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

Erratum: Factors influencing time-location patterns and their impact on estimates of exposure: the Multi-Ethnic Study of Atherosclerosis and Air Pollution (MESA Air)

We assessed time-location patterns and the role of individual- and residential-level characteristics on these patterns within the Multi-Ethnic Study of Atherosclerosis and Air Pollution (MESA Air) cohort and also investigated the impact of individual-level time-location patterns on individual-level estimates of exposure to outdoor air pollution. Reported time-location patterns varied significantly by demographic factors such as age, gender, race/ethnicity, income, education, and employment status. On average Chinese participants reported spending significantly more time indoors and less time outdoors and in transit than white, black, or Hispanic participants. Using a tiered linear regression approach, we predicted time indoors at home and total time indoors. Our model, developed using forward selection procedures, explained 43 percent of the variability in time spent indoors at home, and incorporated demographic, health, lifestyle, and built environment factors. Time-weighted air pollution predictions calculated using recommended time indoors from USEPA(1) overestimated exposures as compared to predictions made with MESA Air participant-specific information. These data fill an important gap in the literature by describing the impact of individual and residential characteristics on time-location patterns and by demonstrating the impact of population-specific data on exposure estimates

Neighborhood environment and incident diabetes, a neighborhood environment-wide association study (‘NE-WAS’): Results from the Hispanic Community Health Study/Study of Latinos (HCHS/SOL)

The prevalence of type 2 diabetes is increasing among the Hispanic/Latino population. Type 2 diabetes incidence rates vary between neighborhoods, but no single aspect of the neighborhood environment is known to cause type 2 diabetes. Using data from the Hispanic Community Health Study/Study of Latinos cohort of 16,415 Hispanic/Latino adults in four major US cities, we conducted a neighborhood environment-wide association study to identify neighborhood measures or clusters of measures associated with diabetes incidence. Two-hundred and four neighborhood measures were calculated at the census tract level or within a 1-km buffer of participants' residential addresses. Independent covariate-adjusted and survey-weighted Poisson regressions were run for each neighborhood measure and incident diabetes. Principal component analysis of neighborhood measures was conducted to reduce dimensionality. No coherent pattern of neighborhood measures or principal component scores were associated with diabetes incidence within the cohort, though established individual-level risk factors such as age and family history were strongly associated with diabetes incidence. Results from our analysis did not indicate specific neighborhood measures, clusters, or patterns. Individual, rather than neighborhood, factors distinguish incident diabetes cases from non-cases

Welche Funktionsbegriffe gab Leonhard Euler?

AbstractLeonhard Euler’s notion of function as an „analytical expression“ occasionally denoted by fx is well-known. But it has gone unnoticed that Euler used a second well-defined notion of function for which he even coined a particular denotation: f:, used as f:x. In fact, this second notion of function is the earlier one, defined as „the ordinate which depends on the abscissa“, given by the curve. Euler argues that this „geometric“ notion of function is more general than the „algebraic“ one. Consequently, Euler relies on this more general notion of function when he integrates functions of several variables