On separating a fixed point from zero by invariants

Assume a fixed point v∈V^G can be separated from zero by a homogeneous invariant f∈𝕜[V]^G of degree p^rd, where p>0 is the characteristic of the ground field 𝕜 and p,d are coprime. We show that then v can also be separated from zero by an invariant of degree p^r, which we obtain explicitly from f. It follows that the minimal degree of a homogeneous invariant separating v from zero is a p-power.     

Moufang sets of mixed type F4

Moufang sets were introduced by Jacques Tits in order to understand isotropic linear algebraic groups of relative rank one, but the notion is more general. We describe a new class of Moufang sets, arising from so‐called mixed groups of type ${\mathsf{F}}_4$ in characteristic 2, obtained as the fixed point set under a suitable involution.     

Observation and mechanistic study of facile C-O bond formation between a well-defined aryl-copper(III) complex and oxygen nucleophiles

A well-defined macrocyclic aryl–Cu(III) complex (2) reacts readily with a variety of oxygen nucleophiles, including carboxylic acids, phenols and alcohols, under mild conditions to form the corresponding aryl esters, biaryl ethers and alkyl aryl ethers. The relationship between these reactions and catalytic C-O coupling methods is demonstrated by the reaction of the macrocyclic aryl–Br species with acetic acid and p-fluorophenol in the presence of 10 mol% Cu(I). An aryl-Cu(III)-Br species 2(Br) was observed as an intermediate in the catalytic reaction. Investigation of the stoichiometric C-O bond-forming reactions revealed nucleophile-dependent changes in the mechanism. The reaction of 2 with carboxylic acids revealed a positive correlation between the log(k(obs)) and the pK(a) of the nucleophile (less-acidic nucleophiles react more rapidly), whereas a negative correlation was observed with most phenols (more-acidic phenols react more rapidly). The latter trend resembles previous observations with nitrogen nucleophiles. With carboxylic acids and acidic phenols, UV-visible spectroscopic data support the formation of a ground-state adduct between 2 and the oxygen nucleophile. Collectively, kinetic and spectroscopic data support a unified mechanism for aryl-O coupling from the Cu(III) complex, consisting of nucleophile coordination to the Cu(III) center, deprotonation of the coordinated nucleophile, and C-O (or C-N) reductive elimination from Cu(III).     

Solvent/base effects in the selective domino synthesis of phenanthridinones that involves high-valent palladium species: experimental and theoretical studies

The domino reaction of o-bromobenzamides 1a-m in the presence of K(2)CO(3) and the [PdCl(2)(PPh(3))(2)] catalyst granted a selective access to phenanthridinones 2 or to the new 1-carboxamide phenanthridinones 3 depending on the solvent, DMF or 1,4-dioxane, respectively. Investigations of the reaction parameters provided the first example of a direct correlation between the base dissociation and the solvent polarity on the selectivity observed. Moreover, mechanistic studies (NMR spectroscopy and ESI-MS monitoring) allowed us to characterize Pd(II) palladacycle 4 and biaryl species as common intermediates for these two domino processes. On that basis, C(sp(2))-C(sp(2)) bond formation is envisaged by generation of a Pd(IV) complex after oxidative addition of 1 into Pd(II) palladacycle 4, a rationale that is supported by DFT calculations. A general catalytic cycle is proposed to account for these observations.     

Catalytic Organometallic Reactions of Ammonia

Until recently, ammonia had rarely succumbed to catalytic transformations with homogeneous catalysts, and the development of such reactions that are selective for the formation of single products under mild conditions has encountered numerous challenges. However, recently developed catalysts have allowed several classes of reactions to create products with nitrogen‐containing functional groups from ammonia. These reactions include hydroaminomethylation, reductive amination, alkylation, allylic substitution, hydroamination, and cross‐coupling. This Minireview describes examples of these processes and the factors that control catalyst activity and selectivity.     

Selectively reductive cleavage of the protected taxol side chain with sodium borohydride

During the course of our studies on the taxol chemistry,we have ever tried thereductive cleavage of the acotyl group at 20-C in compound 1a~(1,2) with the hydroxylgroup assistance at 4-C by sodium borohydride. However, the major product 2a~2(27%)still had the acetyl group at 20-C and the signal for the 10-C proton at 6.61 ppm(s)was shifted to 6.31 ppm(d, J=3.79 Hz). A new doublet signal at 3.62 ppm(d, J=3.80     

Index Reduction Formulas for Twisted Flag Varieties, II

This is a continuation of the determination begun in K-Theory 10 (1996), 517–596, of explicit index reduction formulas for function fields of twisted flag varieties of adjoint semisimple algebraic groups. We give index reduction formulas for the varieties associated to the classical simple groups of outer type A _n-1 and D _n, and the exceptional simple groups of type E ₆ and E ₇. We also give formulas for the varieties associated to transfers and direct products of algebraic groups. This allows one to compute recursively the index reduction formulas for the twisted flag varieties of any semi-simple algebraic group.     

Parameter Determination for Reductive Dechlorination of Chlorinated Solvents

The contamination of subsurface due to the chlorinated solvents such as tetrachlorethylene (PCE) and trichlorethylene (TCE) is one of the most difficult environmental problems to treat. Bioremediation has been shown by many researchers to be a remedial alternative for this type of contamination. Chlorinated solvents are not directly mineralized but rather are transformed by microorganisms into one or more intermediate compounds before converting into a final compound. These sequential reactions, termed “reductive dehalogenation”, consist of replacing a chlorine atom by a hydrogen atom. The pathway of degradation of PCE can be expressed by the following scheme PCE → TCE → DCE → VC → ETH, where dichloroethene (DCE), vinyl chloride (VC) and finally ETH is ethylene. Since the biotransformation rate coefficients of each intermediate compound are different, they have to be determined very precisely to establish an effective treatment operation. The sequential decay can be described by Michaelis–Menten’s kinetics, which constitutes a highly nonlinear system of ordinary differential equations (ODEs). This is very sensitive to the changes of biotransformation rate coefficients. In this study we introduce a methodology how to numerically estimate the rate coefficients for Michaelis–Menten’s equations from the knowledge of the concentrations of PCE, TCE, DCE, VC and ETH. The efficiency of the proposed method is demonstrated on some examples. Estimated biotransformation coefficients are employed to predict the concentrations of chlorinated solvents. Computations and measurements show a very good agreement.     

Electroreductive intramolecular coupling of aromatic δ- and ε-keto esters

Electroreduction of aromatic δ- and ε-keto esters in the presence of chlorotrimethylsilane and triethylamine gave five- and six-membered cyclized products. The products were transformed to the corresponding α-hydroxy ketones.     

Classification of two-orbit varieties

We obtain the classification of two-orbit varieties, i.e. the normal complete complex algebraic varieties on which a reductive complex algebraic group acts with two orbits. We prove also Luna's conjecture saying that these varieties are spherical, i.e. admit a dense orbit of a Borel subgroup. Received: September 1, 2000