Cauchy's infinitesimals, his sum theorem, and foundational paradigms

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy's proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy's proof closely and show that it finds closer proxies in a different modern framework. Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation; uniform convergence; foundational paradigms.Comment: 42 pages; to appear in Foundations of Scienc

Cognitive functions and cognitive reserve in relation to blood pressure components in a population-based cohort aged 53 to 94 years.

In 288 men and women from general population in a cross-sectional survey, all neuropsychological tests were negatively associated with age; memory and executive function were also positively related with education. The hypertensives (HT) were less efficient than the normotensives (NT) in the test of memory with interference at 10 sec (MI-10) (−33%, P = 0.03), clock drawing test (CLOX) (−28%, P < 0.01), and mini-mental state examination (MMSE) (−6%, P = 0.02). Lower MMSE, MI-10, and CLOX were predicted by higher systolic (odds ratio, OR, 0.97, P = 0.02; OR 0.98, P < 0.005; OR 0.95, P < 0.001) and higher pulse blood pressure (BP) (OR 0.97, P = 0.02; OR 0.97, P < 0.01; and 0.95, P < 0.0001). The cognitive reserve index (CRI) was 6% lower in the HT (P = 0.03) and was predicted by higher pulse BP (OR 0.82, P < 0.001). The BP vectors of lower MMSE, MI-10, and CLOX were directed towards higher values of systolic and diastolic BP, that of low CRI towards higher systolic and lower diastolic. The label of hypertension and higher values of systolic or pulse BP are associated to worse memory and executive functions. Higher diastolic BP, although insufficient to impair cognition, strengthens this association. CRI is predicted by higher systolic BP associated to lower diastolic BP

The C825T GNB3 Polymorphism, Independent of Blood Pressure, Predicts Cerebrovascular Risk at a Population Level.

BACKGROUND The role of C825T polymorphism of the candidate GNB3 gene in predicting cerebrovascular outcome has been poorly explored in longitudinal setting at a population level. METHODS In an epidemiological setting, 1,678 men and women from general population were genotyped for C825T polymorphism of GNB3 gene and follow-up for 10 years to detect nonfatal and fatal cerebrovascular events (CE). Established cerebrovascular risk factors were used to adjust the multivariate Cox analysis for confounders. RESULTS Seventy-three nonfatal and 30 fatal CE were recorded. Incidence of CE was higher in TT than in C-carriers (fatal: 2.6 vs. 1.7%, P < 0.03; nonfatal: 7.8 vs. 3.9%, P < 0.03; fatal recurrences: 1.6 vs. 0.6%, P < 0.03). In Cox analysis, the TT genotype predicted nonfatal (hazard ratio 1.99, 95% confidence interval 1.05\u20133.79, P = 0.03), fatal (2.91, 1.05\u20138.12, P = 0.04), and fatal recurrent CE (6.82, 1.50\u201331.1, P = 0.02) also after adjustment for age, gender, systolic and diastolic blood pressure, body adiposity, atherogenetic blood lipids, serum uric acid, diabetes, calories, caffeine and ethanol intake, and coronary events at baseline. Further adjustment for historical CE made the association between TT genotype and incident fatal CE nonsignificant (hazard ratio 2.72, 95% confidence interval 0.96\u20137.22, P = 0.06). CONCLUSIONS The TT genotype of GNB3 gene predicts incident CE independent of blood pressure and other established risk factors at a population level. Further studies are needed to clarify the nature and pathways of this association