The Haar System as a Schauder Basis in Spaces of Hardy–Sobolev Type

We show that, for suitable enumerations, the Haar system is a Schauder basis in the classical Sobolev spaces in \({\mathbb R}^d\) with integrability \(1<p<\infty \) and smoothness \(1/p-1<s<1/p\). This complements earlier work by the last two authors on the unconditionality of the Haar system and implies that it is a conditional Schauder basis for a nonempty open subset of the (1 / p, s)-diagram. The results extend to (quasi-)Banach spaces of Hardy–Sobolev and Triebel–Lizorkin type in the range of parameters \(\frac{d}{d+1}<p<\infty \) and \(\max \{d(1/p-1),1/p-1\}<s<\min \{1,1/p\}\), which is optimal except perhaps at the end-points.     

On MRI turbulence quantification

Turbulent flow, characterized by velocity fluctuations, accompanies many forms of cardiovascular disease and may contribute to their progression and hemodynamic consequences. Several studies have investigated the effects of turbulence on the magnetic resonance imaging (MRI) signal. Quantitative MRI turbulence measurements have recently been shown to have great potential for application both in human cardiovascular flow and in engineering flow. In this article, potential pitfalls and sources of error in MRI turbulence measurements are theoretically and numerically investigated. Data acquisition strategies suitable for turbulence quantification are outlined. The results show that the sensitivity of MRI turbulence measurements to intravoxel mean velocity variations is negligible, but that noise may degrade the estimates if the turbulence encoding parameter is set improperly. Different approaches for utilizing a given amount of scan time were shown to influence the dynamic range and the uncertainty in the turbulence estimates due to noise. The findings reported in this work may be valuable for both in vitro and in vivo studies employing MRI methods for turbulence quantification.     

On an extreme class of real interpolation spaces

We investigate the limit class of interpolation spaces that comes up by the choice θ=0 in the definition of the real method. These spaces arise naturally interpolating by the J-method associated to the unit square. Their duals coincide with the other extreme spaces obtained by the choice θ=1. We also study the behavior of compact operators under these two extreme interpolation methods. Moreover, we establish some interpolation formulae for function spaces and for spaces of operators.     

Pulse wave velocity with 4D flow MRI: Systematic differences and age-related regional vascular stiffness

Purpose

The objective of this study was to compare multiple methods for estimation of PWV from 4D flow MRI velocity data and to investigate if 4D flow MRI-based PWV estimation with piecewise linear regression modeling of travel-distance vs. travel time is sufficient to discern age-related regional differences in PWV.

Methods

4D flow MRI velocity data were acquired in 8 young and 8 older (age: 23 ± 2 vs. 58 ± 2 years old) normal volunteers. Travel-time and travel-distance were measured throughout the aorta and piecewise linear regression was used to measure global PWV in the descending aorta and regional PWV in three equally sized segments between the top of the aortic arch and the renal arteries. Six different methods for extracting travel-time were compared.

Results

Methods for estimation of travel-time that use information about the whole flow waveform systematically overestimate PWV when compared to methods restricted to the upslope-portion of the waveforms (p < 0.05). In terms of regional PWV, a significant interaction was found between age and location (p < 0.05). The age-related differences in regional PWV were greater in the proximal compared to distal descending aorta.

Conclusion

Care must be taken as different classes of methods for the estimation of travel-time produce different results. 4D flow MRI-based PWV estimation with piecewise linear regression modeling of travel-distance vs. travel time can discern age-related differences in regional PWV well in line with previously reported data.     

Numerical Performance of Optimized Frolov Lattices in Tensor Product Reproducing Kernel Sobolev Spaces

Foundations of Computational Mathematics - In this paper, we deal with several aspects of the universal Frolov cubature method, which is known to achieve optimal asymptotic convergence rates in a...     

Possible Tests of Curvature Effects in Weak Gravitational Fields

The possibility of performing in proximity of the Earth experiments to test the predictions of general relativity for the trajectories of particles and photons in weak gravitational fields is studied. Three experiments are analyzed, all based on optical detection methods. The first experiment would allow to detect the tidal acceleration using a freely-falling Michelson interferometer. In a second experiment, based on laser-ranging methods, the relativistic correction to the classical gravity acceleration could be detected. Finally, the possibility of detecting light deflection induced by space curvature in an optical cavity with high-reflectivity mirrors is discussed.     

Spectroscopy of the 689 nm intercombination line of strontium using an extended-cavity InGaP/InGaAIP diode laser

A visible diode laser emitting at 690 nm at room temperature has been frequency-stabilized using a simple scheme based on optical feedback from a diffraction grating. The possibilities offered by these lasers for high-resolution spectroscopy are demonstrated by recording the sub-Doppler signal of the 689 nm intercombination line of strontium and resolving the hyperfine structure of the close P63 R70 (8, 4) transitions of iodine.     

Tunable frequency-controlled laser source in the near ultraviolet based on doubling of a semiconductor diode laser

Continuously tunable ultraviolet laser radiation at 397 nm was generated by doubling the output of a semiconductor diode laser. The fundamental radiation was provided by a 150 mW AlGaAs laser diode injected by a low-power AlGaAs laser diode which was frequency stabilized by optical feedback using a new scheme of a miniature external cavity. Second-harmonic generation was produced in a lithium-triborate crystal placed in a compact enhancement cavity. The fundamental radiation was used for sub-Doppler spectroscopy of the Ar I 4s ³ P ⁰ ₀–4p ¹ P ₁ transition at 795 nm; the second-harmonic radiation was used for spectroscopy of the Ca II 4² S _1/2–4² P _1/2 transition at 397 nm.