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Publication Open Access Rank distributions of graphs over the field of two elements(University of Galway, 2026-01-14)A square matrix M represents a graph Γ if its nonzero off-diagonal entries encode the adjacencies of Γ according to a fixed vertex ordering. Over the field of two el ements, we study the distribution of ranks within the affine space of all matrices representing a particular graph. The motivating question is which graphs of or der n are represented by more matrices of rank n − 1 than of rank n. This reflects the fact that the most frequently occurring rank is not n but n − 1 in the space of all n × n matrices over F2, a property which is exceptional to F2. This thesis focuses on connected graphs that have a path or cycle as a subgraph induced on all but one vertex (called the extra vertex). The path graph Pn serves as the starting point of this study. The path graph is fundamental in the related and widely studied minimum rank problem, and provides a foundation for our later analysis of the set G P of graphs containing an induced path on all but one vertex. A main result is a characterisation of all such graphs that are represented by more matrices of rank n − 1 than rank n over F2. This is achieved by first examining the vectors in the nullspace of each matrix representing Pn. An expression for the difference α(Γ ) between rank n − 1 and rank n representations of a given graph Γ ∈ G P is determined in terms of these nullspace vectors. A recurrence is then established, expressing α(Γ ) in terms of α for graphs in G P for which the extra vertex has lower degree than in Γ . We classify all Γ ∈ G P satisfying α(Γ ) < 0 by first classifying those for which the extra vertex has degree 1, then using that to simplify and classify the degree 2 case, and continuing like this until it is shown that no such graphs exist for degree ⩾ 6. We then turn to the analogous problem for the cycle graph. We show that half of all F2-matrices representing Cn have rank n − 1, approximately one-third have rank n, and approximately one-sixth have rank n − 2 . We then investigate the set of graphs containing an induced cycle on all but one vertex, denoted by G C . Our analysis reveals essential structural contrasts between the classes G P and G C : while the degree of the extra vertex is bounded in the path case, it can be arbitrarily large in the cycle case. An infinite family of graphs called alternat ing wheel graphs demonstrate this contrast, as there exists an alternating wheel graph Γ ∈ G C with an extra vertex of any even degree d ⩾ 4 satisfying α(Γ ) < 0.Publication Open Access Principles of Lipschitz continuity in neural networks(University of Galway, 2026-01-14)Deep learning has achieved remarkable success across a wide range of domains, significantly expanding the frontiers of what is achievable in artificial intelligence. Yet, despite these advances, critical challenges remain --- most notably, ensuring robustness to small input perturbations and generalization to out-of-distribution data. These critical challenges underscore the need to understand the underlying fundamental principles that govern robustness and generalization. This understanding is indispensable for establishing deep learning systems that are: reliable --- performing consistently under expected conditions on in-distribution data; resilient --- capable of recovering from unexpected conditions such as noise or adversarial attacks; and trustworthy --- behaving transparently, ethically, in alignment with intended use, and technically robust, particularly in safety-critical applications. Among the theoretical tools available, Lipschitz continuity plays a pivotal role in governing the fundamental properties of neural networks related to robustness and generalization. It quantifies the worst-case sensitivity of network's outputs to small input perturbations. While its importance is widely acknowledged, prior research has predominantly focused on empirical regularization approaches based on Lipschitz constraints, leaving the underlying principles less explored. This thesis seeks to advance a principled understanding of the principles of Lipschitz continuity in neural networks within the paradigm of machine learning, examined from two complementary perspectives: an internal perspective --- focusing on the temporal evolution of Lipschitz continuity in neural networks during training (i.e., training dynamics); and an external perspective --- investigating how Lipschitz continuity modulates the behavior of neural networks with respect to features in the input data, particularly its role in governing frequency signal propagation (i.e., modulation of frequency signal propagation). Guided by these perspectives, the thesis formulates three primary research questions: (RQ1) State of Knowledge --- what is the state of knowledge of Lipschitz continuity in neural networks? (RQ2) Training Dynamics --- how does Lipschitz continuity in neural networks evolve during the training process? (RQ3) Modulation of Frequency Signal Propagation --- how does Lipschitz continuity modulate frequency signal propagation in neural networks?Publication Embargo Development of a 3D in vitro model to investigate tumour-stromal interactions in metastatic castration resistant prostate cancer(University of Galway, 2026-01-14)Prostate cancer (PCa) is among the most commonly diagnosed malignancies in men worldwide, with metastatic castration-resistant prostate cancer (mCRPC) representing its most aggressive and therapeutically challenging stage. Despite recent advancements in androgen deprivation therapy (ADT) and androgen receptor-targeted agents, mCRPC remains incurable due to the development of resistance and limited therapeutic response. Emerging evidence suggests that the tumour microenvironment (TME), particularly the interaction between tumour cells and mesenchymal stem cells (MSCs), plays a critical role in disease progression and metastasis. Tumour-secreted cytokines can reprogram MSCs into a pro-tumorigenic phenotype that enhances cancer cell invasion and migration. To better understand these interactions, we developed a 3D in vitro model using synthetic and biologically derived hydrogels to co-culture prostate cancer cells (DU145, LNCaP, and PC3) with healthy and patient-derived MSCs. Matrigel-based hydrogels enabled robust spheroid formation, with rheological and SEM characterization confirming optimal porosity and viscoelasticity conducive to nutrient diffusion and cellular remodelling. The incorporation of MSCs, particularly those derived from patient tissue and differentiated along adipogenic and osteogenic lineages, significantly promoted cancer cell migration, invasion, and epithelial-to-mesenchymal transition (EMT). Cytokine profiling revealed elevated expression of IL-6, IL-8, and IL-1β in co-cultures, supporting an inflammatory and metastatic niche. Importantly, therapeutic assessment demonstrated that combining STAT3 inhibitors (Niclosamide, STATTIC) with nitric oxide donors (DETA/NO) elicited synergistic cytotoxicity across both 2D and 3D platforms, effectively overcoming resistance to enzalutamide and docetaxel. PC3 cells, known for their metastatic potential, exhibited the highest invasiveness in response to osteogenic MSCs, further validating the model’s relevance to bone tropism in mCRPC. Overall, this 3D co-culture model recapitulates key features of the prostate TME and offers a translational platform to study tumour-stroma interactions and evaluate novel therapeutic strategies targeting the metastatic cascade. This system holds promise for advancing personalized therapy and preclinical drug screening in advanced PCa.Publication Open Access Farming for soil health: assessing the impact of agricultural management systems of soil biodiversity and functioning(University of Galway, 2026-12-14)This thesis investigated the effects of agricultural management systems on soil health in temperate grasslands. Three major gaps in knowledge were identified: (1) What impacts, if any, do different agricultural management systems (conventional intensive, organic, and extensive) have on physical and chemical health in grassland soils? (2) Does biological soil health differ between management systems? (3) Will the intensity of management, as measured by stocking rate, within these systems create a gradient of soil health in which functional trade-offs will occur? Physical health was not affected by managements system as all sites were well drained and of similar soil types meaning they had similar physical characteristics. Chemical health was influenced by managements systems. Conventional (CON) systems had excesses of nutrients like P and K according to the Teagasc soil indexes, thus requiring other nutrient inputs (e.g. Cu, Mg), to balance chemical health. Extensive (EXT) and organic (ORG) systems had significantly lower pH, P and K than conventional systems but ORG systems were within the optimum soil index ranges while EXT systems were not. Biological health was affected by management systems with increased nitrification gene abundance in CON systems compared to ORG and EXT systems. Fungal gene abundance was increased in EXT systems at depth (15-30 cm) compared to CON systems. Alpha diversity was reduced in EXT systems for both prokaryotes (compared to CON) and fungi (compared to ORG). Beta diversity showed significant differences between systems and this was driven mainly by soil fertility (pH, P and K levels), indicating that nutrient deficiencies as well as excesses affected microbial communities. Functional trade-offs occurred across the systems, however, contrary to the hypothesis it was not a gradient of management intensity, as ORG systems had the best balance of functionality, however they did have slightly reduced soil carbon stocks compared to CON systems.Publication Open Access Wavelet-based time-frequency fingerprinting for feature extraction of traditional Irish music(University of Galway, 2026-01-12)This work presents a wavelet-based approach to time-frequency fingerprinting for time series feature extraction, with a focus on audio identification from live recordings of traditional Irish tunes. The challenges of identifying features in time-series data are addressed by employing a continuous wavelet transform to extract spectral features and wavelet coherence analysis is used to compare recorded audio spectrograms to synthetically generated tunes. The synthetic tunes are derived from ABC notation, which is a common symbolic representation for Irish music. Experimental results demonstrate that the wavelet-based method can accurately and efficiently identify recorded tunes. This research study also details the performance of the wavelet coherence model, highlighting its strengths over other methods of time-frequency decomposition. Additionally, we discuss and deploy the model on several applications beyond music, including in EEG signal analysis and financial time series forecasting.
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