Perspectives on Advanced Mathematical Thinking

This article sets the stage for the following 3 articles. It opens with a brief history of attempts to characterize advanced mathematical thinking, beginning with the deliberations of the Advanced Mathematical Thinking Working Group of the International Group for the Psychology of Mathematics Education. It then locates the articles within 4 recurring themes: (a) the distinction between identifying kinds of thinking that might be regarded as advanced at any grade level, and taking as advanced any thinking about mathematical topics considered advanced; (b) the utility of characterizing such thinking for integrating the entire curriculum; (c) general tests, or criteria, for identifying advanced mathematical thinking; and (d) an emphasis on advancing mathematical practices. Finally, it points out some commonalities and differences among the 3 following articles.     

Large Scale Linear Programs and Heuristics for the Design of Survivable Telecommunication Networks

We consider the problem of providing a minimum cost multi-service network subject to one link failure scenarios. We assume our network is fully meshed and demand is satisfied by using direct or two hop-paths. We provide a large scale linear programming formulation and propose and test two effective heuristics.     

Rational approximation of vertical segments

In many applications, observations are prone to imprecise measurements. When constructing a model based on such data, an approximation rather than an interpolation approach is needed. Very often a least squares approximation is used. Here we follow a different approach. A natural way for dealing with uncertainty in the data is by means of an uncertainty interval. We assume that the uncertainty in the independent variables is negligible and that for each observation an uncertainty interval can be given which contains the (unknown) exact value. To approximate such data we look for functions which intersect all uncertainty intervals. In the past this problem has been studied for polynomials, or more generally for functions which are linear in the unknown coefficients. Here we study the problem for a particular class of functions which are nonlinear in the unknown coefficients, namely rational functions. We show how to reduce the problem to a quadratic programming problem with a strictly convex objective function, yielding a unique rational function which intersects all uncertainty intervals and satisfies some additional properties. Compared to rational least squares approximation which reduces to a nonlinear optimization problem where the objective function may have many local minima, this makes the new approach attractive.     

Time reversal for infinite-dimensional diffusions

Summary In this paper we analize the reversibility of the diffusion property for the solution of certain infinite-dimensional systems of stochastic differential equations. Necessary and sufficient conditions ensuring this reversibility are given. The proofs use the techniques of the stochastic calculus of variations.This work was partly done when the first author was visiting the Centre de Recerca Matemàtica at Barcelona     

Efficient RNase H-directed cleavage of RNA promoted by antisense DNA or 2'F-ANA constructs containing acyclic nucleotide inserts

The ability of modified antisense oligonucleotides (AONs) containing acyclic interresidue units to support RNase H-promoted cleavage of complementary RNA is described. Manipulation of the backbone and sugar geometries in these conformationally labile monomers shows great benefits in the enzymatic recognition of the nucleic acid hybrids, while highlighting the importance of local strand conformation on the hydrolytic efficiency of the enzyme more conclusively. Our results demonstrate that the duplexes support remarkably high levels of enzymatic degradation when treated with human RNase HII, making them efficient mimics of the native substrates. Furthermore, interesting linker-dependent modulation of enzymatic activity is observed during in vitro assays, suggesting a potential role for this AON class in an RNase H-dependent pathway of controlling RNA expression. Additionally, the butyl-modified 2'F-ANA AONs described in this work constitute the first examples of a nucleic acid species capable of eliciting high RNase H activity while possessing a highly flexible molecular architecture at predetermined sites along the AON.     

Specific determination of 20 primary aromatic amines in aqueous food simulants by liquid chromatography-electrospray ionization-tandem mass spectrometry

A multi-analyte method without any pre-treatment steps using reversed-phase liquid chromatography-electrospray ionization-tandem mass spectrometry (LC-ESI-MS/MS) was developed and applied for the determination of 20 primary aromatic amines (PAA) associated with polyurethane (PUR) products or azo-colours. The method was validated in-house for water and 3% acetic acid food simulants using spiked migrates from plastic laminates. Detection limits ranged from 0.27 to 3 microg amine/L food simulants, and RSD values of within-laboratory reproducibility at the 2 microg PAA/L level ranged from 3.9 to 19%. PAA migration from plastic laminates and black nylon cooking utensils were determined with the method, and high levels of 4,4'-methylenedianiline and aniline were found in migrates from about half of the tested cooking utensils. The method fulfils present legislative demands in the EU for screening and verification of PAA migration from food contact materials.     

Rheological properties of multisticker associative polyelectrolytes in semidilute aqueous solutions

Multisticker associative polyelectrolytes of acrylamide (≈86 mol %) and sodium 2‐acrylamido‐2‐methylpropanesulfonate (≈12 mol %), hydrophobically modified with N,N‐dihexylacrylamide groups (≈2 mol %), were prepared with a micellar radical polymerization technique. This process led to multiblock polymers in which the length of the hydrophobic blocks could be controlled through variations in the surfactant‐to‐hydrophobe molar ratio, that is, the number of hydrophobes per micelle (N_H). The rheological behavior of aqueous solutions of polymers with the same molecular weight and the same composition but with two different hydrophobic block lengths (N_H = 7 or 3 monomer units per block) was investigated as a function of the polymer concentration with steady‐flow, creep, and oscillatory experiments. The critical concentration at the onset of the viscosity enhancement decreased as the length of the hydrophobic segments in the polymers increased. Also, an increase in the N_H value significantly enhanced the thickening ability of the polymers and affected the structure of the transient network. In the semidilute unentangled regime, the behavior of the polymer with long hydrophobic segments (N_H = 7) was studied in detail. The results were well explained by the sticky Rouse theory of associative polymer dynamics. Finally, the viscosity decreased with an increase in the temperature, mainly because of a lowering of the sample relaxation time. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1640–1655, 2004     

Electrochemical grafting by reduction of 4-aminoethylbenzenediazonium salt: Application to the immobilization of (bio)molecules

We propose in this study a simple and rapid way to produce stable amino-derivatized conductive surfaces for the subsequent immobilization of (bio)molecules. This was achieved through the use of (4-aminoethyl)benzenediazonium salt (AEBD), which was immobilized on glassy carbon and gold electrodes by its electrochemical reduction. The presence of terminal grafted amino functions was evidenced with XPS by analyzing N1s core level. Besides this conventional surface characterisation, an electrochemical strategy is proposed here to evidence the presence of immobilized amines, in which the chemical reactivity of amines towards 2,4,6-trinitrobenzenesulfonic acid (TNBS) is used. Surface-bound TNBS served as an electrochemical marker and was detected by cyclic voltammetry. Additionally, pre-modified gold electrodes with amino functions can be derivatized with biomolecules such as glutathione (GSH). Glutathione attachment was evidenced by studying the electrochemical behaviour of ferri/ferrocyanide redox before and after its immobilization. The functionalized electrodes were then used for the detection of copper ions in neutral aqueous solutions.