Convergence of multistep discretizations of DAEs

Standard ODE methods such as linear multistep methods encounter difficulties when applied to differential-algebraic equations (DAEs) of index greater than 1. In particular, previous results for index 2 DAEs have practically ruled out the use of all explicit methods and of implicit multistep methods other than backward difference formulas (BDFs) because of stability considerations. In this paper we embed known results for semi-explicit index 1 and 2 DAEs in a more comprehensive theory based on compound multistep and one-leg discretizations. This explains and characterizes the necessary requirements that a method must fulfill in order to be applicable to semi-explicit DAEs. Thus we conclude that the most useful discretizations are those that avoid discretization of the constraint. A freer use of e.g. explicit methods for the non-stiff differential part of the DAE is then possible.Dedicated to Germund Dahlquist on the occasion of his 70th birthdayThis author thanks the Centro de Estadística y Software Matemático de la Universidad Simón Bolivar (CESMa) for permitting her free use of its research facilities.Partial support by the Swedish Research Council for Engineering Sciences TFR under contract no. 222/91-405.     

Preservice Elementary Teachers' Use of Mathematics in a Project‐Based Science Approach

The use of a project‐based science (PBS) approach to teaching encourages students to integrate mathematics and science in meaningful ways as they create projects. As a beginning study of how students use mathematics in such an approach, an analysis of 23 projects developed by preservice elementary teachers enrolled in an elementary science course was conducted. Findings showed that students made a number of different types of mathematical errors and underutilized data representation and summary forms. Implications included the importance of developing methods for supporting the use of mathematical tools in utilizing a project‐based approach and considering ways that such tools mediate scientific thinking.     

Novel Enantioselective Synthesis of Both Enantiomers of Furan‐2‐yl Amines and Amino Acids

A new enantioselective synthesis of furan‐2‐yl amines and amino acids is described, in which the key step is the oxazaborolidine‐catalyzed enantioselective reduction of O‐benzyl (E)‐ and (Z)‐furan‐2‐yl ketone oximes to the corresponding chiral amines. The chirality of the furan‐2‐yl amines is fully controlled by the appropriate choice of the geometrical isomer of the O‐benzyl oxime. Oxidation of the furan ring furnished amino acids in high yields.     

A Study of Amorphous Precursors for PbZrO3-PbTiO3 Based Ceramic Materials

Amorphous precursors for PbZrO₃ and PbTiO₃ ceramics were prepared from lead acetate and the transition metaln-propoxide inn-propanol orn-butoxide inn-butanol and hydrolysed with an excess of water. According to GLC and TGA/EGA analyses, the type of alkoxide group influences distinctly the structure of heterometallic precursors, i.e., oxo or acetate bridging, and the amounts of hydroxyl and organic groups bound to the metal network. The local environments of metal atoms in the amorphous precursors were also studied by EXAFS. The analysis reveals that in Pb−Zr precursors alkoxide groups modify the coordination spheres of the zirconium atoms. Conversely, local environments of both lead and titanium atoms within the analysed range of 3.4 A depend weakly on the type of alkoxide used.     

Novel dye-attached macroporous films for cadmium, zinc and lead sorption: Alkali Blue 6B-attached macroporous poly(2-hydroxyethyl methacrylate)

Alkali Blue 6B-attached poly(2-hydroxyethyl methacrylate) (poly(HEMA)) microporous films were investigated as chelate forming sorbents for heavy metal removal. Poly(HEMA) microporous films were prepared by UV-initiated photo-polymerization of HEMA in the presence of an initiator (azobisisobutyronitrile (AIBN)). Alkali Blue 6B was attached covalently. These films with a swelling ratio of 58%, and carrying 14.8 mmol Alkali Blue 6B m(-2) which were then used in the removal of Cd(II), Zn(II) and Pb(II) from aqueous media. Adsorption rates were very high, equilibrium was achieved in about 30 min. The maximum adsorption of heavy metal ions onto the Alkali Blue 6B-attached films were 41.4 mmol m(-2) for Cd(II), 52.4 mmol m(-2) for Zn(II), and 64.5 mmol m(-2) for Pb(II). When the heavy metal ions competed during the adsorption from a mixture the adsorption values for Cd(II), Zn(II) and Pb(II) were quite close. Heavy metal ions were desorbed by using 0.1 M HNO(3). A significant amount of the adsorbed heavy metal ions (up to 95%) could be desorbed in 30 min. Repeated adsorption/desorption cycles showed the feasibility of these novel dye-attached microporous films for heavy metal removal.     

Intercalation of Cyclic Amines into α-Titanium Phosphate

The intercalation of some amines (aniline, benzylamine, cyclohexylamine,piperidine, pyridine, pyrazine and piperazine) into -titaniumphosphate, Ti(HPO₄)_2×H₂O,has been investigated by the batch method and/or by exposing the host to thevapour of the amines. The changes in the interlayer distance of the solidduring the intercalation process was followed by X-ray powder diffraction.The new intercalates were characterised by chemical and thermal analysis.Materials with a monolaminar and/or bilaminar arrangement of amine moleculesin the phosphate interlayer region are obtained depending on the nature ofthe amine. Due to steric hindrance, saturated phases are not obtained forall amines studied. The thermal decomposition of the intercalates (nitrogenatmosphere), takes place in three stages: dehydration, removal of amines andcondensation of the hydrogenphosphate to pyrophosphate groups.     

Selective adsorption, pre-concentration and matrix elimination for the determination of Pb(II), Cd(II), Hg(II) and Cr(III) using 1,5,9,13-tetrathiacyclohexadecane-3,11-diol anchored poly (p-chloromethylstyrene-ethyleneglycoldimethacrylate) microbeads

Poly(p-chloromethylstyrene-ethyleneglycoldimethacrylate) polymeric microbeads, poly(p-CMS-EGDMA), were synthesized and 1,5,9,13-tetrathiacyclohexadecane-3,11-diol (S4HD) was attached chemically onto the polymeric microbeads. Characterization of all microbeads was done by Fourier transform-infrared spectrometry (FT-IR) and elemental analyzer. The amount of attached 1,5,9,13-tetrathiacyclohexadecane-3,11-diol to the polymer was found to be 2.23 mmol g⁻¹ polymers. The ligand attached microbeads, poly(p-CMS-EGDMA-S4HD), were used to examine the adsorption capacity of Pb(II), Cd(II), Hg(II) and Cr(II) ions for recovery, pre-concentration and the matrix elimination by changing the pH and the initial metal ion concentrations and also adsorption kinetics of the studied metal ions was determined. Pre-concentration factors for the studied toxic metal ions were found to be more than 500-fold and recovery was between 92 and 106%. In the drinking, lake, tap and sea-water samples from water lands, ultra-trace toxic metal ion concentrations were determined easily by using ligand modified microbeads after pre-concentration because of the high pre-concentration factor and easily matrix elimination using ligand modified microbeads. Reference sea-water material was used for the validation of the method and it was found that recovery, pre-concentration and the matrix elimination were performed perfectly. For the desorption of the toxic metal ions, 3 M HCl containing 0.8 M thiourea was used and desorption ratio was obtained more than 96%.