On Differential Rota-Baxter Algebras

A Rota-Baxter operator of weight $\lambda$ is an abstraction of both the integral operator (when $\lambda=0$) and the summation operator (when $\lambda=1$). We similarly define a differential operator of weight $\lambda$ that includes both the differential operator (when $\lambda=0$) and the difference operator (when $\lambda=1$). We further consider an algebraic structure with both a differential operator of weight $\lambda$ and a Rota-Baxter operator of weight $\lambda$ that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objects in the corresponding categories. In the commutative case, the free objects are given in terms of generalized shuffles, called mixable shuffles. In the noncommutative case, the free objects are given in terms of angularly decorated rooted forests. As a byproduct, we obtain structures of a differential algebra on decorated and undecorated planar rooted forests.Comment: 21 page

Reactivity of (3-Methylpentadienyl)iron(1+) Cation: Late-stage Introduction of a (3-Methyl-2Z,4-pentadien-1-yl) Side Chain

The 3-methyl-2Z,4-pentadien-1-yl sidechain is found in various sesquiterpenes and diterpenes. A route for the late stage introduction of this functionality was developed which relies on nucleophilic attack on the (3-methylpentadienyl)iron(1+) cation, followed by oxidative decomplexation. This methodology was applied to the synthesis of the proposed structure of heteroscyphic acid A methyl ester. Realization of this synthesis led to a correction of the proposed structure

Social preferences of future physicians.

We measure the social preferences of a sample of US medical students and compare their preferences with those of the general population sampled in the American Life Panel (ALP). We also compare the medical students with a subsample of highly educated, wealthy ALP subjects as well as elite law school students and undergraduate students. We further associate the heterogeneity in social preferences within medical students to the tier ranking of their medical schools and their expected specialty choice. Our experimental design allows us to rigorously distinguish altruism from preferences regarding equality-efficiency tradeoffs and accurately measure both at the individual level rather than pooling data or assuming homogeneity across subjects. This is particularly informative, because the subjects in our sample display widely heterogeneous social preferences in terms of both their altruism and equality-efficiency tradeoffs. We find that medical students are substantially less altruistic and more efficiency focused than the average American. Furthermore, medical students attending the top-ranked medical schools are less altruistic than those attending lower-ranked schools. We further show that the social preferences of those attending top-ranked medical schools are statistically indistinguishable from the preferences of a sample of elite law school students. The key limitation of this study is that our experimental measures of social preferences have not yet been externally validated against actual physician practice behaviors. Pending this future research, we probed the predictive validity of our experimental measures of social preferences by showing that the medical students choosing higher-paying medical specialties are less altruistic than those choosing lower-paying specialties