A theorem of the alternatives for the equation Ax + B|x| = b

The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax?+?B|x|?=?b has a unique solution for each B with |B|?≤?D and for each b, or the equation Ax?+?B ₀|x|?=?0 has a nontrivial solution for some matrix B ₀ of a very special form, |B ₀|?≤?D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax|?=?λ|x| for some x?≠?0, and we prove that each square real matrix has an absolute eigenvalue.     

Simultaneous determination of eight biologically active thiol compounds using gradient elution-liquid chromatography with Coul-Array detection

The most active form of sulfur in biomolecules is the thiol group, present in a number of biologically active compounds. Here we present a comprehensive study of thiol analysis using flow injection analysis/HPLC with electrochemical detection. The effect of different potentials of working electrodes, of organic solvent contents in the mobile phase, and of isocratic and gradient elution on simultaneous determination of thiol compounds (cysteine, cystine, N-acetylcysteine, homocysteine, reduced and oxidised glutathione, desglycinephytochelatin, and phytochelatins) are described and discussed. These thiol compounds were well separated and detected under optimised HPLC-electrochemical detection conditions (mobile phase: 80 mM trifluoroacetic acid and methanol with a gradient profile starting at 97:3 (TFA:methanol), kept constant for the first 8 min, then decreasing to 85:15 during one minute, kept constant for 8 min, and finally increasing linearly up to 97:3 from 17 to 18 min; the flow rate was 0.8 mL/min, column and detector temperature 25 degrees C, and the electrode potential 900 mV). We were able to determine tens of femtomoles (3 S/N) of the thiols per injection (5 microL), except for phytochelatin5 whose detection limit was 2.1 pmole. This technique was consequently used for simultaneous determination of compounds of interest in biological samples (maize tissue and human blood serum).     

Low-threshold lasing eigenmodes of an infinite periodic chain of quantum wires

We study the lasing eigenvalue problems for a periodic open optical resonator made of an infinite grating of circular dielectric cylinders standing in free space, in the E- and H-polarization modes. If possessing a "negative-absorption" refractive index, such cylinders model a chain of quantum wires made of the gain material under pumping. The initial-guess values for the lasing frequencies are provided by the plane-wave scattering problems. We demonstrate a new effect: the existence of specific grating eigenmodes that have a low threshold of lasing even if the wires are optically very thin.     

A note on generating <Emphasis Type="Italic">P</Emphasis>-matrices

We prove that for any \({A,B\in\mathbb{R}^{n\times n}}\) such that each matrix S satisfying min(A, B) ≤ S ≤ max(A, B) is nonsingular, all four matrices A ^?1 B, AB ^?1, B ^?1 A and BA ^?1 are P-matrices. A practical method for generating P-matrices is drawn from this result.     

Self-assembled molecular rafts at liquid|liquid interfaces for four-electron oxygen reduction

The self-assembly of the oppositely charged water-soluble porphyrins, cobalt tetramethylpyridinium porphyrin (CoTMPyP(4+)) and cobalt tetrasulphonatophenyl porphyrin (CoTPPS(4-)), at the interface with an organic solvent to form molecular "rafts", provides an excellent catalyst to perform the interfacial four-electron reduction of oxygen by lipophilic electron donors such as tetrathiafulvalene (TTF). The catalytic activity and selectivity of the self-assembled catalyst toward the four-electron pathway was found to be as good as that of the Pacman type cofacial cobalt porphyrins. The assembly has been characterized by UV-visible spectroscopy, Surface Second Harmonic Generation, and Scanning Electron Microscopy. Density functional theory calculations confirm the possibility of formation of the catalytic CoTMPyP(4+)/ CoTPPS(4-) complex and its capability to bind oxygen.     

Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method

The simulation of nonlinear ultrasound propagation through tissue realistic media has a wide range of practical applications. However, this is a computationally difficult problem due to the large size of the computational domain compared to the acoustic wavelength. Here, the k-space pseudospectral method is used to reduce the number of grid points required per wavelength for accurate simulations. The model is based on coupled first-order acoustic equations valid for nonlinear wave propagation in heterogeneous media with power law absorption. These are derived from the equations of fluid mechanics and include a pressure-density relation that incorporates the effects of nonlinearity, power law absorption, and medium heterogeneities. The additional terms accounting for convective nonlinearity and power law absorption are expressed as spatial gradients making them efficient to numerically encode. The governing equations are then discretized using a k-space pseudospectral technique in which the spatial gradients are computed using the Fourier-collocation method. This increases the accuracy of the gradient calculation and thus relaxes the requirement for dense computational grids compared to conventional finite difference methods. The accuracy and utility of the developed model is demonstrated via several numerical experiments, including the 3D simulation of the beam pattern from a clinical ultrasound probe.     

Calculation and analysis of vibrational spectra of PbCl2-Sb2O3-TeO2 glass from first principles

Ternary PbCl₂-Sb₂O₃-TeO₂ system provides promising materials for photonic applications. Glasses in this system are thermally stable, their refraction index is about 2.2 and they are transparent from 400 nm up to 6 μm. The ab initio molecular dynamics (MD) simulations of xPbCl₂-10SbO_3/2−(90-x)TeO₂ glasses (x = 10, 20, 30, 40, and 50) were carried out using density functionals, the plane-wave basis set expansion, and the projector-augment waves (PAW). The resulting glassy structures were analyzed with the help of partial radial distribution functions (RDF) and partial coordination numbers. Good agreement with the available experimental data was confirmed. The axial and equatorial oxygen atoms of TeO₄ trigonal bi-pyramids were distinguished. Vibrational frequencies and their corresponding eigen-modes were found by diagonalization of the dynamical matrix. The simulated vibrational spectra were decomposed to determine the contributions from individual atomic species and some more complex structural features and compared with experimental Raman spectra. The spectral decomposition was carried out by projecting vibrational displacement vectors onto various normal modes of typical symmetric structural units. The method proved to be able to successfully interpret the experimental Raman spectra.     

A residual existence theorem for linear equations

A residual existence theorem for linear equations is proved: if ${A \in \mathbb{R}^{m\times n}}$ , ${b \in \mathbb{R}^{m}}$ and if X is a finite subset of ${\mathbb{R}^{n}}$ satisfying ${{\rm max}_{x \in X}p^T(Ax-b) \geq 0}$ for each ${p \in \mathbb{R}^{m}}$ , then the system of linear equations Ax = b has a solution in the convex hull of X. An application of this result to unique solvability of the absolute value equation Ax + B|x| = b is given.     

X-ray spectra and core hole energy curves of some diatomic molecules

Soft X-ray emission spectra of the molecules CO, N₂, NO and O₂ are examined for the purpose of deriving information on their core hole energy curves. Molecular force constant and equilibrium bond lengths are determined for the core hole species C*O and N₂*, and a qualitative analysis is made for CO*, N*O, NO* and O₂*. The results show that differences of equilibrium geometries between the core hole states and the ground states are very well reproduced (better than 1 pm) by SCF calculations within the Hartree-Fock formalism. Inclusion of anharmonicity in the Franck-Condon analysis gives a small but significant effect on the best fitted value for the core hole state bond lengths (about 0.5 pm). Oxygen is binding energies determined from the X-ray spectra are shown to agree with ESCA data, in most cases within a few tenths of an eV. Calculated ΔSCF transition energies reproduce the experimental data within a few eV.