On the invariant measure for the quasi‐linear Lasota equation

The problem of the existence of the invariant measure is important considering its connections with chaotic behaviour. In the papers (Zesz. Nauk. Uniw. Jagiellońskiego, Pr. Mat. 1982; 23 :117–123; Ann. Pol. Math. 1983; XLI :129–137; J. Differential Equations 2004; 196 :448–465) the existence of invariant and ergodic measures according to the dynamical system generated by the Lasota equation was proved, i.e. the equation describing the dynamics and becoming different of the population of cells. In this paper, the existence of such measure for the quasi‐linear Lasota equation is proved. This measure is the carriage of the measure described by Dawidowicz (Zesz. Nauk. Uniw. Jagiellońskiego, Pr. Mat. 1982; 23 :117–123). Copyright © 2006 John Wiley & Sons, Ltd.     

Dual Problems and Separation of Sets: Lagrange and Fenchel Duality

An extremum problem is embedded in a parametric scheme which contains, as a particular case, the classic perturbation function. The introduction of the image of the embedded problem allows one to derive a generalized duality and, in particular, Lagrangian and Fenchel duality.     

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).     

Limit cycles and Lie symmetries

Given a planar vector field U which generates the Lie symmetry of some other vector field X, we prove a new criterion to control the stability of the periodic orbits of U. The problem is linked to a classical problem proposed by A.T. Winfree in the seventies about the existence of isochrons of limit cycles (the question suggested by the study of biological clocks), already answered by Guckenheimer using a different terminology. We apply our criterion to give upper bounds of the number of limit cycles for some families of vector fields as well as to provide a class of vector fields with a prescribed number of hyperbolic limit cycles. Finally we show how this procedure solves the problem of the hyperbolicity of periodic orbits in problems where other criteria, like the classical one of the divergence, fail.     

Ricci curvature and quasiconformal deformations of a Riemannian manifold

The theory of quasiconformal deformations of a Riemannian manifold (M, g) of dimensionn leads in a natural way to the AhlforsS operator being the symmetric and trace free part of the Levi-Civita connection Δ on 1-forms, and to the Ahlfors Laplacian , whereR is the Ricci action. It is well known that there are no conformal deformations on compact Riemannian manifoldsM with negative Ricci curvature. The question arises, how close to being conformal a deformation on suchM can be, i.e. the question on the minimal constant of quasiconformality. Using spectral properties ofL, we derive several lower bounds for the constant of quasiconformality for the normalized deformations of compact manifolds with the positive definite, negative definite, or vanishing Ricci tensorR. As a result, we also obtain that there are no conformal deformations if the Ricci curvature is positive definite but small enough (Corollary 5.17).     

Vapor-particle partitioning of hydrocarbons in Western Mediterranean urban and marine atmospheres

An analytical procedure is described for a comprehensive determination of the composition of hydrocarbons in the atmosphere, based on: (a) aerosol filtration and subsequent adsorption of the vapor phase onto active charcoal and polyurethane foam; (b) GPC and HPLC fractionation of the extracts, and (c) analysis of the fractions by GC-MS under EI and NICI modes. Special emphasis is placed onn-alkanes, PAHs and their oxygenated derivatives.The characterization of samples collected in a coastal urban area (Barcelona city), and far away over the Western Mediterranean, at sea level and at 1100 m of altitude, revealed significant changes in the aerosol composition, mainly attributed to initial vapor-particle partitioning processes, influenced by ambient temperature variations, and to others taking place during long-range atmospheric transport, related with the different compound photoreactivities and with an unexplained source-decoupling phenomenon.     

DABCO-directed self-assembly of bisporphyrins (DABCO=1,4-diazabicyclo[2.2.2]octane)

Three isomeric zinc bisporphyrins have been prepared by covalently linking together two aminoporphyrins with an isophthalic acid derivative. The porphyrins differ in the substitution pattern on the meso phenyl groups, that is, ortho, meta, or para. Titrations carried out by UV-visible and 1H NMR spectroscopy have been used to map out the stabilities and the stoichiometries of the complexes formed with 1,4-diazabicyclo[2.2.2]octane (DABCO) in chloroform. The ortho- and meta-substituted bisporphyrins form 1:1 intramolecular sandwich complexes. The para-substituted bisporphyrin cannot adopt the cofacial conformation required for this type of complex and forms a higher order 2:2 intermolecular assembly, which is stable over a wide range of DABCO concentrations.     

An anion-dependent switch in selectivity results from a change of C-H activation mechanism in the reaction of an imidazolium salt with IrH5(PPh3)2

Changing the counteranion along the series Br, BF4, PF6, SbF6 in their ion-paired 2-pyridylmethyl imidazolium salts causes the kinetic reaction products with IrH5(PPh3)2 to switch from chelating N-heterocyclic carbenes (NHCs) having normal C2 (N path) to abnormal C5 binding (AN path). Computational work (DFT) suggests that the AN path involves C-H oxidative addition to Ir(III) to give Ir(V) with little anion dependence. The N path, in contrast, goes by heterolytic C-H activation with proton transfer to the adjacent hydride. The proton that is transferred is accompanied by the counteranion in an anion-coupled proton transfer, leading to an anion dependence of the N path, and therefore of the N/AN selectivity. The N path goes via Ir(III), not Ir(V), because the normal NHC is a much less strong donor ligand than the abnormal NHC. PGSE NMR experiments support the formation of ion-pair in both the reactants and the products. 19F,1H-HOESY NMR experiments indicate an ion-pair structure for the products that is consistent with the computational prediction (ONIOM(B3PW91/UFF)).     

Protective properties of redox polymer film deposited on stainless steel

We propose a novel composite organic-inorganic coating in the form of a redox polymer film for protection of stainless steel against general corrosion in strong acid medium (2 M H₂SO₄). We utilize an anion exchange polymer, protonated poly(4-vinylpyridine), into which hexacyanoferrate anions have been introduced. Owing to the presence of Fe(CN)₆^3–/4– at the interface formed by the film and the steel, a sparingly soluble metal hexacyanoferrate (mostly Prussian blue, PB) is formed as an overcoating on the steels surface, presumably on the passive (metal oxide) layer. The redox polymer film on the steel seems to act as a composite three-dimensional bilayer-type coating in which hexacyanoferrate(III,II) anions (that are capable of effective charge storage) exist in the outer portions of the film, whereas the inner PB layer improves the systems overall adherence and stability. By analogy to a conducting polymer (e.g. polyaniline, polypyrrole), introduction of the redox polymer composite film leads to stabilization of the steel substrates potential within the passive range.Contribution to the 3rd Baltic Conference on Electrochemistry, Gdansk-Sobieszewo, Poland, 23–26 April 2003Dedicated to the memory of Harry B. Mark, Jr. (28 February 1934–3 March 2003)