A metric approach to asymptotic analysis

We introduce a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. We relate this notion to a concept of “firm” asymptotic function. We use these notions to study boundedness properties which can be applied to continuity questions for some operations on sets and functions. Such questions arise in stability analysis of Hamilton-Jacobi equations. We present some other applications such as an extension of a theorem of Dieudonné and existence results in optimization and fixed point theory.     

Evolution equations for shapes and geometries

The first object of this paper is to introduce a new evolution equation for the characteristic function of the boundary Γ of a Lipschitzian domain Ω in the N-dimensional Euclidean space under the influence of a smooth time-dependent velocity field. The originality of this equation is that the evolution takes place in an L^p-space with respect to the (N − 1)-Hausdorff measure. A second more speculative objective is to discuss how that equation can be relaxed to rougher velocity fields via some weak formulation. A candidate is presented and some of the technical difficulties and open issues are discussed. Continuity results in several metric topologies are also presented. The paper also specializes the results on the evolution of the oriented distance function to initial sets with zero N-dimensional Lebesgue measure.     

Solutions rationnelles de certaines équations fonctionnelles

Dans cet article, nous démontrons essentiellement les deux résultats suivants, qui montrent que les solutions séries formelles à coefficients dans de certaines équations fonctionnelles sont rationnelles. Soient tout d'abords un entier naturel non nul, eta _i ,b _i ,(i = 1, , s), 2s nombres complexes, lesa _i étant non nuls. On définit l'ensembleA comme étant l'intersection des parties de , contenant l'origine et stables par toutes les applicationsg _i (x) = a _i x + b _i . On a alors le résultat suivant: Théorème 1.Soient f, R ₁, ,R _s s + 1 fractions rationnelles de (x), régulières à l'origine, et a_i, b_i (i = 1,, s), 2s éléments de . On suppose que les a_i sont non nuls et de module strictement inférieur à un pour tout i = 1,, s. Soit y(x) un élément de [[x]], vérifiant l'équation fonctionnelle

Photochemical expulsion of the neutral monodentate ligand L in Ru(terpy*)(diimine)(L)2+: a dramatic effect of the steric properties of the spectator diimine ligand

A series of photoreactive complexes of the type Ru(terpy*)(N-N)(L)(2+), where terpy* is 4'-(3,5-ditertiobutylphenyl)-2,2':6',2' '-terpyridine, N-N is the bidentate chelate phen or dmp (phen = 1,10-phenanthroline, dmp = 2,9-dimethyl-1,10-phenanthroline), and L is the monodentate ligand dms, MeBN, or MeOBN (dms = dimethyl sulfide, MeBN = 2,6-dimethyl benzonitrile, MeOBN = 2,6-dimethoxybenzonitrile), has been synthesized and fully characterized by proton NMR spectroscopy, electrospray mass spectrometry, and UV-vis spectroscopy. The X-ray structures of four complexes were also obtained. In neat pyridine, the quantum yields for the photosubsitution of L by pyridine were measured and showed dramatic variations depending on the steric interactions between the spectator bidentate ligand and the leaving monodentate ligand L. The use of dmp instead of phen multiplied the photosubstitution efficiency by a factor of 20-50, depending on L. This effect could be qualitatively correlated to the distortions observed in the X-ray structures of the corresponding complexes. The highly distorted structure of Ru(terpy)(dmp)(dms)(PF(6))(2) showed a very high photosubsitution quantum yield phi = 0.36 in neat pyridine. The high photoreactivity of some of the compounds makes them particularly promising as components of future light-driven molecular machines.     

AB initio study of the regular polyhedral molecules N4, P4, As4, N8, P8 and As8

Ab initio pseudopotential SCF calculations were performed on tetrahedral X₄ molecules using double-zeta basis sets with and without d functions. The inclusion of d orbitals shortens the bond lengths, stabilizes the X₄ structures and intensifies the electron density inside the tetrahedron. The cubic X₈ molecules, calculated without d AOs, are not predicted to be more stable than 2X₄. Repulsions between parallel bonds in X₈ may compensate the lack of ring strain.     

Enzymatic synthesis of neopentylpolyol esters in organic media

Utilization of lipases for synthesis of esters of hydrophilic polyols has been investigated. The choice of a suitable solvent is crucial in this type of reaction. An interesting case is fatty acid esters from neopentylpolyols, such as trimethylolpropane, which are of great interest as high temperature lubricants. Enzymatic synthesis of trimethylolpropane tricaprylate was studied as an alternative to chemical manufacturing. Triester production occurred only if the water produced by esterification was continuously removed from the medium. In these condition, kinetics of appearance and transformation of mono-, di- and triesters were determined in order to define optimal conditions.     

DEUSS: a perdeuterated poly(oxyethylene)-based resin for improving HRMAS NMR studies of solid-supported molecules

A novel resin called DEUSS (perdeuterated poly(oxyethylene)-based solid support) has been prepared by anionic polymerization of deuterated [D4]ethylene oxide, followed by cross-linking with deuterated epichlorohydrin. DEUSS can be suspended in a wide range of solvents including organic and aqueous solutions, in which it displays a high swelling capacity. As measured by proton HRMAS of the swollen polymer, the signal intensity of the oxyethylene protons is reduced by a factor of 110 relative to the corresponding nondeuterated poly(oxyethylene)poly(oxypropylene) (POEPOP) resin, thus facilitating detailed HRMAS NMR studies of covalently linked molecules. This 1H NMR invisible matrix was used for the solid-phase synthesis of peptides, oligoureas, and a series of amides as well as their characterization by HRMAS NMR spectroscopy. On-bead NMR spectra of high quality and with resolution comparable to that of liquid samples were obtained and readily interpreted. The complete absence of the parasite resin signals will be of great advantage, for example, for the optimization of multistep solid-phase stereoselective reactions, and for the conformational study of resin-bound molecules in a large variety of solvents.     

Action de CF3 of sur des aziridines N-non substitutees

Aziridines $\text{I}$ react with trifluoromethyl hypofluorite at ? 40°C to produce mixtures of 1-(aziridine)carbonyl fluoride $\text{II}$ and 1-fluoroaziridine $\text{III}$, the proportions of which depend on steric effects. Several compounds $\text{II}$ react with starting materials to give 1, 1′-(carbonyl)bisaziridines $\text{IV}$. Most compounds $\text{II}$ and all compounds $\text{IV}$ are isolated. Chemical properties and ir and nmr data of $\text{II}$, $\text{III}$, $\text{IV}$ are described; the formation of an isocyanate and a substituted urea from a compound $\text{II}$ is observed. A mechanism for this reaction is suggested.     

Chiral separation of the four stereoisomers of a novel antianginal agent using a dual cyclodextrin system in capillary electrophoresis

Reported here is an analytical method enabling the stereochemical resolution of a new antianginal compound possessing two stereogenic centers, leading to four stereoisomers. Only one of these isomers is currently under development as a novel antianginal agent and consequently, the other three isomers are considered as unwanted chiral impurities. Therefore, an enantioselective method is required in order to check its enantiomeric purity. This paper presents a method exploiting the high efficiency of capillary electrophoresis and the complexing properties of cyclodextrins to achieve the separation of the four stereoisomers of this weakly basic compound (pKa = 7.4). For this purpose, the combination of a neutral cyclodextrin, hydroxypropyl-gamma-cyclodextrin (HP-gamma-CD), and an anionic cyclodextrin, carboxymethyl-beta-cyclodextrin (CM-beta-CD), was added to the separation buffer running in an uncoated silica capillary. After selection of the suitable cyclodextrin system, satisfactorily separation conditions were as follows: 30 mM phosphate buffer (pH 6.4) containing 10 mM of HP-gamma-CD and 10 mM of CM-beta-CD, running voltage +30 kV. The resulting run time and resolutions were respectively about 17 min and between 1.95 and 2.84. Linearity curves (0.993 < r2 < 0.998) are also shown.     

Macrocyclization on the fullerene core: Direct regio- and diastereoselective multi-functionalization of [60]fullerene,and synthesis of fullerene-dendrimer derivatives

The macrocyclization between buckminsterfullerene, C₆₀, and bis-malonate derivatives in double Bingel reaction provides a versatile and simple method for the preparation of covalent bis-adducts of C₆₀ with high regio- and diastereoselectivity. A combination of spectral analysis, stereochemical considerations, and X-ray crystallography (Fig. 2) revealed that out of the possible in-in, in-out, and out-out stereoisomers, the reaction of bis-malonates linked by o-, m-, or p-xylylene tethers afforded only the out-out ones (Scheme 1). In contrast, the use of larger tethers derived from 1,10-phenanthroline also provided a first example, (±)- 19 (Scheme 2), of an in-out product. Starting from optically pure bis-malonate derivatives, the new bis-functionalization method permitted the diastereoselective preparation of optically active fullerene derivatives (Schemes 4 and 5) and, ultimately, the enantioselective preparation (enantiomeric excess ee > 97%) of optically active cis-3 bis-adducts whose chirality results exclusively from the addition pattern (Fig. 6). The macrocyclic fixation of a bis-malonate with an optically active, 9,9′-spirobi[9H-fluorene]-derived tether to C₆₀ under generation of 24 and ent- 24 with an achiral addition pattern (Scheme 4) was found to induce dramatic changes in the chiroptical properties of the tether chromophore such as strong enhancement and reversal of sign of the Cotton effects in the circular dichroism (CD) spectra (Figs. 4 and 5). By the same method, the functionafized bis-adducts 50 and 51 (Schemes 10 and 11) were prepared as initiator cores for the synthesis of the fullerene dendrimers 62 , 63 , and 66 (Schemes 12 and 13) by convergent growth. Finally, the new methodology was extended, to the regio- and diastereoselective construction of higher cyclopropanated adducts. Starting from mono-adduct 71 , a clipping reaction provided exclusively the all-cis-2 tris-adduct (±)- 72 (Scheme 14), whereas the similar reaction of bis-adduct 76 afforded the all-cis-2 tetrakis-adduct 77 (Scheme 15). Electrochemical investigations by steady-state voltammetry (Table 2) in CH₂Cl₂ (+0.1M Bu₄NPF₆) showed that all macroeyciic bis(methano)fullerenes underwent multiple reduction steps, and that regioisomerism was not much influencing the redox potentials, All cis-2 bis-adducts gave an instable dianion which decomposed during the electrochemical reduction. In CH₂Cl₂, the redox potential of the fullerene core in dendrimers 62, 63 , and 66 is not affected by differences in size and density of the surrounding poly(ether-amide) dendrons. The all-cis-2 tris- and tetrakis(meihano)fullercnes (±) -72 and 77 , respectively, are reduced at more negative potential than previously reported all-e tris- and tetrakis-adducts with methane bridges that are also located along an equatorial belt. This indicates a larger perturbation of the original fullerene π-chromophore and a larger raise in LUMO energy in the former derivatives.