Positivity of exponential Runge–Kutta methods

In this article, we study positivity properties of exponential Runge–Kutta methods for abstract evolution equations. Our problem class includes linear ordinary differential equations with a time-dependent inhomogeneity. We show that the order of a positive exponential Runge–Kutta method cannot exceed two. On the other hand there exist second-order methods that preserve positivity for linear problems. We give some examples for the latter.     

Two novel bis­(2,9‐dimeth­yl‐1,10‐phenanthroline)copper(I) complexes: [Cu(dmp)2]2(PF6)2·0.5(bpmh)·CH3CN and [Cu(dmp)2][N(CN)2]

Two new salts of the cation [Cu^I(dmp)₂]⁺ (dmp is 2,9‐dimeth­yl‐1,10‐phenanthroline, C₁₄H₁₂N₂), namely bis­[bis­(2,9‐dimeth­yl‐1,10‐phenanthroline‐κ²N,N′)copper(I)] bis­(hexa­fluorophos­phate) hemi[bis­(4‐pyridylmethyl­idene)hydrazine] acetonitrile solvate, [Cu(C₁₄H₁₂N₂)₂]₂(PF₆)₂·0.5C₁₂H₁₀N₄·C₂H₃N or [Cu(dmp)₂]₂(PF₆)₂·0.5(bpmh)·CH₃CN [bpmh is bis­(4‐pyridylmethyl­idene)hydrazine, C₁₂H₁₀N₄], (I), and bis­(2,9‐dimeth­yl‐1,10‐phenanthroline‐κ²N,N′)copper(I) dicyanamide, [Cu(C₁₄H₁₂N₂)₂](C₂N₃) or [Cu(dmp)₂][N(CN)₂], (II), are reported. The Cu—N bond lengths and the distortion from idealized tetra­hedral geometry of the dmp ligands are discussed and compared with related compounds. The bpmh molecule in (I) is π–π stacked with a dmp ligand at a distance of 3.4 Å, rather than coordinated to the metal atom. The molecule lies across an inversion center in the crystal. In (II), the normally coordinated dicyanamide mol­ecule is present as an uncoordinated counter‐ion.     

A novel scoop-shaped conformation of C-methylcalix[4]resorcinarene in a bilayer structure

A new scoop-shaped conformation of C-methylcalix[4]resorcinarene has been identified; it is a hybrid of the previously observed crown- and flattened cone conformations.     

On CP, LP and other piecewise perturbation methods for the numerical solution of the Schr?dinger equation

The piecewise perturbation methods (PPM) have proven to be very efficient for the numerical solution of the linear time-independent Schr?dinger equation. The underlying idea is to replace the potential function piecewisely by simpler approximations and then to solve the approximating problem. The accuracy is improved by adding some perturbation corrections. Two types of approximating potentials were considered in the literature, that is piecewise constant and piecewise linear functions, giving rise to the so-called CP methods (CPM) and LP methods (LPM). Piecewise polynomials of higher degree have not been used since the approximating problem is not easy to integrate analytically. As suggested by Ixaru (Comput Phys Commun 177:897–907, 2007), this problem can be circumvented using another perturbative approach to construct an expression for the solution of the approximating problem. In this paper, we show that there is, however, no need to consider PPM based on higher-order polynomials, since these methods are equivalent to the CPM. Also, LPM is equivalent to CPM, although it was sometimes suggested in the literature that an LP method is more suited for problems with strongly varying potentials. We advocate that CP schemes can (and should) be used in all cases, since it forms the most straightforward way of devising PPM and there is no advantage in considering other piecewise polynomial perturbation methods.