Mathematical Sciences - Masters by Research Theses

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https://hdl.handle.net/10468/9978

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Linear regression applied to agricultural data to model dry matter intake. Examination of researcher degrees of freedom and impact of choices: a multiverse analysis
(University College Cork, 2024) O'Shea Ryan, Fintan; O'Sullivan, Kathleen (Catherine); O'Mullane, John J.; Fitzgerald, Tony; Buckley, Frank; University College Cork
Multiverse analysis offers a transparent and systematic reporting of alternative results that could be obtained from plausible decisions made during the entire data analysis pipeline from data collection to reporting of results. These decisions are also referred to as the researcher degrees of freedom (DF), the flexibility with which data are collected, analysed and reported. This thesis answers the question what impact do researcher degrees of freedom have on the development of a linear regression model. A mini scoping review of the literature on multiverse analyses of regression modelling was conducted which highlighted a set of frequently investigated researcher DFs and identified previously overlooked researcher DFs. Multiverse analysis first emerged in response to the statistical crisis in psychology, but since then it has been successfully used in the fields of neuroscience, economics and education. This thesis presents the first use of multiverse analysis in agricultural science. The work in this thesis represents a collaboration between researchers at University College Cork and Teagasc, who provided the dataset used in this multiverse analysis. This data was originally collected as part of previous study by Teagasc in which researchers developed and validated a linear regression model for predicting dry matter intake in grazing cows. The potential researcher DFs in this study were identified and used with the set of researchers DFs from the mini scoping review to design this multiverse analysis. Reasonable alternative choices were proposed for each of the researcher DFs included in this multiverse analysis. A total of 2,592 different universes of analysis were included in this multiverse analysis, each the result of a unique combination of choices made at each researcher DF. The choices made at all stages of the data analysis pipeline were found to have had an affect on the results. The results of this multiverse analysis demonstrate the importance of expert knowledge in developing a linear regression model. The range of different results observed highlight the potential dangers of using black box style methods for developing models, which prioritise predictive ability over real world interpretation.
Adaptive mesh construction for the numerical solution of stochastic differential equations with Markovian switching
(University College Cork, 2024) O'Donovan, Kate; Kelly, Conall
In this dissertation we demonstrate an approach to the numerical solution of nonlinear stochastic differential equations with Markovian switching. Such equations describe the stochastic dynamics of processes where the drift and diffusion coefficients are subject to random state changes according to a Markov chain with finite state space. We propose a variant of the Jump Adapted-Adaptive approach introduced by K, Lord, & Sun (2024) to construct nonuniform meshes for explicit numerical schemes that adjust timesteps locally to rapid changes in the numerical solution and which also incorporate the switching times of an underlying Markov chain as mesh-points. It is shown that a hybrid scheme using such a mesh that combines an efficient explicit method (to be used frequently) and a potentially inefficient backstop method (to be used occasionally) will display strong convergence in mean-square of order δif both methods satisfy a mean-square consistency condition of the same order in the absence of switching. We demonstrate the construction of an order δ= 1 method of this type and apply it to generate empirical distributions of a nonlinear SDE model of telomere length in DNA replication.
Modelling the tumour microenvironment and reversible and irreversible chemotherapy resistance
(University College Cork, 2024) Quinlan, Thomas; Wieczorek, Sebastian; Alkhayuon, Hassan; Mulchrone, Kieran F.; University College Cork
The tumour microenvironment (TME) comprises various cell types, including cancer cells, immune cells, and stromal cells, which all engage in complex, dynamical interactions. Understanding these interactions, particularly their role in promoting cancer growth and the emergence of drug resistance, is crucial for developing effective treatment strategies. This project constructs a comprehensive mathematical model of the TME that incorporates the effects of chemotherapy. We systematically derive a series of process-based equations to describe the tripartite interactions between cancer cells, immune cells and stromal cells within the TME, as well as their response to chemotherapy. Throughout the model development, we analyse sub-models to ensure we capture key qualitative behaviours. Finally, we conduct a qualitative study by simulating our model under various hypothetical TME scenarios and different chemotherapy dosing schedules. Our simulations demonstrate how variations in stromal activity and treatment regimens can lead to diverse patient outcomes, highlighting the need for personalised treatment strategies.
Invariant polynomials in harmonic analysis
(University College Cork, 2024) Murphy, Fergal; Spyridon, Dendrinos; Mustata, Andrei; University College Cork; La Trobe University
This thesis presents a methodology for analysing equiaffine invariant measures on surfaces, building on a modified version of a comparability lemma from a 2019 work of P. Gressman. The research focuses on simplifying the process of calculating the equiaffine invariant measure for 2-surfaces in R^n by avoiding the need for a complete set of generators for the algebra of invariant polynomials. Instead, we compute a sufficient set of invariant polynomials whose intersection characterises the set of unstable points under the SL_2(C) action. Our approach is demonstrated on specific surfaces in R^6, R^10, and R^15, where we successfully identify the relevant invariant polynomials and use them, along with the modified lemma, to derive the associated density functions. These density functions can then be used to define the equiaffine invariant measure. The results offer a practical framework for understanding affine invariants in geometric contexts and suggest possible extensions to higher-dimensional surfaces and different group actions. This work aims to provide a foundation for future studies in geometric invariant theory and harmonic analysis, exploring the interplay between algebraic geometry, invariant theory, and analysis.
Dynamics of adaptive recurrent neural networks
(University College Cork, 2023) Fox, David; Amann, Andreas; Keane, Andrew; University College Cork
In this thesis a simple, phenomenological model of a neural network with plasticity is presented in the form of a slow-fast adaptive dynamical recurrent neural network. The plasticity rule is chosen from the class of Hebbian learning rules, in which the synaptic connection between two neurons evolves continuously as a function of their correlation in the recent past. Initially an analysis of networks of two neurons is presented, which exhibit relaxation oscillations in which one neuron switches between an ’off’ state, where it takes a negative value, and an ’on’ state, where it takes a positive value, while the other neuron stays in one on/off state. Then, by means of an example with a nine neuron network, the system is shown to exhibit both stable frequency cluster synchronization and transient frequency cluster synchronization.

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