Genus bounds for minimal surfaces arising from min-max constructions

In this paper we prove genus bounds for closed embedded minimal surfaces in a closed 3-dimensional manifold constructed via min-max arguments. A stronger estimate was announced by Pitts and Rubistein but to our knowledge its proof has never been published. Our proof follows ideas of Simon and uses an extension of a famous result of Meeks, Simon and Yau on the convergence of minimizing sequences of isotopic surfaces. This result is proved in the second part of the paper.Comment: Accepted for publication on Journal for Pure and Applied Mathematic

Direct Experimental Evidence of Metal-Mediated Etching of Suspended Graphene

Atomic resolution high angle annular dark field imaging of suspended, single-layer graphene, onto which the metals Cr, Ti, Pd, Ni, Al and Au atoms had been deposited was carried out in an aberration corrected scanning transmission electron microscope. In combination with electron energy loss spectroscopy, employed to identify individual impurity atoms, it was shown that nano-scale holes were etched into graphene, initiated at sites where single atoms of all the metal species except for gold come into close contact with the graphene. The e-beam scanning process is instrumental in promoting metal atoms from clusters formed during the original metal deposition process onto the clean graphene surface, where they initiate the hole-forming process. Our observations are discussed in the light of calculations in the literature, predicting a much lowered vacancy formation in graphene when metal ad-atoms are present. The requirement and importance of oxygen atoms in this process, although not predicted by such previous calculations, is also discussed, following our observations of hole formation in pristine graphene in the presence of Si-impurity atoms, supported by new calculations which predict a dramatic decrease of the vacancy formation energy, when SiOx molecules are present.Comment: final version accepted in ACS Nano + supplementary info. 22+6 pages, 4+5 figure

Meta-Analysis in Educational Research

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Regional scale effects of the aerosol cloud interaction simulated with an online coupled comprehensive chemistry model

We have extended the coupled mesoscale atmosphere and chemistry model COSMO-ART to account for the transformation of aerosol particles into cloud condensation nuclei and to quantify their interaction with warm cloud microphysics on the regional scale. The new model system aims to fill the gap between cloud resolving models and global scale models. It represents the very complex microscale aerosol and cloud physics as detailed as possible, whereas the continental domain size and efficient codes will allow for both studying weather and regional climate. The model system is applied in a first extended case study for Europe for a cloudy five day period in August 2005. <br><br> The model results show that the mean cloud droplet number concentration of clouds is correlated with the structure of the terrain, and we present a terrain slope parameter TS to classify this dependency. We propose to use this relationship to parameterize the probability density function, PDF, of subgrid-scale cloud updraft velocity in the activation parameterizations of climate models. <br><br> The simulations show that the presence of cloud condensation nuclei (CCN) and clouds are closely related spatially. We find high aerosol and CCN number concentrations in the vicinity of clouds at high altitudes. The nucleation of secondary particles is enhanced above the clouds. This is caused by an efficient formation of gaseous aerosol precursors above the cloud due to more available radiation, transport of gases in clean air above the cloud, and humid conditions. Therefore the treatment of complex photochemistry is crucial in atmospheric models to simulate the distribution of CCN. <br><br> The mean cloud droplet number concentration and droplet diameter showed a close link to the change in the aerosol. To quantify the net impact of an aerosol change on the precipitation we calculated the precipitation susceptibility <i>β</i> for the whole model domain over a period of two days with an hourly resolution. The distribution function of <i>β</i> is slightly skewed to positive values and has a mean of 0.23. Clouds with a liquid water path LWP of approximately 0.85 kg m<sup>−2</sup> are on average most susceptible to aerosol changes in our simulations with an absolute value of <i>β</i> of 1. The average <i>β</i> for LWP between 0.5 kg m<sup>−2</sup> and 1 kg m<sup>−2</sup> is approximately 0.4

Insecurity for compact surfaces of positive genus

A pair of points in a riemannian manifold $M$ is secure if the geodesics between the points can be blocked by a finite number of point obstacles; otherwise the pair of points is insecure. A manifold is secure if all pairs of points in $M$ are secure. A manifold is insecure if there exists an insecure point pair, and totally insecure if all point pairs are insecure. Compact, flat manifolds are secure. A standing conjecture says that these are the only secure, compact riemannian manifolds. We prove this for surfaces of genus greater than zero. We also prove that a closed surface of genus greater than one with any riemannian metric and a closed surface of genus one with generic metric are totally insecure.Comment: 37 pages, 11 figure