Codescente en K-théorie étale et corps de nombres

In this paper, we give a codescent criterion for the higher tame kernelK _2i ^−2/ét O _E for a Galoisp-extensionE/F of algebraic number fields (p odd). As an application, we give fori∈ℤ a “going-up” theorem for certain property called (p, i)-regularity, which allows us in particular to construct examples of number fields verifying “twisted” Leopoldt conjectures.      

Stereoselective synthesis and Diels-Alder reactions of bis-Ether 1,3 Dienes. A further test for cooperativity during cycloaddition reactions.

A stereoselective synthesis of dienes $\text{1a}$ and $\text{1b}$, using butadiene-iron tricarbonyl complexes, is described. Higher diastereoselectivity is observed during Diels-Alder reaction of $\text{1a}$ as compared with $\text{1c}$, affording good evidence for cooperativity in these cycloadditions.     

Rearrangement of a cyclohexyl radical to a cyclopentylmethyl radical on the avermectin skeleton

The rearrangement of a substituted cyclohexyl radical to a cyclopentylmethyl radical on the skeleton of avermectin B1 was observed experimentally and explored computationally. The Stork-Nishiyama methodology was applied to the macrocycle of interest followed by a Tamao oxidation. The expected 5-6 fused ring product was observed in minor amounts. The major product was a 5-5 fused ring resulting from apparent conversion of the initially formed cyclohexyl radical to a cyclopentylmethyl radical. Preliminary computational results indicate that substituents in the macrocycle induce the rearrangement.     

Substituent effects on the rearrangements of cyclohexyl to cyclopentyl radicals involving avermectin-related radicals

The rearrangement of a substituted cyclohexyl radical to a cyclopentylmethyl radical on the skeleton of avermectin B1 has been investigated using density functional (UB3LYP/6-31G(d)) and G3MP2B3 computational methods. The rearrangement is preferred when highly radical stabilizing groups are present at the 2- and 3-positions of the cyclohexyl radical. A substituent on the 3-position of the cyclohexyl radical enables ring-cleavage of the cyclohexyl radical, while a radical stabilizing substituent on the 2-position of the cyclohexyl radical stabilizes the final cyclopentylmethyl radical, enabling the overall rearrangement and reversing the normal thermodynamic preference for the hexenyl radical ring closure.     

Strontium dicobalt orthophosphate

In crystalline SrCo₂(PO₄)₂; triclinicP¯1,M _r =395.4,a=5.014(2),b=8.639(4),c=9.691(1)Å,a=118.04(3),=75.09(4), =86.90(4)°,V=350.4(3)ų,Z=2,D _calc=3.747 g cm^–3, MoK=0.71069 Å,=12.49 mm^–1,F(000)=372,T=298 K,R=4.1% for 1190 obs. reflections; Co²⁺ and PO ₄ ^3– ions are found on planes parallel to (001). Co²⁺ occupies sites of four- and five-(trigonal bipyramidal) coordination with average Co-O distances, 1.948(8) and 2.087(8) Å, respectively. The blue color of solid is due to the tetrahedral coordination geometry of the four-coordinate copper atoms.     

New method based on 1-(trimethysilyl)alk-1-yne to prepare 1,4-skipped diynes

[reaction: see text] We have developed a novel reaction between a terminal TMS-alkyne and a propargyl halide in the presence of a fluoride source and a catalytic amount of copper iodide to prepare 1,4-skipped diynes with good yields and in mild conditions. We have shown that this reaction also works very well with germanium and tin derivatives as an alternative to silicon. This new method can be useful for the synthesis of polyunsaturated fatty acids.     

Sur la constante de Kummer-Leopoldt d’un corps de nombres

Kummers lemma for the cyclotomic field gives sufficient local conditions for a unit to be a global p-th power, and can thus be viewed as an early version of Leopoldts conjecture. To study generalizations and converse statements, we introduce the Kummer-Leopoldt constant (F) of a number field F and compute it in Iwasawa theoretic terms. The vanishing of (F) gives the desired generalization and converse of Kummers lemma.Mathematics Subject Classification(2000): Primary 11R23     

Computation of normal forms of Hamiltonian systems in the presence of Poisson commuting integrals

The aim of this paper is to show the role of first integrals in further reducing the normal form unfolding of Hamiltonian systems. Based on a work by Cicogna and Gaeta, the joint normal form approach for Hamiltonian vector fields is considered. This normal form procedure, couched in a Lie-Poincaré scheme, allows us to see that we can reduce simultaneously the Hamiltonian together with its Poisson commuting integrals to a simplified normal form - a joint normal form - which is given a simple characterization. In this algorithmic procedure, approximate first integrals can be constructed (and used to simplify the normal form) at the same time that we bring the Hamiltonian to normal form. Further, following Walcher, we show that we can derive the joint normal form via a structure preserving transformation (in a sense to be specified). The approach is discussed from an implementational standpoint and illustrated by a Liouville-integrable Hénon-Heiles system. This revised version was published online in June 2006 with corrections to the Cover Date.     

A facile method for the solution and solid-phase synthesis of substituted [3.3.1] bicycles

[reaction: see text] Interest in bicyclic natural products from the Guttiferae classification has led to the development of an improved method for the selenium-mediated cyclization of alkenyl-substituted beta-dicarbonyls (I) to form a variety of bicyclo[3.3.1]nonan-9-ones (II) both in solution and on solid support.