Isoperimetric Inequalities for Minimal Submanifolds in Riemannian Manifolds: A Counterexample in Higher Codimension

For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B.White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four-dimensional ball B with the following properties: (1) B has strictly convex boundary. (2) There exists a complete nonconstant geodesic. (3) There does not exist a closed geodesic in B.Comment: 11 pages, We changed the title and added a section that exhibits the relation between our example and the question posed by Brian White concerning isoperimetric inequalities for minimal submanifold

Operating Characteristics of the Multiple Critical Venturi System and Secondary Calibration Nozzles Used for Weight-Flow Measurements in the Langley 16-Foot Transonic Tunnel

An investigation has been conducted in the Langley 16 Foot Transonic Tunnel to determine the weight flow measurement characteristics of a multiple critical Venturi system and the nozzle discharge coefficient characteristics of a series of convergent calibration nozzles. The effects on model discharge coefficient of nozzle throat area, model choke plate open area, nozzle pressure ratio, jet total temperature, and number and combination of operating Venturis were investigated. Tests were conducted at static conditions (tunnel wind off) at nozzle pressure ratios from 1.3 to 7.0

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

Small-Angle X-ray and neutron scattering from diamond single crystals

Results of Small-Angle Scattering study of diamonds with various types of point and extended defects and different degrees of annealing are presented. It is shown that thermal annealing and/or mechanical deformation cause formation of nanosized planar and threedimensional defects giving rise to Small-Angle Scattering. The defects are often facetted by crystallographic planes 111, 100, 110, 311, 211 common for diamond. The scattering defects likely consist of clusters of intrinsic and impurity-related defects; boundaries of mechanical twins also contribute to the SAS signal. There is no clear correlation between concentration of nitrogen impurity and intensity of the scattering.Comment: 6 pages, 5 figures; presented at SANS-YuMO User Meeting 2011, Dubna, Russi

The comprehensive model system COSMO-ART - Radiative impact of aerosol on the state of the atmosphere on the regional scale

A new fully online coupled model system developed for the evaluation of the interaction of aerosol particles with the atmosphere on the regional scale is described. The model system is based on the operational weather forecast model of the Deutscher Wetterdienst. Physical processes like transport, turbulent diffusion, and dry and wet deposition are treated together with photochemistry and aerosol dynamics using the modal approach. Based on detailed calculations we have developed parameterisations to examine the impact of aerosol particles on photolysis and on radiation. Currently the model allows feedback between natural and anthropogenic aerosol particles and the atmospheric variables that are initialized by the modification of the radiative fluxes. The model system is applied to two summer episodes, each lasting five days, with a model domain covering Western Europe and adjacent regions. The first episode is characterised by almost cloud free conditions and the second one by cloudy conditions. The simulated aerosol concentrations are compared to observations made at 700 stations distributed over Western Europe. <br><br> For each episode two model runs are performed; one where the feedback between the aerosol particles and the atmosphere is taken into account and a second one where the feedback is neglected. Comparing these two sets of model runs, the radiative feedback on temperature and other variables is evaluated. <br><br> In the cloud free case a clear correlation between the aerosol optical depth and changes in global radiation and temperature is found. In the case of cloudy conditions the pure radiative effects are superposed by changes in the liquid water content of the clouds due to changes in the thermodynamics of the atmosphere. In this case the correlation between the aerosol optical depth and its effects on temperature is low. However, on average a decrease in the 2 m temperature is still found. <br><br> For the area of Germany we found on average for both cases a reduction in the global radiation of about 6 W m<sup>2</sup>, a decrease of the 2 m temperature of 0.1 K, and a reduction in the daily temperature range of −0.13 K

The type numbers of closed geodesics

A short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesicsComment: 29 pages, an appendix to the Russian translation of "The calculus of variations in the large" by M. Mors