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Abstract. We propose a finite difference scheme for the Heisenberg equation and the LandauLifshitz equation. These equations have a length-preserving property and energy conservation or dissipation property. Our proposed scheme inherits both characteristic properties. We also show that the boundedness of finite difference solutions and an unique solvability of our scheme. Finally, we show some numerical examples.

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Phosphorus-containing amino acid-type herbicides (PAAHs) and their metabolites in human plasma and whole blood were extracted with titania and determined by capillary electrophoresis (CE). The recoveries of glyphosate (GLYP), aminomethylphosphonic acid (AMPA), gluphosinate (GLUF), and 3-methylphosphonico-propionic acid (MPPA) from human plasma were 84.6, 76.8, 90.4, and 89.6%, respectively. The recoveries of GLYP, AMPA, GLUF, and MPPA from whole blood were 79.6, 84.4, 36.9, and 31.8%, respectively. The low recoveries of GLUF and MPPA from whole blood were improved by the dilution of whole blood with water to 4-fold.

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and bialaphos (L-2-amino-H-[(hydroxyl)(methylphosphinoyl)]butyryl-L-alanyl-L-alanine, BIAL) are classified into nonspecific phosphorus-containing amino acid-type herbicides (cf. Fig. 1). BIAL is a tripeptide composed of one L-GLUF and two L-alanine molecules, and is produced by Streptomyces hygroscopicus. 1 D,L-GLUF and BIAL contain L-GLUF as the active ingredient, which inhibits the action of the enzyme glutamine synthetase, and has herbicidal activity. 2 Since GLUF and BIAL are readily biotransformed to 3-methylphosphinicopropionic acid and L-GLUF, respectively, by microorganisms in the soil, 3 these herbicides have been increasingly used all over the world. So far, the chiral separation and the determination of D,L-GLUF have been reported only in HPLC.2 Therefore, a facile, rapid, and accurate method has to be developed for the determination of not only the total amounts of those herbicides, but also that of the enantiomers.In our recent study, 4 the electrophoretic equilibrium and the chiral separation of D,L-GLUF were investigated by capillary zone electrophoresis (CZE) with β-and γ-cyclodextrin (CD) as a chiral selector, in which D,L-GLUF was derivatized with dansyl chloride (Dns-Cl). Dns-derivatization was employed to enhance the sensitivity in UV absorption and fluorescence detection, and the interaction with CD. 5,6 A favorable chiral separation of Dns-D,L-GLUF was given by γ-CD.In the present study, a novel absolute determination method was developed for the total amount of D,L-GLUF and BIAL and the amount of enantiomeric isomers of D,L-GLUF without any calibration curves and comparison standards. This method is based on a change in the mole ratio of enantiomeric isomers of the analyte after the addition of a known amount of the enantiomeric isomer as a spike. 7 The principle is similar to that of isotope dilution mass spectrometry.Since the physicochemical properties of enantiomeric isomers are exactly equal between them under achiral conditions, the enantiomeric isomer acts as an ideal internal standard as an isotope filled the role of that in isotope dilution. Hence, an enantiomeric isomer A novel absolute determination method using chirality without any calibration curves or comparison standards has been proposed for phosphorus-containing amino acid-type herbicides, glufosinate (D,L-GLUF) and bialaphos (BIAL). This method is based on a change in the enantiomeric ratio after the spiking of a known amount of the enantiomers with different enantiomeric ratios to a sample. D,L-GLUF was determined by adding a known amount of L-GLUF to the sample, derivatizing them with dansyl chloride, and measuring the ratio of the peak area of the D-isomer to that of the Lisomer by means of γ-cyclodextrin modified capillary zone electrophoresis. The accuracy and precision of the method were evaluated using a synthetic sample. The mean values obtained for D-and L-GLUF agreed with the values taken within 1.6%; also the reproducibility was as good as less than 2.8%. The determination of BIAL was achieved by determining G...

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