Synthesis of polysiloxane xerogels with fluorine-containing groups in the surface layer and their sorption properties

Hydrolysis and polycondensation reactions of tetraethoxysilane (TEOS) with 3,3,3-trifluoropropyl-trimethoxysilane (TFMS) or 1H,1H,2H,2H-perfluorooctyltriethoxysilane (PFES) were used to synthesize xerogels functionalized with fluorine-containing groups. According to transmission electron microscopy, the skeletons of synthesized polysiloxane xerogels have globular structures and consist of agglomerates of particles with fluorinated groups on their surfaces. FTIR spectroscopy showed that primary xerogel particles possess spatial polysiloxane networks, with fluorinated groups along with silanol groups and water in the surface layer. According to thermal analyses, the water content was 3–8 wt.%, and it decreased with increasing length of the fluorinated chain. Thermal destruction of the surface layer starts above 300 °C. The sorbents that were obtained were predominantly mesoporous materials with well-developed porous structures (S_BET = 400–960 m² g⁻¹, V_s = 0.66–0.93 cm³ g⁻¹). The influence of the TEOS/functional silane ratio and the natures of the functional groups on the structural and adsorptive properties were shown. The samples synthesized are organophilic. The affinity for n-hexane increases with increasing length of the fluorine-containing chain (PFES) and the content of fluorinated groups in the surface layer. The hybrid organic–inorganic materials that were obtained can be used for adsorption of hydrocarbons, including oil, from water.     

Uniqueness and symmetry results for solutions of a mean field equation on 𝕊2 via a new bubbling phenomenon

Motivated by the study of gauge field vortices, we consider a mean field equation on the standard sphere 𝕊² involving a Dirac distribution supported at a point P ∈ 𝕊². Consistently with the physical applications, we show that solutions “concentrate” precisely around the point P for some limiting value of a given parameter. We use this fact to obtain symmetry (about the axis ) and uniqueness property for the solution. The presence of the Dirac measure makes such a task particularly delicate to handle from the analytical point of view. In fact, the bubbling phenomenon about the singularity allows the existence of solution sequences with a double‐peak profile near P. The new and more delicate part of this paper is to exclude this possibility by using the method of moving planes together with the Alexandrov‐Bol inequality. © 2011 Wiley Periodicals, Inc.     

A coassociativeC *-quantum group with nonintegral dimensions

By weakening the counit and antipode axioms of aC ^*-Hopf algebra and allowing for the coassociative coproduct to be nonunital, we obtain a quantum group, that we call aweak C ^*-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. This amounts to generalizing the Baaj-Skandalis multiplicative unitaries to multipicative partial isometries. Every finite-dimensional weakC ^*-Hopf algebra has a dual which is again a weakC ^*-Hopf algebra. An explicit example is presented with Lee-Yang fusion rules. We briefly discuss applications to amalgamated crossed products, doubles, and quantum chains.Supported by the Hungarian Scientific Research Fund, OTKA T 016 233.Supported by the Hungarian Scientific Research Fund, OTKA-1815.     

Optimization of a 2D photonic crystal add/drop multiplexer based on contra-directional coupling

We present a highly integrated add/drop multiplexer, where the contra-directional coupling is realized by phase matching two photonic crystal waveguides. The device band structure, the corresponding transmission and drop spectra, and the coupling length are carefully analysed. Different device configurations are discussed and by tailoring the coupling factor, we optimize the frequency response of the filter, obtaining a -sized channel selector, characterized by a very high drop efficiency.     

Propagation of Optical Beams in Active Media Inside Gaussian Cavities

The evolution of relevant beam parameters of a light beam propagating through an iterated sequence of spherical lenses and Gaussian apertures immersed in an active medium is investigated. This structure is a classical model for an optical resonator. The effective size of the field distribution, the beam quality factor and the radius of wave front curvature are calculated as functions of the number of transits. Steady-state values of these parameters are also determined and the dependence of the results on the gain of the medium, the aperture width and the focal length is illustrated. A comparison with the case without active medium is also discussed.     

Heterocyclic rearrangements. Phenylhydrazones and N-methyl-N-phenylhydrazones of 3-acylisoxazoles

The reaction of 3-benzoyl-5-phenylisoxazole ( 4 ) and 3-acetyl-5-methylisoxazole ( 5 ) with phenylhydrazine and N-methyl-N-phenylhydrazine has been investigated and the reactivity of (E)- and (Z)-phenylhydrazones and N-methyl-N-phenylhydrazones has been studied.     

On the relations between two types of convergence for convex functions

Theory and applications have shown that there are two important types of convergence for convex functions: pointwise convergence and convergence in a topology induced by the convergence of their epigraphs. We show that these two types of convergence are equivalent on the class of convex functions which are equi-lower semicontinuous. This turns out to be maximal classes of convex functions for which this equivalence can be obtained. We also indicate a number of implications of these results to the convergence of convex sets and the corresponding support functions and to the convergence of the infima of sequences of convex minimization problems.     

Differentiability of solutions of some evolution equations in nonreflexive Banach spaces

Summary The purpose of this paper is to present a regularity result that provides a unified treatment of the Cauchy problem for certain nonlinear partial differential equations that appear in kinetic theory. Part. 1 contain the main theorem, based on the theory of evolution equations. In part 2 it is indicated how these abstract results are applicable to the spatially homogeneous Boltzmann equazion and to the kinetic equation of vehicular traffic. Entrata in Redazione il 28 giugno 1978. Work performed under the auspices of the G.N.A.F.A. of the National Research Couucil.