Elliptic difference equations and interior regularity

Summary Letu be a solution of an elliptic differential equation and letu _h be a solution of a corresponding elliptic difference equation. It is proved that ifu _h converges tou at a certain rate as the mesh-widthh tends to zero, then the appropriate difference quotient ofu h converges to the corresponding derivative ofu at the same rate whenh tends to zero.     

Eigenfunction analysis of stochastic backscatter for characterization of acoustic aberration in medical ultrasound imaging

Presented here is a characterization of aberration in medical ultrasound imaging. The characterization is optimal in the sense of maximizing the expected energy in a modified beamformer output of the received acoustic backscatter. Aberration correction based on this characterization takes the form of an aberration correction filter. The situation considered is frequently found in applications when imaging organs through a body wall: aberration is introduced in a layer close to the transducer, and acoustic backscatter from a scattering region behind the body wall is measured at the transducer surface. The scattering region consists of scatterers randomly distributed with very short correlation length compared to the acoustic wavelength of the transmit pulse. The scatterer distribution is therefore assumed to be delta correlated. This paper shows how maximizing the expected energy in a modified beamformer output signal naturally leads to eigenfunctions of a Fredholm integral operator, where the associated kernel function is a spatial correlation function of the received stochastic signal. Aberration characterization and aberration correction are presented for simulated data constructed to mimic aberration introduced by the abdominal wall. The results compare well with what is obtainable using data from a simulated point source.     

Estimation of ultrasound wave aberration with signals from random scatterers

A method for estimating waveform aberration from random scatterers in medical ultrasound imaging has been derived and its properties investigated using two-dimensional simulations. The method uses a weighted and modified cross-spectrum in order to estimate arrival time and amplitude fluctuations from received signals. The arrival time and amplitude fluctuations were used in a time delay, and a time delay and amplitude aberration correction filter, for evaluation of the retransmitted aberration corrected signal. Different types of aberration have been used in this study. First, aberration was concentrated on the plane of the transmitting/receiving array. Second, aberration was generated with a distributed aberrator. Both conditions emulated aberration from the human abdominal wall. Results show that for the concentrated aberrator, arrival time and amplitude fluctuations were estimated in close agreement with reference values. The reference values were obtained from simulations with a point source in the focal point of the array. Correction of the transmitted signal with a time delay, and a time delay and amplitude filter produced approximately equal correction as with point source estimates. For the distributed aberrator, the estimator performance degraded significantly. Arrival time and amplitude fluctuations deviated from reference values, leading to a limited correction of the retransmitted signal.     

An asymmetric iodolactonization reaction catalyzed by a zinc bis-proline–phenol complex

The intramolecular zinc bis-proline-phenol complex 2a was found to promote enantioselective iodolactonization reactions of both electron-rich and electron-poor 5-aryl-5-hexenoic acids affording δ-iodolactones in good chemical yields with up to 82% enantiomeric excess. The reactions were found to be insensitive to air and moisture, providing an experimentally simple protocol for synthetically useful compounds.     

The role of intermediate chain migration in propene polymerization using substituted {iPr(CpFlu)}ZrCl2/MAO catalysts

Comparison of pentad distributions obtained from NMR spectra and from a molecular mechanics‐based modeling approach is performed for the catalysts {ⁱPr(3‐X‐CpFlu)}ZrCl₂ (X = H, Me, Et, ⁱPr, ^tBu) at a range of different temperatures. In order to model the temperature dependency of the pentad distributions the variation in steric influence along with the change of the rotational energy level for catalysts with substituents displaying relatively low barriers to rotation is treated approximately by calculating energy profiles of 360° rotation of the alkyl groups. The temperature at which intermediate chain migration (back‐skip) or chain epimerization starts to be important seems to be rather constant (30–50°C) among the five catalysts. Even in the case of X = ^tBu, back‐skip seems to be unimportant for explaining the formation of isotactic polymer at room temperature.