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

All Stable Characteristic Classes of Homological Vector Fields

An odd vector field $Q$ on a supermanifold $M$ is called homological, if $Q^2=0$. The operator of Lie derivative $L_Q$ makes the algebra of smooth tensor fields on $M$ into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called characteristic classes. These take values in the cohomology of the operator $L_Q$ and are represented by $Q$-invariant tensors made up of the homological vector field and a symmetric connection on $M$ by means of tensor operations.Comment: 17 pages, references and comments adde

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

Graph complexes in deformation quantization

Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an $L_\infty$-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich's proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich's star-product is described.Comment: 39 pages; 3 eps figures; uses Xy-pic. Final version. Details added, mainly concerning the tree-level approximation. Typos corrected. An abridged version will appear in Lett. Math. Phy

Fluid mixing from below in unconformity-related hydrothermal ore deposits

This research was partly funded by German Research Foundation (DFG) grant BO 1776/8 and was carried out within the framework of DGMK (German Society for Petroleum and Coal Science and Technology) project 718, funded by the companies ExxonMobil Production Deutschland GmbH, GDF SUEZ E&P Deutschland GmbH, RWE Dea AG, and Wintershall Holding GmbH. Assistance by Simone Kaulfuss, Gabi Stoschek, Sara Ladenburger, Mathias Burisch, and Bernd Steinhilber with sample preparation and crush-leach analyses is gratefully acknowledged. We thank Steve Cox and two anonymous reviewers for their critical comments.Peer reviewedPostprin

Hypercommutative operad as a homotopy quotient of BV

We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/$\Delta$ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.Comment: minor corrections added, to appear in Comm.Math.Phy

Formality theorems for Hochschild complexes and their applications

We give a popular introduction to formality theorems for Hochschild complexes and their applications. We review some of the recent results and prove that the truncated Hochschild cochain complex of a polynomial algebra is non-formal.Comment: Submitted to proceedings of Poisson 200

Structure-Based Rationale for Selectivity in the Asymmetric Allylic Alkylation of Cycloalkenyl Esters Employing the Trost ‘Standard Ligand’ (TSL): Isolation, Analysis and Alkylation of the Monomeric form of the Cationic η3-Cyclohexenyl Complex [(η3-c-C6H9)Pd(TSL)]+

The solution-phase structures of the monomeric forms of the cationic Pd-η3-allyl and Pd-η3-cyclohexenyl complexes [Pd(R,R)-1(η3-C3H5)]+ (7+) and [Pd(R,R)-1(η3-C6H9)]+ (8+) bearing the trans-cyclohexylenediamine-based Trost ‘Standard Ligand’ (R,R)-1 have been elucidated by NMR, isotopic labeling and computation. In both complexes, (R,R)-1 is found to adopt a C1-symmetric conformation, leading to a concave shape in the 13-membered chelate in which one amide group in the chiral scaffold projects its NH unit out of the concave surface in close vicinity to one allyl terminus. The adjacent amide has a reversed orientation and projects its carbonyl group out of the concave face in the vicinity of the opposite allyl terminus. Stoichiometric and catalytic asymmetric alkylations of [8+][X−] by MCHE2 (E = ester, M = ‘escort’ counterion, X = Pd allyl counterion) show the same selectivities and trends as have been reported for in situ-generated catalysts, and a new model for the enantioselectivity has been explored computationally. Three factors are found to govern the regioselectivity (pro-S vs pro-R) of attack of nucleophiles on the η3-C6H9 ring in 8+ and thus the ee of the alkylation product: (i) a pro-R torquoselective bias is induced by steric interaction of the η3-C6H9 moiety with one phenyl ring of the ligand; (ii) pro-S delivery of the nucleophile can be facilitated by hydrogen-bonding with the concave orientated amide N−H; and (iii) pro-R delivery of the nucleophile can be facilitated by escort ion (M) binding to the concave orientated amide carbonyl. The latter two opposing interactions lead to the selectivity of the alkylation being sensitive to the identities of X− and M+. The generation of 8+ from cyclohexenyl ester substrate has also been explored computationally. The concave orientated amide N−H is able to activate the leaving group of the allylic ester by hydrogen bonding to its carbonyl group. However, this interaction is only feasible for the (S)-enantiomer of substrate, leading to the prediction of a powerful kinetic resolution (kS kR), as is found experimentally. This new model involving two regiochemically distinct (NH) and (CO) locations for nucleofuge or nucleophile binding, may prove of broad utility for the interpretation of the selectivity in asymmetric allylic alkylation reactions catalyzed by Pd complexes of (R,R)-1 and related ligands.<br/