ABELIANIZED OBSTRUCTION FOR FIXED POINTS OF FIBER-PRESERVING MAPS OF SURFACE BUNDLES

Let f: M -> M be a fiber-preserving map where S -> M -> B is a bundle and S is a closed surface. We study the abelianized obstruction, which is a cohomology class in dimension 2, to deform f to a fixed point free map by a fiber-preserving homotopy. The vanishing of this obstruction is only a necessary condition in order to have such deformation, but in some cases it is sufficient. We describe this obstruction and we prove that the vanishing of this class is equivalent to the existence of solution of a system of equations over a certain group ring with coefficients given by Fox derivatives.

NIELSEN COINCIDENCE THEORY OF FIBRE-PRESERVING MAPS AND DOLD`S FIXED POINT INDEX

Let M -> B, N -> B be fibrations and f(1), f(2): M -> N be a pair of fibre-preserving maps. Using normal bordism techniques we define an invariant which is an obstruction to deforming the pair f(1), f(2) over B to a coincidence free pair of maps. In the special case where the two fibrations axe the same and one of the maps is the identity, a weak version of our omega-invariant turns out to equal Dold`s fixed point index of fibre-preserving maps. The concepts of Reidemeister classes and Nielsen coincidence classes over B are developed. As an illustration we compute e.g. the minimal number of coincidence components for all homotopy classes of maps between S(1)-bundles over S(1) as well as their Nielsen and Reidemeister numbers.

Fixed points on Klein bottle fiber bundles over the circle

The main purpose of this work is to study fixed points of fiber-preserving maps over the circle S(1) for spaces which are fiber bundles over S(1) and the fiber is the Klein bottle K. We classify all such maps which can be deformed fiberwise to a fixed point free map. The similar problem for torus fiber bundles over S(1) has been solved recently.

Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps

We study the 1-parameter Wecken problem versus the restricted Wecken problem, for coincidence free pairs of maps between surfaces. For this we use properties of the function space between two surfaces and of the pure braid group on two strings of a surface. When the target surface is either the 2-sphere or the torus it is known that the two problems are the same. We classify most pairs of homotopy classes of maps according to the answer of the two problems are either the same or different when the target is either projective space or the Klein bottle. Some partial results are given for surfaces of negative Euler characteristic. (C) 2010 Elsevier B.V. All rights reserved.

Behaviour of TiO(2)-SiMgO(x) hybrid composites on the solar photocatalytic degradation of polluted air

The photocatalytic performance of TiO(2)-SiMgO(x) ceramic plates for trichloroethylene abatement in gas phase has been evaluated under sun irradiance conditions. A continuous flow Pyrex glass reactor fixed on the focus of a compound parabolic collector has been used. The performance of the hybrid photocatalyst has been evaluated as the variation of TCE conversion and reaction products formation with the solar irradiance at different total gas flow, TCE concentration, and water vapour content. SiMgO(x) not only provides adsorbent properties to the photocatalyst, but it also allows the effective use of the material during low solar irradiance conditions. The adsorption-desorption phenomena play a pivotal role in the behaviour of the system. Thus, TCE conversion curves present two different branches when the sun irradiance increases (sunrise) or decreases (sunset). CO(2), COCl(2) and DCAC were the most relevant products detected. Meanwhile CO(2) concentration was insensitive to the branch analysed, COCl(2) or DCAC were not indicating the ability of these compounds to be adsorbed on the composite. An increase of the UV irradiation at total TCE conversion promotes the CO(2) selectivity. The excess of energy arriving to the reactor favours the direct reaction pathway to produce CO(2). The photonic efficiency, calculated as a function of the rate of CO(2) formation, decreases linearly with the solar irradiance up to around 2 mW cm(-2), where it becomes constant. For decontamination systems high TCE conversion is pursuit and then high solar irradiance values are required, in spite of lower photonic efficiency values. The present photocatalyst configuration, with only 17% of the reactor volume filled with the photoactive material, allows total TCE conversion for 150 ppm and 1 L min(-1) in a wide sun irradiance window from 2 to 4 mW cm(-2). The incorporation of water vapour leads to an increase of the CO(2) selectivity keeping the TCE conversion around 90%, although significant amounts of COCl(2) were observed. (c) 2010 Elsevier B.V. All rights reserved.

TiO(2)-PHOTOCATALYZED DEGRADATION OF PHENOL IN SALINE MEDIA IN AN ANNULAR REACTOR: HYDRODYNAMICS, LUMPED KINETICS, INTERMEDIATES, AND ACUTE TOXICITY

The photocatalytic degradation of phenol in aqueous suspensions of TiO(2) under different salt concentrations in an annular reactor has been investigated. In all cases, complete removal of phenol and mineralization degrees above 90% were achieved. The reactor operational parameters were optimized and its hydrodynamics characterized in order to couple mass balance equations with kinetic ones. The photodegradation of the organics followed a Langmuir-Hinshelwood-Hougen- Watson lumped kinetics. From GC/MS analyses, several intermediates formed during oxidation have been identified. The main ones were catechol, hydroquinone, and 3-phenyl-2-propenal, in this order. The formation of negligible concentrations of 4-chlorophenol was observed only in high salinity medium. Acute toxicity was determined by using Artemia sp. as the test organism, which indicated that intermediate products were all less toxic than phenol and a significant abatement of the overall toxicity was accomplished, regardless of the salt concentration.