On the numerical treatment of a small divisor problem

Summary Quasiperiodic solutions of perturbed integrable Hamiltonian equations such as weakly coupled harmonic oscillators can be found by constructing an appropriate coordinate transformation which leads to a small divisor problem. However the numerical difficulties are not merely caused by the small divisors but rather by the appearence of ghost solutions, which appear in any reasonable discretization of the problem. Our numerical treatment, based on a Newton-type iteration, guarantees an approximation of the relevant solution of the nonlinear problem. Numerical solutions are found up to a critical value of the coupling constant, which is much larger than the coupling constants allowed by the existence theory available so far.     

Molecular structure and conformation of 2,3-dichloro-1-propene as determined by gas-phase electron diffraction

The molecular structure and conformation of 2,3-dichloro-1-propene have been determined by gas-phase electron diffraction at nozzle temperatures of 24, 90 and 273°C. The molecules exist as a mixture of two conformers with the chlorine atoms anti (torsion angle ∠φ = 0°) or gauche (∠φ = 109°) to each other and with the anti form the more stable. The composition (mole fraction) of the vapor with uncertainties estimated at 2σ was found to be 0.55 (0.08), 0.49 (0.08) and 0.41 (0.10) at 24, 90 and 273°, respectively. These values correspond to an energy difference with estimated standard deviation ΔE° = E°_g-E°_a = 0.7 ± 0.3 kcal mol^?1 and an entropy difference ΔS° = S°_g-S°_a = 0.6 ± 0.9 cal mol^?1 K^?1. Some of the diffraction results, together with spectroscopic observations, permit the evaluation of an approximate torsional potential function of the form 2V = V₁ (1 - cos φ) + V₂ (1 - cos 2φ) + V₃ (1 - cos 3φ); the results are V₁ = 4.4 ± 0.5, V₂ = ?2.9 ± 0.5 and V₃ = 4.8 ± 0.2, all in kcal mol^?1. The results at 24°C for the distance (r_a) and angle (∠_α) parameters, with estimated uncertainties of 2σ, are: r(C_sp²-H) = 1.098(0.020)Å, r(C_sp³-H) = 1.103(0.020)Å, r(CC) = 1.334(0.009)Å, r(C-C) = 1.504(0.013)Å, r(C_sp²-Cl) = 1.752(0.021)Å, r(C_sp³-Cl) = 1.776(0.020)Å, ∠C-CC = 127.6(1.1)°, ∠C_sp³-C_sp²-Cl = 110.2(1.0), ∠C_sp²-C_sp³-Cl = 113.1(1.2)°, ∠H-C_sp³-H = 109.5° (assumed), ∠CC-H = 120.0° (assumed) and ∠φ = 108.9(3.4)°.     

Fractionation and purification of antioxidant peptides from Chinese sturgeon (Acipenser sinensis) protein hydrolysates prepared using papain and alcalase 2.4L

Protein hydrolysates have the potential to be natural and safer sources of bioactive peptides. In this study, two proteases were used to hydrolyze Chinese sturgeon (Acipenser sinensis) protein, and the hydrolysates were then purified to yield antioxidant peptides. The degree of hydrolysis of 23.56 % and 18.14 % was obtained using papain and alcalase 2.4L, respectivly, and hydrolysates had 96.80 % and 87.24 % total amino acid content, respectivly. The papain hydrolysate (PH) and alcalase 2.4L hydrolysate (AH) showed good antioxidant activity against DPPH^? (IC₅₀ of 3.64 and 3.15 mg/mL) and ABTS^?+ (IC₅₀ of 1.92 and 1.58 mg/mL), respectively. The low-molecular-weight (<1000 Da) fraction of both hydrolysates demonstrated the highest antiradical activity (IC₅₀ of 2.59 and 2.31 mg/mL, DPPH) and (IC₅₀ of 1.54 and 1.36 mg/mL, ABTS), respectively. Nine peptides were separated from both hydrolysates using reverse phase high performance liquid chromatography (RP-HPLC). The IC₅₀ for ABTS^?+ scavenging activity of peptide P5 with valine, glycine and asparagine (MW of 282.13 Da) from PH, and peptide P3 with histidine, glycine and alanine (MW of 302.74 Da) from AH was 0.89 and 0.72 mg/mL, respectively. The fractions and purified peptides obtained from Chinese sturgeon hydrolysates could be utilized as natural antioxidant substitutes in pharmaceuticals and food products.     

Two-beam interference detection of the changes in fibre structural parameters during low drawing process

A Pluta polarizing interference microscope with a fibre stretching device attached was used to detect the changes in molecular structure that occur during the fibre cold drawing process. Fibres drawn with low draw ratios show different types of deformation mechanisms. The interference patterns recorded at different draw ratios are used to relate the deformation mechanisms with the measured structural parameters. Some optical parameters are measured such as; refractive indices nand n birefringence n, the polarizability per unit volume , the orientation distribution function f(θ), the angle θ between the stretching direction and the chain axis, and the angle θ_m which the transition dipole moment of the molecular species makes with the molecular axis (chain axis). Three polymeric fibres (Polyethylene terephthalate PET, Polypropylene PP and Polyamide PA) were used in this study and their interferograms are illustrated.     

Growth and characterization of Mn- and Co-doped ZnO nanowires

A flow injection chemiluminescence method is proposed for the determination of cobalt, based on the strong catalytic effect of Cobalt(II) (1,10-phenanthroline)₃ complex on the lucigenin-periodate reaction in alkaline medium. Under the optimum experimental conditions, the chemiluminescence signal responded linearly to the concentration of cobalt(II) in the 1.0 × 10⁻⁹–3.0 × 10⁻⁷ g mL⁻¹ range with a detection limit of 4.4 × 10⁻¹⁰ g mL⁻¹ cobalt(II). The relative standard deviation for the determination of 5.0 × 10⁻⁸ g mL⁻¹ of cobalt was 2.3% in eleven replicated measurements. The method was successfully applied to the determination of cobalt(II) in pharmaceutical preparations.     

Determination of trace metals in industrial boron carbide by solid sampling optical emission spectrometry. Optimization of DC arc excitation (current, atmosphere and chemical modifier)

Graphite arc emission spectrometry has become possible as a result of the invention of novel types of optical spectrometers with Echelle-optics and semiconductor array detectors, and by the application of electronically controlled, high current arc generators. An optimization of the excitation parameters to boron carbide analysis is reported here, measuring background corrected line intensities that were integrated for the time of total evaporation of 5 mg boron carbide sample with or without added chemical modifiers. The following set of experimental conditions were compared with respect of analytical sensitivity and precision: (A) no modifier, Ar + O₂ (20%), 16 A; (B) sample + graphite powder (1 + 1), Ar + O₂ (20%), 16 A; (C) sample + CaF₂ (1 + 1), Ar, 25 A; (D) sample + CaF₂ + graphite powder (1 + 1 + 1), Ar, 25 A. The graphite powder modifier resulted in improved precision in general, and the CaF₂ was effective as a plasma ionization buffer and fluorinating agent. The best compromise was found under conditions B, when oxygen was present in the discharge atmosphere. This is likely due to the stepwise conversion of the boron carbide matrix to the more volatile boron oxide. Under conditions B, detection limits in the ranges of 0.3–9 μg g⁻¹ for Al, Ca, Cr, Cu, Fe, Mg, Mn and Si and that of 18–38 μg g⁻¹ for Ti, W, and Zr were attained. Average RSDs of 10.2 and 9.7% were found, respectively, without and with internal referencing to boron.     

Online preconcentration system for determining ultratrace amounts of Cd in vegetal samples using thermospray flame furnace atomic absorption spectrometry

A system has been developed for online preconcentration and determination of Cd using thermospray flame furnace atomic absorption spectrometry (TS-FF-AAS). It is based on the solid-phase extraction of Cd in an Amberlite XAD-2 minicolumn loaded with 2-(2-thiazolylazo)-5-dimethylaminophenol (TAM), and the metal ion is eluted with 1 mol L⁻¹ hydrochloridric acid, followed by its determination. The online system allows determining Cd with a detection limit of 8.0 ng L⁻¹ and a quantification limit of 25.0 ng L⁻¹. The precision (repeatability), calculated as the relative standard deviation (% RSD) in sample solutions containing 0.5 and 3.0 μg L⁻¹ of Cd, was 6.1 and 4.5%, respectively. The preconcentration factor obtained was 24. The system’s accuracy was confirmed by analyzing the following certified reference materials (CRMs): Rice flour NIES 10b, Apple leaves NIST 1515 and Orchard leaves NBS 1571. This procedure was applied to the determination of Cd in cabbage and powdered guarana samples.     

Determination of trace amounts of cadmium and copper by atomic absorption spectrometry after simultaneous extraction and preconcentration using a new water-soluble polyacrylic acid/alumina sorbent

An analytical method using alumina modified with water-soluble polyacrylic acid polymer for the simultaneous separation and preconcentration of trace Cu and Cd in a column system, and their determination by flame atomic absorption spectrometry was developed. The conditions for coating Al₂O₃ with polyacrylic acid were optimized, then the column was packed with 50 mg of this sorbent. Cd and Cu solution was passed through a glass column at pH 4.5, and elution was carried out with 5 mL of 0.05 M HCl at a flow rate of 1 mL min⁻¹. A sorption capacity of 10 mg Cu and 12 mg Cd was obtained for 1 g sorbent. The detection limit was calculated as 4.5 μg L⁻¹ for Cu and 1.54 μg L⁻¹ for Cd in the final solution. Enrichment factors of 300 for Cd and 400 for Cu were obtained.     

The coulometric titration of bromate with cuprous copper

A coulometric procedure has been developed by which 18 to 240 μg quantities of bromate may be determined to within 0.3 μg. The bromate is allowed to react with bromide in an acid solution and electrolytically generated cuprous copper is used for the titration. A dual platinum electrode indicator system is used in an amperometric end-point procedure     

Study of isomorphism in fluorine-containing antimony(III) complexes

The mutual influence of the atoms on the composition of solid fluorine-containing antimony(iii) complexes formed in aqueous solutions in the (MF) _x −(M′F) _n−x −SbF₃ (M, M′=Na, K, Rb, Cs, and NH₄;n=1, 2;x=0 to 2), (KNO₂) _n −(KY) _n −SbF₃ (Y=F, Cl, SO₄;n=0.5, 1), and K₂SbF₅−K₂SbCl₅ systems was investigated by elemental, X-ray, and thermogravimetric analyses and by IR and^121,123Sb NQR spectroscopy. The isomorphism conditions for fluorine-containing antimony(iii) compounds resulting in the formation of complexes NaM′SbF₅·1.5H₂O (M′=K and Rb), K₂SbF₅·1.5H₂O, NaCs₃Sb₄F₁₆·H₂O, KsbF₃Cl, K₂SbF₂Cl₃ with constant compositions, continuous M _x M′_2−x SbF₅ (0<x<2) and limited M _x M′_1−x SbF₄ (0.25<x<0.75; M, M′=K, Rb, Cs, and NH₄) solid solutions or LiF+MSbF₄ (M=Na, K, Rb, and Cs), M₂SbF₅+Cs₂SbF₅ (M=Na and K) and MSbF₄+NaSbF₄ (M=Rb and NH₄) mechanical mixtures were determined. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 1, pp. 103–108, January, 1999.