Rank-perfect and" weakly rank-perfect graphs

An edge e of a perfect graph G is critical if G−e is imperfect. We would like to decide whether G−e is still “almost perfect” or already “very imperfect”. Via relaxations of the stable set polytope of a graph, we define two superclasses of perfect graphs: rank-perfect and weakly rank-perfect graphs. Membership in those two classes indicates how far an imperfect graph is away from being perfect. We study the cases, when a critical edge is removed from the line graph of a bipartite graph or from the complement of such a graph.     

On rank-perfect subclasses of near-bipartite graphs

Shepherd95 proved that the stable set polytopes of near-bipartite graphs are given by constraints associated with the complete join of antiwebs only. For antiwebs, the facet set reduces to rank constraints associated with single antiwebs by Wagler2004. We extend this result to a larger graph class, the complements of fuzzy circular interval graphs, recently introduced in ChudnovskySeymour2004. Received: November 2004 / Revised version: June 2005     

Resolvent estimates in W−1,p related to strongly coupled‐linear parabolic systems with coupled nonsmooth capacities

We investigate linear parabolic systems with coupled nonsmooth capacities and mixed boundary conditions. We prove generalized resolvent estimates in W^{?1, p} spaces. The method is an appropriate modification of a technique introduced by Agmon to obtain L^p estimates for resolvents of elliptic differential operators in the case of smooth boundary conditions. Moreover, we establish an existence and uniqueness result. Copyright © 2007 John Wiley & Sons, Ltd.     

Synchronized picosecond pulse generation with a synchronously pumped dye laser and a compressed Nd: YAG laser for non-degenerate transient grating experiments

We have set up a laser system which simultaneously provides synchronized picosecond pulses in the visible and at a wavelength of 1.06 m with a repetition rate of 76 MHz. The set-up consists of a dye laser synchronously pumped by the second harmonic of a cw mode-locked Nd:YAG laser and a fiber-grating compressor for the fundamental wavelength of the Nd;YAG laser. Crosscorrelation measurments reveal the time jitter between the two pulse trains to be less than 10 ps. As a first application we have performed non-degenerate transient grating experiments in semiconductors. The non-degenerate technique allows to use excitation energies well above the bandgap energy and to separate non-linear refractive index effects from photoinduced absorption or transmission changes.     

Syntheses of a Cp'Re=S derivative and more complex products

The reaction of Cp'ReCl(2)S(3) (Cp' = Me(4)EtC(5)) with slightly less than 2 equiv of a phosphine reagent results in the formation of [Cp'Re(Cl)(2)(mu-S)](2), 2, which has been characterized by an X-ray diffraction study. Reactions of 2 with nucleophiles did not lead to monomeric derivatives of the type Cp'ReS(Cl)(2)(Nuc). The reaction of Cp'ReCl(2)(SC(2)H(4)S) with (Me(3)Si)(2)S resulted in the formation of three new products: Cp'ReS(SC(2)H(4)S), 4; Cp'Re(S(3))(SC(2)H(4)S), 5; and a tetranuclear derivative, [(Cp'Re)(2)(mu-S)(mu,eta(2)-SC(2)H(4)S)(mu,eta(1)-SC(2)H(4)S](2)Cl(2), 6. Complexes 4 and 6 have been characterized by X-ray diffraction studies. The electrochemical properties of the mononuclear Re=S derivative, 4, are compared with those of Re=O and Re=NR analogues.     

Synthesis and structure of trigonal and tetragonal connectors for a "Tinkertoy" construction set

We describe a convergent synthesis of eight 1,3,5- and 1,2,4,5-substituted benzene derivatives with long rigid arms containing 4-pyridyl, 2,2'-bipyridyl, and 2,2'-bipyrimidyl termini, meant to be used as trigonal or tetragonal connectors for the construction of large molecular structures. The synthesis involved copper-free Pd-mediated coupling of terminal acetylenes to aryl halides. First, one of the termini of 1,3-diethynylbicyclo[1.1.1]pentane was coupled with a brominated aza heterocycle, and second, 3 equiv of the resulting extended arm were coupled with 1,3,5-triiodobenzene or 4 with 1,2,4,5-tetraiodobenzene. An improved large-scale synthesis for 1,3-diethynylbicyclo[1.1.1]pentane is described. The structures of two of the arms were determined by single-crystal X-ray analysis. Several long molecular rods with 4-pyridyl termini were obtained as byproducts, and a single-crystal X-ray structure is reported for the shortest of these.     

Reactions of dithiolate ligands in mononuclear complexes of rhenium(V)

The thermal reactions of the Re(V) dithiolate complex Cp'ReCl2(SCH2CH2S), 1 (where Cp' = EtMe4C5), and related derivatives have been studied. When 1 is heated in toluene in a sealed evacuated tube at 100 degrees C, a dehydrogenation reaction occurs to form a new rhenium complex with a dithiolene ligand, Cp'ReCl2(SCHCHS), 6, in ca. 40% yield. The structure of 6 has been confirmed by an X-ray diffraction study. Under the thermal conditions studied, 1 also undergoes an olefin extrusion reaction. Free ethene is detected in the NMR spectrum of the products, and insoluble rhenium products are also formed. When 1 is reacted with excess ethene under mild conditions, a new organic product, 1,4-dithiane, is formed. Complex 1 is also found to react with oxidants, such as O2 and S8, under mild conditions to form the dehydrogenation product 6. Kinetic studies of the thermal reaction of 1 and related derivatives have been completed, and possible mechanisms for the thermally induced dehydrogenation reaction are discussed.