Packing Non-Zero A-Paths In Group-Labelled Graphs

Let G=(V,E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A⊂V. An A-path is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P. (If Γ is not abelian, we sum the labels in their order along the path.) We are interested in the maximum number of vertex-disjoint A-paths each of non-zero weight. When A = V this problem is equivalent to the maximum matching problem. The general case also includes Mader's S-paths problem. We prove that for any positive integer k, either there are k vertex-disjoint A-paths each of non-zero weight, or there is a set of at most 2k −2 vertices that meets each of the non-zero A-paths. This result is obtained as a consequence of an exact min-max theorem. These results were obtained at a workshop on Structural Graph Theory at the PIMS Institute in Vancouver, Canada. This research was partially conducted during the period the first author served as a Clay Mathematics Institute Long-Term Prize Fellow.     

The change of the superstructure of semicrystalline polymers during deformation: results from small-angle scattering with synchrotron radiation

The change of the superstructure of different polyethylenes during uniaxial deformation is investigated. The method used is small-angle scattering with synchrotron radiation. For branched polyethylene (Lupolen 1840D) the whole deformation range is analyzed. Beginning with superstructure of the lamellar cluster type, the superstructure partly disappears on a time scale of a few minutes and the fibrillar structure is built up. The degree of destruction and rebuilding depends on the drawing temperature. For very high molecular weight polyethylene (GUR) a reversible change of the superstructure at higher deformation ratios and at different temperatures is observed. The superstructure of (ethylene—hexene) copolymers (TIPELIN) at high draw ratios depends on the drawing temperature and is almost independent of the side group content. Interfibrillar microcracks parallel to the draw direction are produced in samples with a low side group content for draw ratios λ ≥ 1.5.     

Least-Squares Fitting of Algebraic Spline Surfaces

We present an algorithm for fitting implicitly defined algebraic spline surfaces to given scattered data. By simultaneously approximating points and associated normal vectors, we obtain a method which is computationally simple, as the result is obtained by solving a system of linear equations. In addition, the result is geometrically invariant, as no artificial normalization is introduced. The potential applications of the algorithm include the reconstruction of free-form surfaces in reverse engineering. The paper also addresses the generation of exact error bounds, directly from the coefficients of the implicit representation.     

Synthesis and Conformational Switching of Partially and Differentially Bridged Resorcin[4]arenes Bearing Fluorescent Dye Labels. Preliminary Communication

We report the synthesis of modified Cram‐type cavitands bearing one or two fluorescent labels for single‐molecule spectroscopic studies of vase? kite conformational switching (Scheme 3). Syntheses were performed by stepwise bridging of the four couples of neighboring H‐bonded OH groups of resorcin[4]arene bowls (Schemes 2 and 3). The new substitution patterns enable the construction of a large variety of future functional architectures. ¹H‐NMR Investigations showed that the new partially and differentially bridged cavitands feature temperature‐ and pH‐triggered vase? kite conformational isomerism similar to symmetrical cavitands with four identical quinoxaline bridges (Table). It was discovered that vase? kite switching of cavitands is strongly solvent‐dependent.     

Selective oxidation of methane by the bis(mu-oxo)dicopper core stabilized on ZSM-5 and mordenite zeolites

This work reports on the capability of the O2-activated Cu-ZSM-5 and Cu-MOR zeolites to selectively convert methane into methanol at a temperature of 398 K. A strong correlation between (i) the activity and (ii) the intensity of the 22 700 cm-1 UV-vis band, assigned to the bis(mu-oxo)dicopper core, is found (i) as a function of the reaction temperature, (ii) as a function of the Cu loading of the zeolite, and (iii) in comparison to other Cu materials. These three lines of evidence firmly support the key role of the bis(mu-oxo)dicopper core in this selective, low-temperature hydroxylation of methane.