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     

Diagrams,orientations, and varieties

One of the central problems in the theory of ordered sets is to describe the orientations of the covering graph of an ordered set. We show that the particular operation called inversion, together with the classical constructions of retraction and product, provide a context for the classification of all such orientations. AMS subject classifications (1980). 06A10, 05C75, 05C20.     

MO-LCAO Calculations of Polymethines

Idealized polymethines are defined as conjugated chain molecules with complete bond equilization and charge alternation. A constant π-bond order along the chain with N atoms has been derived for three types of polymethines which differ by the nature of the heteroatomic end groups. Corresponding to the number of π-electrons they are termined (N + 1)π-, Nπ- or (N ? 1)π-polymethines. This classification embraces closed shell as well as open shell systems. The relations for the energy eigenvalues, coefficients and derived quantities have been obtained in closed form. The results for the different classes of polymethines are strongly related. Their molecular properties should be similar.     

Multiphoton-Ionization-Mass Spectrometry (MUPI-MS) [New Analytical Methods (34)]

Mass spectrometry is one of the most important analytical tools in chemistry, biology, medicine and related areas. During the past 30 years, methods have been developed, both for the qualitative as well as the quantitative analysis of a wide variety of substances. The introduction of lasers into chemistry has also profited mass spectrometry, since its nonlinear properties and its tunability open up fundamentally new frontiers. Multiphoton ionization mass spectrometry combines UV-spectroscopy and mass spectrometry, thus providing a two-dimensional method that enables substance-specific and even state-specific analyses. This progress report presents the fundamentals and possibilities of MUPI mass spectrometry and discusses investigations on amino acids, peptides, chlorophylls and sugars. For each substance, multiphoton ionization can be tuned in such a way that only the molecular ion is formed. An increase of laser intensity induces substance-specific fragmentations of the molecule, thus contributing to a quick and easy identification of the substance.     

A DFT study of the ground state multiplicities of linear vs angular polyheteroacenes

Unrestricted density functional calculations in combination with the broken-symmetry approach and spin-projection methods have been employed to study a series of formally 4n pi antiaromatic linear and angular polyheteroacenes. Calculations show that the linear polyheteroacene molecules have either stable singlet zwitterionic 6-9 or singlet diradical 5 ground states because they sacrifice the aromaticity of the central arene to form two independent cyanines. The corresponding angular compounds 10-14 have robust triplet states, since they cannot create independent cyanines to escape their overall antiaromaticity. An analysis based on the SOMO-SOMO energy splittings, their spatial distributions, and the spin density populations for the triplet states is presented to clarify the factors that determine their ground state multiplicities.