Covering vertices of a graph by k disjoint cycles

Let d, k and n be three integers with k3, d4k−1 and n3k. We show that if d(x)+d(y)d for each pair of nonadjacent vertices x and y of a graph G of order n, then G contains k vertex-disjoint cycles converting at least min{d,n} vertices of G.     

Effect of labile hydrogens on measurement of tritium preserved in hydrogens of cellulose

Various f-elements are separated as anionic complexes from both acidic and alkaline solutions by precipitation with alkylpyridinium salts. The precipitates are also cationic surfactants where the simple counter-ion (e.g. nitrate or chloride) is replaced by the negatively charged complex anion of an actinide or lanthanide. The low solubility of these precipitates is explained by a strong affinity of divalent complex counter-ions of f-elements to the quaternary nitrogen. Precipitations in solutions of nitric acid allow to separate tetravalent f-elements from other metals, in alkaline carbonate solutions tetravalent and hexavalent actinides are precipitated simultaneously. The last procedure yields precipitates, which are very intimate mixtures of hexavalent and tetravalent actinides. This allows to prepare mixed oxides in a simple way.     

X-ray scattering by water molecules studied by using synchrotron radiation

The energy spectra of free water molecules were measured at scattering angles 2θ ranging from 10.5° to 75.7°, using an angle-dispersive-type diffractometer and synchrotron radiation as an X-ray source. A silicon (111) monochrometer was used to obtain incident X-rays with the wavelengths of (1.543/n) Å (n = 1,3,4,5). Observed inelastic scattering peaks are clearly separated from eleastic ones at s values [s = (4π/λ) sin Å] larger than 8 Å^?1. The increase of the separation with an increasing s value was consistent with the classical theory of the Compton shift. The total (elastic plus inelastic) intensities were obtained over a range of s = 0.74–5.0 Å^?1. Experimental difference intensities Δσ_ee and Δσ_ne were obtained separately by combining the X-ray and high-energy electron scattering data. The experimental results are in reasonable agreement with the theoretical intensities calculated from SCF and CI molecular wave functions with a basis set of double-zeta plus polarization functions. © 1994 John Wiley & Sons, Inc.     

One-step furan ring formation: Synthesis of furo[3,2-h]quinolones

The one-pot reaction of ethyl 1-cyclopropyl-6,7,8-trifluoro-1,4-dihydro-4-oxoquinoline-3-carboxylate ( 6 ) with tert-butyl acetoacetate gave 3-tert-butyl 7-ethyl 9-cyclopropyl-4-fluoro-6,9-dihydro-2-methyl-6-oxofuro[3,2-h]quinoline-3,7-dicarboxylate ( 5 ). This regioselective cyclization was rationalized by the Hard and Soft Acids and Bases principle. By use of a similar furan-forming reaction, we prepared 2-(amino-methyl)furo[3,2-h]quinoline-7-carboxylic acid 4 . Compound 4 showed weak antibacterial activity.     

Photocrosslinking of poly(2,3-epithiopropyl methacrylate) by the oxidation of pendant episulfide groups

In the photocrosslinking of poly(2,3-epithiopropyl methacrylate) (PETMA) films the effect of the pendant episulfide group's oxidation on the crosslinking of PETMA was investigated. Thermal crosslinking of PETMA is promoted by peroxides such as benzoyl peroxide and hydrogen peroxide. IR spectrum of the crosslinked PETMA showed that the reaction proceeded through the oxidation of episulfide groups by the peroxides. The anthracene (An) sensitized photocrosslinking of PETMA films also proceeded via the oxidation of episulfide groups by singlet oxygen. It was found that residual tetrahydrofuran (THF) in the films remarkably increased the rate of the photocrosslinking and/or reduced the induction period. From the further investigation concerning casting solvents it was found that residual CS₂, CCl₄, and CHCl₃ in films increased the rate of the photocrosslinking and/or reduced the induction period of the photocrosslinking. The disappearance rate of An in the films was also increased by the presence of residual CS₂, CCl₄, and CHCl₃, differring from the result of THF. These results were explained by a difference in lifetime of singlet oxygen in the films. From the results were explained by a difference in lifetime of singlet oxygen in the films. From the results concerning the effects of hydroperoxides such as THF hydroperoxide and t-butyl hydroperoxide on the photocrosslinking of PETMA films the acceleration effect of the residual THF was deduced to be due to the promotion of singlet oxygen-oxidation of sulfide groups by protic compounds such as THF hydroperoxide and H₂O in the THF.     

A mixed version of Menger's theorem

An(a, b)-n-fan means a union ofn internally disjoint paths. Menger's theorem states that a graphG has an(a, b)-n-fan if and only ifG isn-connected betweena andb. We show thatG contains edge-disjoint(a, b)-n-fans if and only if for anyk withk0min{n–1, |V(G)|–2} and for any subsetX ofV(G)-{a, b} with cardinalityk, G-X is (n-k)-edge-connected betweena andb.     

On component factors

LetK be a connected graph. A spanning subgraphF ofG is called aK-factor if every component ofF is isomorphic toK. On the existence ofK-factors we show the following theorem: LetG andK be connected graphs andp be an integer. Suppose|G| = n|K| and 1 <p < n. Also suppose every induced connected subgraph of orderp|K| has aK-factor. ThenG has aK-factor.