Fine gradings and automorphism groups on associative algebras

First we prove that any inner automorphism in the stabilizer of a graded-simple unital associative algebra whose grading group is abelian is the conjugation by a homogeneous element. Now consider a grading by an abelian group on an associative algebra such that the algebra is graded-simple and satisfies the DCC on graded left ideals. We give necessary and sufficient conditions for the grading to be fine. Then we assume that one of these necessary conditions to be fine is satisfied, and we compute the automorphism groups of the grading; the results are expressed in terms of the automorphism groups of a graded-division algebra. Finally we compute the automorphism groups of graded-division algebras in the case in which the ground field is the field of real numbers, and the underlying algebra (disregarding the grading) is simple and of finite dimension.Comment: 15 page

Gradings on simple real Lie algebras

Español: Esta obra es una tesis doctoral en matemáticas por compendio de cuatro artículos. Aquí explicamos, utilizando un lenguaje lo más sencillo posible, los resultados alcanzados en esos artículos. El objetivo general esla clasificación de las graduaciones en las álgebras de Lie reales simples. El texto está escrito tanto en inglés como en español. Este documento se puede descargar desde la siguiente dirección web: https://arxiv.org/abs/1806.06115English: This work is a doctoral thesis in mathematics by compendium of four articles. Here we explain, using a language as simple as possible, the results achieved in those articles. The general objective is the classification of gradings on simple real Lie algebras. The text is written in both english and spanish. This document can be downloaded from the following web address: https://arxiv.org/abs/1806.06115<br /

Brain clocks capture diversity and disparities in aging and dementia

Brain clocks, which quantify discrepancies between brain age and chronological age, hold promise for understanding brain health and disease. However, the impact of diversity (including geographical, socioeconomic, sociodemographic, sex and neurodegeneration) on the brain-age gap is unknown. We analyzed datasets from 5,306 participants across 15 countries (7 Latin American and Caribbean countries (LAC) and 8 non-LAC countries). Based on higher-order interactions, we developed a brain-age gap deep learning architecture for functional magnetic resonance imaging (2,953) and electroencephalography (2,353). The datasets comprised healthy controls and individuals with mild cognitive impairment, Alzheimer disease and behavioral variant frontotemporal dementia. LAC models evidenced older brain ages (functional magnetic resonance imaging: mean directional error = 5.60, root mean square error (r.m.s.e.) = 11.91; electroencephalography: mean directional error = 5.34, r.m.s.e. = 9.82) associated with frontoposterior networks compared with non-LAC models. Structural socioeconomic inequality, pollution and health disparities were influential predictors of increased brain-age gaps, especially in LAC (R² = 0.37, F² = 0.59, r.m.s.e. = 6.9). An ascending brain-age gap from healthy controls to mild cognitive impairment to Alzheimer disease was found. In LAC, we observed larger brain-age gaps in females in control and Alzheimer disease groups compared with the respective males. The results were not explained by variations in signal quality, demographics or acquisition methods. These findings provide a quantitative framework capturing the diversity of accelerated brain aging.</p

Brain clocks capture diversity and disparities in aging and dementia across geographically diverse populations

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