Elliptic curves and a theorem of Gauss

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Eduardo Soto Ballesteros[en] Just like in life, in mathematics many times we find ourselves seeking for the unknown as are the solutions of an equation. In our case instead of focusing on the solutions we would rather know how many options there are, and so how many solutions we can have in a given equation. The aim of this work is to study some of the properties of elliptic curves, as well as some additional theory related to the p-adic numbers and idèles. Moreover, we will see how a perfect combination of it all can helps to find out how many solutions there are of an elliptic curve over a finite field with some additional conditions

Connectome architecture shapes large-scale cortical alterations in schizophrenia: a worldwide ENIGMA study.

Schizophrenia is a prototypical network disorder with widespread brain-morphological alterations, yet it remains unclear whether these distributed alterations robustly reflect the underlying network layout. We tested whether large-scale structural alterations in schizophrenia relate to normative structural and functional connectome architecture, and systematically evaluated robustness and generalizability of these network-level alterations. Leveraging anatomical MRI scans from 2439 adults with schizophrenia and 2867 healthy controls from 26 ENIGMA sites and normative data from the Human Connectome Project (n = 207), we evaluated structural alterations of schizophrenia against two network susceptibility models: (i) hub vulnerability, which examines associations between regional network centrality and magnitude of disease-related alterations; (ii) epicenter mapping, which identifies regions whose typical connectivity profile most closely resembles the disease-related morphological alterations. To assess generalizability and specificity, we contextualized the influence of site, disease stages, and individual clinical factors and compared network associations of schizophrenia with that found in affective disorders. Our findings show schizophrenia-related cortical thinning is spatially associated with functional and structural hubs, suggesting that highly interconnected regions are more vulnerable to morphological alterations. Predominantly temporo-paralimbic and frontal regions emerged as epicenters with connectivity profiles linked to schizophrenias alteration patterns. Findings were robust across sites, disease stages, and related to individual symptoms. Moreover, transdiagnostic comparisons revealed overlapping epicenters in schizophrenia and bipolar, but not major depressive disorder, suggestive of a pathophysiological continuity within the schizophrenia-bipolar-spectrum. In sum, cortical alterations over the course of schizophrenia robustly follow brain network architecture, emphasizing marked hub susceptibility and temporo-frontal epicenters at both the level of the group and the individual. Subtle variations of epicenters across disease stages suggest interacting pathological processes, while associations with patient-specific symptoms support additional inter-individual variability of hub vulnerability and epicenters in schizophrenia. Our work outlines potential pathways to better understand macroscale structural alterations, and inter- individual variability in schizophrenia

DenseNet and Support Vector Machine classifications of major depressive disorder using vertex-wise cortical features

Major depressive disorder (MDD) is a complex psychiatric disorder that affects the lives of hundreds of millions of individuals around the globe. Even today, researchers debate if morphological alterations in the brain are linked to MDD, likely due to the heterogeneity of this disorder. The application of deep learning tools to neuroimaging data, capable of capturing complex non-linear patterns, has the potential to provide diagnostic and predictive biomarkers for MDD. However, previous attempts to demarcate MDD patients and healthy controls (HC) based on segmented cortical features via linear machine learning approaches have reported low accuracies. In this study, we used globally representative data from the ENIGMA-MDD working group containing an extensive sample of people with MDD (N=2,772) and HC (N=4,240), which allows a comprehensive analysis with generalizable results. Based on the hypothesis that integration of vertex-wise cortical features can improve classification performance, we evaluated the classification of a DenseNet and a Support Vector Machine (SVM), with the expectation that the former would outperform the latter. As we analyzed a multi-site sample, we additionally applied the ComBat harmonization tool to remove potential nuisance effects of site. We found that both classifiers exhibited close to chance performance (balanced accuracy DenseNet: 51%; SVM: 53%), when estimated on unseen sites. Slightly higher classification performance (balanced accuracy DenseNet: 58%; SVM: 55%) was found when the cross-validation folds contained subjects from all sites, indicating site effect. In conclusion, the integration of vertex-wise morphometric features and the use of the non-linear classifier did not lead to the differentiability between MDD and HC. Our results support the notion that MDD classification on this combination of features and classifiers is unfeasible

Neurostructural subgroup in 4291 individuals with schizophrenia identified using the subtype and stage inference algorithm

Abstract Machine learning can be used to define subtypes of psychiatric conditions based on shared biological foundations of mental disorders. Here we analyzed cross-sectional brain images from 4,222 individuals with schizophrenia and 7038 healthy subjects pooled across 41 international cohorts from the ENIGMA, non-ENIGMA cohorts and public datasets. Using the Subtype and Stage Inference (SuStaIn) algorithm, we identify two distinct neurostructural subgroups by mapping the spatial and temporal ‘trajectory’ of gray matter change in schizophrenia. Subgroup 1 was characterized by an early cortical-predominant loss with enlarged striatum, whereas subgroup 2 displayed an early subcortical-predominant loss in the hippocampus, striatum and other subcortical regions. We confirmed the reproducibility of the two neurostructural subtypes across various sample sites, including Europe, North America and East Asia. This imaging-based taxonomy holds the potential to identify individuals with shared neurobiological attributes, thereby suggesting the viability of redefining existing disorder constructs based on biological factors