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     

Theoretical study of single-electron capture in He2+ —H− collision

We present a detailed theoretical treatment of single-electron transfer between He²⁺ and H^?. The total cross section is calculated using stationary molecular states which are appropriate in the energy range covered by the experiments (between 0.5 and 2250 eV in the centre of mass frame). We use an expansion on a two-electron basis built with one-electron diatomic molecule (OEDM) orbitals and including the common translation factor of Errea et al. All coupling terms are calculated explicitly. Because of the small binding energy of H^? compared to that of the ground state of He⁺, capture occurs into highly excited states of He⁺. Results obtained with a straight-line quasiclassical calculation are in good agreement with the experimental data. At low energy, He⁺ (n=5) +H(1s) is the dominant capture channel; at higher energy, the He⁺ (n=4) + H(1s) channel becomes important. The rise in the cross section below 6 eV can be attributed to the Coulomb attraction in the incoming channel. To account for this effect, a fully quantal calculation has been performed. The agreement with the low-energy measurements is then excellent.     

Ni-catalyzed alkylative dimerization of vinyl grignard reagents using alkyl fluorides

Alkyl halides underwent unique cross-coupling reaction with vinylmagnesium chloride in the presence of Ni catalyst to give 2-alkyl-3-butenyl Grignard reagent (1) in high yields. This reaction proceeded efficiently at 25 degrees C in THF using primary and secondary alkyl fluorides. On the other hand, PhCH=CHMgBr gave double alkylative vinyl coupling product 4 in good yield as the sole coupling product. Alkyl fluorides react as the most suitable alkylating reagent in comparison to the corresponding chlorides, bromides, and iodides.     

Oxidative homocoupling of chiral 3-arylpropanoic acid derivatives. Application To asymmetric synthesis of lignans

The oxidative homocouplings of lithium enolates of (4S)-3-(3-arylpropanoyl)-4-isopropyl-2-oxazolidinones and (4R, 5S)-1-(3-arylpropanoyl)-3,4-dimethyl-5-phenyl-2-imidazolidinones gave the corresponding R,R-dimers stereoselectively with TiCl(4), PhI(OAc)(2), or CuCl(2) as an oxidant. The stereoselectivity can be explained by a radical coupling mechanism. Optically active dibenzylbutyrolactone lignans, such as (-)-hinokinin and (-)-dimethylmatairesinol, and dibenzylbutanediol lignans, such as (-)-dihydrocubebin and (-)-dimethylsecoisolariciresinol, were synthesized from the major R,R-dimers. The oxidative coupling of (4R, 5S)-1-(3-arylpropanoyl)-3,4-dimethyl-5-phenyl-2-imidazolidinones with LDA-I(2) gave R,S-dimers mainly, and this result can be explained by an S(N)2 mechanism.