Extrapolation methods in numerical integration

Extrapolation methods have been used for many years for numerical integration. The most well-known of these methods is Romberg integration. A survey by Joyce on the use of extrapolation in numerical analysis appeared in 1971 in which a substantial portion is devoted to numerical integration. In this paper, we shall survey progress made in this field since 1971. The topics surveyed include partition-extrapolation methods for dealing with singular integrands, the work of Lyness and others in generating asymptotic expansions for the error functional in one and several dimensions, the work of de Doncker and others on adaptive extrapolation and the work of Sidi and others on the evaluation of highly oscillatory infinite integrals by extrapolation. Other extrapolation techniques will be mentioned briefly.     

Direct Metalation of 1,3,5,7-Tetranitrocubane with Amides of Tin and Lead. Preparation and Chemistry of o-Nitrostannanes and o-Nitroplumbanes in the Cubyl Series(1)

Trialkyltin and trialkyllead amides react directly and remarkably easily with 1,3,5,7-tetranitrocubane to form mono- to tetrakis(trialkyltin)- and -(trialkyllead) tetranitrocubanes. These are all stable compounds. The X-ray crystallographic properties of some are given. The (trialkylstannyl)cubanes react with electrophiles such as bromine with unexpected cleavage of alkyltin bonds rather than cubyl-tin bonds. On the other hand, the (trialkylplumbyl)cubanes do ultimately undergo cubyl-lead bond cleavage. This provides a useful way to achieve substitution on the cubane nucleus and provides access to compounds such as 1,3,5,7-tetrabromo-2,4,6,8-tetranitrocubane. The lead derivatives of tetranitrocubane are also useful for making 1,2,3,5,7-pentanitrocubane and 1,2,3,4,5,7-hexanitrocubane.     

Duality and Multiplicative Stochastic Processes on Quantum Groups

An analogue of McKean's stochastic product integral is introduced and used to define stochastic processes with independent increments on quantum groups. The explicit form of the dual pairing (q-analogue of the exponential map) is calculated for a large class of quantum groups. The constructed processes are shown to satisfy generalized Feynman-Kac type formulas, and polynomial solutions of associated evolution equations are introduced in the form of Appell systems. Explicit calculations for Gauss and Poisson processes complete the presentation.     

Asymptotic neutrality of diatomic molecules

We use the no-binding theorem of Thomas-Fermi theory to prove that a large diatomic molecule is almost neutral. That is to say, that if the total nuclear charge isZ then the numberN of electrons required for the diatomic molecule to be stable satisfies . In contrast to the atomic case the emphasis here is on the lower bound onN. Our analysis will imply a new bound on the size of the molecule. These results are proved in the Born-Oppenheimer approximation. We also give bounds onN which hold for allZ by a very elementary method, not assuming the Born-Oppenheimer approximation.Work supported by a Danish Research Academy fellowship and U.S. National Science Foundation grant PHY-85-15288-A03Adress from September 1989: Department of Mathematics, The University of Michigan, Ann Arbor, Michigan 48109 USA     

Application of headspace solid-phase microextraction (HS-SPME) and comprehensive two-dimensional gas chromatography (GC x GC) for the chemical profiling of volatile oils in complex herbal mixtures

The coupling of headspace solid-phase microextraction (HS-SPME) with comprehensive two-dimensional gas chromatography (GC x GC) was shown to be a powerful technique for the rapid sampling and analysis of volatile oils in complex herbal materials. When compared to one-dimensional (1-D) GC, the improved analytical capabilities of GC x GC in terms of increased detection sensitivity and separation power were demonstrated by using HS-SPME/GC x GC for the chemical profiling (fingerprinting) of essential/volatile oils contained in herbal materials of increasing analytical complexity. More than 20 marker compounds belonging to Panax quinquefolius (American ginseng) can be observed within the 2-D contour plots of ginseng itself, a mixture of ginseng and another important herb (P. quinquefolius/Radix angelicae sinensis), as well as a mixture of ginseng and three other herbs (P. quinquefolius /R. angelicae sinensis/R. astragali/R. rehmanniae preparata). Such analytical capabilities should be important towards the authentication and quality control of herbal products, which are receiving increasing attention as alternative medicines worldwide. In particular, the presence of Panax in the herb formulation could be readily identified through its specific peak pattern in the 2-D GC x GC plot.