En route from patterns to algebra: comments and reflections

This article is a commentary on the papers in this issue of ZDM on Understanding Generalization in K-12 Algebra. It tries to interpret and understand at least some of the reported phenomena within a constructivist framework. The latter locates meaning not in the external representations but in the individual’s activity on and with them. This activity on the other hand is strongly regulated (but not determined) by social contracts and belief systems. From that, and considerations of the more general mathematical context, various suggestions for further and extended research can be drawn. One special aspect is that generalization processes will have to be complemented by some kind of instruction on the conventional algebraic symbolism and its usage.     

A spline interpolation method for solving boundary value problems of potential theory from discretely given data

An interpolation procedure using harmonic splines is described and analyzed for solving (exterior) boundary value problems of Laplace's equation in three dimensions (from discretely given data). The theoretical and computational aspects of the method are discussed. Some numerical examples are given.     

Energy Transfer from a Lumazine (= Pteridine-2,4(1H,3H)-dione) Chromophore to Bathophcnanthroline-ruthenium(II) Complexes during Hybridization Processes of DNA

We incorporated 6,7-dimethyllumazine chromophores and a [Ru^II(bathophenanthroline)] complex into different oligodeoxynucleotides, the former via the corresponding phosphoramidites of the 2′-deoxynucleosides and the latter specifically by the phosphonate approach at the 5′-end of synthetic oligonucleotides, directly in the course of their synthesis on solid support. Efficiency measurements of this energy-transfer system with the lumazine as energy donor and the Ru complex as energy acceptor allowed us to distinguish between the hybridized and the non-hybridized state of oligonucleotides. Thus, the approach should be of value in DNA probe technology for the detection of single-stranded DNA targets.