Jackson-type inequality on the sphere

In a recent paper, I introduced new moduli of smoothness for functions on the sphere which did not use averages and, as a result, had some interesting properties. The direct, Jackson-type, estimate of the best approximation by spherical harmonics using the new moduli will be proved here. Equivalence with the appropriate K-functionals will be given. Relations with the moduli used earlier will be shown and used to prove new results for these moduli.     

Easy preparation of cobalt corrole and hexaphyrin and isolation of new oligopyrroles in the solvent-free condensation of pyrrole with pentafluorobenzaldehyde

[reaction: see text] Following the discovery that meso-substituted corroles are formed in solvent-free condensation of pyrrole with aldehydes, we demonstrate that a small variation in the methodology is suitable for facile synthesis of cobalt(III) corrole and hexaphyrin. These compounds, as well as three noncyclic products, were fully characterized by spectroscopy and X-ray crystallography.     

Mesomorphism,isomerization, and dynamics in a new series of pyramidic liquid crystals

Nona-alkanoyloxy tribenzocyclononene (CTV-n, where n is the number of carbons in the side chains) were prepared for n = 2 to 14. The homologues of this series appear in two stable isomeric forms, rigid crown and flexible saddle. We report on their isomerization equilibria and dynamics in solution and on their mesomorphic properties in the neat state. The crown-saddle equilibrium and interconversion kinetics of the CTV-8 isomers were studied in dimethyl formamide solutions using high-resolution (1)H NMR in the temperature range from 50 to 130 degrees C. At lower temperatures, the isomerization is too slow to measure. In this range the equilibrium saddle fraction increases from approximately 0.40 to approximately 0.65, whereas the isomerization rate increases from approximately 10(-)(4) to approximately 1 s(-)(1). The saddle isomer undergoes fast pseudorotation at room temperature, but below about -50 degrees C, it becomes slow enough to affect the NMR line width. The rate parameters for this process were estimated from the carbon-13 spectra in methylene chloride solutions to be, k(p)(-100 degrees C) approximately 1.7 x 10(3) s(-)(1) and E(a) approximately 9.6 kJ/mol. The slow crown-saddle isomerization at room temperature (half-life of about one year) allows quantitative separation (by chromatography) of the two isomers and their separate investigation. When the alkanoyloxy side chains are sufficiently long both isomers are mesogenic (n >or= 4 for the saddle and n >or= 5 for the crown), exhibiting hexagonal columnar mesophases. The structure, dynamics, and mesomorphic properties of these mesophase were investigated by X-ray diffraction, optical polarizing microscopy, differential scanning calorimetry, and NMR. The lattice parameters of the crown and saddle mesophases of corresponding homologues are almost identical and increase monotonically with increasing length of the side chains. The clearing temperatures of the saddle isomers are consistently lower than those of the corresponding crowns. Within each series, the clearing temperatures are almost independent of the length of the side chains (156 to 170 degrees C for the crown and 115 to 148 degrees C for the saddle). The thermal and kinetic properties of the neat compounds lead to peculiar phase sequences, as observed in the polarizing microscope and in the DSC thermogram, involving repeated, back and forth, interconversion between the two isomers. Carbon-13 MAS NMR measurements of the crown and saddle mesophases of several homologues were carried out. The spectra of the crown mesophase exhibit dynamic features consistent with planar 3-fold molecular jumps about the column axes. A quantitative analysis for the CTV-8 crown homologue yielded the following Arrhenius parameters, A = 3.1 x 10(22)s(-)(1) and E(a) = 130.1kJ/mol. These unusually high values suggest that the barrier to the jump process is temperature dependent, decreasing with increasing temperature. The rate of this 3-fold jump process is slower for the lower homologues and faster for the higher ones. In contrast, the saddle isomers in the mesophase do not show dynamic effects in their carbon-13 MAS spectra. They do not undergo pseudorotation, and it appears that the molecules remain locked within the columns in a saddle conformation, up to the clearing temperature. However, on (super-)cooling to room temperature and below, selective line broadening is observed in their carbon-13 MAS spectra. This suggests that the saddle conformation is twisted in the mesophase and undergoes fast high-amplitude jumps between the twisted forms. On cooling, these high-amplitude librations freeze out to give an orientationally disordered state. On a very long time scale (of the order of days at 100 degrees C), the saddle mesophase transforms into that of the crown, apparently by sublimation.     

Quantum mechanics versus realism

Quantum theory accepts the point-like indivisible (classical) character of a particle as a mere product of a measuring process, or what has become known as a collapse. Following the notion of empty waves, which accepts the particle as a real existent entity without regard to the measurement process, we propose an experiment that may shed some light on the reality of the particle and the consequences of that reality.     

One-dimensional rigorous hole theory of fluids: Internally constrained ensembles

A hole in a fluid is specified in a well-defined manner. The concentration of holes is a thermodynamic property of the fluid and we derive this concentration in three different ensembles for a one-dimensional fluid of hard rods. The thermodynamics of these rigorously defined holes is developed, and the properties of holes are explored. The ensemble in which the concentration of holes is maintained fixed exhibits dramatic properties. Finally, pair correlation functions for hard rods in the various ensembles are computed. Contrary to a frequently made assumption, the equilibrium number of holes is found to never be proportional to the probability of finding a single hole in the fluid. Constraining the concentration of holes as well as the density leads to dramatic structural effects prominently displayed by the pair correlation function. The ensemble in which the concentration of holes is fixed is an example of an internally constrained metastable system.     

Stöber synthesis of monodispersed luminescent silica nanoparticles for bioanalytical assays

We have developed a simple method to prepare bright and photostable luminescent silica nanoparticles of different sizes and narrow size distribution in high yield. The method is based on the use of St?ber synthesis in the presence of a fluorophore to form bright silica nanoparticles. Unlike micro-emulsion-based methods often used to prepare luminescent silica particles, the St?ber method is a one-pot synthesis that is carried out at room temperature under alkaline conditions in ethanol:water mixtures and avoids the use of potentially toxic organic solvents and surfactants. Our luminescent particles contained the transition metal complex tris(1,10-phenanthroline) ruthenium(II) chloride, [Ru(phen)3]Cl2. They showed higher photostability and a longer fluorescence lifetime compared to free Ru(phen)3 solutions. Leakage of dye molecules from the silica particles was negligible, which was attributed to strong electrostatic attractions between the positively charged ruthenium complex and the negatively charged silica. To demonstrate the utility of the highly luminescent silica nanoparticles in bioassays, we further modified their surface with streptavidin and demonstrated their binding to biotinylated glass slides. The study showed that digital counting of the luminescent nanoparticles could be used as an attractive alternative to detection techniques involving analogue luminescence detection in bioanalytical assays.     

Carbon-carbon and carbon-nitrogen coupling reactions catalyzed by palladium nanoparticles derived from a palladium substituted Keggin-type polyoxometalate

Palladium 15-20 nm particles stabilized by a Keggin-type polyoxometalate were prepared by reduction of K(5)PPdW(11)O(39) with H(2). The nanoparticles were shown to be effective catalysts for Suzuki-, Heck-, and Stille-type carbon-carbon coupling and carbon-nitrogen coupling reactions of bromoarenes in aqueous media. Chloroarenes were also reactive in reaction media without solvent. [reaction: see text]     

Single and double hydrogen rearrangements in the mass spectra of O-ethyl,S-alkyl methanephosphonothioates

Whereas the oxygen analogues of the title series gives rise to double hydrogen rearrangement on electron-impact, alkylthiophosphonates show in addition a McLafferty rearrangement. The quantitative relationship between these two rearrangements in the thio series is dependent upon the chain length and branching on the alkyl group attached to sulphur. Comparison of the mass spectra of these compounds indicates that in the pathway of the double hydrogen rearrangement the hydrogen transferred to the sulphur atom originates primarily from the carbon δ to sulphur.     

Extrusion of halogens from aromatic compounds by transition metal complexes. Reaction of IrCl(CO)(PPh3)2 with halogenated benzaldehydes

Summary The chlorobenzaldehydes, 2-bromo-4-tolualdehyde and 2-iodobenzaldehyde react with IrCl(CO)(PPh₃)₂ at 190 to 220° to give PhIrCl₂(CO)(PPh₃)₂,p-tolyl IrBr₂(CO)(PPh₃)₂ and PhIrCl₂(CO)(PPh₃), respectively. Iodo-and bromobenzene form PhIrCl(Hal)(CO)(PPh₃)₂ complexes, whereas chlorobenzene does not react under these conditions. Halogen transfer from the aromatic to the metal atom in the reaction of halobenzaldehydes, and halogen extrusion in various homogeneously catalyzed processes, are assumed to follow similar reaction patterns.