An envelope for the spectrum of a matrix

We introduce and study an envelope-type region ɛ(A) in the complex plane that contains the eigenvalues of a given n×n complex matrix A. ɛ(A) is the intersection of an infinite number of regions defined by cubic curves. The notion and method of construction of ɛ(A) extend the notion of the numerical range of A, F(A), which is known to be an intersection of an infinite number of half-planes; as a consequence, ɛ(A) is contained in F(A) and represents an improvement in localizing the spectrum of A.     

Characterization and magnetic properties of a "super stable" radical 1,3-diphenyl-7-trifluoromethyl-1,4-dihydro-1,2,4-benzotriazin-4-yl

1,3-Diphenyl-7-trifluoromethyl-1,4-dihydro-1,2,4-benzotriazin-4-yl (4), prepared in high yield via the catalytic oxidation of the corresponding amidrazone 5 by using Pd/C (1.6 mol %) and 1,8-diazabicyclo[5.4.0]undec-7-ene (0.1 equiv) in air, is stable in dichloromethane solutions in the presence of MnO(2) and KMnO(4). Furthermore, radical 4 is thermally stable well past its melting point (160-161 °C) with a decomposition onset temperature of 288 °C. X-ray studies show that radical 4 packs in equidistant slipped π-stacks along the a axis. Cyclic voltammetry shows two fully reversible waves, corresponding to the -1/0, 0/+1 processes. EPR studies indicate that the spin density is mainly delocalized on the triazinyl fragment of the heterocycle. Magnetic susceptibility measurements in the 5-300 K region showed that the radical obeys Curie-Weiss behavior down to 10 K (C = 0.376 emu·K·mol(-1) and θ = +1.41 K) consistent with weak ferromagnetic interactions between S = 1/2 radicals. Subsequent fitting of the magnetic data to a 1D ferromagnetic chain model provided an excellent fit (g = 2.00, J/k = +1.49 K) down to 10 K but failed to reproduce the subsequent decrease in χT at lower temperatures, which has been ascribed to the onset of weaker antiferromagnetic interactions between ferromagnetic chains.     

Studies on the Air Oxidation of Some Arsenic(III) Compounds

The air oxidation of As(III) oxides [(PhAsO) _x and Ph₂As-O-AsPh₂] and thioesters [Ph-As(SPh)₂, Ph₂As-SPh Me-As(SPh)₂, Me₂As-SPh], in chloroform and in methanol was studied. The air oxidation in chloroform was faster probably because the solubility of dioxygen is greater than in methanol, and it is favored by the electron-withdrawing phenyl groups bound to As(III). The products obtained were the arsonic or arsinic acids and diphenyl disulfide. In one case, diphenyl disulfide and thiophenol were produced. The results can be rationalized by assuming first hydrolysis of the As(III) compounds to arsonous or arsinous acids followed by their oxidation to arsonic and arsinic acids, which should involve the binding of dioxygen to As(III). The other hypothesis assumes first the binding of dioxygen to As(III) of these oxides and thioesters followed by the decomposition of the adducts. The binding of the ground state dioxygen to As(III) may have biochemical implications for toxicity or chemotherapy of arsenic(III) compounds.     

Low-frequency dipolar electromagnetic scattering by a solid ellipsoid in lossless environment

Electromagnetic wave scattering phenomena for target identification are important in many applications related to fundamental science and engineering. Here, we present an analytical formulation for the calculation of the magnetic and electric fields that scatter off a highly conductive ellipsoidal body, located within an otherwise homogeneous and isotropic lossless medium. The primary excitation source assumes a time-harmonic magnetic dipole, precisely fixed and arbitrarily orientated that operates at low frequencies and produces the incident fields. The scattering problem itself is modeled with respect to rigorous expansions of the electromagnetic fields at the low-frequency regime in terms of positive integral powers of the real wave number of the ambient. Obviously, the Rayleigh static term and a few dynamic terms are sufficient for the purpose of the present work, as the additional terms are neglected due to their minor contribution. Therein, the classical Maxwell's theory is suitably modified, leading to intertwined either Laplace's or Poisson's equations, accompanied by the impenetrable boundary conditions for the total fields and the limiting behavior at infinity. On the other hand, the complete spatial anisotropy of the three-dimensional space is secured via the introduction of the genuine ellipsoidal coordinate system, being appropriate for tackling incrementally such scattering boundary value problems. The nonaxisymmetric fields are obtained via infinite series expansions in terms of ellipsoidal harmonic eigenfunctions, providing handy closed-form solutions in a compact fashion, whose validity is verified by a straightforward reduction to simpler geometries of the metal object. The main idea is to demonstrate an efficient methodology, according to which the constructed analytical formulae can offer the appropriate environment for a fast numerical estimation of the scattered electromagnetic fields that could be useful for real data inversion.     

Luminescent poly(phenylene vinylene) derivatives with m‐terphenyl or 2,6‐diphenylpyridine kinked segments along the main chain: Synthesis,characterization, and stimulated emission

Two new poly(phenylene vinylene)s containing m‐terphenyl or 2,6‐diphenylpyridine kinked units along the main chain were synthesized and were used as luminescent and laser materials. They were prepared from Heck coupling of 2,5‐didodecyloxy‐1,4‐divinylbenzene with 4,4″‐dibromo‐3′‐phenyl‐m‐terphenyl or 2,6‐di(4‐bromophenyl)‐4‐phenylpyridine. The kinked units along the main chain caused a partial interruption of the conjugation leading to emission at a shorter wavelength as compared with poly(p‐phenylene vinylene). The polymers presented blue‐green emission in solution and green‐yellow emission in the solid state with photoluminescence maxima at 465–497 and 546–550 nm, respectively. Polymer containing 2,6‐diphenylpyridine segments emitted at a longer wavelength than that containing m‐terphenyl and displayed higher quantum yields in solution (0.61 and 0.40, respectively). The influence of the solvent and polymer concentration on the photoluminescence characteristics was investigated. The photoluminescence properties of protonated polymer containing 2,6‐diphenylpyridine segments were investigated both in solution and in film. Amplified spontaneous emission and tunable laser action were also obtained from the two polymers. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 2214–2224, 2004     

Synthesis of poly(n‐hexyl isocyanate‐b‐N‐vinylpyrrolidone) block copolymers by the combination of anionic and nitroxide‐mediated radical polymerizations: Micellization properties in aqueous solutions

A combination of anionic and nitroxide‐mediated radical polymerizations (dual initiator) was employed for the synthesis of poly(n‐hexyl isocyanate‐b‐N‐vinylpyrrolidone) (PHIC‐b‐PNVP) block copolymers. The samples were characterized with a size exclusion chromatograph equipped with refractive‐index and light scattering detectors as well as ¹H NMR spectroscopy. Relatively good control over the molecular weights was achieved. However, rather broad molecular weight distributions were obtained. The micellar properties of the PHIC‐b‐PNVP block copolymers were studied in water, which is a selective solvent for the poly(N‐vinylpyrrolidone) blocks. Static and dynamic light scattering revealed the presence of equilibrium between the micelles and clusters. The clusters partially deaggregated with increasing temperature. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 5719–5728, 2006     

A plausible explanation of the b-value in the Gutenberg-Richter law from first Principles

It is explained, from first Principles, why in the Gutenberg-Richter law (stating that the cumulative number of earthquakes N(>M) with magnitude greater than M is given by N(>M) ~ 10^−bM) the so called b-value is usually found to be around unity varying only slightly from region to region. The explanation is achieved just by applying the analysis in the natural time domain, without using any adjustable parameter.