Chern numbers for two families of noncommutative Hopf fibrations

We consider noncommutative line bundles associated with the Hopf fibrations of SU_q(2) over all Podle? spheres and with a locally trivial Hopf fibration of S³_pq. These bundles are given as finitely generated projective modules associated via 1-dimensional representations of U(1) with Galois-type extensions encoding the principal fibrations of SU_q(2) and S³_pq. We show that the Chern numbers of these modules coincide with the winding numbers of representations defining them. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Global and exploding solutions in a model of self-gravitating systems

We study asymptotic properties of solutions to an extension to arbitrary dimensions of the astrophysical model proposed by Chavanis et al. to explain phenomena of gravitational collapse in clouds of self-gravitating particles. In particular, we show that in the two-dimensional case the solutions can be continued to global ones, while in three space dimensions large data of negative energy blow up in a finite time. Relations between isothermal, Streater's energy-transport and the present models are also studied.     

Invariant measures for equicontinuous semigroups of continuous transformations of a compact Hausdorff space

Equicontinuous semigroups of transformations of a compact Hausdorff space and their sets of all invariant (Borel, regular and probabilistic) measures are studied. Conditions equivalent to the existence of at least one invariant measure are given. The (algebraic and topological) structure of the set of invariant measures is researched.     

On exactness and isoperimetric profiles of discrete groups

We introduce a quasi-isometry invariant related to Property A and explore its connections to various other invariants of finitely generated groups. This allows to establish a direct relation between asymptotic dimension on one hand and isoperimetry and random walks on the other. We apply these results to reprove sharp estimates on isoperimetric profiles of some groups and to answer some questions in dimension theory.     

Initiation process of L‐lactide polymerization carried out with zirconium(IV) acetylacetonate

Lactide polymerization using zirconium(IV) acetylacetonate [Zr(acac)₄] as an initiator was investigated. In the reaction between Zr(acac)₄ and the monomer molecule, lactide deprotonation and the release of acetylacetone occurred. The structures of the obtained complexes were analyzed with high‐resolution NMR spectroscopy. A computational method was used to calculate the hypothetical structures. The role of the obtained complexes in the initiation of polymerization and the reaction of chain growth was proposed. The influence of the reaction temperature on the structures of the complexes was investigated. Polylactide chain growth proceeded by an insertion‐coordination mechanism. The polymer chain grew on one ligand, which was formed in advance from a deprotonated lactide. The molecular masses of the obtained polymers were the same as the theoretical masses and were directly proportional to the reaction conversion. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 1886–1900, 2004     

Long time behaviour of a flow in infinite pipe conforming to slip boundary conditions

The paper analyses long time behaviour of solutions of the Navier–Stokes equations in a two‐dimensional pipe‐like domain. The system is studied with perfect slip boundary conditions with arbitrary inflow conditions at infinity. The main results show the existence of global in time solutions and of an attractor for the dynamical system generated by the model. The paper also establishes an upper bound for the Hausdorff dimension of the attractor. Copyright © 2005 John Wiley & Sons, Ltd.     

Activity of TiO2 deposited by the CVD method on ammoxidized surface of a carbonaceous material in hydrogenation of styrene

Hydrogenation of styrene has been applied as a test reaction to study the catalytic activity of TiO₂ deposited by the CVD (chemical vapour deposition) method on the surface of a carbonaceous material enriched in nitrogen (C_N).     

Synthesis of biodegradable copolymers with low‐toxicity zirconium compounds. V. Multiblock and random copolymers of L‐lactide with trimethylene carbonate obtained in copolymerizations initiated with zirconium(IV) acetylacetonate

Using zirconium(IV) acetylacetonate as an initiator of lactide/trimethylene carbonate copolymerization allowed us to obtain high‐molecular‐weight copolymers with high efficiency. The reactivity ratios of the comonomers were 13.0 for lactide and 0.53 for trimethylene carbonate. Despite the large differences between the values of the reactivity ratios, copolymers with randomized chain structures were obtained. This phenomenon occurred as a result of an intensive intermolecular transesterification process proceeding along with the reaction of copolymer chain growth and modifying its final structure. Conducting the copolymerization at the relatively low temperature of about 110 °C, which minimized the influence of intermolecular transesterification, made it possible to obtain semicrystalline copolymers with multiblock structures. Increasing the temperature of copolymerization up to 180 °C was associated with strong intensification of the transesterification reactions. At this temperature, amorphous copolymers were obtained with identical compositions but highly randomized chain structures. An analysis of the chain microstructures of the obtained copolymers, determining the average length of the blocks, the intermolecular transesterification ratio, and the degree of chain randomization, was conducted by means of NMR spectroscopy. For this purpose, very specific signal assignment in the carbonyl and methylene carbon regions of the ¹³C NMR spectra to appropriate comonomer sequences of polymeric chains was performed. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 3184–3201, 2006     

Synthesis of biodegradable copolymers with low‐toxicity zirconium compounds. IV. Copolymerization of glycolide with trimethylene carbonate and 2,2‐dimethyltrimethylene carbonate: Microstructure analysis of copolymer chains by high‐resolution nuclear magnetic resonance spectroscopy

The results of the copolymerization of glycolide with cyclic trimethylene carbonate and 2,2‐dimethyltrimethylene carbonate are described. The copolymerization was conducted in the presence of low‐toxicity zirconium(IV) acetylacetonate as an initiator. With this kind of initiator, the composition of the comonomer units in the copolymer chains was assumed to be obtained with high efficiency. Despite significant differences in the comonomer reactivity, in copolymers containing comparable amounts of glycolidyl and carbonate sequences, highly randomized chain structures were observed. This effect resulted from strong intermolecular transesterification that proceeded during the studied copolymerization and caused glycolidyl microblock randomization. The assignment of the spectral NMR lines to appropriate comonomer sequences of polymeric chains was performed in the region of methylene protons of glycolidyl units in ¹H NMR spectra of the copolymers and in the carbonyl region of carbon spectra. The equations were formulated for a detailed characterization of the obtained copolymer chains, the average lengths of the blocks, and the transesterification and randomization coefficients. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 98–114, 2006