A note on the complex semi‐definite matrix Procrustes problem

This note outlines an algorithm for solving the complex ‘matrix Procrustes problem’. This is a least‐squares approximation over the cone of positive semi‐definite Hermitian matrices, which has a number of applications in the areas of Optimization, Signal Processing and Control. The work generalizes the method of Allwright (SIAM J. Control Optim. 1988; 26 (3):537–556), who obtained a numerical solution to the real‐valued version of the problem. It is shown that, subject to an appropriate rank assumption, the complex problem can be formulated in a real setting using a matrix‐dilation technique, for which the method of Allwright is applicable. However, this transformation results in an over‐parametrization of the problem and, therefore, convergence to the optimal solution is slow. Here, an alternative algorithm is developed for solving the complex problem, which exploits fully the special structure of the dilated matrix. The advantages of the modified algorithm are demonstrated via a numerical example. Copyright © 2007 John Wiley & Sons, Ltd.     

Preparation and crystal structure of Na3SbO(PO4)2

The single phase compound Na₃SbO(PO₄)₂ was prepared by a solid state reaction. It crystallizes in the orthorhombic system, space group P2₁2₁2₁ with a = 6.964(1) Å, b = 9.284(2) Å, c = 12.425(2) Å, Z = 4. The structure was determined from 968 reflections collected on a Nonius CAD4 automatic diffractometer with MoKα radiation. The final R index and weighted R_w index are 0.031 and 0.039, respectively. The structure is built up from strings of corner-sharing SbO₆ octahedra to which phosphate groups are linked by two of their vertices. These chains, running parallel to the a-axis, are separated from each other by sodium atoms.     

A new formula for the saddle-to-scission time

We propose a new formula for the saddle-to-scission time that is more general that the one based on Kramers' approach. Its validity and applicability is then studied in detail. Such a formula is useful for the evaluation of the fission time of very heavy nuclei.     

Langevin description of the charge distribution of fragments originating from the fission of excited nuclei

The charge distribution of fragments originatingfrom the fission of the ²³⁶U compound nucleus is calculated within a stochastic approach based on Langevin equations. The elongation coordinate, the neck-thickness coordinate, and the charge-asymmetry coordinate are chosen as collective variables. The friction parameter of the charge mode is calculated on the basis of two nuclear-viscosity mechanisms, that of one-body and that of two-body dissipation. It is shown that the Langevin approach is applicable to studying isobaric distributions. In addition, the charge distribution in question is studied as a function of the excitation energy of the compound nucleus and as a function of the coefficient of two-body viscosity.     

Adsorption of nonionic surfactant triton X-100 on solids from aqueous and nonaqueous solutions

The adsorption of nonionic surfactant Triton X-100 on quartz sand and methylated quartz sand from water and toluene was investigated by means of spectrophotometry, the radiotracer technique, and wetting angle measurements.     

Structural study and electrical properties of Zr-doped Nd<Subscript>2</Subscript>Sn<Subscript>2</Subscript>O<Subscript>7</Subscript> pyrochlore compounds

Nd₂Sn₂O₇ pyrochlores with the substitution of Zr⁴⁺ were prepared by conventional ceramic double sintering technique. The single-phase formation was confirmed by X-ray diffraction and neutron diffraction techniques. Relative intensity calculations for X-ray diffraction analysis were performed for oxygen positional parametersx = 0.331 and 0.375, while Rietveld refinements were employed for neutron diffraction data. The neutron diffraction study revealed that there are only two anion sites with 48f and 8b positions. This indicates that the 8a site, i.e. O(3) sublattice, is completely vacant and the structure is a perfect cubic pyrochlore with space group Fd3m (O _h ⁷ ). From the conductivity measurements, it is observed that the electronic conductivity dominates from room temperature up to about 525 K and forT > 525 K, the oxygen ion conduction dominates the charge transport in these compositions. Complex impedance spectroscopy indicates the existence of grain and grain boundary as two separate elements.