Steiner Symmetrization is Continuous in W1,p

We study the continuity, smoothing, and convergence properties of Steiner symmetrization in higher space dimensions. Our main result is that Steiner symmetrization is continuous in W ^{1, p} in all dimensions. This implies that spherical symmetrization cannot be approximated in W ^{1, p} by sequences of Steiner symmetrizations. We also give a quantitative version of the standard energy inequalities for spherical symmetrization. Submitted: September 1996     

Atomic quantum zeno effect for ensembles and single systems

The so-called quantum Zeno effect is essentially a consequence of the projection postulate for ideal measurements. To test the effect, Itanoet al. have performed an experiment on an ensemble of atoms where rapidly repeated level measurements were realized by means of short laser pulses. Using dynamical considerations, we give an explanation why the projection postulate can be applied in good approximation to such measurements. Corrections to ideal measurements are determined explicitly. This is used to discuss how far the experiment of Itanoet al. can be considered as a test of the quantum Zeno effect. We also analyze a new possible experiment on a single atom where stochastic light and dark periods can be interpreted as manifestation of the quantum Zeno effect. We show that the measurement point of view gives a quick and intuitive understanding of experiments of the above type, although a finer analysis has to take the corrections into account.     

X-ray transmission properties of intelligent anvils in diamond anvil cells

High-pressure and/or high-temperature analysis of geo- and material science samples routinely employs diamond anvil cells (DACs) as a research instrument. In particular, DACs allow for various in situ characterizations (e.g. Raman and Fourier transform infra red spectroscopies, X-ray diffraction (XRD), X-ray spectroscopy including fluorescence (XRF) and absorption (XAS)) at elevated pressure and temperature. The measurement of pressure (P) and/or temperature (T) in the sample chamber is crucial, but not always accurate, more specifically in the case of low-pressure applications (a few GPa). The development of modified diamonds (intelligent anvils ‘i-anvils’) adapted to a new generation of DACs (intelligent diamond anvil cells: iDAC) can contribute to solve this problem, as the diamond itself serves as the PT sensor, being prepared, for example, by high-energy ion implantation [H. Bureau, M. Burchard, S. Kubsky et al., This volume (2006).] on a micrometric scale. Several most interesting measurement methods used with DACs are based on X-ray techniques (e.g. XRF, XRD, XAS). We present the first results of X-ray transmission measurements with iDACs, performed at the hard-X-ray microfocus beamline ID22 at the ESRF (European Synchrotron Radiation Facility), Grenoble, France. Sensor response to intense irradiation as a function of X-ray energy (E _ph∈10;18.1 Burchard, M., Zaitsev, A. M. and Maresh, W. 2003. Rev. Sci. Instr., 74: 1263–1266.  [Google Scholar] keV) was investigated. The values of the sensor were found to be independent of the irradiation in the investigated energy range and thus validate the use of these sensors for precise and reliable measurements on a wide range of applications with high-energy synchrotron radiation. No influence of the sensor on the X-ray transmission properties of the anvil has been found.     

Convergence to Equilibrium for a Thin-Film Equation on a Cylindrical Surface

The degenerate parabolic equation u _t + ? _x [u ³(u _xxx  + u _x  ? sin x)] = 0 models the evolution of a thin liquid film on a stationary horizontal cylinder. It is shown here that for each mass there is a unique steady state, given by a droplet hanging from the bottom of the cylinder that meets the dry region with zero contact angle. The droplet minimizes the associated energy functional and attracts all strong solutions that satisfy certain energy and entropy inequalities, including all positive solutions. The distance of solutions from the steady state cannot decay faster than a power law.     

Simulation of tropical-cyclone-like vortices in shallow-water ICON-hex using goal-oriented r-adaptivity

We demonstrate how efficient r-adapted grids for the prediction of tropical cyclone (TC) tracks can be constructed with the help of goal-oriented error estimates. The binary interaction of TCs in a barotropic model is used as a test case. We perform a linear sensitivity analysis for this problem to evaluate the contribution of each grid cell to an error measure correlated with the cyclone positions. This information allows us to estimate the local grid resolution required to minimize the TC position error. An algorithm involving the solution of a Poisson problem is employed to compute how grid points should be moved such that the desired local resolution is achieved. A hexagonal shallow-water version of the next-generation numerical weather prediction and climate model ICON is used to perform model runs on these adapted grids. The results show that for adequately chosen grid adaptation parameters, the accuracy of the track prediction can be maintained even when a coarser grid is used in regions for which the estimated error contribution is low. Accurate track predictions are obtained only when a grid with high resolution consisting of cells with nearly constant size and regular shape covers the part of the domain where the estimated error contribution is large. The number of grid points required to achieve a certain accuracy in the track prediction can be decreased substantially with our approach.     

High-pressure synthesis, crystal structure, and structural relationship of the first ytterbium fluoride borate Yb5(BO3)2F9

Yb₅(BO₃)₂F₉ was synthesized under high-pressure/high-temperature conditions in a Walker-type multianvil apparatus at 7.5 GPa and 1100 °C, representing the first known ytterbium fluoride borate. The compound exhibits isolated BO₃-groups next to ytterbium cations and fluoride anions, showing a structure closely related to the other known rare-earth fluoride borates RE₃(BO₃)₂F₃ (RE=Sm, Eu, Gd) and Gd₂(BO₃)F₃. Monoclinic Yb₅(BO₃)₂F₉ crystallizes in space group C2/c with the lattice parameters a=2028.2(4) pm, b=602.5(2) pm, c=820.4(2) pm, and β=100.63(3)° (Z=4). Three different ytterbium cations can be identified in the crystal structure, each coordinated by nine fluoride and oxygen anions. None of the five crystallographically independent fluoride ions is coordinated by boron atoms, solely by trigonally-planar arranged ytterbium cations. In close proximity to the above mentioned compounds RE₃(BO₃)₂F₃ (RE=Sm, Eu, Gd) and Gd₂(BO₃)F₃, Yb₅(BO₃)₂F₉ can be described via alternating layers with the formal compositions “YbBO₃” and “YbF₃” in the bc-plane.     

Zur Konstruktion komplexer Basen

A method of Stein [17] is generalized to construct in an explicit way complex bases of open holomorphic mappings, defined on normal complex analytic spaces. Using theorems concerning the existence of pushouts, we can sharpen a theorem of Wolffhardt ([22],§4) and generalize the above construction. As a consequence, we can prove for a big class of complex bases b:X→B, that the induced mapping π₁(X)→π₁(B) its surjective. We give a characterisation of locally simple and analytically locally simple analytic equivalence relations.