Begleitfiguren und Invariantensystem minimaler Differentiationsordnung von Kurven im reellenn-dimensionalen affinen Raum

For such curves we know a moving frame and a complete system of invariants (affine curvatures) with respect to the group of volume-conserving transformations (unimodular group) of the order 2n?1 respectively 2n (see [1], p. 171). In this paper we study for a curvec a moving frame and a system of invariants of the minimal ordern+1, respectivelyn+2, by means of a curve with vanishing affine curvatures that has contact of maximal order withc. Forn=3 this is a result ofA. Winternitz ([1], p. 171 or [3], p. 86).     

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.     

Kriterien in der Theorie der Gleichverteilung

It is the aim of this paper to introduce two new notions of discrepancy. They are defined by the formulas $$\begin{gathered} \Delta _N^r \left( {\omega ;f} \right) = \mathop {\sup }\limits_{\left| z \right| = r} \left| {\left( {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}} \right)\sum\limits_{n = 1}^N {f\left( {z e^2 \pi i\omega \left( n \right)} \right)} - f\left( 0 \right)} \right|, and \hfill \\ \delta _N^r \left( {\omega ;f} \right) = \mathop {\sup }\limits_{\left| z \right| = r} \left| {\left( {{1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N}} \right)\sum\limits_{n = 1}^N {f\left( {z \omega \left( n \right)} \right)} \cdot z - \int\limits_0^z {f\left( \zeta \right)d\zeta } } \right|, \hfill \\ \end{gathered} $$ wheref is a holomorphic function defined in the unit disc withf ^(k) (0)≠0 for allk∈?,r<1 is a positive number, and ω is a sequence in [0, 1]. The first of these discrepancies can be generalized for multidimensional sequences. ω is uniform distributed if and only if lim _N→∞ Δ _N ^r (ω;f)=0 resp. lim _N→∞δ _N ^r (ω;f)=0. These results are proved in a quantitative way by estimating the classical discrepancyD _N (ω) by means ofΔ _N ^r (ω;f) and δ _N ^r (ω;f): $$\begin{gathered} \Delta _N^r \left( {\omega ;f} \right) \ll D_N \left( \omega \right) \ll \Phi \left( {\Delta _N^r \left( {\omega ;f} \right)} \right), \hfill \\ \delta _N^r \left( {\omega ;f} \right) \ll D_N \left( \omega \right) \ll \Psi \left( {\delta _N^r \left( {\omega ;f} \right)} \right). \hfill \\ \end{gathered} $$ The functions Φ and Ψ only depend onf andr. These estimations are based on the inequalities ofKoksma-Hlawka andErdös-Turán.     

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.     

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.     

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.