A sharp inequality for Sobolev functionsUne inégalité dans un espace de Sobolev

Let N?5, a>0, $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$, $\text{2}{}^1\text{=}\text{2N}\text{N?2}$, $\text{2}{}^{\mathrm{\#}}\text{=}\text{2(N?1)}\text{N?2}$ and 6u6²=|?u|₂²+a|u|₂². We prove there exists an α₀>0 such that, for all $\text{u∈}\text{H}{}^1\text{(Ω)?{0}}$,

This inequality implies Cherrier's inequality. To cite this article: P.M. Girão, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 105–108     

Novel acid-catalyzed rearrangement of tetrahydro-1,2,3,4-tetrazines: unexpected formation of glycosazones

The present contribution discloses a simple and unexpected acid-catalyzed cleavage of tetrahydrotetrazines leading to 1,2-bis(hydrazones). Incorporation of a chiral fragment derived from carbohydrates enables the rapid preparation of glycosazones, a family of compounds employed by Emil Fischer to elucidate the configuration of sugars. In addition, a mechanistic proposal accounts for experimental observations.     

Reactions of Metal Salts and Alkali Metal Nitrates. Role of the Metal Precursors and Alkali Metal Ions in the Resulting Phase of Zirconia

Zirconia powders are prepared by reaction of a zirconium precursor with an alkali metal nitrate. The major part of the reactions takes place before the melting points and then the reactions go slowly to completion at 450°C in the molten salts. The roles of the precursor and the alkali metal ion are discussed considering the reaction between two precursors, octahydrated zirconium oxychloride and zirconium tetrachloride, and two nitrates, LiNO₃ and NaNO₃, and some resulting physico-chemical differences. The obtained zirconia powders contain very small amounts of alkali metal ions which act as stabilizing agent. Their effect on the balance tetragonal-monoclinic ZrO₂ depends upon the homogeneity of their distribution which is related to their ability to diffuse inside the bulk of particles and their polarizing power when located mainly on the surface.     

Intramolecular dealkylation of chelating diamines with Ru(II) complexes

N,N, N',N'-Tetraethylethylenediamine undergoes simple or double intramolecular dealkylation reactions in the presence of the complexes [RuCl2(diene)]n (diene = cod, nbd) or trans-[RuCl2(diene)(morfoline)2 at 80 degrees C to afford chelating amine ligands which contains one or two N-H functionalities.     

Copper(I)-homoscorpionate catalysts for the preferential, kinetically controlled cis cyclopropanation of alpha-olefins with ethyl diazoacetate

In situ prepared copper catalysts Tp(X)Cu (Tp(X) = homoscorpionate) catalyze the olefin cyclopropanation reaction using ethyl diazoacetate as the carbene source. Very high values of both activity and diastereoselectivity toward the cis isomer have been obtained for styrene, alpha-methylstyrene, 1-hexene, 1-octene, vinyl acetate, n-butyl vinyl ether, 2,5-dimethyl-2,4-hexadiene, and 3,3-dimethyl-1-butene. The effect of the temperature in the diastereoselectivity was almost negligible within the range -10 to +30 degrees C. Kinetic studies have allowed us to propose that the homoscorpionate ligand might act in a dihapto form during the catalytic process. This transformation seems to operate under kinetic control, where the formation of the cis isomer would govern the reaction rate.     

Intermolecular copper-catalyzed carbon [bond] hydrogen bond activation via carbene insertion

A series of catalysts of general formula TpXCu (TpX = homoscorpionate ligand) promote the insertion of :CHCO2Et (ethyl diazoacetate as the carbene source) into the C-H bonds of cycloalkanes and cyclic ethers in moderate to high yield. A correlation between the steric hindrance of these catalysts and the yield of the transformation has been observed.     

Granular segregation as a critical phenomenon

We present the results of an experimental study of patterned segregation in a horizontally shaken shallow layer of a binary mixture of dry particles. An order parameter for the segregated domains is defined and the effect of the variation of the combined filling fraction, C, of the mixture on the observed pattern formation is systematically studied. We find that there is a critical event associated with the onset of segregation, at C(c)=0.647+/-0.049, which has the characteristics of a continuous phase transition, including critical slowing down.     

Design and synthesis of new bicyclic diketopiperazines as scaffolds for receptor probes of structurally diverse functionality

Diketopiperazines (DKPs) are a common motif in various biologically active natural products, and hence they may be useful scaffolds for the rational design of receptor probes and therapeutic agents. We constructed a new bicyclic scaffold that combines a DKP bridged with a 10-membered ring. In this way we obtained a three-dimensional molecular skeleton, with several amendable sites that provide a starting point to design a new combinatorial library having diverse substituent groups. Structural variation is based upon the flexibility of alkylation of the nitrogen atoms of the DKP and on the side-chain olefin. We obtained a 10-membered secondary ring through a ring-closure metathesis reaction using the second generation Grubbs catalyst. Rings containing both O-ethers and S-ethers were compared. N-Alkyl or arylalkyl groups were introduced optionally at the two Nalpha-atoms. This is a general scheme that will allow us to test rings of varying sizes, linkages, and stereochemical parameters. The DKP derivatives were tested for activity in astrocytoma cells expressing receptors coupled to phospholipase C. Inhibitory effects were observed for signaling elicited by activation of human nucleotide P2Y receptors but not m3 muscarinic receptors. Compound 20 selectively inhibited calcium mobilization (IC50 value of 486 +/- 16 nM) and phosphoinositide turnover elicited by a selective P2Y1 receptor agonist, but this compound did not compete for binding of a radiolabeled nucleotide-competitive receptor antagonist. Therefore, the new class of DKP derivatives shows utility as pharmacological tools for P2Y receptors.     

New chromogenic probes into nanoscopic pockets in enhanced sensing protocols for amines in aqueous environments

[structure: see text] A new sensory MCM-41-based material for the colorimetric sensing and discrimination of amines has been developed.     

From labdanes to drimanes. Degradation of the side chain of dihydrozamoranic acid

A new route for the degradation of the saturated side chain of dihydrozamoranic acid has been devised, giving an advanced intermediate, compound 14, useful for the synthesis of insect antifeedants such as warburganal and polygodial.