The Tabular Mode: Not Just Another Way to Represent a Function

Understanding mathematical functions as systematic processes involving the covariation of related variables is foundational in learning mathematics. In this article, findings are reported from two investigations examining students' thinking processes with functions. The first study focused on seven middle school students' explorations with a dynamic physical model. Students were videotaped during the 20‐ to 45‐minute sessions occurring two or three times per week over a period of 2 months, and students' written work was collected. The second investigation included 19 preservice elementary and middle school teachers enrolled in a course focusing on a combination of mathematical content and pedagogy. Participants' written problem‐solving work and reflective writing were collected, and participants were individually interviewed in 50‐minute videotaped sessions. Results from both investigations indicated that students often relied on a table, or some variation of a table, as a cognitive link advancing the development of their reasoning about underlying function relationships.     

Effect of multi-walled carbon nanotubes incorporation into benzyl methacrylate monolithic columns in capillary liquid chromatography

This work describes the preparation of polymer based monolithic materials and their use as stationary phases in capillary liquid chromatography. Multi-walled carbon nanotubes (MWCNT) were incorporated into a mixture containing benzyl methacrylate (BMA) and ethylene dimethacrylate (EDMA) as co-monomers. The optimized porogenic mixture was a ternary solution composed of cyclohexanol, 1,4-butandiol and butanol which resulted in a stable and homogeneous suspension. Six capillary columns with increasing amounts of MWCNT, from 0 to 0.4 mg mL(-1), were prepared by thermal polymerization in 0.32 mm (i.d.) and 150 mm length fused silica tubing. The chromatographic evaluation showed that the synthesized monolithic beds were mechanically stable while their porosity and permeability increased with the MWCNT content. The prepared capillary columns were tested for the separation of mixtures of ketones and phenols at an optimum flow rate of 2 μL min(-1). The results showed that incorporation of MWCNT slightly affected the retention while it enhanced the column efficiency by increasing the column efficiency by a factor of up to 9. This effect corresponded also to an improved resolution and full separation of the solutes.     

Determination of traces of terbium and indium by candoluminescence in various matrices by using a rod technique

The candoluminescence of terbium and indium was studied in various matrices coated on CaO/CaSO₄ rods. Terbium was studied in Y₂O₃, Gd₂O₃, ThO₂ and La₂O₃. Calibration graphs were linear over different ranges, up to 90 ng in some cases, with 0.2–0.3 ng μl^?1 detection limits and 2.8–3.2% relative standard deviations (c.s.d.). The candoluminescence of indium in Y₂O₃, Gd₂O₃ and Al₂O₃ was also examined. Calibration graphs were linear over different ranges, up to 20 ng in some cases, with detection limits of 0.08 ng μl^?1 for Y₂O₃ and Al₂O₃ and 0.2 ng μl^?1 for Gd₂O₃ with r.s.d. of 2.5–2.9%. There are many interferences.     

Enforced helicity: efficient access to self-organized helical molecular strands by the imine route

The condensation of the two oligoheterocyclic aldehydes 8 and 16 with the bis-hydrazine 17 gives the bis-hydrazones 1 and 2. These molecular strands are shown to adopt helical conformations of 1.5 and 2.5 turns, respectively. The helical shape of 1 has been confirmed and structurally characterized by X-ray crystallography. The results indicate that the pyrimidine-hydrazone unit is a satisfatory helicity codon, so that the facile hydrazone formation provides an efficient procedure for generating helical structures. This greatly widens the scope of the methodology based on designed heterocyclic sequences for enforcing helicity in molecular strands, and opens interesting routes towards a variety of derived structures.     

Spectrophotometric study of etodolac complexes with copper (II) and iron (III)

A rapid, simple, and selective method was developed for the determination of etodolac. The method depends on complexation of etodolac with copper (II) acetate and iron (III) chloride followed by extraction of complexes with dichloromethane and then measuring the extracted complexes spectrophotometrically at 684 and 385 nm in case of Cu (II) or Fe (III), respectively. Different factors affecting the reaction, such as pH, reagent concentration, and time, were studied. By use of Job's method of continuous variation, the molar ratio method, and elemental analysis, the stoichiometry of the reaction was found to be in the ratio of 1:2 and 1:3, metal:drug in the case of Cu (II) and Fe (III), respectively. The method obeys Beer's law in a concentration range of 2.00-9.00 and 0.50-2.00 mg/mL in case of Cu (II) and Fe (III), respectively. The stability of the complexes formed was also studied, and the reaction products were isolated for further investigation. The complexes have apparent molar absorptivities of about 32.14 +/- 0.97 and 168.32 +/- 1.12 for Cu (II) and Fe (III), respectively. The suggested procedures were successfully applied to the analysis of pure etodolac and its pharmaceutical formulations. The validity of the procedures was further ascertained by the method of standard additions, and the results were compared with other reported spectrophotometric methods and showed no significant difference in accuracy and precision.     

Reactions of ethyl β-methoxycrotonate with some α,β-unsaturated carbonyl compounds. Synthesis of 6-aroyl-5-aryl-3-methoxycyclohex-2-enones

3-Aryl-1-phenyl-2-propen-1-ones Ia-h and 1-aryl-3-phenyl-2-propen-1-ones Ii-ℓreacted with ethyl β-methoxy-crotonate (II) in the presence of sodium hydride in dry THF at 0° for 10 hours to give the corresponding 6-aroyl-5-aryl-3-methoxy-2-cyclohexen-1-ones III. The structures of the products were confirmed by ir, pmr, ¹³C nmr and uv spectroscopy.     

Helical molecular programming: supramolecular double helices by dimerization of helical oligopyridine-dicarboxamide strands

Helically preorganized oligopyridine-dicarboxamide strands are found to undergo dimerization into double helical supramolecular architectures. Dimerization of single helical strands with five or seven pyridine rings has been characterized by NMR and mass spectrometry in various solvent/ temperature conditions. Solution studies and stochastic dynamic simulations consistently show an increasing duplex stability with increasing strand length. The double helical structures of three different dimers was characterized in the solid phase by X-ray diffraction analysis. Both aromatic stacking and hydrogen bonding contribute the double helical arrangement of the oligopyridinedicarboxamide strand. Inter-strand interactions involve extensive face-to-face overlap between aromatic rings, which is not possible in the single helical monomers. Most hydrogen bonds occur within each strand of the duplex and stabilize its helical shape. Some inter-strand hydrogen bonds are found in the crystal structures. Dynamic studies by NMR as well as by molecular modeling computations yield structural and kinetic information on the double helices and on monomer-dimer interconversion. In addition, they reveal the presence of a spring-like extension/compression as well as rotational displacement motions.     

Characterization of aromatic polymers (benzenoid and heterocyclic) by cross-polarization,magic-angle 13C-NMR spectroscopy

The cross-polarization, magic-angle ¹³C-NMR spectra of solid samples of poly(p-phenylene), its nitro derivative, polymerized biphenyl, poly(p-phenylene sulfide), poly(1-methyl-2,5-pyrrolylene), poly(1-phenyl-2,5-pyrrolylene), poly(2,5-thienylene), and poly(2,5-selenienylene) and the high-resolution ¹³C-NMR spectrum of poly(m-phenylene) in solution are presented and assigned. In all cases the chemical shifts are similar to those of model compounds in solution. In most instances the resonances of the nonprotonated carbon atoms in the solid polymers are more intense than those of the protonated carbons and appear to depend on the charge distribution in the molecular orbitals.     

Liquid chromatographic/electrospray ionization mass spectrometric identification of the oxidation end-products of metformin in aqueous solutions

Metformin is an antihyperglycemic drug that exhibits some antioxidant properties. HO*-induced oxidation of metformin was studied in aqueous solution, in both aerated and deaerated conditions. Gamma radiolysis of water was used to generate HO* free radicals, capable of initiating one-electron oxidation of metformin. Oxidation end-products were identified by direct infusion mass spectrometry (MS) and high-performance liquid chromatography/mass spectrometry (HPLC/MSn): for every product, structure elucidation was based on its mass (simple mass spectra confirmed by HPLC/MS). In addition, fragmentation spectra (MS2, MS3 and MS4) and the determination of deuterium-hydrogen exchange sites provided valuable information allowing the complete identification of some of the end-products. At low radiation dose, four products were identified as primary ones, since they result from the direct attack of HO* radicals on metformin. These primary oxidation end-products were identified respectively as hydroperoxide of metformin, covalent dimer of metformin, methylbiguanide and 2-amino-4-imino-5-methyl-1,3,5-triazine. At high radiation dose, seven other products were identified as secondary ones, resulting from the HO*-induced oxidation of the primary end-products. A reaction scheme was postulated for the interpretation of the results.