On the Smolyak cubature error for analytic functions

We consider Smolyak's construction for the numerical integration over the d‐dimensional unit cube. The underlying class of integrands is a tensor product space consisting of functions that are analytic in the Cartesian product of ellipses. The Kronrod–Patterson quadrature formulae are proposed as the corresponding basic sequence and this choice is compared with Clenshaw–Curtis quadrature formulae. First, error bounds are derived for the one‐dimensional case, which lead by a recursion formula to error bounds for higher dimensional integration. The applicability of these bounds is shown by examples from frequently used test packages. Finally, numerical experiments are reported. This revised version was published online in June 2006 with corrections to the Cover Date.     

Pharmaceutical analysis of 5-aminolevulinic acid in solution and in tissues

Quantification of 5-aminolevulinic acid (ALA) in solution, and methods used to achieve this, have been extensively reported in the literature. However, validated methods have only rarely been presented and never have methods been compared. Due to a necessity in drug delivery research for optimised and validated methods for determination of ALA in solution, this paper compares, for the first time, two such methods validated to International Conference on Harmonisation (ICH) standards. Of major importance, derivatisation of ALA with acetyl acetone and formaldehyde was found to be more suitable for routine fluorimetric HPLC analysis of ALA than derivatisation with o-phthaldialdehyde and 2-mercaptoethanol. This former method was successfully utilised in the comparison of in vitro drug release from a proprietary ALA cream and a novel bioadhesive patch-based system. In addition, determination of ALA in tissue is necessary to compare different topical formulations, in terms of their ability to deliver the drug successfully, and different tissue types, to assess their barrier properties to penetration of the drug. Consequently, this paper also describes the use of liquid scintillation spectroscopy as an analytical tool for rapid, convenient and routine quantification of ALA in tissue and determination of penetration depth following topical application of creams and patches.     

On a conjecture of D. B. Hunter

We consider errors of positive quadrature formulas applied to Chebyshev polynomials. These errors play an important role in the error analysis for many function classes. Hunter conjectured that the supremum of all errors in Gaussian quadrature of Chebyshev polynomials equals the norm of the quadrature formula. We give examples, for which Hunter's conjecture does not hold. However, we prove that the conjecture is valid for all positive quadratures if the supremum is replaced by the limit superior. Considering a fixed positive quadrature formula and the sequence of all Chebyshev polynomials, we show that large errors are rare.     

Principles of verified numerical integration

We describe methods for the numerical calculation of integrals with verified error bounds. The problems range from integration over an interval to integration of parameter-dependent integrands over the whole d-variate space. It is argued, why we use bounds for the integrands in the complex plane as a tool for bounding the error in our own integration software.     

Energy Balance of Low-Voltage Spark Discharges

In this work we are presenting an experimental and theoretical analysis of the energy balance of low-voltage spark discharges. It is pointed out that the greatest part of the energy from the central hot plasma region is lost in the form of radiation, but the radiation from electrode particles (here aluminium) can be important, too.     

Optimal stochastic quadrature formulas for convex functions

We study optimal stochastic (or Monte Carlo) quadrature formulas for convex functions. While nonadaptive Monte Carlo methods are not better than deterministic methods, we prove that adaptive Monte Carlo methods are much better.Supported by a Heisenberg scholarship of the DFG.     

Electron spray ionization mass spectrometry and 2D 31P NMR for monitoring 18O/16O isotope exchange and turnover rates of metabolic oligophosphates

A new method was here developed for the determination of ¹⁸O-labeling ratios in metabolic oligophosphates, such as ATP, at different phosphoryl moieties (α-, β-, and γ-ATP) using sensitive and rapid electrospray ionization mass spectrometry (ESI-MS). The ESI-MS-based method for monitoring of ¹⁸O/¹⁶O exchange was validated with gas chromatography–mass spectrometry and 2D ³¹P NMR correlation spectroscopy, the current standard methods in labeling studies. Significant correlation was found between isotopomer selective 2D ³¹P NMR spectroscopy and isotopomer less selective ESI-MS method. Results demonstrate that ESI-MS provides a robust analytical platform for simultaneous determination of levels, ¹⁸O-labeling kinetics and turnover rates of α-, β-, and γ-phosphoryls in ATP molecule. Such method is advantageous for large scale dynamic phosphometabolomic profiling of metabolic networks and acquiring information on the status of probed cellular energetic system.     

Stieltjes Polynomials and Gauss-Kronrod Quadrature for Jacobi Weight Functions

Summary. Let w _,beta (t)=(1–t)(1+t), ,>–1, denote the Jacobi weight function. For 0,<5/2 it is shown that on compact subintervals of (–1,1), the corresponding Stieltjes polynomials and their derivatives are asymptotically equal to certain Jacobi polynomials. This also leads to asymptotic representations of those weights of Gauss-Kronrod quadrature formulae, which correspond to nodes in a compact subinterval of (–1,1). On the other hand, it is demonstrated that for the parameters satisfying min(,)0 and max(,)>5/2 the Stieltjes polynomial has only few real zeros and that Gauss-Kronrod quadrature is not possible. So far, results of the above type have been known only for ultraspherical weight functions for 02. For >3, the impossibility of Gauss-Kronrod quadrature has been proved by the authors recently. Mathematics Subject Classification (1991):33C10, 33C45, 42C05, 65D32The second author is sponsored by a Heisenberg scholarship of the Deutsche Forschungsgemeinschaft.     

Fast calculation of coefficients in the Smolyak algorithm

For many numerical problems involving smooth multivariate functions on d-cubes, the so-called Smolyak algorithm (or Boolean method, sparse grid method, etc.) has proved to be very useful. The final form of the algorithm (see equation (12) below) requires functional evaluation as well as the computation of coefficients. The latter can be done in different ways that may have considerable influence on the total cost of the algorithm. In this paper, we try to diminish this influence as far as possible. For example, we present an algorithm for the integration problem that reduces the time for the calculation and exposition of the coefficients in such a way that for increasing dimension, this time is small compared to dn, where n is the number of involved function values.