On the abstract theory of additive and multiplicative Schwarz algorithms

Summary. In recent years, it has been shown that many modern iterative algorithms (multigrid schemes, multilevel preconditioners, domain decomposition methods etc.) for solving problems resulting from the discretization of PDEs can be interpreted as additive (Jacobi-like) or multiplicative (Gauss-Seidel-like) subspace correction methods. The key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space into a sum of subspaces and the splitting of the variational problem on into auxiliary problems on these subspaces. In this paper, we propose a modification of the abstract convergence theory of the additive and multiplicative Schwarz methods, that makes the relation to traditional iteration methods more explicit. The analysis of the additive and multiplicative Schwarz iterations can be carried out in almost the same spirit as in the traditional block-matrix situation, making convergence proofs of multilevel and domain decomposition methods clearer, or, at least, more classical. In addition, we present a new bound for the convergence rate of the appropriately scaled multiplicative Schwarz method directly in terms of the condition number of the corresponding additive Schwarz operator. These results may be viewed as an appendix to the recent surveys [X], [Ys]. Received February 1, 1994 / Revised version received August 1, 1994     

Zur Synthese sulfonierter Derivate von 2-Amino-p-xylol

Synthese of sulfonated derivatives of 2-amino-p-xylene Sulfonation of 2-amino-p-xylene (2) gave 2-amino-p-xylene-5-sulfonic acid (1) . The 2-amino-p-xylene-6-sulfonic acid (3) was prepared via three routes: (1) sulfonation of 2-amino-5-chloro-p-xylene (19) to 5-amino-2-chloro-p-xylene-3-sulfonic acid (20) followed by hydrogenolysis; (2) sulfur dioxide treatment of the diazonium salt derived from 2-amino-6-nitro-p-xylene (21) to 2-nitro-p-xylene-6-sulfonyl chloride (11) followed by hydrolysis to 2-nitro-p-xylene-6-sulfonic acid (4) and Béchamp reduction; (3) Béchamp reduction of 2-chloro-3-nitro-p-xylene-5-sulfonic acid (13) to 3-amino-2-chloro-p-xylene-5-sulfonic acid (16) and subsequent hydrogenolysis. Catalytic reduction of 13 in aqueous sodium carbonate solution gave mixtures of 3 and 16 . 2-Amino-p-xylene-3-sulfonic acid (27) was synthesized via two routes: (1) reaction of 19 with sulfamic acid to 2-amino-5-chloro-p-xylene-3-sulfonic acid (26) followed by hydrogenolysis; (2) sulfur dioxide treatment of the diazonium salt derived from 2-amino-3-nitro-p-xylene (28) to 2-nitro-p-xylene-3-sulfonyl chloride (12) , hydrolysis to 2-nitro-p-xylene-3-sulfonic acid (7) and Béchamp reduction.     

Kryoskopische Reinheitsbestimmung kleiner Substanzmengen nach der Methode der konstantgesteuerten aufgepr?gten W?rmerate

Zusammenfassung Zur Reinheitsbestimmung schmelzbarer, vorwiegend organischer Proben wurde eine kryoskopische Apparatur nachSmit gebaut. Sie arbeitet oberhalb 99 Mol-% im Halbmikrobereich, wobei der Absolutwert der Reinheit auf ±0,05 Mol-% genau meßbar ist. Die Probe, die ständig unter strömendem Stickstoff gehalten wird, umgibt als dünne Schicht ein Platin-Widerstandsthermometer, das sich in der Mitte eines elektrisch geheizten Aluminiumzylinders befindet. Konstanz der Wärmezufuhr wird gesichert durch Regelung der Temperaturdifferenz gegen ein gleichartiges zweites Thermometer, das dicht am geheizten Zylindermantel eingesteckt ist. Die Thermometer brauchen nicht geeicht zu werden. Die Regelung wird durch eine Doppel-Wheatstonebrücke vermittelt, deren einer Ausgang auf den Schreiber und deren anderer auf die Heizung wirkt. Die graphische Auswertung ist aus dem Zweiphasen-Diagramm fest-flüssig abgeleitet. Als Bezugsstandard ist eine zweite Probe zu messen, die man durch Verunreinigung der ursprünglichen Substanz um einen bekannten Betrag erhält.

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Summary A cryoscopic apparatus for the purity assay of fusible, mostly organic samples was built afterSmit. It works in the half-micro scale above 99 mole%, where the absolute value of the purity will lie within ±0.05 mole%. The sample, which is always kept under streaming nitrogen, surrounds as a thin layer a platinum resistance thermometer that stands in the center of an electrically heated aluminium cylinder. A constant supply of heat is ensured by controlling the temperature difference against a second thermometer of the same kind, which is inserted near the heated periphery of the cylinder. It is not necessary to calibrate the thermometers. Control is mediated by a double Wheatstone bridge, the one output of which acts on the recording device, the other on the heater. The graphical evaluation of the records derives from the two-phase diagram solid-liquid. As a referential instance, a second sample has to be measured, which is obtained from the specimen through artificial contamination by a known amount.

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Unter Mitarbeit von F.Weigand und R.Baumgart.

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Herrn Dr.Ebinger, AEG-Meßgerätefabrik Berlin-Reinickendorf und Herrn Dipl.-Ing. Lutz, AEG-Büro Mannheim, habe ich für ihre große Bereitwilligkeit zu danken, mit der sie auf die Wünsche zur Bauart der Brücke eingingen. — Für Anregungen danke ich Herrn Dr.Kienitz.     

Untersuchungen zur Physisorption und Chemisorption von Wasser auf Zinkoxid-Oberflächen

The investigation of different zinc oxide samples by means of thermogravimetry and infrared spectroscopy has shown that the surface of the samples is covered by an approximately monoatomic layer of hydroxide groups. Furthermore, varying amounts of carbonate groups are found which are due to the presence of zinc hydroxide carbonate II [Zn₅(OH)₆(CO₃)₂]. Below relative water vapour pressures of p/p₀ = 0.2 (25°C), two hydrogen bridges connect one physisorbed water molecule with two chemisorbed surface hydroxide groups. In addition, about the same amount of water is physically adsorbed between vapour pressures of p/p₀ 0.2 and 0.8. At still higher relative humidity, a multimolecular layer is built up which reaches a thickness of about 200 water molecules at p/p₀ = 1.0. All samples show in the v-OH region of the IR. spectrum a broad absorption with four bands, A, B, C, and D. The position of the bands and the change of their intensities when rising the temperature of the samples up to 600°C indicate that both bands of longer wave lengths, C and D, arise from physically adsorbed water molecules, while the bands A and B are due to hydroxide groups located on the crystallographic faces (0001) and (0001 ), respectively.     

Zur Synthese sulfonierter Derivate von 4-Amino-1,3-dimethylbenzol und 2-Amino-1,3-dimethylbenzol

Syntheses of Sulfonated Derivatives of 4-Amino-1, 3-dimethylbenzene and 2-Amino-1, 3-dimethylbenzene Direct sulfonation of 4-amino-1, 3-dimethylbenzene (1) and sulfonation of 4-nitro-1,3-dimethylbenzene ( 4 ) to 4-nitro-1,3-dimethylbenzene-6-sulfonic acid ( 3 ) followed by reduction yield 4-amino-1,3-dimethylbenzene-6-sulfonic acid ( 2 ). The isomeric 5-sulfonic acid ( 5 ) however is prepared solely by baking the acid sulfate salt of 1 . Reaction of sulfur dioxide with the diazonium chloride derived from 2-amino-4-nitro-1,3-dimethylbenzene ( 7 ) leads to 4-nitro-1,3-dimethylbenzene-2-sulfonyl chloride ( 8 ), which is successively hydrolyzed to 4-nitro-1,3-dimethylbenzene-2-sulfonic acid ( 9 ) and reduced to 4-amino-1, 3-dimethylbenzene-2-sulfonic acid ( 6 ). Treatment of 4-amino-6-bromo-1,3-dimethylbenzene ( 12 ) and 4-amino-6-chloro-1, 3-dimethylbenzene ( 13 ), the former obtained by reduction of 4-chloro-6-nitro-1,3-dimethyl-benzene ( 10 ) and the latter from 4-chloro-6-nitro-1, 3-dimethylbenzene ( 11 ), with oleum yield 4-amino-6-bromo-1,3-dimethylbenzene-2-sulfonic acid ( 14 ) and 4-amino-6-chloro-1,3-dimethylbenzene-2-sulfonic acid ( 15 ) respectively; subsequent carbon-halogen hydrogenolyses of 14 and 15 lead also to 6 (Scheme 1). Baking the acid sulfate salt of 2-amino-1, 3-dimethylbenzene ( 17 ) gives 2-amino-1, 3-dimethylbenzene-5-sulfonic acid ( 16 ), whereas the isomeric 4-sulfonic acid ( 18 ) can be prepared by either of the following three possible pathways: Sulfonation of 2-nitro-1,3-dimethylbenzene ( 20 ) to 2-nitro-1,3-dimethylbenzene-4-sulfonic acid ( 21 ) followed by reduction or sulfonation of 2-acetylamino-1,3-dimethylbenzene ( 19 ) to 2-acetylamino-1,3-dimethylbenzene-4-sulfonic acid ( 22 ) with subsequent hydrolysis or direct sulfonation of 17 . Further sulfonation of 18 yields 2-amino 1,3-dimethylbenzene-4,6-disulfonic acid ( 23 ), the structure of which is independently confirmed by reduction of unequivocally prepared 2-nitro- 1,:3-dimethylbenzene-4,6-disulfonic acid ( 24 )(Scheme 2).     

Thermal reduction and reoxidation of spinel-type Cu1−xZnxAl2O4

The thermochemical reactivity of the spinel-type quaternary metal oxide Cu_1–xZn_xAl₂O₄ has been investigated for different Cu:Zn ratios. In oxygen or inert gas atmospheres no considerable reduction is observed. In molecular hydrogen metal selective reduction of the Cu is found at relatively high temperature. The solid reduction product is made up of sintered, poorly dispersed metallic copper on a Zn-Al-O metal oxide support, a potential catalyst for the methanol synthesis. Owing to the measured high reduction temperature leading to the mentioned sintering of the metallic copper, the activity of this system cannot be high.Financial support by the Swiss National Science Foundation under project Nr. 2027933.89 is gratefully acknowledged.     

Fourier series and the conjugate function in the classes ϕ(L)

Обозначим через ? (L) кл асс всех ?-интегрируе мых 2π-периодических функ ций. При ограничении?(L)?L ¹ устанавливаются необходимые и достат очные условия, которым долж на удовлетворять?(t), что бы для каждой функцииf(x)ε ?(L) а) тригонометрически й ряд Фурье сходился в метрике ?, в) (С, а) средние ряда Фур ье сходились в метрик е ? для некоторого α > 0, с) сопряженная функци я \(\tilde f(x)\) также принадлежал а классу?(L).     

Zur Synthese sulfonierter Derivate von 2,3-Dimethylanilin und 3,4-Dimethylanilin

On the Synthesis of Sulfonated Derivatives of 2,3-Dimethylaniline and 3,4-Dimethylaniline Baking the hydrogensulfate salt of 2,3-dimethylaniline ( 1 ) or of 3,4-dimethylaniline ( 2 ) led to 4-amino-2,3-dimethylbenzenesulfonic acid ( 4 ) and 2-amino-4,5-dimethylbenzenesulfonic acid ( 5 ), respectively (Scheme 1). The sulfonic acid 5 was also obtained by treatment of 2 with sulfuric acid or by reaction of 2 with amidosulfuric acid. 3-Amino-4,5-dimethylbenzenesulfonic acid ( 3 ) and 5-Amino-2,3-dimethylbenzenesulfonic acid ( 6 ) were prepared by sulfonation of 1,2-dimethyl-3-nitrobenzene ( 9 ) to 3,4-dimethyl-5-nitrobenzenesulfonic acid ( 11 ) and of 1,2-dimethyl-4-nitrobenzene ( 10 ) to 2,3-dimethyl-5-nitrobenzenesulfonic acid ( 12 ), respectively, with subsequent Béchamp reduction (Scheme 1). Preparations of 2-amino-3,4-dimethylbenzenesulfonic acid ( 7 ) and of 6-amino-2,3-dimethylbenzenesulfonic acid ( 8 ) were achieved by the sulfur dioxide treatment of the diazonium chlorides derived from 3,4-dimethyl-2-nitroaniline ( 24 ) and from 2,3-dimethyl-6-nitroaniline ( 31 ) to 3,4-dimethyl-2-nitrobenzenesulfonyl chloride ( 29 ) and 2,3-dimethyl-6-nitrobenzenesulfonyl chloride ( 32 ), respectively, followed by hydrolysis to 3,4-dimethyl-2-nitrobenzenesulfonic acid ( 30 ) and 2,3-dimethyl-6-nitrobenzenesulfonic acid ( 33 ), and final reduction (Scheme 3). Compound 7 was also synthesized by reaction of 4-chloro-2,3-dimethylaniline ( 23 ) with amidosulfuric acid to 2-amino-5-chloro-3,4-dimethylbenzenesulfonic acid ( 20 ) and subsequent hydrogenolysis (Scheme 2). 4′-Bromo-2′, 3′-dimethyl-acetanilide ( 13 ) and 4′-chloro-2′, 3′-dimethyl-acetanilide ( 14 ) on treatment with oleum yielded 5-acetylamino-2-bromo-3,4-dimethylbenzenesulfonic acid ( 17 ) and 5-acetylamino-2-chloro-3,4-dimethylbenzenesulfonic acid ( 18 ), respectively. Their structures were proven by hydrolysis to 5-amino-2-bromo-3,4-dimethylbenzenesulfonic acid ( 21 ) and 5-amino-2-chloro-3,4-dimethylbenzenesulfonic acid ( 22 ), followed by reductive dehalogenation to 3 .     

Size-related vaporisation and ionisation of laser-induced glass particles in the inductively coupled plasma

Ongoing discussions about the origin of elemental fractionation occurring during LA-ICP-MS analysis show that this problem is still far from being well understood. It is becoming accepted that all three possible sources (ablation, transport, excitation) contribute to elemental fractionation. However, experimental data about the vaporisation size limit of different particles in the ICP, as produced in laser ablation, have not been available until now. This information should allow one to determine the signal contributing mass within the ICP and would further clarify demands on suitable laser ablation systems and gas atmospheres in terms of their particle size distribution.The results presented here show a vaporisation size limit of laser induced particles, which was found at particle sizes between 90 nm and 150 nm using an Elan 6000 ICP-MS. Due to the fact that the ICP-MS response was used as evaluation parameter, vaporisation and ionisation limits are not distinguishable.The upper limit was determined by successively removing the larger particles from the aerosol, which was created by ablation of a NIST 610 glass standard at a wavelength of 266 nm, using a recently developed particle separation device. Various particle fractions were separated from the aerosol entering the ICP. The decrease in signal intensity is not proportional to the decrease in volume, indicating that particles above 150 nm in diameter are not completely ionised in the ICP. Due to the limited removal range of the particle separation device, which cannot remove particles smaller than 150 nm, single hole ablations were used to determine the lower vaporisation limit. This is based on measurements showing that larger particles occur dominantly during the first 100 laser pulses only. After this period, the ratio of ICP-MS counts and total particle volume was found to be constant while most of the particles are smaller than 90 nm, indicating complete vaporisation and ionisation of these particles.To describe the influence of different plasma forward powers on the vaporisation limit, the range 1000–1600 W was studied. Results indicate that optimum vaporisation and ionisation occurs at 1300 W. However, an increase of the particle ionisation limit towards larger particles was not observed within the accuracy of this study using the full range of parameters available for optimisation on commonly used ICP-MS instruments.     

1H-NMR-spektroskopische Analyse von 2-Chlor-1.3.2-dioxarsolan

The¹H-NMR-spectral data of 2-chloro-1.3.2-dioxarsolane are presented and discussed. The protons of the methylene groups form in concentrated solutions by rapide chlorine exchange anAAAA spin system. In dilute solutions the protons form anAABB spin system, which is changed to anAAAA system by addition of chlorine ions.The vicinal H–C–C–H-coupling constants indicate a twist-envelope conformation.

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Mit 1 Abbildung