An analytical proof of Hardy-like inequalities related to the Dirac operator

We prove some sharp Hardy-type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.     

Optical properties of a high-efficiency glass ceramic X-ray storage phosphor

The turbidity, photoluminescence, and photo-stimulated luminescence (PSL) of fluorozirconate glass containing barium chloride nano- and micro-crystals have been measured for samples prepared by isochronal (70 min) annealing over a temperature range of 220–283°C, and correlated with the microstructure as determined by X-ray diffraction measurements. Crystallization of hexagonal phase barium chloride commences at around 220°C, but until 275°C the material retains excellent transparency although it displays negligible PSL. Between 275°C and 277°C, the hexagonal phase converts to the orthorhombic phase, the transparency abruptly decreases, and the PSL rises to a value of around 13% of that found for the commercial storage phosphor BaFBr:Eu. For a slightly higher temperature of 280°C, new phases appear which correspond to the onset of bulk crystallization, and at 283°C the relative PSL rises to 33%, while the transparency falls further. The trade-off between optical transparency and PSL over this narrow temperature window for X-ray imaging plate applications is briefly discussed.     

New developments in X-ray storage phosphors

We found a significant PSL effect in Eu²⁺-doped fluorozirconate glasses (ZBLAN) which were additionally doped with Br⁻ or Cl⁻ ions. The PSL is attributed to the characteristic emission of Eu²⁺ present in nano-crystallites of BaBr₂ or BaCl₂, which form in the glass upon annealing. The metastable hexagonal form of BaX₂ (X=Br,Cl) is always formed first before it is converted into the stable orthorhombic form. The particle size increases upon annealing and so does the PSL efficiency of the glass ceramic. However, there is a saturation of the PSL efficiency, which is for Br⁻ doping about 9% and for Cl⁻-doping about 80% of the Eu-doped BaFBr standard. The particle size was determined by transmission electron microscopy (TEM). The TEM results show a clear tendency for bigger particles for longer annealing at the expense of its number. The particle size for the most efficient phosphor is about 100 nm.     

Filter products of C0-semigroups and ultraproduct representations for Lie groups

We provide a new approach to filter products of C₀-semigroups and prove a spectral theorem for the generator and its filter product. In a similar fashion, we construct ultraproducts of strongly continuous unitary representations of locally compact groups and study spectral theoretic connections between the representations and their ultraproducts. In the case of Lie groups, our investigations are extended to the infinitesimal representation.     

Unsupervised constrained radar imaging of low resolution targets

A linear spectral estimation technique, the PDFT algorithm, is used as part of a nonlinear iterative reconstruction scheme to obtain improved radar images. The iterative PDFT algorithm is used to address the limited resolution problem inherent to imaging objects buried in soil and hidden under foliage. This is achieved by subsequent application of two properties of the PDFT algorithm: the energy parameter of the PDFT algorithm is used to determine the target shape, while the shape information in turn is used to obtain super-resolved images. We describe algorithms able to exploit both properties automatically and without manual intervention. Two methods are investigated in particular, one iteratively optimizing the constraints by monitoring the energy parameter, the other method computing energy values for all points, from which a weighted prior function is determined. In addition, we discuss variants of both algorithm which provide an optimized trade-off between computation time and performance. Additional attention is given to situations, where a known target is embedded in an unknown background. Imaging results are presented for both synthetic and measured data.     

The emergence of a giant component in random subgraphs of pseudo‐random graphs

Let G be a d‐regular graph G on n vertices. Suppose that the adjacency matrix of G is such that the eigenvalue λ which is second largest in absolute value satisfies λ = o(d). Let G_p with p = α/d be obtained from G by including each edge of G independently with probability p. We show that if α < 1, then whp the maximum component size of G_p is O(log n) and if α > 1, then G_p contains a unique giant component of size Ω(n), with all other components of size O(log n). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004