Multiderivations of Coxeter arrangements

Let V be an ℓ-dimensional Euclidean space. Let G⊂O(V) be a finite irreducible orthogonal reflection group. Let ? be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H∈? choose α _H ∈V ^* such that H=ker(α _H ). For each nonnegative integer m, define the derivation module D ^(m) (?)={θ∈Der _S |θ(α _H )∈Sα ^m _H }. The module is known to be a free S-module of rank ℓ by K. Saito (1975) for m=1 and L. Solomon-H. Terao (1998) for m=2. The main result of this paper is that this is the case for all m. Moreover we explicitly construct a basis for D ^(m) (?). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m−1)h/2)+m _i (1≤i≤ℓ) (when m is odd). Here m ₁≤···≤m _ℓ are the exponents of G and h=m _ℓ+1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat generators by K. Saito (or equivalently the Frobenius manifold structure for the orbit space of G). Some new results concerning the primitive derivation D are obtained in the course of proof of the main result. Oblatum 27-XI-2001 & 4-XII-2001?Published online: 18 February 2002     

WIDE RANGE DETECTOR USING PARABOLIC CYLINDRICAL MIRROR FOR THz APPLICATIONS

A new, wide-band, high-speed and high-sensitivity THz detector has been developed. The prototype detector consists of a parabolic cylindrical mirror, a long wire antenna and a Schottky barrier diode. Direct detection measurements have shown a stable sensitivity of 150 ± 50 V/W for 1–2 THz without any adjustments. The long wire antenna was fixed at the focus of parabolic cylindrical mirror then it has been realized less operation steps, easy coupling to the external THz signals and a dramatic enhancement in the practicality of this system. The optically polished mirror and frosted surface one showed comparable sensitivities, thus easy polishing and less cost mirror fabrication can be applied for this system. The radiation pattern showed a maximum radiation angle of approximately 23° with its dominant main lobe, which was attributed to the wire antenna character and confirmed good agreements with classical antenna theory.     

New Chiral Kemp's Acid Diamides for Chiral Amine Recognition by 1 H NMR

Two Kemp's acid diamides were synthesized and applied to chiral amine recognition using ¹H NMR analysis. One derivative based on 1-(1-naphthyl)ethylamine had good chiral recognition of six amines and was useful to determine the optical purity for three amines, i.e., methylbenzylamine, 1-(1-naphthyl)ethylamine and 1- henylpropylamine,however, the cyclohexylethylamine derivative showed little discrimination for the amines studied. Together with the results for alkylamines, it was shown that aromatic structure was important for aromatic shielding anisotropy and –– interactions between host and guest. The structure of the 1-(1-naphthyl)ethylamine derivative in solution was also considered based on ¹H NMR data and computer simulation.     

Electrode kinetics of the metal complexes with aminopolycarboxylic acids: Part II. Cadmium-ethylenediamine-N,N,N′,N′-tetraacetate (EDTA) complexes

The d.c. polarographic current-potential curves of Cd(II)-EDTA complexes were examined in the pH range 0.5–10.0, to elucidate the mechanism of their electrode processes and to determine the relevant electrochemical kinetic parameters. It was shown that the first wave observed below pH 3 at ?0.58 to ?0.65 V vs. SCE is the reversible reduction wave of Cd(II) aquo-ion with kinetically-controlled limiting current, and the second wave observed above pH 1.5 at ?0.75 to ?1.21 V vs. SCE corresponds to the simultaneous irreversible reduction of four complex species, CdH₃L⁺, CdH₂L, CdHL^? and CdL^2?, where CdH_pL^(p?2)+ and L^4? denote the protonated complex species with p protons and the unprotonated EDTA ion, respectively. Analysis of the dependence of limiting current on the hydrogen ion concentration led to the conclusion that the preceding reaction determining the behaviour of limiting current is CdH₃L⁺?Cd²⁺+H₃L^? with k₃^d=6.3×10² s^?1 and k₃^f=3.3×10⁶ s^?1M^?1, where k₃^d and k₃^f are the dissociation and formation rate constants, respectively. On the other hand, from analysis of the dependence of half-wave potentials of the second wave on the hydrogen ion concentration, the kinetic parameters of the four complex species were evaluated, and are given in Table 1. Further, it was shown that the cathodic rate constants of these four charge transfer processes at some reference potential together with those of Cd(II)-HEDTA complexes fulfil the linear free energy relationship.     

Pyrene-azacrown ether hybrid: cation-π interaction

A pyrene-modified lariat ether was synthesized in order to observe the cation-π interaction in solution.     

Stereoselective synthesis of (25R)-25,26-dihydroxy-23-oxovitamin D3 and its enzymatic conversion to (25R)-1α,25,26-trihydroxy-23-oxovitamin D3, a putative metabolite of vitamin D3

(25$\text{R}$)-25,26-Dihydroxy-23-oxovitamin D₃ was synthesized efficiently and stereoselectively, and it was converted enzymatically to (25$\text{R}$)-1α,25,26-trihydroxy-23-oxovitamin D₃, a putative metabolite of 1α,25-dihydroxyvitamin D₃. The spectral and chemical properties of (25$\text{R}$)-25,26-dihydroxy-23-oxovitamin D₃ and its 1α,hydroxylated derivative disagree with those reported for the isolated metabolite.     

Vascular dilatory action of the Chinese crude drug. II. Effects of scoparone on calcium mobilization

Further experiments were conducted to examine the effect of scoparone (6,7-dimethoxycoumarin), a coumarin derivative found in the Chinese crude drug "Capillaris Flos," on calcium mobilization. Scoparone does not affect Ca2+ influx through the voltage-dependent channel due to membrane depolarization. Its inhibitory action may be dependent not only on the inhibition of Ca2+ release from sarcoplasmic reticulum but also on the inhibition of extracellular Ca2+ influx through the receptor-operated channel.     

The proper structure of cubic ice confined in mesopores

To examine the reason for the formation and the structure of cubic ice in a restricted space, we measured the powder x-ray diffraction patterns of cubic ice formed within the mesopores of porous silicas as a function of pore size (4-70 nm). The results strongly suggest that cubic ice formed in the mesopores does not take a cubic structure as envisaged by Konig. It may be actually composed of very small crystallites of hexagonal ice that contains a large amount of growth faults depending on the crystallite size, that is, ice with disordered stacking sequence. Suppression of crystal growth of ice in the mesopores seems to be a vital factor for the formation and the stability of cubic ice.