Failure detection for transport processes on networks

Diffusion on networks is a convenient framework to describe transport systems of different nature (from biological transport systems to urban mobility). The mathematical models are based on master equations that describe the diffusion processes by means of the weighted Laplacian matrix that connects the nodes. The link weight represent the coupling strength between the nodes. In this thesis we cope with the problem of localizing a single-edge failure that occurs in the network. An edge failure is meant to be as a sudden decrease of its transport capacities. An incomplete observation of the dynamical state of the network is available. An optimal clustering procedure based on the correlation properties among the node states is proposed. The network dimensionality is then reduced introducing representative nodes for each cluster, whose dynamical state is observed. We check the efficiency of the failure localization for our clustering method in comparison with more traditional techniques, using different graph configurations.

A study of brain functional connectivity and cognitive processes in long COVID-19 patients

In this work, a prospective study conducted at the IRCCS Istituto delle Scienze Neurologiche di Bologna is presented. The aim was to investigate the brain functional connectivity of a cohort of patients (N=23) suffering from persistent olfactory dysfunction after SARS-CoV-2 infection (Post-COVID-19 syndrome), as compared to a matching group of healthy controls (N=26). In particular, starting from individual resting state functional-MRI data, different analytical approaches were adopted in order to find potential alterations in the connectivity patterns of patients’ brains. Analyses were conducted both at a whole-brain level and with a special focus on brain regions involved in the processing of olfactory stimuli (Olfactory Network). Statistical correlations between functional connectivity alterations and the results of olfactory and neuropsychological tests were investigated, to explore the associations with cognitive processes. The three approaches implemented for the analysis were the seed-based correlation analysis, the group-level Independent Component analysis and a graph-theoretical analysis of brain connectivity. Due to the relative novelty of such approaches, many implementation details and methodologies are not standardized yet and represent active research fields. Seed-based and group-ICA analyses’ results showed no statistically significant differences between groups, while relevant alterations emerged from those of the graph-based analysis. In particular, patients’ olfactory sub-graph appeared to have a less pronounced modular structure compared to the control group; locally, a hyper-connectivity of the right thalamus was observed in patients, with significant involvement of the right insula and hippocampus. Results of an exploratory correlation analysis showed a positive correlation between the graphs global modularity and the scores obtained in olfactory tests and negative correlations between the thalamus hyper-connectivity and memory tests scores.

Rational algorithms for the decomposition of Feynman integrals via intersection theory

The decomposition of Feynman integrals into a basis of independent master integrals is an essential ingredient of high-precision theoretical predictions, that often represents a major bottleneck when processes with a high number of loops and legs are involved. In this thesis we present a new algorithm for the decomposition of Feynman integrals into master integrals with the formalism of intersection theory. Intersection theory is a novel approach that allows to decompose Feynman integrals into master integrals via projections, based on a scalar product between Feynman integrals called intersection number. We propose a new purely rational algorithm for the calculation of intersection numbers of differential $n-$forms that avoids the presence of algebraic extensions. We show how expansions around non-rational poles, which are a bottleneck of existing algorithms for intersection numbers, can be avoided by performing an expansion in series around a rational polynomial irreducible over $\mathbb{Q}$, that we refer to as $p(z)-$adic expansion. The algorithm we developed has been implemented and tested on several diagrams, both at one and two loops.