Starting in maximum-polynomial-degree Nordsieck-Gear methods

The intent of this paper is to show that the Nordsieck-Gear methods with maximum polynomial degree k+1, first described in [1], admit of matched starting methods which are exact for all polynomials of degree ?k+1. In general, it is shown that these starting methods yield starting errors of the required order, O(h^k+2), for all initial-value problems

$\text{y}{}^{\mathrm{(P)}}\text{(x)=f(x,y,y}{}^{\mathrm{(1)}}\text{,y}{}^{\mathrm{(2)}}\text{,…,y}{}^{\mathrm{(p?1)}}\text{),}$

$\text{y(0)=y}{}_0\text{, y}{}^{\mathrm{i}}\text{(0)=y}{}_0{}^{\mathrm{(i)}}\text{, i=1,2,…,p?1,}$

where f is k+1 times continuously differentiable in a neighborhood of the graph of the exact solution (x,?(x)), x?[0,X]. Two theorems are proved. The first is the constructive existence of an algorithm which requires (k-p+1)(k-p+2)/2 evaluations of the function f to obtain approximations of the method's required higher-order scaled derivatives at the origin:

$\text{h}{}^{\mathrm{p+1}}\text{y}\text{?}{}^{\mathrm{(p+1)}}\text{(0)}\text{(p+1)!}\text{,…,}\text{h}{}^{\mathrm{k}}\text{y}\text{?}{}^{\mathrm{(k)}}\text{(0)}\text{k!}\text{,}$

each of which is accurate to O(h^k+2). The second, less general theorem, shows that when f is a polynomial in x, y, and its higher order derivatives y⁽¹⁾,y⁽²⁾,…,y(^p?1), an algorithm can be constructed for obtaining the higher-order scaled derivatives exactly. These results lay to rest once and for all any heuristic arguments against varying corrector minus predictor coefficients for preserving maximal order (polynomial degree) because starting values are inexact. Furthermore, and perhaps most importantly, the maximum-polynomial-degree Nordsieck-Gear methods are shown to have a unique property of zero starting error for an important class of ordinary differential equations.     

The kinetic study of isothermal solid state decompositions: a combined numerical and experimental approach

This paper identifies and makes quantitative allowance for experimental errors in isothermal weight change data. For experiments which involve a purge gas to remove decomposition products, attention is directed to errors due to variations in the gas flow rate and the need to heat the sample initially to the temperature of study. For experiments carried out at reduced pressure, importance is also attached to the undesirable influence of draughting, buoyancy, pressure and conventional effects. An experimental procedure which minimises the errors due to some of these factors is discussed. The paper also examines how the non-linear least squares method employed to analyse the kinetic data may be extended to allow for error produced by the initial temperature change. This extended method is used to analyse weight change data which are consistent with a contracting-interface mechanism, and the results clearly demonstrate the improved accuracy of the kinetic parameter estimates.     

THE PHOTO-DTMERS OF ANTHRACENE, TETRACENE AND PENTACENE

Abstract— Ultra-violet irradiation of deoxygenated solutions of anthracene, tetracene or pentacene causes the disappearance of the polyacene absorption spectrum and the appearance of an absorption spectrum which is attributed to the photo-dimer of the polyacene. Molecular structures for ditetracene and dipentacene are proposed, based on that of dianthracene.     

Barium manganate in microwave-assisted oxidation reactions: synthesis of solvatochromic 2,4,6-triarylpyrimidines

Synthesis of 2,4,6-trisubstituted pyrimidines by tandem oxidation/heterocyclocondensation of propargylic alcohols and amidines is effected rapidly and efficiently under microwave dielectric heating using barium manganate as oxidant. Irradiation at 150 °C in ethanol–acetic acid for 45 min results in dramatic improvements in yield over the corresponding manganese dioxide-mediated method and establishes a rapid route to triarylpyrimidines in order to investigate their photophysical properties.     

Potassium bis­(carbonato‐O,O′)(ethylenedi­amine‐N,N′)­cobaltate(III) monohydrate at 173 K

The title salt, K[Co(C₂H₈N₂)(CO₃)₂]·H₂O, consists of a distorted octahedral cobalt complex anion and a seven‐coordinate potassium cation. Both metal atoms have crystallographic twofold symmetry, one C₂ axis passing through the Co atom and C—C bond, and another along a short K—O (water) bond of 2.600 Å (corrected for libration). The carbonate is bidentate to both cobalt and potassium and the water forms a hydrogen bond to a carbonate O atom.     

Sensitized near-infrared lanthanide luminescence from Nd(III)- and Yb(III)-based cyclen-ruthenium coordination conjugates

The development of novel mixed lanthanide-transition-metal (f-d) based supramolecular self-assemblies made from neodymium- and ytterbium-based tetraamide-functionalized cyclen complexes bearing a single 1,10-phenanthroline moiety coordinating to a RuII(bipy)2 (bipy = bipyridine) unit is described. Excitation of the Ru(II) metal-to-ligand charge-transfer band in water gave rise to long-wavelength sensitized emission from the Yb(III) or Nd(III) centers, observed in the near-infrared.     

Observation of parity nonconservation in møller scattering

We report a measurement of the parity-violating asymmetry in fixed target electron-electron (M?ller) scattering: A(PV)=[-175+/-30(stat)+/-20(syst)] x 10(-9). This first direct observation of parity nonconservation in M?ller scattering leads to a measurement of the electron's weak charge at low energy Q(e)(W)=-0.053+/-0.011. This is consistent with the standard model expectation at the current level of precision: sin((2)theta(W)(M(Z))((-)MS)=0.2293+/-0.0024(stat)+/-0.0016(syst)+/-0.0006(theory).     

Anti-de Sitter-space/conformal-field-theory Casimir energy for rotating black holes

We show that, if one chooses the Einstein static universe as the metric on the conformal boundary of Kerr-anti-de Sitter spacetime, then the Casimir energy of the boundary conformal field theory can easily be determined. The result is independent of the rotation parameters, and the total boundary energy then straightforwardly obeys the first law of thermodynamics. Other choices for the metric on the conformal boundary will give different, more complicated, results. As an application, we calculate the Casimir energy for free self-dual tensor multiplets in six dimensions and compare it with that of the seven-dimensional supergravity dual. They differ by a factor of 5/4.