Global regular and singular solutions for a model of gravitating particles

The existence of solutions of a nonlinear parabolic equation describing the gravitational interaction of particles is studied for the initial data in spaces of (generalized) pseudomeasures. This approach permits us to relax regularity assumptions on the initial conditions and to prove asymptotic stability results for the above problem.Mathematics Subject Classification (2000):35B40, 35K15, 82C21     

Deuterium-labeling and NMR study of the dearomatization of N-alkyl-N-benzyldiphenylphosphinamides through anionic cyclization: ortho and benzylic lithiation directed by complex-induced proximity effects

The mechanism of the anionic cyclization dearomatizing reaction of N-benzyl-N-methyldiphenylphosphinamide (1) upon treatment with s-BuLi in tetrahydrofuran (THF) at -90 degrees C has been analyzed by deuterium-labeling and natural abundance multinuclear magnetic resonance ((1)H, (2)H, (7)Li, (13)C, (31)P) studies. In the absence of coordinating cosolvents such as hexamethylphosphoramide (HMPA), eight major anionic species were identified, which allowed us to unravel the pathway of the metalation reaction. In agreement with the complex-induced proximity effect (CIPE) mechanism, the sequence of transformations emerging from this study involves the coordination of the lithium base to the P=O group of 1 to give four dimeric precomplexes whose NMR data are consistent with structures Va/Vb and VIIIa/VIIIb. The diastereomers Va/Vb are the precursors of the monomeric benzylic anion II, whereas the VIIIa/VIIIb diastereomers are assumed to undergo ortho deprotonation leading to anions I. Translocation from the ortho anion to the benzylic one is not observed. Intramolecular conjugate addition of anion II to the P-phenyl rings happens in a reversible way, affording the monomeric dearomatized anions III, IV, VI, and VII. The reaction progresses to yield a mixture containing only the species I, III, and IV. HMPA acts as a catalyst for the ortho-to-benzylic translocation and anionic cyclization reactions. Two-dimensional (2D) (7)Li,(31)P[(1)H] shift correlations and (7)Li[(31)P] NMR spectra proved to be crucial for the structural assignment of the anionic species. These techniques also demonstrated the diastereotopicity of the two achiral ligands involved in a dimer with s-BuLi (Vb) owing to the slow configuration inversion of the carbanion center.     

Electrically forced coaxial nanojets for one-step hollow nanofiber design

The outer liquid of a two-liquid coaxial electrified jet is gelled before the onset of natural instabilities to yield hollow nanofibers. By using sol-gel chemistry, innocuous solvents such as glycerol and olive oil, and electrohydrodynamics, it is possible to make such structures in a rather straightforward manner.     

Density estimation using game theory

In this note we show that the mathematical tools of cooperative game theory allow a successful approach to the statistical problem of estimating a density function. Specifically, any random sample of an absolutely continuous random variable determines a transferable utility game, the Shapley value of which proves to be an estimator of the density function of binned kernel and WARPing types, with good computational and statistical properties.Authors acknowledge the financial support of Spanish Ministry for Science and Technology and FEDER through projects BFM2002-03213 and BEC2002-04102-C02-02 and of Xunta de Galicia through projects PGIDT00PXI20104PR and PGIDT03PXIC20701PN. They also thank the comments of two anonymous referees.     

Development of an optical refractometer by analysis of one-dimensional photonic bandgap structures with defects

A theoretical analysis of an optical fiber photonic-bandgap-based refractometer is presented. The design is based on a quarter-wave reflector with one defect. By modifying both the real and the imaginary parts of the index of refraction of the defects we begin to change either the frequency or the amplitude of the localized optical mode. So we could fabricate a specific optical fiber refractometer by combining all the variables: index of refractive index of the defects and the rest of layers, thickness of the defect, number of layers, etc. to yield a large set of design possibilities, for example, detecting wider or thinner ranges of refractive indices, or controlling the detection accuracy. Some rules for the practical implementation of the refractometer are given.     

Isometries of Quaternionic Lie Groups

We study connected Lie groups whose Lie algebra is obtained as the tensor product of a real associative algebra and the algebra of quaternions. It is proved that they carry a natural integrable -structure. We endow such quaternionic Lie groups with a left-invariant Hermitian metric and study the identity connected component of their isometry groups. The determination of such identity connected component is illustrated with a family of examples.