Polynomial operator matrices as generators: The general case

Of concern are systems of linear evolution equations

On a problem proposed by H. Braß concerning the remainder term in quadrature for convex functions

Summary We prove that the error inn-point Gaussian quadrature, with respect to the standard weight functionw1, is of best possible orderO(n ^–2) for every bounded convex function. This result solves an open problem proposed by H. Braß and published in the problem section of the proceedings of the 2. Conference on Numerical Integration held in 1981 at the Mathematisches Forschungsinstitut Oberwolfach (Hämmerlin 1982; Problem 2). Furthermore, we investigate this problem for positive quadrature rules and for general product quadrature. In particular, for the special class of Jacobian weight functionsw _, (x)=(1–x)(1+x), we show that the above result for Gaussian quadrature is not valid precisely ifw _, is unbounded.Dedicated to Prof. H. Braß on the occasion of his 55th birthday     

On perturbations of linear m-accretive operators on reflexive Banach spaces

In this note we prove some results on the m-accretivity of sums and products of linear operators. In particular we obtain the following theorem: LetA, B be two m-accretive operators on a reflexive Banach space. IfA is invertible and (A)^–1 B is accretive thenBA ^–1 andA+B are m-accretive.     

Pseudoreceptor model for ryanodine derivatives at calcium release channels

This paper describes the generation of a pseudoreceptor model for ryanodine receptor (RyR) modulating ryanoids in rabbit skeletal muscle. For this purpose, the molecular modelling software PrGen was applied to correlate experimentally determined and calculated free energies of binding for a set of 15 ryanodine derivatives. The final model indicates a narrow cleft with hydrogen bond donor and acceptor capacities (represented by an Asn) as most crucial for binding the pyrrole carboxylate substituent at C3 of ryanodine. In addition, hydrophobic residues flank the aromatic pyrrole ring (Tyr, Phe, and Ile). Two of those residues (Tyr and Ile) interact with the 2-isopropyl moiety, which seems to contribute to binding. Opposite to the pyrrole locus, a second hydrophobic region (represented by a Leu) restricts ryanodine derivatives in their longitudinal axis and leads to the discrimination of equatorial and axial positioned methyl groups and of polar substituents at C9. Finally, a charged glutamate residue generates strong hydrogen bonding and electrostatic interactions with the hydroxyl groups at C10 and C15. For this binding-site model – composed of six amino acid residues – a correlation for the training set ligands of R = 0.99 (Q² = 0.975) and a root mean square (rms) deviation of 0.568 kcal/mol for the prediction of the binding energies of four test set ligands was obtained. Based on this pseudoreceptor model the putative topology of the real binding site of ryanoids will be discussed.     

On chebyshev quadrature and variance of quadrature formulas

The purpose of this note is to give an example which demonstrates that one can achieve much higher algebraic precision with a quadrature rule with small but not minimal variance than with a Chebyshev rule with minimal variance.     

Positivity and Stability for One-Sided Coupled Operator Matrices

Many evolutionary systems can be described by an abstract Cauchy problem governed by an operator matrix. Assuming this problem to be one-sided coupled and well-posed we study in this paper the positivity and the stability of the associated matrix semigroup. The abstract results are illustrated by several examples.