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