Abstract. Using five basic principles we treat Gerstenhaber/Lie brackets, BV operators and Master equations appearing in mathematical and physical contexts in a unified way. The different contexts for this are given by the different types of (Feynman) graphs that underlie the particular situation.Two of the maxims we bring forth are (1) that extending to the non-connected graphs gives a commutative multiplication forming a part of the BV structure and (2) that there is a universal odd twist that unifies and explains seemingly ad hoc choices of signs, and is responsible for the BV operator being a differential.Our treatment results in uniform, general theorems. These allow us to prove new results and recover and connect many constructions that have appeared independently throughout the literature. The more general point of view also allows us to disentangle the necessary from the circumstantial.

First we argue that many BV and homotopy BV structures, including both familiar and new examples, arise from a common underlying construction. The input of this construction is a cyclic operad along with a Maurer-Cartan element in an associated Lie algebra. Using this result we introduce and study the operad of cyclically invariant operations, with instances arising in cyclic cohomology and S 1 equivariant homology. We compute the homology of the cyclically invariant operations; the result being the homology operad of M 0,n+1 , the uncompactified moduli spaces of punctured Riemann spheres, which we call the gravity operad after Getzler. Motivated by the line of inquiry of Deligne's conjecture we construct 'cyclic brace operations' inducing the gravity relations up-to-homotopy on the cochain level. Motivated by string topology, we show such a gravity-BV pair is related by a long exact sequence. Examples and implications are discussed in course.

We study the relationship between opioid use and child well‐being. We combine data on legal opioid prescriptions, opioid‐related emergency department visits, and opioid‐involved mortality with foster care entrance records and child maltreatment reports. We find that increases in opioid‐related mortality and emergency department visits are associated with increased foster care entry, particularly among young children. We find no significant relationship between legal opioid distribution quantities and home removals. Finally, we examine the relationship between opioid‐related public policies and child welfare outcomes, finding mixed relationships between various policies and removal from the home.

This paper shows that the operad encoding modular operads is Koszul. Using this result, we construct higher composition operations on (hairy) graph homology, which characterize its rational homotopy type.

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