Trees in tournaments

Letf(n) be the smallest integer such that every tournament of orderf(n) contains every oriented tree of ordern. Sumner has just conjectures thatf(n)=2n–2, and F. K. Chung has shown thatf(n)(1+o(1))nlog₂ n. Here we show thatf(n)12n andf(n)(4+o(1))n.     

Sound propagation in metallic glasses—a new look at the tunneling model

A variety of amorphous metals exhibit a characteristic behavior in their acoustic properties at low frequencies which differs from the predictions of the standard tunneling model. We point out that a lower cut-off _min for the tunnel matrix element, which is needed for consistency of the tunneling model, leads to an upper bound on relaxation times induced by the conduction electrons. We derive explicit expressions for the velocity shift and internal friction for the normalconducting and superconducting case. It is shown that a maximum relaxation time plays an essential role at audio frequencies. The corresponding change of the acoustic properties is in qualitative agreement with vibrating reed experiments.     

Wiener-Hope equations with the transmission property

Conditions on the kernel of the classical Wiener-Hopf equation are obtained which provide the same smoothness of any solution in terms of its inclusion in different spaces of smooth functions such as as the function in the right side of the equation.     

Mono- and dinuclear molybdenum complexes with sterically demanding cycloheptatrienyl ligands

Sterically demanding cycloheptatrienylium (tropylium) salts of the type (1,3,5-C7H4R3)BF4 [R = t-Bu, (3a)BF4; R = SiMe3, (3b)BF4] have been prepared from the corresponding 1,3,5-trisubstituted benzene derivatives 1 by ring expansion with diazomethane followed by hydride abstraction with triphenylcarbenium tetrafluoroborate, (Ph3C)BF4. Complexation can be achieved by arene exchange and Mo(CO)3 group transfer employing [(eta6-p-xylene)Mo(CO)3] (4) to yield the cationic complexes (5)BF4. In refluxing mesitylene, [(eta7-C7H4t-Bu3)Mo(CO)3]BF4, (5a)BF4, undergoes CO substitution to furnish the mesitylene sandwich complex (6a)BF4. A cyclic voltammetric study reveals that this complex exhibits a reversible one-electron oxidation to the dicationic 17e complex 6a2+, which can also be accessed by chemical oxidation with AgBF4. On the contrary, the reduction of 6a+ is irreversible and does not yield a stable 19e complex 6a. To study the fate of the reduced 19e form, (5a)BF4 was treated with Na2Hg to diastereoselectively afford the C-C coupled bicycloheptatriene complex 7a. Paramagnetic, dinuclear complexes of the type [(eta7-C7H4R3)Mo(mu-Cl)3Mo(eta7-C7H4R3)] (8) have been obtained from the reaction of (5)BF4 with Me3SiCl. These can be regarded as mixed-valence Mo(0)/Mo(+I) compounds with a metal-metal bond order of 0.5. Cyclic voltammetric studies reveal that both complexes 8a and 8b undergo reversible one-electron oxidation as well as reduction. Treatment with one equivalent of ferrocenium hexafluorophosphate leads to removal of the unpaired electron and formation of the diamagnetic complexes (8)PF6. Theoretical DFT calculations have been carried out to further elucidate the bonding in these systems. In addition, the X-ray crystal structures of (5b)BF4, (6a)BF4 x CH2Cl2, (6a)(BF4)2 x (acetone)2, 7a x CH2Cl2, 8a x 0.5C6H14, and (8a)PF6 x Et2O are reported.     

Pyrrolopyridazine nucleosides. The synthesis of certain 1-β-D-ribofuranosyl-1H-pyrrolo[2,3-d]pyridazin-4(5H)-ones

Nucleosides of pyrrolo[2,3-d]pyridazin-4(5H)-ones were prepared by the single-phase sodium salt glycosylation of appropriately functionalized pyrrole precursors. The glycosylation of the sodium salt of ethyl 4,5-dichloro-2-formyl-1H-pyrrole-3-carboxylate ( 4 ), or its azomethino derivative 7 , with 1-bromo-2,3,5-tri-O-benzoyl-D-ribofuranose in acetonitrile afforded the corresponding substituted pyrrole nucleosides ethyl 4,5-dichloro-2-formyl-1-(2,3,5-tri-O-benzoyl-β-D-ribofuranosyl)-1H-pyrrole-3-carboxylate ( 5 ) and ethyl 4,5-dichloro-2-phenylazomethino-1-(2,3,5-tri-O-benzoyl-β-D-ribofuranosyl)-1H-pyrrole-3-carboxylate ( 8 ), respectively. The latter, upon treatment with hydrazine, afforded the annulated product 2,3-dichloro-1-β-D-ribofuranosyl-1H-pyrrolo[2,3-d]pyridazin-4(5H)-one ( 6 ), in good yield. The unsubstituted analog 1-β-D-ribofuranosyl-1H-pyrrolo[2,3-d]pyridazin-4(5H)-one ( 9 ), was obtained upon catalytic dehalogenation of 6 . This report represents the first example of the synthesis of nucleosides of pyrrolopyridazines.     

Determination of mono-ortho substituted chlorobiphenyls by multidimensional gas chromatography and their contribution to TCDD equivalents

The mono-ortho chlorobiphenyls (CBs) 60, 74, 114, 123, 157, 167 and 189 were determined in Aroclor mixtures and aquatic organisms by multidimensional gas chromatography with electron capture detection (MDGC/ECD), using a combination of an Ultra 2 and an FFAP column. MDGC/ECD is recommended as the most suitable technique for direct determination of these CBs, without a liquid chromatographic (LC) pre-separation of mono-ortho CBs from the other CBs. Dependent, to some extent, on the stationary phase used, single-column determinations of these CBs easily yield too high results due to the presence of interferences. The contribution of the mono-ortho CBs studied to the total 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) equivalents (TEQs) from CBs in fish is highly dependent on the toxic equivalency factors (TEFs) used and varies between 0.02 and 22%. This contribution is mainly due to CBs 74, 114, 157 and 167 which may easily be determined by taking three heart-cuts and combining them into one MDGC/ECD run. The analytical error is negligible compared with the huge uncertainty of the TEFs. A need for more precise TEFs is emphasized.     

Formation of dinuclear titanium and zirconium complexes by olefin metathesis--catalytic preparation of organometallic catalyst systems

The titanium complex [(C(5)H(4)bond;allyl)TiCl(3)] (2) undergoes olefin metathesis coupling when treated with 3 mol % of [Cl(2)(L(1))(L(2))Ru=CHPh] (L(1)=L(2)=PCy(3), 4 a; L(1)=PCy(3), L(2)=(H(2)IMes), 4 b) to yield the dimetallic complex [Cl(3)Ti(C(5)H(4))-CH(2)CH=CHCH(2)-(C(5)H(4))TiCl(3)] (5). The allyl-substituted titanocene complex [Cp(C(5)H(4)bond;allyl)TiCl(2)] (3) analogously yields the dimetallic system 6 when treated with 4. The ansa-zirconocene complex [Me(2)Si(C(5)H(4))(C(5)H(3)bond;allyl)ZrCl(2)] (7) cleanly yields the analogous dimetallic coupling product 8 (>95 % isomerically pure), when treated with catalytic amounts of 4 b in toluene. Complex 8 gives an active homogeneous ethene or propene polymerization catalyst, especially at elevated temperatures, when treated with excess methylalumoxane.     

Purification and characterization of a laccase from the white-rot fungus Trametes multicolor

The wood-degrading fungus Trametes multicolor secretes several laccase isoforms when grown on a simple medium containing copper in the millimolar range for stimulating laccase synthesis. The main isoenzyme laccase II was purified to apparent homogeneity from the culture supernatant by using anion-exchange chromatography and gel filtration. Laccase II is a monomeric glycoprotein with a molecular mass of 63 kDa as determined by sodium dodecylsulfate polyacrylamide gel electrophoresis, contains 18% glycosylation, and has a pI of 3.0. It oxidizes a variety of phenolic substrates as well as ferrocyanide and iodide. The pH optimum depends on the substrate employed and shows a bell-shaped pH activity profile with an optimum of 4.0 to 5.0 for the phenolic substrates, while the nonphenolic substrates ferrocyanide and 2,2′-azino-bis(3-ethylbenzthiazoline-6-sulfonate) show a monotonic pH profile with a rate decreasing with increasing pH.