious Diseases Through Disease Informed Neural Networks (DINNs)
(2025-04-10) Golooba, Nickson; Woldegerima, Woldegebriel Assefa
Deep learning has emerged in many fields in recent times where neural networks are used to learn and understand data. This thesis combines deep learning frameworks with epidemiological models and is aimed specifically at the creation and testing of DINNs with a view to modeling the infection dynamics of epidemics. This research thus trains the DINN on synthetic data derived from an SI-SIR model designed for Avian influenza and shows the model’s accuracy in predicting extinction and persistence conditions. In the method, a twelve hidden layer model was constructed with sixty-four neurons per layer and ReLU activation function was used. The network is trained to predict the time evolution of five state variables for birds and humans over 50,000 epochs. The overall loss minimized to 0.000006, was characterized of the loss of data and physics, which made the DINN follow the differential equations that fundamentally described the disease progression.
Access status: Open Access ,
Investigating The Plethysm Coefficients For Schur Functions
(2025-04-10) Tatsinkou Tenekeu, Roosvel; Zabrocki, Mike
Plethysm is an operation on symmetric functions, which is important in representation theory and algebraic combinatorics. However, efficient methods for finding the coefficients that emerge when plethysms of Schur functions are expanded using the Schur basis remain a challenge. The main goal of this thesis is to explore and suggest solutions to the problem of determining plethysm coefficients. This study aims to introduce methods for computing these plethysm coefficients, offering formulas where feasible and investigating their combinatorial and algebraic characteristics. This research seeks to employ theoretical and computational tools analysis to enhance our comprehension of plethysm and contribute to the wider domain of symmetric function theory.
Access status: Open Access ,
Algebra Structure On Set Partitions
(2025-04-10) Solomon, Yohana; Zabrocki, Mike
The partition algebra is an algebra with a basis of set partitions diagrams. Its subalgebra includes diagram algebras such as the uniform block permutations and the group algebra of the symmetric group. We connect the Hopf algebra of uniform block permutations to the diagram algebra known as the party algebra. This is done by describing a new basis of the partition algebra and looking at the relationship to the basis given for the Hopf algebra of uniform block permutations. The product and coproduct of the Hopf algebra of uniform block permutations are the generalization of the product of the Malvenuto-Reutenauer Hopf algebra of permutations. We connect the product of the uniform block permutations with the bases of the partition algebra. The centralizer algebra has an internal product and we define an external product on the partition algebra. This algebra contains the algebra of uniform block permutations and the algebra of permutations.
Access status: Open Access ,
Selected Computational Problems In Insurance
(2025-04-10) Fleck, Andrew; Furman, Edward
The coming together of digital data sets, computational power and rigorous probability theory has transformed finance and insurance in the last century. Once the purview of heuristics and an almost artisanal knowledge, these fields have increasingly taken on a scientific sophistication in technique. Modern regulations even require firms to retain the mathematical skill necessary to perform complex risk analysis. Mechanical heuristics originally developed in the absence of probabilistic assumptions have been demystified and reworked for novel applications. Complex structured products can be simulated and statistical learning algorithms can be applied to gain insights where none existed before. This dissertation is concerned with such problems. In the realm of property and casualty insurance, this thesis addresses challenges in risk estimation, quantification and allocation when the risk can be modelled by multivariate Stable distributions, which we will argue provide a suitable null model in the case of heavy-tailed losses. Traditionally the lack of means and distribution functions has rendered these distributions difficult to work with. We will sidestep this issue in estimation by using an integral transformation-based method of estimation. For risk quantification, we develop computationally simple and efficient representations of commonly used risk measures. Allocation then follows from our choice of dependence structure. In the area of life contingencies, we will study a relatively new product, the fixed index annuity (FIA). The variety of annuity parameters and the complexity of the underlying index make FIA comparisons very challenging. While still an insurance product, FIAs require sophisticated models of equity indices to analyze. We elect to use machine learning techniques to reproduce FIA-linked equity indices. In order to understand our often surprising results, we make use of a few stochastic volatility models.
Access status: Open Access ,
Multiple Risk Factors Dependence Structures With Applications to Actuarial Risk Management
(2024-11-07) Su, Jianxi; Furman, Edward
Actuarial and financial risk management is one of the most important innovations of the 20th century, and modelling dependent risks is one of its central issues. Traditional insurance models build on the assumption of independence of risks. Criticized as one of the main causes of the recent financial crisis, this assumption has facilitated the quantification of risks for decades, but it has often lead to under-estimation of the risks and as a result under-pricing. Hence importantly, one of the prime pillars of the novel concept of Enterprise Risk Management is the requirement that insurance companies have a clear understanding of the various interconnections that exist within risk portfolios. Modelling dependence is not an easy call. In fact, there is only one way to formulate independence, whereas the shapes of stochastic dependence are infinite. In this dissertation, we aim at developing interpretable practically and tractable technically probabilistic models of dependence that describe the adverse effects of multiple risk drivers on the risk portfolio of a generic insurer. To this end, we introduce a new class of Multiple Risk Factor (MRF) dependence structures. The MRF distributions are of importance to actuaries through their connections to the popular frailty models, as well as because of the capacity to describe dependent heavy-tailed risks. The new constructions are also linked to the factor models that lay in the very basis of the nowadays financial default measurement practice. Moreover, we use doubly stochastic Poisson processes to explore the class of copula functions that underlie the MRF models. Then, motivated by the asymmetric nature of these copulas, we propose and study a new notion of the paths of maximal dependence, which is consequently employed to measure tail dependence in copulas.
Access status: Open Access ,
Results about Proximal and Semi-proximal Spaces
(2024-07-18) Almontashery, Khulod Ali M.; Szeptycki, Paul J.
Proximal spaces were defined by J. Bell as those topological spaces $X$ with a compatible uniformity ${\mathfrak U}$ on which Player I has a winning strategy in the so-called proximal on $(X,{\mathfrak U})$. Nyikos defined the class of semi-proximal spaces where Player II has no winning strategy on $(X,{\mathfrak U})$ with respect to some compatible uniformity. The primary focus of this thesis is to study the relationship between the classes of semi-proximal spaces and normal spaces. Nyikos asked whether semi-proximal spaces are always normal. The main result of this thesis is the construction of two counterexamples to this question. We also examine the characterization of normality in subspaces of products of ordinals, relating it to the class of semi-proximal spaces in finite power of $\omega_1$. In addition, we introduce a strengthening of these classes by restricting the proximal game to totally bounded uniformities. We study connections between the proximal game, the Galvin game, and the Gruenhage game. Further, we explore the relationship between semi-proximality and other convergence properties.
Access status: Open Access ,
Polarization Operators in Superspace
(2024-07-18) Chan, Kelvin Tian Yi; Bergeron, Nantel
The classical coinvariant rings and its variants are quotient rings with rich connections to combinatorics, symmetric function theory and geometry. Studies of a generalization of the classical coinvariant rings known as the diagonal harmonics have fruitfully produced many interesting discoveries in combinatorics including the q, t-Catalan numbers and the Shuffle Theorem. The super coinvariant rings are a direct generalization of the classical coinvariant rings to one set of commuting variables and one set of anticommuting variables. N. Bergeron, Li, Machacek, Sulzgruber, and Zabrocki conjectured in 2018 that the super coinvariant rings are representation theoretic models for the Delta Conjecture at t = 0. In this dissertation, we explore the super coinvariant rings using algebraic and combina- torial methods. In particular, we study the alternating component of the super harmonics and discover a novel basis using polarization operators. We use polarization equivalence to establish a triangularity relation between the new basis and a known basis due to two groups of researchers Bergeron, Li, Machacek, Sulzgruber, and Zabrocki and Swanson and Wallach. Furthermore, we prove a folklore result on the cocharge statistics of standard Young tableaux and propose a basis for every irreducible representation appearing in the super harmonics.
Access status: Open Access ,
One-Parameter Semigroups Generated by Strongly M-Elliptic Pseudo-Differential Operators on Euclidean Spaces
(2024-07-18) Gao, Yaodong; Wong, Man Wah
We begin with a recall of the definitions and basic properties of the standard Hörmander classes of pseudo-differential operators on Rn. Then we introduce a new class of pseudo-differential operators that can be traced back to Taylor, generalized by Garello and Morando and further developed by M. W. Wong. A related class of pseudo-differential operators depending on a complex parameter on an open subset of the complex plane is constructed. We tease out from this related class the strongly M – elliptic pseudo-differential operators and prove that they are infinitesimal generators of holomorphic and hence strongly continuous one-parameter semigroups of bounded linear operators on Lp(Rn), 1

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