http://theses.ncl.ac.uk/jspui/handle/10443/5374 2026-02-09T17:44:18Z http://theses.ncl.ac.uk/jspui/handle/10443/6640 Title: The structure of C*-algebras of product systems Authors: Dessi, Joseph Alexander Abstract: A prevalent trend in the theory of operator algebras is the study of geometric/topological structures via bounded linear operators on a Hilbert space. The goal is to establish a rigid correspondence between such a structure and a C*-algebra, and use the rich theory of the latter to study the former. This approach has been met with much success in recent years, revealing surprising links with quantum mechanics, graphs, groups, dynamics, subshifts and more. Initially these applications were studied individually; however, the introduction of C*-correspondences and product systems within the past thirty years has presented a unifying framework. Broadly speaking, C*-correspondences and their C*-algebras account for low-rank examples (e.g., directed graphs) and are by now well explored. The more general product systems and their C*-algebras account for higher-rank examples (e.g., higher-rank graphs) and less is known in this context. In turn, there is motivation to analyse the structure of C*-algebras of product systems and interpret the results with respect to the applications that these objects encompass. The current work falls within the remit of this programme, and focuses on the gauge invariant ideal structure of C*-algebras associated with the subclass of strong compactly aligned product systems. We parametrise the gauge-invariant ideals of every equivariant quotient of the Toeplitz-Nica-Pimsner algebra (most importantly the Cuntz-Nica-Pimsner algebra) via tuples of ideals of the coefficient algebra. We describe the conditions defining these families via product system operations alone. In the process, we prove a Gauge Invariant Uniqueness Theorem. We characterise the lattice operations on the parametris ing families such that the bijection is a lattice isomorphism. We then interpret the main re sult in the settings of regular product systems, C*-dynamical systems, higher-rank graphs and product systems on finite frames. We close by examining the case of proper product systems in further detail. Description: PhD Thesis 2025-01-01T00:00:00Z http://theses.ncl.ac.uk/jspui/handle/10443/6588 Title: Manifold-adapted models for time series of covariance and correlation matrices with applications to Electroencephalography data Authors: Ding, Tao Abstract: EEG (electroencephalography) is a method for recording the brain’s electrical activity in real-time by placing electrodes on the subject’s scalp or, in some circumstances, surgically within the cranium, allowing for the measurement of neural signals. Analysing dynamic brain patterns with EEG is crucial for diagnosing and treating epilepsy. Statistical analysis of EEG data commonly relies on p × p covariance (or correlation) matrices derived from pre-processed signals, where p represents the number of electrodes. However, the space of covariance (or correlation) matrices is not a vector space with the usual additive structure, and so analysis of samples or time series of covariance (or correlation) matrices must make some geometrical assumptions about the underlying space. Fortunately, both the set of p × p non-singular covariance and correlation matrices form Riemannian manifolds. Riemannian geometry provides a systematic mathematical framework that allows conventional linear statistical methods to be adapted to the non-linear geometrical setting. The number of electrodes p can be large, necessitating dimensional reduction to make analysis more computationally tractable. Moreover, electromagnetic artefacts and high correlations between sets of electrodes can result in rank deficiencies in the observed matrix-valued time series. These singularities make analysis more difficult and represent redundancies in the data. To address this, we employ dimensional reduction techniques by identifying linear combinations of channels and selecting channel subsets. These approaches ensure that the reduced time series of covariance (or correlation) matrices remain strictly positive definite, and the data thereby lie on certain smooth manifolds with the natural Riemannian structure. Our focus is on modelling time series {Si : i = 1, . . . , n} of full-rank covariance (or correlation) matrices. This data can be examined within one of three spaces: C +(p) ⊂ S +(p) ⊂ Sym(p). In this context, Sym(p) comprises symmetric matrices and is equipped with the Frobenius norm for Euclidean geometry. S +(p) represents the space of symmetric positive definite matrices, equipped with an affine invariant metric that preserves invariance under affine transformations, especially for high-magnitude covariance matrices. By factoring out variances from covariance matrices, the set of full-rank correlation matrices is represented in the quotient geometry, denoted as C +(p). Thus, we intrinsically analyse EEG matrix-valued time series data within these three spaces. Although manifold-valued data have gained substantial attention and applications in various fields recently, the literature on manifold-valued time series remains limited. This research aims to address two main objectives. First, we aim to develop manifold-adapted models for time series of matrix-valued EEG data with interpretable parameters for different possible modes of EEG dynamics. The model specifies a distribution for the tangent direction vector at any time point, combining an autoregressive term, a mean-reverting term, and a form of Gaussian noise. This model effectively captures a wide range of potential dynamics governing the evolution of EEG data, from a smooth progression along geodesics to a noisy mean-reverting random walk within the underlying manifold. Secondly, we aim to explore the extent to which modelling results are affected by the choice of the manifold and its associated geometry. Manifold-adapted models are implemented in different tangent spaces of Sym(p), S +(p), and C +(p). This enables modelling time series of covariance matrices in Sym(p) and S +(p), and time series of correlation matrices in Sym(p), S +(p), and C +(p). Note that Sym(p) can be specified as a Riemannian manifold and is convenient to present it in that way for comparison with geometries on S +(p) and C +(p). The comparison of these geometries sheds light on their relative advantages. To handle the potentially large number of parameters, we simplify the general manifoldadapted model to two simpler models with fewer parameters. These simplified coefficients reveal the relative coefficients of each dynamics mode at each time point for each pair of electrodes. Parameter inference is carried out through maximum likelihood estimation. The Mahalanobis distance serves as a metric to gauge the dissimilarity of seizures based on estimated coefficients and their asymptotic covariance matrices. The results effectively discriminate between epileptic ictal (during a seizure) and interictal (between seizures) periods in patients and quantify the dissimilarity among seizures. The affine invariant geometry and quotient geometry also provide a better fit for time series of covariance matrices and correlation matrices, respectively. In this research, we primarily construct manifold-adapted models for time series of covariance and correlation matrices derived from EEG data for epilepsy patients. We also contribute to the research on correlation matrix space by introducing a quotient metric inspired by the affine invariant metric in the covariance matrix space. The geometric concepts within the Riemannian structure of three spaces open avenues for future work related to non-Euclidean statistical models using manifold-valued data. Description: PhD Thesis 2024-01-01T00:00:00Z http://theses.ncl.ac.uk/jspui/handle/10443/6582 Title: Cosmic structure formation in the non-linear regime: beyond Gaussian statistics and standard cosmologies Authors: Gough, Alexander Abstract: The cosmic large scale structure encodes the formation and evolution of a weblike network of dark matter and galaxies within the Universe. The cosmological information is wrapped up in non-Gaussian statistics requiring characterisation beyond two-point correlations. Accurate modelling of these non-Gaussian statistics and the underlying non-linear dynamics of gravitational collapse are key to extracting maximal information from ongoing and upcoming cosmological surveys. This thesis centres on questions relating to clustering statistics, dynamics, and fundamental physics: A. How can we efficiently characterise the statistics of the late time matter field? B. How can we capture the non-linear phase-space dynamics of gravitational collapse? C. How do changes to fundamental physics impact those clustering statistics and dynamics? Specifically we present four aspects addressing these questions: 1. We demonstrate the probability distribution function (PDF) of the matter density can be accurately predicted in modified gravity and dynamical dark energy models, and that it provides good complementarity to standard two-point analyses for detecting these features. 2. We demonstrate the joint PDF of densities in two cells can be used to predict the covariance of the one-point PDF in simple clustering models, providing estimates of the density dependent clustering and super-sample covariance missed in cosmological simulations. 3. We use a wave-based forward model of dark matter to demonstrate its capability to encode the full phase-space dynamics beyond a perfect fluid and determine certain universal scaling features in such models. 4. Using the wave dark matter forward model we analyse one-point statistics to complement existing analytic and numerical approaches in studying fundamentally wavelike dark matter. Description: Phd Thesis 2024-01-01T00:00:00Z http://theses.ncl.ac.uk/jspui/handle/10443/6566 Title: Modelling hydrocarbon concentrations in groundwater monitoring networks Authors: Cowley, Josh Edward Abstract: Groundwater networks provide a critical resource across the world by supplying fresh water for a wide array of scenarios including extraction for drinking water and irrigation. Protection of these naturally occurring geological features is an important component of the wider climate problem. Pollution to groundwater networks can occur in many forms, including nitrates, radioactive material and the focus of this thesis, hydrocarbons. Due to their carcinogenic nature, data is collected at groundwater monitoring sites for regulatory compliance and to ensure safe concentration levels are not exceeded. Data collection involves extraction of a water sample, in situ, to be later analysed in a laboratory capable of measuring hydrocarbon concentrations above a certain “non-detection” limit. This process is less than desirable as our data is left-censored at laboratory-dependent thresholds and it requires the construction of several groundwater monitoring wells. Furthermore, observations may be missed due to faulty wells, unsafe working conditions and other potential obstructions. Hence, the aim of this thesis is to investigate whether statistical modelling of hydrocarbon concentrations based on measurements of predictors that are easier to obtain can provide more insight with less information. Models proposed in this thesis take the form of a regression where the dependent variable is a left-censored analyte of interest and the regressors are indicators of water quality such as temperature, pH and dissolved oxygen that could be more feasibly obtained using sensors and telemetry in the future. An application with such complexity requires an inter-disciplinary approach and this thesis presents an exploratory data analysis, machine learning methods and mechanistic transport models based on physical laws. Following these results, we propose models that avoid replacing censored data with half the detection limit; leverage the high correlation between analytes; apply mixture models to deal with non-linearity and a varying intercept model that makes use of the spatial aspect of the wells from which the data are sampled. Description: PhD Thesis 2024-01-01T00:00:00Z