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.     

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 % ).     

Thermische Zersetzung von Blei(II)-Acetat

Zusammenfassung Die Zersetzung von Pb(CH₃COO)₃ · 3 H₃O wurde im Bereich von Raumtemperatur bis 450° unter N₃ mittels TG, DTA verfolgt und die Produkte wurden analytisch charakterisiert. PbO ist das hauptsächliche feste Endprodukt, neben dem im wesentlichen nur Pb entsteht. Als feste Zwischenprodukte wurden neben Pb(CH₃COO)₃ die basischen Acetate Pb(CH₃COO)₂ · PbO und Pb(CH₃COO)₂ · 2 PbO gefunden und durch Röntgenpulveraufnahmen identifiziert; diese basischen Acetate wurden durch isotherme Zersetzung bei 240 bzw. 305° dargestellt. Unter den durch GC und MS erfaßten und quantitativ bestimmten flüchtigen Zersetzungsprodukten sind CO₂ und Aceton Hauptbestandteile. Verschiedene Bildungsweisen der übrigen Produkte, CH₃COOH, (CH₃CO)₂O, Acetylaceton und des Zwischenproduktes Keten werden aufgezeigt.

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The decomposition of Pb(CH₃COO)₃ · 3 H₂O under N₂ has been studied by TG and DTA between room temperature and 450° and the products have been characterized analytically. PbO is the main solid product, besides which essentially only Pb is produced. As solid intermediates, aside from Pb(CH₃COO)₃ the basic acetates Pb(CH₃COO)₂ · PbO and Pb(CH₃COO)₂ · 2 PbO have been found and identified via their X-ray powder diagrams; these basic acetates have been prepared by isothermal decomposition at 240 and 305°, respectively. Among the volatile decomposition products analyzed by GC and MS and determined quantitatively, CO₂ and acetone are the main products. Different routes for the formation of the other products, CH₃COOH, (CH₃CO)₂O, acetylacetone and the intermediate ketene are shown.

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Résumé On a suivi, par TG et ATD sous N₂, dans l'intervalle de températures allant de l'ambiante jusqu'à 450°, la décomposition de Pb(CH₃COO)₂ · 3 H₂O dont on a caractérisé les produits par des méthodes analytiques. PbO est le principal produit final solide, en dehors duquel il ne se forme essentiellement que du Pb. Comme produits intermédiaires solides, on a trouvé, à part Pb(CH₃COO)₂, les acétates basiques Pb(CH₃COO)₃. PbO et Pb(CH₃COO)₂ · · 2 PbO qui ont été identifiés par analyse de poudres aux rayons X. Ces acétates basiques se sont formés, par décomposition isotherme, respectivement à 240 et à 305°. Parmi les produits de décomposition volatils décelés par GC et MS et dosés par des méthodes quantitatives, CO₃ et l'acétone sont les composants principaux. On montre les différents modes de formation des autres produits, comme CH₃COOH, (CH₃CO)₃O, l'acétylacétone et le cétène formé transitoirement.

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Pb(CH₃COO)₂. 3₂O 450°. . PbO , , . , Pb(CH₃COO)₂, Pb(CH₃COO)₂. PbO Pb(CH₃COO)₂. 2PbO, . , , 240 305°. , -, CO₂ . , CH₃COOH, (CH₃CO)₂O, .

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Dem Fonds der Chemischen Industrie danken wir für finanzielle Unterstützung.     

Generalization of a result of Matsuo and Cherednik to the Calogero-Sutherland-Moser integrable models with exchange terms

A few years ago, Matsuo and Cherednik proved that from some solutions of the Knizhnik-Zamolodchikov (KZ) equations, which first appeared in conformal field theory, one can obtain wave functions for the Calogero integrable system. In the present communication, it is shown that from some solutions of generalized KZ equations, one can construct wave functions, characterized by any given permutational symmetry, for some Calogero-Sutherland-Moser integrable models with exchange terms. Such models include the spin generalizations of the original Calogero and Sutherland ones, as well as that with -function interaction.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995