Dendriform equations

We investigate solutions for a particular class of linear equations in dendriform algebras. Motivations as well as several applications are provided. The latter follow naturally from the intimate link between dendriform algebras and Rota–Baxter operators, e.g. the Riemann integral map or Jackson's q-integral.     

Influence of surface chemistry on the adsorption of oxygenated hydrocarbons on activated carbons

The objective of this work was to study the adsorption of different oxygenated hydrocarbons (methanol, ethanol, 1 and 2-butanol, methyl acetate) on activated carbons from organic mixtures with cyclohexane. Three activated carbons prepared by thermal and chemical treatments of a commercial carbon were employed for this purpose. Their textural properties were found to be similar, whereas their surface chemistries were modified, as shown by temperature-programmed desorption coupled to mass spectrometry (TPD-MS) and X-ray photoelectron spectroscopy (XPS). The adsorption isotherms were obtained by depletion method, and the analysis of adsorbed species was evaluated by TPD-MS to obtain new insight into the interactions between the different hydrocarbons and the carbon surface. Ethanol leads to a high-energy interaction between its hydroxyl function and the oxygenated surface groups and also to a lower energy interaction between the aliphatic part of the molecule and the carbon material. The desorption activation energy for this hydrophilic interaction is high (50 to 105 kJ/mol), and it is related to the nature of the carbon surface groups. The relative importance of these two interactions depend on the size of the alcohol/methanol is similar to ethanol, whereas butanols lead to more dispersive interactions. Methyl-acetate cannot undergo this kind of strong interaction and behaves like cyclohexane, having desorption activation energies ranging between 25 and 45 kJ/mol no matter the molecule and the carbon surface chemistry.     

High-pressure investigations under CO/H2 of rhodium complexes containing hemispherical diphosphites

The two rhodium complexes [Rh(acac)(L(R))] (L(R)=(S,S)-5,11,17,23-tetra-tert-butyl-25,27-di(OR)-26,28-bis(1,1'-binaphthyl-2,2'-dioxyphosphanyloxy)calix[4]arene; 6: R=benzyl, 7: R=fluorenyl), each based on a hemispherical chelator forming a pocket about the metal centre upon chelation, are active in the hydroformylation of 1-octene and styrene. As expected for this family of diphosphanes, both complexes resulted in remarkably high selectivity towards the linear aldehyde in the hydroformylation of 1-octene (l/b≈15 for both complexes). Linear aldehyde selectivity was also observed when using styrene, but surprisingly only 6 displayed a marked preference for the linear product (l/b=12.4 (6) vs. 1.9 (7)). A detailed study of the structure of the complexes under CO or CO/H(2) in toluene was performed by high-pressure NMR (HP NMR) and FT-IR (HP-IR) spectroscopies. The spectroscopic data revealed that treatment of 6 with CO gave [Rh(acac)(CO)(η(1)-L(benzyl))] (8), in which the diphosphite behaves as a unidentate ligand. Subsequent addition of H(2) to the solution resulted in a well-defined chelate complex with the formula [RhH(CO)(2)(L(benzyl))] (9). Unlike 6, treatment of complex 7 with CO only led to ligand dissociation and concomitant formation of [Rh(acac)(CO)(2)], but upon addition of H(2) a chelate complex analogous to 9 was formed quantitatively. In both [RhH(CO)(2)(L(R))] complexes the diphosphite adopts the bis-equatorial coordination mode, a structural feature known to favour the formation of linear aldehydes. As revealed by variable-temperature NMR spectroscopy, these complexes show the typical fluxionality of trigonal bipyramidal [RhH(CO)(2)(diphosphane)] complexes. The lower linear selectivity of 7 versus 6 in the hydroformylation of styrene was assigned to steric effects, due to the pocket in which the catalysis takes place being less adapted to accommodate CO or larger olefins and, therefore, possibly leading to facile ligand decoordination. This interpretation was corroborated by an X-ray structure determination carried out for 7.     

Site-selective lanthanide doping in a nonanuclear yttrium(III) cluster revealed by crystal structures and luminescence spectra

A series of lanthanide-doped nonanuclear yttrium(III) clusters with general formulas (Y(9-x)Ln(x))(acac)(16)(μ(3)-OH)(8)(μ(4)-O)(μ(4)-OH) (Ln = Pr, Eu, Tb, Dy, and Yb) were synthesized. Characterization by single-crystal X-ray diffraction allowed for analysis of relative populations of yttrium (Z = 39) and dopant trivalent lanthanide (Z = 59-70) at every crystallographic metal position. Nonuniform distribution of ions along the three different sites seems to be correlated to the site volume and the ratio of ionic radii. In support, luminescence spectra of europium(III)-doped nonanuclear clusters were measured over a wide range of dopant concentrations. Emission intensities of peaks characteristic of specific sites correlate well with the site population determined through X-ray diffraction.     

The Role of Ion Pairs in the Second‐Order NLO Response of 4‐X‐1‐Methylpiridinium Salts

A series of 4‐X‐1‐methylpyridinium cationic nonlinear optical (NLO) chromophores (X=(E)‐CH?CHC₆H₅; (E)‐CH?CHC₆H₄‐4′‐C(CH₃)₃; (E)‐CH?CHC₆H₄‐4′‐N(CH₃)₂; (E)‐CH?CHC₆H₄‐4′‐N(C₄H₉)₂; (E,E)‐(CH?CH)₂C₆H₄‐4′‐N(CH₃)₂) with various organic (CF₃SO₃^?, p‐CH₃C₆H₄SO₃^?), inorganic (I^?, ClO₄^?, SCN^?, [Hg₂I₆]^2?) and organometallic (cis‐[Ir(CO)₂I₂]^?) counter anions are studied with the aim of investigating the role of ion pairing and of ionic dissociation or aggregation of ion pairs in controlling their second‐order NLO response in anhydrous chloroform solution. The combined use of electronic absorption spectra, conductimetric measurements and pulsed field gradient spin echo (PGSE) NMR experiments show that the second‐order NLO response, investigated by the electric‐field‐induced second harmonic generation (EFISH) technique, of the salts of the cationic NLO chromophores strongly depends upon the nature of the counter anion and concentration. The ion pairs are the major species at concentration around 10^?3 M , and their dipole moments were determined. Generally, below 5×10^?4 M , ion pairs start to dissociate into ions with parallel increase of the second‐order NLO response, due to the increased concentration of purely cationic NLO chromophores with improved NLO response. At concentration higher than 10^?3 M , some multipolar aggregates, probably of H type, are formed, with parallel slight decrease of the second‐order NLO response. Ion pairing is dependent upon the nature of the counter anion and on the electronic structure of the cationic NLO chromophore. It is very strong for the thiocyanate anion in particular and, albeit to a lesser extent, for the sulfonated anions. The latter show increased tendency to self‐aggregate.     

Maxima and minima of the displacement components for the Lamb modes

This paper revisits the vanishing of the transverse component of the particle displacement vector in free surfaces of an isotropic homogeneous plate, for both symmetric and antisymmetric Lamb waves. Drawing on well-known analytical expressions from Viktorov's book [(1967) Rayleigh and Lamb Waves: Physical Theory Applications, Chap. II, pp. 67-121], two distinct frequency-thickness product expressions, in cases where this vanishing occurs, are derived: one for the symmetric modes and another for the antisymmetric modes. At these frequency-thickness products, phase and group velocities have appreciable values which are discussed herein. It appears that these velocities depend on the transverse bulk wave velocity only. This is the specific condition of the Lame? modes. Moreover, theoretical and experimental investigations of displacements in the surface of a plate in air have been carried out. The theoretical part shows that the normal and transverse displacements have, respectively, a local maximum and a local minimum in the vicinity of these frequency-thickness products. The experimental part corroborates the presence of the local maximum of the S(0) Lamb mode for various materials.     

Polarization control in spun and telecommunication optical fibers

We consider the counterpropagating interaction of a signal and a pump beam in a spun fiber and in a randomly birefringent fiber, the latter being relevant to optical telecommunication systems. On the basis of a geometrical analysis of the Hamiltonian singularities of the system, we provide a complete understanding of the phenomenon of polarization attraction in these two systems, which allows to achieve a control of the polarization state of the signal beam by adjusting the polarization of the pump. In spun fibers, all polarization states of the signal beam are attracted toward a specific line of polarization states on the Poincaré sphere, whose characteristics are determined by the polarization state of the injected backward pump. In randomly birefringent telecommunication fibers, we show that an unpolarized signal beam can be repolarized into any particular polarization state, without loss of energy.     

Essential countability of treeable equivalence relations

We establish a dichotomy theorem characterizing the circumstances under which a treeable Borel equivalence relation E is essentially countable. Under additional topological assumptions on the treeing, we in fact show that E   is essentially countable if and only if there is no continuous embedding of _E1${\mathbb{E}}_1$ into E. Our techniques also yield the first classical proof of the analogous result for hypersmooth equivalence relations, and allow us to show that up to continuous Kakutani embeddability, there is a minimum Borel function which is not essentially countable-to-one.