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).     

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.     

Acetylene diethers: Dineopentyloxyethyne

Dineopentyloxyethyne has been synthesized fromtrans-2,3-dichloro-1,4-dioxane according to a general procedure already described. In contrast with alkyl neopentyloxyacetylenes, RC=COCH₂Bu^t, which only polymerize when are strongly heated (150°), dineopentyloxyethyne is a kinetically unstable acetylene diether that polymerizes readily at room temperature. However, the compound has been trapped with Co₂(CO)₈ to give either the corresponding hexacarbonyl complex, m.p. 74–75°, or the cyclic trimer, m.p. 205–206°, depending upon the experimental conditions.     

TEMPO-promoted Pauson-Khand reaction. Single-electron activation of cobalt-carbonyl bonds?

[reaction: see text] The Pauson-Khand reaction is notably accelerated by TEMPO. According to DFT calculations, TEMPO could trigger a radical, low-energy pathway for the reaction by facilitating the decarbonylation of doublet complexes arising either from a CO/nitroxide exchange or from nitroxide addition to a CO ligand.     

Ring-closing metathesis of chiral allylamines. Enantioselective synthesis of (2S,3R,4S)-3,4-dihydroxyproline

The ring-closing metathesis (RCM) of two types of unsaturated chiral allylamines III, easily available from enantiomerically enriched epoxy alcohols, has been studied. Fully protected allylamines IIIa [(1)R = CH(2)-(CH(2))(n)()-CH=CH(2); (2)R = Boc; (3)R = PMB] have been prepared from unsaturated epoxy alcohols, whereas bis-allylamines IIIb ((1)R = Ph, (2)R = allyl,(3)R = Boc or PMB) have been prepared from 2,3-epoxy-3-phenylpropanol. Both types have been subjected to RCM to provide either cyclic allylamine I or II. The synthetic potential of these intermediates has been demonstrated by the enantioselective synthesis of (2S,3R,4S)-3,4-dihydroxyproline.     

Thermal and photochemical rearrangement of bicyclo[3.1.0]hex-3-en-2-one to the ketonic tautomer of phenol. Computational evidence for the formation of a diradical rather than a zwitterionic intermediate

The ground state (S(0)) and lowest-energy triplet state (T(1)) potential energy surfaces (PESs) concerning the thermal and photochemical rearrangement of bicyclo[3.1.0]hex-3-en-2-one (8) to the ketonic tautomer of phenol (11) have been extensively explored using ab initio CASSCF and CASPT2 calculations with several basis sets. State T(1) is predicted to be a triplet pipi lying 66.5 kcal/mol above the energy of the S(0) state. On the S(0) PES, the rearrangement of 8 to 11 is predicted to occur via a two-step mechanism where the internal cyclopropane C-C bond is broken first through a high energy transition structure (TS1-S(0)()), leading to a singlet intermediate (10-S(0)()) lying 25.0 kcal/mol above the ground state of 8. Subsequently, this intermediate undergoes a 1,2-hydrogen shift to yield 11 by surmounting an energy barrier of only 2.7 kcal/mol at 0 K. The rate-determining step of the global rearrangement is the opening of the three-membered ring in 8, which involves an energy barrier of 41.2 kcal/mol at 0 K. This high energy barrier is consistent with the fact that the thermal rearrangement of umbellulone to thymol is carried out by heating at 280 degrees C. Regarding the photochemical rearangement, our results suggest that the most efficient route from the T(1) state of 8 to ground state 11 is the essentially barrierless cleavage of the internal cyclopropane C-C bond followed by radiationless decay to the S(0) state PES via intersystem crossing (ISC) at a crossing point (S(0)()/T(1)()-1) located at almost the same geometry as TS1-S(0)(), leading to the formation of 10-S(0)() and the subsequent low-barrier 1,2-hydrogen shift. The computed small spin-orbit coupling between the T(1) and S(0) PESs at S(0)()/T(1)()-1 (1.2 cm(-)(1)) suggests that the ISC between these PESs is the rate-determining step of the photochemical rearrangement 8 --> 11. Finally, computational evidence indicates that singlet intermediate 10-S(0)() should not be drawn as a zwitterion, but rather as a diradical having a polarized C=O bond.