Euler configurations and quasi-polynomial systems

Consider the problem of three point vortices (also called Helmholtz’ vortices) on a plane, with arbitrarily given vorticities. The interaction between vortices is proportional to 1/r, where r is the distance between two vortices. The problem has 2 equilateral and at most 3 collinear normalized relative equilibria. This 3 is the optimal upper bound. Our main result is that the above standard statements remain unchanged if we consider an interaction proportional to r ^b, for any b < 0. For 0 < b < 1, the optimal upper bound becomes 5. For positive vorticities and any b < 1, there are exactly 3 collinear normalized relative equilibria. The case b = −2 of this last statement is the well-known theorem due to Euler: in the Newtonian 3-body problem, for any choice of the 3 masses, there are 3 Euler configurations (also known as the 3 Euler points). These small upper bounds strengthen the belief of Kushnirenko and Khovanskii [18]: real varieties defined by simple systems should have a simple topology. We indicate some hard conjectures about the configurations of relative equilibrium and suggest they could be attacked within the quasi-polynomial framework.     

Synthesis of periphery-functionalized dendritic molecules using polylithiated dendrimers as starting material

A general method for the functionalization of Si-Cl terminated carbosilane dendritic molecules via organolithium or organomagnesium reagents is described. Quantitative exchange of the bromine atoms of 4-bromophenyl-functionalized dendrimers affords polylithiated species that are valuable starting materials for further functionalization, e.g., into pyridyl alcohols. The latter were successfully applied as catalyst precursors in a ruthenium-mediated ring-closure metathesis reaction.     

X-ray diffuse scattering from incommensurable structures

X-Ray diffuse scattering from a series of examples where the reason for the existence of incommensurable features is fairly well understood, is described. They include in particular non stoichiometric compounds such as [DIPS Φ₄(l₃)0.76] and such as intercalated graphite RbC₂₄. and quasi one dimensional conductors. A particular emphasis is given on various 1 - D conductors and to the relation of the incommensurability to the characteristics of the electron conduction bands.     

Oscillatory melting temperature of the vortex smectic phase in layered superconductors

We report on transport measurements of YBa 2Cu 3O (7-delta) single crystals with different oxygen contents in the geometry B, J ||ab (J perpendicularB). Our data show that the vortices become confined between the Cu-O planes below a well-defined temperature at which the effective size 2xi of the vortex core is approximately equal to the period of the Cu-O layers. This confinement strongly increases the vortex liquid freezing temperature. A new melting line is found separating a vortex liquid and a smectic phase, which shows an oscillatory field dependence reflecting differences between commensurate and incommensurate smectic states.     

Synthesis of Novel Chiral Diol Ligands for the Enantioselective Reactions

The chiral diols ((,(,((,((-Tetraaryl-1,3-dioxolane-4,5-dimethanols) have been used for the formation of cyclic titanates and similar derivatives of Mg, Al and Zr. These complexes of oxophilic metal centers were employed in catalytic and stiochiometric enantioselective reactions such as enantioselective addition reactions of carbon-centered nucleophiles to aldehydes1, [2+2] cycloadditions2, and Diels-Alder reactions3. So far, the two hydrogen atoms of all this type of diols are in trans form…     

The diffraction of elastic waves by an imperfect fusion bond

We consider diffraction by a semi-infinite crack located alonga fusion interface between two differing elastic media. Twotypes of crack, namely open and partially closed cracks, areinvestigated. An open crack is modelled by a stress-free contactboundary condition and a partially closed crack is modelledby a spring contact boundary condition. For the latter, thejump in the stress across the crack is assumed to be proportionalto the jump in the displacement across the crack. This situationarises in, for example, a K-weld where the fine grain of theparent material (for example, ferritic or forged austeniticsteel) is in stark contrast with the coarse-grained weld metal(for example, austenitic weld metal). In the metal weld thedirection of the grain axis varies through the metal. However,diffraction is a local phenomenon and so the austenitic steelis assumed to have a zonal axis so that it may be modelled bya transversely isotropic composite. The ferritic or forged austeniticsteel will be modelled as an isotropic material.     

A First Insight in the Mechanisms Involved in the Self-Assembly of 2D-Hexagonal Templated SiO2 and TiO2 Mesostructured Films During Dip-Coating

Since mesoporous materials can be prepared by combining the sol-gel chemistry and the structuring effect of surfactants, they have attracted attention for application in various high technology fields. The present work deals with the analyses of the mechanisms involved in the formation of SiO₂ and TiO₂ highly organised 2D-hexagonal meso-structured films using Brij 58 as surfactant. The preparation of such films by dip-coating involves rapid evaporation which makes the different steps difficult to control. Simultaneous in-situ SAXS (synchrotron) and interferometry analyses have been performed to get a first understanding of the self-assembly process. SiO₂ and TiO₂ materials have a different chemical reactivity (kinetics and coordination aspects). However, we show that the mechanisms involved during dip-coating are quite similar : the self-assembly leading to the organised phase takes place at a final stage of the drying process, involves the formation of a disorganised intermediate phase and depends also on the presence of micellar interfaces in addition to film/air and film/substrate interfaces.     

Détermination par RMN 1H et 13C de la Conformation Favorisée des Ozonides d'Acènes

The preferred conformation of 9,10-dimethylanthracene ozonide has been established by ¹³C and ¹H NMR. Results hold for other anthracenic ozonides such as those of 9,10-dimethoxyanthracene and anthracene. The latter compound, resulting from the ozonolysis of anthracene, has for the first time been shown to exist in solution.