Mathematics & Statistics

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

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Now showing 1 - 20 of 129

Access status: Open Access ,
Multicomponent Optimal Control of Contamination Flows in Porous Media And Applications
(2026-07-24) Hossain, Khan Enaet; Liang, Dong; Zhu, Hongmei
This thesis develops advanced PDE-constrained optimization frameworks for the control and remediation of multicomponent groundwater contamination in porous media. First, a robust optimal control model is formulated to describe nonlinear multicomponent contaminant transport governed by advection, diffusion, and chemical reactions. Environmental and economic objectives are simultaneously considered through concentration matching and abatement cost minimization. A splitting-based improved upwind finite difference scheme is developed to ensure stable and accurate numerical solutions, while a Differential Evolution (DE) algorithm is employed to determine optimal control strategies. Numerical experiments demonstrate effective contaminant reduction in both idealized and realistic aquifer systems. Next, a parallel DE-PDE optimization framework is developed to address the substantial computational demands of large-scale PDE-constrained optimization. Using the Message Passing Interface (MPI), population evaluations are distributed across multiple processors, enabling efficient parallel solution of the contaminant transport PDEs within the optimization loop. The proposed parallel strategy significantly reduces computational time while maintaining solution quality and convergence performance. Finally, a multi-control optimization framework is proposed that integrates concentration matching, pollutant reduction, and pumping-related operational costs within a unified objective function. This formulation balances environmental effectiveness with economic feasibility and supports practical remediation decision-making. Numerical studies on rectangular and L-shaped aquifers demonstrate that the proposed framework produces cost-effective remediation strategies, while the parallel DE-PDE implementation further enhances computational efficiency and scalability.
Access status: Open Access ,
Enhanced Biologically-Informed Models of Infectious Diseases: Structure and Dynamics
(2026-07-24) Ruma, Mahmuda Binte Mostofa; Heffernan, Jane Marie
Infectious diseases such as HIV, measles, and influenza continue to pose major challenges to global health, particularly in the presence of waning immunity, evolving pathogens, and heterogeneous contact patterns. Understanding the mechanisms driving infection persistence, disease resurgence, and the effectiveness of interventions requires the integration of mathematical modeling. However, an increased level of understanding that modelling can provide here depends on the robustness of the model in (1) it’s representation of the biology and (2) the analyzes and simulation studies that can be conducted to produce quality results. This thesis explores the effects of incorporating biological mechanisms into models of infectious diseases that are often ignored. In the first project we incorporate the virus loss term into a model of pathogen dynamics in-host in order to determine if such a term affects the model dynamics and bifurcation. In the second project we analyze the effects of waning immunity on the probability and severity of measles infections in populations with varying degrees of vaccine-induced immunity. We then extend this waning immunity framework to a study of seasonal influenza. Finally, we explore the effects of environmental reservoirs on the transmissibility of COVID-19, influenza, measles and norovirus in a defined spatial location – a theme park. In all projects we consider deterministic and/or stochastic modelling outcomes, and sensitivity analyses to study the model population dynamics and bifurcations and determine model parameters that most affect population and infection outcomes. By integrating theoretical analysis, numerical simulation, and sensitivity-based methods, this research provides a unified understanding of infection dynamics from within-host to population scales. The findings offer valuable information for designing effective medical and public health intervention strategies.
Access status: Open Access ,
Orthogonal Factors of Operators on the Rosenthal Spaces and the Bourgain-Rosenthal-Schechtman Spaces
(2026-07-24) Konstantos, Konstantinos; Motakis, Pavlos
We study factorization properties of bounded linear operators on the Rosenthal spaces $X_{p,w}$ and the Bourgain-Rosenthal-Schechtman spaces $R_{\alpha}^{p,0}$, $1 \leq \alpha < \omega_{1}$. Specifically, we prove that the Rosenthal spaces $X_{p,w}$ and the limit Bourgain-Rosenthal-Schechtman spaces $R_{\alpha}^{p,0}$, equipped with their natural bases, have the factorization property, and that the isomorphic spaces $R_{\omega}^{p,0}$ and $X_{p,w}$ have the primary factorization property. We establish the factorization property of the spaces $R_{\omega}^{p,0}$ and $X_{p,w}$ with their respective bases separately. For the spaces $X_{p,w}$, the proof relies on the notion of a strategically reproducible basis. In contrast, for $R_{\omega}^{p,0}$, the proof is based on an approximate orthogonal reduction to a diagonal operator: every bounded linear operator on $R_{\omega}^{p,0}$ can, via distributional embedding and up to arbitrary precision, be reduced to a diagonal operator. Moreover, we show that $R_{\omega}^{p,0}$ has the primary factorization property, which immediately implies the same property for $X_{p,w}$. The proof of the aforementioned reduction relies on an approximate orthogonal reduction to a scalar multiple of the identity operator. In addition, we establish the factorization property for the limit spaces $R_{\alpha}^{p,0}$ with their standard martingale difference sequence bases. Although this proof is derived from an approximate orthogonal reduction to a scalar finite-dimensional decomposition (FDD)-diagonal operator, the construction is significantly more involved than in the case $\alpha = \omega$. For every $1 \leq \alpha < \omega_1$, we construct an explicit unconditional FDD $(X_\lambda)_{\lambda \in \mathcal{T}_\alpha}$ of the Bourgain-Rosenthal-Schechtman space $R_\alpha^{p,0}$. This FDD has strong reproducing properties and supports a theory of distributional representations between the spaces $R_\alpha^{p,0}$, for $1 \leq \alpha < \omega_1$. We use this framework to establish the approximate orthogonal reduction to a scalar FDD-diagonal operator.
Access status: Open Access ,
Multi-Algebra Independence and Bi-Free Probability
(2026-07-24) Pepper, Daniel Eric; Skoufranis, Paul
Non-commutative probability is an abstraction of classical probability that arises from considering random variables that do not commute in multiplication. In this non-commutative setting, five different types of independence between variables arise that are unique but closely parallel the classical independence. For example, each type of independence has its own theory of probability associated including central limit theorems and stochastic processes. More recently an independence between pairs of variables known as bi-free independence has received some attention. It is a remarkable fact that each of the five main types of independence can be embedded into this bi-free independence. Shortly after the inception of bi-free independence, a generalization was introduced, spawning simultaneously several notions of independence between tuples of variables. This wave of new types of independence appears to have dramatically increased the potential avenues of research. One is left with the question of how these can be corralled, if possible. Given the embeddability of the five main types of independence into bi-free probability, one naturally extends the question of embeddability to the more recent multi-algebra independences. Can these too be studied from the context of bi-free probability? The main results of this work will be an affirmative answer for a small collection of these, most notably the free-Boolean and free-free-Boolean independences for pairs and triples of algebras respectively. Some other simple embeddings are shown, as well as some partial negative results.
Access status: Open Access ,
Multi-Point Pade Approximation and Its Applications in Probability Theory and Finance
(2026-07-24) Mohammadioroojeh, Armin; Kuznetsov, Alexey
This thesis proposes a computational framework for constructing and applying multi-point Padé approximations to study Stieltjes functions, distribution functions including generalized gamma convolutions (GGCs), and other special functions through their Laplace transforms. We establish algorithmic foundations for multi-point Padé approximation via both matrix-based and continued-fraction approaches, with particular emphasis on an efficient reduced-parameter continued-fraction scheme. Numerical implementations are performed using high-precision arithmetic to ensure the accuracy of results. The practical applicability of the method is demonstrated through a range of examples, including the approximation of Gaussian, lognormal, and gamma densities, as well as completely monotone functions, hockey-stick functions, and unit step functions by sums of exponentials. As a consequence, we introduce a new approach for approximating cumulative distribution functions (cdfs) through the Laplace transform of the underlying distribution. We further propose approximation methods for several well-known risk measures widely used in insurance and finance, based on the approximations of hockey-stick and unit step functions. The examples include expected shortfall, individual economic capital allocation, tail standard deviation and expectiles. The thesis also investigates dependence structures of GGCs through their lower and upper tail dependence coefficients, and finally, we present a new Laplace inversion relation for GGCs.
Access status: Open Access ,
High-Dimensional Covariate-Dependent Gaussian Graphical Models
(2026-03-10) Wang, Jiacheng; Gao, Xin
In this dissertation, we propose a covariate-dependent Gaussian graphical model (cdexGGM) for capturing network structure that varies with covariates through a novel parameterization. Utilizing a likelihood framework, our methodology jointly estimates all edge and vertex parameters. We further develop statistical inference procedures to test the dynamic nature of the underlying network. Concerning large-scale networks, we perform composite likelihood estimation with an $\ell_1$ penalty to discover sparse covariate-dependent graph structures. We establish the estimation error bound in $\ell_2$ norm and validate the sign consistency in the high-dimensional context. We apply our method to an influenza vaccine data set to model the gene network that evolves with time. We also investigate a Down syndrome data set to model the protein network, which varies with several covariates under a factorial experimental design. These applications demonstrate the applicability and effectiveness of the proposed model. Moreover, to further address the limitations of GGMs in capturing heterogeneous networks with known structural constraints, we introduce a covariate-dependent colored Gaussian graphical model (CD-CGGM). This model incorporates covariate effects and structured sparsity (through colorings) to model dynamic conditional dependencies. We perform model estimation using penalized composite likelihood, employing coordinate descent and Broyden’s method for optimization under different scenarios. We provide theoretical results ensuring both parameter and sign consistency of the proposed estimator. The method is applied to the same influenza vaccine dataset, where it effectively models the time-evolving gene regulatory network under symmetry constraints, thereby demonstrating its empirical performance and interpretability.
Access status: Open Access ,
Reconfiguration of Hamiltonian Cycles and Paths in Rectangular Grid Graphs
(2026-03-10) Kazazi, Albi; Madras, Neal N.
An m×n grid graph is the induced subgraph of the square lattice whose vertex set consists of all integer grid points {(i, j) : 0 ≤ i < m, 0 ≤ j < n}. Let H and K be Hamiltonian cycles in an m × n grid graph G. We study the problem of reconfiguring H into K using a sequence of local transformations called moves. A box of G is a unit square face. A box with vertices a, b, c, d is switchable in H if exactly two of its edges belong to H, and these edges are parallel. Given such a box with edges ab and cd in H, a switch move removes ab and cd, and adds bc and ad. A double-switch move consists of performing two consecutive switch moves. If, after a double-switch move, we obtain a Hamiltonian cycle, we say that the double-switch move is valid. We prove that any Hamiltonian cycle H can be transformed into any other Hamiltonian cycle K via a sequence of valid double-switch moves, such that every intermediate graph remains a Hamiltonian cycle. This result extends to Hamiltonian paths. In that case, we also use single-switch moves and a third operation, the backbite move, which enables the relocation of the path endpoints.
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.

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