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.     

Size‐Tunable Micron‐Bubbles Based on Fluorous–Fluorous Interactions of Perfluorinated Dendritic Polyglycerols

This paper describes the behavior of various generations of polyglycerol dendrimers that contain a perfluorinated shell. The aggregation in organic solvents is based on supramolecular fluorous–fluorous interactions, which can be described by means of ¹⁹F NMR spectroscopy. In order to study the interaction and aggregation phenomena of dendrimers with perfluorinated shell and perfluoro‐tagged guest molecules we investigated [G3.5]‐dendrimer with a perfluorinated shell in the presence of perfluoro‐tagged disperse red. Noteworthy, the interaction intensities varied in an unexpected manner depending on the equivalents of perfluoro‐tagged guest molecules added to the dendrimers in solution which then formed supramolecular complexes based on fluorous–fluorous interactions. We found that these complexes aggregated around residual air in the solvent to form stable micron‐sized bubbles. Their sizes correlated with the interaction intensities measured for certain dendrimer–guest molecule ratios. Degassing of the solutions led to a quasi phase separation between organic and fluorous phase, whereby the dendrimers formed the fluorous phases. Regassing the sample with air afforded bubbles of the initial size again.     

Novel silver‐tubing method for quantitative introduction of water into high‐temperature conversion systems for stable hydrogen and oxygen isotopic measurements

A new method to seal water in silver tubes for use in a TC/EA (thermal conversion/elemental analyzer) reduction unit using a semi‐automated sealing apparatus can yield reproducibilities (1 standard deviation) of δ²H and δ¹⁸O measurements of 1.0‰ and 0.06‰, respectively. These silver tubes containing reference waters may be preferred for the calibration of H‐ and O‐bearing materials analyzed with a TC/EA reduction unit. The new sealing apparatus employs a computer‐controlled stepping motor to produce silver tubes identical in length. The reproducibility of the mass of water sealed in tubes (in a range of 200–400 µg) can be as good as 1%. Approximately 99% of the sealed silver tubes are satisfactory (leak free). Although silver tubes sealed with reference waters are robust and can be shaken or heated to 110°C with no loss of integrity, they should not be frozen because the expansion during the phase transition of water to ice will break the cold seals and all the water will be lost. The tubes should be shipped in insulated containers. This new method eliminates air inclusions and isotopic fractionation of water associated with the loading of water into capsules using a syringe. The method is also more than an order of magnitude faster than preparing water samples in ordinary Ag capsules. Nevertheless, some laboratories may prefer loading water into silver capsules because expensive equipment is not needed, but users of this method are cautioned to apply the necessary corrections for evaporation, back exchange with laboratory atmospheric moisture, and blanks. Copyright © 2010 John Wiley & Sons, Ltd.     

Solution of porous medium type systems by linear approximation schemes

Summary The convergence of semi-discrete and discrete linear approximation schemes is analysed for nonlinear degenerate parabolic systems of porous medium type. The enthalpy formulation and variational technique are used. The semi-discretization used reduces the original parabolic P.D.E. to linear elliptic P.D.E. The algebraic correction arising from nonlinearities is treated by Newton-like iterations in finite steps. Some numerical experiments are discussed and compared with the analytical solutions.Supported by the Alexander von Humboldt-Foundation in 1989, and by SFB 123, University Heidelberg     

Accurate distance determination of nucleic acids via Förster resonance energy transfer: implications of dye linker length and rigidity

In Fo?rster resonance energy transfer (FRET) experiments, the donor (D) and acceptor (A) fluorophores are usually attached to the macromolecule of interest via long flexible linkers of up to 15 ? in length. This causes significant uncertainties in quantitative distance measurements and prevents experiments with short distances between the attachment points of the dyes due to possible dye-dye interactions. We present two approaches to overcome the above problems as demonstrated by FRET measurements for a series of dsDNA and dsRNA internally labeled with Alexa488 and Cy5 as D and A dye, respectively. First, we characterize the influence of linker length and flexibility on FRET for different dye linker types (long, intermediate, short) by analyzing fluorescence lifetime and anisotropy decays. For long linkers, we describe a straightforward procedure that allows for very high accuracy of FRET-based structure determination through proper consideration of the position distribution of the dye and of linker dynamics. The position distribution can be quickly calculated with geometric accessible volume (AV) simulations, provided that the local structure of RNA or DNA in the proximity of the dye is known and that the dye diffuses freely in the sterically allowed space. The AV approach provides results similar to molecular dynamics simulations (MD) and is fully consistent with experimental FRET data. In a benchmark study for ds A-RNA, an rmsd value of 1.3 ? is achieved. Considering the case of undefined dye environments or very short DA distances, we introduce short linkers with a propargyl or alkenyl unit for internal labeling of nucleic acids to minimize position uncertainties. Studies by ensemble time correlated single photon counting and single-molecule detection show that the nature of the linker strongly affects the radius of the dye's accessible volume (6-16 ?). For short propargyl linkers, heterogeneous dye environments are observed on the millisecond time scale. A detailed analysis of possible orientation effects (κ(2) problem) indicates that, for short linkers and unknown local environments, additional κ(2)-related uncertainties are clearly outweighed by better defined dye positions.