Periodic solutions of some infinite-dimensional Hamiltonian systems associated with non-linear partial difference equations. II

This is our second paper devoted to the study of some non-linear Schrödinger equations with random potential. We study the non-linear eigenvalue problems corresponding to these equations. We exhibit a countable family of eigenfunctions corresponding to simple eigenvalues densely embedded in the band tails. Contrary to our results in the first paper, the results established in the present paper hold for an arbitrary strength of the non-linear (cubic) term in the non-linear Schrödinger equation.     

Correlation length bounds for disordered Ising ferromagnets

Thed-dimensional, nearest-neighbor disordered Ising ferromagnet: $$H = - \sum {J_{ij} \sigma _i \sigma _j }$$ is studied as a function of both temperature,T, and a disorder parameter,λ, which measures the size of fluctuations of couplingsJ _ij ≧0. A finite-size scaling correlation length,ζ _f (T, λ), is defined in terms of the magnetic response of finite samples. This correlation length is shown to be equivalent, in the scaling sense, to the quenched average correlation lengthζ(T, λ), defined as the asymptotic decay rate of the quenched average two-point function. Furthermore, the magnetic response criterion which definesζ _f is shown to have a scale-invariant property at the critical point. The above results enable us to prove that the quenched correlation length satisfies: $$C\left| {\log \xi (T)} \right|\xi (T) \geqq \left| {T - T_c } \right|^{ - {2 \mathord{\left/ {\vphantom {2 d}} \right. \kern-\nulldelimiterspace} d}}$$ which implies the boundv≧2/d for the quenched correlation length exponent.     

Transfer hydrogenation using recyclable polyurea-encapsulated palladium: efficient and chemoselective reduction of aryl ketones

A robust and recyclable palladium catalyst [Pd0EnCat] has been prepared by ligand exchange of polyurea-encapsulated palladium(II) acetate with formic acid, resulting in deposition of Pd(0) in the support material; Pd0EnCat is shown to be a highly efficient transfer hydrogenation catalyst for chemoselective reduction of a wide range of aryl ketones to benzyl alcohols.     

Reactions of dimethyldivinylsilane,dimethyldivinyltin and allyltrimethyltin with diethylene (tertiary) phosphine)platinum complexes

The compounds [Pt(C₂H₄)₂(PR₃)] [PR₃ = P-tBu₂Me, P(C₆H₁₁)₃, PPh₃] react dimethyldivinylsilane or dimethyldivinyltin to give chelate complexes [Pt{(CH₂CH)₂MMe₂} (PR₃)] (M = Si or Sn). allyltrimethyltin reacts with various diethylene (tertiary phosphine)platinum compounds with cleavage of the allyl group to afford complexes [Pt(SnMe₃)(η³-C₃H₅)(PR₂)]. The NMR spectra (¹³C, ¹H and ³¹P) of the new compounds have been recorded, and the data are discussed in terms of the structures proposed.     

Towards novel difluorinated sugar mimetics; syntheses and conformational analyses of highly-functionalised difluorinated cyclooctenones

Highly-functionalised difluorinated cyclooctenones were synthesised from trifluoroethanol using either metallated difluoroenol acetal or carbamate chemistry, followed by a [2,3]-Wittig rearrangement or aldol reaction. Efficient RCM reactions afforded the title compounds which showed rather restricted fluxional behaviour by VT (19)F NMR. Topological characterisation by molecular modelling and NOESY/ROESY experiments offered a number of challenges, but allowed the identification of two favoured boat-chair conformers which interconverted by pseudorotation with relatively large activation barriers.     

Generation of 3-pyridyl biaryl systems via palladium-catalyzed Suzuki cross-couplings of aryl halides with 3-pyridylboroxin

The synthesis of 3-pyridyl biaryl systems can be readily achieved by means of palladium-catalyzed Suzuki cross-coupling reactions between aryl halides and 3-pyridylboroxin. A series of cross-couplings were conducted in order to investigate the scope and limitations of this protocol.     

Improved preparation and new reactions of β -(1-phenylthio) cyclopropyl enones

β-(1-Phenylthio)cyclopropyl enones can be conveniently prepared via reaction of the lithium salts of α-hydroxymethylene ketones with 1-lithio-1-phenylthiocyclopropane and are converted efficiently by treatment with aqueous acid to γ-keto cyclobutanones and less successfully by thermolysis to -γ-keto phenylthiocyclopentenes.     

Hydroformylation of cyclic dienes catalysed by acetatocarbonylbis(triphenylphosphine)rhodium(I)

The hydroformylation of cyclic dienes in benzene using Rh(CO₂Me)(CO)(PPh₃)₂ as catalyst was studied. The formation of mono- or di-aldehydes was critically dependent on ring size, the smaller cyclic dienes favouring dihydroformylation. Intermediate alkene-aldehydes could be isolated under appropriate conditions, indicating that initial attack on the diene was by hydroformylation. Under hydroformylation conditions, these intermediates underwent either hydroformylation or hydrogenation, depending on the ring size. Linear dienes gave complex mixtures of products.     

Positive Lyapunov exponents for a class of deterministic potentials

LetV() be a smooth, non-constant function on the torus and letT be a hyperbolic toral automorphism. Consider a discrete one dimensional Schrödinger operatorH, whose potential at sitej is given bygV _j =gV(T ^j ). We prove that wheng0 is small andg ^1/2 |E|2–g ^1/2 , the Lyapunov exponent for the cocycle generated byH-E is proportional tog ². The proof relies on a formula of Pastur and Figotin and on symbolic dynamics.