Maurer-Cartan Elements and Cyclic Operads

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 cyclically invariant 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 $\mathcal{M}_{0,n+1}$, 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.Comment: revised version to appear in the Journal of Noncommutative Geometr

Feynman Categories

In this paper we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs, modular operads, twisted (modular) operads, properads, hyperoperads, their colored versions, as well as algebras over operads and an abundance of other related structures, such as crossed simplicial groups, the augmented simplicial category or FI--modules. The usefulness of this approach is that it allows us to handle all the classical as well as more esoteric structures under a common framework and we can treat all the situations simultaneously. Many of the known constructions simply become Kan extensions. In this common framework, we also derive universal operations, such as those underlying Deligne's conjecture, construct Hopf algebras as well as perform resolutions, (co)bar transforms and Feynman transforms which are related to master equations. For these applications, we construct the relevant model category structures. This produces many new examples.Comment: Expanded version. New introduction, new arrangement of text, more details on several constructions. New figure

Stirling Decomposition of Graph Homology in Genus 1

We prove that commutative graph homology in genus $g=1$ with $n\geq 3$ markings has a direct sum decomposition whose summands have rank given by Stirling numbers of the first kind. These summands are computed as the homology of complexes of certain decorated trees. This paper was written with a non-expert audience in mind, and an emphasis is placed on an elementary combinatorial description of these decorated tree complexes.Comment: to appear in Contemp. Mat

Koszul Feynman Categories

A cubical Feynman category, introduced by the authors in previous work, is a category whose functors to a base category $\mathcal{C}$ behave like operads in $\mathcal{C}$. In this note we show that every cubical Feynman category is Koszul. The upshot is an explicit, minimal cofibrant resolution of any cubical Feynman category, which can be used to model $\infty$ versions of generalizations of operads for both graph based and non-graph based examples.Comment: final version with minor revision

Carbon Dioxide Gas Sensors and Method of Manufacturing and Using Same

A gas sensor includes a substrate and a pair of interdigitated metal electrodes selected from the group consisting of Pt, Pd, Au, Ir, Ag, Ru, Rh, In, and Os. The electrodes each include an upper surface. A first solid electrolyte resides between the interdigitated electrodes and partially engages the upper surfaces of the electrodes. The first solid electrolyte is selected from the group consisting of NASICON, LISICON, KSICON, and .beta.''-Alumina (beta prime-prime alumina in which when prepared as an electrolyte is complexed with a mobile ion selected from the group consisting of Na.sup.+, K.sup.+, Li.sup.+, Ag.sup.+, H.sup.+, Pb.sup.2+, Sr.sup.2+ or Ba.sup.2+). A second electrolyte partially engages the upper surfaces of the electrodes and engages the first solid electrolyte in at least one point. The second electrolyte is selected from the group of compounds consisting of Na.sup.+, K.sup.+, Li.sup.+, Ag.sup.+, H.sup.+, Pb.sup.2+, Sr.sup.2+ or Ba.sup.2+ ions or combinations thereof

CO2 Sensors Based on Nanocrystalline SnO2 Doped with CuO

Nanocrystalline tin oxide (SnO2) doped with copper oxide (CuO) has been found to be useful as an electrical-resistance sensory material for measuring the concentration of carbon dioxide in air. SnO2 is an n-type semiconductor that has been widely used as a sensing material for detecting such reducing gases as carbon monoxide, some of the nitrogen oxides, and hydrocarbons. Without doping, SnO2 usually does not respond to carbon dioxide and other stable gases. The discovery that the electrical resistance of CuO-doped SnO2 varies significantly with the concentration of CO2 creates opportunities for the development of relatively inexpensive CO2 sensors for detecting fires and monitoring atmospheric conditions. This discovery could also lead to research that could alter fundamental knowledge of SnO2 as a sensing material, perhaps leading to the development of SnO2-based sensing materials for measuring concentrations of oxidizing gases. Prototype CO2 sensors based on CuO-doped SnO2 have been fabricated by means of semiconductor-microfabrication and sol-gel nanomaterial-synthesis batch processes that are amendable to inexpensive implementation in mass production