A smoothing-regularization approach to mathematical programs with vanishing constraints

We consider a numerical approach for the solution of a difficult class of optimization problems called mathematical programs with vanishing constraints. The basic idea is to reformulate the characteristic constraints of the program via a nonsmooth function and to eventually smooth it and regularize the feasible set with the aid of a certain smoothing- and regularization parameter t>0 such that the resulting problem is more tractable and coincides with the original program for t=0. We investigate the convergence behavior of a sequence of stationary points of the smooth and regularized problems by letting t tend to zero. Numerical results illustrating the performance of the approach are given. In particular, a large-scale example from topology optimization of mechanical structures with local stress constraints is investigated.     

Implicit QR algorithms for palindromic and even eigenvalue problems

In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques. D. S. Watkins partly supported by Deutsche Forschungsgemeinschaft through Matheon, the DFG Research Center Mathematics for key technologies in Berlin.     

Nanosized aerosols from consumer sprays: experimental analysis and exposure modeling for four commercial products

Consumer spray products are already on the market in the cosmetics and household sector, which suggest by their label that they contain engineered nanoparticles (ENP). Sprays are considered critical for human health, because the lungs represent a major route for the uptake of ENP into the human body. To contribute to the exposure assessment of ENP in consumer spray products, we analyzed ENP in four commercially available sprays: one antiperspirant, two shoe impregnation sprays, and one plant-strengthening agent. The spray dispersions were analyzed by inductively coupled plasma mass spectrometry (ICPMS) and (scanning-) transmission electron microscopy ((S)TEM). Aerosols were generated by using the original vessels, and analyzed by scanning mobility particle sizer (SMPS) and (S)TEM. On the basis of SMPS results, the nanosized aerosol depositing in the respiratory tract was modeled for female and male consumers. The derived exposure levels reflect a single spray application. We identified ENP in the dispersions of two products (shoe impregnation and plant spray). Nanosized aerosols were observed in three products that contained propellant gas. The aerosol number concentration increased linearly with the sprayed amount, with the highest concentration resulting from the antiperspirant. Modeled aerosol exposure levels were in the range of 10¹⁰ nanosized aerosol components per person and application event for the antiperspirant and the impregnation sprays, with the largest fraction of nanosized aerosol depositing in the alveolar region. Negligible exposure from the application of the plant spray (pump spray) was observed.     

A note on medieval microfabrication: the visualization of a prayer nut by synchrotron‐based computer X‐ray tomography

One of the most fascinating objects in the Rijksmuseum (Amsterdam, The Netherlands) is an early 16th century prayer nut. This spherical wooden object measures 4 cm in diameter and consists of two hemispheres connected with a small hinge so that it can be opened. The interior of the nut holds wood carvings with scenes from the life of Christ. These miniature reliefs show an incredible degree of finish with carving details well beyond the millimetre scale. In the present paper it is shown how synchrotron‐based computer X‐ray tomography revealed the structure and fabrication method of the bead. The central part of the relief was cut from a single piece of wood, rather than assembled from multiple components, underlining the extraordinary manual dexterity of its maker. In addition, a piece of fibrous material contained in the inner structure of the bead is revealed. This may have served as a carrier for an odorous compound, which would be in line with the religious function of the prayer nut.