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.     

Extending an edge-coloring

When can a k-edge-coloring of a subgraph K of a graph G be extended to a k-edge-coloring of G? One necessary condition is that for all X ? E(G) - E(K), where μ_i(X) is the maximum cardinality of a subset of X whose union with the set of edges of K colored i is a matching. This condition is not sufficient in general, but is sufficient for graphs of very simple structure. We try to locate the border where sufficiency ends.     

Packing directed circuits fractionally

LetG be a digraph, and letk1, such that no fractional packing of directed circuits ofG has value >k, when every vertex is given capacity 1. We prove there is a set ofO (k logk logk) vertices meeting all directed circuits ofG.     

The use of stable carbon isotope analysis to detect the abuse of testosterone in cattle.

The use of stable carbon isotope analysis to detect the administration of anabolic steroids to cattle was investigated. Samples were extracted by solid-phase extraction on C18 cartridges. Stable isotope ratios (13C:12C) were measured by gas chromatography-isotope ratio mass spectrometry (GC-IRMS) of the underivatised extracts. A programmed temperature vaporiser (PTV) injector was installed in the GC-IRMS system, which conferred a number of advantages. First, it allowed large volumes of sample to be injected whilst the injector liner was cool. The solvent was subsequently vented to the atmosphere prior to transfer of the sample to the GC column. Thus a significantly greater amount of sample could be presented for analysis, thereby increasing the sensitivity. Second, by this means virtually all the solvent could be removed prior to analysis. This eliminates solvent peak tailing, which can be a major problem in GC-IRMS. Finally, the PTV allowed the use of higher initial GC oven temperatures, which in turn facilitated the analysis of underivatised steroids by reducing the GC run time and improving the chromatographic peak shape. The carbon isotope composition of 5 beta-androstane-3 alpha,17 alpha-diol, the major metabolite of testosterone found in bovine bile, was measured in bile samples from untreated cattle and from cattle injected intramuscularly with testosterone or a mixture of testosterone esters. There was considerable inter-animal variation in the values obtained and there was no significant difference between samples from treated and untreated animals. However, when the isotopic composition of the metabolite was normalised with respect to that of an endogenous reference compound (cholesterol) in the same sample, the difference between treated and untreated animals become statistically significant.     

Quantum efficiency of the 2537 A. Photolysis of a mixed phenyl–methyl polysiloxane

The ultraviolet (λ = 2537 A.) photolysis of a degassed mixed phenyl and methyl polysiloxane liquid is examined in terms of gas and crosslinking yields. Results are compared to the published values obtained by ionizing irradiation of this type of molecule. It is shown that ultraviolet radiation is less efficient by two orders of magnitude in producing decomposition (i.e., gaseous products) than is ionizing radiation. The comparisons for crosslinking efficiencies are less certain, but the yields seem to have much more similar values in this case based on a spectroscopic estimation of crosslinking (i.e., analysis for substituted phenylcyclohexadiene formation). The gas quantum yields were ?H₂ = 2.6 × 10^?5, ?CH₄ = 0.63 × 10^?5, ?C₂H₆ ≈ 0.12 × 10^?5, and ?C₂H₂ ≈ 0.06 × 10^?5.     

Stereoisomeric effects on mass spectra—I. A Review

The unimolecular reactions that give rise to mass spectra are controlled by spatial relationships and energy considerations. In molecules that contain a heteroatom, elimination reactions, involving bond-making as well as breaking, are often prominent, e.g. loss of water from alcohols. The ease of such reactions depends on spatial relationships in the molecule, and the resultant ion intensities in the spectra of geometric isomers can consequently be correlated with differences in geometry and hence furnish a basis for assigning structures. Processes that do not involve bond-making do not have such rigorous geometric requirements, but depend rather on attainment of a transition state defined in terms of a minimum energy content. Common product ions from stereoisomeric hydrocarbons seem often to arise via a common transition state. When this condition holds, the difference between the enthalpies of the isomers is reflected in the relative appearance potentials and–though the cause-and-effect relationship here is less direct and more readily obscured by other factors–relative intensities of the common product derived from the isomers. In both classes of processes, the spectra of stereoisomers can be simplified and made more distinctive by lowering source temperature and ionizing voltage.