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.     

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     

Antiferromagnetic three-dimensional order induced by carboxylate bridges in a two-dimensional network of [Cu3(dcp)2(H2O)4] trimers

A new Cu(II) complex, [Cu(3)(dcp)(2)(H(2)O)(4)](n), with the ligand 3,5-pyrazoledicarboxylic acid monohydrate (H(3)dcp) has been prepared by hydrothermal synthesis, and it crystallizes in the monoclinic space group P2(1)/c with a = 11.633(2) A, b = 9.6005(14) A, c = 6.9230(17) A, beta = 106.01(2) degrees, and Z = 2. In the solid state structure of [Cu(3)(dcp)(2)(H(2)O)(4)](n), trinuclear [Cu(3)(dcp)(2)(H(2)O)(4)] repeating units in which two dcp(3-) ligands chelate the three Cu(II) ions with the central Cu(II) ion, Cu(1) (on an inversion center), link to form infinite 2D sheets via syn-anti equatorial-equatorial carboxylate bridges between Cu(2) atoms in adjacent trimers. These layers are further linked by syn-anti axial-equatorial carboxylate bridging between Cu(1) atoms in adjacent sheets resulting in the formation of a crystallographic 3D network. A detailed analysis of the magnetic properties of [Cu(3)(dcp)(2)(H(2)O)(4)](n) reveals that the dcp(3-) ligand acts to link Cu(II) centers in three different ways with coupling constants orders of magnitude apart in value. In the high temperature region above 50 K, the dominant interaction is strongly antiferromagnetic (J/k(B) = -32 K) within the trimer units mediated by the pyrazolate bridges. Below 20 K, the trimer motif can be modeled as an S = 1/2 unit. These units are coupled to their neighbors by a ferromagnetic interaction mediated by the syn-anti equatorial-equatorial carboxylate bridge. This interaction has been estimated at J(2D)/k(B) = +2.8 K on the basis of a 2D square lattice Heisenberg model. Finally, below 3.2 K a weak antiferromagnetic coupling (J(3D)/k(B) = -0.1 K) which is mediated by the syn-anti axial-equatorial carboxylate bridges between the 2D layers becomes relevant to describe the magnetic (T, H) phase diagram of this material.     

Surprising selectivity in the transformation of dimethoxy azaindoles

Transformation of 4,7-dimethoxy-6-azaindole into 4-hydroxy-7-methoxy-6-azaindole or 7-hydroxy-4-methoxy-6-azaindole can be readily controlled by careful selection of a reagent. Treatment with concentrated HCl results in hydrolysis at the 4-position exclusively, while TMS-I provides demethylation at the 7-position only. Products were unambiguously identified by single crystal X-ray crystallography.     

GC-MS and GC-MS-MS analysis of a complex essential oil

Summary The alcoholic portion of avetiver oil (Bourbon), the gas chromatogram of which shows more than 100 components, was analyzed by combined GC-MS with different ionization methods. This enabled us to identify major components. The specificity of the method was improved by using combined gas chromatography-tandem mass spectrometry in the collision-activated-dissociation mode.     

Direct determination of the enantiomeric purity of oxyphenonium using chiral HPLC with post-column extraction detection

Summary A method is described for the determination of the enantiomeric purity (enantiomeric excess) of the anticholinergic drug oxyphenonium. The method for this quaternary ammonium compound is based on the direct HPLC analysis with a chiral stationary phase. Two kinds of 1-acid glycoprotein-bonded phases were used.For the detection a post-column extraction with fluorescence detection of the ion-pair counter ion dimethoxyantracene sulphonate was used.     

The synthesis and studies towards the self-replication of bis(capped porphyrins)

Connecting two facially-protected porphyrins was expected to lead to an equal mixture of laterally-bridged doubly-protected bis-porphyrins; one in which the two porphyrin units were protected on the same face (syn) and one with the two prophyrin units protected on opposite faces (anti). Addition of a co-factor (bidentate ligand) was expected to lead predominantly to the syn-bis-porphyrin by a templated self-replication process. This concept was explored using Baldwin's capped porphyrin. Bis(capped porphyrins) were synthesised in several steps starting from zinc(II) capped porphyrin 2. Nitration of 2 followed by reduction and photo-oxidation yields a mixture of zinc(II) porphyrindiones 7 and 8 that can separated by HPLC. The condensation of 2 molar eq. of zinc(II) porphyrin-7,8-dione 8 with 1,2,4,5-benzenetetramine leads to the formation of a 1:1 mixture of syn- and anti-dizinc(II) bis(7,8-capped porphyrins), 11 and 12, respectively, that have almost identical spectroscopic properties. These two geometric isomers were distinguished by significant differences in their molecular recognition properties. Likewise the syn- and anti-dizinc(II) bis(2,3-capped porphyrins), 9 and 10, respectively, are synthesised from the related zinc(II) capped porphyrin-2,3-dione 7, and were also identified using molecular recognition studies. The molecular recognition properties of these bis(capped porphyrins) were utilised in studies of self-replicating porphyrin systems. The results show that tetraazaanthraceno-bis-porphyrins 9-12 can catalyse their own formation but self-replication was not observed. These results highlight the potential that these interesting hosts have as templates in supramolecular chemistry, synthesis and catalysis.