Up to isometries,a deformation is a continuous function of its metric tensorAux isométries près,une déformation est une fonction continue de son tenseur métrique

If the Riemann–Christoffel tensor associated with a field of class $\text{C}{}^2$ of positive definite symmetric matrices of order three vanishes in a connected and simply connected open subset $\text{Ω?}\text{R}{}^3$, then this field is the metric tensor field associated with a deformation of class $\text{C}{}^3$ of the set $\text{Ω}$, uniquely determined up to isometries of $\text{R}{}^3$. We establish here that the mapping defined in this fashion is continuous, for ad hoc metrizable topologies. To cite this article: P.G. Ciarlet, F. Laurent, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 489–493.     

Multiple Coverings of the Plane with Triangles

We prove that any 43-fold covering of the plane with translates of a triangle can be decomposed into two coverings.     

The remainder in Weyl's law for random two-dimensional

We prove that the average order of the remainder in counting the number of points of a random lattice inside a disc of radius . Our proof is spectral in nature. Submitted: July 2001, Revised: August 2001, Revised: March 2002.     

A Law of Large Numbers for the Zeroes of Heine–Stieltjes Polynomials

We determine the limiting density of the zeroes of Heine–Stieltjes polynomials (or of any set of points satisfying the conclusion of Heine–Stieltjes Theorem) in the thermodynamic limit and use this to prove a strong law of large numbers for the zeroes.     

Synthesis and Odor of Chiral Partial Structures of Khusimone. Part 1

Khusimone (1), one of the main odor-donating compounds of vetiver oil is subject of the following study on structure/odor relationship. The omittance of the ethano bridge of the tricyclic khusimone leads to a bicyclic system. The stereoselective approach to this degraded structure is described, and the olfactory properties are studied. The key step of the synthesis of the hydrindane nucleus is based on a highly diastereoselective conjugate addition to a chiral oxo-cyclopentene-2-carboxylate.     

A new approach to phosphoserine and phosphothreonine analysis in peptides and proteins: chemical modification,enrichment via solid-phase reversible binding,and analysis by mass spectrometry

beta-Elimination of the phosphate group on phosphoserine and phosphothreonine residues and addition of an alkyldithiol is a useful tool for analysis of the phosphorylation states of proteins and peptides. We have explored the influence of several conditions on the efficiency of this PO(4)(3-) elimination reaction upon addition of propanedithiol. In addition to the described influence of different bases, the solvent composition was also found to have a major effect on the yield of the reaction. In particular, an increase in the percentage of DMSO enhances the conversion rate, whereas a higher amount of protic polar solvents, such as water or isopropanol, induces the opposite effect. We have also developed a protocol for enrichment of the modified peptides, which is based on solid-phase covalent capture/release with a dithiopyridino-resin. The procedure for beta-elimination and isolation of phosphorylated peptides by solid-phase capture/release was developed with commercially available alpha-casein. Enriched peptide fragments were characterized by MALDI-TOF mass spectrometric analysis before and after alkylation with iodoacetamide, which allowed rapid confirmation of the purposely introduced thiol moiety. Sensitivity studies, carried out in order to determine the detection limit, demonstrated that samples could be detected even in the low picomolar range by mass spectrometry. The developed solid-phase enrichment procedure based on reversible covalent binding of the modified peptides is more effective and significantly simpler than methods based on the interaction between biotin and avidin, which require additional steps such as tagging the modified peptides and work-up of the samples prior to the affinity capture step.     

Infrared spectra of M+NO 3 − contact ion pairs in aprotic solvents and matrix isolated in the corresponding glasses

Studies of the vibrational spectra of matrix-isolated M⁺NO ₃ ^– ion pairs have been extended to glassy aprotic solvents. The deuterated form of the solvents DMSO, THF, and ACN have windows through the 7- nitrate ionv ₃(e) mode infrared region, so it was possible to clearly observe the splitting of the degeneracy of this mode,v ₃, produced by the contacting, but solvated, alkali metal cation. Primary attention has been directed to the extent to which this splitting is reduced relative to the argon matrix values. This reduction, which reflects electron-density transfer from the solvating molecules to the ion pairs, is comparable to that observed for H₂O and NH₃ matrices as the splitting is reduced to 20–35% of the argon-matrix values. The extent of reduction ofv ₃ for the different solvents has been related to Gutmann's donicity number scale with the correlation holding well for solvent molecules of comparable size, DMSO, THF and DMF, but breaking down for the smaller linear ACN, apparently because of more molecules in the cation solvation sphere. The matrix data have also been used, through comparison with spectra for saturated liquid solutions of Li⁺NO ₃ ^– , to show that the contact ion pair is the dominant species in liquid THF and ACN, whereas the ions are largely solvent separated in DMSO.     

Fibre optic fluorometric enzyme sensors for hydrogen peroxide and lactate,based on horseradish peroxidase and lactate oxidase

An optical biosensor for the determination of hydrogen peroxide based on immobilized horseradish peroxidase is described. The fluorescence of the dimeric product of the enzyme catalysed oxidation of homovanillic acid is utilized to determine the concentration of H₂O₂. The membrane-bound enzyme is attached to a bifurcated fibre bundle permitting excitation and detection of the fluorescence by a fluorometer. The response of the sensor is linear from 1 to 130 M hydrogen peroxide; the coefficient of variation is 3%. The sensor is stable for more than 10 weeks. The operating pH for maximal sensor response is 8.15. This allows the sensor to be used in combination with oxidase reactions producing hydrogen peroxide, as is demonstrated with a co-immobilized lactate oxidase-horseradish peroxidase optode for the determination of L-lactate. The fluorescence intensity of this sensor depends linearly on the concentration of lactate between 3 and 200 M and a throughput of 10 samples per hour is possible. The precision is in the same range as that of the monoenzyme optode. The lifetime of the bienzyme sensor for lactate is considerably shorter than that of the peroxidase sensor; it is limited by the stability of the immobilized lactate oxidase enzyme. The sensor has been applied to the determination of lactate in control serum.     

4,4-Disubstituierte Imidazol-Derivate aus der Umsetzung von 3-Amino-2H-azirinen mit Salicylamid

4,4-Disubstituted Imidazole Derivatives from the Reaction of 3-Amino-2H-azirines with Salicylamide Reaction of 3-amino-2H-azirines 1a–c with salicylamide ( 7 ) in MeCN leads to imidazoles 10 and 11 in different rates, depending on the conditions. In the case of 1a and 1b, 11a and 11b , respectively, have been obtained as the main product at 50°; in reactions at 80°, 10a and 10b are the favored products (Tables 1 and 2). 2,2-Dimethyl-3-(N-methyl-N-phenylamino)-2H-azirine ( 1c ) reacts with 7 in MeCN mainly to 2-(2-hydroxyphenyl)-5,5-dimethyl-3,5-dihydroimidazol-4-one ( 10a ); in boiling toluene, 11c is formed with low preference (Table 3). The structure of the products has been established by spectroscopic means, and in the case of 10b and 11c , by X-ray crystallography. Two different reaction mechanisms for the formation of the products are discussed (Scheme 2).