Shannon sampling II: Connections to learning theory

We continue our study [S. Smale, D.X. Zhou, Shannon sampling and function reconstruction from point values, Bull. Amer. Math. Soc. 41 (2004) 279–305] of Shannon sampling and function reconstruction. In this paper, the error analysis is improved. Then we show how our approach can be applied to learning theory: a functional analysis framework is presented; dimension independent probability estimates are given not only for the error in the L² spaces, but also for the error in the reproducing kernel Hilbert space where the learning algorithm is performed. Covering number arguments are replaced by estimates of integral operators.     

Prediction of yield and long‐term failure of oriented polypropylene: Kinetics and anisotropy

The time‐dependent yield and failure behavior of off‐axis loaded uniaxially oriented polypropylene tape is investigated. The yield and failure behavior is described with an anisotropic viscoplastic model. A viscoplastic flow rule is used with an equivalent stress, based on Hill's anisotropic yield criterion, and the Eyring flow theory combined with a critical equivalent strain definition. This model is based on factorization of the rate and draw ratio dependence and is capable of quantitatively predicting the rate, angle and draw ratio dependence of the yield stress as well as time‐to‐failure in various off‐axis tensile loading conditions characterized solely from the transverse direction. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2026–2035, 2009     

The signal/noise of an HMBC spectrum can depend dramatically upon the choice of acquisition and processing parameters

The effect of various acquisition and processing parameters on the sensitivity of HMBC spectra for typical organic molecules has been systematically investigated. For molecules in the 200–600 molecular weight range, an acquisition time of 0.2 to 0.4 s, a recycle time of no more than 1.0 s, optimization for ⁿJ_CH = 8 Hz and 512 time increments (with two‐ to fourfold linear prediction) are recommended. Some form of sine bell weighting along f₂ and either Gaussian or sine bell weighting along f₁ is suggested. The use of a 0.1‐s acquisition time and/or Gaussian or exponential weighting along f₂ can result in dramatic sensitivity loss, particularly for correlation peaks involving protons with complex splitting patterns, and should be avoided. Copyright © 2009 John Wiley & Sons, Ltd.     

Selbstheilende Polymere

Even though the field of self‐healing is rarely known so far – self healing materials are already present at our market. Nevertheless just due to modern scientific concepts we are now able to understand the basic mechanistic steps in a more detailed way. Further progress on this field will open access to materials with a wide range of adjustable properties. Therefore, applications of such self healing materials are not limited – assuming the market‐price is competitive and the elongated lifetime delivers an appropriate advantage. Already demonstrated for concrete and clear coatings for cars, the investigations done so far have generated materials with improved properties and prolonged durability.     

Measurement of turbulent wall velocity gradients using cylindrical waves of laser light

A new optical instrument has been developed for direct measurement of instantaneous velocity gradients at the bounding wall. Light emerging from two tiny optical slits in the surface is used to form a fan of fringes in the region very near the wall. Doppler frequency of the light scattered by the seed particles is directly proportional to the velocity gradient. The system has been used to measure the statistics of the streamwise and spanwise velocity gradients in a turbulent boundary layer. The streamwise and spanwise rms fluctuations were found to be 38% and 11% of the mean streamwise value respectively. The latter result is subject to a large uncertainty.List of symbols a slit width - B transfer function of the instrument - B ^* normalized transfer function - path-averaged value of the normalized transfer function - c constant in logarithmic velocity profile - C _f skin friction coefficient - d _f fringe spacing - f ₁,f₂ frequencies at the downstream and upstream slits resp. - f _d heterodyne Doppler frequency of the signal - g(t) instantaneous wall velocity gradient - G Clauser shape factor - mean wall velocity gradient - g rms value of the wall velocity gradient - H boundary layer shape factor - i, j, k unit vectors along x, y, z axes - wavenumber of laser light - L major axis of the elliptic cross-section of the laser sheet at the slit - l length of each slit - N number of cycles in a signal - N ₀ number of cycles without frequency-shifting - n difference of the unit vectors u ₁and u ₂ - P power transmitted through a slit - P _o power incident on a slit - Re ₁ Reynolds number based on displacement thickness and free-stream velocity - Re ₂ Reynolds number based on momentum thickness and free-stream velocity - S spacing between the slits - S ^* normalized spacing between the slits - u streamwise velocity - u ₁,u₂ unit vectors along the local directions of propagation of the two cylindrical waves - u _l linear term in the streamwise velocity profile - u _nl nonlinear terms in the streamwise velocity - u _nl ^* normalized value of nonlinear streamwise velocity - u _nl ^* mean streamwise velocity - u friction velocity - u⁺ mean velocity normalized with friction velocity - v velocity component normal to the wall - v ^* normal velocity normalized with streamwise velocity - V velocity vector - w spanwise component of velocity - W minor axis of the elliptic cross-section of the laser sheet at the slit - x streamwise distance - ± x _m limiting values of streamwise distance for a signal - x ^* normalized streamwise distance - x ^* normalized value of x _m - y normal distance - y ⁺ normal distance normalized with friction length scale - z spanwise distance - z ⁺ spanwise distance normalized with friction length scale - half-spreading angle of the cylindrical waves - boundary layer thickness in Coles' profile - ₁ displacement thickness of the boundary layer - ₂ momentum thickness of the boundary layer - ₃ energy thickness of the boundary layer - constant in logarithmic velocity profile - wavelength of laser light - kinematic viscosity - coefficient of wake function in Coles' profile Currently at LSTM, Universitat Erlangen-Nürnberg, Cauerstraße 4, W-8520 Erlangen, BRD