Contractions of low-dimensional nilpotent Jordan algebras

In this paper we classify the laws of three-dimensional and four-dimensional nilpotent Jordan algebras over the field of complex numbers. We describe the irreducible components of their algebraic varieties and extend contractions and deformations among them. In particular, we prove that J2 and J3 are irreducible and that J4 is the union of the Zariski closures of two rigid Jordan algebras.Comment: 12 pages, 3 figure

On the structure of maximal solvable extensions and of Levi extensions of nilpotent algebras

We establish an improved upper estimate on dimension of any solvable algebra s with its nilradical isomorphic to a given nilpotent Lie algebra n. Next we consider Levi decomposable algebras with a given nilradical n and investigate restrictions on possible Levi factors originating from the structure of characteristic ideals of n. We present a new perspective on Turkowski's classification of Levi decomposable algebras up to dimension 9.Comment: 21 pages; major revision - one section added, another erased; author's version of the published pape

Mortality prediction in chronic obstructive pulmonary disease comparing the GOLD 2015 and GOLD 2019 staging: a pooled analysis of individual patient data

In 2019, The Global Initiative for Chronic Obstructive Lung Disease (GOLD) modified the grading system for patients with COPD, creating 16 subgroups (1A–4D). As part of the COPD Cohorts Collaborative International Assessment (3CIA) initiative, we aim to compare the mortality prediction of the 2015 and 2019 COPD GOLD staging systems. We studied 17 139 COPD patients from the 3CIA study, selecting those with complete data. Patients were classified by the 2015 and 2019 GOLD ABCD systems, and we compared the predictive ability for 5-year mortality of both classifications. In total, 17 139 patients with COPD were enrolled in 22 cohorts from 11 countries between 2003 and 2017; 8823 of them had complete data and were analysed. Mean±sd age was 63.9±9.8 years and 62.9% were male. GOLD 2019 classified the patients in milder degrees of COPD. For both classifications, group D had higher mortality. 5-year mortality did not differ between groups B and C in GOLD 2015; in GOLD 2019, mortality was greater for group B than C. Patients classified as group A and B had better sensitivity and positive predictive value with the GOLD 2019 classification than GOLD 2015. GOLD 2015 had better sensitivity for group C and D than GOLD 2019. The area under the curve values for 5-year mortality were only 0.67 (95% CI 0.66–0.68) for GOLD 2015 and 0.65 (95% CI 0.63–0.66) for GOLD 2019

Large-scale external validation and comparison of prognostic models: an application to chronic obstructive pulmonary disease

Background: External validations and comparisons of prognostic models or scores are a prerequisite for their use in routine clinical care but are lacking in most medical fields including chronic obstructive pulmonary disease (COPD). Our aim was to externally validate and concurrently compare prognostic scores for 3-year all-cause mortality in mostly multimorbid patients with COPD. Methods: We relied on 24 cohort studies of the COPD Cohorts Collaborative International Assessment consortium, corresponding to primary, secondary, and tertiary care in Europe, the Americas, and Japan. These studies include globally 15,762 patients with COPD (1871 deaths and 42,203 person years of follow-up). We used network meta-analysis adapted to multiple score comparison (MSC), following a frequentist two-stage approach; thus, we were able to compare all scores in a single analytical framework accounting for correlations among scores within cohorts. We assessed transitivity, heterogeneity, and inconsistency and provided a performance ranking of the prognostic scores. Results: Depending on data availability, between two and nine prognostic scores could be calculated for each cohort. The BODE score (body mass index, airflow obstruction, dyspnea, and exercise capacity) had a median area under the curve (AUC) of 0.679 [1st quartile-3rd quartile = 0.655-0.733] across cohorts. The ADO score (age, dyspnea, and airflow obstruction) showed the best performance for predicting mortality (difference AUC(ADO) - AUC(BODE) = 0.015 [95% confidence interval (CI) = - 0.002 to 0.032]; p = 0.08) followed by the updated BODE (AUCBODE updated - AUCBODE = 0.008 [95% CI = -0.005 to +0.022]; p = 0.23). The assumption of transitivity was not violated. Heterogeneity across direct comparisons was small, and we did not identify any local or global inconsistency. Conclusions: Our analyses showed best discriminatory performance for the ADO and updated BODE scores in patients with COPD. A limitation to be addressed in future studies is the extension of MSC network meta-analysis to measures of calibration. MSC network meta-analysis can be applied to prognostic scores in any medical field to identify the best scores, possibly paving the way for stratified medicine, public health, and research

On the cohomology of frobeniusian model Lie algebras

We compute the first and second cohomology groups with coefficients in the adjoint module of frobeniusian model algebras whose parameters move in a dense open subset of C p−1, and obtain upper bounds for the dimension of cohomology groups of frobeniusian Lie algebras. Moreover, it is shown that for a dense open subset of C p−1 the deformations of model algebras also belong to the family. Therefore any frobeniusian non-model algebra contracts on some element of the model whose parameters move on a finite union of hyperplanes. Further applications as the nullity of Rim’s quadratic map sq1 are obtained

Algèbres de Lie résolubles réelles algébriquement rigides

We present all real solvable algebraically rigid Lie algebras of dimension lower or equal than eight. We point out the differences that distinguish the real and complex classification of solvable rigid Lie algebra