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.     

Strukturviskosität von verdünnten Lösungen von Biopolymeren

The importance of the rheological behaviour of solutions of macromolecules is briefly evaluated. The viscosity of the solutions depends on concentration, shear rate and time of shear, this relation being determined by the structure of the dissolved molecules. In dilute solutions shear dependence of viscosity is very frequently caused by the preferential orientation of anisotropic molecules. In such a case the particle dimensions can be calculated from the true limiting viscosity number, an anisotropy factor, the rotational diffusion constant and the effective particle density. These numbers can be derived from the flow curve, which has been extrapolated to zero concentration. It is necessary to measure the flow curve at shear gradients, which are sufficiently low to allow for an extrapolation to vanishing shear rate. By comparing the experimental flow curve with a choice of theoretical ones, the rotational diffusion constant and the anisotropy factor (axial ratio) can be found. From the limiting viscosity number and the axial ratio, the particle density can be calculated.     

Thermisches Verhalten von 1,2-Dipropenylbenzolen

1-cis, 2-cis-Dipropenylbenzene (cis, cis- 1 ) isomerises thermally at 215–235° with 1st order kinetics to give trans, cis- 1 and vice versa. At equilibrium 89% trans, cis- and 11% cis, cis- 1 are present. It is shown by thermal rearrangement of cis, cis-2′, 2″-d₂- 1 that the isomerisation is attributable to aromatic [1, 7a]-sigmatropic H-shifts. trans, trans- 1 rearranges thermally at 225–245° to yield 2, 3-dimethyl-1, 2-dihydronaphthalene ( 2 ). The formation of 2 can be visualized by disrotatory ring closure followed by an aromatic [1, 5s]-sigmatropic H-shift. 2 is also formed when, cis, cis- or trans, cis- 1 are heated for 153 h at 225°. Besides 2 a small amount (3%) of 1-ethyl-1, 2-dihydronaphthalene ( 5 ) is formed. The rearrangement of trans, trans- 1 and trans, trans-2′, 2″-d₂- 1 shows a secondary isotope effect k_H/k_D = 0,90.     

Komplexe von 2,6-Dimethylpyridin-1-oxid mit Lanthanidjodiden

Complexes of 2,6-dimethylpyridine 1-oxide with lanthanide iodides of the formulaeLn(2,6-LTNO)₅I₃ whereLn=La, Tb and Yb,Ln(2,6-LTNO)₄I₃ whereLn=Pr and Nd and Er(2,6-LTNO)_4.5I₃ have been prepared and characterised by chemical analysis, infrared and conductance studies. Infrared and conductance data have been interpreted in terms of dimeric (or polymeric) structures involving bridging amine oxide groups.

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Mit 2 Abbildungen     

Über die Synthese von 5-(o-Trifluormethylphenyl)-1H-thieno[3,4—e]-1,4-diazepin-2(3H)-onen

The synthesis of 5-(o-trifluoromethylphenyl)-1H-thieno-[3,4-e]1,4-diazepin-2(3H)-one (7) and its nitration and chlorination in pos. 8 are described.     

Potential non-steroidal estrogens and antiestrogens,IV Organic azides in heterocyclic synthesis,part 13: Synthesis of aza- and diazacoumestrols via azido derivatives

Summary 4-Chloro-3-aryl-coumarins and quinolones2 a–e undergo thermolytic ring closure by reaction with sodium azide in refluxing dimethyl formamide to yield indolo[3,2-c]coumarins and indolo[3,2-c]quinolin-6(5H)-ones6 a–e. In the case of the coumarin2 a the azido coumarin5 can be isolated. The mono- and diazacoumestrol-dimethylethers6 a–c are converted into the coumestrol analogues7 a–c and their diacetyl derivatives8 a–c.

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Potentielle nichtsteroidale Östrogene und Antiöstrogene, 4. Mitt.: Organische Azide in der Heterocyclensynthese, Teil 13: Synthese von Aza- und Diazacumöstrolen über Azidzwischenstufen

Zusammenfassung 4-Chlor-3-arylcumarine und-chinolone2 a–e reagieren thermolytisch mit Natriumazid in siedendem Dimethylfomamid unter Ringschluß zu Indolo[3,2-c]cumarinen und Indolo[3,2-c]chinolin-6(5H)-onen6 a–e. Nur aus dem Cumarinderivat2 a kann das zwischenzeitlich gebildete Azidocumarin5 isoliert werden. Die so erhaltenen Mono- und Diazacumöstroldimethylether6 a–c werden in die entsprechenden Cumöstrole7 a–c und ihre Diacetylderivate8 a–c umgewandelt.

     

Studies of the Reaction Behavior of Nitryl Compounds Towards Azides: Evidence for Tetranitrogen Dioxide,N4O2

The reaction behavior of NaN₃, AgN₃, and Me₃SiN₃ towards FNO₂, CINO₂, NO₂SbF₆, and NO₂BF₄ was investigated. At -30°C or below in a solvent-free system sodium azide did not react with CINO₂, NO₂BF₄, or NO₂SbF₆. Below -30°C silver azide did not react either with neat C1NO₂. Treatment of Me₃SiN₃ with pure C1NO₂ led to the formation of C1N₃, N₂O, and Me₃SiOSiMe₃. A mechanism for this reaction has been proposed. Pure chlorine azide was isolated by fractional condensation and identified by its low-temperature Raman spectrum (liquid state). The reaction of Cp₂Ti(N₃)₂ with C1NO₂ also yielded C1N₃ as the only azide-containing reaction product. Treatment of FNO₂ with NaN₃ at temperatures as low as -78°C always ended in an explosion which was probably due to the formation of FN₃ as one of the reaction products. The reaction of NO₂SbF₆ with NaN₃ in liquid CO₂ (-55°C· T· -35°C) as the solvent afforded a new azide species which was stable at low temperature in solution only and was investigated by means of low-temperature Raman spectroscopy. The obtained vibrational data give strong evidence for the presence of tetranitrogen dioxide, N₄O₂, which can be regarded as nitryl azide (NO₂N₃). The structure and vibrational frequencies of N₄O₂ were computed ab initio at correlated level (MP2/6-31 + G^*). In liquid xenon (-100°C· T· -60°C) NaN₃ did not react with NO₂SbF₆. A previous literature report on the preparation of N₄O₂ could not be established.     

Berechnung von mittleren Schwingungsamplituden einiger tetraedrischer Moleküle und Ionen nach der Methode der charakteristischen Schwingungen

Mean amplitudes of vibration of a series of tetrahedralXY ₄ molecules and ions (hydrides, halides, oxides and oxoanions) have been calculated using the “Method of the Characteristic Vibrations” ofA. Müller. The results indicate that this method leads to very good values for most of the investigated species, and especially in the cases of highM _X/M_Y mass ratio.     

Supramolecular framework adducts constructed of hexamethylenetetramine,aquated alkali metal cationic clusters,and hexacyanoferrates(II/III)

The reaction of alkali metal hexacyanoferrate(II/III) with (CH₂)₆N₄ (hexamethylenetetramine, abbreviated HMT) in an acidic medium yielded crystalline compounds of stoichiometries HK₂[Fe¹¹¹(CN)₆]·2HMT·4H₂O, H₂K₂[Fe¹¹(CN)₆]·2HMT·4H₂O, and HNa₂[Fe¹¹¹(CN)₆]· 2HMT·5H₂O. Their crystal structures are based on a packing of three molecular components: neutral and/orprotonated HMT, hexacyanoferrate, and an alkali metal ion-water cluster. The resulting three-dimensional supramolecular framework is constructed from the coordination of the alkali metal ion by aqua ligands as well as [Fe(CN)₆]{n–} and HMT units, and further stabilization is achieved by hydrogen bonding between water molecules and the noncoordinated nitrogen atoms of HMT and hexacyanoferrate.     

Matrix-dependent instability of selenium(IV) stored in teflon containers

Solutions of selenium(IV) standards with different acid matrices were stored in containers constructed of boosilicate glass, conventional polyethylene, and fluorinated ethylenepropylene (teflon FEP). After 50 days of storage in FEP, there were highly significant losses of Se(IV) from standards in either 5% HCl/5% H₂SO₄, or 5% H₂SO₄. Increasing the hydrochloric acid concentration, e.g., 15% HCl/5% H₂SO₄, greatly reduced this loss. Addition of selenium-75 (selenate-free) indicated that the losses did not result from physical adsorption onto container surfaces. It is shown that the losses were caused by oxidation of Se(IV) to Se(VI).