Remarks on the Cauchy functional equation and variations of it

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the acknowledgments section in the official paper exists, but it appears before the appendix and not before the references as in the arXiv version); correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of Theorem 2.1; a few small improvements in various sections; added thank

Cytokines and Neurodevelopmental Outcomes in Extremely Low Birth Weight Infants

OBJECTIVE: To determine if selected pro-inflammatory and anti-inflammatory cytokines/mediators of inflammation reported to be related to development of cerebral palsy predict neurodevelopmental outcome in extremely low birth weight infants. STUDY DESIGN: Infants with birth weights ≤ 1000 g (n=1067) had blood samples collected at birth and on days 3±1, 7±1, 14±3, and 21±3 to examine the association between cytokines and neurodevelopmental outcomes. The analyses were focused on five cytokines (IL-1β, IL-8, TNF-α, RANTES, and IL-2) reported to be most predictive of CP in term and late preterm infants. RESULTS: IL-8 was higher on days 0–4 and subsequently in infants who developed CP compared with infants who did not develop CP in both unadjusted and adjusted analyses. Other cytokines (IL-12, IL-17, TNF-β, SIL-rα, MIP-1β) were found to be altered on days 0–4 in infants who developed CP. CONCLUSIONS: CP in former preterm infants may, in part, have a late perinatal and/or early neonatal inflammatory origin