On the singular graph and the Weyr characteristic of anM-matrix

LetA be anM-matrix in standard lower block triangular form, with diagonal blocksA _ii irreducible. LetS be the set of indices such that the diagonal blockA is singular. We define the singular graph ofA to be the setS with partial order defined by > if there exists a chain of non-zero blocksA _i, A_ij, , A_l.Let ₁ be the set of maximal elements ofS, and define thep-th level _p ,p = 2, 3, , inductively as the set of maximal elements ofS \( _{1 p-1}). Denote by _p the number of elements in _p . The Weyr characteristic (associated with 0) ofA is defined to be (A) = ( ₁, ₂,, _h ), where ₁ + + _p = dim KerA ^p ,p = 1, 2, , and _h > 0, _h+1 = 0.Using a special type of basis, called anS-basis, for the generalized eigenspaceE(A) of 0 ofA, we associate a matrixD withA. We show that(A) = ( ₁, , _h) if and only if certain submatricesD _p,p+1 ,p = 1, , h – 1, ofD have full column rank. This condition is also necessary and sufficient forE(A) to have a basis consisting of non-negative vectors, which is a Jordan basis for –A. We also consider a given finite partially ordered setS, and we find a necessary and sufficient condition that allM-matricesA with singular graphS have(A) = ( ₁, , _h). This condition is satisfied ifS is a rooted forest.The work of the second-named author was partly supported by the National Science Foundation, under grant MPS-08618 A02.     

On cubature formulae with a minimal number of knots

Summary In this paper an approach is outlined to the two-dimensional analogon of the Gaussian quadrature problem. The main results are necessary and sufficient conditions for the existence of cubature formulae which are exact for all polynomials of degree m and which have a minimal number of 1/2k(k+1) knots,k=[m/2]+1. Ifm is odd, similar results are due to I.P. Mysovskikh ([5, 6]) which will be derived in a new way as a special case of the general characterization given here. Furthermore, it will be shown how this characterization can be used to construct minimal formulae of even degree.     

Isolierung von 10′-Apo-β-carotin-10′-ol und (3R)-10′-Apo-β-carotin-3,10′-diol (Galloxanthin) aus Rosenblüten. 5. Mitteilung über Rosenfarbstoffe

Isolation of 10′-Apo-β-carotene-10′-ol and (3R)-10′-Apo-β-carotene-3,10′-diol (Galloxanthin) from Rose Flowers The novel (all-E)-10′-apol-β-carotene-10′-ol ( 2 ) and (all-E,3R)-10′-apo-β-carotene-3,10′-diol ( 5 ) have been isolated from petals of one yellow species and various whitish or yellow blend varieties of rose cultivars. Each (all-E)-compound is accompanied by a (Z)-isomer, probably the (9Z)-isomer. Diol 5 proved to be identical with galloxanthin, an apo-10′-carotenol previously isolated from the retina of chicken.     

Studies of the reactions between indole-2,3-diones (isatins) and 2-aminobenzylamine

Reflux of equimolecular amounts 2-aminobenzylamine and isatins in acetic acid produced indolo[3,2-c]quinolin-6-ones in good yields. A proposed mechanism involving initial formation of a spiro compound is given. This isolable intermediate subsequently rearranges via a sequential isocyanate ring opening and a cyclisation process to a urea derivative which finally cyclized to the indolo[3,2-c]quinolin-6-ones. The urea derivative could be prepared separately and cyclized selectively to indolo[3,2-c]quinolin-6-one. Reaction of N-acetylisatin with 2-aminobenzylamine at room temperature yielded the 1,4-benzodiazepinone 3-(2-acetamidophenyl)-1,5-dihydro-1,4-benzodiazepin-2-one whereas its isomer 2(2-acetamidophenyl)-4,5-dihydro-1,4-benzodiazepin-3-one was obtained from 2-(2-acetylaminophenyl)-N-(2-aminobenzyl)-2-oxoacetamide in acetic acid at room temperature.The previously unknown linear isomer of indolo[3,2-c]quinolin-6-one, i.e. indolo[2,3-b]quinolin-11-one, has been prepared by thermal (260°C) cyclization of methyl 2-phenylamino indole-3-carboxylate, which in turn was prepared in two steps from methyl indole-3-carboxylate.     

Übergangsmetallphosphidokomplexe. VIII. Strukturuntersuchungen an Übergangsmetallphosphor-Vier- und Sechsringkomplexen. Die Strukturen von [(CO)4MnPH2]2, [(CO)4MnPH2]3 und [cpNiPH2]3

Transition Metal Phosphido Complexes. VIII. X-Ray Diffraction Studies of Transition Metal Phosphorus Four- and Six-Membered Ring Complexes. Structures of [(CO)₄MnPH₂]₂, [(CO)₄MnPH₂]₃, and [cpNiPH₂]₃ [(CO)₄MnPH₂]₂ 1 crystallizes triclinic in the space group P1 with a = 680.4 pm, b = 706.4 pm, c = 919.1 pm, α 110.5°, β = 91.92°, γ 115.65°, and Z = 1 formula unit. The molecule exhibits a centrosymmetrical structure. The bond angles within the planar four-membered (Mn? P)₂-ring are 76.1° at the Mn atoms and 103.9° at the P atoms, respectively. The average Mn? P bond distance is found to be 235.1 pm. [(CO)₄MnPH₂]₃ 2 crystallizes monoclinic in the space group P2/n with a = 905.2 pm, b = 974.8 pm, c = 1264.2 pm, β = 109.1°, and Z = 2 formula units. The framework of the six-membered (Mn? P)₃-ring can be described as having a twist boat conformation. The average endocyclic bond angles are with 89.1° at the Mn atoms and 130.1° at the P atoms, respectively, largely widened compared to 1 . The average Mn? P bond distance, which is found to be 238.5 pm, is also slightly increased compared to 1 . [cpNiPH₂]₃ 3 crystallizes rhombohedral in the space group R3. The cell constants (hexagonal setting) are a = b = 1686.1 pm, c = 561.1 pm and Z = 3 formula units. The six-membered (Ni? P)₃-ring exhibits a chair conformation. The endocyclic bond angles are with 92.3° at the Ni atoms and 124.3° at the P atoms, respectively, comparable with those of the six-membered ring compound 2 . The Ni? P bond distance is found to be 215.2 pm. The eyclopentadienyl ligands are disordered and have been refined as rigid groups.     

Photoinduced electron transfer at liquid|liquid interfaces. Part VII. Correlation between self-organisation and structure of water-soluble photoactive species

The self-organisation of a variety of dyes at the water|1,2-dichloroethane interface was studied by admittance measurements, photocurrent–potential curves and light polarisation anisotropy of the photocurrent. The heterogeneous photo-oxidation of ferrocene was studied at interfaces sensitised by Sn(IV) meso-tetra-(4-carboxyphenyl) porphyrin dichloride (SnTPPC), chlorin e-6, protoporphyrin IX (protoIX) and Fe(III) protoporphyrin IX chloride (Fe-protoIX). Cyclic voltammograms and capacitance voltage curves exhibit different features associated with the self-assembly of the dye species at the liquid|liquid boundary. In the case of SnTPPC, the capacitance curves displayed the characteristic responses commonly associated with the specific adsorption of ionic species. On the other hand, chlorin e-6, protoIX and Fe-protoIX show rather complex behaviour suggesting not only changes in the excess charge but also in the dielectric permittivity of the interface. Differences in the photocurrent efficiency were also observed under the same experimental conditions. The relative magnitude of the photocurrent responses were rationalised in terms of the phenomenological electron transfer rate constant, the photon capture cross-sections and the lifetime of the triplet state as obtained from nanosecond flash photolysis. Finally, the average molecular orientation of the adsorbed photoactive species was estimated from the photocurrent dependence on the angle of light polarisation in total internal reflection. The results show a clear correlation between the orientation of the transition dipole and the distribution of the peripheral carboxyl groups responsible for the hydrophilic nature of the dyes.     

R?ntgenspektrometrische Goldbestimmung in sulfidischen Erzen nach chemischer Anreicherung

Zusammenfassung Es wird eine Goldbestimmung an sulfidischen Erzen für den Bereich von 0,05–5 ppm bei 20 g Einwaage beschrieben. Das Gold wird durch Salpetersäurebehandlung von den übrigen Metallen getrennt, mit Salzsäure/ Salpetersäure gelöst und mit Selen als innerem Standard und Tellur als Spurenfänger mittels Zinn(II)-chlorid gefällt. Der Niederschlag wird auf ein Membranfilter aufgezogen. Die Goldbestimmung erfolgt über den Impulsratenquotienten der Gold L- und Selen K-Linien. Die störende Molybdän K-Strahlung und die Comptonstreuung werden durch ein Zirkoniumblech auf dem Röhrenfenster unterdrückt. Der Variationskoeffizient beträgt 20% für 0,2–5 ppm und 53,6% für 0,05 ppm.

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Summary A method is described for the determination of gold in sulphide ores within the concentration range of 0.05–5 ppm with sample weights of 20 g. Gold is separated from the other metals by treatment with nitric acid, dissolved by hydrochloric acid, containing some nitric acid, and is finally precipitated by stannous chloride applying selenium as internal standard and tellurium as collecting agent. The precipitate is mounted on a diaphragm filter and the determination of gold is carried out by X-ray analysis suppressing the disturbing Mo _K radiation and compton scattering by a zirconium sheet-metal on the X-ray tube. The pulse rate quotient of the lines Au _L /Se _K is a function of the concentration of gold. The variation coefficient is 20% for 0.2–5 ppm and 53.6% for 0.05 ppm.

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Herrn Prof. Dr.-Ing. habil. J. Hesemann zum 65. Geburtstag gewidmet.     

Über eine Klasse von nichtlinearen Randwertproblemen: Änderung des kritischen Parameters bei Symmetrisierungen

Zusammenfassung Seif(t) positiv und wachsend. Es wird gezeigt, dass die Stabilität kleiner wird für die nichtlinearen Randwertprobleme (1) bzw. (1) bei Steiner-bzw. Schwarz-Symmetrisierung.

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Summary We show, forf(t) positive and increasing, that stability of (1) resp. (1) is diminished by Steiner's symmetrization resp. Schwarz's symmetrization.

     

Struktur von S-9,10-Dimethyl-1,3,5,7-tetraarsa-2,4,6,8-tetraoxaadamantan und 9,10-Diethyl-1,3,5,7-tetraarsa-2,4,6,8-tetraoxaadamantan

Structure of S-9,10-Dimethyl-1,3,5,7-tetraarsa-2,4,6,8-tetraoxaadamantane and 9,10-Diethyl-1,3,5,7-tetraarsa-2,4,6,8-tetraoxaadamantane S-9,10-Dimethyl-1,3,5,7-tetraarsa-2,4,6,8-tetraoxaadamantane ( 1 ) and 9,10-diethyl-1,3,5,7-tetraarsa-2,4,6,8-tetraoxaadamantane ( 2 ) have been prepared by the reaction of propionic acid, propionic anhydride and butyric acid, butyric anhydride, respectively, with arsenic(III)-oxide. The crystals of 1 are rhombic, a = 6.902(4), b = 11.121(5), c = 13.988(8), space group P2₁2₁2₁. The crystals of 2 are monoclinic, a = 11.757(10), b = 11.255(10), c = 18.631 (18), β = 91.78(7), space group P2₁/n. The mean bond lengths and angles in 1 are AsO = 1.790 Å, AsC = 1.959 Å, OAsO = 100.60°, CAsO = 99.65°, AsOAs = 128.77°, AsCAs = 118.73°, and in 2 they are AsO = 1.780 Å, AsC = 1.978 Å, OAsO = 101.45°, CAsO = 99.55°, AsOAs = 129.64°, AsCAs = 117.72°.     

Strukturchemische Untersuchungen von Monammin-Kupfer(I)-Komplexen: [CuNH3]2[Pt(CN)6], [CuNH3]2[Pt(CN)4] und Cu3[Co(CN)6]-2 NH3

Structurally Chemical Investigation of Monoammin Copper (I) Complexes : [CuNH₃]₂[Pt(CN)₆], [CuNH₃]₂[Pt(CN)₄] and Cu₃[Co(CN)₆] · 2NH₃ The preparation and the properties of [CuNH₃]₂[Pt(CN)₆], [CuNH₃]₂[Pt(CN)₄] and Cu₃[Co(CN)₆] · 2NH₃ are described. I.R. and Raman spectra have been recorded and assigned. According to X-ray powder diagrams, [CuNH₃]₂[Pt(CN)₆] crystallizes in the trigonal space group D–P3 ml, a = 7.771, c = 5.988 Å, Z = 1. According to the spectroscopic and crystallographic data, it is concluded that the Cu_I ion is coordinated with one NH₃ group and with the N atoms of the cyanometallate anions. The coordination number of the Cu⁺ is 4 in [CuNH₃]₂[Pt(CN)₆] and 3 in [CuNH₃]₂[Pt(CN)₄]. In the Cu₃[Co(CN)₆] · 2 NH₃ complex two Cu atoms have the coordination number 2, the third Cu atom 4.