Dynamics of a quasi-quadratic map

We consider the map \cchi:\Q\to\Q given by \cchi(x)= x\ceil{x}, where \ceil{x} denotes the smallest integer greater than or equal to $x$, and study the problem of finding, for each rational, the smallest number of iterations by \cchi that sends it into an integer. Given two natural numbers $M$ and $n$, we prove that the set of numerators of the irreducible fractions that have denominator$M$ and whose orbits by \cchi reach an integer in exactly $n$ iterations is a disjoint union of congruence classes modulo $M^{n+1}$. Moreover, we establish a finite procedure to determine them. We also describe an efficient algorithm to decide if an orbit of a rational number bigger than one fails to hit an integer until a prescribed number of iterations have elapsed, and deduce that the probability that such an orbit enters $\Z$ is equal to one.Fundação para a Ciência e a Tecnologia (FCT

Multihierarchical electrodes based on titanate nanotubes and zinc oxide nanorods for photoelectrochemical water splitting

Studies involving water splitting to form hydrogen and oxygen have attracted attention because H2 is considered the fuel of the future. Photoelectrocatalysts have been widely used for this application, and several metal oxides can be applied as catalysts. Among them, we highlight zinc oxide nanorods (ZnONRs) and titanate nanotubes (TiNTs); however, their individual nanostructures exhibit disadvantages. For example, ZnONR shows rapid recombination of the photogenerated charges, and TiNT gives rise to randomly orientated films; these disadvantages limit their application as photoanodes. In this study, for the first time, we present a new class of multihierarchical electrodes based on TiNT-decorated ZnONR films that exhibited superior results to the individual species. The TiNTs are homogenously dispersed over the surface of the rods without forming agglomerates, giving rise to a heterojunction that exhibits lower recombination rates. It was found that the results are better when the contents of TiNT in the electrode are higher; thus, glycine was successfully used as a bridge to link both of the structures, increasing the amount of TiNT decorating the rods. As a result, the photocurrent generated with these multihierarchical electrodes is higher than that obtained for pure ZnONR electrodes (0.9 mA and 0.45 mA, respectively), and the electrode potentials for O2 evolution is lower than that observed for pure TiNT electrodes (0 V and 0.8 V vs. ERHE, respectively). The IPCE values are also higher for the multihierarchical electrodes

Radiative decays of light vector mesons in a quark level linear sigma model

We calculate the P0 to gamma gamma, V0 to P0 gamma and V0to V'0 gamma gamma decays in the framework of a U(3)xU(3) linear sigma model which includes constituent quarks. For the first two decays this approach improves results based on the anomalous Wess-Zumino term, with contributions due to SU(3) symmetry breaking and vector mixing. The phi to (omega,rho) gamma gamma decays are dominated by resonant eta' exchange . Our calculation for the later decays improves and update similar calculations in the -closely related- framework of vector meson dominance. We obtain BR(phi to rho gamma gamma)=2.5x10^{-5} and BR(phi to omega gamma gamma)=2.8x10^{-6} within the scope of the high-luminosity phi factories.Comment: 8 pages, submitted to Phys. Rev.

An assessment of Evans' unified field theory I

Evans developed a classical unified field theory of gravitation and electromagnetism on the background of a spacetime obeying a Riemann-Cartan geometry. This geometry can be characterized by an orthonormal coframe theta and a (metric compatible) Lorentz connection Gamma. These two potentials yield the field strengths torsion T and curvature R. Evans tried to infuse electromagnetic properties into this geometrical framework by putting the coframe theta to be proportional to four extended electromagnetic potentials A; these are assumed to encompass the conventional Maxwellian potential in a suitable limit. The viable Einstein-Cartan(-Sciama-Kibble) theory of gravity was adopted by Evans to describe the gravitational sector of his theory. Including also the results of an accompanying paper by Obukhov and the author, we show that Evans' ansatz for electromagnetism is untenable beyond repair both from a geometrical as well as from a physical point of view. As a consequence, his unified theory is obsolete.Comment: 39 pages of latex, modified because of referee report, mistakes and typos removed, partly reformulated, taken care of M.W.Evans' rebutta

Planck 2015 results. XXVII. The Second Planck Catalogue of Sunyaev-Zeldovich Sources

We present the all-sky Planck catalogue of Sunyaev-Zeldovich (SZ) sources detected from the 29 month full-mission data. The catalogue (PSZ2) is the largest SZ-selected sample of galaxy clusters yet produced and the deepest all-sky catalogue of galaxy clusters. It contains 1653 detections, of which 1203 are confirmed clusters with identified counterparts in external data-sets, and is the first SZ-selected cluster survey containing > $10^3$ confirmed clusters. We present a detailed analysis of the survey selection function in terms of its completeness and statistical reliability, placing a lower limit of 83% on the purity. Using simulations, we find that the Y5R500 estimates are robust to pressure-profile variation and beam systematics, but accurate conversion to Y500 requires. the use of prior information on the cluster extent. We describe the multi-wavelength search for counterparts in ancillary data, which makes use of radio, microwave, infra-red, optical and X-ray data-sets, and which places emphasis on the robustness of the counterpart match. We discuss the physical properties of the new sample and identify a population of low-redshift X-ray under- luminous clusters revealed by SZ selection. These objects appear in optical and SZ surveys with consistent properties for their mass, but are almost absent from ROSAT X-ray selected samples