Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I₂(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G₂, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in B_n-, D_n- and G₂-bracket algebras, and as an application, discover a Pieri-type formula in the B_n-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary B_n-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.     

Constructive and Destructive Two-Pulse Excitation Investigated with a White-Light Michelson Interferometer

Linear propagation of two pulses through methanol solution of aluminum phthalocyanine chloride is investigated using a modified white-light Michelson interferometer. The observed coherence time of the white light is 6 fs, and the separation between the two-excitation pulses is set to about 10 fs. The excitation is dependent on the phase-relation between the two pulses. We have observed an enhancement of the excitation when the two pulses are in-phase and strong suppression of the excitation when the two pulses are out of phase by π.     

Investigation of the drying conditions for amidosulfuric acid.

The drying conditions for primary standards of volumetric analysis have a significant effect on the titration results due to changes in the purity, stability and homogeneity. Amidosulfuric acid, a strong acid used as a reference material for volumetric analysis in Japan, was dried in a vacuum desiccator or heated at different temperatures, and then measured by Karl-Fischer titration, thermogravimetry/mass spectroscopy (TG-MS), ion chromatography and coulometric titration. The optimum drying conditions were at 50 degrees C for 24 h with crushing.     

In Situ Immobilization of Ultrafine Particles Synthesized in a Water/Oil Microemulsion

We investigated the in situ immobilization of ultrafine particles synthesized in a water/oil (w/o) microemulsion to silica for its possible application to supported metal catalysts. ZnS particles immobilized to silica by the ME method were consistent with those synthesized in a w/o microemulsion. Therefore, ZnS particles in a w/o microemulsion could be immobilized to silica without aggregation by this method. The relationship between the method of synthesizing Rh ultrafine particles in a w/o microemulsion and the diameter and diameter distribution of Rh particles immobilized to silica was studied. Rh-SiO(2) catalysts with a sharp diameter distribution could be prepared by immobilizing Rh-hydrazine complex particles because these complex particles would be very stable in a w/o microemulsion. The Rh particle diameters of Rh-SiO(2) catalysts prepared by changing the amount of silica produced were almost identical. Accordingly, the Rh particle diameter of Rh-SiO(2) catalysts could be controlled independently of Rh content by the ME method. Copyright 2001 Academic Press.     

Magnetic interactions in a series of paramagnetic Ln(III) complexes with a chelated imino nitroxide radical

The magnetic interactions in a new series of isostructural imino nitroxide radical lanthanide(III) complexes, [Ln(hfac)₃(IM2py)] (Ln = Gd–Yb: IM2py = 2-(2′-pyridyl)-4,5-dihydro-4,4,5,5-tetramethyl-1H-imidazoline-1-oxy; hfac = 1,1,1,5,5,5-hexafluoro-2,4-pentanedione), are examined by considering the intrinsic paramagnetic contribution of the Ln(III) ion from the corresponding [Ln(hfac)₃(pybzim)] with a diamagnetic pybzim(2-(2-pyridyl)benzimidazole) ligand; the Ln(III)–IM2py interaction being antiferromagnetic for the 4f⁷ to 4f¹³ Ln(III) complexes and negligibly small for the other complexes. This series is the first example reverse to the previous cases for the series of Ln–Cu or Ln–aminoxyl(NIT) radical (4,5-dihydro-4,4,5,5-tetramethyl-1H-imidazoline-3-oxide-1-oxy) complexes, other than only a few examples of semiquinone Ln complexes. This reverse nature of the magnetic interaction, as compared with the NIT complexes, validates the empirical approach by O. Kahn et al. [Inorg. Chem. 38 (1999) 3692; J. Am. Chem. Soc. 122 (2000) 3413] in the spin-coupled systems for a series of Ln(III) complexes.     

Motion of redox molecules in solution monitored by the highly-sensitive EQCM technique

The behavior of redox molecules in solution that was not detected by electrochemical techniques was measured by a highly-sensitive electrochemical quartz crystal microbalance (EQCM) technique that has been improved in this study to obtain a high sensitivity of EQCM measurement in solution. The improved EQCM technique allowed to monitor the motion of a redox molecule, that is an access of the molecule to an electrode surface and repulsion from the surface during redox. An EQCM technique currently in use has measured adsorption of redox molecules on an electrode surface or polymerization on the surface caused by a chemical reaction following redox, which exhibits an enough large mass change response to detect with an EQCM measurement. However, access and repulsion of redox molecule, which is a slight motion of the molecule near on electrode surface, has not been detected and investigated by an EQCM technique, because the mass change response seems to be very small. In this study, the redox behavior of methyl viologen on a bare gold surface, pyridinethiol surface and methylpyridinethiol surface was investigated. Although the three electrodes give the same cyclic voltammogram of methyl viologen, the three are different in QCM response recorded at the same time as the voltammetry. Access/repulsion of methyl viologen within an electrical double layer was monitored by the highly-sensitive EQCM technique.     

Rotational isomerism involving an acetylenic carbon. Part 5: Restricted rotation about acetylenic axis in sterically crowded bis(1-phenyl-9-anthryl)ethynes

The title sterically crowded di-9-anthrylethyne derivative with 3,5-ⁱPr₂-phenyl groups at 1-position showed a barrier to rotation about the acetylenic axis of 18.0 kcal mol⁻¹ based on a dynamic NMR study, which is an extremely high value for acyclic diarylethynes. The mechanism of the dynamic stereochemistry and the substituent effect on the rotational barrier are discussed.     

Acidity distribution of carboxyl groups in Loy Yang brown coal: its analysis and the change by heat treatment

Brown coals have a considerable number of acidic functional groups of which the main component is carboxyl groups, and the acidity has a wide distribution. In this paper, changes of the acidity distribution were examined by aqueous titration when brown coal was heat-treated to control its acidity distribution. For Loy Yang brown coal from Australia dried at 50 degrees C under vacuum (LY50), the acid dissociation constant, Ka, was distributed over a wide pKa range between 2 and 9. Then, using Gaussian functions, the acidity distribution was divided into four groups, which were characterized by average pKa values: average pKa value of 3.8 (hereafter referred to as Group A), 5.2 (Group B), 6.8 (Group C), and 8.3 (Group D). Among them, Groups A, B, and C were assigned to carboxyl groups. From the changes of the number of carboxyl groups when brown coal was heat-treated up to 400 degrees C, it was found that the way of decrease was different among these acidic groups. The decrease of the amount of carboxyl groups in Group C was significant, and at 325 degrees C most of them disappeared. On the other hand, the carboxyl groups in Group A remained even at a high temperature of 400 degrees C. We estimated approximately the structures around carboxyl groups for LY50 and their structural changes by heat treatment using the known pKa values for simple carboxylic acids and the pKa values calculated by the MOPAC program for complicated carboxylic acids.