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].     

Magnetic anisotropy of textured CrO2 thin films investigated by X-ray magnetic circular dichroism

Highly a-axis-textured CrO₂ films have been deposited on Al₂O₃ (0001) substrates by chemical vapor deposition. CrO₂ has been found to have highly a-axis (010)-oriented columnar growth on a Cr₂O₃ (0001) initial layer. The six-fold surface symmetry of the Cr₂O₃ initial layer leads to three equivalent in-plane orientations of the a-axis-oriented CrO₂ unit cell. We report Cr L_2,3 X-ray magnetic circular dichroism data along the surface normal and at 60° off-normal sample orientation. For a 60° sample alignment, a strong increase of the projected orbital moment could be observed for unoccupied majority t_2g states using moment analysis. Therefore, the c axis is identified as the intrinsic magnetic easy axis of CrO₂. In addition, a small spin moment and a very strong magnetic dipole term T_z have been found. Received: 8 January 2002 / Accepted: 8 January 2002     

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.     

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 % ).