Mathematical Sciences - Doctoral Theses
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Item Lebesgue space bounds for 2-surfaces in convolution and decoupling(University College Cork, 2023) Meade, Conor; Dendrinos, Spyridon; Carroll, Thomas; University College CorkThis thesis is a collection of proofs of theorems in the field of Harmonic Analysis. The through-line that connects these theorems is the estimation of L^p norms of operators relating to polynomial 2-surfaces, with the curvature of these surfaces playing a key role in each of the results. The operators that define these theorems are most generally parameterised by a choice of manifold in R^n. Typically, the cases where the manifold is of dimension 1 or of dimension n-1 have been well studied, but intermediate dimensions remain difficult to obtain results for. In this thesis, we aim to extend techniques used for 1-dimensional manifolds to obtain results for some 2-dimensional manifolds.Item 2D image classification and alignment in single-particle Cryo-EM(University College Cork, 2024) ZiJian, Bai; Huang, Jian; O'Sullivan, Finbarr; China Scholarship CouncilBackground and Objective: Single-particle Cryo-EM is a powerful tool for structural biologists to determine structures of macro-molecules at atomic resolution. In single-particle Cryo-EM, many individual biological molecular particles with identical structures are rapidly frozen at unknown, random orientations, and then imaged by transmission electron microscopy to generate a set of two-dimensional (2D) projection images. These generated images are treated as projections of one particle from various directions and used to reconstruct the 3D structure of the particle. Reconstruction in single-particle Cryo-EM is extremely challenging due to the lack of knowledge on the projection directions and low signal to noise ratios (SNR). To tackle this difficulty, most methods start by estimating an ab-initio model from class-averaged particle images. Then, this initial model is refined iteratively until a high resolution map is obtained, a task named ‘3D refinement’. There are two challenges in reconstructing a 3D particle structure from projection images in single-particle Cryo-EM. One is that as the particles are oriented randomly within the ice, the projection angle of the image is unknown. Another is the very low SNR, it is extremely hard to estimate projection angles. The main reason for the low SNR is the low electron doses allowed to be used for imaging. To improve the SNR in single-particle Cryo-EM, projection images are classified and averaged to generate the class-average images. This thesis focus on identifying and developing methods that can accurately and effectively compute class-averages. Methodology: Two important steps in class averaging are: (1) classifying images with similar viewing directions but with different in-plane rotations; (2) rotation alignment of images in each class. To perform rotation-invariant classification and rotation alignment, we first propose a non-uniform discrete Fourier transform (NUDFT) to calculate the Fourier transform of an image in polar coordinates. Based on the proposed NUDFT, we develop a rotation-invariant feature extraction algorithm. After using principal component analysis to reduce the dimension of the extracted rotation-invariant features and defining a distance between images using their corresponding features, K-means is employed to classify images. We also investigate combining spectral clustering with our NUDFT based rotation-invariant features for the image classification and compare its performance with the K-means. We build an algorithm for estimation of rotation between a pair of images on the base of proposed NUDFT. We also develop a novel spectral clustering method to improve the accuracy of image alignment in the same class. To demonstrate the efficiency of the algorithms, extensive simulations and experiments are performed to compare with rotation-invariant classification and rotation alignment based conventional FT and some existing algorithms in single-particle Cryo-EM. Results and conclusions: The results for the first set of simulation studies show that combining NUDFT with K-means has superior performance in rotation-invariant classification compared with K-means using features extracted by the classical FT and CL2D, displaying enhanced noise resistance, particularly in very low signal-to-noise ratio environments. The second set of simulation studies show that combining NUDFT with spectral clustering can further improve classification performance. In third set of simulation studies, the results show that NUDFT has superior performance in image rotation estimation compared with classical FT, it shows enhanced noise resistance in different SNR levels. The fourth set of simulation studies show that applying spectral clustering and Z-score method on the frequency information generated from NUDFT can further enhanced alignment performance. The overall studies show that NUDFT is effective and efficient in 2D class-averaging, and proposed algorithms can realize the potential of NUDFT and improve its performance.Item Statistical methods for mapping kinetics together with associated uncertainties in long field of view dynamic PET studies(University College Cork, 2024) Wu, Qi; O'Sullivan, Finbarr; Huang, Jian; University College Cork; Science Foundation Ireland; National Cancer InstitutePositron Emission Tomography (PET) is an essential diagnostic imaging technique in clinical care settings, as well as in medical research. It plays a crucial role in diagnosis, prognosis, treatment planning, and clinical decision-making. PET imaging, a well-established radio-tracer imaging technique, involves injecting a radio-tracer to analyze in-vivo metabolic processes. Dynamic PET scanning provides multiple time frames, offering more detailed metabolic information. However, traditional methods like Patlak and compartmental modeling are commonly used in data obtained from conventional scanners. The use of constant or exponential residue functions may be limited in complex environments, such as diverse tissues or multiple organs. This thesis aims to develop statistical approaches for enhancing and assessing parametric imaging from dynamic PET scans. The Non-Parametric Residue Mapping (NPRM) procedure is established as an entirely automatic process that integrates data-driven segmentation, non-parametric residue analysis, and voxel-level kinetic mapping. A model-based image-domain bootstrapping method is developed with the objective to generate reliable uncertainty estimates, which are crucial for accurate data interpretation and subsequent treatment decisions. This method uses an empirical distribution of re-scaled data and a non-parametric approach for analysis of the spatial correlation structure. Numerical simulations using both direct Filtered Backprojection (FBP) and iterative Maximum Likelihood (ML) reconstructions are considered. Illustrative examples on conventional scanners and Long Axial Field-of-View (LAFOV) PET scanners are conducted. A short-duration dynamic scanning protocol is proposed to enhance the quantitation of a shortened dataset specifically. This protocol utilizes NPRM and machine learning techniques to aim at making short dynamic acquisition protocols clinically feasible.Item Rate-induced tipping to metastable Zombie fires(University College Cork, 2023) O'Sullivan, Eoin; Mulchrone, Kieran F.; Wieczorek, Sebastian; ATSR Ltd.Surface wildfires are generally believed to be the cause of so-called Zombie fires observed in peatlands, that disappear from the surface, smoulder underground during the winter, and ''come back to life" in the spring. Here, we propose rate-induced tipping (R-tipping) to a subsurface hot metastable state in bioactive peat soils as a main cause of Zombie fires. Our hypothesis is based on a conceptual soil-carbon model subjected to realistic changes in weather and climate patterns, including global warming scenarios and summer heatwaves. Mathematically speaking, R-tipping to the hot metastable state is a nonautonomous instability, due to crossing an elusive quasithreshold, in a multiple-timescale dynamical system. The instability is {\em reversible}, in the sense that the system eventually returns to its base state. To explain this instability, we provide a framework that combines a special compactification technique with concepts from geometric singular perturbation theory. This framework allows us to reduce a reversible R-tipping problem due to crossing a quasithreshold to a heteroclinic orbit problem in a singular limit. Thus, we identify generic cases of such R-tipping via: (i) unfolding of a codimension-two heteroclinic folded saddle-node type-I singularity for global warming, and (ii) analysis of a codimension-one saddle-to-saddle hetroclinic orbit for summer heatwaves, which in turn reveal new types of excitability quasithresholds.Item Nonlinear dynamics & stochastic processes in cybersecurity applications(University College Cork, 2023) Ryan, Pierce; Amann, Andreas; Healy, Sorcha; Irish Research Council for Science, Engineering and Technology; McAfeeThe Internet is an extremely complex system which has a significant impact on the world we live in. In this thesis, we formalise Internet-based problems as mathematical models to better understand their dynamics. Modelling these problems requires dynamical features such as time delay, periodic forcing, switching and stochasticity. We study several dynamical systems which employ a combination of these features from Internet applications, including targeted ransomware, data networks, and signal processing. We also study a climate science system which shares features with the signal processing system and exhibits similar dynamics. Stochasticity is found to be critical in the modelling of the negotiations involved in targeted ransomware, while time delay is a crucial feature in the modelling of data networks. The signal processing and climate science systems give rise to extremely rich dynamics, which we are able to study analytically due to the presence of switching. This yields further insights into related smooth systems.