A symmetric numerical range for matrices

Summary For each normv on ⁿ, we define a numerical rangeZ _v, which is symmetric in the sense thatZ _v=Z_vD, wherev ^D is the dual norm.We prove that, fora ⁿⁿ,Z _v(a) contains the classical field of valuesV(a). In the special case thatv is thel ₁-norm,Z _v(a) is contained in a setG(a) of Gershgorin type defined by C. R. Johnson.Whena is in the complex linear span of both the Hermitians and thev-Hermitians, thenZ _v(a),V(a) and the convex hull of the usualv-numerical rangeV _v(a) all coincide. We prove some results concerning points ofV(a) which are extreme points ofZ _v(a).Part of this research was done while the authors were at the Mathematische Institut, Technische Universität, München, West Germany. The first author presented these results at the Seminar on Matrix Theory (Positivity and Norms) held in Munich in December, 1974. The second author also acknowledges support from the National Science Foundation under grant GP 37978X.     

Small-angle X-Ray scattering from polyurethane elastomers

Small-angle X-ray scattering is shown to be an effective technique, complementary to mechanical and thermal methods, for the study of structural ordering in polyurethane elastomers. Wide variations in intensity were found for different sample compositions and are interpreted in terms of the degree of hard segment aggregation. The results support previous conclusions regarding the organization in these samples based on analysis by scanning thermal methods. The polyether polyurethanes studied showed a higher degree of organization than the equivalent polyesters, and longer urethane segments enhance the extent of domain structure. Studies of the temperature dependence of scattering strengthen the previous assignment of a transition at about 150°C to the disordering of domain structure.     

Lanthanide Identity Governs Guest-Induced Dimerization in LnIII[15-MC N(L-pheHA)-5])3+ Metallacrowns

Series of lanthanide-containing metallic coordination complexes are frequently presented as structurally analogous, due to the similar chemical and coordinative properties of the lanthanides. In the case of chiral (Ln^III[15-MC _N(L-pheHA)-5])³⁺ metallacrowns (MCs), which are well established supramolecular hosts, the formation of dimers templated by a dicarboxylate guest (muconate) in solution of neutral pH is herein shown to have a unique dependence on the identity of the MC's central lanthanide. Calorimetric data and nuclear magnetic resonance diffusion studies demonstrate that MCs containing larger or smaller lanthanides as the central metal only form monomeric host-guest complexes whereas analogues with intermediate lanthanides (for example, Eu, Gd, Dy) participate in formation of dimeric host-guest-host compartments. The driving force for the dimerization event across the series is thought to be a competition between formation of highly stable MCs (larger lanthanides) and optimally linked bridging guests (smaller lanthanides).     

Reversible heat generation rates of blended insertion electrodes

Journal of Solid State Electrochemistry - The blending of lithium ion insertion compounds is a novel and promising approach to design advanced electrodes for future secondary batteries. Recently,...     

Temperature and Solvent Effects on the Europium(III)-catalyzed Hydrolysis of Bis-(nitrophenyl) Phosphate

The hydrolysis of bis-(nitrophenyl) phosphate (BNPP) as model for secondary phosphate esters is analyzed at six different concentrations of Eu(III) ions and four temperatures between 303 and 343 K. Eyring plots yield activation enthalpy parameters which with, e.g. [Eu 3+ ] between 0.40 and 10.0 mM drop from 130 to 74 kJ mol m 1 , respectively, with a relatively small drop in the opposing entropy contributions. The observed saturation profiles allow for the first time to evaluate the influence of j H and j S separately on the Michaelis-Menten values K M and k cat , showing that the catalytic metal-ion effects are largely due to changes in k cat and dominated by favorable j H changes. Preliminary studies of the solvent influence show a surprising difference between water mixtures with either ethanol or DMSO. With, e.g. 40% ethanol, one observes a doubling of the rate constant, with 40% DMSO an almost 10-fold rate decrease. In both cases, a linear correlation with the solvent polarity parameter E T is found.     

Amine-Directed Palladium-Catalyzed C−H Halogenation of Phenylalanine Derivatives

An efficient primary-amine-directed, palladium-catalyzed C−H halogenation (X=I, Br, Cl) of phenylalanine derivatives is reported on a range of quaternary amino acid (AA) derivatives thanks to suitable conditions employing trifluoroacetic acid as additive. The extension of this original native functionality-directed ortho-selective halogenation was even demonstrated with the more challenging native phenylalanine as tertiary AA.     

A parameter-free smoothness indicator for high-resolution finite element schemes

This paper presents a postprocessing technique for estimating the local regularity of numerical solutions in high-resolution finite element schemes. A derivative of degree p ≥ 0 is considered to be smooth if a discontinuous linear reconstruction does not create new maxima or minima. The intended use of this criterion is the identification of smooth cells in the context of p-adaptation or selective flux limiting. As a model problem, we consider a 2D convection equation discretized with bilinear finite elements. The discrete maximum principle is enforced using a linearized flux-corrected transport algorithm. The deactivation of the flux limiter in regions of high regularity makes it possible to avoid the peak clipping effect at smooth extrema without generating spurious undershoots or overshoots elsewhere.     

The shadow of a collapsing dark star

The shadow of a black hole is usually calculated, either analytically or numerically, on the assumption that the black hole is eternal, i.e., that it has existed for all time. Here we ask the question of how this shadow comes about in the course of time when a black hole is formed by gravitational collapse. To that end we consider a star that is spherically symmetric, dark and non-transparent and we assume that it begins, at some instant of time, to collapse in free fall like a ball of dust. We analytically calculate the dependence on time of the angular radius of the shadow, first for a static observer who is watching the collapse from a certain distance and then for an observer who is falling towards the centre following the collapsing star.