À propos de l’indice de composition des nombres

 We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,

Cold collision frequency shifts in a 87Rb atomic fountain

We present measurements of cavity frequency pulling and collisional frequency shifts in a 87Rb fountain with a frequency resolution of 3x10(-16). Agreement with theory is found for the cavity pulling and the measured collisional shifts. The clock shift is found at least 50 times smaller than in 133Cs.     

Crystallization and stereocomplexation behavior of poly(D‐ and L‐lactide)‐b‐poly(N,N‐dimethylamino‐2‐ethyl methacrylate) block copolymers

The thermal properties, crystallization, and morphology of amphiphilic poly(D ‐lactide)‐b‐poly(N,N‐dimethylamino‐2‐ethyl methacrylate) (PDLA‐b‐PDMAEMA) and poly (L ‐lactide)‐b‐poly(N,N‐dimethylamino‐2‐ethyl methacrylate) (PLLA‐b‐PDMAEMA) copolymers were studied and compared to those of the corresponding poly(lactide) homopolymers. Additionally, stereocomplexation of these copolymers was studied. The crystallization kinetics of the PLA blocks was retarded by the presence of the PDMAEMA block. The studied copolymers were found to be miscible in the melt and the glassy state. The Avrami theory was able to predict the entire crystallization range of the PLA isothermal overall crystallization. The melting points of PLDA/PLLA and PLA/PLA‐b‐PDMAEMA stereocomplexes were higher than those formed by copolymer mixtures. This indicates that the PDMAEMA block is influencing the stability of the stereocomplex structures. For the low molecular weight samples, the stereocomplexes particles exhibited a conventional disk‐shape structure and, for high molecular weight samples, the particles displayed unusual star‐like shape morphology. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 49: 1397–1409, 2011     

Preparation of both enantiomers of a synthon for novel nucleoside analogs by enzymatic desymmetrization of a meso-diol with a methylene cyclopropane skeleton

The enzymatic desymmetrization of methylenecyclopropane diol or its corresponding diacetate derivative, generated from a [2+1] cycloaddition between dioxepin and methylchlorocarbene, is described. After screening five commercial lipases, the two enantiomers of acetic acid 2-hydroxymethyl-3-methylene-cyclopropylmethyl ester are obtained in high yields and excellent enantioselectivities by using PFL or LPP in organic solvent. The stereostructure of the desymmetrization products was established by X-ray analysis. We also reported a new example with this non racemic chiral building block where the sign of optical rotation is dramatically solvent dependent and inverted. Using these enantiopure building blocks, a synthesis of novel nucleoside analogs is also presented.     

Synthesis and characterization of hexasubstituted azacryptands

Using an imine condensation strategy, hexaazacryptands were prepared from an aromatic triamine and various dialdehydes. The global shape of these cages can be modified by simple alkylation of the six nitrogen atoms. The cavity is thus elongated and rendered hydrophobic.     

Imaginary Kählerian Killing spinors I

We describe and to some extent characterize a new family of Kähler spin manifolds admitting non-trivial imaginary Kählerian Killing spinors.     

Resonance ionization mass spectrometry for precise measurements of isotope ratios

Resonance ionization mass spectrometry offers extremely high sensitivity and elemental selectivity in microanalysis, but the isotopic precision attainable by this technique has been limited. Measured isotope ratios are sensitive to small fluctuations in the pointing, pulse timing, and wavelength of the resonance lasers. We show that, by minimizing these fluctuations using feedback controls and by power-broadening the optical transitions, we are able to measure chromium isotope ratios with statistics-limited precision better than 1%. Small additional improvements in reproducibility come from careful shaping of the electric field in the region where atoms are photoionized and from minimizing pulse-to-pulse variations in the time-of-flight mass spectrometer through which the photoions travel. The increased reproducibility of isotopic measurements on standard materials has enabled us to detect anomalous chromium isotopic abundances in presolar SiC grains extracted from primitive meteorites.     

Possible use of DFT approaches for the determination of double exchange interactions

DFT calculations are performed on a model mixed-valence system presenting a double exchange phenomenon. Due to the intrinsic multireference character of the lowest Ms components of the spin states, it is shown that the interactions involved in the double-exchange model cannot be simply extracted from the DFT energies as it is sometimes done. It is, however, possible to extract from different DFT single determinant energies the interactions of a generalized Hubbard Hamiltonian, from which, in a second step, the double-exchange spectrum may be evaluated. The problems generated by the charge and spin polarization are discussed in both symmetric and non symmetric geometries, and the sensitivity of the results to the choice of the density functional is illustrated.