Higher Poincare Lemma and Integrability

We prove the non-abelian Poincare lemma in higher gauge theory in two different ways. The first method uses a result by Jacobowitz which states solvability conditions for differential equations of a certain type. The second method extends a proof by Voronov and yields the explicit gauge parameters connecting a flat local connective structure to the trivial one. Finally, we show how higher flatness appears as a necessary integrability condition of a linear system which featured in recently developed twistor descriptions of higher gauge theories.Comment: 1+21 pages, presentation streamlined, section on integrability for higher linear systems significantly improved, published versio

Interval total colorings of graphs

A total coloring of a graph $G$ is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. An \emph{interval total $t$-coloring} of a graph $G$ is a total coloring of $G$ with colors $1,2,\...,t$ such that at least one vertex or edge of $G$ is colored by $i$, $i=1,2,\...,t$, and the edges incident to each vertex $v$ together with $v$ are colored by $d_{G}(v)+1$ consecutive colors, where $d_{G}(v)$ is the degree of the vertex $v$ in $G$. In this paper we investigate some properties of interval total colorings. We also determine exact values of the least and the greatest possible number of colors in such colorings for some classes of graphs.Comment: 23 pages, 1 figur

Arc Operads and Arc Algebras

Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identified with open subsets of a compactification due to Penner of a space closely related to Riemann's moduli space. Algebras over these operads are shown to be Batalin-Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy, and it similarly acts on the loop space of any topological space. New operad structures on the circle are classified and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper15.abs.htm

A quantitative comparison of in-line coating thickness distributions obtained from a pharmaceutical tablet mixing process using discrete element method and terahertz pulsed imaging

The application of terahertz pulsed imaging (TPI) in the in-line configuration to monitor the coating thickness distribution of pharmaceutical tablets has the potential to improve the performance and quality of the spray coating process. In this study, an in-line TPI method is used to measure coating thickness distributions on pre-coated tablets during mixing in a rotating pan, and compared with results obtained numerically using the discrete element method (DEM) combined with a ray-tracing technique. The hit rates (i.e. the number of successful coating thickness measurements per minute) obtained from both terahertz in-line experiments and the DEM/ray-tracing simulations are in good agreement, and both increase with the number of baffles in the mixing pan. We demonstrate that the coating thickness variability as determined from the ray-traced data and the terahertz in-line measurements represents mainly the intra-tablet variability due to relatively uniform mean coating thickness across tablets. The mean coating thickness of the ray-traced data from the numerical simulations agrees well with the mean coating thickness as determined by the off-line TPI measurements. The mean coating thickness of in-line TPI measurements is slightly higher than that of off-line measurements. This discrepancy can be corrected based on the cap-to-band surface area ratio of the tablet and the cap-to-band sampling ratio obtained from ray-tracing simulations: the corrected mean coating thickness of the in-line TPI measurements shows a better agreement with that of off-line measurements

The homotopy theory of simplicial props

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.Comment: Final version, to appear in Israel J. Mat

Geometric combinatorial algebras: cyclohedron and simplex

In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the Malvenuto-Reutenauer algebra of permutations and the Loday-Ronco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra (one-sided) with basis all the faces of the simplices.Comment: 23 figures, new expanded section about Hopf algebra of simplices, with journal correction