Notes on the non-autonomous Lyness equation

We prove that the solutions of the non-autonomous Lyness equation x_n+1=(x_n+b_n)/x_n−1, b_n>0, x₀>0, x₁>0, are bounded away from zero and infinity if the sequence b_n is monotonic. We also give an example to show that the solutions need not be bounded even when the sequence b_n attains only two values.     

Enantioselective synthesis of preclavulone A and its methyl ester

A highly enantioselective synthesis of the (8S,12S)-enantiomer of preclavulone A and its methyl ester is described featuring the Julia protocol for installing the (Z)-double bond in the lower chain. This procedure is suitable for the preparation of labeled preclavulone analogues for biosynthetic studies on marine clavulones.     

The distribution of nickel ions among octahedral and tetrahedral sites in NiAl2O4MgAl2O4 solid solutions

The cation distribution in NiAl₂O₄ and in the solid solutions Ni_xMg_1?xAl₂O₄ with x ranging from 0.01 to 0.55 has been studied by X-ray analysis, magnetic susceptibility and reflectance spectroscopy.The relative X-ray intensities of various reflections for each compound were measured and compared with intensities calculated for various models of cation distribution. Lattice parameters, magnetic moments of Ni²⁺, and the position of nickel absorption bands have also been measured. The results show that all specimens have a predominantly octahedral distribution for their nickel ions, and that there is a small change towards a random distribution when the quenching temperature is increased from 1273 to 1673°K.Both X-ray and magnetic results show that the fraction, α, of Ni²⁺ ions on tetrahedral sites varies with nickel content. For samples quenched from 1273°K, NiAl₂O₄ is 78% inverse, but between x = 1.0 and x = 0.25 the value of α decreases from 0.22 towards 0.1. Thus, the fraction of nickel on octahedral sites in the solid solutions is higher than in NiAl₂O₄. The effect is linked with an increase in Dq for the Ni²⁺ ion. Below x = 0.25, there is tentative evidence that α may increase. In nickel-dilute solutions the matrix as a whole is nearly normal, and it is suggested that Madelung energy may then be more effective in counteracting the octahedral stabilization of nickel. The same trend is observed in the specimens quenched from 1673°K.     

Reactivity of methyl mandelate-Ti(IV)-enediolate: oxidative homocoupling versus aldol and direct Mannich-type syn-diastereoselective condensation

[reaction: see text] Methyl mandelate undergoes quantitative oxidative homocoupling on treatment with TiCl4/amine at room temperature. In the presence of ArCHO, quantitative syn-diastereoselective aldol condensation takes over the dimerization, whereas exclusive Mannich-type syn-diastereoselective reaction is observed in the presence of both ArCHO and PhNH2. The subsequent reactions of the title intermediate do not depend on how it is generated.     

Synthesis of protected derivatives of O-phosphotyrosine incorporation in a heptapeptide

The solid phase synthesis of the phosphopeptide Leu-Arg-Arg-Ala-Tyr(P)-Leu-Gly is reported via the stepwise incorporation of the protected phosphoamino acid Nα -tert-butyloxycarbonyl-O-dimethyl-phosphono-L-tyrosine.     

1,2-Oxopalladation versus pi-allyl palladium route. A regioconvergent approach to a key intermediate for cyclopentanoids synthesis. New insights into the Pd(II)-catalyzed lactonization reaction

Regioconvergent synthesis of the key lactone 1 from an equimolar mixture of the two olefins 4 and 5 was achieved by unique Pd(II) chemistry. The synthetic versatility of lactone 1 has been demonstrated in the synthesis of iridoids and of the endo-Corey lactone 2, which is a key intermediate for the F(2)-isoprostane synthesis. Upon exposure of the sodium salts of 4 and 5 to a catalytic amount of Pd(OAc)(2) under oxygen, in the presence of AcOH, an isomeric lactone 12 was obtained in addition to the title compound 1. The Pd(II) lactonization was optimized by fine-tuning all the factors participating in the catalytic cycle: solvent, oxidant, co-oxidant, and Pd(II) source. The Hosokawa's heterobimetallic couple emerged as the catalyst of choice. With a Cu(II)-Pd(II) couple, the redox process was transferred to copper, and the formal oxidation state of palladium remained constant during the reaction. By virtue of this new methodology, lactone 1 was obtained in a rewarding 60% yield, along with isomeric lactone 12 in 30% yield. A detailed mechanistic study was carried out in order to elucidate the formation of lactones 1 and 12. Lactone 1 was formed from either olefin 8 or olefin 10; on the other hand, lactone 12 was formed exclusively from olefin 10. An intramolecular 1,2-acyloxypalladiation was invoked for the transformation of 8 into 1, whereas the pi-allyl complexes 13 and 11 were involved in the transformation of olefin 10 into 12 and 1, respectively.     

X-ray and mössbauer studies of tricyclohexyltin(IV) halides. The crystal structures of (cyclo-C6H11)3SnX (X = F,Br and I)

The structures of tricyclohexyltin fluoride (I), bromide (II) and iodide (III) have been determined by X-ray analysis. Compound I crystallizes in the space group P2₁/m with a = 10.422(6), b = 17.238(9), c = 5.769(3) Å, β = 104.6(1)° and Z = 2. Compounds II and III crystallize in the space group Pcmn with a = 10.427(6), b = 16.914(9), c = 11.366(6) Å, Z = 4; and a = 10.400(6), b = 16.900(10), c = 11.400(4) Å, Z = 4, respectively. All three structures consist of discrete tetrahedral (cyclo-C₆H₁₁)₃SnX units.The temperature dependence of the Mössbauer resonance areas has been examined in order to obtain information about the relationship between chemical structure and lattice dynamics.     

Long-range dispersion coefficients for like centrosymmetric linear molecules and an application to H2-H2

The real spherical tensor theory of long-range intermolecular coefficients developed in previous papers is applied to derive explicit formulae for the first three dispersion coefficients for like centrosymmetric linear molecules. The expansion of angle-dependent coefficients in associated Legendre polynomials allows one to identify the isotropic and anisotropic components of the dispersion interaction in terms of London dispersion constants, the treatment of higher coefficients being simplified by the coupling of the elementary (l, l′)-polarizations to resultant angular momenta L_A and L_B onto each molecule. The contributions from all coupling schemes are given explicitly for C₆, C₈, C₁₀, and numerical results are presented for H₂-H₂ using two-term reduced spectra values from the Kaiserlautern group.     

On the supposed equilibrium Ph6Pb2Ph4Pb+Ph2Pb in solution

The reaction between pentadeuteriophenyllithium and hexaphenyldilead combined with new measurements of isotopic exchange between Ph₆Pb₂ (labelled with RaD) and Ph₄Pb, have shown that the dilead compound is undissociated in solution. These results cast doubt upon the previously reported equilibrium: Ph₆Pb₂Ph₄Pb+Ph₂Pb.