Existence globale pour l'équation de Smoluchowski continue non homogène et comportement asymptotique des solutions

We prove global existence of solutions to the continuous nonhomogeneous Smoluchowski equation for coagulation rates satisfying a more general structure condition than the Galkin–Tupchiev monotony hypothesis considered in (Ph. Laurençot, S. Mischler, Arch. Rational Mech. Anal. 162 (1) (2002) 45–99). The Smoluchowski coagulation rate fulfils this condition as well as some rates which vanish on the diagonal. Under the condition of positivity of the coagulation rate outside of the diagonal we prove that solutions tend to 0 in the large time asymptotic. These results depend on a new estimate from below for the dissipation rate of the L^p-norm, p>1. To cite this article: S. Mischler, M. Rodriguez Ricard, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A note about the Gaussian property of the Brownian bridge

The Gaussian property of the Brownian bridge is characterized as an application of Ramachandran's theorem in terms of the independence of the random variables that appear in the Karhunen-Loéve expansion of the process. A reference about the construction of the Brownian bridge by means of functional transformations is also included.     

Cartesian currents,weak diffeomorphisms and existence theorems in nonlinear elasticity

Summary In this paper we introduce some new classes of functions, among these a class of weak diffeomorphisms. In these classes we prove by direct methods the existence of minimizers for several kinds of variational integrals. In particular, we prove the existence of one-to-one orientation-preserving maps that minimize suitable energies associated with hyperelastic materials. The minimizers are also proved to satisfy equilibrium equations. Finally radial deformations are discussed in connection with cavitation.     

The hydroazocine route to highly functionalized pyrrolizidines

Studies of the preparation of 1,8-dihydroazocines and transannular cyclization of hydroazocines to produce functionalized pyrrolizidines are described. Results are presented which demonstrate that unsymmetrically substituted acetylenes bearing at least one electron withdrawing groups undergo efficient cycloaddition to 1 - β - styryl - 1,2 - dihydropyridine producing in a regio-selective fashion 3,4 - disubstituted - 1,8 - dihydroazocines. The dihydroazocines generated in this manner can be converted to 1 - formyl - Δ^4,5 - epoxyazocines which undergo interesting rearrangement reactions to form pyrrolizidines when subjected to methoxide deformylation followed by acid treatment. In addition, 1,6,7,8 - tetrahydroazocines can be converted to pyrrolizidines under bromination conditions. The intriguing chemical process which occur under the conditions outlined above are described.     

The di-π-methane rearrangement : The effect of 1,4-diene structure and conformation on the reaction efficiency and stereochemistry

The photochemistry of the stereoisomeric 1,3-dimethyl-3(2-phenylethenyl)cyclohexenes has been explored Direct irradiation of the cis- and trans-β-styrylcyclohexenes leads to di-π-methane rearrangement, producing the endo- and exo-3,7-dimethyl-8-phenybicyclo[5.1.0]oct-2-enes, and cis -trans isomerization, interconverting the 1,4-diene containing substrates. Triplet sensitized photolysis of both substituted cyclohexenes leads exclusively to cis-trans isomerization. Results from low conversion direct irradiations of the cis- and trans-β-styrylcyclohexenes indicate that the singlet rearrangements are stereospecific, and lead to formation of the 8-exo and 8-endo-phenylbicyclic octenes, respectively. The relationship between di-π-methane structure and triplet reaction efficiency, and the effect of conformation on the rearrangement stereochemistry are discussed.     

Vilsmeier formylation of 5,10,15-triphenylcorrole: expected and unusual products

5,10,15-Triphenylcorrole (1) reacts with the Vilsmeier reagent (POCl(3)/DMF) to give the corresponding 3-formyl derivative 3 as the major product. The regioselectivity of the reaction was proven by X-ray crystallography and only traces of the 2-formyl isomer were observed. A more polar product is also observed and this compound becomes the major product when an excess of DMF is used for the preparation of the Vilsmeier reagent, while the formation of the 3-formyl isomer is almost completely suppressed. X-ray crystallography allowed us to identify this compound as the fully substituted N-ethane bridged derivative 4, formed from the attack of the Vilsmeier reagent at the inner core of the macrocycle. This compound is unique among porphyrinoid macrocycles, and further confirms the peculiarity of corrole chemistry.     

Preparation and self-assembly of chiral porphyrin diads on the gold electrodes of quartz crystal microbalances: a novel potential approach to the development of enantioselective chemical sensors

Porphyrin diad 1 was synthesized by reaction of the acyl chloride of porphyrin 2 and trans-1,2-dithiane-4,5-diol. The Co complex of this diad was studied as a potential enantioselective receptor for chiral recognition in solution and in the solid state. In solution both enantiomers of limonene induce significant changes in the visible and circular dichroism (CD) spectra of [Co2(1)], while a different behavior is observed in the case of the enantiomeric pair of trans-1,2-diaminocyclohexane. A different efficiency of [Co2(1)] chiral recognition is obtained for these compounds, with a remarkable degree of enantiodiscrimination observed in the case of limonene. Self-assembled monolayers of [Co2(1)] were deposited onto the gold electrodes of quartz crystal microbalances to be used as sensing materials of nanogravimetric sensors operating in the gas phase. The enantiodiscrimination properties of these sensors towards the enantiomeric pairs of chiral analytes have been studied. While in the case of analytes bearing donor ligand atoms we did not observe a remarkable enantioselectivity, a significant degree of chiral discrimination was observed in the case of limonene; this result is particularly encouraging for the potential development of enantioselective chemical sensors for use in an array configuration.     

Exploratory Investigations Probing a Preparatively Versatile, Pyridinium Salt Photoelectrocyclization-Solvolytic Aziridine Ring Opening Sequence

A novel pyridinium salt photoelectrocyclization-nucleophilic bicyclic aziridine ring opening reaction sequence has been investigated in order to determine its preparative potential. N-Alkylpyridinium perchlorates were found to undergo photoinduced electrocyclization upon irradiation in nucleophilic solvents, such as H(2)O and MeOH, to efficiently produce 6-alkyl-6-azabicyclo[3.1.0]hex-2-en-4-yl alcohols and ethers. The bicyclic aziridine photoproducts react with a number of different nucleophiles (e.g., H(2)O, MeOH, AcOH, AcSH) under acid-catalyzed conditions to produce 5-(nucleophile-substituted)-4-(alkylamino)cyclopenten-3-yl alcohols and ethers. The aziridine ring opening processes are both regioselective and stereoselective, yielding trans,trans-trisubstituted cyclopentenes exclusively, apparently as a consequence of the operation of an SN(2) mechanism. The effects of C-alkyl substitution on the regiochemistry of the pyridinium cation photocyclization reaction were briefly probed, and a method was developed to produce trans,cis-trisubstituted cyclopentenes by use of this tandem preparative sequence.     

Solidification of metallic tin in dispersed phase

Differential scanning calorimetry (DSC) and particle size measurements were carried out on disproportionation products of pure SnO to investigate the fusion and solidification behaviour of Sn droplets and their catalytic nucleation on Sn oxides. If disproportionation reaction takes place at T ≥ 798 K, the products are metallic Sn and SnO₂; but for 523 < T < 798 K, SnO₂ is replaced by an intermediate oxide (IO) Sn_xO_(1+x). On melting, samples with IO show a drop of melting point of metallic tin due to Gibbs–Thomson effect; no lowering of melting point was observed in samples with SnO₂. On the other hand, if solidification occurs in the presence of IO, Tin droplets always displayed three distinct exothermic solidification peaks, but if it takes place in the presence of SnO₂, only one exothermic peak is observed. Undercooling values and contact angles were determined for each of the heterogeneous nucleation processes. The different behaviour of metallic Tin droplets was related to the different lattice symmetry of SnO₂ and IO, which act as nucleation catalysts.