Searching for extensible Korobov rules

Extensible lattice sequences have been proposed and studied in [F.J. Hickernell, H.S. Hong, Computing multivariate normal probabilities using rank-1 lattice sequences, in: G.H. Golub, S.H. Lui, F.T. Luk, R.J. Plemmons (Eds.), Proceedings of the Workshop on Scientific Computing (Hong Kong), Singapore, Springer, Berlin, 1997, pp. 209–215; F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138; F.J. Hickernell, H.Niederreiter, The existence of good extensible rank-1 lattices, J. Complexity 19 (2003) 286–300]. For the special case of extensible Korobov sequences, parameters can be found in [F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C.Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138]. The searches made to obtain these parameters were based on quality measures that look at several projections of the lattice. Because it is often the case in practice that low-dimensional projections are very important, it is of interest to find parameters for these sequences based on measures that look more closely at these projections. In this paper, we prove the existence of “good” extensible Korobov rules with respect to a quality measure that considers two-dimensional projections. We also report results of experiments made on different problems where the newly obtained parameters compare favorably with those given in [F.J. Hickernell, H.S. Hong, P. L’Ecuyer, C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2001) 1117–1138].     

Indecomposable modules over the Virasoro Lie algebra and a conjecture of V. Kac

We consider a class of indecomposable modules over the Virasoro Lie algebra that we call bounded admissible modules. We get results concerning the center and the dimensions of the weight spaces. We prove that these modules always contain a submodule with one-dimensional weight spaces. From this follows the proof of a conjecture of V. Kac concerning the classification of simple admissible modules.Preprint Université de Bourgogne-mai 1990     

Two Other Interpretations of the Entringer Numbers

The Entringer's integers were introduced to enumerate the down-up (or up-down) permutations according to the first term. In a previous paper these integers are shown to enumerate André's trees according to two parameters. In this paper we prove that they enumerate the direct alternative permutations according to the absolute value of the first two terms and also according to a specified term.     

Wavelength independence of Eu quantum yields

A quantitative comparison between the excitation and absorption spectra of Eu³⁺ in aqueous solution is given which leads to the conclusion that fluorescence quantum yields are independent of the excitation wavelength. Use is made therefore of the series of homologous lines ⁵D₃, ⁵D₂, ⁵D₁, ⁵D₀ → ⁷F₁.     

Percolative phenomena in lecithin reverse micelles: the role of water

 The role played by the solvation water molecules on the macroscopically observed sol–gel transition in lecithin/cyclohexane/water reverse micelles is investigated by quasielastic neutron scattering, dielectric relaxation and conductivity measurements. The experimental results are juxtaposed to those from spherical Aerosol OT reverse micelles. It is shown how the results from lecithin-based system can be interpreted only assuming that, in contrast to Aerosol OT systems, the water molecules are entrapped at the interfaces without coalescing into an inner water pool. It is suggested that, in the case of lecithin, the solvation water can induce a change in the surface curvature, in such a way promoting the formation of branch points. Such a hypothesis is supported by the temperature dependence of the conductivity which agrees with the hypothesis of an intermicellar bond percolation. The investigation of the structures imposed by an external electric field is also studied. The observed electrorheological behaviour seems to confirm the existence of a percolated transient network in the gel phase. Received: 21 March 2001 Accepted: 24 August 2001     

Carbonylation of (+)-2-carene induced by iron-pentacarbonyl.

(+)-2-carene heated neat with iron pentacarbonyl leads to α-phellandrene-Fe(CO)₃ complex ( ～15 % ), p. cymene ( ～15 % ), (-)-(1S)-3,8,8-trimethylbicyclo (4.1.1) oct-3-ene-7-one ( ～50 % ) and (+)-(1S,7S)-3,8,8-trimethylbicyclo (4.1.1.) oct-3-ene-7-ol ( ～20 % ).